Response function of hot asymmetric nuclear mattercnedres.org/literature/Phys.Lett.B/Phys.Lett.B...

14 January 1999

Ž .Physics Letters B 446 1999 1–8

Response function of hot asymmetric nuclear matter

Fabio L. Braghin 1´Nuclear Theory and Elementary Particle Phenomenology Group, Instituto de Fısica, UniÕersidade de Sao Paulo, C.P. 66.318, CEP´ ˜

05315-970, Sao Paulo SP, Brazil˜

Received 4 September 1998; revised 20 November 1998Editor: W. Haxton

Abstract

An approximated expression for the response function of hot asymmetric nuclear matter to small external perturbationsŽ .was calculated. A general expression for the four channels scalar, isovector, spin and spin-isovector was obtained and the

results for the isovector channel are presented in this letter. Two possible prescriptions for the neutron and proton densityfluctuations are considered leading to different results. In the case of a realistic Skyrme-type force, SGII, for an asymmetrycoefficient corresponding to that of the lead nucleus, a damped zero sound was found at zero temperature for small

Ž .momentum transfer in the case of the first second prescription. For a higher asymmetry coefficient the mode becomeseither more or less collective, depending on the ansatz for the density fluctuations. Although both prescriptions yield strengthfunctions which satisfy the energy-weighted sum rule sufficiently well, it is argued that the prescription leading to a morecollective mode as the asymmetry increases has a sounder physical meaning. As temperature increases collectivity weakensmaking the zero sound disappear whenever it occurs. q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 21.30.qy; 21.60.Jz; 21.65.q f; 24.30.CzKeywords: Asymmetry; Nuclear matter; Response function; Zero sound

The linear response method for non relativistic nuclear matter at zero and finite temperatures has provided aw xreasonable large amount of interesting results concerning fundamental properties of this system 1–5 . For

w xinstance, in Refs. 2–4 , there were motivations from the possibility of getting information for the effectivenucleon-nucleon interaction and for the theoretical descriptions of the nuclear isovector dipole giant resonancesŽ .IVDGR with increasing excitation energy. In these references, a zero sound collective mode was found at lowtemperatures. The model worked out in these references was based on the Steinwedel-Jenssen one, consideringthat the transferred momentum between protons and neutrons is related to the mass number of the analysednucleus. In particular, for heavy nuclei one should consider small momentum transfer q. The resulting collectivemode is then undamped at zero temperature for Skyrme effective interactions which contain density dependentterm. The zero sound has the following dispersion relation: v sc q, where c is the zero sound velocity.res 0 0

With the increase of temperature the collective mode couples to the particle-hole spectrum making zero sound

1 E-mail: [email protected]

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01508-1

( )F.L. BraghinrPhysics Letters B 446 1999 1–82

damped until its disappearance by the time when the temperature is of the order of a few MeV. This limitingtemperature above which there is no more zero sound depends on the used effective interaction. An

w xinvestigation of the other three channels of the response function was done by Hernandez et al. in 5 .´In this paper a nearly exact expression for the response function of asymmetric hot nuclear matter is

calculated and the results for the isovector channel are analysed. This calculation provides interesting qualitativeinformation on nuclear matter and heavy nuclei. For heavy nuclei there is a non negligible asymmetry in theneutron-proton number and it is interesting to evaluate its influence on the collective motion as well as to getinformation for the effective interaction. Besides that, largely asymmetric nuclear matter or neutron matter have

w xbeen extensively studied with several motivations 6 . Moreover, the isovector dipole giant resonance has beenw x 122observed in light exotic nuclei 7 , and an halo region has been recently predicted for the exotic Zr nucleus

w x8 . We can then hope the IVDGR to take place in exotic heavier nuclei which would possess a neutron halo.The basis of this linear response method is the time dependent Hartree- Fock equation for the density of

Ž .nucleons r which, in natural units, is given by:

w xiE rs W ,r , 1Ž .t

Ž .where W is the Hartree-Fock energy of protons or neutrons. An external field V with small amplitude e ise

added to the Hartree-Fock equation in such a way to produce density fluctuations around the static solution soŽ .that Eq. 1 can be linearized. A suited form for this external field in an infinite medium is:

ˆ yi qPr yiŽvqih . tV se Oe e . 2Ž .e

In the above equation h is an infinitesimal number which corresponds to the adiabatic switching on of theˆexternal source. The operator O produces different density fluctuations for different cases: 1,t ,s ,t s ,3 3 3 3

respectively for the scalar, isovector, spin and spin-isovector channels. In the following, the calculation of theresponse function for the isovector channel with an asymmetry in the neutron-proton densities is showed, but theprocedure is the same for the other channels.

Ž .The mean energy for Skyrme- type effective interaction is expressed in terms of the nucleon density r ,Ž . Ž . w xkinetic energy density t and momentum density j 9 . The resulting equation for the total fluctuation density

Ž .in momentum space drsr yr is:n p

² < < X: X X X ² < < X: X ² < < X:iE k dr k s 1qac P e k ye k k dr k q2 f k y f k k d W k q2 f kŽ . Ž . Ž . Ž . Ž . Ž .ŽŽ . Ž .t 0 p 0 p p p p n

X ² < < X: X X X yiŽvqih . tyf k k d W k q2e f k y f k q f k y f k d k ykyq e ,Ž . Ž . Ž . Ž . Ž . Ž .. Ž .n n n n p p

3Ž .X Ž . 2Ž . ) ) Ž .where e k sk 1qac r2m , and m is the proton effective mass. In this expression f k is the0 p p p q

Ž .fermionic occupation number q stands for neutrons or protons- n, p and also three asymmetry coefficientshave been used:

m) r drp 0 n nas y1, bs y1, cs . 4Ž .

)m r drn 0 p

ŽThe coefficient b is related to another frequently used asymmetry coefficient, as2 r yr , by: bs2ar 1y0 n 0.a .For the induced total fluctuations we consider the following time-dependent ansatze, analogously to the form¨

of V :e

² < < : yi qPr yiŽvqih . t ² < < : yi qPr yiŽvqih . t ² < < : yi qPr yiŽvqih . tr dr r sa e e , r dt r sbe e , r d j r sg qe e . 5Ž .These ansatze are required to satisfy the following equalities:¨

d3k 1² < < :a ,b ,g s 1,kP kqq , 2 kqq Pq k dr ts0 kqq . 6Ž . Ž . Ž . Ž . Ž .H 3 2ž /q2pŽ .

( )F.L. BraghinrPhysics Letters B 446 1999 1–8 3

Ž . Ž .From here on the index j,t for each spin, isospin channel is used. The deviation in the energy density ofŽ .neutrons and protons can be written as:

W yW sd W s2 V 0,1 qV 0,1 dr r ,t q2V 0,1=Pdr r ,t =q2V 0,1dt r ,tŽ . Ž . Ž .Ž .n 0 n n 0 2 n 1 n 1 n

q2 iV 0,1 =Pd j qd j P= . 7Ž . Ž .1 n n

The functions V 0,1 are related to the parameters of the Skyrme interaction via the following expressions:i

t t q20 31 10,1 aV sy x q y x q r y 3t 1q2 x q t 1q2 x ,Ž . Ž .Ž .Ž . Ž .0 0 3 0 1 1 2 22 22 12 16

10,1V s t 1q2 x y t 1q2 x ,Ž . Ž .Ž .1 2 2 1 116

10,1 ay1V s t qx r cr y cy1 r r12, 8Ž . Ž .Ž . Ž .2 3 3 0 n0 p02

where r , r and r are the proton, neutron and total equilibrium densities of asymmetric nuclear matter.0 n 0 p 0

To conclude this calculation another asymmetry coefficient is needed to be defined:

bnds . 9Ž .

bp

Ž Ž ..Two prescriptions were considered for coefficients c from expression 4 and d. They were taken either to beŽ . Žequal to 1r2 prescription A or calculated in terms of the saturation densities at zero temperature prescription

.B . In this case it was assumed that the density fluctuations are proportional to the equilibrium densities, forŽ . Ž . Ž 2 .2r3 5r3which r s 1qb r 2qb r . For the coefficient d, the zero temperature expression t s3 3p r was0 n 0 n n

linearized, considering that dtst yt . This yielded:n p

1qb 2r3cs , ds1r 1q 1qb . 10Ž . Ž .Ž .2qb

Ž .From Eq. 3 it is straightforward to show that the matrix elements at ts0 satisfy:

10,1 0,1 0,1² < < :k dr ts0 kqq s 4 V qV qV kP kqqŽ . Ž .� Ž .X X 0 2 1

e kqq ye k qvq ihŽ . Ž .p p

=a d f cyd f cy1 q2e d f yd f qV 0,1b d f dyd f dy1Ž . Ž .Ž .Ž . Ž .n p n p 1 n p

yV 0,1g 2 kqq Pq d f cXX yd f cXX y1 q2e f kqq y f k . 11Ž . Ž . Ž . Ž . Ž .Ž . 4Ž .1 n p n n

Ž X. Ž . XXIn this expression d f s f k y f k , and another asymmetry coefficient is used: c sg rg . Its value wasq q q n p

considered to be cXX sc and the resulting response function does not depend sensitively on cXX.Ž . Ž . Ž .Multiplying Eq. 11 respectively by 1, kP kqq and 2 kqq Pq and integrating them over k we obtain a

set of linear equations for a , b and g . The retarded response function is the polarizability, i.e., the ratio of theŽ .density fluctuation to the field strength: P v,q sare , which is obtained by solving the set of linear

equations, resulting the following expression for the polarizability:

P 0,1 v ,qŽ .R

1 1n p n pP qP 1yV P q V P qP PŽ .Ž . Ž .0 0 1 2 d 1 2 2 0 d2 2s 20,1 0,1 0,1 0,1 0,11yV P yV P qP qV V P P yP P q V P P yP PŽ . Ž . Ž .Ž .0 0,c 1 2,c 2,d 1 0 0,c 2,d 2,c 0,d 1 2,c 2,d 4,c 0,d

12Ž .) ) Ž .In the above expression we have used M sm r 1qac and:p p

P sÕP n v ,q q 1yÕ P p v ,q . 13Ž . Ž . Ž . Ž .2 i ,Õ 2 i 2 i


In this expression Õsc,d,cXX and is0,1,2. The functions P q are referred to as generalized Lindhard2 N

functions. They are defined as:

4 f kqq y f kŽ . Ž .q q Nq q 3P sP v ,q s d k kP kqq , 14Ž . Ž . Ž .Ž .H X X2 N 2 N 3 vq ihye k qe kqqŽ . Ž .2pŽ . p p

q Žq .Ž . w xwhere the limit h™0 is implicit. Expressions for P v,q were given in appendix B of Ref. 4,3 in the2 N

case of a symmetric nuclear matter. However, several corrections were needed in order to take into accountw x Ž .asymmetry effects. These modified expressions will be shown elsewhere 10 . In Eq. 12 we have used a

0,1modified coefficient V defined by:02

) 0,12 M v 2Vp 10,1 0,1 0,1V sV qV y . 15Ž .0 0 2 0,1 )ž /q 1y4V m r1 p 0 p

This modified coefficient arises because the change in the momentum density induced by the external field andŽ . w xalso it is also due to the asymmetry between protons and neutrons. Expression 12 generalizes that of 3,4 to

which it reduces when considering a symmetric nuclear matter.Ž .The strength distribution per unit volume, S v,q , corresponds to real transitions of the system and is

directly related to the photoabsorption strength distribution S . Both are proportional to the imaginary part ofabs

the polarizability:1

yv r TS v ,q s 1ye S sy I P v ,q . 16Ž . Ž . Ž . Ž .absp

The denominator of the polarizability determines the existence of a collective mode. When there is a pole inŽ .P v,q , the real part of the denominator gives the energy of the resonance while its imaginary part is directly

related to the width.

Ž . Ž y2 . Ž .Fig. 1. Distribution of strength per unit volume for the operator exp iqPr in fm as a function of the energy v in MeV for amomentum qs0.23 fmy1 and for different values of the temperature T s0,2,4,6 MeV, in the case of the interaction SGII with bs.54.Using prescription A.


w xIt is worth to recall that in the model worked out in 2,4 , based on the Steinwedel-Jenssen model, themomentum transfer q is directly related to the radius a nucleus. For a heavy nucleus the momentum transfer issmall, in particular, for the nucleus of lead one has considered qs .23 fmy1.

Ž .It is important to check the results by considering the energy weighted sum rule EWSR . For the asymmetricnuclear matter it results:

` r r0 p 0 n2 0,1 ) 0,1 )m s dvS v ,q vsq 1y2V m q 1y2V m . 17Ž . Ž .Ž .Ž .H1 1 p 1 n) )ž /m m0 p n

w xIn the four cases analysed bellow, the SGII Skyrme interaction was used 11 . In Fig. 1 the strengthŽ .distribution for an asymmetry coefficient bs .54 which corresponds to that of the nucleus of lead, i.e. a, .23

using prescription A for the coefficients c and d is shown. At low temperatures there is a damped collectivemode whose strength weakens with temperature increasing. We notice that the particle- hole spectrum is spreadto higher energies making the zero sound damped. In Fig. 2 the same strength is shown for the prescription Bfor the coefficients c and d. The collectivity is stronger than for the first prescription, but the othercharacteristics are similar. In both cases the resonance energy is slightly higher than in the symmetric nuclear

w x Ž 3.matter which was shown in 3,4 and the value of the function V ,260 MeV fm is slightly higher. This0

quantity together with the effective mass are the most relevant for the collectivity. The sum rule is sufficientlywell satisfied in all cases and the disagreements between the integration of the strength and the analyticalexpression of the sum rule are of the order of 1% to nearly 3%. These values are very reasonable considering

Ž .the approximations which have been done. In Table 1 the analytical values from expression 17 , m , are1

compared to the energy weighted integral over all spectrum. The zero sound contribution is of the order of halfof the total integration. The zero sound, whose dispersion relation is v sc q as written above, disappears if0 0

one considers higher momentum transfer q. According to our model it would correspond to smaller nuclei.

Fig. 2. Same as Fig. 1, with bs.54 for prescription B.


Table 1Ž . Ž y3 .Right hand side RHS of the energy weighted sum rule MeV = fm compared to the integrated value of the strength m for T s 0, 31

and 6 MeV

Ž . Ž . Ž .RHS m T s0 m T s3 m T s61 1 1

Ž .SGII bs.54 -A 53 53 52 53Ž .SGII bs.54 -B 53 52 53 53Ž .SGII bs8. -A 53 52 51 52Ž .SGII bs8. -B 53 54 54 54

In Fig. 3 a very asymmetric nuclear matter was considered with prescription A, for an asymmetry coefficientbs8. This means a nuclear matter with 90% of neutrons. In this case, there is no more coherent effect and thestrength has a weaker temperature dependence. The strength corresponds nearly to that of a gas of non-inter-acting particles, as if the parameters V were zero. The strength for prescription B is showed in Fig. 4. Iti

Žexhibits a much stronger zero sound at low temperatures with a higher value of the function V ,313 MeV03.fm . The zero sound frequency is higher and this prevents the coupling to the particle hole spectrum. Therefore

it starts damping at higher temperatures, approximately 6 MeV. It is interesting to note the very small relativeenhancement in the low energy part of the particle-hole spectrum until T,3 MeV. It is important to stress thatthe density of nuclear matter was kept constant for all the analysed cases with different asymmetries at finitetemperature.

As we have just shown, if prescription A were correct it would cause the progressive disappearance of thecollective mode as more asymmetric nuclear matter is considered. However, experimentally it is observed that ifa nucleus has a bigger neutron-proton asymmetry, the collectivity in the isovector vibrational mode is stronger.Consequently the amplitudes of the proton and neutron density fluctuations seem to be proportional to the

w xdensity of protons and neutrons, as it has been assumed for prescription B. It has been noticed in 4 that the

Fig. 3. Same as Fig. 1, with bs8 for prescription A.


Fig. 4. Same as Fig. 3, with bs8 for prescription B.

zero sound is undamped at zero temperature due to the repulsive character of the proton-neutron effectiveinteraction. Considering that in asymmetric cases the collectivity of the resonance is stronger, the nucleon-nucleoneffective interaction seems to depend on the proton-neutron asymmetry.

To summarize, we have seen that the asymmetric nuclear matter exhibits stronger collective behaviour thanthe symmetric nuclear matter for small momentum transfer. The zero sound is slightly damped at zerotemperature for small proton-neutron asymmetries because the particle-hole spectrum is spread to higherfrequencies. With the increase of the temperature, the behaviour of the collective mode is the same of that found

w x Ž .with the response function of symmetric nuclear matter 3,4 . For a very high asymmetry coefficient bs8 , thestrength shows a more collective zero sound placed at a still higher energy. In this case, the difference betweenits frequency and the region of the particle-hole spectrum augments preventing the collective mode fromdamping at low excitation energies. Its was also noticed the existence of a small relative enhancement in thelower frequency part of the spectrum. We argued that the effective interaction should depend on theneutron-proton asymmetry as far as the collective mode properties are determined by its characteristics.

Acknowledgements

The author would like to thank C.L. Lima for a reading of the manuscript and M.S. Hussein for indicatingw xRef. 8 . He wishes to thank FAPESP, Brazil, for the financial support.

References

w x Ž . Ž .1 C. Garcia-Recio, J. Navarro, N. Van Giai, L.L. Salcedo, Ann. Phys. NY 214 1992 293.w x Ž .2 F.L. Braghin, D. Vautherin, Phys. Lett. B 333 1994 289; A. Abada, F.L. Braghin, D. Vautherin, in: Proceedings of the third IN2P3-

Riken Symposium on Heavy Ion Collision, Shinrin-Koen, Saitama, 1995, World Scientific, Singapore, in press.


w x3 F.L. Braghin, Ph.D. thesis, University of Paris XI, France, 1995, unpublished.w x Ž .4 F.L. Braghin, D. Vautherin, A. Abada, Phys. Rev. C 52 1995 2504.w x Ž .5 E.S. Hernandez, J. Navarro, A. Polls, J. Ventura, Nucl. Phys. A 597 1996 1.´w x Ž .6 H.A. Bethe, Ann. Rev. Nucl. Part. Sci. 38 1988 1.w x Ž .7 M.S. Hussein, C-Y. Lin, A.F. R de Toledo Piza, Z. fur Phys. A 355 1996 165; H. Sagawa, N. Van Giai, N. Takigawa, M. Ishihara, K.

Ž . Ž .Yazaki, Z. fur Phys. 351 1995 385; C.A. Bertulani, L.F. Canto, M.S. Hussein, Phys. Rep. 226 1993 281.w x Ž .8 J. Meng, P. Ring, Phys. Rev. Lett. 80 1998 460.w x Ž . Ž .9 D. Vautherin, D. Brink, Phys. Rev. C 5 1972 626; Y.M. Engel, D.M. Brink, K. Goeke, D. Vautherin, Nuc. Phys. A 249 1975 215.

w x10 F.L. Braghin, in preparation.w x Ž .11 N. Van Giai, H. Sagawa, Phys. Lett. B 106 1981 379.

14 January 1999


Critical scaling at zero virtuality in QCD

Romuald A. Janik a,c, Maciej A. Nowak b,c, Gabor Papp d,e, Ismail Zahed f´a SerÕice de Physique Theorique, CEA-Saclay, F-91191, Gif-sur-YÕette, France´

b GSI, Planckstr. 1, D-64291 Darmstadt, Germanyc Department of Physics, Jagellonian UniÕersity, 30-059 Krakow, Poland

d ITP, UniÕ. Heidelberg, Philosophenweg 19, D-69120 Heidelberg, Germanye Institute for Theoretical Physics, EotÕos UniÕersity, Budapest, Hungary¨ ¨

f Department of Physics and Astronomy, SUNY, Stony Brook, NY 11794, USA

Received 21 October 1998Editor: J.-P. Blaizot

Abstract

We show that at the critical point of chiral random matrix models, novel scaling laws for the inverse moments of theeigenvalues are expected. We evaluate explicitly the pertinent microscopic spectral density, and find it in agreement withnumerical calculations. We suggest that similar sum rules are of relevance to QCD at the critical temperature, and evenabove if the transition is amenable to a Ginzburg-Landau description. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 11.30.Rd; 11.38.Aw; 64.60.Fr

1. A large number of physical phenomena can bew xmodeled using random matrix models 1 . An impor-

tant aspect of these models is their ability to capturethe generic form of spectral correlations in the er-godic regime of quantum systems. This regime isreached by electrons traveling a long time in disor-

w xdered metallic grains 2 or virtual quarks moving aw xlong proper time in a small Euclidean volume 3 .

In QCD, the ergodic regime is characterized by ahuge accumulation of quarks eigenvalues near zerovirtuality. This is best captured by the Banks-Casherw x <² : < Ž .4 relation qq 'Sspr 0 , where the nonvan-ishing of the chiral condensate in the vacuum signals

Ž .a finite quark density r ls0 /0 at zero virtuality.This behavior is at the origin of spectral sum rulesw x5 , which are reproduced by chiral random matrix

w xmodels 6 . These sum rules reflect on the distribu-w xtion of quark eigenvalues and correlations 7 .

If QCD is to undergo a second or higher orderchiral transition, then at the critical point there is adramatic reorganization of the light quark states nearzero virtuality as the quark condensate vanishes. InSection 2, we suggest that such a reorganization isfollowed by new scaling laws, which are captured bya novel microscopic limit. In Section 3, we use achiral random matrix model with a mean-field transi-tion to illustrate our point. In Section 4, we explicitlyconstruct the pertinent microscopic spectral distribu-tion in the quenched case and compare it to numeri-cal calculations. In Section 5 we argue that if QCD isto be characterized by mean-field universality thenthe present matrix model results are applicable. In

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01498-1

( )R.A. Janik et al.rPhysics Letters B 446 1999 9–1410

Section 6, we suggest that spectral correlations per-sist in the vicinity of the critical point from abovewith new sum rules. Our conclusions are in Section7.

2. The nonvanishing of S in the QCD vacuumimplies that the number of quark states in a volume

Ž . Ž .V, N E sVHdlr l in the virtuality band E aroundŽ .0 grows linearly with E, that is N E ;EV. As a

result, the level spacing DsdErdN;1rV for N;

1, and the eigenvalues of the Dirac operator obeyw xspectral sum rules 5 . During a second or higher

order phase transition S vanishes in the chiral limit.Scaling arguments give S;m1r d at the critical point,

w xwhere m is the current quark mass 8 . It followsw xagain from the Banks-Casher relation 4 , that for

Ž . < <1r dsmall virtualities l, r l ; l to leading order inw x Ž . 1q1rdthe current quark mass 9 . Hence, N E ;VE ,

and the level spacing is now D ;1rV drŽdq1. at)

N;1.For a mean-field exponent ds3, and we have

D sVy3r4, which is intermediate between Vy1 in)

the spontaneously broken phase and Vy1r4 in freespace. At the critical point there are still level corre-lations in the quark spectrum except for the freelimit, corresponding formally to ds1r3. We nowconjecture that at the critical point, the rescaling ofthe quark eigenvalues through l™lrD , yields

)

new spectral sum rules much like the rescaling withw xDs1rV in the vacuum 5 . The master formula for

Žthe diagonal sum rules is given by the dimension-.less microscopic density of states

n s s lim VD r sD 1Ž . Ž . Ž . Ž .) ) )

V™`

and similarly for the off-diagonal sum rules in termsof the microscopic multi-level correlators. Since welack an accurate effective action formulation of QCD

Žat TsT a possibility based on mean-field univer-c.sality is discussed below , the nature and character of

these sum rules is not a priori known, but couldeasily be established using lattice simulations inQCD.

Could these sum rules be shared by random ma-trix models? We will postpone the answer to thisquestion till Section 5, and instead show in whatfollows that the present scaling laws hold at zerovirtuality for chiral random matrix models withmean-field exponents.

3. Consider the set of chiral random plus deter-ministic matrices

im tqAMMs 2Ž .†ž /tqA im

where A is an N=N complex matrix with Gaussianweight, m a ‘mass’ parameter, and t a ‘temperature’parameter. Such matrices or variant thereof havebeen investigated by a number of authors in the

w xrecent past 10 . Their associated density of states is

1² :r l s Tr d lyMM 3Ž . Ž . Ž .

2 N

where the averaging is carried using the weightassociated to the following partition function

w x † Nf yN Tr A A†Z m ,t s dAdA det MM e 4Ž .H

Ž .For tsms0 the density of states is r 0 s1rp ,Ž .while zero for tG1 and ms0. At ts1, r l s

< <1r3l , which indicates that the t-driven transition is

Ž .mean-field with r 0 as an order parameter. In par-ticular, at ts1 the level spacing is D sNy3r4

)

near zero.Ž .Standard bosonization of the partition function 4

yields

w † 2 x †Ž .† N logdet mqP mqP qt yN TrP PŽ .w xZ m ,t s dPdP eH5Ž .

where P is an N =N complex matrix. We mayf f

shift P by the mass matrix P™QsPqm and get

N† † 2w xZ m ,t s dQdQ det QQ q tŽ .H= yN TrQQ†ym Tr QqQ† qm 2w xŽ .e 6Ž .

dropping the irrelevant normalization factor. We willnow specialize to the critical temperature ts1, anddenote the rescaled mass by xs imrD and eigen-

)

value by sslrD . This suggests the rescaling Q)

˜ 1r4™QsN Q, so that1 .† 2 †˜ ˜ ˜ ˜Ž† y Tr QQ i x TrŽQqQ .˜ ˜w xZ x s dQdQ e e 7Ž .2H

reducing to1` 4y rw xZ x s rdr e J 2 xr 8Ž . Ž .2H 0

0

( )R.A. Janik et al.rPhysics Letters B 446 1999 9–14 11

for one flavor. Expanding this integral in powers ofx,

2 kk` 2 mrD kq1Ž .)w x w xZ m sZ 0 1q GÝ 2 ž /' 2k! pŽ .ks1

9Ž .

gives rise to spectral sum rules for the moments ofthe reciprocals of the eigenvalues of MM by matchingthe mass power in the spectral representation of thepartition function,

m2

w x w xZ m rZ 0 s 1q 10Ž .Ł 2¦ ;ž /ll )0 kk 0

where the averaging is done over the Gaussian ran-domness with the additional measure Ł l2.l ) 0 kk

Matching the terms of order m2 yields

1 2s 11Ž .Ý 2 2¦ ; 'l p Dk )l )0k 0

and matching the terms of order m4 gives

21 1 1

y s 12Ž .Ý Ý2 4 4¦ ;¦ ;ž /l l Dk k )l )0 l )0k k 00

Ž . Ž .The relations 11 – 12 are examples of microscopicsum rules at the critical point ts1.

One should note here that the preceding calcula-tion has been performed for the gaussian matrixmodel. It turns out that only if we were to add termsof the form

eyN g 0Tr ŽP P †.2yN g1ŽTrP P †.213Ž .

Ž .to the measure in 5 , they would influence thespectral sum rules. However for N s1 this additionf

Žamounts just to a global shift x™xr 1q2 g q0.2 g which sets the normalization in the sum rules.1

Higher powers of PP † are subleading after rescalingand do not affect the sum rules in the large N limit.

4. The diagonal moments of the reciprocals of therescaled eigenvalues are generated by the micro-

Ž .scopic density 1 , in the limit N™` and l™0 butŽ .sslrD fixed. The microscopic density 1 for N

) f

flavours in a fixed topological sector n: n , can) , N ,nf

w xbe evaluated using supersymmetric methods 1,11 .For example,

1

42 8n s sy k s j s qk s j sŽ . Ž . Ž . Ž . Ž .Ž

) ,0 ,0 2 0 0 22p

yk s j s 14Ž . Ž . Ž ..1 1

Ž . Ž .where k s and j s are given byn n

` s 42 nq1 yz r2j s s dz z J 2 z e 15Ž . Ž .Hn 0 3r4ž /20

s 42 nq2 z r2k s s dz z K 2 z e 16Ž . Ž .Hn 1 3r4ž /2C

and where the integration contour C is the sum ofw Ž . x w Ž . xtwo lines: Cs y 1q i `,0 j 0, 1y i ` . In this

Ž .spirit, the first sum rule 11 reads

` 1 2n s dss 17Ž . Ž .H

) ,1 ,02 's p0

Ž .The second sum rule 12 involves the 2-level micro-scopic correlator for N s1 and ns0 which can bef

obtained using a similar reasoning.Ž .In Fig. 1 we compare the quenched N s1 leftf

Ž .and unquenched N s1 right results to the numeri-f

cally generated microscopic spectral density at ts1using Ns100 size matrices. The agreement sug-gests that the present method of finding the scalingproperties of microscopic spectral distributions canbe used to accurately determine the value of thecritical exponent d in lattice simulations.

5. In QCD the character of the finite temperaturetransition depends crucially on the number of flavors

Ž .N and the fate of the U 1 quantum breaking. Forf A

Ž .Fig. 1. left: n s at ts1 and N s0 for matrices of size) ,0,0 f

Ž . Ž . Ž .Ns100 dots and the theoretical prediction 14 solid line .Ž .right: n s at ts1 and N s1 for matrices of size Ns20

) ,1,0 fŽ . Ž . Ž .dotted , Ns50 dashed and Ns100 solid .


two light flavors, we may assume with Pisarski andw x Ž .Wilczek 12 that the transition is from an SU 2

Ž . Ž .spontaneously broken phase to Z =SU 2 =SU 22Ž . w x;O 4 13 . Choosing the order parameter FssqŽ .itPp vacuum analogous to a ferromagnet , implies

for the Ginzburg-Landau potential

VV F sqm Tr F † qF qg T Tr F †FŽ . Ž . Ž . Ž .0

2†qg T Tr F F q . . . 18Ž . Ž . Ž .Ž .1

at zero vacuum angle u . The dots refer to marginalor irrelevant terms. For a second order transition,Ž .g T ;TyT , which is negative below T and0 c c

Ž .positive above. The O 4 critical exponents follow-Ž .ing from 18 hold within the pertinent Ginzburg-

w x Ž .Landau window 14 and 18 implies a specific setof microscopic sum rules which are not amenable tothe mean-field matrix model we have discussed.They may be readily established using Dirac spectrafrom Lattice QCD simulations.

Outside the Ginzburg-Landau window the criticalw xexponents are in general mean-field 14 and our

matrix model arguments hold. Indeed, for TsT thecŽ .potential 18 when reduced to the space of constant

w xmodes, is reminiscent of the one discussed above 9 .If we note that the measure on the manifold withrestored symmetry is eyb V3 VV 'eyV VV , we conclude

Ž .that 18 is enough to accommodate for the leveldrŽdq1. Ž .spacing D s1rV with ds3 mean-field .

)

Ž .After the rescaling xs imrD we recover 7 with)

g s1r2 and the proper identification of the mani-1

fold.

6. For temperatures near T from above, we cancŽ .use 18 to define new sum rules for the quark

eigenvalues in the sector with zero winding numberw x Ž .5 . In particular N s2f

1 1 1 du2p

sÝ H2 2¦ ; 2 N 2pV l 0fkl )0k 0

=

2iuiuNf † yTr e F qF e Nfž /ž /− <

19Ž .

The rhs measures the variance in the scalar directionŽ . yV VVon an invariant O 4 manifold with e as a

measure. As T™T from above, the scalar suscepti-cŽ Ž ..bility averaged over ‘u-states’ rhs of 19 diverges

since s and p become degenerate. These modes arethe analogue of the ones originally discussed by

w xHatsuda and Kunihiro 15 using an effective modelof QCD.

In the case where the fluctuations are not impor-Ž . Ž .tant mean-field , then 19 can be readily assessed

'by rescaling F™ V F and noting that the quarticŽ .contribution in 18 becomes subleading in large V.

Hence,

1 1 1s 20Ž .Ý 2¦ ;V 2 g Tl Ž .0kl )0k 0

ŽNear T from above it is seen to diverge as 1r Tyc. Ž .T with the critical exponent gs1 mean-field .c

Ž .In 20 the eigenvalues are of order 1 due to thegap in the spectrum, hence 1rV normalizes a sum ofV terms and the sum rules carries information on theentire spectrum, not just the microscopic part. Theyare not universal in the sense of matrix theory.

The possibility of sum rules in the vicinity of thecritical temperature from above reflects on the per-sistence of the level correlations in the Dirac spec-

Ž .trum in relation to the O 4 manifold despite the gapdeveloping in the eigenvalue density. The latter isdue to the fact that near zero virtuality the accumula-tion of eigenvalues is not commensurate with thevolume V. At high temperature, the quark eigenval-ues are typically of order T , and both p and s

correlations are dissolved in the ‘plasma’ with trivialsum rules. In QCD we expect this to take place at a

w xtemperature T;3T 16 .c

These ideas can be tested in the context of amean-field transition using again a chiral randommatrix model with one flavor. We may use the

Ž .bosonized form of the partition function 6 , but now' 'instead introduce the variable ys N im; N l, and

˜ 'rescale Q by Q™QsQ N . This leads to thefollowing expression for the partition function

1` 2 22 N y 1y r yž /Z y s t rdre J 2 yr eŽ . Ž .t 2H 00

2 N 2 1p t t 2y ys P e 21Ž .t 2y12N t y1

Expanding the partition function in powers of y

( )R.A. Janik et al.rPhysics Letters B 446 1999 9–14 13

leads again to modified sum rules, the simplestexample being

1 1s 22Ž .Ý 2 2¦ ;Nl t y1kl )0k 0

which is seen to diverge at ts1 as expected.The comparison to numerical simulation is shown

in Fig. 2 using a random set of matrices MM dis-tributed with a Gaussian measure. For high tempera-tures or large matrix sizes the agreement is good. Atthe critical temperature, the finite size effects areimportant. Amusingly, we note the drop by twoorders of magnitude at t;3s3t .c

ŽThe Green’s function both for quenched the.mass playing the role of an external parameter and

one flavor chiral random matrix model in the rescaledvariables may be readily obtained,

1 1 1 yG y s sy 23Ž . Ž .Ý 2' ' ' t y1N yy N l Nii

Since for T)T the eigenvalue spectrum develops ac

gap, there are no eigenvalues for small values of y,hence the resolvent is purely real. For finite sizeshowever, y may get out of the gap and our resultbreaks down as well as the scaling arguments. Thisis seen in Fig. 3, where for temperatures slightlyabove the critical one we are entering the nonzeroeigenvalue density part of the spectra. This effect isshifted for higher values with increasing temperatureand matrix size.

7. Using arguments based on universality we havesuggested that the QCD Dirac spectrum may exhibit

Ž . Ž .Fig. 2. The result 22 solid line checked against numericalŽ . Ž .simulations using Ns20 open squares , Ns50 open circles

Ž .and Ns100 open triangles random matrices.

Ž . Ž .Fig. 3. Scaled and quenched resolvent 23 solid line in compari-Ž .son to a numerical simulation with Ns20 long dashes , Ns50

Ž . Ž .short dashes and Ns100 dotted line random matrices.

universal spectral correlations at TsT that reflectc

on the nature of the chirally restored phase. We notethat these correlations are different from the ones

w xconsidered in 17 , which are valid for T-T butc

not in the vicinity of TsT where they vanish. Toc

probe the correlations around the chiral phase transi-tion, one has to consider the new scaling regimedescribed in this paper. To emphasize its novelty, wewill refer to it as ‘‘critical scaling’’. To illustrate ourpoints, we have used a chiral random matrix model.Although the matrix model is based on a Gaussianweight, we provided qualitative arguments for whythe results are insensitive to the choice of the weightat the critical point. In QCD this can be readily

w xchecked using our recent arguments 3 at finitetemperature, since the closest singularity to zero inthe virtuality plane is persistently ‘pionic’ for TFTcw x8 .

The existence of a microscopic spectral density atTsT for QCD opens up the interesting possibilityc

of measuring both the critical temperature T and thec

critical exponent d by simply monitoring the perti-Ž .nently rescaled distribution n s of eigenvalues in

)

lattice QCD simulation. We have also suggested thatspectral correlations persist near T from above.c

Clearly, most of our observations are subject to finitesize effects whose analysis is beyond the scope ofthis work.

Acknowledgements

This work was supported in part by the US DOEgrant DE-FG-88ER40388, by the Polish Government

Ž .grant Project KBN grants 2P03B04412 and2PB03B00814 and by the Hungarian grants FKFP-0126r1997 and OTKA-T022931. RAJ is supported

Ž .by the Foundation for Polish Science FNP .


Note addedAfter completing this work we noticed the paper

w xby Brezin and Hikami 18 , where the issue regard-´ing a new universality class at the closure of the gapin the eigenvalue distribution is also discussed usingdifferent arguments. Their results are carried fornon-chiral ensembles, and hence different from ours.However, the similarities between say our resultsŽ . Ž .15 , 16 and their integral forms suggest relation-ships between the chiral and non-chiral results, thatare worth unravelling.

References

w x1 K. Efetov, Supersymmetry in Disorder and Chaos, Camb.UP, NY, 1997, and references therein.

w x Ž .2 D.J. Thouless, Phys. Rep. 13 1974 93.w x3 R.A. Janik, M.A. Nowak, G. Papp, I. Zahed, Phys. Rev. Lett.

Ž .81 1998 264.

w x Ž .4 T. Banks, A. Casher, Nucl. Phys. B 169 1980 103.w x Ž .5 H. Leutwyler, A. Smilga, Phys. Rev. D 46 1992 5607.w x Ž .6 E. Shuryak, J. Verbaarschot, Nucl. Phys. A 560 1993 306.w x Ž .7 J. Verbaarschot, I. Zahed, Phys. Rev. Lett. 70 1993 3852.w x Ž .8 A. Kocic, J. Kogut, M. Lombardo, Nucl. Phys. B 398 1993

376.w x Ž .9 R.A. Janik, M.A. Nowak, I. Zahed, Phys. Lett. B 392 1997

155.w x Ž .10 E. Brezin, S. Hikami, A. Zee, Phys. Rev. E 51 1995 5442;

Ž .A. Jackson, J. Verbaarschot, Phys. Rev. D 53 1996 7223;T. Wettig, A. Schafer, H. Weidenmuller, Phys. Lett. B 367¨Ž .1996 28; M.A. Nowak, G. Papp, I. Zahed, Phys. Lett. B

Ž .389 1996 137.w x Ž .11 K. Efetov, Adv. Phys. 32 1983 53.w x Ž .12 R. Pisarski, F. Wilczek, Phys. Rev. D 29 1984 338.w x Ž .13 The Z factor is from the persistent U 1 breaking effects.2 Aw x14 M. Rho, M.A. Nowak, I. Zahed, Chiral Nuclear Dynamics,

WS 1996; and references therein.w x Ž .15 T. Hatsuda, T. Kunihiro, Phys. Rev. Lett. 55 1985 158.w x Ž .16 I. Zahed, Act. Phys. Pol. B 25 1994 99.w x17 A.D. Jackson, M.K. Sener, J.J.M. Verbaarschot, Nucl. Phys.

Ž . Ž .B 479 1996 707; Nucl. Phys. B 506 1997 612; T. Guhr,Ž .T. Wettig, Nucl. Phys. B 506 1997 589.

w x18 E. Brezin, S. Hikami, cond-matr9804023.´

14 January 1999


Pions in isospin asymmetric matter and nuclear Drell-Yanscattering

C.L. Korpa a, A.E.L. Dieperink b

a Department of Theoretical Physics, Janus Pannonius UniÕersity, Ifjusag u. 6, 7624 Pecs, Hungaryb Kernfysisch Versneller Instituut, Zernikelaan 25, NL-9747AA Groningen, The Netherlands

Received 20 August 1998; revised 9 November 1998Editor: J.-P. Blaizot

Abstract

Using a self-consistent delta-hole model the pion propagation in isospin asymmetric nuclear matter is studied. Inneutron-rich matter, corresponding to heavy nuclei, a significant difference in positive and negative pion light-conedistributions is obtained leading to a nuclear enhancement of up antiquark distribution compared to the down antiquark one.This means that the sea-quark asymmetry in the free nucleon cannot be extracted model independently from an experimenton a neutron-rich nucleus. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 21.65.q f; 24.85.qp; 13.40.-f

Keywords: Isospin-asymmetric medium; Drell-Yan scattering; Antiquark flavour asymmetry

1. Introduction

The meson-cloud model plays an important rolein dealing with non-perturbative Quantum Chromo-dynamics effects in the nucleon. It has been used byseveral groups to interpret momentum distributions

w xof sea quarks in the nucleon 1–5 , measured in deepinelastic scattering. The standard approach is based

w xupon the Sullivan process 6 , in which the onlyessential parameter is the cut-off in the pion-nucleon-nucleon vertex.

Originally the emphasis was mainly on the de-scription of the isoscalar uqd distributions. Morerecently, since the observed violation of the Gottfriedsum rule, showing an excess of d over u in proton,

Ž . Ž .the interest focussed on the properties of u x yd xin the nucleon, whose x dependence was measured

w xrecently 7 . The asymmetry in antiquark distribu-tions has been interpreted mostly in terms of the

Ž w xpion-cloud model a review is given in Ref. 8 , seew x.also 9 , and also in a soliton model in the large-Ncw xlimit 10 .

Since pion properties are strongly affected by thenuclear medium, the pion cloud plays also an impor-tant role in modelling nuclear effects on deep inelas-tic processes. It was used in the past in connection

w xwith the EMC effect 11–13 , leading to a fewpercent enhancement of the structure-functions ratioaround xs0.2. Similarly, most approaches pre-dicted a nuclear enhancement of the pion field and


( )C.L. Korpa, A.E.L. DieperinkrPhysics Letters B 446 1999 15–2116

Ž . Ž .hence the u x and d x distributions, leading to aŽ .noticeable increase in the predicted Drell-Yan DY

cross-section ratio. On the other hand experimentallyw xthe DY scattering on nuclear targets 14 did not

show much evidence for enhancement over nucleontargets. Several papers have dealt with this discrep-ancy, pointing out different mechanisms leading to

w xreconciliation with the measurements 4,5,15,16 .For the purpose of presenting our results we find

it convenient to separate them into two aspects. Thecalculation of experimentally measured proton-nucleus to proton-deuteron DY cross-section ratio isour first aim. The isospin asymmetric medium af-fects the various isospin states of the nucleon’s pioncloud differently, leading to an excess of up anti-

Žquarks over down ones even if the distributions in.free proton are identical for nuclei with more neu-

trons than protons. Since the charges of up and downquarks are different, the effect also shows up in thenuclear DY scattering cross section. In addition topresenting the DY ratio, one can also investigate

Ž .possible nuclear effects on the difference u x yŽ .d x for a nucleon in the medium. It has been

pointed out by Kumano that in a neutron-excessnucleus there could be medium effects contributingto this difference, coming from flavor-asymmetricparton recombination in the small Bjorken-x regionw x17 . The mechanism he considered contributes only

Ž .at relatively small x values x-0.1 and the effectŽis also rather small 2–10% of the observed asym-

.metric sea in the nucleon .In the present calculation we extend the effective

field theory approach to compute pion properties inw xisospin-symmetric nuclear medium 18 to the case

of arbitrary proton and neutron densities. Then wew xuse the pion-cloud model 16 to compute the

medium-modified quark and antiquark distributionsand the proton-nucleus DY cross section. We assumethat the medium effects solely originate from a mod-ification of the pion cloud 1. The nucleons are treated

1 w xIt is known 19 that the pion and other mesons cloud cannotaccount completely for the antiquark distribution of the nucleon.For example, gluon splitting gives a sizable contribution at smallx and large Q2, but this contribution is approximately flavor

w xsymmetric 19 and thus should not modify our results signifi-cantly.

in mean field approximation, while the delta-isobarand the pion are dressed self-consistently. The ef-fects of short-range baryon repulsion are included

X Ž X X X .through Migdal’s g parameters g , g , g .NN ND DD

Sensitivity of the pion distribution to the delta-holew xself-energy was already recognized in Ref. 11 , thus

we carried out the computation with the self-con-sistently determined in-medium delta-isobar spectral

w xfunction 18 .We compute the pion light-cone momentum dis-

tributions separately for the three charge states as aŽfunction of the asymmetry parameter b' Ny

. yZ rA. For b)0 the p distribution exceeds that ofp 0, which in turn is larger than the pq distribution,as expected on the basis of particle-hole self-energyrelationships.

2. Drell-Yan cross section

The Drell-Yan cross section for the process pqAq y Ž 2™m m X is given by suppressing the Q depen-.dence

4paK x , xŽ .1 22 2d ss e q x q xŽ . Ž .Ý f f 1 f 29sx x1 2 f

qq x q x dx dx , 1Ž . Ž . Ž .f 1 f 2 1 2

where the sum is over all flavors, and x , x are the1 2

longitudinal momentum fractions carried by quark ofthe beam and target nucleons, respectively. By asuitable selection of kinematics the values of x , x1 2

w xcan be deduced from experiment 14 .If we consider the region x )0.3 when the1

antiquarks in the projectile play a negligible role, theratio of the proton-nucleus to proton-deuteron DYcross-sections takes on the form

2 ds pA u x qd xŽ . Ž . Ž .Nr A 2 Nr A 2R ' sA d A ds pdŽ . u x qd xŽ . Ž .p 2 p 2

u x yd xŽ . Ž .Nr A 2 Nr A 2q f x , 2Ž . Ž .1 u x qd xŽ . Ž .p 2 p 2

Ž . Ž Ž . Ž .. Ž Ž .where f x ' 4u x y d x r 4u x q1 p 1 p 1 p 1Ž ..d x is close to unity. Here u and d are anti-p 1 p p

( )C.L. Korpa, A.E.L. DieperinkrPhysics Letters B 446 1999 15–21 17

quark distributions in the free proton, while uNr A

and d are the antiquark distributions per nucleonNr A

in the nucleus, differing from the free-nucleon distri-butions by the medium modified pion-cloud contri-bution. Denoting the latter by d u and d d thep r A p r A

DY ratio becomes

1R s1q d u xŽ .½A d p r A 2u x qd xŽ . Ž .p 2 p 2

qd d x q b f x d x yu xŽ . Ž . Ž . Ž .p r A 2 1 p 2 p 2

q d u x yd d x rb . 3Ž . Ž . Ž .5ž /p r A 2 p r A 2

We see from the above expression that apart fromnuclear effects, leading to nonzero d u andp r A

w xd d , for b/0 there is a nucleonic one 4 , stem-p r A

ming from the nonzero value of the antiquark-distri-bution difference d yu in the free proton. Thisp p

underlines the necessity to use parton distributions inŽaccordance with latest d yu observations for dis-p p

.cussion on this point see Section IV . For the rela-tively small asymmetries of interest for stable nuclei

Ž .the ratio d u yd d rb appearing in thep r A p r A

above expression is practically independent of b ,leading to a linear dependence of R on it.A d

For the change of antiquark distributions due topion-cloud modification we use the convolution for-

w xmula 16

dyA a ap r A pd q x s d f y q xry ,Ž . Ž . Ž .ÝHf ,p r A b fyxa

p aŽ .where d f y , given byb

1qba a ap p r n r A p r nd f y s f y y f yŽ . Ž . Ž .Ž .b 2

1yb a ap r pr A p r pq f y y f y ,Ž . Ž .Ž .2

4Ž .

represents the change of the pion light-cone-momentum distribution per nucleon in the medium.

p a r n r A Ž p a r pr A. p a r n Ž p a r p.f f and f f denote thea Ž .distribution of p per neutron proton in medium

and in free space, respectively. They are discussed inthe next section.

3. Pions in isospin asymmetric nuclear medium

We consider a model consisting of pions, nucle-ons and delta-isobars in an infinite, spatially uniformsystem at zero temperature. The proton and neutrondensities are given through their chemical potentialsm and m . The equilibrium conditions for nucleons,p n

delta-isobars and pions imply that the chemical po-tential for the neutral pion is zero, while those ofcharged pions are: m ysym qsm ym . The an-p p n p

tiparticles of nucleons and isobars are neglected, buta relativistic kinematics is used. The nucleons arefurther treated in the mean-field approximation, with

Žmomentum-independent mass M yM , M y) p p ) n

. Ž .M and energy shifts c ,c modelling their bind-n p n

2 2Ž . (ing, i.e. E p s M qp qc for neutrons andn ) n n

2 2Ž .E p s M q p q c for protons. The(p ) p p

Schwinger-Dyson equations without vertex correc-tions are then solved self-consistently for the delta-

w xisobar and pion 18 . The pion self-energy consists ofthe particle-hole and delta-hole contributions, withboth imaginary and real parts taken into account,assuring correct analytical properties. Short-rangebaryon repulsion is taken into account throughMigdal’s gX , gX , gX parameters. The sum-rulesNN ND DD

for spectral functions of pions and deltas are checkedand found to be satisfied to 1–2%.

The pion light-cone momentum distributions arethen calculated, using the in-medium pion self-en-ergy. Direct calculation of the diagrams correspond-ing to the pion emitted by the in-medium nucleon

w xproceeds analogously to Ref. 16 , but separately forthe three charge states of the pion and taking into

Žaccount different neutron and proton densities andthus different M , M effective masses and c ,c

) p ) n p n.energy shifts .

Ž .It is well known that f y can be expressedcompactly in terms of an integral over the spin-iso-spin response function or imaginary part of the pionpropagator. It is also possible to compute it directly

w xfrom summing contributing diagrams 16 , whichrepresent emission of a dressed pion by a nucleon.The latter procedure we prefer for numerical reasons,


since it needs computation of the pion self-energy insmaller energy-momentum region. The schematic ex-pression for the pion light-cone distribution per nu-cleon has the form

2 d3p d3pX

p r pr Af y sŽ . H Hb 3 3 Xr 2p 2p 2 E pŽ . Ž . Ž .

=k 0 qk 3

2p< <d yy X k ,Ž .bž /M

where Xp represents the sum of relevant diagrams,b

Ž . Ž X. Xr the nucleon density, k sE p yE p ,kspyp0

Ž X. Ž .and p p is the momentum of incoming outgoingnucleon.

The full expression generalizing the case forw x Žisospin-symmetric medium from 16 reads for the

q .special case of p on proton :

3 yg 2 Mq p NN )p r pr Af y sŽ .b 32 2p p MŽ .F p

=2 2p p ypFp ( Fp 3

dp p dpH H3 H Hyp 0Fp

` 2pX X 2 2= p dp du k F kŽ .H HH H p NNXp 0Hmin

=1

2˜< <qD k ,k , 5Ž . Ž .X p 0z

where p is the Fermi momentum of the proton, MF pŽ .the nucleon’s mass, M ' M qM r2, g

) ) p ) n p NN0 Ž .the p NN coupling and F k the form factor,p NN

X ˜ qwhile u is the angle between p and p . D isH H p

the full pion propagator, corrected for the presenceof four-fermion couplings through Migdal’s gX pa-

˜ Xw xrameters, given as D in Ref. 16 , p sp N H minŽ X Ž 2 2 .1r2 Ž X 2 . 2 .1r22 z M M qp y z q1 M if the

) ) n F n ) p

argument of the square root is positive, otherwise isX Ž Ž . .zero, and z ' yMyqp qE p yc rM . The3 n )

expression for py is obtained by swapping theindices n and p and for neutral pions the totaldistribution is a sum of two terms corresponding toemission by proton or neutron.

The difference in distributions for the variouscharge states of pion basically comes from twofactors. One is the Pauli blocking of the outgoingnucleon, which in neutron-rich medium restricts

q Žemission of p from a proton, creating a neutron in. y Žthe final state more than the emission of p since

.a proton appears in this case in the final state . Theother effect is the dressing of the pion propagator, inwhich the particle-hole and delta-hole self-energiesenter. Since the dominant contribution comes from

Žthe particle-hole contribution for N)Z neutron. ydensity larger than the proton one the p propaga-

q Žtor is affected more than the one of the p more.details are given in the next section .

While the delta has been shown to play an impor-tant role in the asymmetry for free nucleons, the

w xmedium effects are negligible 16 and not includedhere, which is partially a consequence of the use of asoft pion-nucleon-delta form-factor, as obtained from

w xa fit to pion-nucleon scattering 18 . Since the isobar’scontribution for b/0 to d u yd d is of the oppo-p p

Ž .site sign compared to the contribution of Eq. 5 , asdiscussed in the next section, at small x inclusion ofthis term might result in a small decrease of thecalculated isospin-asymmetry effect.

4. Results and discussion

For numerical calculation we used the proton toneutron ratio of tungsten, for which there are mea-

w xsurements of the DY cross section 14 . The asym-metry parameter in this case is bs0.196 and for theFermi momenta of protons and neutrons we chosep s238 MeV and p s272 MeV, correspondingF p F n

to total nucleon density slightly below the saturationdensity. To take into account the different binding ofprotons and neutrons for the energy shifts we takec s40 MeV, c s42 MeV, with the common effec-p n

tive mass M sM s0.85 GeV, thus assuring the) n ) p

correct asymmetry energy of 28 MeV. For the pion-nucleon-nucleon vertex we use a dipole form-factorwith cut-off Ls1 GeV. We checked that varyingthe cut-off in the range 0.9–1.1 GeV does not changeappreciably the medium effect on the pion.

In Fig. 1 we present the pion light-cone-momen-tum distributions for nucleons in medium and in free

Ž .space upper four curves , as well as the excess pionŽ . q Ž y.distributions lower three curves . The p p

Ž .distributions are per proton neutron , while that ofp 0 is per average nucleon in the medium. Theneutral pions see little difference between an


Fig. 1. Pion light-cone-momentum distributions. Full line is forfree nucleon, long dashed for pq, short-dashed for py, dot-dashedfor p 0, with gX s0.6, gX s gX s0.3. The three lower curvesNN ND DD

show the excess distributions with respect to the free nucleon.

isospin-symmetric and an isospin-asymmetric nu-clear medium, as long as the total nucleon density is

Žthe same actually, the different mean-field shifts for.proton and neutron may lead to a very small effect .

However, for the case of neutron excess, the py

distribution per neutron is larger than the pq distri-bution per proton. This comes from a difference inparticle-hole self energies, which is mainly responsi-ble for the light-cone-momentum distributions. It iseasy to understand the difference if we look at theimaginary part of the self energy as a function ofpion’s energy. If the energy is positive, the pion canexcite a neutron from the Fermi sea to become a

Ž .proton above the proton Fermi sea if its charge ispositive. A negative pion can make from a proton inthe Fermi sea a neutron above the neutron Fermi sea.Since there are more neutrons than protons in theFermi sea, the absolute value of the imaginary partof the pq self energy will be larger than that of py.

Ž .Bearing in mind that in expression 5 the pionŽenergy is negative for the dominating part of the

.integraton region and using the relation thatŽ . Ž .q yIm P v,k s Im P yv,k we arrive at the in-p p

pyr n r AŽ . pqr pr AŽ .equality f y ) f y , in accordancewith the numerical computation. We mention that forthe delta-hole self-energies the relationship betweenthe py and pq is the opposite to that of particle-hole

y Žone, i.e. for positive energy that of p is larger in. qabsolute value than the self-energy of p ; a conse-

quence of different isospin factors in the pion-nucleon-delta vertex. We see that for the studiedasymmetry there is a significant difference in thedistributions of three charge states. The distributionin isospin symmetric medium is very close to the p 0

distribution.The valence distribution of negative pions con-

tains up antiquarks and thus in neutron-rich matterthey outnumber the down antiquarks present in thepositive pions, due to nuclear effects on the pionclouds. A comparison of this effect to the recently

w x Ž . Ž .measured 7 d x yu x difference for proton isp pŽpresented in Fig. 2. It shows the quantity d u yp r A

. Ž .d d rb compared to the mentioned d x yp r A pŽ . Ž .u x , to which it is added in the expression 3 forp

Ž .the DY ratio R . The quantity d u yd d rbA d p r A p r A

does not change appreciably with b up to its valueof 0.2, but it is sensitive to the gX parameters.Existing experimental and theoretical informationw x X15 suggests values of 0.55–0.7 for g and 0.3–0.4NN

for gX and gX . These values give results consis-ND DD

tent with observed DY scattering for the isospin-w xsymmetric calculation 16 , and we use these values

also in the present case. Taking momentum depen-X Ž w x.dent g parameters as suggested in Ref. 15 could

change the details of our results. Exploration of thisand other effects we leave for a future publication.

ŽFig. 2. The antiquark-distribution difference, d u yp r AX.d d r b per nucleon in the medium. Full line is for g sp r A NN

0.6, gX s gX s0.3, dashed line for gX s0.5, gX s gX s0.3ND DD NN ND DD

and dash-dot line for gX s0.5, gX s gX s0.4. The dotted lineNN ND DD

shows d yu for comparison.p p


Ž .Fig. 3. DY cross-section ratio. Full dash-dot line is for asymmet-X Ž X .ric nuclear medium with b s0.196 and g s0.6 g s0.5 ,NN NN

dotted line for symmetric matter with gX s0.6; in both casesNN

gX s gX s0.3. The dashed line is for b s0.196, but withoutND DD

w xnuclear pion effects. Experimental results from ref. 14 for W areshown as points with error bars.

The free nucleon parton distributions are takenw xfrom Ref. 20 . They fit the d yu and d ru ofp p p p

w xRef. 7 quite well up to xs0.15. However, atlarger x the measured difference d yu and espe-p p

w xcially the ratio d ru are poorly fitted 7 . Top p

correct for this discrepancy, which would cause asignificant increase in calculated R , we impose aA d

Ž . Ž . Žconstraint d x su x for xG0.3 by using thep p.arithmetic mean value and interpolate linearly be-

tween xs0.15, when the unmodified distributionsare used, and the point xs0.3. In this way, theemployed distributions fit the measurements of Ref.w x7 very well. From Fig. 2 we see that the two terms,

Ž .whose sum appears in the square bracket of Eq. 3 ,are of comparable magnitude, i.e. the nuclear effectof the isovector part of the DY ratio plays as impor-tant role as the nucleon antiquark asymmetry.

Since the squared charge of up antiquarks is fourtimes that of down antiquarks, the former enter in theDY cross section with correspondingly larger weight.This implies an enhancement of the DY cross-sec-tion ratio for neutron-rich medium compared to theisospin symmetric case. In Fig. 3 the ratio of cross-

Ž .sections per nucleon is shown as a function of x ,2

where both in the numerator and denominator inte-gration over x is performed for x )x q0.2, cor-1 1 2

w xresponding to the experimental situation of Ref. 14 .Ž .The full line dash-dot line corresponds to bs0.196

X Ž X .and g s0.6 g s0.5 , dotted line is for bs0,NN NN

while the dashed line is for asymmetric mediumŽ .bs0.196 , but without medium effect on the pion

w xcloud. Measurements from Ref. 14 for W are shownas points with error bars. The errors are too large forany definite statement to be made on the isospin-asymmetry effect.

We mention that the difference for larger x2

values between the bs0 case of Fig. 3 in thew xpresent work and Fig. 8 of Ref. 16 is a consequence

of the use of a more realistic dipole pion-nucleon-nucleon form-factor in the present calculation, and toa smaller extent due to a different set of parton

w xdistributions 20 .We remark that the small pion excess probability

found in the present work would also lead to a pioncontribution to the EMC ratio for x;0.1 which is

w xsmaller than in some other approaches 13 . How-ever, this ratio is mostly sensitive to the nucleonself-energy in the medium and the role of pions isdifficult to isolate.

.We conclude: i a simultaneous experiment on aneutron-rich nucleus and an NfZ nucleus with a5% accuracy should in principle be able to isolate

Žthe see up-down asymmetry term proportional to b

Ž .. .in Eq. 3 ; ii about 50% of the uyd difference in anucleus comes from nuclear effects. Therefore theassumption that nuclear effects are negligible as in

w xRef. 4 cannot be justified.

Acknowledgements

We acknowledge useful discussions with S. Ku-mano. This research was supported in part by an

Ž .NWO Netherlands fellowship and the HungarianŽ .Research Foundation OTKA grant T16594. This

work is also part of the research program of the‘‘Stichting voor Fundamenteel Onderzoek der Ma-

Ž .terie’’ FOM with financial support from the‘‘Nederlandse Organisatie voor Wetenschappelijk

Ž .Onderzoek’’ NWO .

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Ž .Phys. A 593 1995 295.w x Ž .16 A.E.L. Dieperink, C.L. Korpa, Phys. Rev. C 55 1997 2665.w x Ž .17 S. Kumano, Phys. Lett. B 342 1995 339.w x Ž .18 C.L. Korpa, R. Malfliet, Phys. Rev. C 52 1995 2756.w x19 W. Koepf, L.L. Frankfurt, M. Strikman, Phys. Rev. D53

Ž .1996 2586.w x Ž .20 H.L. Lai, Phys. Rev. D 55 1997 1280.

14 January 1999


Single particle signatures in high-spin, quasicontinuum states in193,194Hg from g-factor measurements

L. Weissman a,1, R.H. Mayer b,2, G. Kumbartzki b, N. Benczer-Koller b,3, C. Broude a,J.A. Cizewski b, M. Hass a, J. Holden b, R.V.F. Janssens c, T. Lauritsen c, I.Y. Lee d,

A.O. Macchiavelli d, D.P. McNabb b,4, M. Satteson b,5

a Weizmann Institute of Science, RehoÕot 76100, Israelb Department of Physics and Astronomy, Rutgers UniÕersity, New Brunswick, NJ 08903, USA

c Physics DiÕision, Argonne National Laboratory, Argonne, IL 60439, USAd Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Received 17 November 1998; revised 25 November 1998Editor: J.P. Schiffer

Abstract

The average g factors of high spin, high-excitation energy, quasicontinuum structures in 194,193Hg were measured byobserving the precessions of the angular distributions of g-ray transitions in several normal-deformation bands that coalescein the decay of the entry distribution of states. The average g factors of the states leading to the three main bands in the193,194 ² Ž193 .: Ž . ² Ž194 .: Ž .Hg isotopes were: g Hg sq0.19 1 and g Hg sq0.26 1 , respectively. These average g factors aresmaller than the average of the g factors of the high energy states in the three superdeformed bands of 194Hg,² Ž 194 .: Ž .g SD; Hg sq0.41 8 . While the nucleus in the superdeformed well behaves like a rigid rotor, the present resultsdemonstrate the important role played by multiple, quasiparticle neutron configurations in the structure of normal-deforma-tion, highly-excited nuclear states. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 21.10.Ky; 27.80.qw

Keywords: g factors; High spin; Normal deformation structures; 193,194 Hg

1 Present address:Instituut voor Kern- en Stralingsfysica,Katholieke Universiteit, Leuven, B-3001 Belgium.

2 Present address: School of Chemical Engineering, PurdueUniversity, West Lafayette, IN 47907.

3 E-mail: [email protected] Present address: Lawrence Livermore National Laboratory,

Livermore, CA 94551.5 Present address: Interventional Innovations Corp., St. Paul,

MN 55113.

The light 190 – 196 Hg isotopes exhibit a variety ofcoexisting nuclear shapes ranging from weakly-de-formed oblate spheroids to superdeformed prolateellipsoids. Both non-collective neutron excitationsand high-J, high-K proton-hole configurations have

w xbeen observed 1–8 . Most of these structures decayby E2 transitions, but the highest energy level se-quences consist of mixed dipole and quadrupole


( )L. Weissman et al.rPhysics Letters B 446 1999 22–27 23

transitions. These observations have been confirmedby cranked Nilsson-Strutinsky calculations and totalRouthian surface calculations based on a deformedWoods-Saxon potential and the Strutinsky shell-model correction formalism with a monopole-pairing

w xinteraction 9,10 . The highest energy bands ob-w xserved 5,6 associated with the presence of quasipar-

ticle states involving valence neutrons in i , f13r2 5r2

and p orbitals and protons in h and d3r2 11r2 3r2

orbitals, support the view that multiple particle exci-tations play a dominant role, even at very high spins.However, the nature of unresolved states at yethigher excitation energy, decaying via a quasicontin-uum g spectrum has been characterized in a few

w xcases only 11 . In general, the quasicontinuum spec-tra exhibit strong E2 characteristics and are thoughtto be mainly collective. In the mass As150 region,these strong quadrupole collective excitations feedinto states decaying via a mix of dipole and

w xquadrupole transitions 11 . Fluctuation analysis ofg–g energy spectra have provided evidence of

w xdamped rotational motion at very high energy 12 .However, g-factor measurements can most sensi-tively determine the single-particle couplings to thecollective motions, as g factors distinguish uniquelyproton and neutron contributions to the total angularmomentum.

Magnetic moment determinations of discrete, pi-cosecond, high spin states in nuclei produced infusion-evaporation reactions are scarce because ofthe experimental difficulties associated with these

w xmeasurements 13–16 . Measurements of magneticmoments of states in the pre-yrast continuum are

w xeven scarcer 13,17–20 . The experimental signal, inthe form of a precession of the angular distributionof decay g rays, is very small. For such cases, theprecession can be detected via the use of the tran-sient magnetic field which originates from rapidlyfluctuating hyperfine fields acting on swift ions

w xtraversing a ferromagnetic material 21 . The mainprocesses which take place after a fusion evaporationreaction together with a schematic view of the kine-matics of the reaction products inside the target aresketched in Figs. 1 and 2.

Under the experimental conditions presented here-after, the final nucleus is populated, after fast evapo-ration of a few neutrons, in states that exhibit a broaddistribution in excitation energy and spin and that,

Fig. 1. Schematic illustration of the g –decay history of 194 Hgreaction products.

furthermore, are spin-aligned by the reaction. Thevery high level density in the vicinity of the entrypoint results in a quasicontinuum spectrum of unre-solved g-ray cascades that proceed via numerousdecay paths. These paths eventually merge into a fewdistinct bands and finally end up decaying into theyrast levels. The higher, unresolved states are popu-lated during the traversal through the gadoliniumferromagnetic layer, where the transient field is oper-ational. The precessions of the nuclear spins in thetransient field are subsequently probed by observingthe precession of the discrete, low-lying transitionsthat carry the final alignment of the nucleus as itemerges from the ferromagnetic layer. It is possible,by using a large Ge array and applying selectedgating conditions to high-multiplicity events, to iso-late specific decay paths in the g cascades anddetermine the average g factors of the states leadingto these lower-energy discrete bands. The absolutevalues of resulting g factors depend on the

w xparametrization of the transient field 21 , but therelatiÕe g factors of different ensembles of states areindependent of the uncertainties in the transient hy-perfine field.

( )L. Weissman et al.rPhysics Letters B 446 1999 22–2724

Fig. 2. Schematic time history of g –decays inside the target.

This paper presents the first experimental mea-surement of the g factors of the high spin, quasicon-tinuum distribution of states in 193,194 Hg. A precursorexperiment designed to test the feasibility of tran-sient field measurements with large g-ray arrays wascarried out on 193Hg at the Eurogam array. Theanalysis of data, albeit of considerably less statisticalsignificance than the data obtained in this experi-ment, confirmed the applicability of the techniquew x22 .

The experiment was carried out at the 88’’ Cy-clotron at Lawrence Berkeley National Laboratory

w xwith the Gammasphere array 23 which had, at thetime, ninety Compton-suppressed Ge detectors. The

150 Ž48 .193,194reaction Nd Ca,5n-4n Hg at 203 MeV wasused to populate high-spin levels and to provide therecoiling Hg nuclei with sufficient velocity to tra-verse the ferromagnetic layer of gadolinium. Thetarget consisted of 1 mgrcm2 150 Nd evaporated on2.15 mgrcm2 rolled and annealed gadolinium. Alayer of 40 mgrcm2 of gold was evaporated on thedownstream side of the gadolinium layer to providea perturbation free environment for the stopped Hg

ions. The excited nuclei enter the ferromagnetic foilwith an average initial energy of 40 MeV and veloc-

Ž . Ž .ity ÕrÕ s2.9 2.1%c , and exit from the foil0 inŽ . Ž .with an average velocity ÕrÕ s2.0 1.4%c ,0 out

where Õ is the Bohr velocity. In this target, the Hg0

ions enter the gadolinium foil at an average time of0.1 ps after production and exit on average 0.5 pslater. During the experiment, the beam intensity waskept at 2 pnA. The target was cooled to 77 K by aliquid-nitrogen transfer line which fits into the enclo-sure of Gammasphere. The gadolinium foil was po-larized by an external field of 0.06 T produced in thegap of a small electromagnet mounted outside thevacuum chamber. The current in the electromagnetwas reversed every five minutes. The magnetizationof the gadolinium foil, measured independently as afunction of temperature in an ac magnetometer, was

Ž .Ms0.182 4 T.The sorting of the multi-parameter coincidence

events for a magnetic moment experiment requiresthat the spectra of g-ray transitions be generated foreach detector individually, gated by transitionsrecorded in all other detectors, as a function of themagnetic field direction, ‘‘up’’ or ‘‘down’’. Theangular distributions of the discrete transitions stud-ied can be determined by summing the spectra ofdetectors located in rings at the same angle withrespect to the beam direction. However, precessionmeasurements require the analysis of the spectra inindiÕidual detectors located at the Euler angles u

and f with respect to the direction of the externalmagnetic field. Coincidence spectra were sorted sep-arately, for 36 detectors, by gating on two transitionswithin one of three bands corresponding to the maindecay branches ABC, ABE q ABCDE, ABF qABCDF for 193Hg and AB q ABCD, AF q ABCFand AE q ABCE for 194 Hg, where the notation

w xrefers to the cranked shell model calculations 24w xused by Hubel et al. 3 . Specifically the following¨

gating transitions were used in 193Hg: 617.9, 738.8,193.5, 480.6, 704.3 and 807.7 keV for the ABCband; 745.5, 205.1, 487.7, 639.6, 660.3, 512.8 and651.1 keV for the ABFqABCDF bands and 606.4,857.7, 375.3, 302.8, 573.0, 653.3, 523.2 and 873.1keV for the ABEqABCDE bands. The correspond-ing gating transitions in 194 Hg were: 565.0, 412.9,643.0, 743.6, 710.4 and 592.8 keV for the ABqABCD bands; 232.9, 544.6, 706.2, 485.2, 235.5 and


665.8 keV for the AEqABCE bands and 227.6,423.8, 611.2, 574.7, 236.3, 305.9 and 606.8 keV forthe AFqABCF bands.

The coincidence events were ‘‘unpacked’’ as de-w xscribed in 25–28 and background spectra were

determined for each band and subtracted by thew xtechnique described by Crowell et al. 26 . The angu-

lar distributions of the g rays in the selected bandswere obtained from the analysis of the efficiency-corrected sum spectra of individual detectors locatedat the same angle with respect to the beam. Typicalspectra are shown in Fig. 3a and Fig. 3b.

The detailed account of these measurements canw xbe found in 29 . The E2 transitions from different

bands in both nuclei exhibit similar angular correla-tion coefficients A and A since a typical stretched2 4

g-ray cascade is governed by the initial nuclear spinalignment and results in identical angular distribu-tions for all members of the cascade. The fitted

Ž . Ž .coefficients, A s0.31 1 and A sy0.15 2 , were2 4

used for all E2 transitions in the precession analysis.The angular precessions Du were determined fol-lowing the standard procedure described inw x13,21,30 .

The measured precessions of individual transi-tions were averaged over nine symmetrical combina-tions of four detectors. The g factors were extractedfrom the precessions through the relation,

Dusy g m r" B Õ t ,Z dtŽ . Ž .Ž .HN

where the integration was carried over the time the193,194 Hg ion spends in the gadolinium foil. TheChalk River parametrization of the transient fieldw x Ž . Ž .31 , B Õ , Z s 154.7 = Z = ÕrÕ = exp0Ž .y0.135ÕrÕ =M and the Ziegler stopping powers0w x32 were used. Table 1 displays the resulting preces-sions for each of the groups of states decaying intoone of the main bands together with the correspond-ing average g factors.

The region of excitations probed under the condi-tions of this experiment, determined from Monte

Fig. 3. Spectra of g-rays in coincidence with two cascade radia-. 193 .tions specifying the six bands of interest in a Hg and b in

194 Hg, observed in one detector, in one field direction.

( )L. Weissman et al.rPhysics Letters B 446 1999 22–2726

Table 1Results of the precession measurements and average g factors for

Ž . 193,194 Žthe main normal-deformation ND bands in Hg present. Ž . 194 wresults , and the three superdeformed SD bands in Hg Ref.

w xx30

Ž . ² :Bands Du mrad g193ND bands in Hg

ABC 18.4"1.4 0.188"0.014ABEqABCDE 17.0"1.4 0.176"0.014ABFqABCDF 19.3"1.7 0.200"0.018

194ND bands in HgABqABCD 24.6"1.5 0.255"0.016AEqABCE 24.9"2.5 0.258"0.026AFqABCF 26.0"2.1 0.270"0.022

194SD bands in HgSD1 33"9 0.36"0.10SD2 36"18 0.41"0.20SD3 73"24 0.72"0.26

Carlo simulations similar to the ones carried out in152 w x w x 192 w xDy 11,13 , Hf isotopes 17,18 , and Hg 33 ,lies 2–3 MeV above the yrast line, at spins If30y50". In both 193Hg and 194 Hg, the decay toward theyrast states proceeds via a variety of pathways. Theband selectiÕity aspect of the present measurementallows a comparison of the average g factors ofdifferent subsets of states in the quasicontinuumdistribution. Thus, the structural differences in en-sembles of excited states in this spin and energyregion can be specifically assessed.

Three main conclusions can be drawn from the² :present measurements. First, the g factors corre-

sponding to the different subsets of states within anyone nucleus are found to be essentially the same.This observation leads to the conclusion that, eitherthe entry distribution of states feeds uniformly thethree main lower paths and, hence, the average gfactors measured belong to one ensemble of states,or, that there are three distinct distributions of statesat high spin and energy which happen to have thesame average g factor. Assuming that the feedinginto the various bands is a statistical process, theformer interpretation is preferred.

Second, the present results yield average g fac-tors in the odd 193Hg nucleus which are about 30%smaller than the ones in 194 Hg and highlight theimportance of the additional neutron alignment in the

high-spin region. This result is consistent with the² Ž163,165 . :observed average g factors, g Hf s

Ž . ² Ž162,164,166 . : Ž .q0.15 2 and g Hf sq0.22 1 , in thew xodd and even Hf nuclei, respectively 17,18 . Fur-

thermore, the average g factors of the high spinstates in the even isotopes are also smaller than thevalue of the collective rotational estimate gfZrAŽ .Fig. 4 . This reduction of the g factors can beinterpreted in the context of the pattern of proton andneutron quasiparticle excitations predicted in various

w xcranking calculations 1–4,9,10,34 . At moderate ro-tational frequencies, the Ns114 quasineutron Rou-thians indicate that a large number of neutron or-bitals are available for multi-quasiparticle excita-tions. In contrast, because of the proximity of theZs82 shell closure, the Zs80 quasiproton Routhi-

Ž .ans exhibit a large energy gap f 1 MeV in thesame frequency regime, making proton multi-particleexcitations less probable.

Third, the average g factor of the highest-spinstates in the three superdeformed bands in 194 Hg,² Ž .: Ž .g SD sq0.41 8 has been determined in a study

w x Žcarried on the same data set 30 Table 1 and Fig..4 . This value is significantly larger than the average

g factors of the normal-deformation states in thequasicontinuum distribution. This result also agreeswith theoretical calculations that predict that thesuperdeformed nucleus is likely to behave like a

w xrigid rotator with a g;ZrA 30 . Thus, the compar-ison of the present data for superdeformed and nor-mal deformation nuclei indicates that, while a nu-cleus in the superdeformed well resembles a classical

Fig. 4. Summary of measured g factors for normal-deformedstates in 193Hg, and both normal-deformed and superdeformedw x 19430 states in Hg.


rotational system, the structure of the nucleus inhigh-spin, normal-deformation states is governed bydistinct single-particle configurations.

The continuous development and improved per-formance of large Ge arrays provides the possibilityof systematic studies of these phenomena. Futureexperiments using targets with a gap between a Ndtarget and a gadolinium foil in a plunger arrange-

w xment 16 will allow variations in the transient-fieldtime window and, hence, to probe the evolution ofthe nuclear structure as the nucleus decays fromregions of high spin and excitation to the yrast line.

Acknowledgements

The authors would like to thank J. Dudek and W.Nazarewicz for stimulating discussions. Thanks aredue to L. Sapir and J. Greene for target preparation,to B. Elkonin for the design of the cooling system, toR. McLeod for assistance with the data acquisitionsystem and to the staff of the 88’’ Cyclotron for theirhelp during the experiment. This work was supportedin part by the US National Science Foundation, theIsrael – US Binational Science Foundation and theUS Department of Energy, Nuclear Physics Division,

Ž .under contract Nos. W-31-109-ENG-38 ANL andŽ .DE-AC03-765F00098 LBNL .

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14 January 1999


Nucleosynthesis bounds in gauge-mediated supersymmetrybreaking theories

T. Gherghetta 1, G.F. Giudice 2, A. Riotto 3

Theory DiÕision, CERN, CH-1211 GeneÕe 23, Switzerland

Received 27 August 1998Editor: R. Gatto

Abstract

In gauge-mediated supersymmetry breaking theories the next-to-lightest supersymmetric particle can decay during orafter the nucleosynthesis epoch. The decay products such as photons and hadrons can destroy the light element abundances.Restricting the damage that these decays can do leads to constraints on the abundance and lifetime of the NLSP. Wecompute the freezeout abundance of the NLSP by including all coannihilation thresholds which are particularly important inthe case in which the NLSP is the lightest stau. We find that the upper bound on the messenger scale can be as stringent as1012 GeV when the NLSP is the lightest neutralino and 1013 GeV when the NLSP is the lightest stau. Our findings disfavourmodels of gauge mediation where the messenger scale is close to the GUT scale or results from balancing renormalisableinteractions with non-renormalisable operators at the Planck scale. When combined with the requirement of no gravitinooverabundance, our bound implies that the reheating temperature after inflation must be less than 107 GeV. q 1999 ElsevierScience B.V. All rights reserved.

1. Gauge-mediated supersymmetry breaking theo-ries provide an interesting alternative to the usualgravity-mediated scenarios in transmitting supersym-

w xmetry breaking effects to the low energy world 1Ž w x.for a review, see Ref. 2 . Among the most attrac-tive features of these theories is the natural suppres-sion of flavour-violating interactions which is guar-anteed by the gauge symmetry. In addition low-en-ergy phenomenology is now governed by the factthat the gravitino is the lightest supersymmetric par-ticle and this may lead to interesting collider signals.

1 E-mail: [email protected] On leave of absence from INFN, Sezione di Padova, Italy.3 On leave of absence from Oxford University, Department of

Theoretical Physics, Oxford, U.K. E-mail: [email protected]

The next-to-lightest supersymmetric particleŽ .NLSP also plays an important role in low-energyphenomenology. Since the soft mass of a supersym-metric particle is determined by its gauge quantumnumbers, the NLSP will be either a neutralino N or1

the lightest, mainly right-handed stau t , depending˜1Žon the choice of parameters the possibility of a.sneutrino NLSP is now marginal . The relevant pa-

rameters which define the gauge-mediated model areŽthe messenger index N twice the sum of the Dynkin

.indices of the messenger gauge representations , andthe supersymmetric and supersymmetry-breakingmessenger mass parameters M and F. We alsodistinguish the supersymmetry breaking scale F feltby the messenger fields and the fundamental scale ofsupersymmetry breaking F which determines the0


( )T. Gherghetta et al.rPhysics Letters B 446 1999 28–36 29

gravitino mass and couplings by defining k'FrF0

F1. In our analysis, we will trade F for the NLSPmass which is a parameter of more direct physicalmeaning. The Higgs mass parameters m and B arem

new independent inputs, which can be determined byimposing electroweak symmetry breaking, and there-fore be fixed in terms of tanb and the algebraic signof m. As a result, the lightest neutralino turns out tobe mainly B-ino. More details on the definition of

w xthe parameters can be found in Ref. 2 .The NLSP is not stable and eventually will decay

into the gravitino. If these decays occur during thenucleosynthesis epoch the light element abundances

w xcan be drastically altered 3–9 . For example, if theNLSP is a neutralino the dominant decay modeproduces a photon which affects the primordial 4 Heabundance. On the other hand hadronic decays canalso prove dangerous for the nucleosynthesis prod-ucts. In the case of the neutralino NLSP, the photoncan hadronise or, in the case of a stau NLSP, thesemileptonic decay of a tau lepton can producehadronic showers which leads to energetic nucleons.Even though nucleosynthesis may be over, theseenergetic nucleons destroy 4 He or synthesise 3Heand tritium leading to the overproduction of the lightelements. Alternatively if the NLSP actually decayshadronically during nucleosynthesis these nucleonswill instead establish thermal equilibrium with thesurrounding plasma by colliding with the ambientprotons and 4 He leading to the eventual increase ofthe nrp ratio. This results in a greater abundance of4 He, 3He and deuterium D.

In order to avoid the destructive effects on thenucleosynthesis products the lifetime of the NLSPmust be restricted so as to decay sufficiently wellbefore it can interfere with the nucleosynthesis prod-ucts or if it decays during nucleosynthesis that theenhanced light element abundances are consistentwith astrophysical observations. In addition theabundance of the NLSP at the time of decay willalso be important. Since the abundance and lifetimeare related to the messenger scale M, an upper

Žbound can be placed on the messenger scale for a.typical set of the other gauge-mediated parameters ,

depending on whether the neutralino or stau is theNLSP. These bounds will be shown to be fairlygeneric for most of the parameter space of gauge-mediated theories.

2. The damaging effects of the NLSP decay prod-ucts during the nucleosynthesis epoch constrains theabundance and lifetime of the decaying particle. Inorder to obtain a bound on the NLSP lifetime adetailed calculation of the NLSP abundance at thetime of decay must be performed. This amounts tocalculating the NLSP abundance at the time of freezeout when the NLSP is no longer in chemical equilib-rium.

We will consider the two separate cases of aneutralino and stau NLSP with mass m . ForNLSP

moderate values of tanb , t is lighter than the˜1

neutralino whenever

66N) ,

5 13jy2Ž .a 2 mŽ .1 NLSP

j' 2a MŽ .1

222 mNLSPs 1q a m ln . 1Ž . Ž .1 NLSP4p M

For large tanb this region becomes slightly largerbecause of the stau left-right mixing. In order todetermine the NLSP abundance we consider all rele-vant channels which change the number of NLSP’s.When the NLSP is a neutralino the most relevantannihilation channels are those consisting of fermionsand gauge bosons as depicted in Table 1. The anni-hilation channels into Higgs bosons are suppressedbecause the lightest neutralino is mainly B-ino. Forthe same reason, the neutralino NLSP generically isnot degenerate in mass with other particles, so it will

Table 1Final state annihilation channels including coannihilations inparentheses

Initial state Final state

q yN N ZZ,W W , ff1 1

q y q y q y q yŽ .t t , m m ,e e ZZ,W W ,gg ,Zg ,hh,˜ ˜ ˜ ˜ ˜ ˜1 1 1 1 1 1

g h,Zh, ff" " " " " " " " " " " "Ž . Ž .t t , m m ,e e t t , m m ,e e˜ ˜ ˜ ˜ ˜ ˜1 1 1 1 1 1

" " " " " " " " " " " "Ž . Ž .t m ,t e ,m e t m ,t e ,m e˜ ˜ ˜ ˜ ˜ ˜1 1 1 1 1 1

" . " . " . " .Ž . Ž .t m ,t e ,m e t m ,n n ,n n ,˜ ˜ ˜ ˜ ˜ ˜1 1 1 1 1 1 t m t m" .Ž .t e ,n n ,n n ,t e t e" .Ž .m e ,n n ,n nm e m e

( )T. Gherghetta et al.rPhysics Letters B 446 1999 28–3630

not be important to consider coannihilations for thiscase. We can also neglect annihilation channels whichinvolve gravitino vertices since these lead to scatter-ing amplitudes which are suppressed by a factorm2 rF . Since we are interested in NLSP lifetimesNLSP 0

of the order of the nucleosynthesis timescale, thissuppression makes all gravitino-emission processesnegligible.

On the other hand when the NLSP is the stau, thelightest smuon and selectron have a small massdifference with the NLSP:

22m ym m mtanbyAŽ .m ,e t t t˜ ˜ ˜1 1 1 , y1 . 2Ž .2 2 2m 2m m ym˜ ˜ ˜t ll˜ ll ll1 R L R

Here A is the supersymmetry-breaking trilinear termt

and m are the flavour-independent contributionsllL, R

Žto the left and right slepton masses including the.D-term contribution . Since in gauge-mediated theo-

2 2 Ž .ries m )m , the mass difference in Eq. 2 isllL

always positive. Because of the approximate massdegeneracy among the sleptons, the calculation ofthe stau relic abundance must include all coannihila-tion processes listed in Table 1. We also need tocompute the smuon and selectron density at thedecoupling time, since these ‘‘co-NLSP’s’’ are alsoresponsible for producing damaging decay products.

The NLSP abundance is determined by consider-ing the evolution of the number density n of particlei

i which is governed by the Boltzmann equation. Inthe presence of coannihilations the Boltzmann equa-

w xtion can be written as 10

dn2 2² :sy3Hny s Õ n yn 3Ž .Ž .eff eqdt

where H is the Hubble expansion parameter, nsÝ n and the thermal average of the effective crossi i

section is defined as

neq neqi j² : ² :s Õ s s Õ . 4Ž .Ýeff i j i j eq eqn nij

The individual cross sections s include the pro-i j

cesses listed in Table 1, Õ is the relative velocityi j

and neq is the equilibrium number density.The cross sections for all annihilation channels

are numerically evaluated using the CompHEPsoft-w xware package 11 . After performing the thermal

average and including all relevant coannihilation

thresholds, the ratio Y eq of the equilibrium numberdensity neq to the entropy density s is given by

neq 45x 2 m2i

Y T ' s gŽ . Ýeq i4 2s 4p g T mŽ .eff NLSPi

=mi

K x , 5Ž .2 ž /mNLSP

where xsm rT , g is the number of internalNLSP i

degrees of freedom and g s81. The NLSP abun-eff

dance Y at the time of nucleosynthesis is givenNLSPŽ .by Y 'Y T , where T is the freezeout tem-NLSP eq F F

perature. The freezeout temperature is determined² : Ž .from the condition n s Õ sH T .eq eff

Thus in the case of the neutralino NLSP theabundance Y is just simply Y while for theNLSP N1

stau NLSP it will be Y sY qY qY which isNLSP t m e˜ ˜ ˜the sum of all the ‘‘co-NLSP’’ abundances. Theeffect of including the coannihilating channels be-tween the stau and the ‘‘co-NLSP’s’’ changes thepure stau abundance by up to ;50%. This can beseen in Fig. 1 where the stau abundance with andwithout coannihilations is shown. Notice the impor-tant reduction of the relic abundance for values ofm close to half the Higgs mass, caused by thet1

resonant annihilation channel. It should be noted,however that given the present LEP bound on stablet , this possibility is no longer realistic for the stau.˜1

Fig. 1. Comparison of the stau abundance with and withoutŽ .coannihilation lower and upper curves, respectively . The gauge-

mediated parameters are Ms1013 GeV, Ns12, tanb s1.1 andsgnms1.


3. Let us now consider the effect of the NLSPdecaying during the nucleosynthesis epoch. Supposefirst that the NLSP is a neutralino. The dominantdecay mode of the neutralino is into a photon and

w xgravitino with a decay rate given by 2,12,13

k 2k m5g N1˜G N ™g G s 6Ž .Ž .1 216p F

< < 2where k s N cosu qN sinu and N areg 11 W 12 W 11,12

the NLSP gaugino components in standard notation.There are also other decay modes which involve theZ or a Higgs boson, but these modes are suppressedby a b 8 phase-space factor. The photon of thedominant decay mode has a dramatic effect on thelight element abundance 4. In the radiation domi-nated thermal background high energy photons initi-ate electromagnetic cascades and create many low-energy photons capable of photodissociating the light

w xelements 3 . These photodissociation effects causedby the photon of the decaying neutralino can beparametrised by a ‘‘damage’’ factor

YN1d 'm 7Ž .g N1 h

where h is the baryon-to-photon number densityratio. This quantity d is constrained by astrophysi-g

cal observations of light elements. Recently newobservations of the D and 4 He primordial abundancehave led to a reanalysis of the constraints on the

w xabundance and lifetime of long-lived particles 9 . Inparticular, controversy in the measurement of pri-mordial deuterium have led to two different values

Žfor the deuterium abundance X normalised toD. w xhydrogen . In Ref. 14 a high value of the deuterium

Ž . y4abundance X s 1.9"0.5 =10 is quoted whileDw x Ž . y5in Ref. 15 a low value X s 3.39"0.25 =10D

is obtained.The neutralino abundance Y depends on theN1

details of the sparticle spectrum and can be calcu-lated for a particular choice of the parameters N, M,

w xF, tanb and sgnm. Using the constraints in Ref. 9we show in Fig. 2 the bound on the messenger scalefor a representative neutralino NLSP scenario in

4 Notice that the gravitinos released in the NLSP decays do notthermalise and their present contribution to the energy density ofthe Universe is negligible.

Fig. 2. Upper bound on the messenger scale M as a function ofthe neutralino NLSP mass m from photodissociations whereN1

ks1, Ns2, tanb s2 and sgnms1. Both bounds use a 4 HeŽ4 .abundance Y He s0.234 but the dashed line uses a low valuep

of the deuterium abundance while the solid line uses the highŽ .value of the deuterium abundance see text .

which Ns2, tanbs2 and sgnms1, for both highand low values of the measured X . We have alsoD

assumed that ks1 and the limiting value of Mgrows approximately linearly with k. Notice that thehigh deuterium value gives a slightly more stringentbound than the low deuterium value, because forlifetimes less than 106 seconds deuterium is effec-tively photodissociated.

The bound will change slightly for different val-ues of the gauge-mediated supersymmetry breakingparameters, but typically for the neutralino NLSPscenario lifetimes greater than 105 seconds are ruledout by photodissociations.

Note that photodissociations are only importantŽafter nucleosynthesis is over which corresponds to

4 .times later than 10 seconds . This is because duringearlier times when thermal photons are more ener-getic and numerous, photon-photon interactions aremuch more probable than photon-nucleus interac-tions. Consequently stronger constraints can only beobtained if we consider processes that affect nucle-osynthesis at times earlier than 104 seconds.

However before discussing the relevant processesfor times earlier than 104 seconds it is possible toobtain stronger constraints than those from photodis-sociations by considering hadronic decays of the

w xNLSP 8 . If the NLSP decays hadronically during


times R104 seconds then the photodissociationbound needs to be reconsidered since while D isdestroyed by photodissociation it can be produced bythe hadronic showers. In fact even if the NLSPdecays exclusively into photons it is possible thathadronic showers will be generated. The main effectof hadronic showers is not only to increase theabundance of D but also the other light nuclei 3He,6 Liand 7Li. In particular D and 3He arise mainly from‘‘hadrodestruction’’ processes, which occur when anenergetic nucleon breaks up the ambient 4 He nucleusinto D and 3He. The other light nuclei, 6 Li and 7Liarise from ‘‘hadrosynthesis’’ of 3He, T or 4 He. More

w xcomplete details can be found in Ref. 5 . Again the‘‘damage’’ of the hadronic decays on the primordialnuclear abundances can be parametrised as

d 'd B) 8Ž .B g h

) Ž .where B s n r5 B FF is an effective baryonich B h

branching ratio which depends on the true baryonicbranching ratio B , the baryonic multiplicity n andh B

a factor FF representing the dependence of the yieldson the energy of the primary shower baryons.

One of the contributions to the neutralino hadronic˜branching ratio comes from the decay N ™ZG1

42mZ2B ,0.7tan u 1y 9Ž .h W 2ž /mN1

where the hadronic branching ratio of the Z-boson is0.7, and we have assumed a pure B-ino. Althoughthis contribution, and the analogous one from Higgsdecay, can be forbidden by phase space, there isalways at least a contribution to B of order a fromh

photon conversion into a qq pair. The explicit ex-Ž .pression for B can be obtained from Eq. 13 ofh

w xRef. 16 . Using the corresponding value of d , oneB

finds that the overproduction of 7Li constrains thelifetime of the neutralino to be shorter than 104

seconds. This is also true if the NLSP is the stau.The stau decays predominantly into a tau and grav-itino and since the tau has a large hadronic branchingratio, the corresponding value of d is much largerB

than that for the neutralino. Consequently the life-time of the stau must also be less than 104 seconds if

7 w xthe overproduction of Li is to be avoided 8 .It is clear that in order to obtain precise con-

straints on the abundance and lifetime of the NLSP

we need to consider processes occurring at timesearlier than 104 seconds. At these times the maindecay products that interfere with nucleosynthesisare hadrons. Hadronic showers induce interconver-sions between the ambient protons and neutrons thus

w xchanging the equilibrium nrp ratio 4 . In particularduring the lifetime interval t ;1–100 secondsNLSP

the overall effect of the hadronic decays is to convertprotons into neutrons. The additional neutrons thatare produced are all synthesised into 4 He and thushadronic decays during this time interval are con-strained by the observational upper bound on the

Ž4 .primordial helium abundance Y He .p

Eventually the neutron fraction falls to zero be-cause all neutrons are contained in the 4 He nucleiand the remaining neutrons created by the NLSPdecay increase the deuterium D abundance. Further-more for t ;100–1000 seconds the 3He abun-NLSP

dance is also increased by D-D burning. After t NLSP

R104 seconds all the neutrons arising from NLSPdecay will themselves decay before forming D. Thusin the interval 102–104 seconds the appropriate con-straint on hadronic decays arises from the observa-

Ž 3 .tional bounds on Dq He rH.The overall effect of these hadronic decays has

w xbeen considered in Ref. 4 , where the constraints onthe abundance and lifetime of the decaying particleare parametrised by

² :N B n EŽ .jet h jetfs 10Ž .² :2 n 33 GeVŽ .where N is the number of jets, B is the hadronicjet h

² Ž .:branching ratio and n E is the average chargejet

multiplicity for a jet with energy E and is given byjet

² :n EŽ .jet

Ejets1q0.0135 exp 1.9 2ln .( ž /0.15 GeV

11Ž .

Since at the parton level the neutralino NLSP decaysinto three particles we will assume that E sjet

m r3 and N is the number of quarks at theNLSP jet

parton level. In the case of the stau in which the taudecays semi-leptonically we assume E sm r6.jet NLSP

The most stringent constraints on the NLSP abun-dance and lifetime in the interval 102–104 seconds


come from the primordial deuterium abundance. Pre-vious constraints on Y f and the lifetime wereNLSP

y4 y10 y9 w xobtained for X -10 and hs3=10 ,10 4 .D

In order to use the more recent measurements, werescale the previous constraints for the new values

Ž . y4 Ž .X s 1.9"0.5 =10 or X s 3.39"0.25 =D D

10y5. This rescaling can be done analytically byw xusing the change in the deuterium abundance 4

DYNLSPDX s e a , 12Ž .D D nYH

where Y is the hydrogen density, e is the fractionH D

of injected neutrons that end up in deuterium and an

is the number of nn pairs per NLSP decay. Thisequation is valid as long as X <e , which conve-D D

niently holds whenever the deuterium constraint isimportant. Since the limit comes from the overpro-duction of deuterium and the standard X predictionD

decreases with h, we make the most conservativechoice of hs6=10y10, which is the largest valuecompatible with 4 He and 6 Li abundances.

When the high value of the deuterium abundanceis used, one obtains no constraints in the region102–104 seconds. All values of the NLSP abundanceand lifetime are consistent with high X value andD

consequently there is no improvement on the lifetimeupper bound of 104 seconds obtained from the over-abundance of 7Li.

On the other hand stringent constraints in theinterval 102–104 seconds are obtained when the lowvalue of the deuterium abundance is used. Let usconsider the two NLSP scenarios separately. First,when the NLSP is the neutralino the scaling parame-ter f is determined using N s2 and B s10y2 .jet h

w xRescaling the solid curve of Fig. 4 in Ref. 4 enablesone to determine the constraint arising from the lowvalue of X and hs6=10y10. Thus calculatingD

the value for the neutralino number density for ageneric set of gauge-mediated supersymmetry break-ing parameters leads to constraints on the abundanceand lifetime of the neutralino which can be ex-pressed as a bound on the messenger scale M, seeFig. 3. The bound on M does not significantlydepend on the value of tanb , but grows approxi-mately linearly with N.

Let us now discuss the case of the stau NLSP. Atthe decoupling time, the abundances of smuons andselectrons are comparable to the stau abundance. The

Fig. 3. Upper bound on the messenger scale M as a function ofthe neutralino NLSP mass m from hadronic decays whereN1

Ž .ks1, Ns2, tanb s2 and sgnms1. The solid dashed lineŽ .corresponds to the bound assuming a low high deuterium mea-

surement. More specifically the dashed line represents the 104

second lifetime contour which arises from the 7Li overabundance.

cosmological fate of the frozen-out smuons and se-Ž .lectrons depends on the mass difference in Eq. 2 . If

Ž .tanb is large, then m ym )m ym llse,mll t t ll˜11˜and the decay ll ™t t ll is kinematically open. The˜11

value of tanb for which this transition occurs de-pends on the parameter choice, but it is typically of

Ž .order 10, as it can be estimated from Eq. 2 . If thismode is accessible, it dominates over the two-bodydecay into gravitino. This can be simply understood

w xfrom the decay rate expression 17 in the limit ofvanishing lepton masses and small slepton massdifference,

y y q yG ll ™t t ll˜ž /11

4G2 M 45F W4, tan u m ym , 13Ž .˜Ž .W ll t3 4 1115p mN1

y q y yG ll ™t t ll˜ž /11

4G2 M 45F W4, tan u m ym . 14Ž .˜Ž .W ll t3 2 2 1115p m m ˜N ll1 1

Here m is the mass of the B-ino which mediatesN1

the decay, and is assumed to be much larger thanm . In this case, smuons and selectrons decay soonll1

after decoupling, with each one producing a stau inthe final state. Therefore, the relevant stau abun-


dance at the nucleosynthesis epoch is determined bythe sum over the three slepton abundances at thedecoupling time. The stau hadronic branching frac-tion comes from the semileptonic tau decay, and it isgiven by B ,0.65.h

For smaller values of tanb the neutralino-media-ted three-body process is forbidden, and the rates ofthe two competing slepton decay modes are

k 2 m5ll1˜ ˜G ll ™ ll G s 15Ž .ž /1 216p F

G2 M 45F Wy y˜G ll ™t n n , m ym˜ ˜Ž .1 ll t ll tž /1 ˜3 4 11

q15p mx

=sin2u sin2u . 16Ž .˜t ll˜

Ž .qHere m is the chargino gaugino mass whichx

mediates the decay, and u , u are the left-right˜t ll˜mixing angles in the slepton system. The process in

Ž .Eq. 15 is suppressed by the large value of Frequired in our study of nucleosynthesis and the

Ž .process in Eq. 16 is suppressed by the small mixingangles proportional to the corresponding leptonmasses. It turns out that in the small tanb regimeand for m ;100 GeV, the two rates are compara-ll1

9'ble when F ;10 GeV, which is just the value of'F necessary to have a stau decay during nucleosyn-thesis. Therefore, depending on the particular choiceof the gauge mediation parameters, we can encounter

˜Ždifferent situations. The first possibility is that G ll 1y y˜˜. Ž .™ ll G )G ll ™t n n , in which all light slep-˜1 ll t1

tons decay directly into gravitinos around the sametime. In this case smuons and selectrons do notsignificantly contribute to the effective hadronicbranching fraction, because real electrons and muonscannot decay into hadrons. As tanb is increased andm is decreased, we go first to a regime in whichll1 y y y˜ ˜Ž . Ž . ŽG m ™ t n n ) G ll ™ ll G ) G e ™˜ ˜ ˜1 1 m t 11

y .t n n and then in a regime in which the three-body˜1 e t

decays dominate for both m and e . The two regimes˜ ˜1 1Ž .are possible because the decay process in Eq. 16

depends on the lepton mass through the left-rightmixing angle. In the first regime, only the smuoncontributes to the effective hadronic branching frac-tion, while in the second one both the smuon andselectron contribute.

With respect to a neutralino NLSP of equal mass,a stau NLSP has a larger hadronic branching ratio

B , but a smaller relic abundance because of theh

additional annihilation channels. The two effectsroughly compensate each other, although the boundon the NLSP lifetime is slighly weaker in the t 1

case. When expressed in terms of M, the boundappears even weaker because we have to choose avery large value of N in order to satisfy the require-

Ž .ment of Eq. 1 for a stau NLSP. The bound on themessenger mass for Ns12 and tanbs1.1 is shownin Fig. 4. For this choice of parameters, the selectronand the smuon dominantly decay directly into grav-itinos. In the figure we only show the bound arisingfrom the low deuterium value. The bound for thehigh deuterium value corresponds to a lifetime whichcannot be achieved for this choice of N.

4. Although the mass spectrum of supersymmetrictheories with gauge mediation can be predicted interms of a few parameters, the experimental andcosmological features of the theory can drasticallychange as the messenger mass M is varied in theallowed range between about 100 TeV and 1015

GeV. It is thus very important to study possibleconstraints on the value of M. The requirement thatthe NLSP decay products do not upset the successfulpredictions of standard big-bang nucleosynthesisprovides such a constraint. Here we have studied indetail the effects of the decaying NLSP on thelight-element primordial abundances and shown thatthe most dangerous damages come from the hadronic

Fig. 4. Upper bound on the messenger scale M as a function ofthe stau NLSP mass m from hadronic decays where ks1,t

Ns12, tanb s1.1 and sgnms1.


decay modes. After computing the NLSP relic abun-dance, including coannihilation effects, we concludethat the injection of hadronic jets in the Universe attimes later than 104 seconds grossly overproduces7Li. This is true for both neutralino and stau NLSP.At earlier times, the most relevant bounds come fromdeuterium overproduction. At present, there is adisagreement on the extracted observational value of

Ž .the primordial deuterium. Using X s 3.39"0.25Dy5 w x=10 15 , no further limit on the NLSP is de-

Ž . y4 w xrived, while with X s 1.9"0.5 =10 14 theD

limit from 7Li can be improved. The correspondinglimits on the messenger mass M are shown in Figs.3 and 4 for a representative choice of gauge media-tion parameters, in the case of neutralino and stauNLSP respectively.

It should be noticed that the bounds on the mes-senger mass presented here can be evaded in thepresence of other interactions which can lead to afast NLSP decay, e.g. R-parity violation. Neverthe-less these bounds have significant implications formodel building. In particular they disfavour modelsin which M is close to the GUT scale or models inwhich the messenger scale results from balancingrenormalisable interactions with non-renormalisable

w xoperators at the Planck scale 18,19 .It is also interesting to notice that the nucleosyn-

thesis bound discussed here is complementary to thebound obtained from gravitino overabundance. In-deed when m is larger than about a keV, gravitinosG

which were in thermal equilibrium at early times,give a contribution to the present energy densitylarger than the critical value. It is then necessary toassume that gravitinos have been diluted by somemechanism. Let T be the temperature at whichmax

the ordinary radiation-dominated Universe begins.This corresponds to either the reheating temperatureafter an inflationary epoch or to the temperatureassociated with significant entropy production. Therequirement that gravitinos do not overclose the Uni-

w xverse gives the following constraints on T 20,21max

T Q100 GeV–1 TeVmax

for 2h2 keVQm Q100 keV, 17Ž .G

2m TeVG2T Q10 TeV hmax ž / ž /100 keV mg

for m R100 keV, 18Ž .G

where h is the Hubble constant in units of 100 kmsecy1 Mpcy1 and m is the gluino mass. To com-g

pare this bound with the nucleosynthesis result, it isconvenient to express it in terms of the messengermass M, the messenger index N, and the pure B-inomass m . We findN1

T Q100 GeV–1 TeVmax

mM N18 2 9for 10 h GeVQ Q5=10 GeV,ž /kN 100 GeV19Ž .

M 100 GeVy6 2T Q5=10 hmax ž /kN mN1

mM N1 9for R5=10 GeV. 20Ž .ž /kN 100 GeV

This shows that there is no gravitino problem as long8 2 Ž .as MQ10 h GeV kN 100 GeVrm . For largerN1

values of M, there is a very stringent constraint onT which requires inflation at particularly lowmax

temperatures. As M grows this limit becomesweaker, but eventually the nucleosynthesis bound onthe messenger mass sets in. When the two boundsare combined, we find that the reheating temperatureafter inflation should be less than about 107 h2 GeVwhich is stronger than the bound ;1010 GeV usu-

w xally obtained in gravity-mediated scenarios 5,6,20 .Our limit is valid for gauge-mediated theories inwhich the LSP gravitino is heavier than a keV and inwhich there are no new interactions uncorrelatedwith the supersymmetry-breaking scale mediating

Ž .the NLSP decay like R-parity violating interactions .

Acknowledgements

We thank T. Moroi and C. Wagner for discus-sions and are especially indebted to Sasha Pukhovfor help in installing and running the CompHEPsoftware package.

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14 January 1999


Supersymmetry in the AdSrCFT correspondence

Madoka Nishimura a, Yoshiaki Tanii b,1

a Department of Physics, Ochanomizu UniÕersity, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-0012, Japanb Physics Department, Faculty of Science, Saitama UniÕersity, Urawa, Saitama 338-8570, Japan

Received 12 November 1998Editor: M. Cvetic

Abstract

Ž .We study how local symmetry transformations of p,q anti de Sitter supergravities in three dimensions act on fields onthe two-dimensional boundary. The boundary transformation laws are shown to be the same as those of two-dimensionalŽ .p,q conformal supergravities for p,qF2. Weyl and super Weyl transformations are generated from three-dimensionalgeneral coordinate and super transformations. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

w xIt was conjectured in Ref. 1 that the stringrM-Ž . Ž .theory in dq1 -dimensional anti de Sitter AdS

space times a compact space is equivalent to aŽ .d-dimensional conformal field theory CFT . More

precise form of this AdSrCFT correspondence wasw x w xgiven in Refs. 2,3 . According to Refs. 2,3 the

CFTs are defined on the boundary of the AdS spaceŽ .and the generating functional of operators OO x in

the boundary CFTs is given by the partition functionof the stringrM-theory. When the stringrM-theoryis represented by a low energy effective supergravityand the partition function is approximated by a sta-tionary point of the supergravity action S , oneSUGRA

obtains a relation

d w xexp i d x f x OO x sexp iS f .Ž . Ž . Ž .H 0 SUGRA¦ ;ž /CFT

1Ž .


Here, f on the left hand side are arbitrary functions0

defined on the d-dimensional boundary while f onthe right hand side are the solutions of field equa-tions in the bulk satisfying boundary conditions fsf . For fields satisfying the first order field equa-0

tions such as a spinor field one should imposeboundary conditions on only half of the components

w xof the fields 4,5 .The purpose of this paper is to study how local

symmetry transformations in the bulk theories act onthe fields f on the boundary. We are especially0

interested in how local supertransformations act onf . The fields f on the boundary are expected to0 0

form multiplets of d-dimensional conformal super-w x Ž .gravities 6,7 . We consider three-dimensional p,q

w xAdS supergravities of Achucarro and Townsend 8´in the bulk as simple examples. The AdSrCFTcorrespondence for three-dimensional AdS space waspreviously discussed from other points of view in

w xRefs. 9–12 .We partially fix the gauge for the local symme-

tries in the bulk and obtain residual symmetries,


( )M. Nishimura, Y. TaniirPhysics Letters B 446 1999 37–4238

which preserve the gauge fixing conditions. Theseresidual symmetry transformations act on the fieldsnon-locally in the bulk. However, they can act on theboundary fields f locally. It is shown that the0

transformations of the boundary fields have a localform for p,qF2. In particular, the supertransforma-tions in the bulk become two-dimensional super andsuper Weyl transformations on the boundary, whilegeneral coordinate transformations in the bulk be-come general coordinate and Weyl transformationson the boundary. These transformation laws areshown to be exactly the same as those of two-dimen-

Ž .sional p,q conformal supergravities, i.e., confor-mal supergravities with p supersymmetries of posi-tive chirality and q of negative chirality.

2. Three-dimensional AdS supergravities

Ž .The field content of the three-dimensional p,qw x AAdS supergravity 8 is a dreibein e , MajoranaM

i iX Ž .Rarita-Schwinger fields c , c and SO p =M MŽ . i j ji iX jX

SO q Chern-Simons gauge fields A syA , AM M M

s yA jX iX

, where i, j, . . . s 1, . . . , p; iX, jX, . . . sM

1, . . . ,q. We denote three-dimensional world indicesas M, N, . . . s0,1,2 and local Lorentz indices asA, B, . . . s0,1,2. Our conventions are as follows.

Ž .The flat metric is h sdiag y1,q1,q1 and theA B

totally antisymmetric tensor e A BC is chosen as e 012

� 4sq1. 2=2 gamma matrices g satisfy g ,g sA A B

2h . g ’s with multiple indices are antisymmetrizedA B

products of gamma matrices with unit strength. Inparticular, we have g A BC sye A BC in three dimen-sions. The Dirac conjugate of a spinor c is defined

† 0as csc ig . All components of gamma matricesare chosen to be real and Majorana spinors have tworeal components.

The Lagrangian is given by1

1 12 M NP i iLLs eRq4m eq ie c DD cM N P2 28p GX X1 1i M N i M NP i iq imec g c q ie c DD cM N M N P2 2

1X X1 i M N i M NP i j jiy imec g c y e A E AŽM N M N P2 4m1 X X X X2 i j jk k i M NP i j j iq A A A q e A E A. ŽM N P M N P3 4m

X X X X X X2 i j j k k iq A A A , 2Ž ..M N P3

where m is a positive constant. The cosmologicalconstant is proportional to m2. In the following wewill put the gravitational constant as 8p Gs1. Ourconventions for the curvature tensors are

Rse Me NR A B ,A B M N

R A B sE v A B qv A v C B y MlNŽ .M N M N M C N

3Ž .and the covariant derivatives are defined as

1i A B i i j jDD c s E q v g c qA c ,Ž .M N M M A B N M N4X 1 X X X Xi A B i i j jDD c s E q v g c qA c . 4Ž .Ž .M N M M A B N M N4

Ž . Ž .The covariant derivatives without SO p = SO qconnection terms are denoted as D . The spin con-M

nection is given by1A B A B i i i iv sv e q i c g c yc g cŽ . žM M M A B M B A4

X X X Xi i i i i iqc g c qc g c yc g cA M B M A B M B A

X Xi iqc g c , 5Ž ./A M B

A BŽ .where v e is the spin connection without tor-M

sion. If v A B is treated as an independent variableMŽ .in the Lagrangian 2 , its field equation is solved by

Ž .Eq. 5 .Ž .The Lagrangian 2 is invariant under the follow-

Ž .ing local transformations for arbitrary p,q up tototal derivative terms:

d e A sj NE e A qE j Ne A yl A e BM N M M N B M

X X1 i A i i A iq i e g c qe g c ,Ž .M M2

1i N i N i A B i i j jdc sj E c qE j c y l g c yu cM N M M N A B M M4

qDD e i qmg e i ,M MX X X 1 X X X Xi N i N i A B i i j jdc sj E c qE j c y l g c yu cM N M M N A B M M4

qDD e iX

ymg e iX

,M M

i j N i j N i j i j w i j xd A sj E A qE j A qDD u q2 ime c ,M N M M N M M

d AiX jX

sj NE AiX jX

qE j NAiX jX

qDD u iX jX

M N M M N MX Xw i j xy2 ime c . 6Ž .M

M Ž . A BŽ .The transformation parameters j x , l x ,i jŽ . iX jXŽ . iŽ . iXŽ .u x , u x and e x , e x represent general

Ž . Ž .coordinate, local Lorentz, SO p = SO q gaugeand local super transformations respectively. Theparameters e i, e iX

are Majorana spinors and l A B sylB A, u i j syu ji, u iX jX

syu jX iX

. The commutator

( )M. Nishimura, Y. TaniirPhysics Letters B 446 1999 37–42 39

algebra of these transformations closes for arbitraryp, q modulo the field equations.

3. Boundary behaviors of the fields

It is convenient to partially fix the gauge for theŽ .local symmetries 6 . We represent the three-dimen-

sional AdS space as a region x 2 )0 in R3. Theboundary of the AdS space corresponds to a planex 2 s0 and a point x 2 s`. We choose the gaugefixing condition as

1As2 a As2e s , e s0, e s0,Ms 2 Ms2 m22mx

c i s0, c iX

s0, Ai j s0, AiX jX

s0, 7Ž .2 2 2 2

where m,n , . . . s0,1 and a,b, . . . s0,1 are two-di-mensional world indices and local Lorentz indicesrespectively. The metric in this gauge has a form

1M N 2 2 m ndx dx g s dx dx qdx dx g .ˆž /M N mn222mxŽ .

8Ž .

Ž .The SO 2,2 invariant AdS metric corresponds to thecase g sh but we consider the general g . Weˆ ˆmn mn mn

define e a by g se ae bh .ˆ ˆ ˆ ˆm mn m n ab

Let us obtain asymptotic behaviors of the fieldsfor x 2 ™0. We assume that the dreibein e a be-m

Ž 2 .y1 Ž .haves as x just as in the SO 2,2 invariantcase. Asymptotic behaviors of other fields are deter-mined by field equations. The field equations of theRarita-Schwinger fields near x 2 s0 are

12x E " g c s0, 9Ž .Ž .2 2 m2

where q is for c sc i and y is for c sc iX

. Them m m m1 X 1i 2 . i 2 "2 2Ž . Ž .solutions behave as c ; x , c ; xm" m "

for x 2 ™0, where the suffices " here denote eigen-values of g 2, i.e., chiralities in two-dimensionalsense. The field equations which determine theboundary behavior of the gauge fields are

E A s0 10Ž .2 m

for both of A sAi , AiX

. The solutions are indepen-m m m

dent of x 2.

w xAccording to the prescription in Refs. 2,3 onehas to impose boundary conditions on the fields. Asfor gravity we require that the zweibein e a definedm

Ž .below Eq. 8 approaches a given functionaŽ 0 1.e x , x at the boundary. Since the Rarita-0m

Schwinger fields and the Chern-Simons gauge fieldshave field equations which are first order in deriva-tives, one should impose boundary conditions on

w xonly half of their components 4,5 . For the Rarita-Schwinger fields we impose boundary conditions onthe components which become larger for x 2 ™0,i.e., c i and c iX

. For the gauge fields all themq mycomponents become independent of x 2 and one canchoose either A se mA or A se mA . Here,y y m q q m

the suffices " denote the light-cone directions e m"

1 m mŽ .s e "e . We impose boundary conditions0 1'2

on Ai j and AiX jX

. This choice is required by super-y qsymmetry as we will see later. To summarize weimpose boundary conditions on e a, c i , c iX

, Ai jm mq my y

and AiX jX

. The boundary behaviors of these fields areq

1yy1 2a 2 a i 2 ie ™ 2mx e , c ™ 2mx c ,Ž . Ž .m 0m mq 0mq

1yX X2i 2 i i j 2 i jc ™ 2mx c , A ™2mx A ,Ž .my 0my y 0y

AiX jX

™2mx 2AiX jX

, 11Ž .q 0q

where the fields with the suffix 0 are fixed functionson the boundary. Other components of the fields onthe boundary are non-local functionals of the fields

Ž .in Eq. 11 , which are obtained by solving the fieldequations. We also introduce notations c i , c iX

,0my 0mqX 1i i i 2 i2Ž .A , A defined by c ™ 2mx c , etc.0q 0y my 0my

4. Local symmetries on the boundary

Let us study how the fields on the boundary inŽ .Eq. 11 transform under the residual symmetry

transformations after the gauge fixing. The residualŽ .symmetries, which preserve the gauge conditions 7 ,

are obtained by solving

12 2 m a2 mE j y j s0, E j yl e s0,ˆ2 2 a2x

X X2 a 2 2 i 2 i i 2 il e yE j y imx e g c qe g c s0,ˆ ž /a m m m m


D e i qmg e i qE j mc i s0,2 Ms2 2 m

D e iX

ymg e iX

qE j mc iX

s0,2 Ms2 2 m

E u i j qE j mAi j s0, E u iX jX

qE j mAiX jX

s0.2 2 m 2 2 m

12Ž .These equations except the third one determine x 2-dependence of the transformation parameters. Thethird equation fixes la2. The general solution of Eq.Ž . 212 near the boundary x s0 is

j 2 syx 2L x 0 , x1 ,Ž .0

2m m 0 1 2j sj x , x qOO x ,Ž . Ž .ž /0

lab slab x 0 , x1 qOO x 2 , la2 sOO x 2 ,Ž . Ž . Ž .01. 2i 2 i 0 1 2e s 2mx e x , x qOO x ,Ž . Ž . Ž ." 0 "

1X X" 2i 2 i 0 1 2e s 2mx e x , x qOO x ,Ž . Ž . Ž ." 0 "

2i j i j 0 1 2u su x , x qOO x ,Ž . Ž .ž /0

X X X X 2i j i j 0 1 2u su x , x qOO x , 13Ž . Ž . Ž .ž /0

where L , j m, lab, e i , e iX

, u i j and u iX jX

are0 0 0 0 " 0 " 0 00 1 Ž 2 .arbitrary functions of x and x . Order OO x and

ŽŽ 2 .2 .OO x terms are non-local functionals of thesefunctions and the fields. For instance, the orderŽŽ 2 .2 . i j x 2 2 m i jOO x term in u is given by yH dx E j A .0 2 m

Thus, the residual symmetry transformations of thefields in the bulk of the AdS space are non-local.

However, the transformations of the fields on theŽ .boundary in Eq. 11 can be local. Substituting Eqs.

Ž . Ž . Ž .11 and 13 into Eq. 6 and taking the limitx 2 ™0 we find the bosonic transformations of thefields on the boundary as

d e a sj nE e a qE j ne a qL e a0m 0 n 0m m 0 0n 0 0m

yla e b ,0 b 0m

1i n i n i idc sj E c qE j c q L c0mq 0 n 0mq m 0 0nq 0 0mq2

1 ab i i j jy l g c yu c ,0 ab 0mq 0 0mq4

X X X 1 Xi n i n i idc sj E c qE j c q L c0my 0 n 0my m 0 0ny 0 0my2

1 X X X Xab i i j jy l g c yu c ,0 ab 0my 0 0my4

d Ai j sj nE Ai j yL Ai j yl yAi j qDD u i j ,0y 0 n 0y 0 0y y 0y 0y 0

d AiX jX

sj nE AiX jX

yL AiX jX

yl qAiX jX

qDD u iX jX

.0q 0 n 0q 0 0q q 0q 0q 0

14Ž .

We see that the transformations with the parametersj m, L , lab and u i j, u iX jX

represent general coordi-0 0 0 0 0Ž . Ž .nate, Weyl, local Lorentz and SO p = SO q

gauge transformations in two dimensions respec-tively. In particular, the general coordinate transfor-mation in the direction Ms2 became two-dimen-sional Weyl transformation. Weights of the Weyltransformation are determined by the powers of x 2

appearing in the boundary behaviors of the fieldsŽ .11 .

On the other hand, in the limit x 2 ™0 thefermionic transformations of the fields on the bound-

Ž .ary in Eq. 11 become

X X1a i a i i a id e s i e g c qe g c ,ž /0m 0q 0mq 0y 0my2

dc i sD e i qAi j e j q2mg e i ,0mq 0m 0q 0m 0q 0m 0y

dc iX

sD e iX

qAiX jX

e jX

y2mg e iX

,0my 0m 0y 0m 0y 0m 0q

w x w xi j m ic j ic j0mq 0myd A s2 ime e qeŽ .0y 0y 0y 0q

qd e mAi j ,0y 0m

X X X X X Xw x w xi j m i c j i c j0mq 0myd A sy2 ime e qeŽ .0q 0q 0y 0q

qd e mAiX jX

. 15Ž .0q 0m

The transformation of e a is that of the two-dimen-0m

Ž .sional p,q supergravities, i.e., supergravities withp supersymmetries of positive chirality and q of

Ž w x .negative chirality See, e.g., Ref. 13 . . However,the transformations of other fields have differentforms from those of the two-dimensional supergravi-ties. Furthermore, they contain c i , c iX

, Ai ,0my 0mq qAiX

, which are non-local functionals of the fields inyŽ .Eq. 11 . We shall try to rewrite these transforma-

tions in a local form by using field equations.Using an identity

1 12h s h qe g q g g 16Ž .Ž .ab ab ab a b2 2

in the second terms, the transformations of theŽ .Rarita-Schwinger fields in Eq. 15 can be rewritten

as

dc i sD e i qe yAi j e j qg h i ,0mq 0m 0q 0m 0y 0q 0m 0y

dc iX

sD e iX

qe qAiX jX

e jX

qg h iX

, 17Ž .0my 0m 0y 0m 0q 0y 0m 0q

( )M. Nishimura, Y. TaniirPhysics Letters B 446 1999 37–42 41

where1i i i j q jh s2me q A g e ,0y 0y 0q 0q2

X X 1 X X Xi i i j y jh sy2me q A g e . 18Ž .0q 0q 0y 0y2

If we regard h i , h iX

as independent transforma-0y 0qŽ .tion parameters, Eq. 17 do not contain non-local

functionals anymore.As for the gauge fields we use field equations of

the Rarita-Schwinger fields to eliminate c i and0myc iX

. First, the first term of the transformation of0mqi j Ž .A in Eq. 15 can be rewritten as0y

w i n j x w i n j xim e g g c qe g g c . 19Ž .Ž .0q y 0 0ny 0y 0 y 0nq

From the Rarita-Schwinger field equations we obtain1 1n i n i n i j jmg g c s g c y g g Au c ,0m 0 0ny 0 0mnq 0m 0 0 0nq2 4

20Ž .where

c i sD c i qe yAi j c j y mln ,Ž .0mnq 0m 0nq 0m 0y 0nq

c iX

sD c iX

qe qAiX jX

c jX

y mln .Ž .0mny 0m 0ny 0m 0q 0ny

21Ž .Ž iX . Ž .c is for later use. Substituting Eq. 20 into0mny

Ž . i jEq. 19 we obtain an expression for d A indepen-0ydent of c i . Similarly we obtain an expression for0myd AiX jX

independent of c iX

. Thus the transforma-0mq 0mqtions of the gauge fields become

1 1i j w i n j x m w i m j xd A s ie g c e q ih g g c0y 0q 0 0mnq 0y 0y 0 y 0mq2 2

X X1 k y k m i jy ie g c e A0y 0my 0y 0y2

3 w i j k x q k my iA e g c e0q 0q 0mq 0y4

3 k q w i jk x my ie g c A e ,0q 0mq 0q 0y4

X X X X X X1 1i j w i n j x m w i m j xd A s ie g c e q ih g g c0q 0y 0 0mny 0q 0q 0 q 0my2 2

X X1 k q k m i jy ie g c e A0q 0mq 0q 0q2

X X X X3 w i j k x y k my iA e g c e0y 0y 0my 0q4

X X X X3 k y w i j k x my ie g c A e . 22Ž .0y 0my 0y 0q4

The last two terms in these transformations stillcontain the fields Ai j or AiX jX

, which are non-local0q 0yfunctionals of the boundary fields. These terms van-ish for p,qF2 since three indices i, j,k or iX, jX,kX

are antisymmetrized. Therefore, we have a localform of fermionic transformations only for p, qF2.

5. Comparison with two-dimensional conformalsupergravities

Let us compare the above fermionic transforma-Ž . Ž . Ž .tions of the boundary fields 15 , 17 , 22 obtained

from the AdSrCFT correspondence with those in theŽ .two-dimensional p,q conformal supergravities for

p, qF2. We begin with the case psqs2. TheŽ . w xtwo-dimensional 2,2 conformal supergravity 14

contains a zweibein e a, Majorana Rarita-Schwingerm

˜ i i jŽ .fields c is1,2 and a real vector field A . Theirm m

fermionic transformations are

1a i a i i i i j j i˜ ˜ ˜ ˜d e s i e g c , dc sD e qA e qg h ,˜ ˜ ˜ ˜ ˜ ˜m m m m m m2

1 x xi j w i rs j j k k˜ ˜ ˜ ˜ ˜d A s i e g g D c qA c˜ ˜ ˜ ž /m m r s r s2

1 w i n j x˜q i h g g c , 23Ž .˜ ˜ m n2

where the transformation parameters e i and h i are˜ ˜Majorana spinors and represent the supertransforma-tion and the super Weyl transformation respectively.

Ž .By identifying the fields 11 with these fields as

i j y i j q iX jX ˜ i i iX

A se A qe A , c sc qc ,m 0m 0y 0m 0q m 0mq 0my

e i se i qe iX

,˜ 0q 0yX 1 X X 1 Xi i i i j j i j jh sh qh y Au e y Au e , 24Ž .˜ 0y 0q 0 0q 0 0y2 2

where iX s i, jX s j, the transformations obtained fromthe AdSrCFT correspondence can be shown to re-

Ž .produce the fermionic transformations in Eq. 23 .Ž .The fermionic transformations of the 2,1 confor-

mal supergravity can be obtained from those of theŽ . Ž .2,2 theory 23 by a truncation

&1212 12 2 2 1˜ ˜c s0, A s0, e s0, h s Au e .˜ ˜ ˜my q y q y2

25Ž .The transformations of the remaining fields e a,m

˜ i ˜ 2 12c , c , A are exactly the same as those ob-mq my ytained from the AdSrCFT correspondence by an

Ž .obvious identification of the fields. The 1,1 theorya ˜ 1w x15 contains e , c , whose fermionic transforma-m m

Ž .tions are obtained from the 2,2 theory by a trunca-˜ 2 12 2 2tion c s0, A s0, e s0, h s0. On the other˜ ˜m m

Ž . w x Ž .hand, the 2,0 theory 16 is obtained from the 2,2˜ i 12 itheory by a truncation c s0, A s0, e s0,˜my q y

i Ž . w xh s0. The 1,0 theory 17 is obtained from the˜q˜ 2 12Ž .2,0 theory by further truncation c s0, A s0,mq y


e 2 s0, h 2 s0. The fermionic transformations of˜ ˜q ythese theories coincide with those obtained from theAdSrCFT correspondence.

Thus, for all p,qF2 the fermionic transforma-tions of the boundary fields are locally realized andare exactly the same as the super and the super Weyl

Ž .transformations of two-dimensional p,q conformalsupergravities. For p)2 or q)2 the fermionictransformations of the gauge fields are non-local anda relation to two-dimensional conformal supergravi-ties is not clear. We note here that the construction

Ž .of the two-dimensional p, p conformal supergravi-Ž .ties based on the super Lie algebra OSp 2, p [

Ž . w xOSp 2, p in Ref. 16 also failed for p)2. It wouldbe interesting to see a relation between this construc-tion and the AdSrCFT correspondence.

References

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14 January 1999


Quarkonium mass splitting revisited: effects of closed mesonicchannels

M. ShmatikovRussian Research Center ‘‘KurchatoÕ Institute’’, 123182 Moscow, Russia

Received 30 May 1998Editor: P.V. Landshoff

Abstract

Modifications of the quarkonium mass spectrum induced by its virtual dissociation into a pair of heavy mesons areconsidered. Coupling between quark and mesonic channels results in noticeable corrections to spin-dependent mass splitting.In particular, accounting for the dissociation effect allows one to reproduce the observable hierarchy of mass splittings in thex , x and x

X multiplets. q 1999 Elsevier Science B.V. All rights reserved.c b b

PACS: 13.20.Gd; 12.39.Pn; 12.40.Yx

Quarkonium is a testing ground for various theo-retical approaches and, especially, of potential mod-els. Analysis of the quarkonium mass spectrum isused for investigating properties of the confiningpotential. Phenomenological potentials were reason-ably successful in describing gross features of thequarkonia spectra: the level spacing and ordering.More subtle details of the mass spectra, in particular,spin-dependent mass splitting still remains a chal-lenging problem. Time-honored approach to theanalysis of spin-dependent effects in quarkonia con-sists in introducing an interquark QED-type potential

w xbut with more generic vertex structure 1 :

˜ 2VsG V q G 1Ž .Ž .i i

˜ 2Ž .where V q is the propagator of the exchangedŽparticle and G is the coupling vertex equal e.g. to 1i

or g m in the case of scalar and vector-type coupling.respectively . In the nonrelativistic limit, expansion

Ž .of 1 in the inverse powers of the heavy-quark mass

yields the sum of central and spin-dependent compo-nents

VsV q S qS L V qS PVŽ .c 1 2 LO 12 T

q S PS V 2Ž . Ž .1 2 SS

with V , V , V standing for the spin-orbit, tensorLO T SSŽ .and spin-spin potentials respectively. In Eq. 2 S1,2

are the quarks’ spins, L is their orbital momentumŽ .Ž . Ž .and S s12 S Pr S Pr y4 S PS . The spin-ˆ ˆ12 1 2 1 2

Ž .structure of the interquark potential 2 can be trans-lated in the mass splitting of the triplet P states asfollows:

3m P smy2 ay4bŽ .0

3m P smyaq2bŽ .1

3m P smqay2br5 3Ž .Ž .2

Žwhere m is the c.o.g. mass of the triplet weighted.with 2 Jq1 factor , a and b are the averaged values


( )M. ShmatikoÕrPhysics Letters B 446 1999 43–4744

of the V and V potential components respec-LO T

tively. The V component of the potential controlsSS

mass splitting between the triplet and singlet Pstates. The combined action of spin-orbit and tensorforces is parameterized in the form of the ratio

m 3P ym 3PŽ . Ž .2 1Rs 4Ž .3 3m P ym PŽ . Ž .1 0

The masses of the 3P charmonium states as deter-Jw xmined by Crystal Ball 2 yield

R x s0.478"0.01 5Ž . Ž .c

with corresponding ratios for 13P and 23P states ofJ Jw xthe bb-quarkonium reading respectively 2

R x s0.664"0.038 6Ž . Ž .b

and

R xX s0.576"0.014 7Ž . Ž .b

Note the hierarchy of R’s as given by the measuredquarkonia masses:

Rexp x R Rexp xX )Rexp x 8Ž . Ž . Ž . Ž .b b c

At the same time potential models involving the sumŽ . Žof the the scalar confining and vector residual. Ž Ž ..one-gluon exchange potentials cf. 1 yielded con-

Ž . Ž .troversial results predicting R x )R x andb cŽ X .R x slightly larger or, at best, approximately equalb

Ž . Ž .to R x in the qualitative disagreement with 8bŽ w x .see 3 for the details :

R xX R R x )R x 9Ž . Ž . Ž . Ž .b b c

Observed disagreement hints at the existence of addi-tional effects. A solution to the problem was soughtby introducing some unconventional interactions, inparticular, a pseudoscalar component of the quark-

5 Ž . w xantiquark potential with Gsg in 1 4 . Anotherissue might be an account of relativistic effectswhich are more pronounced in cc as compared totheir bb counterparts. Currently no convincing solu-tion to the problem in the framework of the poten-tial-based approach seems to exist. The controversycasts doubts on the universally adopted conclusionabout scalar Lorentz-transformation properties of theconfining potential.

In the present paper we consider an alternativemechanism affecting mass-splitting in the quarkoniasystems which might give an insight to the solution

of the challenging problem. It is coupling of quarko-nia states to pairs of heavy quark-mesons HH. Inves-tigation of the effects emerging from such coupling

w xwas pioneered by 5 , where a unified description ofthe heavy quark-antiquark binding and of thequarkonium coupling to heavy mesons was given.However, the calculations made based heavily on aspecific form of the confining potential. The latterwas assumed to have Lorentz-vector transformationproperties in disagreement with the current experi-mental data on both the spectroscopy of cc states

w xand on their decay amplitudes 6 . The latter circum-stance substantiates revisiting effects which emergefrom coupling between quarkonia states and mesonicchannels.

Ž .Coupling of quarkonium QQ bound states toŽ .heavy-meson pairs HH results, because of the

virtual dissociation, in their mass shift

QQ™ Qq q Qq ™QQ 10Ž . Ž .Ž .

The mass shift emerges as a matter of fact due to anŽ .interaction between heavy QQ and virtual light qq

quark pairs. The situation, however, is somewhattricky. Indeed, in the adiabatic approximation the

Žpresence of the light-quark pairs does not affect at.least the long-range part of the potential operating

between heavy quark and antiquark. More precisely,its linear form is maintained, and only the tension isaltered. The latter, however, is a phenomenologicalparameter determined from the experimental dataimplying that in the m ™` limit coupling toQ

mesonic channels does not result in an observablew xeffect 7 . This conclusions refers to the spectrum

gross features controlled by the string tension. At thesame time spin-dependent mass splitting is scaled bya 1rm factor, and non-vanishing effect are possi-Q

ble.Consider in more detail the mechanism which

produces effective mass splitting of the P triplet.JŽ .The lightest Qq states and their antiparticles mate-

rialize in the form of pseudoscalar and vector mesonsŽ ) .denoted hereafter H and H respectively . Be H

)and H mass-degenerate, coupling between QQ andqq would not result in mass splitting of quarkoniumstates. However, because of the spin Qyq interac-tion the H ) vector meson proves to be heavier thanits pseudoscalar H counterpart. It implies that the

( )M. ShmatikoÕrPhysics Letters B 446 1999 43–47 45

mass shift of the quarkonium states is scaled by thewidth of a HH continuous spectrum band up to the

Ž .)effective mass R m ym . The contributionH H

coming from HH pairs with masses exceeding mH)

)qm is canceled by similar contribution of HHH) )and H H heavy-meson states.

A quantum-mechanical consideration of the prob-lem allows making some qualitative conclusionsconcerning the expected effect. First, quarkoniumstates, depending on their spin-parity quantum num-bers, couple to different HH states whereas some of

3them do not have a HH match at all. Indeed, the P03and P quarkonium states couple to the HH pairs2

with Jp s0q and 2q respectively, while the 3P1Ž .level is uncoupled from the pseudoscalar mesonic

1sector. The P state does not couple to HH mesons1

either. Selective coupling results in different massshifts of the QQ states, and, eventually, in their masssplitting. Second, the HH continuous spectrum liesabove the quarkonium bound states. Spin-indepen-dent potentials reproduce the gross features of thespectrum accurately enough to assume that the spin-dependent forces may be treated as a perturbation.

w xThe second-order perturbation in energy 8 equals

2VnmŽ2.E s 11Ž .Ýn Ž0. Ž0.E yEn mm/n

where V is a perturbing interaction and EŽ0. is thenm nŽ . Ž .unperturbed energy of the n-th state. Eq. 11

Žshows that the mass of the lowest state E i.e. the0.quarkonium decreases because of channel coupling

Ž Ž0. Ž0..since E -E .0 m

Combining these observations we can concludethat the masses of both the 3P and 3P states will0 2

diminish because of their coupling to mesonic HHstates, whereas the mass of the 3P state will remain1

Ž .unaffected. Inspection of 4 shows immediately thatthe considered mechanism results in the decrease ofthe R ratio.

Next observation is that the magnitude of theeffect depends on the width of the energy gap sepa-rating quarkonium states from the continuous HHspectrum: the smaller is the gap, the more pro-nounced will be the mass shift. We infer then that

Ž . Ž 3 .the R x ratio 1 P bottomonium states will beb JŽ X . Ž 3affected less than the R x ratio 2 P bottomo-b J

.nium states . Comparing these qualitatiÕe predic-

Ž .tions to the experimental data 8 we conclude thatŽthe considered mechanism coupling between.quarkonium and mesonic states gives a welcome

trend to the modified theoretical results.We proceed now to quantitative estimates of the

effect. To this end a dynamical model describing thecoupling mechanism under consideration is to devel-

Ž . Ž .oped. We treat quarkonium QQ and mesonic HHstates as the components of a two-channel systemlabeling them channels 1 and 2 respectively. Thenthe Schrodinger equations describing the dynamics¨of the system for each Jp state can be written asfollows

T f qV f qV f sE f1 1 11 1 12 2 1 1

T f qV f sE f 12Ž .2 2 21 1 2 2

Here T is the kinetic energy in the correspondingchannels and V is the interaction potential. The V11

Ž .component is a linear q Coulombic potential oper-ating between heavy quarks. Interaction between H-mesons is neglected: near-threshold phenomena un-der consideration involve small relative momentaand, hence, large inter-particle distances. At the sametime, the long-range interaction – p-meson ex-change – between two pseudoscalar particles is ab-sent. Anyway, this approximation does not alter thequalitative conclusions made above. Finally, E and1

E are the channel energies, differing by the mass2

gap in the corresponding state: E yE s2m y2 1 H

m . The non-diagonal component of the potentialQQŽ .V sV describes the dynamics of the quarko-12 21

Ž .nium dissociation QQ ™HH, or, stated differently,the dynamics of the qq production in the field ofheavy quarks. Forbidding complexity of the problemnecessitates introducing some simplifying approxi-mations.

Given the large mass difference between heavyand light quarks, we make use of the adiabaticapproximation. The latter molds in the followingassumptions. First, the position of the heavy quarksis not affected by the process of the qq production.Second, the combined spin of heavy quarks in a pairof heavy mesons is equal to zero. It implies that inall of the considered transitions the spin of the QQpair is subject to one and the same change fromSs1 to Ss0. Finally, c.o.g. of a heavy mesoncoincides with good accuracy with the position of

( )M. ShmatikoÕrPhysics Letters B 446 1999 43–4746

the heavy meson residing in it. All the assumptionsmade above can be expressed concisely in the fol-lowing form of the non-diagonal component:

V sb d ryr X 13Ž . Ž .12

where b is a coupling strength constant, r and r X arethe separation between heavy quarks prior and after

Žthe qq production i.e. within the quarkonium state.and in the HH pair respectively . The light-quark

spin operator and coordinate variables are assumedto be present and are dropped for notational brevity.

In the framework of the model the strength con-stant b can be determined without going into detailsof the complicated mechanism of the light-quark pairproduction. The same coupling between channelsgoverns decays of the quarkonium states lying aboÕe

Ž . Ž .the HH threshold, i.e. c 3770 and F 10580 in thecase of charmonium and bottomonium respectively.They are known to decay predominantly into the DDand BB pairs with the widths equal to 83.9"2.4and 21"4 MeV respectively. The quantum numbers

Ž p y.of the bottomonium state are known J s13translating into the 4 S state of the bb pair. The1

Ž .c 3770 quantum numbers are not determined; fol-lowing the potential model prescriptions we adopt

3 w xthat it is the 1 D state of the cc pair 5,9 . Using the1Ž .channel-coupling potential 13 we calculate the value

of the b strength coupling constant. It proves to beequal to 153 MeV and 114 MeV for the charmoniumand bottomonium systems respectively. The agree-ment between two values of b is reasonably goodsignaling the validity of the approximations made.Indeed, the paradigm of the non-perturbative QCDassumes the light-quark pair production as a result ofthe string break-up. Flavor-blindness of the confine-

Ž .ment forces materialized as a string prompts thenthat in the infinite heavy-quark mass limit the valuesof b for two types of the quarkonium should thesame. Given possible 1rm corrections, the b val-Q

ues may be considered as coinciding in line with thetheoretical expectation.

Having determined the potential V connecting12

the quarkonium and the mesonic channels we pro-ceed to calculating the matrix element of the transi-tion between them. The matrix element controls the

Ž Ž ..value of the mass shift see 11 . Within the approx-imations made it reduces to the overlap integral of

QQthe quarkonium wave function c and of the waveHHfunction c of two non-interacting heavy mesons:n

QQ HHV sb c r Pc r dr 14Ž . Ž . Ž .Hn n

The subscript n labels the states of the HH continu-ous spectrum, to this end we use the c.m.s. momen-

Ž .tum of heavy mesons n'k . Then the expressionŽ .for the quarkonium-state mass shift 10 can be cast

in the form2

VkDm s dk 15Ž .HQQ m ymQQ HH

where m s2m qk 2rm . Formal convergenceH H H HŽ .of the integral is ensured by slow decrease of the

V matrix element due to diminishing overlap be-k

tween the quarkonium bound-state wave functionand the oscillating wave function of two heavymesons in the continuous spectrum. However, theintegration over k which may be translated in theintegration over the mass of the HH pair is to becut-off at much smaller m values. Indeed, theHH

considered mechanism produces mass splitting of the3P states because of the mass difference betweenJ

the pseudoscalar H-meson and its vector counterpart.We incur then that the scale of integration range isset by the m ) ym mass difference.H H

Results of numerical calculations confirm thequalitative conclusions drawn above. To investigatesensitivity of the effect we cut off the integration inŽ . Ž .15 at some m mG2m . Dependence of theH

R-ratio for the 2 P and 1P states of the bottomoniumand for the 1P states of the charmonium are dis-played in Fig. 1 as a function of the reduced mass

Ž Ž ..Fig. 1. Dependence of the quarkonium mass ratio R Eq. 4 onŽ .the reduced width of the mesonic mass band see text .

( )M. ShmatikoÕrPhysics Letters B 446 1999 43–47 47

Ž . Ž .)band rs mr2ym r m ym . As the initialH H H

values for R’s we used those calculated in a potentialw x Žmodel 10 using other predictions of potential mod-

els does not change qualitatively the obtained re-. Ž . Ž .sults . Note the fast onset of the R 2 P QR 1P

regime: already for rR0.2 the hierarchy of the Rvalues corresponds to the experimental observationsŽ .8 and it maintains with further increase of r. Theconsidered mechanism does not yield quantitativeagreement with the experimental data: the overall setof data is shifted downwards with respect to experi-

Ž .mental values shown by error bars . A possibleexplanation may be that parameters of the spin-de-pendent potential determining the ‘‘initial’’ values ofR were varied already in an attempt to reproduceexperimentally observed regularities. It looks plausi-ble that the consistent description of spin-dependentmass splitting in the quarkonium systems requiresthe account of both the spin-dependent part of theQyQ potential and of the bound-state coupling tomesonic channels. The investigation of the problemfollowing these lines is in progress.

Ž .Summarizing, the process of virtual light-quarkpair production considerably affects masses of thequarkonium states. Account of the emerging masssplitting of the 3P triplet allows to reproduce quali-J

tative features of the experimentally observedquarkonium level spacing.

Acknowledgements

The author is indebted to F. Lev and S. Romanovfor helpful comments.

References

w x1 J.F. Donogue, E. Golowich, B.R. Holstein. Dynamics of theStandard Model, Cambridge University Press, 1992.

w x Ž .2 Particle Data Group, Phys. Rev. D 54 1996 1.w x Ž .3 K.K. Seth, in: J. Kirkby, R. Kirkby Eds. , Proc. of the Third

Workshop on the Tau-Charm Factory, Mirabelle, Spain, June1993, Editions Frontieres, p. 461.

w x Ž .4 J. Lee-Franzini, P.J. Franzini, in: J. Kirkby, R. Kirkby Eds. ,Proc. of the Third Workshop on the Tau-Charm Factory,Mirabelle, Spain, June 1993, Editions Frontieres, p. 447.

w x5 E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.-M.Ž . Ž .Yan, Phys. Rev. D 17 1978 3090; D 21 1980 203.

w x6 T. Barnes, in: M.C. Birse, G.D. Lafferty, J.A. McGovernŽ .Eds. , Proc. of Hadron’95, The Sixth Intern. Conf. onHadron Spectroscopy, Manchester, UK, July 1995, WorldScientific, p. 33.

w x Ž .7 P. Geiger, N. Isgur, Phys. Rev. D 41 1990 1595.w x8 L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Nonrela-

tivistic Theory, 3rd ed., Pergamon Press, Oxford - NewYork, 1977.

w x Ž .9 S. Godfrey, N. Isgur, Phys. Rev. D 32 1985 189.w x10 S.N. Gupta, S.F. Radford, W.W. Repko, Phys. Rev. D 26

Ž . Ž .1982 3305; D 34 1986 201.

14 January 1999


t-dependences of vector meson diffractive production in epcollisions

M.G. Ryskin 1, Yu.M. Shabelski 2, A.G. Shuvaev 3

Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188350, Russia

Received 11 March 1998; revised 17 November 1998Editor: P.V. Landshoff

Abstract

We calculate p -dependences of diffractive vector meson photo- and electroproduction in electron-proton collisions withT

and without proton dissociation at the small momenta transfers. The calculated slopes are in good agreement with the data.q 1999 Published by Elsevier Science B.V. All rights reserved.

The goal of the present paper is the selfconsistentapproach in the framework of the general ideas of

Ž .Additive Quark Model AQM to the t-behaviour ofvector meson photo- and electroproduction. We as-

Ž .sume a hadron to be a bound state of two meson orŽ .three baryon constituent quarks, the hadron interac-

tion with small momentum transfer being treated inthe impulse approximation as the interaction of asingle quark.

In the framework of AQM we can considerdiffractive production of vector meson in the highenergy ep collisions as following. A virtual photoncan turn with the probability a s 1r137 into aem

Ž ) .composite hadronic state virtual vector meson, Vwhich constituent interacts with the constituent ofthe target proton. After that we obtain the real vector

Ž . Ž .meson, V, without Fig. 1a or with Fig. 1b target

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

proton dissociation. The cross section of ‘‘elastic’’Ž .without proton dissociation vector meson photo- orelectroproduction with small momentum transfer tothe proton has the form

ds g p™VpŽ . 22 2 2 2< <sF t F t ,Q A s,t ,Q ,Ž . Ž . Ž .p Vdt1Ž .

Ž .where A s,t is the amplitude of constituent interac-tion, and the form factors of the proton and vector

Ž . Ž .meson, F t and F t respectively, account for thep V

probabilities of their ‘‘elastic’’ production. This ex-pression is one of the results of additive quark modelw x1 , although only the assumption that pomerons andphotons interact with hadrons in a similar wayŽ .pomeron-photon duality is really needed here.

ŽThe similar cross section of ‘‘inelastic’’ with.proton dissociation vector meson production does

not contain the proton form factor

ds g p™VYŽ . 22 2 2< <sF t ,Q A s,t ,Q . 2Ž .Ž . Ž .Vdt


( )M.G. Ryskin et al.rPhysics Letters B 446 1999 48–52 49

Ž . Ž .Fig. 1. Diagrams for ‘‘elastic’’ a and ‘‘inelastic’’ b diffractivevector meson photo- and electroproduction.

The proton form factor at moderate values of t isparametrized usually by the dipole expression

1F t s 3Ž . Ž .p 2t

1y 2ž /m

with m2 s 0.71 GeV 2.The vector meson form factors are unknown ex-

perimentally. It seems to be reasonably to assumethey to be similar to the form factors of p and Kmesons, which have a monopole form resulting intothe asymptotic large t behaviour, which is in a goodaccordance with the quark counting rules. The radius

2 2of qq pair produced at large Q is given by R ;VŽ 2 2 .y1 w xQ qm 2,3 and this is the radius that deter-V

Ž .mines the vector meson with mass m wave func-V

tion. Thus we parametrize the vector meson formfactor as

12F t ,Q s 4Ž .Ž .V 2 < <Q q t

1q 2ž /MV

where M is the vector meson mass. The argumentV2 < <Q q t allows to describe simultaneously both the

Ž 2 . 2photoproduction Q s0 and the high-Q electro-production. The value F coincides with a conven-V

tional monopole form factor for the first case. The4 Ž . Ž .factor 1rQ appearing in Eqs. 1 and 2 for theŽ 2 2 < <.second case for Q 4M , t corresponds to theV

propagator of V ) meson in Figs. 1a and 1b.The quark–quark elastic scattering amplitude

Ž 2 .A s,t,Q at high energies can be taken in ReggeŽform as an effective one-pomeron vacuum singular-

. Ž .aP Ž t .ity exchange A sAP srs . The intercept ofqq 0Ž .the pomeron trajectory, a 0 is not important forp

the t-slope value and we will not specify its particu-lar value here.

The numerical value of the slope of pomerontrajectory is known not well enough. Pure phe-nomenologically it is chosen in many papers asa

X; 0.25 GeVy2 . From more theoretical point ofP

Ž . Xview asymptotically at s™` a ™0 for BFKLPw x w xpomeron 4 . In the colour dipole BFKL approach 3

the effective value of aX is assumed to a

X s 0.072P P

GeVy2 . There is also an indication in favour of aŽ X .small slope a consistent with zero in the case ofP

w xelastic Jrc photoproduction 5 . In agreement withw xthe result of 6 obtained from the precise fit of

small-t elastic cross section data with the inclusionof non-vacuum reggeon we have chose the effectiveslope value a

X f0.15 GeVy2 .P

Now the quark–quark elastic scattering amplitudetakes the form

g2m2A s,t ,Q sA s exp B tŽ . Ž .Ž . 0 P2 2ž /Q qMV

g2m sXsA s exp a t ln ,Ž .0 P2 2 ž /ž / sQ qM QV

5Ž .Ž .where A s behaviour is determined by the inter-0

cept of the vacuum exchange, the effective slopeX y2 w x 2a f0.15 GeV 6 , ssW is the interactionP

energy square and s increases with Q2. This effect,QŽ .which reduces the total slope of dsrdt in Eqs. 1

Ž . 2and 2 for the case of large photon virtuality QŽ .andror large mass of produced vector meson hasthe following origin. The transverse size of thepomeron near the photon-vector meson vertex shouldbe very small; the virtuality and the transverse mo-menta k of gluons forming this part of the pomeronT

2 2 2 2 Ž 2 2 .are rather large, k ;Q qM 4m m ;m . OnT V r

the other hand the value of aX is determined by theP

typical gluon transverse momenta, aX Aa rk 2 , andP s T

the pomeron, build up only by high k gluons,T

would have 4 negligibly small value of aX.

4 w xAs was shown in 8 , the slope of the trajectory is propor-tional to the squared inverse transverse momenta of the secon-daries just due to dimensional counting. This question is consid-

w xered in detail in 4 for the case of perturbative QCD.

( )M.G. Ryskin et al.rPhysics Letters B 446 1999 48–5250

ŽThus an appropriate ‘‘time’’ some interval d y of.the rapidity ys ln1rx is needed for the evolution

before the initial small size component of thepomeron develops into the normal equilibrium statewith nonzero value of a

X Aa rm2.P s

Another consequence of the same effect is the2 2 2 g Ž .factor m r Q qM included in Eq. 5 . ItŽ .Ž .V

reflects the fact that the amplitude originated fromthe small size component of the pomeron is less thanthe normal one. The suppression factor is controlledby the anomalous dimension g .

The precise formulae for this ‘‘hard’’ pomeronare based on the BFKL equation for the non-forward

w xQCD pomeron 7 . Unfortunately, the presented ex-pressions are too complicated and it is not the sub-ject of this paper to discuss the structure of suchexpressions. On the other hand as a rule this effect isnumerically small. Therefore we will use the simpli-fied estimations of the effect just replacing the value

1of ln by thex

1 Q2 qM 2 sm2V

ln y ln s ln2 22 2x m Q qMŽ .V

X 1in formula B sa ln for the slope of the ampli-P P x

tude 5.

5 To give some impression about the origin of this extra termQ2 M 2

Vq 2yln we have to recall the structure of k integration inT2m

the BFKL equation. According to the Lipatov log k 2 diffusion, theT

variation of lnk 2 at each step of BFKL evolution is Dlnk 2 f2.T T

To be more precise one has to write Dlnk 2 s1rg , where theT

anomalous dimension g , corresponding to the extremum of theBFKL solution, tends at high energies to the value g s1r2. Bythe same way the rapidity interval D ys Dln1r x needed for one

Ž .step of evolution diffusion can be expressed in term of theŽ .pomeron intercept ls a 0 y1. Numerically for the leadingP

order BFKL pomeron Dln1r xs1rlf2. It means that in orderto diminish the initial large value of k 2 ;Q2 q M 2 downT Ž in . V

to the typical value for the soft processes k 2 ;m2, correspondingT

to the ‘‘equilibrium’’ state of the pomeron with nonzero slopea

X f 0.15 GeVy2 , one has to spend an intervalP

k 2 2 2 2 2Dln1r x g Q q M Q q MT Ž in . V Vd yf ln s ln f ln .

2 2 2 2lDlnk m m mT

Note that the experimental ratio of the typical values of anoma-Ž Ž 2 . yl Ž 2 .g .lous dimension g F x,Q ; x Q and intercept l is2

close to g rlf1 within the HERA domain, where Q2-depen-dence is predicted.

Thus we can write the slope of the pomeronŽ .amplitude of Eq. 5 in the form

1 Q2 qM 2 sm2VX XB sa ln y ln sa lnP P P2 22 2ž /x m Q qMŽ .V

6Ž .2 2 Ž 2 2 .2 2with m sm ; i.e. s s Q qM rm .r Q V

Ž .ds g p™ VpThe experimental cross section is usu-dt

ally parametrized as ebt or ebtqct 2. If we write

ds g p™VpŽ .b P tVAe , 7Ž .

dtd dswe obtain bs log . In our parametrization fordt dt

Ž . Ž .the case of ‘‘elastic’’ production, using Eqs. 1 , 3Ž .and 4

4 2elb t s q q2 B 8Ž . Ž .V P2 2 2< < < <m q t M qQ q tV

and for the case of ‘‘inelastic’’ production

2inelb t s q2 B . 9Ž . Ž .V P2 2 < <M qQ q tV

Note that for hadron-hadron interactions the expres-Ž .sion similar to Eq. 8 provides a good description

w x Ž .1 of the experimental data for the slopes b t ofelastic p p and pp scattering at p s 200 GeVrcl ab

and different t.Let us now compare the theoretical predictions,

Ž . Ž .Eqs. 8 and 9 with the experimental data, see

Table 1Ž .The comparison of predictions of Eq. 8 for the slopes of

Ž . Ž y2 .‘‘elastically’’ produced vector mesons, b 0 in GeV , with theexperimental data

2 ² : Ž . Ž .Reaction Q , W , b 0 , b 02 Ž . Ž .GeV GeV Eq. 8 exp

w xg p™ r p 0 70 11.7 11.6"0.2 9w xg p™ r p 10 80 6.9 7.8"1.0"0.7 10w xg p™ r p 20 80 6.4 5.7"1.3"0.7 10w xg p™ v p 0 70 11.6 10.0"1.2"1.3 11

a w xg p™f p 0 70 7.8 7.3"1.0"0.8 12w xg p™f p 10 100 7.0 5.2"1.6"1.0 13

X w xg p™ r p 10 80 6.8 5.5"1.9"0.5 14b w xg p™ Jrc p 0 90 4.7 4.0"0.2"0.2 15c q2.0 w xg p™ Jrc p 12 90 4.1 3.8"1.2 10y1 .6

a ² < <: 2 b ² < <: 2 c ² < <:At t s 0.3 GeV . At t s 0.5 GeV . At t s 0.5GeV 2.

( )M.G. Ryskin et al.rPhysics Letters B 446 1999 48–52 51

Table 2Ž .The comparison of predictions of Eq. 9 for the slopes ofŽ . Ž y2 .‘‘inelastically’’ produced vector mesons, b 0 in GeV , with

the experimental data2 ² : Ž . Ž .Reaction Q , W , b 0 , b 0

2 Ž . Ž .GeV GeV Eq. 9 exp

w xg p™ rY 0 70 6.1 5.3"0.3"0.7 9w xg p™ rY 10 100 1.4 2.1"0.5"0.5 13

a w xg p™ Jrc Y 0 100 1.45 1.6"0.3"0.1 13

a ² < <: 2At t s 0.5 GeV .

Tables 1 and 2. In the cases, where there are a lot ofŽ Ž .experimental points say, for r 0 photo- and elec-

.troproduction we present the data having the small-est experimental errors. The most accurate value ofthe slope is obtained for r 0 ‘‘elastic’’ photoproduc-

w xtion 9 , where the error is only about 2% and theŽ .dependence of b t on the t region is evident. The

Ž .agreement with Eq. 8 is very good for such asimple theoretical estimations. The predicted value

Ž .of ds g p™r p rdt normalized to the data with theŽ .help of parameter A of Eq. 5 is compared with the0

w xexperimental data of Ref. 9 in Fig. 2a. One can see,that the agreement is quite good.

In the case of r 0 elastic electroproduction thevalues of the slopes decrease significantly in agree-ment with our predictions.

w xThe result for v photoproduction 11 is of theorder of the r 0 case.

As experimentally the slope in f photoproduction2 < <was measured in the region 0.1 GeV - t - 0.5

GeV5, we present the theoretical value of the slopeat t s 0.3 GeV 2. The slope values here alsodecrease for the electroproduction.

Ž . Ž .Eqs. 4 and 6 show that the slopes decreasewhen the mass of produced vector meson increases.This effect should be more significant for the small-Q2 interactions. It can seen for the example of r

X

electroproduction as well as of Jrc ‘‘elastic’’ photo-and electroproduction. In Fig. 2b our predictions for

Ž .ds g p™Jrc p rdt normalized to the data arew xcompared with the experimental points of Ref. 15 .

One can see that the predicted slope is really slightlytoo large as it is presented in Table 1.

Now let us consider the slopes in the ‘‘inelastic’’photo- and electroproduction, i.e. with diffractivedissociation of the proton. We compare our predic-

Ž .tions given by Eq. 9 with the data in Table 2. In allcases the calculated values of the slopes are in goodagreement with the data.

ŽAs a result we can claim that the simplest based.on AQM estimations of the slopes in the processes

of vector meson photo- and electroproduction are insurprising agreement with all available experimentaldata.

The slightly different expression for differentialcross sections of vector meson photo- and electro-

w xproduction was presented in 16 . Our results are inw xqualitative agreement with the model 17 .

o Ž . Ž .Fig. 2. Differential cross sections of r a and Jrc b elasticphotoproduction. The predicted cross sections are normalized tothe data.

( )M.G. Ryskin et al.rPhysics Letters B 446 1999 48–5252

Acknowledgements

This work is supported by Volkswagen Stiftungand, in part, by Russian Fund of Fundamental

Ž .Reswarch 98-02-17629 , INTAS grand 93-0079 andNATO grand OUTR. LG 971390.

References

w x1 V.V. Anisovich, M.N. Kobrinski, J. Nyiri, Yu.M. Shabelski.Quark Model and High Energy Collisions, World Scientific,Singapore, 1985.

w x2 B.Z. Kopeliovich, J. Nemchick, N.N. Nikolaev, B.G. Za-Ž .kharov. Phys. Lett. B 309 1993 179.

w x3 N.N. Nikolaev, B.G. Zakharov, V.R. Zoller, Phys. Lett. BŽ .366 1996 337.

w x Ž .4 E.M. Levin, M.G. Ryskin, Yad. Fiz. 50 1989 1417.

w x Ž .5 A. Levy, Phys. Lett. B 424 1998 191.w x Ž .6 J.P. Burq et al., Phys. Lett. B 109 1982 124; Nucl. Phys. B

Ž .217 1983 285.w x Ž .7 L.N. Lipatov, Sov. Phys. JETP 63 1986 904; ZhETF 90

Ž .1986 1536.w x Ž .8 E.L. Feinberg, D.S. Chernavski, Usp. Fiz. Nauk 82 1964

Ž .41; V.N. Gribov, Yad. Fiz. 9 1969 640.w x9 The ZEUS Collaboration PA02-050, XXVIII Int. Conf. on

High Energy Physics, Warsaw, Poland, July 25–31, 1996.w x10 S. Aid et al., H1 Collaboration Preprint DESY 96-023, 1996.w x Ž .11 The ZEUS Collaboration, Z. Phys. C 73 1996 73.w x Ž .12 The ZEUS Collaboration, Phys. Lett. B 377 1996 259.w x Ž .13 C. Adloff et al., H1 Collaboration, Z. Phys. C 75 1997 607.w x14 The H1 Collaboration PA01-088, XXVIII Int. Conf. on High

Energy Physics, Warsaw, Poland, July 25–31, 1996.w x Ž .15 The H1 Collaboration, Nucl. Phys. B 472 1996 603.w x16 L.P.A. Haakman, A. Kaidalov, J.H. Koch, Phys. Lett. B 365

Ž .1996 411.w x Ž .17 E. Gotsman, E. Levin, U. Maor, Phys. Lett. B 403 1997

120.

14 January 1999


Primordial fluctuations from inflation:a consistent histories approach

David PolarskiLab. de Mathematique et Physique Theorique, EP93 CNRS, UniÕersite de Tours, Parc de Grandmont, F-37200 Tours, France´ ´ ´

Departement d’Astrophysique RelatiÕiste et de Cosmologie, ObserÕatoire de Paris-Meudon, 92195 Meudon Cedex, France´

Received 9 October 1998Editor: L. Alvarez-Gaume

Abstract

We show how the quantum-to-classical transition of the cosmological fluctuations produced during an inflationary stagecan be described using the consistent histories approach. We identify the corresponding histories in the limit of infinitesqueezing. To take the decaying mode into account, we propose an extension to coarse-grained histories. q 1999 ElsevierScience B.V. All rights reserved.

PACS: 04.62.qv; 98.80.Cq

1. Introduction

w xIn the inflationary paradigm 1 , all inhomo-geneities in the universe originated from primordial

Ž .quantum fluctuations of some scalar field s , theŽ .so-called inflaton s . It is then possible to calculate

w xthese fluctuations 2 , to track their evolution untiltoday and to make predictions, for each given model,concerning the formation of large-scale structures inthe universe and the anisotropy of the CosmologicalMicrowave Background. The latter will be measured

Žwith high accuracy till small angular scales see forw x.example 3 A comprehensive understanding of the

quantum-to-classical transition of these fluctuationsis required is order to explain the obviously classicalnature of the inhomogeneities, both in matter andradiation, observed on cosmological scales. It wasalready shown that the fluctuations undergo a quan-tum-to-classical transition as a result of their peculiar

dynamics due to the accelerated expansion of thew xuniverse during the inflationary stage 4 . This dy-

w xnamics leads to a very highly squeezed state 5–7 ,< < < <where the squeezing parameter r satisfies r 41.k k

This is expressed in the Heisenberg representation bya vanishingly small decaying mode, nowadays oncosmological scales, compared to the growing mode.This is why inflationary fluctuations that are inprinciple observable today, after they reenter theHubble radius, have a stochastic amplitude but afixed phase. Note that random amplitude and fixedphase does not point necessarily to a quantum ori-

w xgin, but rather to a primordial origin 8,6 of thefluctuations. The inflationary paradigm explains ele-gantly this primordial origin. This has observationalconsequences, for instance the existence of sec-

Ž .ondary acoustic Sakharov peaks in the small-angleCosmological Microwave Background anisotropy.The amplitudes have, in most models where the


( )D. PolarskirPhysics Letters B 446 1999 53–5754

fluctuations arise from vacuum initial states, aGaussian distribution. However the above results canbe extended to a non-Gaussian distribution whenfluctuations are produced in non vacuum initial satesw x9 . In other words, this quantum-to-classical transi-tion is essentially independent of the initial state. Thesensitivity of this process to the environment was

w xalso considered 10 and the result was found that fora wide class of interactions this transition still holds,independently of the fluctuations initial state at theinflationary stage, the amplitude basis defining thepointer basis. This quantum-to-classical transitionthat takes place solely as a result of the dynamics isvery intriguing. The remaining coherence of theinitial quantum state can be described to very highaccuracy with classical stochastic terms. In the cos-mological context one is certainly willing to acceptthe probabilistic description of the fluctuations dueto an indeterminacy in the initial conditions. Thisfact really becomes non-trivial when the fluctuationsare quantized. In that case, a deterministic evolutionof the wave function leads effectively to a classicalevolution with stochastic initial conditions. Thenon-relativistic free quantum particle at very latetimes provides yet another surprising example where

w xan analogous transition is taking place 11 . Wepresent here a way to describe this transition using

w x Ž w xthe consistent histories approach 12 see also 13.for a review . We feel that the consistent histories

approach is conceptually very appropriate for thedescription of the quantum-to-classical transition tak-ing place for the primordial fluctuations of inflation-ary origin. Furthermore, the Heisenberg picture which

Ž .is used for the description of the second quantizedfluctuations, enters in a natural way in this picturetoo. We first briefly review this approach and weshow how, for the inflationary fluctuations, historiescan be defined.

2. Decoherent histories

In this formalism, the evolution of a quantumsystem is given by specifying a set of alternativehistories. These are defined by a set of Heisenberg

i Ž .projection operators P t , at a sequence of times ta i ii

with t - t - . . . - t . A single history corresponds1 2 n

to a particular choice a for each time t , i.e. to ai i

� 4 � 4particular set of indices a ' a ,a , . . . ,a . The1 2 ni Ž .Heisenberg projection operators P t satisfy thea ii

following conditions

P i t P iX t sP i t d X , 1Ž . Ž . Ž . Ž .a i a i a i a ai i i i i

and

P i s Id . 2Ž .Ý a iai

Ž .Condition 1 , which says that the projection opera-� 4tors are orthogonal, means that the histories a are

Ž .exclusive, while condition 2 , which says that thei Ž .projection operators P t form a complete set ata ii

each time t , means that the set of alternative histo-i

ries is exhaustive. We can now define a historyoperator Ca

C 'P n P ny1 . . . P 2 P1 , 3Ž .a a a a an ny1 2 1

which gives the branch state vector

< :C C , 4Ž .a

when applied to the initial state vector C , where weŽassume that we start with a pure state the case of

.relevance for inflation . Sets of alternative historieswith vanishing interference are said to decohere. Inthat case, it is possible to attach consistently a proba-

Ž . � 4bility p a to each individual history a

² < † < :p a s C C C C . 5Ž . Ž .a a

Interference is measured by the decoherence func-Ž X.tional D a a

X ² < † < :XD a a s C C C C . 6Ž . Ž .a a

Hence we have in the case of decoherent histories

D a aX sp a d X . 7Ž . Ž . Ž .aa

These definitions are straightforwardly generalizedwhen one starts with a mixed state and the corre-sponding density matrix r. In that case we have

D a aX sTr C r C†

X sp a d X . 8Ž . Ž . Ž .Ž .a a a a

3. Quantum fluctuations from inflation

Let us consider now the primordial quantum fluc-tuations produced during the inflationary stage. Re-garding the quantum-to-classical transition, the ten-

Ž .sorial fluctuations or gravitational waves can serve

( )D. PolarskirPhysics Letters B 446 1999 53–57 55

as the paradigm for both types of fluctuations, scalarand tensorial, of inflationary origin. The physics ofthese quantum fluctuations was already studied indepth, we give now the essential results using the

w xnotations of 4 . The quantities y , resp. p , are thek k

Fourier transforms of the amplitude y, resp. theconjugate momentum p, all these quantities being

t X Ž X.time-dependent. The conformal time h'H dt ra tis used.

The dynamics of the system is particularly trans-parent in the Heisenberg representation. We have,

Ž .using the amplitude field modes f h with R f 'k k'Ž . Žf and I f ' f , f h s1r 2k we adopt ak1 k k 2 k 0.similar notation for all quantities

y k , h ' f h a k ,h q f ) h a† yk ,hŽ . Ž . Ž . Ž . Ž .k 0 k 0

s f h e k y f h e k 9Ž . Ž . Ž . Ž . Ž .k1 y k 2 p

Ž . Ž .and the momentum field modes g h , g h sk k 0'kr2 ,

p k , hŽ .) †'yi g h a k ,h yg h a yk ,hŽ . Ž . Ž . Ž .k 0 k 0

sg h e k qg h e k . 10Ž . Ž . Ž . Ž . Ž .k1 p k 2 y

Ž . 'The time independent operators e k ' 2rkp

'Ž . Ž . Ž .p k,h , resp. e k ' 2k y k,h , satisfy0 y 0

² † X :e k e kŽ . Ž .y y

² † X : Ž3. Xs e k e k sd kyk ,Ž . Ž . Ž .p p

e† k se yk . 11Ž . Ž . Ž .y , p y , p

They obey the commutation relationsX X† Ž3.e k , e k s2 i d d kyk , i , jsy , p .Ž . Ž . Ž .i j i j

12Ž .ŽWhen the initial state is the vacuum state, all non

.vanishing correlation functions are derived fromŽ .11 . The field modes can be parametrized by threeparameters, the squeezing parameter r , the squeez-k

ing angle f and the rotation angle u .k k

If l is the physical wavelength of the perturba-Ž .tion, R 'ara in units for which cs1 the Hub-˙HŽ .ble radius and a t the scale factor of the FRW

metric then, time evolution in the regime l4RH

leads to an extreme squeezing which persists whenthe perturbation reenters the Hubble radius, i.e. forl-R . As a result, it is possible to take f ™0H k 2

Ž w x.and g ™0 see e.g. 4 . The quantum coherence isk1

then expressible in classical stochastic terms: for agiven‘‘realization’’ y of the fluctuation field, wek

Ž .have for its canonical momentum p , g rf yk k 2 k1 k

'p , the classical momentum for large squeezing,k ,cl< <i.e., for r ™`. This almost perfect classical corre-k

lation is nicely exhibited with the help of the WignerŽ .function – a well-known example of quasi proba-

w xbility density in phase-space 4,14,15 .We consider now the operators y separately fork

each k but, for brevity of notation, we will drop inŽ .the following the subscript k or k . In the limit of a

perfect classical correlation, the system is describedby a set of decoherent histories, to each singlehistory a probability can be assigned, these probabili-ties add to one. The projection operators defining thehistories in that case are given by

i < : ² <P t s y y , 13Ž . Ž .a i i ii

where y stands for a particular value of the operatoriŽ .y t at time t . We can now consistently assign ai

Ž .probability density to each individual trajectoryŽ Ž . Ž . Ž ..y t y t . . . y t in amplitude space, wheren ny1 1

Ž . Ž < <all the y t lie on the classical trajectory for r ™i k. Ž .` that goes through y t at time t . Indeed, proba-1 1

bility is conserved along classical trajectories passingŽ .through y at some late time t , withi i

< < 2p a ' C y ,h y s f h y 'y ; i .Ž . Ž . Ž .0 0 i k1 i 0 i ,cl

14Ž .In the limit where the decaying mode is negligible,

Ž .i.e. f ™0, 14 holds for all realizations y at timek 2 i

t of the amplitude. This is the highly non-trivialiŽ .property of our isolated pure state allowing for its

description in classical terms. In particular, we canconsistently define the joint probability

W y ,t ; y ,t ; . . . ; y ,t ; y ,t dy . . . dyŽ .n n ny1 ny1 2 2 1 1 n 1

15Ž .to find the amplitude y between y and y qdy ati i i

each late time t . Of course this joint probability isi

Markovian and satisfies

W y ,t ; y ,t ; . . . ; y ,t ; y ,tŽ .n n ny1 ny1 2 2 1 1

sp y ,t ; y ,tŽ .n n ny1 ny1

=p y ,t ; y ,t . . .Ž .ny1 ny1 ny2 ny2

=p y ,t ; y ,t PP y ,t , 16Ž . Ž . Ž .2 2 1 1 1 1

( )D. PolarskirPhysics Letters B 446 1999 53–5756

Ž .where p y ,t ; y ,t is the conditional probabil-i i iy1 iy1Ž .ity density to have y t sy provided we hadi i

Ž . Ž .y t sy . Further, PP y ,t is the probabilityiy1 iy1 1 1Ž .for y t sy and it satisfies1 1

< < 2PP y ,t s C y ,h sp aŽ . Ž . Ž .1 1 0 0

y s f h y . 17Ž . Ž .1 k1 1 0

Actually, we have very generally

f tŽ .1 ip y ,t ; y ,t sd y t y y 1F jFn .Ž .Ž .i i j j i jž /f tŽ .1 j

18Ž .Ž .Eq. 18 is valid in the limit that the decaying mode

is negligible. The consistent histories picture exhibitsnicely the classical stochastic nature of the fluctua-

< <tions: in the limit r ™` one should not expect tok

find the system on one single classical trajectory.Rather, it can be found on any classical trajectorywith a certain probability. Each of these classicaltrajectories constitutes one of the possible alternativehistories to which a probability can be assigned.Usually only probability amplitudes can be assigned

w xto trajectories due to quantum interference 16 . Whilethe precise physical meaning of the individual trajec-

Ž .tories is a subject of endless debate, the crucialpoint here is the ability to consistently assign a

Ž .probability to each history ‘‘classical trajectory’’ .For large enough, though nevertheless finite

< <squeezing parameter r , which is the case relevantk

for fluctuations of inflationary origin, one may wishto take the decaying mode into account. In that casewe suggest a more realistic description of our systemin terms of coarse-grained histories corresponding tocoarse-grained trajectories in amplitude space. In-stead of assigning a certain probability to the historyŽ Ž . Ž . Ž ..y t y t . . . y t , we now assign a probabilityn ny1 1

Ž Ž . Ž . Ž ..to the history D y t D y t . . . D y t wheren ny1 1i Ž .the projection operator P t is now given bya ii

i < : ² <P t s D y D y , 19Ž . Ž .a i i ii

< :where D y corresponds to a state for which theiŽ .amplitude y t at time t is localized within thei

interval D y . The corresponding decoherence func-iŽ X .tional D a a can then be written in the followingD

suggestive way

D a aX s DD y DD yX D y t , yX t , 20Ž . Ž . Ž . Ž .Ž .H HD

Xa a

Ž Ž . Ž X..where D y t , y t corresponds to the historiesconstructed with the projection operators< Ž .: ² Ž . < � 4 Ž .y t y t and a ' D y , . . . ,D y . Decoher-n 1

ence is achieved when

D a aX f0 . 21Ž . Ž .D

We can think of a particular history as a series ofslits trough which our system has to pass at the

Ž .successive times t , the condition 22 then expressesi

the almost complete absence of diffraction. Thisalmost complete absence of diffraction was shown

Ž . w xwith a slit thought experiment in 15 . Note that theŽ .integrals appearing in 21 are really functional inte-

w xgrals, integrals over all the possible paths 17 that� 4can belong to a given coarse-grained history a , not

just classical paths. In inflation we will have thatD y is tremendously small, far beyond our observa-i

tional capabilities. This is where the smallness of thedecaying mode enters.

To summarize, we have shown how the quantum-to-classical transition of quantum fluctuations of in-flationary origin can be nicely described in the con-sistent histories picture. We have identified the histo-ries and the corresponding projection operators in the

< <limit r ™` and we have extended our description,k

using coarse grained histories, to the more realisticcase where the decaying mode, though tremendouslytiny compared to the growing mode, is neverthelesspresent. In the case of fluctuations arising frominflation, the decaying mode is definitely negligiblecompared to the growing mode so that the limit< <r ™` safely applies for most purposes. However,k

depending on the level of precision with which oneis willing, or has, to describe the fluctuations onemay wish to take the decaying mode into account.We propose to use coarse-grained histories for thispurpose. In a sense, it amounts to describe theevolution of our system in terms of classical trajecto-ries with a very tiny quantum noise which is due to

Ž .that piece of the amplitude ‘‘position’’ operatorconnected to the decaying mode. It is also possible toconsider decoherence in the system-environment

w xcouple in the consistent histories picture 18 and weleave these points for further investigation. The con-sistent histories picture is known to be very fruitfulin many problems related to decoherence in particu-lar, and to the interpretation of quantum mechanics

( )D. PolarskirPhysics Letters B 446 1999 53–57 57

in general. It is therefore gratifying that this ap-proach can be used in the context of inflation aswell. In particular all the interpretational questionsarising in this picture can be considered for theparticular, but all-important, case of fluctuations ofinflationary origin.

References

w x Ž .1 A. Linde, Rep. Prog. Phys. 47 1984 925; Particle physicsand inflationary cosmology, Harwood, New York, 1990; E.Kolb, M. Turner, The Early Universe, Addison-Wesley, Red-wood City, 1990.

w x Ž .2 S.W. Hawking, Phys. Lett. B 115 1982 295; A.A. Starobin-Ž .sky, Phys. Lett. B 117 1982 175; A.H. Guth, S-Y. Pi, Phys.

Ž .Rev. Lett. 49 1982 1110.w x3 http:rrastro.estec.esa.nlrSA-generalrProjectsrPlanckr

http:rrmap.gsfc.nasa.govrw x4 D. Polarski, A.A. Starobinsky, Class. Quantum Grav. 13

Ž .1996 377.w x Ž .5 L.P. Grishchuk, Y.V. Sidorov, Phys. Rev. D 42 1990 3413.w x6 A. Albrecht, P. Ferreira, M. Joyce, T. Prokopec, Phys. Rev.

Ž .D 50 1994 4807.

w x7 A. Albrecht, Report hep-thr9402062.w x Ž .8 D. Polarski, A.A. Starobinsky, Phys. Lett. B 356 1995 196.w x9 J. Lesgourgues, D. Polarski, A.A. Starobinsky, Nucl. Phys. B

Ž .497 1997 479.w x10 C. Kiefer, D. Polarski, A.A. Starobinsky, Int. Journ. Mod.

Ž .Phys. D 7 1998 455.w x Ž .11 C. Kiefer, D. Polarski, Ann. Physik 7 1998 137, Report

gr-qcr9805014.w x Ž .12 R. Griffiths, J. Stat. Phys. 36 1984 219; Phys. Rev. A 54

Ž . Ž .1996 2759; R. Omnes, J. Stat. Phys. 53 1988 893; The`interpretation of Quantum Mechanics, Princeton UniversityPress, Princeton, 1994; M. Gell-Mann, J. Hartle, Complexity,Entropy and the Physics of Information, SFI Studies in the

Ž .Sciences of Complexity, vol. VIII, W. Zurek Ed. , AddisonWesley, Reading, 1990, p. 425.

w x13 D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.O. Stamatescu,H.D. Zeh, Decoherence and the Appearance of a ClassicalWorld in Quantum Theory, Springer, 1996.

w x14 J. Lesgourgues, D. Polarski, A.A. Starobinsky, Class. Quan-Ž .tum Grav. 14 1997 881.

w x15 C. Kiefer, J. Lesgourgues, D. Polarski, A.A. Starobinsky,Ž .Class. Quantum Grav. 15 1998 L67.

w x16 R.P. Feynman, A.R. Hibbs, Quantum Mechanics and PathIntegrals, McGraw-Hill, 1965.

w x Ž .17 J. Hartle, Phys. Rev. D 10 1991 3173.w x Ž .18 J. Finkelstein, Phys. Rev. D 47 1993 5430.

14 January 1999


Phase transition of a scalar field theory at high temperatures

Hidenori Sonoda 1

Physics Department, Kobe UniÕersity, Kobe 657-8501, Japan

Received 18 September 1998; revised 29 November 1998Editor: H. Georgi

Abstract

At high temperatures a four dimensional field theory is reduced to a three dimensional field theory. In this letter weconsider the f 4 theory whose parameters are chosen so that a thermal phase transition occurs at a high temperature. Usingthe known properties of the three dimensional theory, we derive a non-trivial correction to the critical temperature. q 1999Elsevier Science B.V. All rights reserved.

PACS: 11.10.Wx; 11.10.Kk

At very high temperatures the thermal propertiesof a 3q1 dimensional field theory are given by an

w xeffective three dimensional field theory. 1,2 In thisletter we use an effective field theory to discuss thethermal phase transition of the f 4 theory at a hightemperature. The idea is simple: we choose theparameters of the f 4 theory in such a way that thephase transition occurs at a high temperature forwhich the three dimensional reduction is a goodapproximation. We can then use the known proper-ties of the three dimensional f 4 theory to discussthe physics near and at the transition temperature. In

w xRef. 3 three-loop calculations were done to com-pute the free energy density of the massless theory athigh temperatures, but the thermal phase transition ofa massive theory was not discussed. The method ofeffective field theory has already been used exten-sively to discuss the thermal phase transitions of the

w x 4electroweak theory. 4 Applied to the simpler f


theory, the method can give non-trivial results moreeasily because the three dimensional f 4 theory ismuch better understood than the gauge theories.

The four dimensional theory is defined by thelagrangian

1 m2 l2 4LLs E fE fq f q f qcounterterms ,m m2 2 4!

1Ž .

where the counterterms are chosen in the MS scheme.We will measure all dimensionful quantities in unitsof the appropriate power of the renormalization massscale m, and this means in particular ms1. Wechoose m2 to be negative so that the Z symmetry is2

broken spontaneously at zero temperature. Then acontinuous transition occurs approximately at tem-

2 w x(perature T , y24m rl 5,6 . The Z symmetryc 2

is restored at temperatures T)T . Observe that wec

can choose ym2 to be of order T and l of orderc

1rT . If we take T large, i.e., the coupling l small,c c

so that the physical mass m is much smaller thanph


( )H. SonodarPhysics Letters B 446 1999 58–61 59

T , we can reduce the four dimensional theory to anc

effective three dimensional theory, and we can usethe well-known results on the phase transition of thethree dimensional f 4 theory. Near the transition,m is small, and the condition m <T is boundph ph c

to be satisfied.The three-dimensional f 4 theory is defined by

the lagrangian

1 m2 l d m23 3 32 4 2LL s E FE Fq F q F q F ,eff m m2 2 4! 2

2Ž .

where the spatial dimension is 3ye , and the renor-malization is done in the MS scheme with

12 2d m s Cl , 3Ž .3 32e

and

1Csy . 4Ž .26 4pŽ .The parameters m2 and l are related to the temper-3 3

ature T and the parameters of the four dimensionalw x 2theory as follows 3 :

l sT lqO l2 5Ž . Ž .Ž .3

l2 2 2m sm 1q lnTqk qO lŽ . Ž .3 2ž /4pŽ .

lT 2 lq 1q ylnTqky2 DŽ .2ž24 4pŽ .

qO l2 , 6Ž . Ž ./where the constants are given by

ks ln4pyg ,

zX y1Ž .

Ds ln4py1y ,y.454 . 7Ž .z y1Ž .

Ž Ž .g, .577 is Euler’s constant, and z z is Riemann’s.zeta function. Note that with the choice

m2 sO T , lsO Ty1 , 8Ž . Ž . Ž .

2 w xThe calculations in Ref. 3 largely depend on the earlierw xcalculations in Ref. 7 .

Žthe above approximations tree for l , one-loop for32 2 2 .the m term in m , and two-loop for the T term3

give all the contributions which survive the limitT™`: the higher order corrections vanish as T™`.

Ž .A comment is in order on our convention. In Eq. 6we find logarithms of the temperature T. The readermight wonder if it is legitimate to have a dimension-ful quantity as the argument of a logarithm, but wehave decided to measure all mass dimension 1 quan-tities in units of the renormalization mass scale m,and lnT should be read as lnTrm.

The parameters l and m2 of the three dimen-3 3

sional theory satisfy the following renormalizationŽ .group RG equations:

d d2 2 2l sl , m s2m qCl . 9Ž .3 3 3 3 3dt dt

Ž . Ž .Note that these are consistent with Eqs. 5 and 6and the one-loop RG equations of the four-dimen-sional theory:

d 3l2 d l2 2l,y , m , 2y m .2 2ž /dt dt4p 4pŽ . Ž .

10Ž .

ŽŽ . .drdt TsT by convention.Let us summarize what is known about the phase

4 Žtransition of the three dimensional f theory. Seew x .8 , especially chapter 8. Given l , the Z symme-3 2

try is exact for m2 )m2 , and it is spontaneously3 3c

broken for m2 -m2 . Whether the symmetry is bro-3 3c

ken or not must be determined by a RG invariantcriterion. Using l and m2 , we can construct only3 3

one independent RG invariant which can be chosenas

m2 yCl2 lnl3 3 32R m ,l ' , 11Ž .Ž .3 3 2l3

Ž .where the constant C is given by Eq. 4 . It is trivialŽ .to check the RG invariance of R using Eq. 9 .

Using R we can rephrase the criterion for the transi-tion: the symmetry is exact for R)R , and brokenc

Ž .for R-R , where R is a constant Fig. 1 . Thisc c

implies that

m2 sl2 R qC lnl . 12Ž . Ž .3c 3 c 3

The constant R has not been calculated analytically.c

( )H. SonodarPhysics Letters B 446 1999 58–6160

Fig. 1. RG flows of the three dimensional theory.

The critical temperature T of the four dimen-c

sional theory can be obtained from the condition2 2 Ž .m sm . Substituting Eq. 6 into this, we obtain3 3c

the following expression for the critical temperature:2y24m l

2 2T s 1q yln y24m y3lnlŽ .Žc 2l 4pŽ .

2 2q24 4p R q2 D qO l , 13Ž . Ž . Ž ..c

Ž .where D is given in Eq. 7 . We observe the non-trivial logarithmic dependence on l. 3

For completeness let us add a few commentsŽabout the critical behavior of the theory. Again a

w x .good reference is Ref. 8 . Clearly the critical ther-mal behavior of the four dimensional theory is thesame as the critical behavior of the three dimensionaltheory. In particular, all the critical exponents are thesame. For example, near TsT the physical massc

m of the theory behaves asph

2

yE2 2m ,constPl RyRŽ .ph 3 c

22 TyT yc E2y 2,constPl T , 14Ž .y cE ž /Tc

where y is the scale dimension of the relevantE

parameter at the non-trivial fixed point. The approxi-

3 w xThis was expected in Ref. 1 . The dependence on the loga-rithm of l occurs also in the three dimensional massless theory3w x9 .

mate value of y has been calculated by variousE

methods: for example, the one-loop Callan-Symanzikw xequation gives 8

5y , . 15Ž .E 3

Similarly, at the critical temperature, the two-pointŽ .thermal correlation function Matsubara function of

f behaves as

1

Tc2 2² : ² :dt f r ,t f 0,0 , F r F 0Ž . Ž . Ž . Ž .TsT m smH c 3 3 ,c

0

lyh Tyhc

,constP ,1qhr16Ž .

where the anomalous dimension h is about .05.How can we improve the high temperature ap-

proximation? At higher orders we must not onlyŽ .calculate the next loop order terms in Eqs. 5 and

Ž . 46 , but also we must introduce irrelevant terms6 Ž . 2 Žsuch as F dimension three and F E FE F di-m m

.mension four in the three dimensional lagrangian.The calculations will be significantly more compli-cated. A simple analysis shows that the parameter ofF 6 is of order 1rT 3, and that of F 2E FE F is ofm m

order 1rT 2. Therefore, we still do not need tointroduce irrelevant terms at the next order, but at thenext next order we must introduce the termF 2E FE F .m m

The above calculation of the critical temperatureŽ .can be easily extended to the four dimensional O N

linear sigma model whose lagrangian is given by

N 2 N1 mI I I ILLs E f E f q f fÝ Ým m2 2Is1 Is1

2NlI Iq f f qcounterterms . 17Ž .Ýž /8 Is1

4 Irrelevant with respect to the fixed point at l s m2 s0 as3 3

opposed to the non-trivial infrared fixed point.

( )H. SonodarPhysics Letters B 446 1999 58–61 61

The critical temperature is obtained as2y24m 3l

2 2T s 1q yln y24mŽ .�c 2Nq2 lŽ . 4pŽ .2y3ln Nq2 l q8 Nq2 4p RŽ . Ž . Ž .Ž . N ,c

2q2 D qO l , 18Ž . Ž .4where the constant R is the value of the RGN,c

invariant

m232R m ,l ' yC ln Nq2 lŽ .Ž .Ž .N 3 3 N 32 2Nq2 lŽ . 3

19Ž .Ž Ž .Ž .2 .at the critical point C 'y1r2 Nq2 4p . InN

Ž .2the large N limit we find R s1r 8p .N,c

In this letter we have computed the critical tem-perature of the four dimensional f 4 theory to the

next leading order in the small coupling constant l

using the effective three dimensional theory.

References

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phr9501375.w x4 K. Farakos, K. Kajantie, K. Rummukainen, M. Shaposhnikov,

Ž .Nucl. Phys. B 425 1994 67. hep-phr9404201.w x Ž .5 L. Dolan, R. Jackiw, Phys. Rev. D 9 1974 3320.w x Ž .6 S. Weinberg, Phys. Rev. D 9 1974 3357.w x Ž .7 P. Arnold, C. Zhai, Phys. Rev. D 50 1994 7603. hep-

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14 January 1999


Search for Leptoquarks and FCNC in eqey annihilations at(s s183 GeV

DELPHI Collaboration

P. Abreu u, W. Adam ax, T. Adye aj, P. Adzic k, T. Aldeweireld b, G.D. Alekseev p,R. Alemany aw, T. Allmendinger q, P.P. Allport v, S. Almehed x, U. Amaldi i,

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K-H. Becks az, M. Begalli f, P. Beilliere h, Yu. Belokopytov i,1, A.C. Benvenuti e,C. Berat n, M. Berggren y, D. Bertini y, D. Bertrand b, M. Besancon am, F. Bianchi as,

M. Bigi as, M.S. Bilenky p, M-A. Bizouard s, D. Bloch j, H.M. Blom ad,M. Bonesini aa, W. Bonivento aa, M. Boonekamp am, P.S.L. Booth v,

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a Department of Physics and Astronomy, Iowa State UniÕersity, Ames, IA 50011-3160, USAb Physics Department, UniÕ. Instelling Antwerpen, UniÕersiteitsplein 1, BE-2610 Wilrijk, Belgium, and IIHE, ULB-VUB, Pleinlaan 2,

BE-1050 Brussels, Belgium, and Faculte des Sciences, UniÕ. de l’Etat Mons, AÕ. Maistriau 19, BE-7000 Mons, Belgiumć Physics Laboratory, UniÕersity of Athens, Solonos Str. 104, GR-10680 Athens, Greeced Department of Physics, UniÕersity of Bergen, Allegaten 55, NO-5007 Bergen, Norway´

e Dipartimento di Fisica, UniÕersita di Bologna and INFN, Via Irnerio 46, IT-40126 Bologna, Italy`f Centro Brasileiro de Pesquisas Fısicas, rua XaÕier Sigaud 150, BR-22290 Rio de Janeiro, Brazil, and Depto. de Fısica, Pont. UniÕ.´ ´

Catolica, C.P. 38071 BR-22453 Rio de Janeiro, Brazil, and Inst. de Fısica, UniÕ. Estadual do Rio de Janeiro, rua Sao Francisco XaÕier´ ´ ˜524, Rio de Janeiro, Brazil

g Comenius UniÕersity, Faculty of Mathematics and Physics, Mlynska Dolina, SK-84215 BratislaÕa, SloÕakiah College de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, FR-75231 Paris Cedex 05, France`

i CERN, CH-1211 GeneÕa 23, Switzerlandj Institut de Recherches Subatomiques, IN2P3 - CNRSrULP - BP20, FR-67037 Strasbourg Cedex, France

k Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greecel FZU, Inst. of Phys. of the C.A.S. High Energy Physics DiÕision, Na SloÕance 2, CZ-180 40 Praha 8, Czech Republic

m Dipartimento di Fisica, UniÕersita di GenoÕa and INFN, Via Dodecaneso 33, IT-16146 GenoÕa, Italy`n Institut des Sciences Nucleaires, IN2P3-CNRS, UniÕersite de Grenoble 1, FR-38026 Grenoble Cedex, France´ ´

o Helsinki Institute of Physics, HIP, P.O. Box 9, FI-00014 Helsinki, Finlandp Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, RU-101 000 Moscow, Russian Federation

q Institut fur Experimentelle Kernphysik, UniÕersitat Karlsruhe, Postfach 6980, DE-76128 Karlsruhe, Germany¨ ¨r Institute of Nuclear Physics and UniÕersity of Mining and Metalurgy, Ul. Kawiory 26a, PL-30055 Krakow, Polands UniÕersite de Paris-Sud, Lab. de l’Accelerateur Lineaire, IN2P3-CNRS, Bat. 200, FR-91405 Orsay Cedex, France´ ´ ´ ´ ˆ

t School of Physics and Chemistry, UniÕersity of Lancaster, Lancaster LA1 4YB, UKu LIP, IST, FCUL - AÕ. Elias Garcia, 14-1o, PT-1000 Lisboa Codex, Portugal

v Department of Physics, UniÕersity of LiÕerpool, P.O. Box 147, LiÕerpool L69 3BX, UKw ( )LPNHE, IN2P3-CNRS, UniÕ. Paris VI et VII, Tour 33 RdC , 4 place Jussieu, FR-75252 Paris Cedex 05, France

x Department of Physics, UniÕersity of Lund, SolÕegatan 14, SE-223 63 Lund, Swedenÿ UniÕersite Claude Bernard de Lyon, IPNL, IN2P3-CNRS, FR-69622 Villeurbanne Cedex, France´

z UniÕ. d’Aix - Marseille II - CPP, IN2P3-CNRS, FR-13288 Marseille Cedex 09, Franceaa Dipartimento di Fisica, UniÕersita di Milano and INFN, Via Celoria 16, IT-20133 Milan, Italy`

ab Niels Bohr Institute, BlegdamsÕej 17, DK-2100 Copenhagen Ø, Denmarkac NC, Nuclear Centre of MFF, Charles UniÕersity, Areal MFF, V HolesoÕickach 2, CZ-180 00 Praha 8, Czech Republic

ad NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlandsae National Technical UniÕersity, Physics Department, Zografou Campus, GR-15773 Athens, Greece

af Physics Department, UniÕersity of Oslo, Blindern, NO-1000 Oslo 3, Norwayag Dpto. Fisica, UniÕ. OÕiedo, AÕda. CalÕo Sotelo srn, ES-33007 OÕiedo, Spain

ah Department of Physics, UniÕersity of Oxford, Keble Road, Oxford OX1 3RH, UKai Dipartimento di Fisica, UniÕersita di PadoÕa and INFN, Via Marzolo 8, IT-35131 Padua, Italy`

aj Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UKak Dipartimento di Fisica, UniÕersita di Roma II and INFN, Tor Vergata, IT-00173 Rome, Italy`

al Dipartimento di Fisica, UniÕersita di Roma III and INFN, Via della Vasca NaÕale 84, IT-00146 Rome, Italyàm DAPNIArSerÕice de Physique des Particules, CEA-Saclay, FR-91191 Gif-sur-YÕette Cedex, France

an ( )Instituto de Fisica de Cantabria CSIC-UC , AÕda. los Castros srn, ES-39006 Santander, Spainao Dipartimento di Fisica, UniÕersita degli Studi di Roma La Sapienza, Piazzale Aldo Moro 2, IT-00185 Rome, Italy`

ap ( )Inst. for High Energy Physics, SerpukoÕ P.O. Box 35, ProtÕino Moscow Region , Russian Federationaq J. Stefan Institute, JamoÕa 39, SI-1000 Ljubljana, SloÕenia and Department of Astroparticle Physics, School of EnÕironmental Sciences,

KostanjeÕiska 16a, NoÕa Gorica, SI-5000 SloÕenia, and Department of Physics, UniÕersity of Ljubljana, SI-1000 Ljubljana, SloÕeniaar Fysikum, Stockholm UniÕersity, Box 6730, SE-113 85 Stockholm, Sweden

as Dipartimento di Fisica Sperimentale, UniÕersita di Torino and INFN, Via P. Giuria 1, IT-10125 Turin, Italyàt Dipartimento di Fisica, UniÕersita di Trieste and INFN, Via A. Valerio 2, IT-34127 Trieste, Italy, and Istituto di Fisica, UniÕersita di` `

Udine, IT-33100 Udine, Italyau UniÕ. Federal do Rio de Janeiro, C.P. 68528 Cidade UniÕ., Ilha do Fundao BR-21945-970 Rio de Janeiro, Brazil˜

av Department of Radiation Sciences, UniÕersity of Uppsala, P.O. Box 535, SE-751 21 Uppsala, Swedenaw ( )IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, AÕda. Dr. Moliner 50, ES-46100 Burjassot Valencia , Spain

ax Ïnstitut fur Hochenergiephysik, Osterr. Akad. d. Wissensch., Nikolsdorfergasse 18, AT-1050 Vienna, Austriaäy Inst. Nuclear Studies and UniÕersity of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Poland

az Fachbereich Physik, UniÕersity of Wuppertal, Postfach 100 127, DE-42097 Wuppertal, Germany

Received 26 November 1998Editor: L. Montanet


Abstract

A search for events with one jet and at most one isolated lepton used data taken at LEP-2 by the DELPHI detector. Thesedata were accumulated at a center-of-mass energy of 183 GeV and correspond to an integrated luminosity of 47.7 pby1.Production of single scalar and vector leptoquarks was searched for. Limits at 95% confidence level were derived on the

Ž 2 2 .masses ranging from 134 GeVrc to 171 GeVrc for electromagnetic type couplings and couplings of the leptoquarkq yŽ .states. A search for top-charm flavour changing neutral currents e e ™ tc or charge conjugate used the semileptonic

Ž .decay channel. A limit on the flavour changing cross-section via neutral currents was set at 0.55pb 95% confidence level .q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

In eqey colliders such as LEP searches for newphysics can be made with high sensitivity in places

Ž .where the expected Standard Model SM contribu-tions are small. Events where all or most particlesare grouped in one direction in space, in a mono-jet

Žtype topology, with one isolated lepton charged or.neutral , are a good example of such processes. SM

extensions related to leptoquark models or single topproduction via Flavour Changing Neutral Currentscan have such a signature. In this paper we report ona topological search for events in these two channels.

Leptoquarks are coloured spin 0 or spin 1 parti-cles with both baryon and lepton quantum numbers.These particles are predicted by a variety of exten-sions of the SM, including Grand Unified Theoriesw x w x w x1 , Technicolor 2 and composite models 3 . Theyhave electric charges of "5r3, "4r3, "2r3 and"1r3, and decay into a charged or neutral leptonand a quark, L ™ l "q or L ™n q. Two hypothesesq q

are considered in this paper, one where only theŽcharged decay mode is possible charged branching

.ratio Bs1.0 , and one, for leptoquark charges be-low 4r3, where both charged and neutral decaymodes are equally probable. If the leptoquark does

Ž .not couple to the charged decay mode Bs0 thenthese leptoquarks can not be produced singly ineqey collisions. Leptoquarks may be producedsingly or in pairs at eqey colliders. For singleproduction, leptoquark mass limits can be set up toalmost the kinematical limit. For this reason only

1 On leave of absence from IHEP Serpukhov.2 Now at University of Florida.

single leptoquark production is considered in thisanalysis. The largest contribution to the productioncross-section at LEP is predicted to come from pro-cesses involving hadrons coming from resolved pho-

w xtons 4 , radiated from the incoming beams, whichare treated using the Weizacker-Williams approxima-tion. The corresponding Feynman diagram is shownin Fig. 1 a. Decays of singly produced high massleptoquarks to a charged lepton are characterised bya high transverse momentum jet recoiling against alepton. In the decay to a neutrino only the jet isdetected. The initial electron which scatters off thequasi real photon is assumed to escape detectiondown the beam pipe. Below the TeV mass range andfor couplings of the order of the electromagneticcoupling, the leptoquarks should not couple to di-quarks in order to prevent proton decay. They shouldalso couple chirally to either left or right handedquarks but not to both, and mainly diagonally. Thisimplies that they should couple to a single leptonicgeneration and to a single quark generation andhence this measurement searches only for decays toe and n .

The properties of leptoquarks are indirectly con-w xstrained by experiments at lower energy 5 , by

w xprecision measurements of the Z width 6 , and byw xdirect searches at higher energies 7–10 . The mass

of scalar leptoquarks decaying to electron plus jetwas constrained to be above 225 GeVrc2 using

w xTevatron data 7 . Limits on leptoquark masses andy w xcouplings were set at HERA using the e p data 8 ,

giving M )216–275 GeVrc2. An excess of eventsLq

was found in the eqp data. The H1 collaborationmeasured a jet-lepton invariant mass of these eventsranging from 187.5 GeVrc2 up to 212.5 GeVrc2.Rare processes, which are forbidden in the SM, also


Ž .Fig. 1. a The resolved photon contribution for single leptoquarkŽ . q yproduction and b single top production via FCNC in e e

collisions.

w xprovide strong bounds on the lrm ratio 11 ,Lq

where l is the leptoquark-fermion Yukawa typecoupling and m is the leptoquark mass.Lq

In the SM, Flavour Changing Neutral CurrentsŽ .FCNC are absent at tree level. Neutral currents

q y Ž .such as e e ™ tc tu can be present at the one loopw xlevel, but the rates are severely supressed 12 .

Flavour changing vertices are present in manyw xextensions of the SM like supersymmetry 13 ,

w xmulti-Higgs doublet models 14 and anomalous t-w xquark production 15 , which could enhance the pro-

duction of top quarks. For instance, in the SM thet™cZ branching ratio is around 10y13 while in thecontext of a two Higgs doublet model without natu-ral flavour conservation the rates can be higher by

w xmore than six orders of magnitude 14 , dependingon the chosen parameters. At tree level, single topproduction is possible via FCNC anomalous cou-

q yŽ . w xplings e e ™ tc 15 . The corresponding Feyn-Ž .man diagram is shown in Fig. 1 b . The t™cZ and

t™cg vertices are described by two anomalouscoupling constants k and k respectively. PresentZ g

w x Žconstraints from LEP–2 data were set 15 at m st2 .175 GeVrc :

k 2 -0.176 , k 2 -0.533g Z

Ž .In single top production at LEP, the tc tu , pairshould be produced almost at rest as the top mass isclose to the centre-of-mass energy. The top quarkdecays subsequently to a b quark and a W. Only

leptonic decays of the W are searched for in thisletter. It is an almost background free signaturecharacterised by one energetic mono-jet and oneisolated charged lepton.

2. The DELPHI detector and data samples

A detailed description of the DELPHI detector, itsperformance, the triggering conditions and the read-

w xout chain can be found in Ref. 16 . This analysisrelies on the charged particle detection provided bythe tracking system and energy reconstruction pro-vided by the electromagnetic and hadronic calorime-ters.

The main tracking detector of DELPHI is theTime Projection Chamber, which covers the angularrange 208-u-1608, where u is the polar angledefined with respect to the beam direction. Otherdetectors contributing to the track reconstruction are

Ž .the Vertex Detector VD , the Inner and Outer De-tectors and the Forward Chambers. The VD consistsof three cylindrical layers of silicon strip detectors,each layer covering the full azimuthal angle.

Electromagnetic shower reconstruction is per-formed in DELPHI using the barrel and the forwardelectromagnetic calorimeters, including the STICŽ .Small angle TIle Calorimeter , the DELPHI lumi-nosity monitor.

The energy resolutions of the barrel and forwardelectromagnetic calorimeters are parameterized re-

'Ž .spectively as s E rE s 0.043 [ 0.32r E and'Ž .s E rEs0.03[0.12r E [0.11rE, where E is

expressed in GeV and the symbol ‘[’ implies addi-tion in quadrature.

The hadron calorimeter covers both the barrel andforward regions. It has an energy resolution of

'Ž .s E rEs0.21[1.12r E in the barrel.The effects of experimental resolution, both on

the signals and on backgrounds, were studied bygenerating Monte Carlo events for the possible sig-nals and for the SM processes, and passing themthrough the full DELPHI simulation and reconstruc-tion chain.

The leptoquark signal was generated for differentw xmass values using the PYTHIA generator 17 . The

leptoquark production cross-section was taken fromw x18 .


Ž .The tc u signal was implemented in the PY-w x Ž .THIA generator 17 by producing a top and c u

quark pair and allowing the top quark to decay into ab quark and a W boson. A singlet colour string was

Ž .formed between the b and c u quarks.Bhabha events were simulated with the Berends,

w xHollik and Kleiss generator 19 . PYTHIA was usedto simulate eqey™tqty, eqey™Zg , eqey™

WqWy, eqey™W "e .n , eqey™ZZ, and eqey

™Zeqey events. In all four fermion channels, stud-w xies with the EXCALIBUR generator 20 were also

Ž .performed. The two-photon ‘‘gg ’’ physics eventsw xwere simulated using the TWOGAM 21 generator

for quark channels and the Berends, Daverveldt andw xKleiss generator 22 for the electron, muon and tau

channels.Data corresponding to an integrated luminosity of

47.7 pby1 were collected at a centre-of-mass energy's of 183 GeV.

3. Event selection

This analysis looks for events with one energeticmono-jet. Leptoquark decays to a charged lepton andtc decays also require an isolated charged lepton.

Ž .The recoil electron in Fig. 1 a is expected to passundetected down the beam pipe while the products

Ž . Ž .of the recoil X in Fig. 1 a and the c-quark in Fig.Ž .1 b are of low energy and are absorbed into the

mono-jet or lepton.Charged particles were considered only if they

had momentum greater than 0.1 GeVrc and impactparameters in the transverse plane and in the beamdirection below 4 cm and 10 cm respectively. Neu-tral clusters were defined as energy depositions inthe calorimeters unassociated with charged particle

Ž .tracks. All electromagnetic hadronic neutrals ofŽ .energy above 100 MeV 1 GeV were selected. In

the present analysis the minimum required chargedmultiplicity was six.

Charged particles were considered isolated if, in adouble cone centred on their track with internal andexternal half angles of 58 and 258, the total energyassociated to charged and neutral particles was be-low 1 GeV and 2 GeV respectively. The energy ofthe particle was redefined as the sum of the energiesof all the charged and neutral particles inside the

inner cone. This energy was required to be greaterthan 4 GeV. No other charged particle was allowedinside the inner cone.

Energy clusters in the electromagnetic calorime-ters were considered to be from photons if therewere no tracks pointing to the cluster, there were nohits inside a 28 cone in more than one layer of theVertex Detector and if at least 90% of any hadronicenergy was deposited in the first layer of the hadroncalorimeter. Photons were considered to be isolatedif, in a double cone centred on the cluster and havinginternal and external half angles of 58 and 158, thetotal energy deposited was less than 1 GeV. Theenergy of the photon was redefined as the sum of theenergies of all the particles inside the inner cone andno charged particles above 250 MeVrc were al-lowed inside this cone.

ŽAll charged and neutral particles excluding any.isolated charged lepton, if present were forced into

w xone jet using the Durham jet algorithm 23 . The jetwas classified as charged if it contained at least onecharged particle.

A detailed description of the basic selection crite-w xria can be found in Ref. 24 . Isolated charged parti-

cles were identified as electrons if there were noassociated hits in the muon chambers, if the ratio ofthe energy measured in the electromagneticcalorimeters, E, to the momentum measured in thetracking chambers, p, was larger than 0.2 and if theenergy deposited in the electromagnetic calorimetersby the lepton candidate was at least 90% of the totalenergy deposited in both electromagnetic andhadronic calorimeters.

The following criteria were applied to the eventsŽ .level 1 :Ø the total visible energy was required to be larger

'than 0.2 s ;Ø events with isolated photons were rejected;Ø the momentum of the monojet was required to be

larger than 10 GeVrc;Ø in channels with one isolated charged particle its

momentum had to be greater than 10 GeVrc; forthe leptoquark search exactly one isolated chargedparticle was required in the event; for the FCNCsearch at least one charged isolated particle wasrequired.After this selection, more specific criteria were

Ž .applied level 2 :


Ø Events were required to have only one jet withŽ . w xthe Durham resolution variable y 23 in thecut

transition from one to two jets smaller than 0.09.Ø The monojet polar angle had to be between 308

Ž . Ž .208 and 1508 1608 for the leptoquark searchŽ .for the FCNC search .

Ø The ratio between the monojet electromagneticenergy and its total energy had to be smaller than0.9. This removes most Bhabha events.

Ø The sum of the transverse momentum of theŽcharged particles in the jet relative to the event

.thrust axis normalized to the total visible mo-

Ž . Ž . Ž .Fig. 2. Leptoquark search: a the y variable distribution for neutral decays level 1 , b the y variable distribution for charged decayscut cutŽ . Ž .level 1 , c the ratio between the energy deposited in the electromagnetic calorimeters by the lepton candidate and the total energy

Ž . Ž . Ž . Ž .deposited in both electromagnetic and hadronic calorimeters level 2 , d the lepton polar angle level 2 and e the angle between the jetŽ .and the lepton level 2 . The dots show the data and the shaded region shows the SM simulation. The dark region is the expected signal

behaviour for a leptoquark mass of 120 GeVrc2. The vertical arrows show the cut used to select events. The accepted or rejected region isalso shown.


Table 1Number of selected data events and expected SM contributions forthe charged and neutral decay modes at different levels of selec-tion criteria

Leptoquark FCNC

charged decay neutral decay charged decayŽ . Ž . Ž .data SM data SM data SM

Ž . Ž . Ž .level 1 537 501"12 3159 2917"28 572 542"12Ž . Ž . Ž .level 2 76 64"4 4 2.6".7 101 96"5Ž . Ž . Ž .level 3 1 1.1".5 1 1.0".4 0 1.1".4

mentum had to be lower then 0.17. This cutreduces the contamination from semileptonic de-cays of WW pairs.In the case of the leptoquark neutral decays the

y criterion is the most effective for distinguishingcut

signal from background. This is illustrated in Fig. 2Ž .a where the dots show the data, the shaded regionthe SM simulation and the dark region the expectedsignal behaviour. The same distributions are shown

Ž .in Fig. 2 b for the leptoquark charged decays.Ž .Additional criteria level 3 were applied in order

to reduce the contamination from background events,mostly qq and WW. These criteria were different forthe different channels:- For the leptoquark charged decay mode it was

required that:Ž .i the lepton was identified as an electron and its

polar angle had to be between 308 and 1508;Ž .ii the angle between the electron and the mono-

jet had to be larger than 908.- For the leptoquark neutral decay mode, where the

contamination of qq is higher, all particles were

Ž . Ž . Ž . Ž . Ž .Fig. 3. FCNC search: a the lepton polar angle, b the jet-lepton angle, c the b-tag variable see text and d the missing momentumpolar angle. The dots show the data and the shaded region shows the SM simulation. The dark region is the expected signal behaviour. Thevertical arrows show the cut used to select events. The accepted or rejected region is also shown.


also forced into two jets, and the following addi-tional criteria were applied:Ž .i the angle between the two jets had to be

smaller than 1558;Ž .ii the momentum of the second jet had to be

smaller than 10 GeVrc, whenever the anglebetween the two jets was larger than 608.

- For the single top production:Ž .i the polar angle of the most energetic lepton

had to be between 208 and 1608, and theangle between the lepton and the monojethad to be between 158 and 1658;

Ž .ii events with a B hadron decay were selectedw xby requiring the b-tag variable 25 to be

below 0.06;Ž .iii the polar angle of the missing momentum

had to be between 208 and 1608.In Table 1 the number of events which survived

the different levels of selection is shown, togetherwith the expected SM background. The WW and qqevents are the main source of background. At level 3the expected background contribution from WW andqq events is: for the leptoquark neutral decay mode,0.12"0.12 and 0.46"0.33 respectively; for theleptoquark charged decay mode 0.12"0.12 and0.69"0.4 respectively; for the FCNC 0.49"0.25

Ž . Žand 0.23"0.23 respectively. Fig. 2 c shows at.level 2 , for the leptoquark search, the ratio between

the energy deposited in the electromagneticcalorimeters by the lepton candidate and the totalenergy deposited in both electromagnetic and

Ž .hadronic calorimeters, d the lepton polar angle andŽ .e the angle between the jet and the lepton. The dotsshow the data and the shaded region shows the SMsimulation. The dark region is the expected signalbehaviour for a 120GeVrc2 leptoquark mass. Noupper bound was imposed in the jet lepton angle to

Žallow good signal efficiency up to threshold where.the jet and the lepton are essentially back to back .

Ž .However the selection on Fig. 2 c removes almostŽ .all the SM background on Fig. 2 e .

Ž . Ž .Fig. 3 shows at level 2 , for the FCNC search, aŽ . Ž .the lepton polar angle, b the jet-lepton angle, c

w x Ž .the b-tag variable 25 and d the missing momen-tum polar angle. The dots show the data and theshaded region shows the SM simulation. The darkregion is the expected signal behaviour. A goodagreement is observed.

4. Results for leptoquarks

Only first-generation leptoquarks were searchedŽ " .for in this analysis L ™e q, L ™n q . As dis-q q e

cussed previously, the highest contribution to theproduction cross-section relevant for this searchcomes from the resolved photon contribution. The

w xGluck-Reya-Vogt parameterization 26 of the parton¨distribution was used. Since the photon has differentu-quark and d-quark contents and the production

Ž .2 Žcross-section is proportional to 1qq where q is.the leptoquark charge , leptoquarks of charge qs

Žy1r3 and qsy5r3 as well as leptoquarks of.charge qsy2r3 and qsy4r3 have similar pro-

w xduction cross-sections 18 . The cross-sections usedhere were calculated within the assumption above.

4.1. Charged decay mode

In this channel one event was found in the data at's s183 GeV and the expected SM background was1.1"0.5.

The leptoquark invariant mass estimated from theenergies and directions of the jet and lepton is 89.9GeVrc2. The mass resolution ranges from 15GeVrc2 to 25 GeVrc2 for leptoquark masses from100 GeVrc2 up to the kinematical limit.

Within the low statistics there is good agreementbetween data and SM predictions.

The efficiency was found to be between 22% and30% for leptoquark masses in the range from 100GeVrc2 up to the kinematic limit.

4.2. Neutral decay mode

In this channel one event was found and theexpected SM background was 1.0"0.4.

The leptoquark invariant mass estimated from themonojet transverse momentum is 72.1 GeVrc2. Themass resolution ranges from 20 GeVrc2 to 34GeVrc2 for leptoquark masses from 100 GeVrc2

up to the kinematical limit. Within the low statisticsthere is good agreement between data and SM pre-dictions.

The efficiency was found to be between 20% and41% for leptoquark masses in the range from 100GeVrc2 up to the kinematic limit.


4.3. Leptoquark mass and coupling limits

Limits were set on the leptoquark coupling pa-w xrameter l 4 . These limits, which depend on the

leptoquark mass, are shown in Fig. 4 for both scalarand vector leptoquarks of different types and forcharged decay branching ratios Bs1 and Bs0.5.For Bs1 the invariant mass plot for the chargeddecay mode was used to set the limits. For Bs0.5the invariant mass plots of the charged and theneutral decay modes were combined to set the limits.Different values of the charged decay branching ratioB, although theoretically not motivated, would implysimilar limits.

The lower limits at 95% confidence level on themass of a first generation leptoquark for a couplingparameter ls 4pa are given in Table 2, where( em

different leptoquark types and branching ratios arew xconsidered 27 . These limits are expected to change

at the level of some percent depending on the differ-

Fig. 4. 95% confidence level upper limits on the coupling l as aŽ . Ž .function of the leptoquark mass for a scalar and b vector

Žleptoquarks B is the branching ratio of the leptoquark to charged.leptons and q is the leptoquark charge .

Table 2Ž 2 .Lower limits in GeVrc at 95% confidence level on the the

mass of a first generation leptoquark for a coupling parameter ofls 4pa' em

Bs0.5 Bs1.0

qs1r3 qs2r3 qs1r3,5r3 qs2r3,4r3

scalar 161 – 161 134vector – 149 171 150

ent theoretical predictions for the total productionw xcross section 28 .

5. Results for top-charm FCNC

In the present analysis no events were foundwhile the expected SM background is 1.1"0.4. Thedetection efficiency, including the W leptonic

Ž .branching ratio, is 11.5"2.0 %.With the present luminosity of 47.7 pby1, an

q yupper limit on the e e ™ tc Flavour ChangingNeutral Current total cross-section can be set at 0.55

Ž .pb 95% confidence level .This value can be translated into a limit on the

anomalous coupling constants k and k , accordingg Zw xto the parametrization described in Ref. 15 . It was

q y q yassumed that both channels e e ™ tc and e e ™

tu contributed to the total cross-section. With aluminosity of 47.7 pby1 the 95% confidence levelupper limit on k is 2, for a k value of zero, andg Z

the corresponding upper limit on k is 1.5, for a kZ g

value of zero. The results are not yet competitivew xwith other experimental results 29 .

6. Conclusions

A search for first generation leptoquarks wasperformed using the data collected by the DELPHI

'detector at s s183 GeV. Both neutral and chargeddecay modes of scalar and vector leptoquarks weresearched for. No evidence for a signal was found inthe data. Limits on leptoquark masses were set rang-ing from 134 GeVrc2 to 171 GeVrc2 at 95% confi-dence level, assuming electromagnetic type cou-plings.


A search for tc flavour changing neutral currentswas also performed. No signal was found in the data.A limit on the FCNC cross-section was set at 0.55 pbŽ .95% confidence level .

Acknowledgements

We would like to thank M. Doncheski and C.Papadopoulos for the very useful discussions on theleptoquark production. We would also like to thankD. Atwood, L. Reina and A. Soni for the on-goingdiscussion relative to the two Higgs doublet model.

We are greatly indebted to our technical collabo-rators, to the members of the CERN-SL Division forthe excellent performance of the LEP collider, and tothe funding agencies for their support in building andoperating the DELPHI detector.

We acknowledge in particular the support ofØ Austrian Federal Ministry of Science and Traf-

fics, GZ 616.364r2-IIIr2ar98,Ø FNRS–FWO, Belgium,Ø FINEP, CNPq, CAPES, FUJB and FAPERJ,

Brazil,Ø Czech Ministry of Industry and Trade, GA CR

202r96r0450 and GA AVCR A1010521,Ø Danish Natural Research Council,

ŽØ Commission of the European Communities DG.XII ,

Ø Direction des Sciences de la Matiere, CEA,`France,

Ø Bundesministerium fur Bildung, Wissenschaft,¨Forschung und Technologie, Germany,

Ø General Secretariat for Research and Technology,Greece,

Ž .Ø National Science Foundation NWO and Founda-Ž .tion for Research on Matter FOM , The Nether-

lands,Ø Norwegian Research Council,Ø State Committee for Scientific Research, Poland,

2P03B06015, 2P03B03311 and SPUBrP03r178r98,

Ø JNICT–Junta Nacional de Investigacao Cientıfica˜ ´e Tecnologica, Portugal,´

Ø Vedecka grantova agentura MS SR, Slovakia, Nr.95r5195r134,

Ø Ministry of Science and Technology of the Re-public of Slovenia,

Ø CICYT, Spain, AEN96–1661 and AEN96-1681,Ø The Swedish Natural Science Research Council,Ø Particle Physics and Astronomy Research Coun-

cil, UK,Ø Department of Energy, USA, DE–FG02–

94ER40817.

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14 January 1999


Search for charginos, neutralinos and gravitinos in eqey

(interactions at s s183 GeV

DELPHI Collaboration

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J. Dolbeau h, K. Doroba ay, M. Dracos j, J. Drees az, M. Dris ae, A. Duperrin y,J-D. Durand y,i, G. Eigen d, T. Ekelof av, G. Ekspong ar, M. Ellert av, M. Elsing i,

J-P. Engel j, B. Erzen aq, M. Espirito Santo u, E. Falk x, G. Fanourakis k,



D. Fassouliotis k, J. Fayot w, M. Feindt q, P. Ferrari aa, A. Ferrer aw, E. Ferrer-Ribas s,S. Fichet w, A. Firestone a, P.-A. Fischer i, U. Flagmeyer az, H. Foeth i, E. Fokitis ae,F. Fontanelli m, B. Franek aj, A.G. Frodesen d, R. Fruhwirth ax, F. Fulda-Quenzer s,

J. Fuster aw, A. Galloni v, D. Gamba as, S. Gamblin s, M. Gandelman au, C. Garcia aw,J. Garcia an, C. Gaspar i, M. Gaspar au, U. Gasparini ai, Ph. Gavillet i, E.N. Gazis ae,

D. Gele j, L. Gerdyukov ap, N. Ghodbane y, I. Gil aw, F. Glege az, R. Gokieli ay,B. Golob aq, G. Gomez-Ceballos an, P. Goncalves u, I. Gonzalez Caballero an,G. Gopal aj, L. Gorn a,2, M. Gorski ay, Yu. Gouz ap, V. Gracco m, J. Grahl a,

E. Graziani al, C. Green v, H-J. Grimm q, P. Gris am, K. Grzelak ay, M. Gunther av,J. Guy aj, F. Hahn i, S. Hahn az, S. Haider i, A. Hallgren av, K. Hamacher az,

F.J. Harris ah, V. Hedberg x, S. Heising q, J.J. Hernandez aw, P. Herquet b, H. Herr i,T.L. Hessing ah, J.-M. Heuser az, E. Higon aw, S-O. Holmgren ar, P.J. Holt ah,

D. Holthuizen ad, S. Hoorelbeke b, M. Houlden v, J. Hrubec ax, K. Huet b,G.J. Hughes v, K. Hultqvist ar, J.N. Jackson v, R. Jacobsson i, P. Jalocha i, R. Janik g,

Ch. Jarlskog x, G. Jarlskog x, P. Jarry am, B. Jean-Marie s, E.K. Johansson ar,P. Jonsson y, C. Joram i, P. Juillot j, F. Kapusta w, K. Karafasoulis k, S. Katsanevas y,

E.C. Katsoufis ae, R. Keranen q, B.P. Kersevan aq, B.A. Khomenko p,N.N. Khovanski p, A. Kiiskinen o, B. King v, A. Kinvig v, N.J. Kjaer ad, O. Klapp az,

H. Klein i, P. Kluit ad, P. Kokkinias k, M. Koratzinos i, V. Kostioukhine ap,C. Kourkoumelis c, O. Kouznetsov p, M. Krammer ax, C. Kreuter i, E. Kriznic aq,

J. Krstic k, Z. Krumstein p, P. Kubinec g, W. Kucewicz r, J. Kurowska ay,K. Kurvinen o, J.W. Lamsa a, D.W. Lane a, P. Langefeld az, V. Lapin ap,

J-P. Laugier am, R. Lauhakangas o, G. Leder ax, F. Ledroit n, V. Lefebure b,L. Leinonen ar, A. Leisos k, R. Leitner ac, J. Lemonne b, G. Lenzen az, V. Lepeltier s,

T. Lesiak r, M. Lethuillier am, J. Libby ah, D. Liko i, A. Lipniacka ar, I. Lippi ai,B. Loerstad x, J.G. Loken ah, J.H. Lopes au, J.M. Lopez an, R. Lopez-Fernandez n,

D. Loukas k, P. Lutz am, L. Lyons ah, J.R. Mahon f, A. Maio u, A. Malek az,T.G.M. Malmgren ar, V. Malychev p, F. Mandl ax, J. Marco an, R. Marco an,

B. Marechal au, M. Margoni ai, J-C. Marin i, C. Mariotti i, A. Markou k,C. Martinez-Rivero s, F. Martinez-Vidal aw, S. Marti i Garcia i,

N. Mastroyiannopoulos k, F. Matorras an, C. Matteuzzi aa, G. Matthiae ak, J. Mazik ac,F. Mazzucato ai, M. Mazzucato ai, M. Mc Cubbin v, R. Mc Kay a, R. Mc Nulty v,

G. Mc Pherson v, C. Meroni aa, W.T. Meyer a, A. Miagkov ap, E. Migliore as,L. Mirabito y, W.A. Mitaroff ax, U. Mjoernmark x, T. Moa ar, R. Moeller ab,

K. Moenig i, M.R. Monge m, X. Moreau w, P. Morettini m, G. Morton ah,U. Mueller az, K. Muenich az, M. Mulders ad, C. Mulet-Marquis n, R. Muresan x,

W.J. Murray aj, B. Muryn n,r, G. Myatt ah, T. Myklebust af, F. Naraghi n,F.L. Navarria e, S. Navas aw, K. Nawrocki ay, P. Negri aa, S. Nemecek l, N. Neufeld i,


N. Neumeister ax, R. Nicolaidou n, B.S. Nielsen ab, M. Nikolenko j,p,V. Nomokonov o, A. Normand v, A. Nygren x, V. Obraztsov ap, A.G. Olshevski p,A. Onofre u, R. Orava o, G. Orazi j, K. Osterberg o, A. Ouraou am, M. Paganoni aa,S. Paiano e, R. Pain w, R. Paiva u, J. Palacios ah, H. Palka r, Th.D. Papadopoulou ae,K. Papageorgiou k, L. Pape i, C. Parkes ah, F. Parodi m, U. Parzefall v, A. Passeri al,

O. Passon az, M. Pegoraro ai, L. Peralta u, M. Pernicka ax, A. Perrotta e, C. Petridou at,A. Petrolini m, H.T. Phillips aj, F. Pierre am, M. Pimenta u, E. Piotto aa,

T. Podobnik aq, M.E. Pol f, G. Polok r, P. Poropat at, V. Pozdniakov p, P. Privitera ak,N. Pukhaeva p, A. Pullia aa, D. Radojicic ah, S. Ragazzi aa, H. Rahmani ae,

D. Rakoczy ax, P.N. Ratoff t, A.L. Read af, P. Rebecchi i, N.G. Redaelli aa,M. Regler ax, D. Reid i, R. Reinhardt az, P.B. Renton ah, L.K. Resvanis c,

F. Richard s, J. Ridky l, G. Rinaudo as, O. Rohne af, A. Romero as, P. Ronchese ai,E.I. Rosenberg a, P. Rosinsky g, P. Roudeau s, T. Rovelli e, Ch. Royon am,

V. Ruhlmann-Kleider am, A. Ruiz an, H. Saarikko o, Y. Sacquin am, A. Sadovsky p,G. Sajot n, J. Salt aw, D. Sampsonidis k, M. Sannino m, H. Schneider q,

Ph. Schwemling w, U. Schwickerath q, M.A.E. Schyns az, F. Scuri at, P. Seager t,Y. Sedykh p, A.M. Segar ah, R. Sekulin aj, R.C. Shellard f, A. Sheridan v,

M. Siebel az, L. Simard am, F. Simonetto ai, A.N. Sisakian p, T.B. Skaali af,G. Smadja y, O. Smirnova x, G.R. Smith aj, A. Sokolov ap, O. Solovianov ap,

A. Sopczak q, R. Sosnowski ay, T. Spassov u, E. Spiriti al, P. Sponholz az,S. Squarcia m, D. Stampfer ax, C. Stanescu al, S. Stanic aq, S. Stapnes af,

K. Stevenson ah, A. Stocchi s, J. Strauss ax, R. Strub j, B. Stugu d,M. Szczekowski ay, M. Szeptycka ay, T. Tabarelli aa, F. Tegenfeldt av,

F. Terranova aa, J. Thomas ah, A. Tilquin z, J. Timmermans ad, N. Tinti e,L.G. Tkatchev p, S. Todorova j, D.Z. Toet ad, B. Tome u, A. Tonazzo aa, L. Tortora al,

G. Transtromer x, D. Treille i, G. Tristram h, M. Trochimczuk ay, C. Troncon aa,A. Tsirou i, M-L. Turluer am, I.A. Tyapkin p, S. Tzamarias k, B. Ueberschaer az,

O. Ullaland i, V. Uvarov ap, G. Valenti e, E. Vallazza at, C. Vander Velde b,G.W. Van Apeldoorn ad, P. Van Dam ad, W.K. Van Doninck b, J. Van Eldik ad,

A. Van Lysebetten b, I. Van Vulpen ad, N. Vassilopoulos ah, G. Vegni aa,L. Ventura ai, W. Venus aj, F. Verbeure b, M. Verlato ai, L.S. Vertogradov p,V. Verzi ak, D. Vilanova am, L. Vitale at, E. Vlasov ap, A.S. Vodopyanov p,

C. Vollmer q, G. Voulgaris c, V. Vrba l, H. Wahlen az, C. Walck ar, C. Weiser q,D. Wicke az, J.H. Wickens b, G.R. Wilkinson i, M. Winter j, M. Witek r, G. Wolf i,

J. Yi a, O. Yushchenko ap, A. Zalewska r, P. Zalewski ay, D. Zavrtanik aq,E. Zevgolatakos k, N.I. Zimin p,x, G.C. Zucchelli ar, G. Zumerle ai

a Department of Physics and Astronomy, Iowa State UniÕersity, Ames, IA 50011-3160, USAb Physics Department, UniÕ. Instelling Antwerpen, UniÕersiteitsplein 1, BE-2610 Wilrijk, Belgium, and IIHE, ULB-VUB, Pleinlaan 2,

BE-1050 Brussels, Belgium, and Faculte des Sciences, UniÕ. de l’Etat Mons, AÕ. Maistriau 19, BE-7000 Mons, Belgium´


c Physics Laboratory, UniÕersity of Athens, Solonos Str. 104, GR-10680 Athens, Greeced Department of Physics, UniÕersity of Bergen, Allegaten 55, NO-5007 Bergen, Norway´

e Dipartimento di Fisica, UniÕersita di Bologna and INFN, Via Irnerio 46, IT-40126 Bologna, Italy`f Centro Brasileiro de Pesquisas Fısicas, rua XaÕier Sigaud 150, BR-22290 Rio de Janeiro, Brazil, and Depto. de Fısica, Pont. UniÕ.´ ´

Catolica, C.P. 38071 BR-22453 Rio de Janeiro, Brazil, and Inst. de Fısica, UniÕ. Estadual do Rio de Janeiro, rua Sao Francisco XaÕier´ ´ ˜524, Rio de Janeiro, Brazil

g Comenius UniÕersity, Faculty of Mathematics and Physics, Mlynska Dolina, SK-84215 BratislaÕa, SloÕakiah College de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, FR-75231 Paris Cedex 05, France`

i CERN, CH-1211 GeneÕa 23, Switzerlandj Institut de Recherches Subatomiques, IN2P3 - CNRSrULP - BP20, FR-67037 Strasbourg Cedex, France

k Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greecel FZU, Inst. of Phys. of the C.A.S. High Energy Physics DiÕision, Na SloÕance 2, CZ-180 40 Praha 8, Czech Republic

m Dipartimento di Fisica, UniÕersita di GenoÕa and INFN, Via Dodecaneso 33, IT-16146 GenoÕa, Italy`n Institut des Sciences Nucleaires, IN2P3-CNRS, UniÕersite de Grenoble 1, FR-38026 Grenoble Cedex, France´ ´

o Helsinki Institute of Physics, HIP, P.O. Box 9, FI-00014 Helsinki, Finlandp Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, RU-101 000 Moscow, Russian Federation

q Institut fur Experimentelle Kernphysik, UniÕersitat Karlsruhe, Postfach 6980, DE-76128 Karlsruhe, Germany¨ ¨r Institute of Nuclear Physics and UniÕersity of Mining and Metalurgy, Ul. Kawiory 26a, PL-30055 Krakow, Polands UniÕersite de Paris-Sud, Lab. de l’Accelerateur Lineaire, IN2P3-CNRS, Bat. 200, FR-91405 Orsay Cedex, France´ ´ ´ ´ ˆ

t School of Physics and Chemistry, UniÕersity of Lancaster, Lancaster LA1 4YB, UKu LIP, IST, FCUL - AÕ. Elias Garcia, 14-1o, PT-1000 Lisboa Codex, Portugal

v Department of Physics, UniÕersity of LiÕerpool, P.O. Box 147, LiÕerpool L69 3BX, UKw ( )LPNHE, IN2P3-CNRS, UniÕ. Paris VI et VII, Tour 33 RdC , 4 place Jussieu, FR-75252 Paris Cedex 05, France

x Department of Physics, UniÕersity of Lund, SolÕegatan 14, SE-223 63 Lund, Swedenÿ UniÕersite Claude Bernard de Lyon, IPNL, IN2P3-CNRS, FR-69622 Villeurbanne Cedex, France´

z UniÕ. d’Aix - Marseille II - CPP, IN2P3-CNRS, FR-13288 Marseille Cedex 09, Franceaa Dipartimento di Fisica, UniÕersita di Milano and INFN, Via Celoria 16, IT-20133 Milan, Italy`

ab Niels Bohr Institute, BlegdamsÕej 17, DK-2100 Copenhagen Ø, Denmarkac NC, Nuclear Centre of MFF, Charles UniÕersity, Areal MFF, V HolesoÕickach 2, CZ-180 00 Praha 8, Czech Republic

ad NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlandsae National Technical UniÕersity, Physics Department, Zografou Campus, GR-15773 Athens, Greece

af Physics Department, UniÕersity of Oslo, Blindern, NO-1000 Oslo 3, Norwayag Dpto. Fisica, UniÕ. OÕiedo, AÕda. CalÕo Sotelo srn, ES-33007 OÕiedo, Spain

ah Department of Physics, UniÕersity of Oxford, Keble Road, Oxford OX1 3RH, UKai Dipartimento di Fisica, UniÕersita di PadoÕa and INFN, Via Marzolo 8, IT-35131 Padua, Italy`

aj Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UKak Dipartimento di Fisica, UniÕersita di Roma II and INFN, Tor Vergata, IT-00173 Rome, Italy`

al Dipartimento di Fisica, UniÕersita di Roma III and INFN, Via della Vasca NaÕale 84, IT-00146 Rome, Italyàm DAPNIArSerÕice de Physique des Particules, CEA-Saclay, FR-91191 Gif-sur-YÕette Cedex, France

an ( )Instituto de Fisica de Cantabria CSIC-UC , AÕda. los Castros srn, ES-39006 Santander, Spainao Dipartimento di Fisica, UniÕersita degli Studi di Roma La Sapienza, Piazzale Aldo Moro 2, IT-00185 Rome, Italy`

ap ( )Inst. for High Energy Physics, SerpukoÕ P.O. Box 35, ProtÕino Moscow Region , Russian Federationaq J. Stefan Institute, JamoÕa 39, SI-1000 Ljubljana, SloÕenia and Department of Astroparticle Physics, School of EnÕironmental Sciences,

KostanjeÕiska 16a, NoÕa Gorica, SI-5000 SloÕenia, and Department of Physics, UniÕersity of Ljubljana, SI-1000 Ljubljana, SloÕeniaar Fysikum, Stockholm UniÕersity, Box 6730, SE-113 85 Stockholm, Sweden

as Dipartimento di Fisica Sperimentale, UniÕersita di Torino and INFN, Via P. Giuria 1, IT-10125 Turin, Italyàt Dipartimento di Fisica, UniÕersita di Trieste and INFN, Via A. Valerio 2, IT-34127 Trieste, Italy, and Istituto di Fisica, UniÕersita di` `

Udine, IT-33100 Udine, Italyau UniÕ. Federal do Rio de Janeiro, C.P. 68528 Cidade UniÕ., Ilha do Fundao, BR-21945-970 Rio de Janeiro, Brazil˜

av Department of Radiation Sciences, UniÕersity of Uppsala, P.O. Box 535, SE-751 21 Uppsala, Swedenaw ( )IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, AÕda. Dr. Moliner 50, ES-46100 Burjassot Valencia , Spain

ax Ïnstitut fur Hochenergiephysik, Osterr. Akad. d. Wissensch., Nikolsdorfergasse 18, AT-1050 Vienna, Austriaäy Inst. Nuclear Studies and UniÕersity of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Poland

az Fachbereich Physik, UniÕersity of Wuppertal, Postfach 100 127, DE-42097 Wuppertal, Germany



Abstract

An update of the searches for charginos and neutralinos is presented, based on a data sample corresponding to the 53.9pby1 recorded by the DELPHI detector in 1997, at a centre-of-mass energy of 183 GeV. No evidence for a signal wasfound. The lower mass limits are 4–5 GeVrc2 higher than those obtained at a centre-of-mass energy of 172 GeV. TheŽ .m,M domain excluded by combining the neutralino and chargino searches implies a limit on the mass of the lightest2

neutralino which, for a heavy sneutrino, is constrained to be above 29.1 GeVrc2 for tanbG 1. q 1999 Elsevier ScienceB.V. All rights reserved.

1. Introduction

In 1997, the LEP centre-of-mass energy reached183 GeV, and the DELPHI experiment collected anintegrated luminosity of 53.9 pby1. These data havebeen analysed to search for the supersymmetric part-ners of Higgs and gauge bosons, the charginos,neutralinos and gravitinos, predicted by supersym-

Ž . w xmetric SUSY models 1 .This paper presents an update of the results de-

w xscribed in 2 which contains a detailed description'of the analysis of s s161–172 GeV data. The

methods used to search for charginos and neutralinosw xpresented in 2 have remained almost unchanged

and only the differences from the previous analysisare described here. A description of the parts of theDELPHI detector relevant to the present paper can

w xbe found in 2 , while the complete descriptions arew xgiven in 3 .

It is assumed that R-parity is conserved, implyingŽ .a stable lightest supersymmetric particle LSP . As in

the previous paper the two cases where either the0 ˜Ž . Ž .lightest neutralino x or the gravitino G is the˜1

LSP are considered. In the former case, events arecharacterised by missing energy carried by the escap-ing neutralinos, while in the latter case the decay

0 ˜ w xx ™Gg is possible 4–6 . If the gravitino is suffi-˜1Ž 2ciently light with a mass below about 10 eVrc

w x.6 , this decay takes place within the detector. Asgravitinos escape detection, the typical signature ofthese SUSY events is missing energy and isolatedphotons. The mass difference DM plays an impor-

1 On leave of absence from IHEP Serpukhov.2 Now at University of Florida.

tant role in the analysis, as the missing transversemomentum and visible mass depend strongly on thisvariable. DM is defined as the difference M ySUSY

M 0 where M is the mass of the particle searchedx SUSY˜1

for, the chargino or the neutralino.The Minimal Supersymmetric Standard Model

Ž .MSSM scheme with universal parameters at thehigh mass scale typical of Grand Unified TheoriesŽ . w xGUTs is assumed 1 . The parameters of this modelrelevant to the present searches are the masses M1

Žand M of the gaugino sector which are assumed to25 2satisfy the GUT relation M s tan u M f0.5M1 W 2 23

.at the electroweak scale , the universal mass m of0

the scalar lepton sector, the Higgs mass parameter m,and the ratio of vacuum expectation values of thetwo Higgs doublets, tanb.

2. Data samples and event generators

The total integrated luminosity collected by DEL-PHI during 1997 at E s183 GeV was 53.9 pby1.cm

This luminosity was used in the chargino analysis fortopologies with a stable neutralino, while 50.6 pby1

was used in topologies with an unstable neutralinodue to a temporary problem in the read-out of the

Ž .barrel electromagnetic calorimeter HPC . The lumi-nosity used in the neutralino analysis was 47.3 pby1

due to different quality selection criteria.To evaluate the signal efficiencies and back-

ground contaminations, events were generated usingseveral different programs. All relied on JETSET

w x w x7.4 7 , tuned to LEP 1 data 8 , for quark fragmenta-tion.

w xThe program SUSYGEN 9 was used to generateneutralino and chargino signal events in both theneutralino and gravitino LSP scenarios, and to calcu-


late cross-sections and branching ratios. Details ofthese signal samples are given in Section 4.

q y Ž .The background process e e ™ qq ng wasw x w xgenerated with PYTHIA 5.7 7 , while DYMU3 10

w x q yŽ .and KORALZ 11 were used for m m g andq yŽ . w xt t g , respectively. The generator of Ref. 12

was used for eqey™ eqey events. Processes lead-Ž .) Ž .)ing to four-fermion final states, Zrg Zrg ,

WqWy, Wen and Zeqey, were also generatede

using PYTHIA. The calculation of the four-fermionbackground was verified using the program EXCAL-

w xIBUR 13 which consistently takes into account allamplitudes leading to a given four-fermion finalstate, but does not include the transverse momentumof initial state radiation when electrons are present inthe final state.

The VDM and QCD components of the two-pho-ton interactions leading to hadronic final states were

w xgenerated using TWOGAM 14 . The generators ofw xBerends, Daverveldt and Kleiss 15 were used for

the QPM component and for leptonic final states.The generated signal and background events were

passed through the detailed simulation of the DEL-w xPHI detector 3 and then processed with the same

reconstruction and analysis programs as the real data.The numbers of simulated events from differentbackground processes were several times the num-bers in the real data.

3. Event selections

The criteria used to select events were defined onthe basis of the simulated signal and backgroundevents. The selections for charged and neutral parti-

w xcles were the same as those presented in 2 . Onenew criterion was introduced in order to reject neu-tral clusters coming from the intrinsic radioactivityŽ .mainly a of the barrel electromagnetic calorimeter.Showers with less than 15 GeV of energy and withmore than 90% of the energy deposited in one singlecalorimeter layer have been rejected. Fig. 1 showsthe distribution of variables relevant for the selectionof chargino topologies with a stable neutralino forreal and simulated events. The agreement is satisfac-tory, the normalization is absolute.

w xAs in 2 , the particle selection was followed bydifferent event selections in the different topologies.

Some of the selection criteria were slightly changedin order to maintain a good signal-to-backgroundratio. In fact the average value of some variablesused to enhance the signal changed due to the in-

'creased s .One example is the missing transverse momentum

for two-photon interactions which increases on aver-age, while it doesn’t change for a degenerate charginoclose to the kinematic limit.

Other selection criteria were re-tuned due to theq y Žincreased W W production cross-section the cri-

.terion on the visible mass or due to a better under-standing of the signal. Some new analyses have beenadded in order to have a better efficiency for smallvalues of DM. The differences in the selection crite-

'ria between the present analysis and the one at s sw x161 and 172 GeV 2 are in general quite small, apart

from the addition of some new analyses, but aredetailed below for completeness.

3.1. Chargino analysis

The remnants of the chargino decays weresearched for in the following topologies: jets and one

Ž .or more isolated charged lepton jj ll , jets only,Ž .purely leptonic final states ll ll and anything plusŽ .one or more isolated photons gg X .

ŽFor the jj ll topology in the degenerate case DM2 .F10 GeVrc , the minimal missing transverse mo-

mentum was changed from 3 to 4 GeVrc and a newrequirement was introduced: the polar angles of bothjet axes were required to be in the range 248-u-

1568.For the jets topology in the non-degenerate case

Ž 2 .DM)10 GeVrc , the maximum allowed energyin the forward and backward 208 cones was changedfrom 30% of the visible energy to 50%, the maxi-mum visible mass was changed to 65 GeVrc2, and anew requirement of visible mass above 15 GeVrc2

was added. In the degenerate case, the jet polarangles were required to be in the same range as inthe jj ll topology.

For the ll ll topology the different multiplicity-de-pendent criteria on the missing transverse momen-tum were changed to a single minimum value of 5.5GeVrc for the non-degenerate case and 4 GeVrcfor the degenerate case.


w xFig. 1. Distributions of total multiplicity and scaled acoplanarity after hadronic topology selection 2 . Distributions of missing transversew xmomentum and percentage of electromagnetic energy after the semileptonic topology selection 2 . Points show distributions for real data

events, histograms for simulated events. The normalization of the histograms is absolute.

The selection criteria used in the gravitino LSPscenario are also used in the radiative topologyŽ .gg X for the neutralino LSP scenario, when the

" 0 0cascade decay x ™x ff™x g ff is possible for˜ ˜ ˜i 2 1

low values of tanb and M ;ym. The requirement2

that the total visible mass should exceed 20 GeVrc2

was added in the non-degenerate case. In the degen-Ž 2erate case re-defined to be 5 GeVrc -DMF


Table 1The numbers of events observed in data and the expected numbers of background events in the different chargino search channels under the

Ž .hypothesis of a stable neutralino Section 3.1

Ž .Chargino channels stable neutralino

Non-degenerate case Degenerate case

topology: jj ll jets ll ll gg X jj ll jets ll llobs. events: 0 11 7 3 4 3 1background: 1.0 " 0.8 7.6 " 0.9 8.8 " 1.0 4.9 " 0.8 3.3 " 0.9 5.1 " 0.9 1.9 " 0.8

total:obs. events: 21 8background: 22.3 " 1.8 10.3 " 1.1

2 . w x10 GeVrc , the minimum scaled acoplanarity 2was changed from 58 to 108 and the momentum ofthe most energetic charged particle was required notto exceed 15 GeVrc, rather than 30 GeVrc. A new

w xselection for the ultra-degenerate case 16 was in-troduced to increase the signal efficiency for DMF5 GeVrc2. In this case, the energy of the mostenergetic isolated photon had to lie between 20 GeVand 60 GeV, and the momentum of the most ener-getic charged particle was required to be smallerthan 10 GeVrc. The visible mass of the event,excluding the isolated photons, had to be smallerthan 40 GeVrc2, and the scaled acoplanarity greaterthan 58. Finally, the same criteria as in the degener-ate case were applied to the missing transverse mo-mentum, to the percentage of energy in the forwardand backward 258 cones and to the total electromag-netic energy, excluding the energy of the most ener-getic isolated photon.

3.2. Neutralino analysis

Neutralino final states were searched for in purelyhadronic and purely leptonic topologies.

A new criterion on the charged multiplicity wasadded in the selection of the jj topology: at least fourwell reconstructed charged particles, including onewith a transverse momentum exceeding 1 GeVrc,were required. Other new selection criteria wereadded: the sum of the absolute values of the mo-menta of well reconstructed charged particles had tobe greater than 4 GeVrc; the transverse energy ofthe event was required to be greater than 4 GeV; nocharged particle was allowed to have a momentum

greater than 20 GeVrc; and finally, the most iso-lated identified electron or muon with an isolationangle to the nearest jet greater than 208was requiredto have momentum less than 10 GeVrc. The maxi-mum allowed fraction of the total energy of particlesemitted within 308 of the beam was changed from40% to 60%. The last step in the selection consistedof the logical OR of three sets of cuts, optimised fordifferent regions of DM:

Ž 2 .Ø For low DM ;10 GeVrc , minimum require-Ž .ments on the scaled acoplanarity 408 and on the

Ž .missing transverse momentum 7.0 GeVrc wereadded.

Ž 2 .Ø For intermediate DM ;40 GeVrc , the selec-tion criteria on the multiplicity, on the total en-ergy in the forward and backward 308 cones, andon the polar angle of the missing momentumwere removed. The invariant mass range of visi-

2'ble particles was required to be between 0.1 s rc2'and 0.3 s rc , the minimum missing mass value

2 2' 'was changed from 0.5 s rc to 0.6 s rc .Ž 2 .Ø For large DM ;90 GeVrc all selection crite-

ria are new: the invariant mass of visible particleswas required to be greater than 0.3 and less than

Table 2The numbers of events observed in data and the expected numbersof background events in the different neutralino search channelsŽ .Section 3.2

Neutralino channels

topology: jj ll llobs. events: 6 5background: 8.2 " 1.2 6.9 " 0.6


2'0.5 s rc , and the missing mass had to exceed2'0.45 s rc . The scaled acoplanarity had to ex-

ceed 258 and the missing transverse momentumhad to exceed 8 GeVrc and be less than 35GeVrc. It was also required that the componentof the missing momentum in the beam directionshould be less than 35 GeVrc.The analysis which selected neutralino decays

Ž .giving di-leptons ll ll was re-optimised to copewith the increased WqWy background and its dif-ferent kinematic characteristics. This analysis se-lected events with exactly two isolated, oppositelycharged particles reconstructed with momenta above1 GeVrc where the isolation criteria requires thatthere should be no more than 2 GeV of chargedenergy in a 108 cone around the track. The totalmultiplicity of the event should not exceed five. Itwas also required that both selected tracks shouldhave hits in at least four pad rows in the Time

Projection Chamber and with at least one associatedhit in the vertex detector. Finally, either both parti-

Žcles had to be identified as electrons or muons loosew x.tag 17 , or one of them had to satisfy stricter

Ž .electron identification criteria standard tag . Toreject eqey™ eqey and Zg events, the acoplanaritybetween the two selected particles was required toexceed 108. Events produced in two-photon interac-tions were rejected by demanding that the direction

< <of the missing momentum satisfy cosu -0.9,pmiss

and that its transverse component be greater than 5GeVrc . It was also required that the energy in the308 cone around the beam be less than 70% of thevisible energy, that no neutral cluster with energyabove 15 GeV and no more than 1 GeV of energywas detected in the very forward and backward

Ž .electromagnetic calorimeter STIC . To reduce thenumber of events from leptonic decays ofWqWypairs, events were rejected if one particle

Ž . . . . .Fig. 2. Chargino pair production detection efficiencies % for the 4 topologies a jj ll , b jets, c ll ll and d gg X, at 183 GeV in theŽ ." 0M , M plane. A stable neutralino is assumed.x x˜ ˜


Ž .was identified loose tag as an electron and theother as a muon. In a similar manner as in the jjtopology the different regions of DM were selected:

Ž 2 .Ø Low DM ;10 GeVrc : the invariant mass ofvisible particles was required to be less than

2'0.1 s rc , the missing mass had to exceed2'0.7 s rc , the acoplanarity had to exceed 408 and

the missing transverse momentum had to exceed8.0 GeVrc.

Ž 2 .Ø Intermediate DM ;40 GeVrc : the invariantmass of visible particles was required to be be-

2 2' 'tween 0.1 s rc and 0.3 s rc , and the missing2'mass had to exceed 0.45 s rc . The acoplanarity

had to exceed 258 and the missing transversemomentum had to exceed 10 GeVrc.

Ž 2 .Ø Large DM ;90 GeVrc : the invariant mass ofvisible particles was required to be between

2 2' '0.3 s rc and 0.55 s rc , and the missing mass2'had to exceed 0.4 s rc . The acoplanarity had to

exceed 158 and the missing transverse momentumhad to exceed 12 GeVrc.

4. Results in case of a stable neutralino

4.1. Efficiencies and selected eÕents

The total number of background events expectedin the different topologies is shown in Tables 1 and2, together with the number of events selected in thedata.

The efficiencies of the chargino selections in Sec-tion 3.1 were estimated using 37 combinations ofx " and x 0 masses in four chargino mass ranges˜ ˜1 1Ž 2 ."M f 91, 85, 70 and 50 GeVrc and with DMx1

ranging from 1 GeVrc2 to 70 GeVrc2. A total of74000 chargino events was generated and passedthrough the complete simulation of the DELPHIdetector. A special study was carried out in the

q y q y Ž .0 0Fig. 3. Neutralino pair production detection efficiency for the jj, e e and m m topologies at 183 GeV in the M , M plane. Thex x˜ ˜2 1

shaded areas are regions where the efficiency is lower than 5%.


< < w xregion of low m and M as described in 2 . The2

efficiencies for the four different topologies areshown in Fig. 2. For the gg X topology, the effi-ciency depends mainly on DMsM "yM 0 and isx x˜ ˜ 2

practically independent of the photon energy.For the neutralino analysis, a total of 130000

x 0 x 0 events was generated for 42 different combi-˜ ˜1 2

nations of M 0 and M 0 masses with M 0 rangingx x x˜ ˜ ˜2 1 1

from 10 GeVrc2 to 85 GeVrc2, with M 0 yM 0x x˜ ˜2 1

ranging from 5 GeVrc2 to the kinematic limit, and0 0 q y 0Žfor different x decay modes qq x , m m x ,˜ ˜ ˜2 1 1

q y 0.e e x . The efficiencies for the neutralino selec-˜1

tions in Section 3.2 are shown in Fig. 3.The numbers of selected events are compatible

with the expectation from the background simulationin all the channels considered. As no evidence for asignal is found, exclusion limits are set.

4.2. Limits

4.2.1. Limits on neutralino productionLimits on neutralino production in the case of a

stable x 0 were derived from the parametrized effi-˜1

ciencies of Section 3.2 and the observed numbers ofw xevents, as described in 2 . The limits obtained for

the x 0 x 0 production cross-section are shown in˜ ˜1 2

Figs. 4a,b and c assuming 100% branching ratio forthe hadronic or leptonic decay modes. The limit

0 0obtained assuming that the x ™ x ff decay is˜ ˜2 1

mediated by a Z) , including both leptonic andhadronic modes and 20% of invisible final states, isof about 1 pb as seen in Fig. 4d. These limits also

0 0 Ž . 0apply to x x ks3,4 production with x decay-˜ ˜ ˜1 k k

ing into x 0.˜1

0 0 'Fig. 4. Contour plots of upper limits on the cross-sections at the 95% confidence level for x x production at s s 183 GeV. In each˜ ˜1 2. .plot, the different shades correspond to regions where the cross-section limit in picobarns is below the indicated number. For figures a , b ,

0 0 0 q y 0 q y 0 0. .c , x decays into x qq, x e e , and x m m , respectively, were assumed to dominate. In d , the x was assumed to decay into x ff˜ ˜ ˜ ˜ ˜ ˜2 1 1 1 2 1

with the same branching ratios into different fermion flavours as the Z. The dotted lines indicate the kinematic limit and the defining relationw x0 0M )M , and the dashed lines indicate the kinematic limit of the search at 172 GeV 2 .x x˜ ˜2 1


Ž .Fig. 5. Expected cross-sections in pb at 183 GeV dots versus the. Žchargino mass in a in the non-degenerate case DM) 10

2 . . Ž 2 .GeVrc and in b in the degenerate case DM F 5 GeVrc .The spread in the dots originates from the random scan over the

Ž 2 .parameters m and M . A heavy sneutrino m ) 300 GeVrc2 n

. 2 .has been assumed in a and m )41 GeVrc in b . The 95%n

C.L. limits on the cross-sections corresponding to the mass limitsare indicated by the horizontal lines.

4.2.2. Limits on chargino productionThe simulated data points were used to

parametrize the efficiencies of the chargino selectioncriteria described in Section 3.1 in terms of DM andthe mass of the chargino. Then a large number ofSUSY points were investigated and the values ofDM, the chargino and neutralino masses and thevarious decay branching ratios were determined foreach point. By applying the appropriate efficiencyand branching ratios for each channel the number ofexpected signal events for a given cross-section canbe calculated. Taking also the expected backgroundand the number of events actually observed intoaccount, certain cross-sections, or the corresponding

Ž .points in the MSSM parameter space m, M , tanb ,2

may be excluded.Fig. 5 shows the chargino production cross-sec-

'tions obtained in the MSSM at s s183 GeV fordifferent chargino masses for the non-degenerate anddegenerate cases. The parameters M and m were2

varied randomly in the ranges 0 GeVrc2 -M -2

3000 GeVrc2 and y400 GeVrc2 -m- 400GeVrc2 for three different values of tanb , namely1, 1.5 and 35. Two different cases were considered

2 Žfor the sneutrino mass: M )300 GeVrc in then

. 2 Žnon-degenerate case and M )41 GeVrc in then

.degenerate case .Applying the appropriate efficiencies and branch-

ing ratios for each of the points shown, the minimumnon-excluded M " was determined. In the deriva-x1

tion of the chargino mass limits, constraints on theprocess Z™x 0 x 0 ™x 0 x 0g were also included.˜ ˜ ˜ ˜1 2 1 1

These were derived from the DELPHI results onw xsingle-photon production at LEP 1 18 . The chargino

mass limits are summarized in Table 3. The tablealso gives, for each case, the minimum excludedMSSM cross-section corresponding to the limit onthe chargino mass. These cross-section values arealso displayed in Fig. 5.

In the non-degenerate case with a large sneutrinoŽ 2 .mass ) 300 GeVrc , the lower limit for the

2 Žchargino ranges between 89.4 GeVrc for a mostly. 2 Žhiggsino-like chargino and 90.8 GeVrc for a

.mostly wino-like chargino . The minimum excluded'MSSM cross-section at s s183 GeV is 0.82 pb,

corresponding to the limit on the chargino mass.

Table 395% confidence level limits for the chargino mass, the corre-sponding pair production cross-sections at 183 GeV and the 95%confidence level upper limit on the number of observed events, forthe non-degenerate case and for a highly degenerate case. The

0 0 ˜scenarios of a stable x and x ™ Gg are considered˜ ˜1 1

excl excl"Case m M s Nn x 95%˜

2 2Ž . Ž . Ž .GeVrc GeVrc pb

stable neutralino2DM)10 GeVrc ) 300 89.4 0.82 10.14

2DMs5 GeVrc ) 41 88.8 0.95 6.39

unstable neutralino2DM)10 GeVrc ) 300 90.5 0.49 4.69

2DMs1 GeVrc ) 41 90.6 0.45 4.36


Ž 2 .In the degenerate case DMs5 GeVrc , thecross-section does not depend significantly on thesneutrino mass, since the chargino is higgsino-likeunder the assumption of gaugino mass unification.The lower limit for the chargino mass, shown in Fig.5, is 88.8 GeVrc2. The minimum excluded MSSMcross-section is in this case 0.95 pb.

4.2.3. Limits on MSSM parameters and neutralinomass

Using the limits on chargino production and onLEP 1 single-photon production the excluded regionin the plane of neutralino mass versus chargino masswas determined, as shown in Fig. 6. A heavy sneu-trino was assumed. The small region outside thechargino kinematic limit derives from the single-pho-ton search.

Ž .The exclusion regions in the m, M plane are2

shown in Fig. 7 for different values of tanb , assum-Žing a heavy sneutrino and a heavy selectron m s0

2 .1 TeVrc . When different event selections con-tributed to the same physical production channel theefficiency and background of a logical OR of the

w xchannels was used, otherwise the method of Ref. 19was used to combine the selections. These limits,

'based on data taken at s s183 GeV, improve onprevious limits at lower energies, and represent asignificant increase in range as compared to LEP 1

w xresults 20 . The neutralino analysis independentlyexcludes a substantial part of the region covered bythe chargino search, and marginally extends thisregion at low tanb for large M and negative m.2

Under the assumption that M rM R0.5, the ex-1 2Ž .clusion regions in the m, M plane can be trans-2

lated into the limit on the mass of the lightest

Fig. 6. Region excluded at 95% confidence level in the plane of the mass of the lightest neutralino versus that of the lightest chargino underthe assumption of a heavy sneutrino, for tanbs1.0, 1.5 and 35. The thin lines show the kinematic limits in the production and the decay.

Ž .The dotted line sometimes hidden under the real exclusion limit shows the expected exclusion limit. The lightly shaded region is notallowed in the MSSM. The limit applies in the case of a stable neutralino. The mass limit on the lightest neutralino is indicated by thehorizontal dashed line. The excluded region outside the kinematic limit is obtained from the limit on x 0 x 0 production at the Z resonance˜ ˜1 2

derived from the single-photon search.


Ž . 2Fig. 7. Regions in the m, M plane excluded at 95% confidence level for different values of tanb , assuming m s1 TeVrc . The lightly2 0w xshaded areas are those excluded by lower energy LEP 1 results 20 . The intermediate shading shows regions excluded by the chargino

search at 183 GeV. The dark shaded areas show the regions excluded by the neutralino search at these energies. With the exception of anarrow strip at negative m for tanb ; 1 the regions excluded by the neutralino results are also excluded by the chargino search.

neutralino shown in Fig. 6. A lower limit of 29.1GeVrc2 on the lighest neutralino mass is obtained,valid for tanb G 1, using the obtained charginoexclusion regions and including the DELPHI resultsw x 0 0 0 018 on the process Z™x x ™x x g . The lower˜ ˜ ˜ ˜1 2 1 1

mass limit is obtained for tanbs1, msy62.3GeVrc2, M s46.0 GeVrc2.2

5. Results in case of an unstable neutralino

5.1. Efficiencies and selected eÕents

The efficiency of the chargino selection for anunstable neutralino decaying into a photon and a

gravitino was calculated from a total of 78000 eventsgenerated using the same combinations of M " andx1

M 0 as in the stable neutralino scenario. As men-x1

w xtioned in 2 , the same selection applies to all topolo-gies. The efficiency, shown in Fig. 8, varies onlyweakly with DM and is around 50%. Note that, dueto the presence of the photons from the neutralino

Ždecay, the region of high degeneracy down to DM2 .s1 GeVrc is fully covered.

The total number of background events expectedin the different DM ranges is shown in Table 4,together with the number of events selected in thedata. 4 events were found in the data, with a totalexpected background of 6.3"0.9. The 3 events


Ž . Ž ." 0Fig. 8. Chargino pair production detection efficiency % at 183 GeV in the M , M plane. An unstable neutralino is assumed.x x˜ ˜

selected in the non-degenerate case are the same asthe ones selected in the chargino gg X topology incase of a stable neutralino. No signal is found, andexclusion limits have been set.

5.2. Limits

The chargino cross-section limits correspondingto the case where the neutralino is unstable and

˜decays to Gg have been computed as explained inSection 4.2 and are shown in Fig. 5 and in Table 3.In the non-degenerate case the chargino mass limit at95% confidence level is 90.5 GeVrc2 for a heavy

Žsneutrino, while in the ultra-degenerate case DMs12 . 2GeVrc the limit is 90.6 GeVrc . The minimum

MSSM cross-section excluded by the above masslimits are 0.49 pb in the non-degenerate case and0.45 pb in the ultra-degenerate case.

Table 4The number of events observed and the expected number of background events in the different DM cases under the hypothesis of an

Ž .unstable neutralino Section 3.1

Ž .Chargino channels unstable neutralino

non-degenerate selection degenerate selection ultra-degenerate selection

Obs. events: 3 0 1Background: 4.9 " 0.8 0.9 " 0.3 0.5 " 0.2


6. Summary

'Searches for charginos and neutralinos at s s183 GeV allow the exclusion of a large domain of

Ž .SUSY parameters at the 95% confidence level .Assuming a difference in mass between chargino

and neutralino, DM, of 10 GeVrc2 or more, and asneutrino heavier than 300 GeVrc2, the existence ofa chargino lighter than 89.4 GeVrc2 is excluded. Ifa gaugino-dominated chargino is assumed in addi-tion, the mass is above 90.8 GeVrc2 . If DM isbetween 5 GeVrc2 and 10 GeVrc2, the lower limiton the chargino mass becomes 88.8 GeVrc2, inde-pendent of the sneutrino mass.

Limits on the cross-section for x 0 x 0 production˜ ˜1 2

of about 1 pb are obtained, and the excluded regionŽ .in the m, M plane is extended by the combined2

use of the neutralino and chargino searches. A spe-< <cial study of the low m , M , tanb region gives a2

limit on the mass of the lightest neutralino, valid inthe case of large m , of 29.1 GeVrc2.0

The search for xqxy , assuming the lightest neu-˜ ˜1 1

tralino decayed into photon and gravitino, gavesomewhat more stringent limits on cross-sectionsand masses than in the case of a stable x 0: 90.5˜1

GeVrc2 for large DM and 90.6 GeVrc2 for DMs1GeVrc2.

Acknowledgements

We are greatly indebted to our technical collabo-rators, to the members of the CERN-SL Division forthe excellent performance of the LEP collider, and tothe funding agencies for their support in building andoperating the DELPHI detector.

We acknowledge in particular the support ofØ Austrian Federal Ministry of Science and Traf-

fics, GZ 616.364r2-IIIr2ar98,Ø FNRS–FWO, Belgium,Ø FINEP, CNPq, CAPES, FUJB and FAPERJ,

Brazil,Ø Czech Ministry of Industry and Trade, GA CR

202r96r0450 and GA AVCR A1010521,Ø Danish Natural Research Council,

ŽØ Commission of the European Communities DG.XII ,

Ø Direction des Sciences de la Matiere, CEA,`France,

Ø Bundesministerium fur Bildung, Wissenschaft,¨Forschung und Technologie, Germany,

Ø General Secretariat for Research and Technology,Greece,

Ž .Ø National Science Foundation NWO and Founda-Ž .tion for Research on Matter FOM , The Nether-

lands,Ø Norwegian Research Council,Ø State Committee for Scientific Research, Poland,

2P03B06015, 2P03B03311 and SPUBrP03r178r98,

Ø JNICT–Junta Nacional de Investigacao Cientıfica˜ ´e Tecnologica, Portugal,´

Ø Vedecka grantova agentura MS SR, Slovakia, Nr.95r5195r134,

Ø Ministry of Science and Technology of the Re-public of Slovenia,

Ø CICYT, Spain, AEN96–1661 and AEN96-1681,Ø The Swedish Natural Science Research Council,Ø Particle Physics and Astronomy Research Coun-

cil, UK,Ø Department of Energy, USA, DE–FG02–

94ER40817.

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21 January 1999


Collective description of nuclear double beta decay transitions

D.R. Bes a,c, O. Civitarese b, N.N. Scoccola a

a ( )Departamento de Fısica, CNEA, AÕ. Libertador 8250, 1429 Buenos Aires, Argentina´b ( )Departamento de Fısica, UNLP, C.C. 67, 1900 La Plata, Argentina´

c ( )UniÕersidad FaÕaloro, Solıs 453, 1078 Buenos Aires, Argentina´

Received 11 September 1998; revised 20 November 1998Editor: W. Haxton

Abstract

A consistent treatment of the intrinsic and collective coordinates relevant for the calculation of matrix elementsdescribing nuclear double beta decay transitions is introduced. The method, which was originally developed for the case ofnuclear rotations, is adapted to include isospin and number of particles degrees of freedom. To illustrate its main features weapply the formalism to the case of Fermi transitions in a simplified model. From the corresponding results we conclude thatthe uncertainties found in many existing double beta decay calculations might be largely due to the mixing of physical andspurious effects in the treatment of isospin dependent interactions. q 1999 Published by Elsivier Science B.V. All rightsreserved.

PACS: 23.40.Bw; 23.40.Hc; 21.60.n

Keywords: Nuclear structure; Double beta decay transitions; Collective coordinates

One can hardly overestimate the importance ofthe double beta decay as a process explicitly linkingthe physics of neutrinos with the nuclear structurew x1–3 . Nuclear double beta decays are described assecond order processes which involve the elec-troweak decay of two nucleons. These transitions areallowed by the Standard Model if they proceed viathe emission of two electron-antineutrino pairsŽ .2nbb and totally forbidden if they proceed throughlepton number violating decays, as the neutrinoless

Ž .mode 0nbb which is a unique test of the proper-w x Ž .ties of the neutrino 2,3 . The 2nbb transitionsw xhave been observed 3 . Their correct theoretical

description is a necessary step towards the under-standing of the neutrinoless mode.

Ž .Earlier calculations of 2nbb were performedw xwithin limited shell model spaces 1 . Since the

possible double beta decays emitters are heavy nu-clei, full scale shell model calculations are unfeasi-ble, and one has to resort to mean field treatmentssuch as the BCSqRPA. Within such approach, itwas shown that the inclusion of pairing-type proton-neutron interactions resulted in the suppression of

Ž . w xthe 2nbb matrix elements 4,5 . Although thissupression was also obtained within several other

w xapproaches 6 and it was confirmed by the few

0370-2693r99r$ - see front matter q 1999 Published by Elsivier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01520-2

( )D.R. Bes et al.rPhysics Letters B 446 1999 93–9894

w xavailable shell model calculations 7 , the reliabilityof the theoretical predictions has been hampered byunstabilities in the BCS q RPA treatments. Analternative approach based on group theoreticalmethods has confirmed the existence of a zero-en-ergy state for certain values of the strength of theproton-neutron, particle-particle, effective interactionw x8,9 . The appearance of such a state has been inter-

w xpreted as a signature of a phase transition 10 .Here we take a point of view based on the fact

that the zero-energy state is a consequence of thebreakdown of the isospin symmetry implicit in theŽ . Ž . Ž .separate neutron n and proton p BCS solutionsw x11 . As similar to the case of deformed nuclei, thesymmetry may be restored in the laboratory framethrough the introduction of collective coordinates.There is, however, a complication due to the fact thatin many existing double beta decay calculations thestrength of the proton-neutron interactions is taken asan adjustable parameter. As a consequence of thisthe resulting effective nuclear hamiltonian does not,in general, conserve isospin. In this paper we do notattempt to discuss the derivation of this effective

Žinteraction from first principles Coulomb effects, np.mass differences, etc. . However, given such interac-

tion, a central many-body problem that must besolved is to disentangle unphysical isospin violations

Žintroduced by the formalism i.e. BCS approximation.for separate protons and neutrons from those isospin

violations produced by isotensor components of theeffective nuclear hamiltonian.

The basic aim of this letter is to introduce aformalism which solves this problem through anexact, albeit perturbative, way. Such formalism isbased on the treatment of collective coordinates andcan be applied to a general realistic nuclear hamilto-nian. In order to simplify the presentation, however,we will apply it here to the case of particles movingin a single j-shell and coupled through a chargedependent monopole pairing force. This exactly solu-

w xble model has been used 9 for the description ofŽ .2nbb transitions of the Fermi-type and, moreover,it already involves all the complications associatedwith the collective treatment.

The corresponding hamiltonian is

1q qHs e t yg S S y g S S 1Ž .Ž .Ý Õ Õ Õ Õ Õ H H H2Õ

where Õsp,n. Use is made of the operators

q q q q q q q qS s c c ; S s c c qc c ,Ž .Ý ÝÕ Õm Õm H pm nm nm pmm)0 m)0

1 1 w xt s t qt ; t s t yt ; t ,t st ,Ž . Ž .A p n 0 p n 1 1 02 2

11T s T qT ; T s T yT ;Ž . Ž .A p n 0 p n22

T ,T syT1 1 0

Ž .where t are the number operators and t 1sy1Õ "1

the rising and lowering isospin operators in thespherical representation. The T ,T are the corre-Õ "1

sponding generators of collective rotations in gauge-and isospace. The parameter g plays the role ofHthe renormalization factor g introduced in thep p

w xliterature 4,5 .The introduction of collective degrees of freedom

is compensated through the appearance of the con-straints

t yT s0 ; zsn , p ,"1 , 2Ž . Ž .z z

which express the fact that we can rotate the intrinsicsystem in one direction or the body in the opposite

w xone without altering the physical situation 12 . Phys-ical states should be annihilated by the four con-straints and physical operators should commute withthem.

The collective Hilbert space appropriate for anisospin conserving pairing interaction was originally

w xintroduced in Refs. 11,13,14 . The states may be< :labeled by the four quantum numbers T ,T ,m,k ,A

where T is the total number of pairs of particles.A1 1Ž . ŽThe quantum numbers m' TqM and k' T2 2

.qT determine M and T , the isospin projections0 0

in the laboratory and intrinsic frame, respectively.We focus on states such that m<T and ks0.Hereof we drop the label k from the collectivestates.

Unphysical violations of the isospin symmetry areallowed in the intrinsic frame. Such frame may bedefined, for instance, by the condition S s0, whereH

w xthe bar denotes the g.s. expectation value 14 . Thiscondition is precisely satisfied by performing theusual separate Bogoliubov transformation for protonsand neutrons. The rotations in isospace and gaugespace restore the symmetries which are present in thelaboratory frame. Thus the np-pairing becomes ef-

( )D.R. Bes et al.rPhysics Letters B 446 1999 93–98 95

fectively incorporated, as well as the pairing betweenidentical particles.

However, as different from previous cases wherecollective coordinates have been used, we are deal-ing here with an interaction which, in general, does

Ž .not conserve isospin. Namely, the hamiltonian 1 isnot generally an isoscalar. As in most collectivetreatments, physical isotensor operators must betransformed from the laboratory frame to the intrin-sic frame. In the case of the single-particle andpairing hamiltonian this procedure yields

Ž lab. 1 1 1H se t qe D t qD t qD tŽ .sp A A 0 00 0 01 1 01 1

1Ž lab. q q qH syg S S qS S q S SŽ .pair ,0 0 p p n n H H2

Ž lab. 1 q qH syg D S S yS SŽ .pair ,1 1 00 p p n n

D101 q qy S S qS SŽ .p H H n'2

1D01 q qq S S qS SŽ .n H H p'2

H Ž lab. syg D2 SqS qSqS ySq SŽ .½pair ,2 2 00 p p n n H H

3 2 q qy D S S yS S( Ž .01 H n p H2

2 q qqD S S yS SŽ .01 H p n H

2 q 2 q'q 6 D S S qD S S 3Ž .5ž /02 p n 02 n p

Here the subindices 0,1,2 on the l.h.s. denoteisoscalar, isovector and isoquadrupole components,ande se qe , e se ye ,A p n 0 p n

g qg qg g ygp n H p ng s , g s ,0 13 2

g qg y2 gp n Hg s .2 6

It is easy to verify that the four components of theŽ . Ž .hamiltonian 3 commute with the constraints 2 and

are therefore physical operators.Ž .Up to now the hamiltonian 3 together with the

Ž .constraints 2 constitute an exact reformulation ofthe original problem, since the introduction of addi-tional collective coordinates is compensated by the

presence of the constraints. Systems of this type canbe treated in a perturbative way within an expansiongiven by the inverse order parameter 1rS , for in-Õ

w xstance through the BRST procedure 15 , as appliedw xto many-body problems in 12 , and to the particular

w xcase of high angular momentum in 16 . There is,however, a new feature in the present case, namelythe presence of the rotational matrices Dl in themn

hamiltonian. This extra complication can be over-come by means of Marshalek’s generalization of the

w xHolstein-Primakoff representation 17 , which isamenable to an expansion in powers of Ty1. In whatfollows we will keep only the lowest order terms in

Ž . Ž .such an expansion, assuming O S sT and O gÕ ny1 Ž .sT . Such terms include the two pp and nn

pairing hamiltonians in a single j-shell e t yÕ Õ

g SqS , which are separately treated within the BCSÕ Õ Õ

approximation. In doing so, Lagrange multiplierŽ .terms yl t yT have to be added. This treatmentÕ Õ Õ

yields the independent quasi-particle energies EÕ1s V g , where V is half the value of the shellÕ2

degeneracy.The spectrum of the system is ordered into collec-

tive bands, each one carrying as quantum numbersŽ .the total number of particles and the isospin TFT .A

The properties of these bands are obtained by addingŽ .the remaining leading order terms in 3 to the

independent np quasi-particle energy terms. To lead-ing order in Ty1

Ž lab. qH sHqv d dqH ,Žspqpair . d 2

2Hs e t yg S ,Ž .Ý Õ Õ Õ Õ

Õ

g 3g1 22 2 2 2v se q S yS q S qS ,ž / ž /d 0 p n p nT T

² < < :T ,Ty2,my2 H T ,T ,mA 2 A

3g2 (sy S S m my1 , 4Ž . Ž .p nT

Ž .plus null terms proportional to 2 . The boson cre-ation operator d† increases in one unit the value of

w xm 17 .The energy of the band head is given by the BCS

expectation value H. The different members of eachband are labeled by the quantum number m and areseparated by the distance v , which includes thed


difference between the proton and the neutron sin-gle-particle energies e . Our strategy has been to0

restore the number of particles T and the isospin TA

as good quantum numbers and, within such a basis,to construct the interband interaction H , which2

allows for the possibility of double-beta decay. Insuch a way we have been able to disentangle thephysical isospin violations from the unphysical ones.

Both within the simple model or in the realisticcase, the t mode disappears 1 from the final"1

Ž . Žphysical hamiltonian 4 to become part of theŽ .. Žconstraints 2 . This is precisely the unrenormaliz-

.able phonon that yields a zero frequency root forw xisoscalar hamiltonians within a naive RPA 9 . From

Ž .the practical point of view it is as if this unphysicalRPA boson becomes substituted by the collectiveboson dq,d, which is well behaved in the limit ofzero frequency. In realistic cases this structure is alsomaintained, but superimposed to the excitations of

Ž .the other physical RPA modes. This substitutionalso becomes apparent in the expression for thestrong current that appears in the weak hamiltonian,which is proportional to the isospin operator, namely

Ž lab. 1 1 1' 'b sy 2 t sy 2 D t qD t qDŽ .y 1 11 1 10 0 11

q'f 2T d qnull operator. 5Ž .From the point of view of the expansion in powersof Ty1, the interband interaction H is of the same2

Ž Ž ..order O 1 as the distance between the states thatare mixed by it. Nevertheless, in the following wecontinue applying perturbation theory by requiring

< <that g -g .2 Õ

Let us proceed now with the discussion of somecalculations. We assume g sg sg. The excitationp n

9energy v is displayed in Fig. 1 for the cases js ,d 2

T s5, Ts3, e s0.8 MeV, gs0.4 MeV and jA 019s , T s10, Ts4, e s0.63 MeV, gs0.2 MeV,A 02

Ž .as function of the ratio g rg upper boxes . We2

predict the exact results for g s0 and very satisfac-2

tory ones for the other values, in spite of the fact thatfor these results we have neglected the interbandinteraction. The matrix element of double beta decay

1 To leading order, the isospin operators have a boson structurew xsince t ,t fT and t annihilates the state with t syT. In1 1 1 0

higher orders of the expansion in powers of Ty1 this modew xappears explicitly 12 .

Fig. 1. Excitation energy and transition matrix elements. ExactŽ . Ž .solid lines and collective dotted lines results for the excitation

Ž .energy upper boxes and transition matrix elements M and M1 2Ž .lower boxes corresponding to the two different sets of parame-

Ž .ters js9r2 and js19r2 discussed in the text.

transitions, which for the present case correspond toŽ w x.pure Fermi transitions cf. 9 , is proportional to the

product of the two matrix elements

'² < < :M s T ,T ,1 b T ,T ,0 f 2T1 A y A

² < < :M s T ,Ty2,0 b T ,T ,12 A y A

' ² < < :2 T T ,Ty2,0 H T ,T ,2A 2 Afy . 6Ž .

H T ,T ,2 yH T ,Ty2,0Ž . Ž .A A

These matrix elements are displayed in the lowerboxes of Fig. 1 for the same parameters as in theupper boxes. The expression for the interband matrix

Ž .element in 4 does not distinguish whether the r.h.s.should be calculated for the initial or the final valueof T , since it is valid for T41. Therefore, theeffective interband matrix element has been chosenas the geometric average of the values obtained foreach of the two connected bands.

( )D.R. Bes et al.rPhysics Letters B 446 1999 93–98 97

Fig. 2 displays Fermi double beta decay matrixelements, corresponding to transitions from the ini-tial to the final ground states. It has been calculatedusing the expression

M M1 2M s 7Ž .2 Õ

v qDd

where the energy released D is taken to be 0.5 MeV,w xas in 9 . In addition to the exact and collective

values of these matrix elements, we have included inthis figure the results obtained by using some otherapproximations. As expected from the fact that Fermitransitions only connect states with the same value ofisospin, the exact result shows the suppression of thematrix element around the point where the strengthof the np symmetry breaking interaction approachesthe value of the fully symmetric interaction. Thisresult is reproduced both in the naive QRPA and inthe collective approach. The other approximation

Fig. 2. Matrix elements for Fermi double beta decay transitionscalculated in several different approximations. The meaning of theQRPA and RQRPA approximations is explained in the text.

badly misses this cancellation. A detailed compari-son between the results of exact, naive QRPA and

Ž .renormalized QRPA RQRPA calculations can bew xfound in 9 . It is worth to note that in the collective

Žapproach the corresponding sum rule Ikeda’s sum.rule is exactly observed. This is not the case of

other approaches, as the RQRPA. The collectiveapproach, as seen in Figs. 1 and 2, not only repro-duces exact results very satisfactorily but it alsogives some insight about the mechanism responsiblefor the suppression of the matrix elements. As foundin the calculations, the value of the matrix elementM depends critically on the strength of the physical2

symmetry breaking term H . On the other hand, the2

values of M are not very much dependent on this1

interaction. Finally, it should be observed that thepoint where the excitation energy vanishes and thepoint where the symmetry is completely restored are

Ž .different cf. Fig. 1 . This result, also obtained in theexact diagonalization of the full hamiltonian, cannot

w xbe reproduced by other means as shown in 9 .In conclusion, it is found that a correct treatment

of collective effects induced by isospin dependentresidual interactions in a superfluid system is feasi-ble: physical effects due to the isospin symmetry-breaking terms in the hamiltonian are obtained evenin the presence of the BCS mean field built uponseparate proton and neutron pairing interactions. Theinterplay of intrinsic and collective coordinates guar-antees that the isospin symmetry is restored and thatspurious contributions to the wave functions aredecoupled from physical ones. Particularly, the prob-lem of the unstabilities found in the standard npQRPA are avoided by the explicit elimination of thezero frequency mode from the physical spectrumŽ .but keeping it in the perturbative expansion . Theappearance of this mode cannot be avoided by theinclusion of higher order terms in the QRPA expan-sion or by any other ad-hoc renormalization proce-dure, like the RQRPA, once the BCS procedure isadopted for the separate treatment of pp- and nn-

w xpairing correlations 9 .From the point of view of the expansion in pow-

ers of Ty1, the results shown in this letter areencouraging, in spite of the fact that we have notused very large values of T. Further details will bepresented in a longer publication, in which the case

Ž .T<O S will also be treated. We will also reportÕ


there on the extension of the formalism to includeany number of non-degenerate j-shells as well as theeffects of the Ss1,Ts0 pairing interaction on theGamow-Teller transitions. In spite of these complica-tions, the main features of the formalism remainessentially the same, albeit the expressions becomemore cumbersome to handle.

Acknowledgements

We would like to thank E. Maqueda and M.Saraceno for fruitful discussions and comments. Theauthors are fellows of the CONICET, Argentina.This work was partially by supported by a grant ofthe ANPCYT, Argentina. DRB and NNS also ac-knowledge grants from the Fundacion Antorchas.´

References

w x1 W.C. Haxton, G.J. Stephenson Jr., Prog. Part. Nucl. Phys. 12Ž .1984 409.

w x2 F. Bohm, P. Vogel, Physics of Massive Neutrinos, Cam-bridge Univ. Press, 2nd edition, Cambridge, 1992.

w x Ž .3 M. Moe, P. Vogel, Ann. Rev. of Nucl. Part. Sci. 44 1994247.

w x Ž .4 P. Vogel, M.R. Zirnbauer, Phys. Rev. Lett. 57 1986 3143.w x5 O. Civitarese, A. Faessler, T. Tomoda, Phys. Lett. B 194

Ž .1987 11.w x Ž .6 J. Suhonen, O. Civitarese, Phys. Rep. 300 1998 123.w x7 E. Caurier, F. Nowacki, A. Poves, J. Retamosa, Phys. Rev.

Ž .Lett. 77 1996 1954.w x8 J. Engel, S. Pittel, M. Stoitsov, P. Vogel, J. Dukelsky, Phys.

Ž .Rev. C 55 1997 1781.w x Ž .9 J. Hirsch, P.O. Hess, O. Civitarese, Phys. Rev. C 56 1997

199.w x Ž .10 O. Civitarese, P.O. Hess, J. Hirsch, Phys. Lett. B 412 1997

1.w x11 A. Bohr, B.R. Mottelson. Nuclear Structure, vol II, Ben-

jamin, Massachusetts, 1975.w x12 D.R. Bes, J. Kurchan, The treatment of collective coordinates

in many-body systems, World Scientific, Singapore, 1990.w x Ž .13 J. Ginocchio, J. Weneser, Phys. Rev. 170 1968 859.w x Ž .14 G.G. Dussel et al., Nucl. Phys. A 175 1971 513; G.G.

Ž .Dussel, R.P.J. Perazzo, D.R. Bes, Nucl. Phys. A 183 1972Ž .298; D.R. Bes et al., Nucl. Phys. A 217 1973 93.

w x Ž .15 C. Becchi, A. Rouet, S. Stora, Phys. Lett. B 52 1974 344.w x16 J. Kurchan, D.R. Bes, S. Cruz Barrios, Nucl. Phys. A 509

Ž .1990 306; J.P. Garrahan, D.R. Bes, Nucl. Phys. A 573Ž .1994 448.

w x Ž .17 E.R. Marshalek, Phys. Rev. C 11 1975 1426.

21 January 1999


A quantum Monte Carlo method for nucleon systems

K.E. Schmidt a, S. Fantoni b,c

a Department of Physics and Astronomy, Arizona State UniÕersity, Tempe, AZ, USAb International School for AdÕanced Studies, SISSA, Via Beirut 2r4, I-34014 Trieste, Italy

c International Centre for Theoretical Physics, ICTP, Strada Costiera 11, I-34014 Trieste, Italy

Received 7 November 1998Editor: J.-P. Blaizot

Abstract

We describe a quantum Monte Carlo method for Hamiltonians which include tensor and other spin interactions such asthose that are commonly encountered in nuclear structure calculations. The main ingredients are a Hubbard-Stratonovichtransformation to uncouple the spin degrees of freedom along with a fixed node approximation to maintain stability. Weapply the method to neutron matter interacting with a central, spin-exchange, and tensor forces. The addition of isospindegrees of freedom is straightforward. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Much of nuclear physics can be described by amodel of mesons and nucleons. For many problemsin nuclear structure the mesons are integrated out toproduce a nuclear Hamiltonian which has differentinteractions in different angular momentum channels.Simplified interactions that have most of the mainfeatures of the nucleon-nucleon interaction are given

w xby potential models such as the Urbana Õ 1 or14w xArgonne Õ 2 potentials. These potentials have the18

form

MŽ p.Vs Õ r O i , j 1Ž . Ž . Ž .Ý Ý p i j

i-j ps1

where i and j label the two nucleons, r is thei j

distance separating the two nucleons, and the OŽ p.

include spin, isospin, and spin orbit operators, and MŽis the maximum number of operators i.e. 14 in Õ14

.models .

The inclusion of the spin and isospin operatorsmakes standard quantum Monte Carlo methods muchmore difficult. The number of spin-isospin states fora nucleus with Z protons and A nucleons is

A!A2 2Ž .

Z! AyZ !Ž .

which quickly becomes intractable as A gets large.Green’s function Monte Carlo methods that includeall the spin-isospin states have been carried out only

Ž . w xfor light nuclei AQ8 3 .Nuclear quantum Monte Carlo not only has a

continuum of spatial positions, but also has a‘‘lattice’’ of spin configurations with nonlocal prop-agators. A similar situation occurs in quantum MonteCarlo calculations with explicit lattice Hamiltonians.Recently, constrained path quantum Monte Carlomethods have given good results for these lattice

w xsystems 4,5 . Our method is to apply these same


( )K.E. Schmidt, S. FantonirPhysics Letters B 446 1999 99–103100

constrained path techniques for the spin lattice toboth deal with the large number of spin degrees offreedom and to stabilize the algorithm so that thewell known fermion sign problem can be controlledw x6 . The spatial part of the Hamiltonian is done withstandard Green’s function or diffusion Monte Carlow x7,8 . For the case where the interaction is spin andisospin independent, our method reduces to the stan-

w xdard fixed node method 8,9 .

2. The method for the z potential6

While we believe that our method can be appliedat least approximately to spin orbit interactions, wehave not verified this in detail, and in this paper wewill talk specifically about spin-isospin dependentinteractions that are present in Õ models. Since6

these models include the full spin-isospin problem ofthe more detailed models, this truncation is not afundamental limitation, and the Õ models contain6

most of the physics of the nucleon-nucleon interac-Ž .tion. These potentials are given by Eq. 1 with the

nŽ .operators, O i, j ; 1 is the central operator, and thespin isospin operators t Pt , s Ps , s Ps t Pt ,i j i j i j i j

t , and t tPt , where t is the tensor operatori j i j j i j

3s Pr s Pr ys Ps . The t and s are theˆ ˆi i j j i j i j i i

Pauli matrices for the isospin and spin of particle i.For N particles, the Õ interaction can be written6

as

6Ž p.Vs Õ r O i , jŽ . Ž .Ý Ý p i j

i-j ps1

sV qVc nc

1Žs .sV q s A sÝc i ,a i ,a , j ,b j ,b2 i ,a , j ,b

1Žst .q s A s t PtÝ i ,a i ,a , j ,b j ,b i j2 i ,a , j ,b

1Žt .q A t Pt 3Ž .Ý i , j i j2 i , j

where V is the sum of the central interactionsc

V s Õ r , 4Ž . Ž .Ýc 1 i ji-j

and the noncentral potential V contains the operatornc

dependence. The roman indices, i, j, etc. indicateparticle number and the greek indices the cartesiancomponent of the operators. In our example calcula-tions we deal only with neutron matter. In this case,the t Pt isospin operator evaluates to 1, the ÕŽ2.i j

term is included in V and only the As term isc

needed to give the noncentral interactions.Ž .We define the A matrices in Eq. 3 so that they

are zero when is j and take them to be symmetric.While many other choices are possible, with thischoice, all the A matrices are real and symmetricand have real eigenvalues and eigenvectors. Wedefine these eigenvectors and eigenvalues as

AŽs . c s j Px slŽs .c s i Px 5Ž . Ž . Ž .ˆ ˆÝ i ,a , j ,b n b n n a

j,b

and similarly for the st and t cases.The matrices can be written in terms of their

eigenvectors and eigenvalues to give the noncentralpotential

1Žs . Žs . Žs .V s s Pc i l c j PsŽ . Ž .Ýnc i n n n j2 i , j ,n

1Žst . Žst . Žst .q s Pc i l c j Ps t PtŽ . Ž .Ý i n n n j i j2 i , j ,n

1Žt . Žt . Žt .q t Pt c i l c j 6Ž . Ž . Ž .Ý i j n n n2 i , j ,n

These can be rewritten as3N1 2Žs . Žs .V s O lŽ .Ýnc n n2 ns1

3 3N1 2Žst . Žst .q O lŽ .Ý Ý na n2as1 ns1

3 N1 2Žt . Žt .q O l 7Ž .Ž .Ý Ý na n2as1 ns1

with

OŽs .s s Pc Žt . i ,Ž .Ýn i ni

OŽst .s t s Pc Žst . i ,Ž .Ýna i a i ni

OŽt .s t c Žt . i 8Ž . Ž .Ýna i a ni

( )K.E. Schmidt, S. FantonirPhysics Letters B 446 1999 99–103 101

We can now use the Hubbard-Stratonovich methodto write the exponential of the potential multipliedby the time step that is needed in Green’s function or

w xdiffusion Monte Carlo 10 . The Hubbard-Stratono-vich transformation is given by the Gaussian integra-tion,

11 1< < `Dt l2 22n < <y l O y D t l x yD t sl O xD tn n n n ne s dxe2 2Hž /2p y`

9Ž .

where s is 1 for l-0, and s is i for l)0.Our O don’t commute, so we need to keep then

time steps small so that the commutator terms can beignored. Each of the O is a sum of 1-body opera-n

tors, so we can easily operate them on our spinstates.

A discrete version of this transformation is givenw xby Koonin et al. 11 . It is essentially a three point

Simpson’s rule integration of the gaussian centeredon the origin with the step size selected to optimizethe order. Modified for our notation, it is

1 `2y l O t yD t sl O x 3n n n ne D s dxf x e qOrder Dt ,Ž . Ž .2 Hy`

10Ž .

where

1f x s d xyh q4d x qd xqh ,Ž . Ž . Ž . Ž .6

13 2

hs . 11Ž .ž /< <l Dtn

We require 3N Hubbard-Stratonovich variablesfor the s terms, 9N variables for the st terms, and3N variables for the t terms. Each time step re-quires the diagonalization of two 3N by 3N matricesand one N by N matrix. Many other breakups arepossible. We could, for example, break up each pairpotential exactly as above, we would not need thematrix diagonalizations, but we would require moreHubbard-Stratonovich variables. We have not yetinvestigated any of these other algorithms. Inclusionof approximate importance sampling in theHubbard-Stratonovich variables is straightforward,but we have not included importance sampling in theHubbard-Stratonovich variables here.

To apply the Monte Carlo method, we use walk-ers that are given by the 3N spatial coordinates ofthe particles, and N 2-component complex spinorswhich give the spin state of each particle.

We apply the imaginary time propagator

exp y HyE Dt 12Ž . Ž .Ž .Trial

to our walker and sample the final configuration.This is done by dividing the propagator into a centralpart and a noncentral part. For small Dt, the commu-tator terms can be ignored, and the central propaga-tor can be sampled to give new positions of theparticles and a weight factor for the configuration. Inaddition, we write the noncentral part using theHubbard-Stratonovich method. Currently, we samplethe Hubbard-Stratonovich variables using the Kooninet al. approximation with probability 2r3 a variableis zero, and with probability 1r6 it is "h, where h

Ž .is defined for each variable by Eq. 11 . Importancesampling could be applied to the sampling of thesevariables to improve the variance.

We use the simplest possible trial function. It is aproduct of central two-body correlations and a Slaterdeterminant of space and spin orbitals. We evaluatethe wave function at the new space and spin posi-tions, and apply the constrained path method bysetting the weight of a walker to zero if the real partof the trial wave function is negative at the newspace-spin configuration.

3. Neutron matter calculations

As a check on our method we calculated theground state of 2 neutrons in a periodic box and in aspin singlet. Our trial function is also a spin singlet.We evaluate the potential using the nearest imageconvention. In this case, the tensor interaction givesno contribution and s Ps evaluates to y3. Ouri j

fixed node wave function should give the correctsinglet node. A direct diagonalization or a Green’sfunction Monte Carlo calculation with a purely cen-tral potential with the interaction given by droppingthe tensor terms and setting s Ps to y3 shouldi j

therefore give the same result as our full operatorresult. For a box of volume 10 fm3 using the Õ6

( )K.E. Schmidt, S. FantonirPhysics Letters B 446 1999 99–103102

components of the Argonne Õ potential, we find18

the correct answer to be y29.3 MeV, while orHubbard Stratonovich method where all the opera-tors are included gives y29.2"0.2 MeV.

We have applied this method to neutron matter.As mentioned above, t Pt s1 in this case, and wei j

can combine these isospin terms with the corre-sponding term without isospin. Only the AŽs . isneeded to describe the potential. We truncate theUrbana Õ potential by setting the terms Õ through14 7

Õ to zero. We have done calculations on 38 and 5414

neutrons at several densities. Our trial function con-tains a Slater determinant of orbitals of the form

sin k PrŽ .m iJ i 13Ž . Ž .s½ 5cos k PrŽ .m i

where the space orbital is either a sine or cosine, thek are the lowest magnitude k vectors that fit in them

Ž .simulation cube, and J i is either an up or downs

spinor for the particle. The determinant of orbitals ismultiplied by a product of central two-body factors

f r . 14Ž . Ž .Ł i ji-j

Ž .We take f r as the central part of the correlationw xoperator obtained as in Ref. 12 . To evaluate C , weT

overlap the orbitals spinors with the walker spinors,and evaluate the spatial orbitals and the two-bodyfactors at the walker particle positions. The gradientsof the C and the evaluation of the full noncentralT

potential to calculate the energy are done similarly.Results for some model neutron matter calcula-

tions are shown in Table 1. For comparison we showfermi hypernetted chain single operator chainŽ .FHNCrSOC calculations using the method of Ref.w x12 . We see the agreement is fairly good showingthat our method is giving reasonable physical corre-lations even though our trial function has no operatorcorrelations built in. The Monte Carlo results inTable 1 have the potential set to zero at a distance ofrsLr2 where L is the side of the box. No correc-tions to this number have been made. Assuming aconstant density of particles for r)Lr2, gives thecentral tail correction shown in the table. We havenot attempted to compensate for the finite size ef-fects from the kinetic energy terms. We have made

Table 1The energy per particle in MeV for neutron matter using theUrbana Õ potential, i.e. Urbana Õ truncated at the Õ level, at6 14 6

the densities r shown. E are the constrained path resultsMC

described in the text using N particles. The tail correction isdescribed in the text. E are results using the fermiFHNC r SOC

w xhypernetted chain single operator chain approximation of Ref. 12for comparison

y3Ž .r fm N E Tail EMC FHNC r SOC

0.10 38 8.8 " 0.1 y0.15 10.920.10 54 8.6 " 0.1 y0.07 10.920.15 38 11.8 " 0.1 y0.52 13.460.15 54 12.0 " 0.1 y0.26 13.460.20 38 15.5 " 0.1 y1.17 15.940.20 54 15.1 " 0.1 y0.61 15.94

runs at multiple time steps. The values in the tableare from runs where the extrapolated time step errorsare less than the statistical errors.

4. Conclusion

In conclusion, we have given an algorithm thatcan allow fixed node calculations of the groundstates of many nuclear systems with fairly realisticpotentials. Although our trial wave functions aresimple single Slater determinants with orbitals thatfactor into space and spin components, there is nodifficulty in using good J orbitals or including moreSlater determinants to improve the nodes if thatproves necessary. The algorithm requires order N 3

operations at each time step, that is the same order asa straightforward implementation of the fixed nodemethod for central potentials. The feasibility of cal-culating reasonable sized systems was demonstratedby our calculations for 54 neutrons.

Similar methods should allow the inclusion ofsome dynamical meson degrees of freedom. TheHubbard-Stratonovich method is identical to a partic-ular formalism we would get if we included a veryfast meson degree of freedom that coupled to thenucleons. This meson could be integrated out to givea potential by using the Born-Oppenheimer approxi-mation and solving for its ground-state energy withthe nucleon coordinates fixed. The meson ‘‘coordi-nate’’ then corresponds to the Hubbard-Stratanovich

( )K.E. Schmidt, S. FantonirPhysics Letters B 446 1999 99–103 103

variables. More realistic mesons could also be in-cluded.

The solution of such a coupled nucleon-mesonsystem can be handled similarly to our calculations.For example we could include pions in our neutronmatter calculations. In outline, each of the pionmodes that fit in the periodic simulation cell wouldbe given a dynamical variable that gives the magni-tude of the mode. These variables satisfy a harmonicoscillator Schroedinger equation with frequency cor-responding to the mode energy. These variableswould correspond to the Hubbard-Stratonovich vari-ables in our formalism above. The nucleons wouldcouple to the pion modes proportional to

s = sin k Pr t PT 15Ž . Ž .i i m i i m

where the sine could also be cosine and gives thespatial mode of the pion. A form factor would bealso need to be included. The T is the isospin of them

mode. Notice that the eigenvectors of the A matricesŽ .in Eq. 3 are analogous to the gradient of the meson

modes. Initial calculations on 1 and 2 nucleonswould be required to extract the corresponding self-energy, mass correction, and static nucleon-nucleoninteraction. The bulk of the interaction could beretained in the Hubbard-Stratonovich form. Thepropagation of this field theory is no harder inprinciple than our Hubbard-Stratonovich method.Semi-realistic two-pion exchange three-body interac-tions would be automatically included, as wouldretardation effects. The fixed node approximationcould be used exactly as we do now. By using thefirst quantized harmonic oscillator Hamiltonian forthe pion mode amplitudes, the bose character of themesons is automatically included. We are currentlypursuing these ideas.

Acknowledgements

We would like to thank Adelchi Fabrocini forcalculating the FHNC results. Portions of this workwere supported by MURST–National Research Pro-jects, and NSF grant CHE94-07309. Some of thecalculations were done on the Arizona State Univer-sity Beowulf cluster. Some preliminary results of this

w xwork were reported in Ref. 13 .

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21 January 1999


Relativistic particle simulation of a coloured parton plasma

Sudip Sen Gupta a,1, Predhiman K. Kaw a, Jitendra. C. Parikh b

a Institute for Plasma Research, Bhat, Gandhinagar 382 428, Indiab Physical Research Laboratory, NaÕrangpura, Ahmedabad 380 009, India

Received 25 April 1998; revised 12 October 1998Editor: J.-P. Blaizot

Abstract

Ž .We propose a relativistic particle-in-cell PIC simulation method for investigating the non-perturbative dynamics andcollective interactions in a classical coloured parton plasma. The method has the advantage of including quarks and bothshort and long scale gluons non-perturbatively, and may be used for the study of a number of conceptual questions relevantto evolution of a quark gluon plasma formed in ultrarelativistic heavy ion collisions. We illustrate the simulation method bya detailed study of the thermalization of a highly nonequilibrium coloured parton plasma due to collective non-abelianinteractions. q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 12.38.Mh; 05.45.qb; 11.15.Kc

Pre-equilibrium evolution of a dense colouredparton plasma, created in ultrarelativistic heavy ioncollisions, is a topic of intense current research. It iswidely recognised that since an exact non-perturba-tive quantum field theoretic treatment of the problemis unavailable, the underlying physics can only beunderstood by recourse to computer simulations.Current simulations of this problem are based on twoexisting models which describe the formation andsubsequent evolution of the quark-gluon plasma cre-ated in ultrarelativistic heavy ion collisions in arelatively incomplete manner. These are the stringmodel and the parton cascade model. In the former

w xmodel 1,2 , the energy of colliding nuclei are de-


Žposited into coherent glue field configurations col-.our flux tubes which get spontaneously converted

into a far from equilibrium distribution of quark-anti-quark pairs and short wavelength gluons by

w xSchwinger type creation-annihilation processes 3 .The entropy associated with a thermal state is gener-ated in the course of pair creation. The string pictureis developed from models of soft hadron-hadron

Ž .interactions P -L and hence its applicabilityT QCDŽbecomes questionable at higher energies P 4T

.L . In the complementary approach based on theQCDw xparton-cascade model 4–7 , collisions of heavy ions

is pictured as an interaction between two counterstreaming fluxes of partons which lead to thermaliza-tion of the resulting dense partonic matter by hard

Ž .and semi-hard collisions P )L . These colli-T QCD

sions are treated by the methods of perturbativeQCD, which is strictly not applicable for coherent


( )S. Sen Gupta et al.rPhysics Letters B 446 1999 104–110 105

long wavelength gluon components. It is clear that acomplete treatment will involve a simultaneous de-scription of the long and short wavelength gluoncomponents in the parton plasma evolution. In thisletter we propose a relativistic particle-in-cell simula-tion method which partly fulfills this need. Ourtreatment is classical and follows the conventionalparticle-in-cell simulation methods extensively usedin electrodynamic plasmas. We base our simulations

w x w xon the work of Wong 8 , Elze and Heinz 9 andothers who treat quarks, anti-quarks and incoherent

Ž .small wavelength P )L gluons as colouredT QCD

partons whose motion is followed in self-consistentlyŽ . Žgenerated long scale P -L glue fields meanT QCD

.fields created via the appropriate Yang-Mills fieldequations. At present our simulation method is re-stricted to coloured plasma evolution in 1q1 di-mensions. However, when the method is extended to3q1 dimensions, it can be effectively used to inves-tigate a number of conceptual and detailed questionsrelated to non-perturbative aspects of colour dynamicevolution in classical coloured parton plasmas –questions such as colour diffusion in quark gluon

w xplasmas 10,11 , dynamic screening of a moving ccw xpair 12,13 , non-perturbative electric and magnetic

screening effects, dilepton production in the pre-w xequilibrium phase etc. 14 .

In this letter we concentrate on the application ofPIC simulation method to the problem of thermaliza-tion in a coloured parton plasma. It is important toestablish the time scale and physical processes re-sponsible for this thermalization process. This isbecause beyond the thermalization time the space-time evolution of the plasma could be convenientlydescribed by the well known Landau-Bjorken hydro-

w xdynamics 15 and because the detailed physicalprocesses may have significant implications for sig-nals of quark gluon plasma. Earlier simulations ofthe thermalization problem may be classified under

Ž .two broad categories: 1 Perturbative QCD calcula-tions done by taking semi-hard collisional processes

Ž w xamong partons into account Shuryak 16 , Geigerw x.and Muller 7 ; these calculations have a limited¨

range of validity because of the use of perturbativeŽ .methods. 2 Non-perturbative calculations which in-

clude classical calculations of collective effects doneŽthrough the use of colour hydrodynamics Bhatt

w x. Ž17 , classical lattice simulations Muller and¨

w x w x.Tryanov 18 , Gong 19 and calculations usingw xstring model 20,21 ; these calculations are incom-

plete because they fail to take account of either shortscale gluons or quarks.

w xWe now describe our work 22 on relativisticparticle simulation of parton thermalization by col-lective effects. The basic objective of our work, is to

w xextend the work of Muller and Tryanov 18 and¨w xGong 19 on pure gluon fields, to a situation in

which both the quark and gluon sectors are treatedsimultaneously. The introduction of colour dynamicsand Yang-Mills field equations into a relativisticparticle simulation code makes our work, a first oneof its kind.

As stated earlier our treatment is purely classical.Unlike the quantum mechanical models, where non-perturbative effects are included phenomenologicallyŽ .strings, flux tubes, etc. , in the classical approachthe non-perturbative aspects come from the QCDLagrangian. We consider quarks, antiquarks and in-coherent small wavelength gluons on the same foot-ing, all being treated as coloured partons. In this waywe can simultaneously incorporate the particle andfield aspect of gluons, a feature that is an intrinsicpart of quantum field theory, into a classical frame-work. This has recently also been discussed by Hu

w xand Muller 23 . Although the separation of gluonfields into long and short wavelength scales is impor-

Žtant as even the latter act as source terms to gener-.ate collective long scale fields , our simulation re-

sults are insensitive to the separation scale as shownlater in our paper.

Our simulation studies are based on a relativisticŽ . w xparticle-in-cell code PIC 24 . In the code we fol-

low the motions of a collection of coloured particlesinteracting via self-consistently generated non-abeliangauge fields. The equations of motion of a classicalcoloured particle having momentum p m and colour

w xcharge I are given by 8a

dp m gIa mns F u 1Ž .a ndt "c

dI ga msy f A I u 2Ž .abc b c mdt "c

where t is the proper time. g is the coupling con-stant, u is the four velocity and f abc are them

structure constants of the relevant gauge group. The

( )S. Sen Gupta et al.rPhysics Letters B 446 1999 104–110106

dynamics of the long scale collective fields F mn isa

governed by the Yang-Mills equation.

g jnamn mnE F q f A F s 3Ž .m a abc m b c

"c c

where

gmn m n n m m nF sE A yE A q f A A 4Ž .a a a abc b c

"c

w xFollowing Elze and Heinz 9 , we include the currentcontributions due to quarks, antiquarks and shortwavelength gluons in jn and retain the long-wave-a

length contributions on the left hand side of equationŽ . 33 . We work in the gauge A s0 and restrict oura

Ž .simulations to SU 2 group symmetries and 1q1dimensions. Thus our treatment includes only longi-tudinal interactions. It should be pointed out thatalthough we have chosen a particular gauge, weevaluate only gauge independent quantities.

In the PIC method of simulation, the whole sys-tem is divided into cells and all the field quantitieslike electric field, potential, charge density etc. are

Ž .calculated at the cell centres grid points . An infinitesystem is modelled using periodic boundary condi-tions. Particles are initially given a perturbation incolour space to set up the colour charge density andthe associated longitudinal colour fields. Using theseinitial and boundary conditions parton momentum

Ž . Ž .and colour are updated using Eqs. 1 and 2 and theŽ . Ž .colour fields are updated using Eqs. 3 and 4 . The

coupling between the fields and particles is intro-duced through the calculation of jn which is evalu-a

ated from the knowledge of particle trajectories ob-Ž . Ž .tained from Eqs. 1 and 2 . The fields evaluated

using this jn as source in turn influences the particlea

trajectories. The details of the numerical schemesw xused are discussed in Sengupta et al. 24 and in the

w xRefs. 25,26 . A point to be noted here is that theŽ n .calculation of charge densities j at the grid pointsa

from the particle positions and the calculation offorces on the particles due to fields evaluated at thegrid points, require particle to grid and grid to parti-

Ž .cle interpolation schemes weighting schemes . Moreclearly, the charge density at a grid point X is givenj

by

1r s W X yx I 5Ž .Ž .Ýja j i i a

Dx i

where x and I are respectively the position andi i a

charge of colour ‘a’ on the ith particle. Dx is theŽ .width of a cell. The weighting function W X yxj i

is colour independent and hence does not alter thegauge covariance properties of the governing Eqs.Ž . Ž .1 – 4 . In our simulation we have used a quadraticspline weighting scheme. Another point to be em-phasized here is that weighting schemes used forcharge density calculation and for force calculationshould be identical. This is important because it can

w xbe shown 27 that for identical charge assignmentand force interpolation schemes and correctly space-centred difference approximations for derivatives,the self force on a particle arising due to variousparticle to grid and grid to particle interpolations iszero and total momentum of the system is conserved.

We now report the main results of our simulation.We first emphasise that collective effects are domi-nant over collisional ones when the plasma parame-

Žter L 'nl i.e. the number of particles in aD D. w xDebye sphere 41 28 . In the simulation we choose

Žparameter nl ,13 this roughly corresponds to aD

3-D parameter L ;2=103, which is much greaterD

than unity and hence we are justified in neglecting.collisions . In our simulation we treat those gluons

which have wavelength F grid size as particles andthose which have wavelength 4 grid size as waves.We initiate the simulation of the scaled form of Eqs.Ž . Ž .1 – 4 with the imposition of a coherent colour

Ž Ž ..charge perturbation I s I 1qdcos k xqu ona 0 L a

a zero temperature parton gas at ts0. This is ourmethod of initializing the plasma in a state which isfar from thermodynamic equilibrium. The results

2depend on two parameters d and asg n m r"k( 0 L

and are essentially independent of other details ofinitialization. The parameter d measures the plasmanonlinearity and a is a measure of the strength ofnon-abelian coupling. The mass m of quarks used inthe parameter a is their current mass whereas for theshort scale gluons we take an effective mass of thesame order of magnitude. It should be noted howeverthat the mass in the calculations is actually domi-nated by relativistic kinetic energy effects, initially inthe coherent sloshing motion in the wave fields andeventually in the temperature of the partonic plasma.For as0 and ds0.001, we find that the systemexhibits usual plasma oscillations at the plasma fre-quency. This is expected since non-abelian physics is


absent and conventional plasma non-linearities areŽ .negligible. For small values of a f0.5 and ds

0.01 the system continues to be coherent and exhibitscharacteristic ‘ear-phone’ oscillations in which aslower frequency modulates the usual plasma oscilla-

w xtions 24,29 .Fig. 1 shows the characteristic time evolution for

as10 and ds0.1. Now the field energy densityshows irregular chaotic oscillations. To examine thenature of this chaos we define a gauge invariant

w xmetric as in Refs. 18,19,30,31 .

NGy12X2 2D t s E yE 6Ž . Ž .Ý ai ai(

is0

This is nothing but the distance between two pointsin a NG-dimensional space, where each point repre-sents the field energy configuration of the whole

Ž Ž ..system. Fig. 2 shows the variation of ln D t versustime. The linear rise shows that the ‘distance’ be-tween two neighbouring configurations increases ex-ponentially in time. The saturation, as also observed

w xin earlier work 18,19 , is because of the compactspace used for description of fields. The slope of the

Ž .linear part gives the largest Lyapunov exponent hw xwhich is related to the rate of entropy rise 18,19 .

The inverse of the Lyapunov exponent gives anestimate of the entropy production time scale or the

Fig. 1. Evolution of field energy density measured at the centre ofthe system for a s10.0 and d s0.1.

Ž .Fig. 2. Evolution of the distance D t between neighbouringrandom gauge field configurations for a s10.0 and d s0.1.

thermalization time. In our case the estimated ther-malization time turns out to be of order 0.2 plasmaperiods. We expect that in a full 3q1 dimensionalsimulation, because of increase in the dimension ofphase phase, there would be a general enhancementof the chaotic behaviour of the system. Hence, theestimate of thermalization time given here should beviewed as an upper limit. This result is independentof the grid size and hence of the coarse graining dueto computational procedure which in turn implies

Žthat the simulation results like final temperature,.thermalization time etc. are insensitive to the separa-

tion of gluon scales.We next verify the thermalization of the plasma

by looking at the phase space distribution and alsothe momentum distribution. Compared to the initial

Ž .cold Ts0 parton distribution, we observe a con-Ž .siderably broadened phase space Fig. 3 . The space

average distribution of particle momenta is shown inŽ .Fig. 4 . It shows that the distribution approaches aJuttner form¨

2px1q( 2 21 m c

exp y2mcK 1rj jŽ .1


Fig. 3. Distribution of particle momenta as function of x forŽ .a s10.0 and d s0.1 at time normalized to plasma period TP

trT s3.2.P

with j'Trmc2 ;3.107=103. In Fig. 4 the jaggedcurve shows the simulation result and the matchingsolid curve shows the theoretical distribution ob-

² 2 2 2:tained by measuring p rm c from the simula-xtion data, calculating j from the equation

Ž .2 2 2 K 1rj2² :p rm c sj and substituting in the Juttner¨xŽ .K 1rj1

form.The mechanism by which cold plasma particles

get heated up is related to a ‘phase mixing’ typedamping of the longitudinal fields. The energy whichis initially loaded in the lowest k flows into high k’s

Fig. 4. Space average distribution of particle momenta for a s10.0Ž .and d s0.1 at time normalized to plasma period T trT s3.2.P P

Ž .through nonlinear non-abelian interactions. Theresonant particle effects associated with the highk-modes then lead to thermalization of the plasma. Inthe thermalized classical plasma we expect equiparti-tion of energy between the particles and collectivemodes. We have verified that kinetic energy per

Žparticle approaches T as it should in a plasma which.is ultrarelativistic and the ‘mean energy’ per mode

with thermal and fat particle correction, which is² 2 :Ž 2 2 Ž .. w xgiven by Ý E Lr4p 1qk l f k 32,33a ak Deff

Ž Ž . Ž 2 .approaches 3T for SU N it should be N y1 Tw x. Ž .34 in the simulations Fig. 5 . The straight linehere shows the position 3T in our simulation units.Here L is the system length and l is the effec-Deff

tive screening length containing non-abelian and rel-ativistic corrections and it turns out to be ;3lD

2 2 Ž .(where l s Trg I n ; and f k is the correctionD 0 0Žfactor for fat particles which for our case i.e. trian-

a2 k2 2. Ž .gular shaped particles is rsin kar4 where16

‘a’ is particle width. The time taken by the partondistribution to take the thermalised Juttner form is¨typically several Lyapunov exponent times.

We now summarise our conclusions and commenton the limitations of our treatment. We have demon-strated that relativistic particle simulations ofcoloured parton plasma, using a PIC code, can beused to answer meaningful conceptual questions re-garding the evolution of coloured partonic matter

Fig. 5. Field energy vs mode number for a s10.0 and d s0.1 atŽ .time normalized to plasma period T trT s3.2.P P


formed in heavy ion collisions. As an illustration westudied the collective thermalization process in aclassical coloured parton plasma in 1q1 dimensions

Ž .and SU 2 group symmetry, and showed that anarbitrary initial configuration of particles and fieldsthermalizes in a few plasma periods. It should beemphasised that such collective effects dominateconventional collisional equilibration effects when

Žthe plasma parameter L snl i.e. number ofD D.particles in a Debye length 41. A consequence of

one-dimensionality is that chromomagnetic effectsare ignored in our calculations. We have also ignoredthe creation and annihilation of particles which canbe justified for collective thermalization when theplasma frequency is large compared to the creationand annihilation rates. Since we restrict our calcula-tions to classical relativistic dynamics, the partondistribution acquires the Juttner form rather than¨Fermi-Dirac or Bose-Einstein form expected in quan-tum physics. The collective modes acquire equiparti-tion with 3T per mode. This simulation in which thelong scale gluons acquire a thermal distribution andthe short wavelength gluons an exponentially fallingJuttner form is the closest that a classical calculation¨can come to model the true quantal Planck distribu-tion law. Although the Wong’s equations used in ourrelativistic particle-in-cell code are purely classical,it is possible to judge when quantum effects becomeimportant by considering corrections to the Wong’sequations. In the derivation of Wong’s equations,one starts with the full QCD Lagrangian and writesHeisenberg equation of motion for all the quantum

Ž .operators like position, momentum etc. . In the finalstep all the operators are replaced by their expecta-

Ž .tion values, which gives the classical Wong’s equa-tions. By expanding the equation of motion about themean position of a particle, it is possible to estimate

Ž .the higher order quantum corrections. It turns outthat quantum corrections are small when kThermal

Ž y14k where k is the thermaldeBroglie max Thermal deBroglie

deBroglie wavelength and k can be taken as themax.inverse of interparticle distance . The question of

quantum corrections to classical results was alsow xconsidered by Gong et al. 35 and a similar condi-

tion was derived by them. In our simulation interpar-ticle distance ;0.2 fm and l ;0.01Thermal deBroglie

fm. So the quantum corrections to our classicalequations are of order ;10%.

Acknowledgements

We thank the referees for constructive criticism.

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w x Ž .29 J.R. Bhatt, P.K. Kaw, J.C. Parikh, Phys. Rev. D 39 1989646.


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21 January 1999


Long range forces induced by neutrinos at finite temperature

F. Ferrer, J.A. Grifols, M. NowakowskiGrup de Fısica Teorica and Institut de Fısica d’Altes Energies, UniÕersitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain´ ` ´ `

Received 7 July 1998Editor: R. Gatto

Abstract

We revisit and extend previous work on neutrino mediated long range forces in a background at finite temperature. ForDirac neutrinos, we correct existing results. We also give new results concerning spin-independent as well as spin-dependentlong range forces associated to Majorana neutrinos. An interesting outcome of the investigation is that, for both types ofneutrinos whether massless or not, the effect of the relic neutrino heat bath is to convert those forces into attractive ones inthe supra-millimeter scale while they stay repulsive within the sub-millimeter scale. q 1999 Elsevier Science B.V. All rightsreserved.

Neutrinos mediate long-range forces betweenw xmacroscopic bodies 1–4 . Indeed double neutrino

exchange among matter fermions generates spin-in-dependent forces that extend coherently over macro-scopic distances. The effect, however, is extremelyweak, much too tiny to be experimentally detectedwith present day technology. Compared to their grav-itational pull, the force between two nucleons 1 cmapart is about 10y28 times weaker. Not only theircoupling strength is very small but also their decaywith distance is fast. Indeed the potential drops asry5 so that the effects die off correspondingly. Phe-nomenological surveys on forces with this particulardistance behaviour have been conducted in the litera-

Ž w x.ture see e.g. 5 over the whole span of distancesfrom astronomical down to the micron scale. Also,the case of a system with high density of matter suchas the core of a neutron star, has recently received

w xconsiderable attention 6–11 .In a neutrino populated medium, such as the

cosmic neutrino background or the hot core of a

supernova, the helicity flip produced by single neu-trino exchange can be balanced by the neutrinos inthe medium and, as a consequence, a spin-indepen-dent interaction takes place that leads to a coherenteffect over many particles in bulk matter.

The neutrino long-range forces in the presence ofa neutrino thermal bath have been explored in Ref.w x12 in the Dirac neutrino case. The long range forcesmediated by Majorana neutrinos, on the other hand,have been studied only in the zero temperature casew x4 . Here we wish to extend the nonzero temperatureresults to the Majorana case. Because the distinctionbetween Dirac and Majorana neutrinos is superfluousfor massless neutrinos, we shall consider the generalm/0 case.

w xWe shall adopt the notation in 12 and write,

d3QV r sy exp iQPr T Q 1Ž . Ž . Ž . Ž .H 32pŽ .

Ž .where T Q is the nucleon-nucleon elastic scatteringŽ .amplitude Fig. 1 in the static limit, i.e. momentum


( )F. Ferrer et al.rPhysics Letters B 446 1999 111–116112

Fig. 1. Lowest order Feynman diagram for two neutrino exchangein the four fermion effective theory.

Ž .transfer Q, 0,Q , where matter is supposed to beat rest in the microwave background radiationŽ .MWBR frame. It can be cast in the form

m nX X X2T Q sy2 iG g ,y2 g S g ,y2 g S IŽ . Ž . Ž .F V A V A mn

2Ž .

with

d4kI s Tr g OiS k g OiS kyQ . 3Ž . Ž . Ž .Hmn m T n T42pŽ .

1 ŽThe operator O is the left-handed projector 1y21'.g for Dirac neutrinos and 2 g for Majorana5 52

neutrinos. The temperature dependent propagator ST

has the explicit form

y12 2S k s ku qm k ym q ieŽ . Ž . Ž .T

q2p id k 2 ym2 u k 0 nŽ . Ž .Ž q

0qu yk n 4Ž . Ž ..y

where n and n are Fermi-Dirac distribution func-q ytions for particle and antiparticle, respectively. As

w xdiscussed in 12 , Fig. 1 evaluated with this propaga-tor taken together with the usual Feynman rules is

Ž .sufficient to calculate the potential. In Eq. 2 , gV , A

are composition-dependent weak vector and axial-vector couplings. We focus first on the spin-indepen-dent potential, that is the g gX component ofV V

Ž .Eq. 2 .

Ž .Use of the first piece in Eq. 4 gives the zerow xtemperature vacuum results 4,6 ,

G2 m3g gXF V V

V r s K 2mr 5Ž . Ž . Ž .Dirac 33 24p r

and

G2 m2 g gXF V V

V r s K 2mr 6Ž . Ž . Ž .Majorana 23 32p r

in terms of the modified Bessel functions K .2,3Ž .At very large distances mr41 , i.e. much larger

than the Compton wavelength of the neutrino, thesepotentials exhibit the asymptotic behaviour

2 X 5r2G g g mF V V y2 m rV r , e 7Ž . Ž .Dirac ž /8 p r

and

1r2X2 3G g g mF V V y2 m rV r , e . 8Ž . Ž .Majorana 5 7ž /4 p r

Ž Ž . Ž ..Of course, both potentials Eqs. 5 and 6 coincidewhen ms0. They give the well known Feinbergand Sucher result:

G2 g gXF V V

V r s . 9Ž . Ž .3 54p r

In a neutrino background, a contribution to the longrange force can arise because a neutrino in thethermal bath may be excited and de-excited back toits original state in the course of the double scatter-ing process. This effect is described by the crossedterms contained in I that involve the thermal piecemn

of one neutrino propagator along with the vacuumpiece of the other neutrino propagator. This thermalcomponent of the tensor I can be written asmn

4d kmn 2 2 0I syp i d k ym u k nŽ . Ž .HT , D q42pŽ .

m nTr g ku qQu g kuŽ .0qu yk nŽ . y 2 2kqQ ym q ieŽ .

m nTr g kug ku yQuŽ .q 10Ž .2 2kyQ ym q ieŽ .

( )F. Ferrer et al.rPhysics Letters B 446 1999 111–116 113

in the Dirac case, and

d4kmn 2 2I syp i d k ym nŽ .HT , M 42pŽ .

=

m nTr g ku qQu qm g ku ymŽ .Ž .2 2kqQ ym q ieŽ .

m nTr g ku qm g ku yQu ymŽ . Ž .q 11Ž .2 2kyQ ym q ieŽ .

for Majorana neutrinos, where in this latter case weput n sn sn since the chemical potential van-q y

Ž .ishes. Note that in 10 there is no component pro-portional to e mna b since after the integration over kthe only four-vector available is Qa. As a resultthere will be no parity violating potentials.

ŽFar from degeneracy i.e. for chemical potential.m<T , as is probably the case for cosmological

neutrinos, we can consider the neutrinos to be Boltz-mann distributed, that is we take

0< <n sexp "my k rT . 12Ž .Ž ."

With this approximation, the integrations involved inthe calculation of potentials can be easily done byconveniently choosing the order in which they areperformed. The results can be expressed again in

Žterms of Bessel functions. Indeed, starting with in< <.the formulae below Q' Q :

2 X`iG g gF Õ ÕDiracV r s dQ Q I Q sin QrŽ . Ž . Ž .HT 002p r 0

13Ž .

Ž .inserting the value of I taken from 10 :00

2 X`

2G g g dkkF Õ ÕDiracV r s cosh mrTŽ . Ž .HT 4 2 2'2p r 0 k qm

=1 22 2'exp y k qm rT dz 2kzŽ .Ž .H

y1

` Qsin QrŽ .2 2y2m y4k dQH 220 Q y 2kzŽ .

14Ž .

and performing first the integration over Q, thenintegrating over z, we obtain:

G2 g gXF Õ ÕDiracV r s cosh mrTŽ . Ž .T 3 42p r

=2y 1q mr I m ,r ,TŽ . Ž .Ž .

dI m ,r ,TŽ .qr 15Ž .

dr

Ž w x.where see 13 :

` dkk2 2'I m ,r ,T s exp y k qm rTŽ . Ž .H

2 2'0 k qm

=2 rTm

sin 2kr sŽ .2(1q 2 rtŽ .

=m 2(K 1q 2 rT . 16Ž . Ž .1 ž /T

Putting things together we get for the Dirac case:

G2 m4 g gXF V VDiracV r sy cosh mrTŽ . Ž .T 3p r

=K r 4K rŽ . Ž .1 2

q 17Ž .2r r

where we have definedm 2(r' 1q 2 rT . 18Ž . Ž .T

Ž .Taking now I from 11 we obtain the correspond-00

ing result for the Majorana neutrino case:

4G2 m4 g gX K rŽ .F V V 2MajoranaV r sy . 19Ž . Ž .T 3 2p r r

Ž .For massless neutrinos and ms0 both poten-tials collapse to

8G2 g gX T 4F V V

V r sy 20Ž . Ž .T 3 22 2p r 1q4r TŽ .w xwhich is the result given in Ref. 12 . Because the

Ž .y1neutrino background temperature is T; 1.2 mm ,we see that for distances much larger than

Ž . Ž .1 mm i.e. rT41 the potential in Eq. 20 reads

G2 g gXF V V

V r ,y . 21Ž . Ž .T 3 52p r


Ž .When added to the vacuum result 9 , the totalpotential is

G2 g gXF V V

V r ,y 22Ž . Ž .tot 3 54p r

that is, in the presence of the cosmic neutrino back-ground the original Feinberg-Sucher force switchessign, i.e. a repulsive force turns into an attractiveone. On the other hand, well within the sub-millime-

Ž .ter domain rT<1 , the temperature dependent po-Ž .tential 20 behaves as follows

8G2 g gX T 4F V V

V r ,y 23Ž . Ž .T 3p r

which is negligible compared to the vacuum contri-Ž .bution in Eq. 9 .

In the general m/0 case, we shall study theŽ . Ž .Dirac and Majorana potentials, Eqs. 17 and 19

respectively, in various physically interesting limits.Consider first the cases where r<my1

<Ty1 ormy1

<r<Ty1. Performing the relevant expan-Ž . Ž .sions of the Bessel functions in 17 and 19 leads

to

G2 m5r2 g gXF V VDirac 3r2V r ,y TŽ .T 1r2 5r22 p r

=cosh mrT eym r T 24Ž . Ž .

and

23r2 G2 m3r2 g gXF V VMajorana 5r2 ym r TV r ,y T e .Ž .T 5r2p r

25Ž .

Hence, thermal effects are exponentially damped inboth distance domains.

A different behaviour is obtained for distancesmuch larger than any inverse energy scale in theproblem, i.e. for r4Ty1

4my1 or r4my14

Ty1. Indeed, now we have

2 X 5r2G g g mF V VDiracV r ,yŽ .T ž /4 p r

=cosh mrT ey2 m r 26Ž . Ž .

and

1r2X2 3G g g mF V VMajorana y2 m rV r ,y e .Ž .T 5 7ž /2 p r

27Ž .

Both expressions exhibit the characteristic Yukawaexponential damping associated to two-particle ex-change. These results when added to their vacuum

Ž . Ž .counterparts, Eqs. 7 and 8 , produce the inversionphenomenon already noticed in the massless case. Atasymptotically large distances the resulting potentialis equal in strength as it would be in vacuum but,contrary to what happens in vacuum, it is attractiveinstead.

There is no exponential suppression only whenr<Ty1

<my1 or Ty1<r<my1, where one es-

Ž .sentially recovers the massless cases, Eq. 23 or Eq.Ž .21 , respectively. Indeed, for Majorana neutrinosone gets these equations as they stand, and for Diracneutrinos both equations should be multiplied by the

Ž .factor cosh mrT for non-zero chemical potential.Ž .Let us note that the results given in Eqs. 24 and

Ž .26 for the m/0 Dirac case disagree with thew xcorresponding results given in Ref. 12 . Indeed,

their formulae do not show the Boltzmann or Yukawasuppression factors that enter the asymptotic expan-sions of the Bessel functions and which are bound tobe there on physical grounds.

Up to this point all calculations refer to spin-inde-pendent potentials, those that can coherently addover macroscopic samples of unpolarized matter. Letus, for the sake of completeness, consider briefly thequestion of potentials that depend on spin. Now weshould focus on the spatial indices of the tensor Imn

Ž .appearing in the scattering amplitude in Eq. 2 . TheŽ .Fourier transformation 1 is in this case somewhat

more involved than before because the amplitudewill depend on the components of the 3-momentumtransfer. We decompose the spatial part of I as:mn

T Q s SPQ SXPQ t Q q SPSX t QŽ . Ž . Ž . Ž . Ž . Ž .1 2

28Ž .

Ž . Ž .where the functions t Q and t Q depend only on1 2ˆ< <Q and we perform first the angular Q integration in

( )F. Ferrer et al.rPhysics Letters B 446 1999 111–116 115

Ž .Eq. 1 . The rest of the calculation goes along similarlines as in the spin-independent part above. We get:

X2 44G m g gF A A XspinV r sy SPS F rŽ . Ž . Ž .T 3p r

XSPr S PrŽ . Ž .q2 G r cosh mrTŽ . Ž .2r

29Ž .

where

K r K r K rŽ . Ž . Ž .1 2 32 2F r 'a q2 y8m rŽ . 2 3r r r

30Ž .

and

K r K rŽ . Ž .2 32 2G r '7 y4m r 31Ž . Ž .2 3r r

with as1 for Dirac neutrinos and as2 for Majo-rana neutrinos and S2 s3r4. Of course, in theMajorana case we must put ms0.

Both cases above, i.e. Dirac and Majorana, lead tothe potential

16G2 g gX T 4F A AspinV r syŽ .T 33 2 2p 1q4r T rŽ .

=X 2 2SPS 1y12 r TŽ . Ž .

XSPr S PrŽ . Ž .2 2q 7q12 r T 32Ž . Ž .2r

Ž .for ms0 and ms0. This result, Eq. 32 , should bew xthen added to the vacuum result 1

G2 g gXF V VspinV r sŽ . 3 52p r

=

XSPr S PrŽ . Ž .X5 y3 SPS . 33Ž . Ž .2r

The various regimes explored before can be studiedalso for the spin-dependent forces. The discussioninvolves the various asymptotic forms of the samemodified Bessel functions and will lead to the same

exponential damping whenever the temperature orthe mass is the relevant energy parameter. Sincethese forces will be even more difficult to detect thanthe spin-independent ones, for they do not add upcoherently in bulk matter, we do not bother here todisplay the explicit form for the different limits.

We end this paper with a short summary. DoubleŽ .neutrino exchange mediates extremely feeble long

range forces. In vacuum these forces have beenŽ .known at least for Dirac neutrinos for quite some

time. Recently, it has been realised that a neutrinobackground will also induce long range interactionsamong bulk matter. The results were given for Diracneutrinos. We have extended the work of Horowitz

w xand Pantaleone 12 to include the case of Majorananeutrinos and, furthermore, we have derived theexact form of the potentials in either case, i.e. Diracand Majorana, and explored physically relevant dis-tance and energy scales. In so doing we have found

Žimportant discrepancies with previous work m/0,.Dirac case . Since matter is embedded in the cosmic

neutrino background, a consequence of our analysisis that the forces are repulsive in the sub-millimeterscale and attractive for distances well beyond 1 mm

Ž .for any kind of neutrino massless or not . In fact, onŽthe small scale the vacuum result Feinberg and

.Sucher dominates whereas on the larger scale therelic neutrino background is responsible for the dom-inant effect. This means that by experimentally de-

Žtecting admittedly a highly improbable event for.laboratory experiments such forces in both different

regimes one would, not only establish these neutrinointeractions, but one would in addition detect therelic neutrino background. Actually it is the neutrino

Ž y1 .background temperature T ;1.2 mm which setsthis 1 mm distance scale. Incidentally, the sub-milli-meter scale has been subject recently of renewed

w xtheoretical as well as experimental interest 14 . Foran experimental point of view of the actual possibledetection of very weak long range forces we refer

w xthe reader to Refs. 15 .

Acknowledgements

Work partially supported by the CICYT ResearchProjects AEN95-0815 and AEN95-0882. F.F. ac-


knowledges the CIRIT for financial support. M.N.would like to thank the Spanish Ministerio de Educa-cion y Ciencia.´

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21 January 1999


Direct search for light gluinos

NA48 Collaboration

V. Fanti a, A. Lai a, L. Musa a,1, D. Marras a, A. Nappi a, B. Hay b, R.W. Moore b,2,K.N. Moore b,2, D.J. Munday b, M.D. Needham b, M.A. Parker b, T.O. White b,

S.A. Wotton b, J. Andersen c, G. Barr c, G. Bocquet c, J. Bremer c, A. Ceccucci c,3,D. Cundy c, N. Doble c, G. Fischer c, W. Funk c, L. Gatignon c, A. Gianoli c,

A. Gonidec c, G. Govi c, P. Grafstrom c, W. Kubischta c, S. Luitz c, G. Kesseler c,¨J.P. Matheys c, A. Norton c, S. Palestini c,3, B. Panzer-Steindel c, D. Schinzel c,

H. Taureg c, M. Velasco c,4, O. Vossnack c, H. Wahl c, G. Wirrer c, V. Kekelidze d,A. Mestvirishvili d, I. Potrebenikov d, G. Tatichvili d, A. Tkatchev d, A. Zinchenko d,O. Boyle e, V.J. Martin e, I.G. Knowles e, H.L.C. Parsons e, P. Dalpiaz f, J. Duclos f,

P.L. Frabetti f, M. Martini f, F. Petrucci f, M. Porcu f, M. Savrie f, A. Bizzeti g,´M. Calvetti g, G. Graziani g, E. Iacopini g, M. Lenti g, A. Michetti g, H.G. Becker h,

H. Blumer h,1, P. Buchholz h,5, D. Coward h,6, C. Ebersberger h, H. Fox h,¨A. Kalter h, K. Kleinknecht h, U. Koch h, L. Kopke h, B. Renk h, J. Scheidt h,¨

J. Schmidt h,V. Schonharting h, I. Schue h, R. Wilhelm h, A. Winharting h,¨ ´M. Wittgen h,J.C. Chollet i, S. Crepe i, L. Iconomidou-Fayard i,´ ´

L. Fayard i, J. Ocariz i,7, G. Unal i,D. Vattolo i, I. Wingerter i, G. Anzivino j,P. Cenci j,P. Lubrano j, M. Pepe j,B. Gorini k,1, P. Calafiura k, R. Carosi k,

C. Cerri k, M. Cirilli k, F. Costantini k,R. Fantechi k, S. Giudici k, I. Mannelli k,V. Marzulli k, G. Pierazzini k, M. Sozzi k,J.B. Cheze l, J. Cogan l, M. De Beer l,P. Debu l, A. Formica l,R. Granier-De-Cassagnac l, P. Hristov l,8, E. Mazzucato l,B. Peyaud l, S. Schanne l, R. Turlay l, B. Vallage l, I. Augustin m, M. Bender m,

M. Holder m, M. Ziolkowski m, R. Arcidiacono n, C. Biino n, F. Marchetto n,E. Menichetti n, J. Nassalski o, E. Rondio o, M. Szleper o, W. Wislicki o,

S. Wronka o, H. Dibon p, M. Jeitler p, M. Markytan p, I. Mikulec p, G. Neuhofer p,M. Pernicka p, A. Taurok p

a Dipartimento di Fisica dell’UniÕersita e Sezione dell’INFN di Cagliari, I-09100 Cagliari, Italy`b CaÕendish Laboratory, UniÕersity of Cambridge, Cambridge, CB3 0HE, UK 9

c CERN, CH-1211 GeneÕa 23, Switzerlandd Joint Institute for Nuclear Research, Dubna, Russian Federation


( )V. Fanti et al.rPhysics Letters B 446 1999 117–124118

e Department of Physics and Astronomy, UniÕersity of Edinburgh, JCMB King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK 9

f Dipartimento di Fisica dell’UniÕersita e Sezione dell’INFN di Ferrara, I-44100 Ferrara, Italy`g Dipartimento di Fisica dell’UniÕersita e Sezione dell’INFN di Firenze, I-50125 Firenze, Italy`

h Institut fur Physik, UniÕersitat Mainz, D-55099 Mainz, Germany 10¨ ï Laboratoire de l’Accelerateur Lineaire, IN2P3-CNRS, UniÕersite de Paris-Sud, F-91406 Orsay, France 11´ ´ ´ ´

j Dipartimento di Fisica dell’UniÕersita e Sezione dell’INFN di Perugia, I-06100 Perugia, Italy`k Dipartimento di Fisica, Scuola Normale Superiore e Sezione INFN di Pisa, I-56100 Pisa, Italy

l DSMrDAPNIA - CEA Saclay, F-91191 Gif-sur-YÕette, Francem Fachbereich Physik, UniÕersitat Siegen, D-57068 Siegen, Germany 12¨

n Dipartimento di Fisica Sperimentale dell’UniÕersita e Sezione dell’INFN di Torino, I-10125 Torino, Italyò Soltan Institute for Nuclear Studies, Laboratory for High Energy Physics, PL-00-681 Warsaw, Poland 13

p Osterreichische Akademie der Wissenschaften, Institut fur Hochenergiephysik, A-1050 Wien, Austria¨

Received 28 October 1998Editor: K. Winter

Abstract

We present the results for a direct search for light gluinos through the appearance of h™3p 0 with high transversemomentum in the vacuum tank of the NA48 experiment at CERN. We find one event within a lifetime range of 10y9–10y3

s and another one between 10y10–10y9 s. Both events are consistent with the expected background from neutrons in thebeam, produced by 450 GeV protons impinging on the Be targets, which interact with the residual air in the tank. From thesedata we give limits on the production of the hypothetical gg bound state, the R0 hadron, and its R0 ™hg decay in the R0˜ ˜mass range between 1 and 5 GeV. q 1999 Elsevier Science B.V. All rights reserved.

1 Present address: CERN, CH-1211 Geneva 23, Switzerland.2 Present address: Physics-Astronomy Building, Michigan State

University, East Lansing, MI 48824, USA.3 Permanent address: Dipartimento di Fisica Sperimentale

dell’Universita e Sezione dell’INFN di Torino, I-10125 Torino,Ìtaly.

4 Corresponding author. E-mail: [email protected] Present address: Institut fur Physik, Universitat Dortmund,¨ ¨

D-44221 Dortmund, Germany.6 Present address: SLAC, Stanford, CA 94309, USA.7 Permanent address: Departamento de Fıisica, Universidad de´

los Andes, Merida 5101-A, Venezuela.´8 Permanent address: Joint Institute for Nuclear Research,

Dubna, Russian Federation.9 Funded by the UK Particle Physics and Astronomy Research

Council.10 Funded by the German Federal Minister for Research and

Ž . Ž .Technology BMBF under contract 7MZ18P 4 -TP2.11 Funded by Institut National de Physique des Particules et de

Ž .Physique Nucleaire IN2P3 , France.´12 Funded by the German Federal Minister for Research and

Ž .Technology BMBF under contract 056SI74.13 Ž .Funded by the Committee for Scientific Research KBN ,

grant 2 P03B07615.

w xRecent theoretical work 1,2 has proposed a classŽ .of supersymmetric models in which the gluino g

Ž .and the photino g are expected to have small˜masses and the photino is stable and an ideal candi-date for dark matter. In such models there is a

Ž .hypothetical spin-1r2 gluon-gluino gg bound state,˜the R0 hadron. This strongly interacting particle isexpected to have a mass of a few GeV and a lifetimebetween 10y10 and 10y6 s. For these reasons the

w x Ž .NA48 experiment 3 see Fig. 1 , designed to mea-Ž X .sure the CP violation parameter R e re using high

intensity K and K beams, is a suitable experimentL S

to look for R0’s produced by a 450 GeV protonbeam impinging on a Be target.

We have searched for R0 ™hg through the ap-˜pearance of h™3p 0 with high transverse momen-tum in the decay volume of this experiment, underthe assumption that the g is not detectable. Neutral˜kaons do not decay into h’s because they are 50MeV heavier, therefore h’s are not expected to befound in decay volume of this experiment. The data

( )V. Fanti et al.rPhysics Letters B 446 1999 117–124 119

Fig. 1. Schematic drawing of the NA48 beamline and detector. In this detector photons are identified with a high resolution liquid kryptonelectromagnetic calorimeter. Extra photon activity is detected by a set of anti-counters located along the vacuum tank. Also shown is thecharged spectrometer consisting of a magnet and wire chamber planes, complemented by a muon veto system, a hadronic calorimeter, andhodoscope triggering planes. See text for the definition of the longitudinal vertex position, d.

were collected in about 3 weeks of data takingduring 1997.

As shown in Fig. 1, the NA48 experiment has twonearly collinear K and K beams which operateS L

w xconcurrently 4 . The beams are produced from 1.1=1012 and 3.4=107 protons impinging on the 40cm long K and K Be targets, respectively, everyL S

14.4 s in a burst that is 2.4 s long. Decays occurringin the K and K beamline are distinguished by aS L

tagging scintillator hodoscope which is positioned inthe proton beam producing the K beam by measur-S

ing the time of flight between the tagging scintillatorhodoscope and the main detector. This dual beamline

Ž y10 y3 .design offers a wide lifetime range 10 –10 sin the R0 search.

A dedicated trigger derived from the eXre neutral

trigger, based on the liquid krypton electromagneticŽ . w xcalorimeter LKr 5 information, was implemented

in order to select 3p 0 events with high transverse

Ž .momentum high-P . The complete ‘neutral’ triggerTw xsystem is described in Ref. 6 . The high-P triggerT

decision was based on the calculated total electro-magnetic energy E , the first moment of the en-LKr

ergy m , the energy center-of-gravity COG, and the1

number of clusters in each projection. These quanti-ties are calculated from:

E s0.5= m qm ,Ž .LKr 0 x 0 y

2 2m s m qm ,(1 1 x 1 y

COGsm rE ,1 LKr

where m sÝ E , m sÝ E , m sÝ x E , m0 x i i 0 y j j 1 x i i i 1 y

sÝ y E , are the calculated moments in each pro-j j j

jection, and the i and j indices denote summationŽover energies and position of the vertical x-projec-

. Ž .tion and horizontal y-projection strips of the LKrŽ .calorimeter. The point xs0, ys0 is defined at the


center of the calorimeter. The resulting high-P trig-T

ger had a high background rejection power withlosses less than 25% of the geometrically acceptedsignal. This was achieved by requiring E G40LKr

GeV, m G1500 GeV cm, COGG20 cm, and four1

or more distinguishable clusters in at least one of thetwo projections. The COG requirement limits thetransverse momentum to more than 0.15 GeV for hsof momentum greater then 75 GeV, while the firstmoment requirement rejects backgrounds from K L

™3p 0’s, where some of the photons escape detec-Ž .tion. The largest measured loss 15% for fully con-

tained events is due to the overlapping of clusters inboth projections. For the final trigger to be issued,the neutral trigger signal required to be in anti-coin-cidence with the muon veto and the ring-shape arrayof photon detectors appearing as ‘‘anti-counters’’ inFig. 1. The high-P trigger was downscaled by aT

factor of two. The resulting trigger rate was below100 triggersrburst out of a total of 13,000triggersrburst handled by the data acquisition sys-tem during the data taking using 1.5=1012 pro-tonsrburst on the K target.L

The six photons in an event are used to recon-struct three p 0’s that have to come from a commonvertex. The photons must have energies above 2GeV, a time difference between them which issmaller than 1.5 ns, to be within the defined LKrfiducial volume, and to have no track in the driftchambers. In addition, it is required that there is noactivity in the hadronic calorimeter and that theenergy of the h’s is greater than 95 GeV. The

0 Ž .reconstructed p masses, m d , are used as con-i

straints in a fit to minimize the x 2 as a function ofthe longitudinal vertex position, d, without an as-sumption on the mass of the parent particle, that is,

2Ž . 3 Ž Ž ..2 20x d sÝ m ym d rs . The typical erroris1 p i i

on the p 0 mass s is around 1.2 MeV. We will seti

a mass window "6 MeV wide, which corresponds0 2Ž .to approximately "3s for 3p events. The x d

was required to be smaller than 8, which correspondsŽ .to a confidence level CL larger than 98.2%.

The masses of the selected high-P events areT

shown in Fig. 2 as a function of the best fit longitu-dinal vertex position. We find 152 K ™3p 0 eventsL

and 31 h™3p 0 events in the d-region betweeny300 and 9600 m. The integrated beam correspondsto 1.2=1017 and 2.1=1012 protons impinging on

Fig. 2. Reconstructed high transverse momentum K and hL

particle decays into 3p 0 in the vacuum region. The two horizon-tal lines define the mass window for h candidates. The insertedplot includes the h events produced by beam interactions in

Ž .elements of the beamline see Fig. 3 and Fig. 4 before theallowed fiducial decay volume.

the K and K targets, respectively. The COGL S

distribution for the K ™3p 0 events is consistentL

with simulations made for elastic and quasi-elasticinteractions in the AKS and beam cleaning collima-tors shown in Fig. 3.

Since the h has a very short lifetime, its decayvertex practically coincides with the position at whichit was produced. Therefore, the expected R0 signa-ture is an h with high-P in the vacuum region rightT

after the last collimators and the AKS counter. Thefiducial region begins around 6 m downstream of theK target, and ends at around 96 m downstream atS

the Kevlar window, see Fig. 1. The vertex resolutionfor h™3p 0 events is about 70 cm. Therefore, inorder to reduce the background from h’s producedin the collimators and the AKS counter, only eventswith a vertex which is at least 200 cm away from theAKS counter position were accepted. As shown inFig. 2, there are three events that survive all theabove cuts and they are in the mass window of "6MeV. of the mass resolution for 3p 0 events. Thevertex of the most downstream event is consistentwith the position of the Kevlar window, and there-fore is excluded from the analysis. The two remain-ing events are identified as one particle coming fromthe K Be target and the other one from the K BeL S

target, by comparing the time of the event as defined


Fig. 3. Schematic drawing of the last set of beam collimators for the K and K beamline.S L

by the LKr system and the K proton tagging systemSw x7 . A time difference smaller than 1.5 ns is requiredfor an association with the K target. We assign oneS

event produced in the K beamline and the otherL

one in the K beamline. The tagging system has anS

efficiency greater than 99.9%, but the rate of protonsin the tagger is of the order of 30 MHz, which givesa probability greater than 10% of having an eventfrom the K beamline identified as coming from theL

K due to an accidental coincidence.S

The main background is due to diffractive neutroninteractions in the remaining air in the 6–9=10y5

mbar vacuum region. There are also about 109 pho-tonsrburst coming from the K beamline, but theyL

do not contribute to the background because theirmean energy is only 30 GeV. The expected meanenergy for the neutrons in the K and K beamlinesL S

is around 190 GeV and 100 GeV, and the expectedrates are 2=108 and 1.5=104 per burst, respec-tively.

The background estimates are based on a specialrun taken with a charged 75 GeV pion beam wherepyN ™ hX events were recorded, and whereŽ y .s p N™hX was measured from eight hours of

data taking in which 107 pionsrburst hit a 6 cmthick CH -target at the nominal SPS cycle time. In2

these data only h™3p 0 events with h energiesabove 95 GeV had trigger requirements similar tothose of the high-P 3p 0 trigger. For this reason, asT

already mentioned a minimum energy requirement of95 GeV in the analysis was applied for the eventsshown in Fig. 2. However, we do not see additionalhigh-P h events if this requirement is relaxed.T

The estimates for high-P h production in inter-T

actions of neutrons in the vacuum tank are foundŽ yfrom the ratios of cross sections s p N ™

. Ž .hX rs nN™hX . We find that our sample shouldŽ y5 .contain about 0.4 and 1.0= 10 h’s in the K L

and the K beamline, respectively. The backgroundS

estimates for the K beamline can be cross checkedS


using the h events produced in the collimators andŽ . w xthe 2 mm Iridium crystal AKS 8 . The AKS is

located around 6 m after the K target, and is usedS

to detect K decaying before this point. As shown inS

the inserted plot in Fig. 2 and in more detail in Fig.4, there are 31 events produced in the region of theAKS and collimators. According to the tagging sys-tem 13 of them are in the K beamline whichS

implies an expected background of 0.4=10y5 h

events produced by all particles in that beamline.This is consistent with the p N estimates.

As discussed above, there is a 10% probabilitythat an event produced in the K beamline is as-L

signed to the K instead, and a 40% probability ofS

having an h event in our data sample that wasproduced by neutrons in the K beamline. As aL

consequence, the probability of having two K eventsL

and that one of them is tagged as from the K S

beamline is 1.2%. Therefore, we conclude that allevents, both in the K and the K beamline, areS L

consistent with background expectation.As shown in Fig. 5, the detector acceptance for

0Ž . 0h™3p BR,32% events produced from R ™hg

decaying within the fiducial volume, shows a strongdependence on the mass ratio between the R0 andthe g , rsm 0rm . At long lifetimes andror for˜ R g

ry14m rm the acceptance becomes almost in-h g

Fig. 4. Vertex distribution for h particles produced in the beamcollimators region, see Fig. 3. Based on the vertex position, wecan conclude that most of the events coming from the K S

beamline were produced at the AKS. As in the analysis, the zerovertex position is defined at the location of the AKS Iridiumcrystal.

Fig. 5. Total selection efficiency for several values of the massratio r s m 0 rm and R0 lifetime as a function of the R0 mass.R g

dependent of r. We have assumed that the R0 energyspectrum is the same as that of L production mea-

w xsured by this experiment 9 . The sensitivity to theenergy spectrum is weak, and it is seen only at shortlifetimes, where the sensitivity drops very quickly asa function of R0 mass.

The angular acceptance in the K and the KL S

beamline is 0.15 and 0.375 mrad, respectively. Themaximum difference in the angular acceptance be-

" "tween neutral kaons and p ,K ,p,p was found us-w xing the results from Ref. 10 and used as an upper

estimate of the expected difference in ‘collimator’acceptance between R0 and neutral kaons. We con-cluded that the R0 ‘collimator’ acceptance will besmaller than that of kaons by about 4% and 23% inthe K and the K beamline, respectively.L S

The expected interaction rate for R0 N is expectedto be between 10% to 100% of the pN cross sectionw x11 . This means that the ratio of the absorption

Fig. 6. Upper limits at 95% confidence level on the flux ratio ofR0 and K production in pq-Be interactions assuming a 100%L

branching ratio for this decay mode. A small improvement in theexclusion at small lifetime is obtained from K events.S


Fig. 7. Upper limits at 95% CL on the flux ratio between R0 andK production in p-Be interactions for r s2.2. The second plotL

shows the 10y6 contours limits given by the analysis presentedhere and by the KTeV collaboration. In both cases a 100%branching ratio in the analysed decay mode R0 ™hg was as-˜sumed.

probabilities in the Be target of R0 and kaons can bebetween 0.75 and 1.

It is more conservative to not apply a backgroundsubtraction, but to evaluate the limits based on onesignal event in each beam. In addition, it is assumedthat the branching ratio of R0 ™hg is equal to˜100%. After taking all the above in considerationand since most of our data comes from the K L

beamline, we first consider the ration of R0 and K L

fluxes. The resulting limits on the flux ratio betweenR0 ™hg and K production at a 95% CL are shown˜ L

Ž . Ž .in Fig. 6 a , b . The fall-off in the sensitivity for0 Ž .small R mass on the left side of plot a is due to

the loss in phase space for the h to be produced,Ž .while the drop in the right side in b is due to the

decrease in detected events as the lifetime increases.The numbers of expected K and K at the exitL S

of the last collimator are 2=107 and 2=102, re-spectively. This means that there are also enoughprotons in the K target in order to improve theS

limits at low lifetimes. The K data gives an upperS

bound of 10y5 and 10y4 for rs2.5 and rs1.3,respectively, based on one event.

These results can be summarized for rs2.2 byplotting the contours of the upper limits at 95% CL.These are shown in Fig. 7 where they are comparedwith the best limit at 90% CL from the direct search

0 q y w xfor R ™p p g by the KTeV Collaboration 12 .˜The two searches are complementary and not neces-sarily comparable because of the different decaymodes. Both analyses show their results by settingthe indicated branching ratio equal to 100%. The

two-body R0 decays are suppressed due to approxi-w xmate C invariance in SUSY QCD 2 , while the

three-body decays are not. The p 0, h and R0 haveCsq1, while Csy1 for photinos. Nevertheless,our limits on the R0rK flux ratio are stringent onL

the R0 production even if the branching ratio ofR0 ™hg is of the order of 10y2 .˜

Though R0 production cross sections are quitemodel dependent, and the theoretical uncertainties inthe estimates of the R0 branching ratio into R0 ™hg

and the productions cross sections of R0’s are ratherlarge. Nevertheless, the available perturbative QCD

w x 0calculations 13 imply that the R rK flux ratioL

goes as 0.14ey2.7mR0. This implies that R0s with lowmass are excluded by these results even if thebranching ratio for R0 ™hg is at the level of 1%, as˜shown in Fig. 7 for an R0 lifetime of 10y8.

In conclusion, limits are given on the upper val-ues for the R0rK flux ratio in a region of R0 massL

and lifetime between 1–5 GeV and 10y10–10y3 s,respectively. As shown in Fig. 6, depending on thevalue for the branching ratio of R0 ™hg , the 95%˜CL on the upper value on the R0rK flux ratioL

could be as low as 6=10y9 for a R0 with a mass of1.5 GeV, a lifetime of 6=10y9 s and rs2.5.

Acknowledgements

We would like to thank the technical staff of theparticipating Laboratories, Universities and affiliatedcomputing centres for their efforts in the construc-tion of the NA48 detector, their help in the operationof the experiment, and in the processing of the data.The Cambridge and Edinburgh groups thank the UKParticle Physics and Astronomy Research Councilfor financial support.

References

w x Ž .1 G. Farrar, Phys. Rev. D 51 1995 3904.w x Ž .2 G. Farrar, Phys. Rev. Lett. 76 1996 4111.w x3 NA48 Collaboration, G. Barr et al., Proposal for a precision

Measurement of eXre in CP Violating K 0 ™2p Decays,

CERNrSPSCr90-22, 1990.w x4 C. Binno, N. Doble, L. Gatignon, P. Grafstrom, H. Wahl,

The simultaneous Long- and Short-lived neutral kaon beamsŽ .for experiment NA48, CERN-SL-98-033 EA .


w x5 NA48 Collaboration, G. Barr et al., Nucl. Instr. and Meth. AŽ .370 1996 413.

w x Ž .6 B. Gorini et al., IEEE Trans. Nucl. Sci. 45 1998 1771; G.Fischer et al., Presented to Vienna Wire Chamber Confer-ence, 1998.

w x Ž .7 T. Beier et al., Nucl. Instr. and Meth. A 360 1995 390.w x8 C. Biino, Workshop on Channelling and Other Coherent

Crystal Effects at Relativistic Energies, Aarhus, Denmark,July, 1995.

w x9 NA48 Collaboration, V. Fanti et al., A Measurement of the

Transversal Polarization of L-Hyperons Produced in Inelas-tic pN-Reactions at 450 GeV Proton, submitted to Euro.Phys. Jour. C.

w x10 H.W. Atherton et al., CERN Yellow Report 80-07, 1980.w x Ž .11 S. Nussinov, Phys. Rev. D 57 1998 7006.w x Ž .12 KTeV Collaboration, J. Adams, Phys. Rev. Lett. 79 1997

4083.w x Ž .13 S. Dawson, E. Eichten, C. Quigg, Phys. Rev. D 37 1985

1581; C. Quigg, private communication; S.V. Somalwar,private communication.

21 January 1999


Exclusive charmless B hadronic decays into hX and hs

B. Tseng 1

Department of Physics, National Cheng-Kung UniÕersity, Tainan, 700 Taiwan, ROC

Received 28 July 1998; revised 15 November 1998Editor: H. Georgi

Abstract

Using the next-to-leading order QCD-corrected effective Hamiltonian, charmless exclusive nonleptonic decays of the Bs

meson into h or hX are calculated within the generalized factorization approach. Nonfactorizable contributions, which can be

parameterized in terms of the effective number of colors N eff for PP and VP decay modes, are studied in two differentcŽ . Ž eff . Ž eff . Ž eff . Ž .schemes: i the one with the ‘‘homogeneous’’ structure in which N f N f PPP f N are assumed, and iic 1 c 2 c 10

effŽ . effŽ . effŽ .the ‘‘heterogeneous’’ one in which the possibility of N VqA /N VyA is considered, where N VqA denotesc c cŽ .Ž . effŽ . Ž .Ž .the effective value of colors for the VyA VqA penguin operators while N VyA for the VyA VyA ones. Forc

Xeff Ž .Ž .processes depending on the N -stable a such as B ™ p ,r h ,h , the predicted branching ratios are not sensitive to thec i s Xeff Ž .factorization approach we choose. While for the processes depending on the N -sensitive a such as B ™vh , there is ac i s

wide range for the branching ratios depending on the choice of the N eff involved. We have included the QCD anomalycX X XŽ . Ž . Ž .effect in our calculations and found that it is important for B ™h h . The effect of the cc ™h mechanism is found tos

be tiny due to a possible CKM-suppression and the suppression in the decay constants except for the B ™fh decay withins

the ‘‘naive’’ factorization approach, where the internal W diagram is CKM-suppressed and the penguin contributionscompensate. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Motivation

Stimulated by the recent observations of the large inclusive and exclusive rare B decays by the CLEOw x w xCollaboration 1 , there is considerable interest in the charmless B meson decays 2 . To explain the abnormally

X w xlarge branching ratio of the semi-inclusive process B™h qX, several mechanisms have been advocated 3–6w xand some tests of these mechanisms have been proposed 7 . It is now generally believed that the QCD anomaly

w x X3–5 plays a vital role. The understanding of the exclusive B™h K , however, relies on several subtle points.w x Ž .Ž .First, the QCD anomaly does occur through the equation of motion 8,9 when calculating the SyP SqP

Xpenguin operator and its effect is found to reduce the branching ratio. Second, the mechanism of cc™h ,w xalthough proposed to be large and positive originally 10,11 , is now preferred to be negative and smaller than

w x w xbefore from a recent theoretical recalculation 12 and several phenomenological analyses 9,13 . Third, the



( )B. TsengrPhysics Letters B 446 1999 125–134126

Ž .Ž . Ž .running strange quark mass which appears in the calculation of the SyP SqP penguin operator, the SU 3breaking effect in the involved h

X decay constants and the normalization of the B™h ŽX . matrix elementinvolved raise the branching ratio substantially. Finally, nonfactorizable contributions, which are parametrized

eff X Ž w xby the N , gives the final answer for the largeness of exclusive B™h K. We refer the reader to 14,15 forc.details.

It is very interesting to see the impacts of these subtleties mentioned above on the the exclusive charmless Bs

decays to an hX or h. In addition to the essential and important QCD penguin contribution as discussed in

w x w x Ž .Ž .16,17 , it is found that the EW penguin contribution is important for some processes 18 , e.g. B ™ p ,r h,fs

which the QCD penguin does not contribute to. However, the effects of QCD anomaly on the B decay are nots

discussed in these earlier papers. This motivates us to consider the contributions of anomaly effects in the BsXdecays. Another interesting topic we would like to study is the importance of the the mechanism cc™h .

Besides, the running quark mass, the h ŽX . decay constant and the normalization of the matrix element involvingh ŽX . are carefully taken care of in this Letter.

2. Theoretical framework

We begin with a brief description of the theoretical framework. The relevant effective DBs1 weakHamiltonian is

10GF) u u ) c c )HH DBs1 s V V c O qc O qV V c O qc O yV V c O qh.c., 1Ž . Ž .Ž . Ž . Ýeff ub uq 1 1 2 2 cb cq 1 1 2 2 t b tq i i'2 is3

where qsd,s, andu cO s ub qu , O s cb qc ,Ž . Ž . Ž . Ž .Vy A VyA VyA VyA1 1

u cO s qb uu , O s qb cc ,Ž . Ž . Ž . Ž .Vy A VyA VyA VyA2 2

X X X XO s qb q q , O s q b q q ,Ž . Ž . Ž . Ž .Ž .Ý ÝVy A VqAVyA3Ž5. 4Ž6. a b b a Ž .VyA VqAVyAX Xq q

X X X X3 3X XO s qb e q q , O s q b e q q , 2Ž . Ž . Ž .Ž . Ž .Ž .Ý ÝVq A VyAVyA7Ž9. q 8Ž10. a b q b a2 2 Ž .VqA VyAVyA

X Xq q

Ž . Ž . Ž .with q q 'q g 1"g q . In Eq. 2 , O are QCD penguin operators and O are electroweak1 2 V " A 1 m 5 2 3 – 6 7 – 10Ž .penguin operators. C m are the Wilson coefficients, which have been evaluated to the next-to-leading orderi

Ž . w xNLO 19,20 . One important feature of the NLO calculation is renormalization-scheme and -scale dependenceŽ w x.of the Wilson coefficients for a review, see 21 . In order to ensure the m and renormalization scheme

independence for the physical amplitude, the matrix elements, which are evaluated under the factorizationhypothesis, have to be computed in the same renormalization scheme and renormalized at the same scale asŽ . w x ² :c m . However, as emphasized in 14 , the matrix element O is scale independent under the factorizationfacti

² Ž .:approach and hence it cannot be identified with O m . Incorporating QCD and electroweak corrections to theŽ .² Ž .: eff² : efffour-quark operators, we can redefine c m O m sc O , so that c are renormalization scheme andtreei i i i i

scale independent. Then the factorization approximation is applied to the hadronic matrix elements of theoperator O at tree level. The numerical values for ceff are shown in the last column of Table 1, wherei

Ž5. 2 2Ž . w xmsm m , L s225 MeV, m s170 GeV and k sm r2 are used 14 .b b MS t b

In general, there are contributions from the nonfactorizable amplitudes. Because there is only one single formŽ . Ž . Žfactor or Lorentz scalar involved in the decay amplitude of B D ™PP, PV decays P: pseudoscalar meson,

. eff w xV: vector meson , the effects of nonfactorization can be lumped into the effective parameters a 22 :i

1 1eff eff eff eff eff effa sc qc qx , a sc qc qx , 3Ž .2 i 2 i 2 iy1 2 i 2 iy1 2 iy1 2 i 2 iy1ž / ž /N Nc c

( )B. TsengrPhysics Letters B 446 1999 125–134 127

Table 1Numerical values of effective coefficients a at N eff s2,3,5,`, where N eff s` corresponds to aeff sceff. The entries for a , . . . ,a have toi c c i i 3 10

be multiplied with 10y4

eff eff eff effN s2 N s3 N s5 N s`c c c c

a 0.986 1.04 1.08 1.151

a 0.25 0.058 y0.095 y0.3252

a y13.9y22.6 i 61 120q18i 211q45.3i3

a y344y113i y380y120 i y410y127i y450y136 i4

a y146y22.6 i y52.7 22q18i 134q45.3i5

a y493y113i y515y121i y530y127i y560y136 i6

a 0.04y2.73i y0.7y2.73i y1.24y2.73i y2.04y2.73i7

a 2.98y1.37i 3.32y0.9 i 3.59y0.55i 48

a y87.9y2.73i y91.1y2.73i y93.7y2.73i y97.6y2.73i9

a y29.3y1.37i y13y0.91i y0.04y0.55i 19.4810

where ceff are the Wilson coefficients of the 4-quark operators, and nonfactorizable contributions are2 i,2 iy1

characterized by the parameters x and x . We can parametrize the nonfactorizable contributions by2 i 2 iy1eff w x eff Ž .defining an effective number of colors N , called 1rj in 23 , as 1rN ' 1rN qx . Different factorizationc c c

approach used in the literature can be classified by the effective value of colors N eff. The so-called ‘‘naive’’c

factorization discards all the nonfactorizable contributions and takes 1rN eff s1rN s1r3, whereas thec cw x eff‘‘large-N improved’’ factorization 24 drops out all the subleading 1rN terms and takes 1rN s0. Inc c c

principle, N eff can vary from channel to channel, as in the case of charm decay. However, in the energeticceff w x efftwo-body B decays, N is expected to be process insensitive as supported by data 25 . If N is processc c

independent, then we have a generalized factorization. In this paper, we will treat the nonfactorizableŽ .contributions with two different phenomenological ways: i the one with ‘‘homogenous’’ structure, which

Ž eff . Ž eff . Ž eff . Ž .assume that N f N f PPP f N , and ii the ‘‘heterogeneous’’ one, which considers thec 1 c 2 c 10effŽ . effŽ .possibility of N VqA /N VyA . The consideration of the ‘‘homogenous’’ nonfactorizable contribu-c c

w xtions, which is commonly used in the literature, have its advantage of simplicity. However, as argued in 14 ,due to the different Dirac structure of the Fierz transformation, nonfactorizable effects in the matrix elements ofŽ .Ž . Ž .Ž . Ž . Ž .VyA VqA operators are a priori different from that of VyA VyA operators, i.e. x VqA /x VyA .Since 1rN eff s1rN qx , theoretically it is expected thatc c

N eff VyA ' N eff f N eff f N eff f N eff f N eff f N eff ,Ž . Ž . Ž . Ž . Ž . Ž . Ž .c c c c c c c1 2 3 4 9 10

N eff VqA ' N eff f N eff f N eff f N eff . 4Ž . Ž .Ž . Ž . Ž . Ž .c c c c c5 6 7 8

Ž . Ž . w xTo illustrate the effect of the nonfactorizable contribution, we extrapolate N VyA f2 from B™Dp r 26c

to charmless decay. The N eff-dependence of the effective parameters a ’s are shown in Table 1, from which wec i

see that a ,a ,a and a are N eff-stable, and the remaining ones are N eff-sensitive. We would like to remark1 4 6 9 c ceff Ž .that a and a are N -sensitive separately, the combination of a ya is rather stable under the variation of3 5 c 3 5

the N eff within the ‘‘homogeneous’’ picture and is still sensitive to the factorization approach taken in thec

‘‘heterogeneous’’ scheme. This is the main difference between the ‘‘homogeneous’’ and ‘‘heterogeneous’’approach. While a ,a can be neglected, a ,a and a have some effects on the relevant processes depending7 8 3 5 10

on the choice of N eff.c

Before carrying out the phenomenological analysis, we would like to discuss the dynamical mechanismsw xinvolved. We first come to the QCD anomaly effect. As pointed out by 8,15 , the QCD anomaly appears

through the equation of motiona sm mn˜E sg g s s2m sig sq G G . 5Ž .Ž .m 5 s 5 mn4p


Neglecting the u and d quark masses in the equations of motion leads toa sX u 2˜² < < : X Xh GG 0 s f m 6Ž .h h4p

and hence

m X2hX s u² < < : X Xh sg s 0 syi f y f . 7Ž .Ž .5 h h2ms

To determine the decay constant f Xq , we need to know the wave functions of the physical h

X and h states whichh

Ž .are related to that of the SU 3 singlet state h and octet state h by0 8

hX sh sinuqh cosu , hsh cosuyh sinu , 8Ž .8 0 8 0

X' '< : Ž . < : < : Ž . < :with h s 1r 3 uuqddqss , h s 1r 6 uuqddy2 ss and ufy208. When the hyh mixing0 8X w xangle is y19.58, the h and h wave functions have simple expressions 2 :

1 1X< : < : < : < :h s uuqddq2 ss , h s uuqddyss . 9Ž .' '6 3

1u s Ž .X XAt this specific mixing angle, f s f in the SU 3 limit. Introducing the decay constants f and f byh h 8 02

² < 0 < : ² < 8 < :0 A h s if p , 0 A h s if p 10Ž .m 0 0 m m 8 8 m

u s w xX Xthen f and f are related to f and f by 31h h 8 0

f f f f8 0 8 0u sX Xf s sinuq cosu , f sy2 sinuq cosu . 11Ž .h h' ' ' '6 3 6 3

Likewise, for the h mesonf f f f8 0 8 0u sf s cosuy sinu , f sy2 cosuy sinu . 12Ž .h h' ' ' '6 3 6 3

X X w xFrom a recent analysis of the data of h,h ™gg and h,h ™pgg 32 , f and u have been determined to be8Ž0.

f f8 0 0s1.38"0.22, s1.06"0.03, usy 22.0"3.3 , 13Ž . Ž .f fp p

which lead to

f u s99 MeV, f s sy108 MeV, f Xu s47 MeV, f X

s s131 MeV. 14Ž .h h h h

w xFor the u and d quarks involved, we follow 14 and useX X X² < < : ² < < : ² < < :Xh ug u 0 s h dg d 0 sr h sg s 0 , 15Ž .5 5 h 5

with r ŽX . being given byh

2 2 2 2' '( (2 f y f 2 f y fcosuq 1r 2 sinu 1 cosuy 2 sinuŽ .0 8 0 8Xr s , r sy . 16Ž .h h2 2 2 2' '2cosuy 2 sinu cosuq 1r 2 sinuŽ .( (2 f y f 2 f y f8 0 8 0

XŽ .We next discuss the cc™h mechanism. This new internal W-emission contribution will be importantwhen the mixing angle involved is V V ) , which is as large as that of the penguin amplitude and yet itscb cs

eff Žcc.Xeffective parameter a is larger than that of penguin operators. The decay constant f , defined as2 h

X Žcc.² < < : w xX0 cg g c h s if q , has been determined from the theoretical calculations 10–12 and the phenomenolog-m 5 h mX X w xical analysis of the data of Jrc™h g , Jrc™h g and of the hg and h g transition form factors 9,13 . In thec

presence of the charm content in the h , an additional mixing angle u neededs to be introduced:0 c

1 1< : < : < : < : < : < :h s cosu uuqddqss qsinu cc , h sy sinu uuqddqss qcosu cc . 17Ž .0 c c c c c' '3 3


Then f Xc scosu tanu f and f c sysinu tanu f , where the decay constant f can be extracted fromh c h h c h hc c cX w xh ™gg , and u from Jrc™h g and Jrc™h g 9 . In the present paper we shall usec c c

f Xc sy6 MeV, f c sytanu f X

c sy2.4 MeV, 18Ž .h h h

for usy228, which are very close to the values

f Xc sy 6.3"0.6 MeV, f c sy 2.4"0.2 MeV 19Ž . Ž . Ž .h h

w xobtained in 13 .In the following we will show the input parameters we used. One of the important parameters is the running

Ž .Ž .quark mass which appears in the matrix elements of SyP SqP penguin operators through the use ofequations of motion. The running quark mass should be applied at the scale m;m because the energy releaseb

w xin the energetic two-body charmless decays of the B meson is of order m . In this paper, we use 27b

m m s3.2 MeV, m m s6.4 MeV,Ž . Ž .u b d b

m m s105 MeV, m m s0.95 GeV, m m s4.34 GeV, 20Ž . Ž . Ž . Ž .s b c b b b

in ensuing calculation, where we have applied m s150 MeV at ms1 GeV.s

It is convenient to parametrize the quark mixing matrix in terms of the Wolfenstein parameters: A, l, r andw xh, where As0.81 and ls0.22 28 . A recent analysis of all available experimental constraints imposed on the

w xWolfenstein parameters yields 29

rs0.156"0.090 , hs0.328"0.054, 21Ž .2 2l lŽ . Ž .where rsr 1y and hsh 1y , and it implies that the negative r region is excluded at 93% C.L.. In2 2

this paper, we employ the representative values: rs0.16 and hs0.34, which satisfies the constraint2 2(r qh s0.37.Under the factorization approach, the decay amplitudes are expressed as the products of the decay constants

w xand the form factors. We use the standard parametrization for decay constants and form factors 23 . For valuesof the decay constants, we use f s132 MeV, f s160 MeV, f s210 MeV, f ) s221 MeV, f s195 MeVp K r K v

and f s237 MeV. Concerning the heavy-to-light mesonic form factors, we will use the results evaluated in thef

w xrelativistic quark model 23,30 , by directly calculating B ™P and B ™V form factors at time-likesŽu,d . sŽu,d .B hs sŽ .momentum transfer. Denoting h sss, the explicit values for the form factors involved are F 0 s0.48,s 1,0

BshXsŽ . Bs K ) Ž . w x 2F 0 s0.44, and A 0 s0.28, which are larger than BSW model’s results 16 . The q dependence ofŽ0,1. 0

the matrix element, parametrized under the pole dominance ansatz, are found to have a dipole behaviour forA ,F , and a monopole one for F . In the following, we will use the exact value calculated at the relevant0 1 0

kinematical point in this paper. Note that these matrix elements should be used with a correct normalizationX XB h B h B h B hs s s s s s' 'w x Ž . Ž . Ž . Ž . Ž . Ž .14 , for which to a good approximation, we take F 0 s y1r 3 F 0 and F 0 s 2r 6 F 0 .1,0 1,0 1,0 1,0

3. Phenomenology

We are now ready to discuss the phenomenology of exclusive charmless rare B decays. To illustrate theseff Ž .issue of N dependence which means the different factorization approach of theoretical predictions, we willc

0 ŽX . ŽX . ŽX .Ž . Ž . Ž .begin with B ™ p ,r,v h . Unlike the case of B™ p ,r h , B ™ p ,r,v h do not receive the anomalys sŽ .Ž . Ž .contribution from the SyP SqP penquin operators due to the particle content of B and p r,v . Thes0 ŽX .decay amplitude for B ™ph readss

G XX .ŽF 3Ž . ) ) ŽB h ,p .sA B ™h p s V V a yV V ya qa X , 22Ž . Ž .� 4Ž .s ub u s 2 t b t s 7 9 u2'2

whereX f X. X .Ž ŽpŽB h ,p . 0 Ž . 2 2 Bh 2s X² < < :² < < : .ŽX ' p uu 0 h sb B syi m ym F m . 23Ž . Ž . Ž .Ž .Ž .Vy A VyA s B h 0 ps'2


Since the internal W-emission is CKM-suppressed and the QCD penguins are canceled out in these decayŽ . ŽX .modes, B ™p r h are dominated by the EW penguin diagram. The dominant EW penguin contributions

proportional to a is N eff-stable, whereas the internal W contribution a is N eff-sensitive. Within the9 c 2 c

‘‘heterogeneous’’ nonfactorizable picture, a is fixed and thus the predicted branching ratio is rather stable2

under the variation of N eff as shown in the last four columns in Table 2. However, a varies within thec 2

‘‘homogeneous’’ nonfactorizable scheme and thus the predicted branching ratios do show the N eff dependence.cŽ . ŽX .We would like to emphasize that although B ™p r h are dominated by the EW penguin diagram, thes

internal W diagram makes some contributions to this decay mode. Since a changes sign from N eff s2,3 to2 c

N eff s5,`, the interference pattern between the internal W diagram and the EW penguin diagram will changec

from the destructive to the constructive one. It is thus easy to see that for N eff s2 there is a larger destructivec

interference between the internal W diagram and the EW penguin contribution and the predicted branching ratiois the smallest one among the first four columns in Table 2, whereas for N eff s` constructive interference takesc

the role and the branching ratio increases.Ž . ŽX . ŽX .While QCD penguin diagrams are canceled out in B ™ p ,r h , B ™vh gets enhanced from the QCDs s

penguin diagram. The decay amplitude for B ™vh ŽX . iss

G XX .ŽF 1Ž . ) ) ŽB h ,v .sA B ™vh s V V a yV V 2 a qa q a qa X , 24Ž . Ž . Ž .� 4Ž .s ub u s 2 t b t s 3 5 7 9 u2'2

whereX X. X .Ž ŽŽB h ,v . Ž . Bh 2s '² < < :² < < :X ' v uu 0 h sb B s 2 f m F m ´Pp . 25Ž . Ž . Ž . Ž .Ž .Vy A VyA s v v 1 v Bs

From Table 2, we see that there is a wide range of predictions for the branching ratios. This process is QCDŽ effŽ . effŽ .. Ž .penguin dominated, except when the ‘‘naive’’ factorization is used or N VyA , N VqA s 2,5 wherec cŽ .there are large cancellations between the QCD penguin contributions i.e. a q a . The largest branching3 5

ration predicted for B ™vh ŽX . occurs when we use the ‘‘large-N improved’’ factorization, where the EWs c

penguin and QCD penguin have constructive interference.

Table 2X Ž y6 . 2 2Average Branching ratios for charmless B decays to h and h in units of 10 . Predictions are for k s m r2, hs0.34, r s0.16. Is b

effŽ . effŽ . Ž . effdenotes the ‘‘homogeneous’’ nonfactorizable contributions i.e. N V y A s N V q A and a,b,c,d represent the cases for N sc c cŽ . effŽ . effŽ . Ž X X X .`,5,3,2 . II denotes the ‘‘heterogeneous’’ nonfactorizable contributions, i.e. N V y A / N V q A and a ,b ,c represent the casesc c

effŽ . Ž . effŽ . Ž .for N V q A s 3,5,` , where we have fixed N V y A s2 see the textc c

X X XDecay I I I I II II IIa b c d a b c

XB ™ph 0.25 0.17 0.13 0.11 0.11 0.11 0.10s

B ™ph 0.16 0.11 0.08 0.07 0.07 0.068 0.067sXB ™ rh 0.70 0.47 0.36 0.30 0.30 0.30 0.31s

B ™ rh 0.45 0.30 0.24 0.19 0.19 0.19 0.20sXB ™ vh 6.9 0.9 0.012 2.14 0.48 0.03 0.83s

B ™ vh 4.45 0.63 0.008 1.39 0.31 0.02 0.54sX 0B ™h K 1.25 1.07 1.01 1.00 1.27 1.51 1.90s

0B ™hK 1.35 0.81 0.68 0.76 0.75 0.74 0.72sX

) 0B ™h K 0.49 0.35 0.32 0.26 0.49 0.60 0.80s) 0B ™hK 0.45 0.05 0.02 0.24 0.24 0.24 0.25s

XB ™hh 47.4 41.8 38.3 34.4 39.5 44.1 51.5sX XB ™h h 26.6 24.9 23.8 22.4 33.8 43.9 62.2s

B ™hh 20.3 17.1 15.1 12.8 11.6 10.7 9.1sXB ™fh 0.44 0.59 2.29 6.20 4.41 3.11 1.66s

B ™fh 0.04 0.91 2.29 4.92 2.28 0.92 0.10s


X0 0Next, we discuss B ™h K decay, which has the decay amplitudes

G X XFX0 0 ) B K ,h ) B K ,hŽ . Ž .s sA B ™K h s V V a X qV V a XŽ .s ub ud 2 u cb cd 2 c½'2

2m XK1) B h , KŽ .syV V a y a q 2 a ya XŽ .t b td 4 10 6 82ž /m qm m ymŽ . Ž .s d b d

X1 1

XŽB K ,h . ŽB K ,h .s sq a ya ya qa X q a ya q a y a XŽ . Ž .3 5 7 9 u 3 5 7 9 s2 2

X1 1 1ŽB K ,h .sq a ya ya qa X q a qa ya q a y a y aŽ .3 5 7 9 c 3 4 5 7 9 102 2 2ž

2 sX XXm fh h1 ŽB K ,h .sXq a y a y1 r X , 26Ž .Ž .6 8 h d2 u 5ž /X /m m ym fŽ .s b s h

whereX X

XŽB h , K . 0 0 2 2 B h 2s sX² < < :² < < :X s K sd 0 h db B syif m ym F m ,Ž . Ž . Ž .Ž .Vy A VyA s K B h 0 Ks

XqXŽB K ,h . 0 0 2 2 B K 2s sX X² < < :² < < :X s h qq 0 K sb B syif m ym F m . 27Ž . Ž . Ž .Ž .Ž .Vy A VyAq s h B K 0 hs

Ž s u . ŽBs K ,hX .X XDue to the QCD anomaly, there is an extra f rf y1 term multiplied with a in X , which ish h 6 d

Ž .Ž .necessary in order to be consistent with the chiral-limit behaviour of the SyP SqP penguin matrixw xelements 14 . Though penguin diagrams play the dominant role, the internal W diagram and the mechanism of

X eff effthe cc pair into the h do have some nonnegligible effects when N s2 and N s` where a gets the largerc c 2

values. Due to the large cancellation, the EW penguin has only tiny effect and can be neglected. The monotonicdecrease of the branching ratio from N eff s` to N eff s2 within the ‘‘homogeneous’’ nonfactorizable picturec c

can be understood from the behaviour of the QCD penguin i.e. the destructive interference between a andŽ3,5.a : as N eff decreases, a contributions increase and hence the branching ratios decrease. As we alreadyŽ4,6. c Ž3,5.mentioned before, a and a are N eff-sensitive while a ya is stable under the variation of N eff and then the3 5 c 3 5 c

predicted branching ratio is N eff-stable within the ‘‘homogeneous’’ factorization approach.c

There exist some general rules for the derivation of the formula from B™P P to its correspondinga bŽ . ŽBPa, Pb. ŽBVa, Pb.B™V P and B™P V . These general rules can be written as: i For X to X , replace the terma b a b

2 wŽ .Ž .x 2 wŽ .Ž .x Ž .m r m qm m ym by ym r m qm m qm and the index P by V , ii discard theP 1 2 3 4 P 1 2 3 4 a ab b

Ž .Ž . ŽBs Pa,Vb.SyP SqP contribution associated with X and a ™ya if they contribute. Thus, TheŽ5,7. Ž5,7.X

)Xfactorizable amplitude of B ™h K can be obtained from the B ™h K one and readss s

XG X) )FX ) 0 ) B K ,h ) ŽB K ,h .Ž .s sA B ™h K s V V a X qV V a XŽ .Ž .s ub ud 2 u cb cd 2 c½'2

X)1 1 1 1 1) B h , KŽ .syV V a y a X q a qa ya q a y a y a y a y aŽ . Ž .t b td 4 10 3 4 5 7 9 10 6 82 2 2 2 2ž

=m2 X f Xs

X X .) ) Žh h ŽB K ,h . ŽB K ,h .s sXy1 r X q a ya ya qa XŽ .h d 3 5 7 9 uuXž / /m m qm fŽ .s b s h

X X. .) Ž ) Ž1 1 ŽB K ,h . ŽB K ,h .s sq a ya q a y a X q a ya ya qa X , 28Ž . Ž .Ž .3 5 7 9 s 3 5 7 9 c2 2 5


withŽX . )

X ŽX .ŽB h , K . )y Ž . Bh 2s ² < < :² < < :) ) )X ' K su 0 h ub B s2 f m F m ´Pp ,Ž . Ž . Ž .Ž .Vy A VyA s K K 1 K Bs

) ŽX . X)ŽB K ,h . Ž . )y q BK 2s X X² < < :² < < : . .Ž ) ŽX ' h qq 0 K sb B s2 f m A m ´Pp . 29Ž . Ž . Ž . Ž .Ž .Vy A VyAq s h K 0 h Bs

It is interesting to see that since there is no a term in X ŽBs K ) ,h X ., the penguin contribution is reduced6

substantially and so does the branching ratio. With a reduced penguin contribution, the involved internal WX Xdiagram and the mechanism of the cc pair into the h become more important than those of the B ™Kh . Thes

larger branching ratios in columns denoted by I and II are due to the constructive interference between thea b

internal W diagram and the penguin contribution.XWe are now coming to the most complicated process B ™hh , which has the decay amplitudes

G X X X XFX) B h ,h B h ,h ) B h ,h B h ,hŽ . Ž . Ž . Ž .s s s sA B ™hh s V V a X qa X qV V a X qa XŽ . Ž . Ž .s ub u s 2 u 2 u cb cs 2 c 2 c½'2

2 uX Xm fh h1 1 1 1)yV V a qa ya q a y a y a q a y a 1yŽ .t b t s 3 4 5 7 9 10 6 82 2 2 2 sž /Xž /m m ym fŽ .s b s h

=X 1 1 X XŽB h ,h . ŽB h ,h . ŽB h ,h .s s sX q 2 a y2 a y a q a X q a ya ya qa XŽ .Ž .s 3 5 7 9 u 3 5 7 9 c2 2

m2 f uXh h1 1 1 1 ŽB h ,h .sq a qa ya q a y a y a q a y a 1y XŽ .3 4 5 7 9 10 6 8 s2 2 2 2 sž /ž /m m ym fŽ .s b s h

X X1 1 ŽB h ,h . ŽB h ,h .s sq 2 a y2 a y a q a X q a ya ya qa X , 30Ž . Ž .Ž .3 5 7 9 u 3 5 7 9 c2 2 5with

X XŽB h ,h . q 2 2 B h 2s s² < < :² < < : X XX ' h qq 0 h sb B syif m ym F m ,Ž . Ž . Ž .Ž .Vy A VyAq s h B h 0 hsX XXŽB h ,h . q 2 2 B h 2s s² < < :² < < : XX ' h qq 0 h sb B syif m ym F m .Ž . Ž . Ž .Ž .Vy A VyAq s h B h 0 hs

X X XŽB h ,h . ŽB h ,h . Ž .s sThe destructive interference between X and X makes the internal W-emission, cc™h and theqs Žu,c. qsŽu,c.corresponding penguin contributions smaller. The EW penguin is smaller than the QCD penguin by an order ofmagnitude at the amplitude level and can be neglected. The dominant QCD penguin contributions are governed

ŽBsh,hX . ŽBshX ,h . Ž .by X and X , which have the constructive interference. The a ya term is positive, contrary tos s 3 5

the negative a and a terms, and becomes smaller when N eff increases within the the ‘‘homogeneous’’4 6 c

nonfactorizable structure. Thus a monotonic increase of the branching ratio when N eff increases comes mainlyc

from this reduced, destructive interference, within the the ‘‘homogeneous’’ nonfactorizable structure. Similararguments are also valid for the ‘‘heterogeneous’’ structure.

XŽ .Ž . Ž .With the general rules i and ii mentioned before, the decay amplitude for B ™fh can be easilysXobtained from B ™hh :s

XG XX Ž .F Ž .Ž . ) B f ,h ) B f ,hŽ .Ž .s sA B ™fh s V V a X qV V a XŽ .s ub u s 2 u cb cs 2 c½'2

2 uXXŽ . Ž .m fh h1 1 1 1)yV V a qa ya q a y a y a y a y a 1yŽ .t b t s 3 4 5 7 9 10 6 82 2 2 2 s Xž /.ž /Žm m qm fŽ .s b s h

=ŽX . 1 1 ŽX . ŽX .ŽB f ,h . ŽB f ,h . ŽB f ,h .s s sX q 2 a y2 a y a q a X q a ya ya qa XŽ .Ž .s 3 5 7 9 u 3 5 7 9 c2 2

X .Ž1 1 1 ŽB h ,f .sq a qa qa y a y a y a X , 31Ž .Ž .3 4 5 7 9 102 2 2 5


with

ŽX . XŽB f ,h . Ž . q B f 2s s X² < < :² < < : X .ŽX ' h qq 0 f sb B s2 f m A m ´Pp ,Ž . Ž . Ž .Ž .Vy A VyAq s h f 0 h Bs

ŽX . X ŽX .ŽB h ,f . Ž . B h 2s s² < < :² < < :X ' f qq 0 h sb B s2 f m F m ´Pp .Ž . Ž . Ž .Ž .Vy A VyAq s f f 1 f Bs

XŽ .While the internal W diagram is subject to the CKM-suppression, the cc™h mechanism suffers from theXŽ .suppression in the decay constant and thus B ™fh is dominated by the penguin contribution. Due to thes

Ž .cancellation among the different a is3,4,5,6 ’s, effect of the QCD penguin, though still dominant, arei

reduced substantially. Within the ‘‘homogeneous’’ nonfactorizable picture, we find a monotonic decrease of thebranching ratios when N eff increases, which comes from a monotonic decrease of the QCD penguinc

contributions as N eff increases. Since the QCD penguin contributions are reduced, the EW penguin contribu-c

tions become important. It is found that the interference pattern between the QCD and EW penguin isdestructive except for the ‘‘large-N improved’’ factorization approach, where a constructive interference existsc

and a very dramatically suppressed QCD penguin contribution appears. The strength of the destructiveinterference depends on N eff and its effect is to reduce the QCD penguin contribution without changing thec

trend of reduced penguin when N eff increases. A dramatic destructive interference among the penguincXŽ . eff eff eff effŽ . Ž . Ž Ž . Ž ..contributions occurs for B ™fh when N VyA sN VqA s` and N VyA , N VqA ss c c c cXŽ .Ž .2,` , thus the internal W diagram and the cc™h mechanism contribution becomes important relative to theother cases and the branching ratio is the smallest in these situations.

4. Summary and discussions

To summarize, we have studied charmless exclusive nonleptonic B meson decay into an h or hX within thes

generalized factorization approach. Nonfactorizable contributions are parametrized in terms of the effectivevalue of colors N eff and predictions of the different factorization approach are shown with the N eff dependence.c cXeff Ž .Ž .It is found that for processes depending on the N -stable a ’s such as B ™ p ,r h , the branching ratios arec i s

not sensitive to the factorization approach we used. While for the processes depending on the the N eff-sensitivecXŽ .a ’s such as the B ™vh , the predicted branching ratios have a wide range depending on the choice of thei s

factorization approach. The effect of the QCD anomaly, which is not discussed in the earlier literature, is foundX X XŽ . Ž . Ž .to be important for the B ™h h . We also found that the mechanism cc ™h , in general, has smallers

effects due to a possible CKM-suppression and the suppression in the decay constants except for the B ™fhs

under the ‘‘large-N improved’’ factorization approach, where the internal W diagram is CKM-suppressed andc

the penguin contributions compensate.In this Letter, we, following the standard approach, have neglected the W-exchange and the space-like

penguin contributions. Another major source of uncertainties comes from the form factors we used, which arelarger than the BSW model’s calculations. Although the CKM matrix element ranges from the negative r topositive one, we have ‘‘fixed’’ it to our representative values. The interference pattern between the internal Wdiagram and the penguin contributions will change when we take the different sign of r. We will study theseform factor- and CKM- dependence involved and all the B ™PP,VP,VV decay modes in a separates

publication.

Acknowledgements

We are very grateful to Prof. H.Y Cheng for helpful discussions. This work is supported in part by theNational Science Council of the Republic of China under Grant NSC87-2112-M006-018.


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21 January 1999


Large N-wormhole approach to spacetime foam

Remo Garattini 1

Mecanique et GraÕitation, UniÕersite de Mons-Hainaut, Faculte des Sciences, 15 AÕenue Maistriau, B-7000 Mons, Belgium´ ´ ´( )Facolta di Ingegneria, UniÕersita degli Studi di Bergamo, Viale Marconi, 5, 24044 Dalmine Bergamo , Italy` `

Received 6 August 1998Editor: L. Alvarez-Gaume

Abstract

A simple model of spacetime foam, made by N wormholes in a semiclassical approximation, is taken under examination.We show that the qualitative behaviour of the fluctuation of the metric conjectured by Wheeler is here reproduced. q 1999Published by Elsevier Science B.V. All rights reserved.

1. Introduction

One of the most fascinating problem of our century is the possibility of combining the principles of QuantumMechanics with those of General Relativity. The result of this combination is best known as Quantum Gravity.However such a theory has to be yet developed, principally due to the UV divergences that cannot be kept under

w xcontrol by any renormalization scheme. J.A. Wheeler 1 was the first who conjectured that fluctuations of themetric have to appear at short scale distances. The collection of such fluctuations gives the spacetime a kind offoam-like structure, whose topology is constantly changing. In this foamy spacetime a fundamental lengthcomes into play: the Planck length. Its inverse, the Planck mass m , can be thought as a natural cut-off. It isp

believed that in such spacetime, general relativity can be renormalized when a density of virtual black holes isw xtaken under consideration coupled to N fermion fields in a 1rN expansion 2 . It is also argued that when

gravity is coupled to N conformally invariant scalar fields the evidence that the ground-state expectation valuew xof the metric is flat space is false 3 . However instead of looking at gravity coupled to matter fields, we will

consider pure gravity. In this context two metrics which are solutions of the equations of motion without acosmological constant are known with the property of the spherical symmetry: the Schwarzschild metric and theFlat metric. We will focus our attention on these two metrics with the purpose of examining the energycontribution to the vacuum fluctuation generated by a collection of N coherent wormholes. A straightforwardextension to the deSitter and the Schwarzschild-deSitter spacetime case is immediate. The paper is structured as

w xfollows, in Section 2 we briefly recall the results reported in Ref. 5 , in Section 3 we generalize the result ofSection 2 to N wormholes. We summarize and conclude in Section 3.w



( )R. GarattinirPhysics Letters B 446 1999 135–142136

2. One wormhole approximation

The reference model we will consider is an eternal black hole. The complete manifold MM can be thought ascomposed of two wedges MM and MM located in the right and left sectors of a Kruskal diagram whose spatialq yslices S represent Einstein-Rosen bridges with wormhole topology S2 =R1. The hypersurface S is divided intwo parts S and S by a bifurcation two-surface S . We begin with the line elementq y 0

dr 22 2 2 2 2 2 2ds syN r dt q qr du qsin u df 1Ž . Ž .Ž .2m

1yr

and we consider the physical Hamiltonian defined on S

13 i

q yH sHyH s d x NHHqN HH qH qHŽ .HP 0 i ES ES2l Sp

1 2 23 i 2 0 2 0' 's d x NHHqN HH q d x s kyk y d x s kyk , 2Ž . Ž . Ž .Ž .H H Hi2 2 2l l lS S Sp p pq y

where l 2 s16p G. The volume term contains two constraintsp

2° 'l gp Ž .i j k l 3HHsG p p y R s0i jk l 2~ ž / ž /l'g , 3Ž .p

i i j¢HH sy2p s0< j

1 Ž .3where G s g g qg g yg g and R denotes the scalar curvature of the surface S. By using theŽ .i jk l i k jl i l jk i j k l2

expression of the trace

1m'ksy h n , 4Ž .Ž . ,m'h

with the normal to the boundaries defined continuously along S as n m s h y y 1r2d m. The value of k dependsŽ . y

on the function r , where we have assumed that the function r is positive for S and negative for S . We, y , y q yobtain at either boundary that

y2 r, yks . 5Ž .

r

The trace associated with the subtraction term is taken to be k 0 sy2rr for B and k 0 s2rr for B . Thenq ythe quasilocal energy with subtraction terms included is

E sE yE s r 1y r y r 1y r . 6Ž .Ž . Ž .quasilocal q y , y , yysy ysyq y

Note that the total quasilocal energy is zero for boundary conditions symmetric with respect to the bifurcationsurface S and this is the necessary condition to obtain instability with respect to the flat space. A little comment0

on the total Hamiltonian is useful to further proceed. We are looking at the sector of asymptotically flat metricsincluded in the space of all metrics, where the Wheeler-DeWitt equation

HHCs0 7Ž .is defined. In this sector the Schwarzschild metric and the Flat metric satisfy the constraint equations 3 . HereŽ .we consider deviations from such metrics in a WKB approximation and we calculate the expectation value

( )R. GarattinirPhysics Letters B 446 1999 135–142 137

following a variational approach where the WKB functions are substituted with trial wave functionals. Then theHamiltonian referred to the line element 1 isŽ .

2 'l gp Ž .3 i j k l 3Hs d x G p p y R .H i jk l 2ž / ž /l'gS p

Instead of looking at perturbations on the whole manifold MM, we consider perturbations at S of the typeŽ . w xg sg qh . g is the spatial part of the background considered in Eq. 1 In Ref. 5 , we have defined DE mŽ .i j i j i j i j

as the difference of the expectation value of the Hamiltonian approximated to second order calculated withrespect to different backgrounds which have the asymptotic flatness property. This quantity is the naturalextension to the volume term of the subtraction procedure for boundary terms and is interpreted as the Casimirenergy related to vacuum fluctuations. Thus

Schw . Flat² :C H yH C C H C² :quasilocalDE m sE m yE 0 s q . 8Ž . Ž . Ž . Ž .

< <² : ² :C C C C

By restricting our attention to the graviton sector of the Hamiltonian approximated to second order, hereafterreferred as H , we define<2

H 1 HC H C¦ ;<2E s ,<2 H H<² :C C

where

1 HH H y1² :C sC h sNNexp y gyg K gyg .Ž . Ž . x , yi j 2½ 54 lp

After having functionally integrated H , we get<2

1 a3 i jk l y1H H'H s d x g G K x , x q ^ K x , x . 9Ž . Ž . Ž . Ž .i jk lH j i ak l<2 224 l Sp

The propagator K H x , x comes from a functional integration and it can be represented asŽ . i ak l

hH x hH yŽ . Ž .i a k lHK x , y :s , 10Ž . Ž .Ýi ak l 2l pŽ .NN

where hH x are the eigenfunctions ofŽ .i a

a a a^ :sy^d q2 R . 11Ž . Ž .j2 j j

This is the Lichnerowicz operator projected on S acting on traceless transverse quantum fluctuations and l pŽ .NŽ .are infinite variational parameters. ^ is the curved Laplacian Laplace-Beltrami operator on a Schwarzschild

background and Ra is the mixed Ricci tensor whose components are:j

y2m m maR sdiag , , . 12Ž .j 3 3 3½ 5r r r


After normalization in spin space and after a rescaling of the fields in such a way as to absorb l 2, E becomesp <2

in momentum space

2` 2 `V E p ,m ,lŽ .i2E m ,l s dpp l p q , 13Ž . Ž . Ž .Ý Ý H<2 i2 l p2p Ž .0 ils0 is1

where

l lq1 3mŽ .2 2E p ,m ,l sp q . 14Ž . Ž .1,2 2 3r r0 0

and V is the volume of the system. r is related to the minimum radius compatible with the wormhole throat.0

We know that the classical minimum is achieved when r s2m. However, it is likely that quantum processes0

come into play at short distances, where the wormhole throat is defined, introducing a quantum radius r )2m.02 Ž .(The minimization with respect to l leads to l p ,l ,m s E p ,m ,l and Eq. 13 becomesŽ . Ž .i i

` 2 `V2 2(E m ,l s2 dpp E p ,m ,l , 15Ž . Ž . Ž .Ý Ý H<2 i22p 0ls0 is1

Ž .2 l lq1 3mwith p q ) . Thus, in presence of the curved background, we get2 3r r0 0

` `V 12 2 2 2 2( (E m s dpp p qc q p qc , 16Ž . Ž .Ý H ž /<2 y q2 22p 0ls0

where

l lq1 3mŽ .2c s . ,. 2 3r r0 0

Ž .2 l lq1while when we refer to the flat space, we have ms0 and c s , with2r0

` `V 12 2 2(E 0 s dpp 2 p qc . 17Ž . Ž .Ý H ž /<2 2 22p 0ls0

Since we are interested in the UV limit, we will use a cut-off L to keep under control the UV divergence

` dp dx LLrc; ; ln , 18Ž .H H ž /p x c0 0

where LFm . Note that in this context the introduction of a cut-off at the Planck scale is quite natural if wep

look at a spacetime foam. Thus DE m for high momenta becomesŽ .2 3 2V 3m 1 r L0

DE m ;y ln . 19Ž . Ž .2 3 ž /ž / 16 3m2p r0

&We now compute the minimum of DE m sE 0 yE m syDE m . We obtain two values for m: m s0,Ž . Ž . Ž . Ž . 1& & 4V L2 y1r2 3i.e. flat space and m sL e r r3. Thus the minimum of DE m is at the value DE m s . RecallŽ . Ž . 22 0 2 64p e

that msMG, thus

MsGy1L2ey1r2 r 3r3. 20Ž .0


ŽWhen L™m , then r ™ l . This means that an Heisenberg uncertainty relation of the type l m s1 inp 0 p p p.natural units has to be satisfied, then

mp2 y1r2 y1Msm e m r3s . 21Ž .p p '3 e

3. N wormholes approximationw

Suppose to consider N wormholes and assume that there exists a covering of S such that Ssj Nw S ,w is1 iœ 2 1 "with S lS s0 when i/ j. Each S has the topology S =R with boundaries ES with respect to eachi j i i

bifurcation surface. On each surface S , quasilocal energy givesi

2 22 0 2 0' 'E s d x s kyk y d x s kyk , 22Ž . Ž . Ž .H Hi quasilocal 2 2l lS Sp piq iy

and by using the expression of the trace

1m'ksy h n , 23Ž .Ž . ,m'h

we obtain at either boundary that

y2 r, yks , 24Ž .

r

where we have assumed that the function r is positive for S and negative for S . The trace associated with, y iq iythe subtraction term is taken to be k 0 sy2rr for B and k 0 s2rr for B . Here the quasilocal energy withiq iysubtraction terms included is

E sE yE s r 1y r y r 1y r . 25Ž .Ž . Ž .i quasilocal iq iy , y , yysy ysyiq iy

Note that the total quasilocal energy is zero for boundary conditions symmetric with respect to each bifurcationsurface S . We are interested to a large number of wormholes, each of them contributing with a Hamiltonian of0, i

Ž .the type H . If the wormholes number is N , we obtain semiclassically, i.e., without self-interactions<2 w

H Nw sH 1 qH 2 q . . . qH Nw . 26Ž .tot

Thus the total energy for the collection is

E tot sN H .<2 w <2

The same happens for the trial wave functional which is the product of N t.w.f. Thusw

1 HH H H H y1² :C sC mC m . . . C s NNexpN y gyg K gygŽ . Ž . x , ytot 1 2 N w 2w ½ 54 lp

H1 y1² :sNNexp y gyg K gyg ,Ž . Ž . x , y½ 54

2where we have rescaled the fluctuations hsgyg in such a way to absorb N rl . Of course, if we want thew p

trial wave functionals be independent one from each other, boundaries ES " have to be reduced with the


enlarging of the wormholes number N , otherwise overlapping terms could be produced. Thus, for N -worm-w w

holes, we obtain

2 'l gp Ž .tot 3 i j k l 3H sN Hs d x G p p N y N RHw i jk l w w 2ž / ž /l'gS p

2 'l gNw3 i j k l 2 Ž3.s d x G p p y N R ,H i jk l w 2ž / ž /l'gS Nw

where we have defined l 2 s l 2 N with l 2 fixed and N ™`. Thus, repeating the same steps of Section 2 forN p w N ww w

N wormholes, we obtainw

2 3 2V 3m 1 r L02DE m ;yN ln . 27Ž . Ž .N w 2 3w ž /ž / 16 3m2p r0

Then at one loop the cooperative effects of wormholes behave as one macroscopic single field multiplied byN 2; this is the consequence of the coherency assumption. We have just explored the consequences of this resultw

w xin Ref. 5 . Indeed, coming back to the single wormhole contribution we have seen that the black hole paircreation probability mediated by a wormhole is energetically favored with respect to the permanence of flatspace provided we assume that the boundary conditions be symmetric with respect to the bifurcation surfacewhich is the throat of the wormhole. In this approximation boundary terms give zero contribution and thevolume term is nonvanishing. As in the one-wormhole case, we now compute the minimum of&

2 y1r2 3DE m s E 0 yE m syDE m . The minimum is reached for msL e r r3. Thus theŽ . Ž . Ž . Ž .Ž . N 0N N ww w

minimum is

V L4&

2DE m sN . 28Ž . Ž .w 2 e64p

The main difference with the one wormhole case is that we have N wormholes contributing with the samew

amount of energy. Since msMN GsMl 2 , thusw Nw

y12 2 y1r2 3Ms l rN L e r r3. 29Ž .Ž .N w 0w

When L™m , then r ™ l and l m s1. Thusp 0 p p p

y12 y1l rN m mŽ .N w p Nw wMs sN . 30Ž .w' '3 e 3 e

So far, we have discussed the stable modes contribution. However, we have discovered that for one wormholew xalso unstable modes contribute to the total energy 4,5 . Since we are interested to a large number of wormholes,

the first question to answer is: what happens to the boundaries when the wormhole number is enlarged. In theone wormhole case, the existence of one negative mode is guaranteed by the vanishing of the eigenfunction ofthe operator D at infinity, which is the same space-like infinity of the quasilocal energy, i.e. we have the ADM2

positive mass M in a coordinate system of the universe where the observer is present and the anti-ADM mass ina coordinate system where the observer is not there. When the number of wormholes grows, to keep thecoherency assumption valid, the space available for every single wormhole has to be reduced to avoidoverlapping of the wave functions. This means that boundary conditions are not fixed at infinity, but at a certainfinite radius and the ADM mass term is substituted by the quasilocal energy expression under the condition ofhaving symmetry with respect to each bifurcation surface. As N grows, the boundary radius r reduces morew

and more and the unstable mode disappears. This means that there will exist a certain radius r where belowc


which no negative mode will appear and there will exist a given value N above which the same effect will bewc

œproduced. In rigorous terms: ;NGN ' r s.t. ; r FrFr , s D s0. This means that the system begins toŽ .w c 0 c 2c

w xbe stable. In support of this idea, we invoke the results discovered in Ref. 6 where, it is explicitly shown thatthe restriction of spatial boundaries leads to a stabilization of the system. Thus at the minimum, we obtain thetypical energy density behavior of the foam

DE2 4;yN L . 31Ž .wV

4. Conclusions and outlooks

According to Wheeler’s ideas about quantum fluctuation of the metric at the Planck scale, we have used asimple model made by a large collection of wormholes to investigate the vacuum energy contribution needed tothe formation of a foamy spacetime. Such investigation has been made in a semiclassical approximation where

Ž .the wormholes are treated independently one from each other coherency hypothesis . The starting point is thesingle wormhole, whose energy contribution has the typical trend of the gravitational field energy fluctuation.The wormhole considered is of the Schwarzschild type and every energy computation has to be done having inmind the reference space, i.e. flat space. When we examine the wormhole collection, we find the same trend inthe energy of the single case. This is obviously the result of the coherency assumption. However, the singlewormhole cannot be taken as a model for a spacetime foam, because it exhibits one negative mode. This

Ž .negative mode is the key of the topology change from a space without holes flat space to a space with an holeŽ .inside Schwarzschild space . However things are different when we consider a large number of wormholes N .w

Let us see what is going on: the classical vacuum, represented by flat space is stable under nucleation of a singleblack hole, while it is unstable under a neutral pair creation with the components residing in different universesdivided by a wormhole. When the topology change has primed by means of a single wormhole, there will be aconsiderable production of pairs mediated by their own wormhole. The result is that the hole production willpersist until the critical value N will be reached and spacetime will enter the stable phase. If we look at thiswc

scenario a little closer, we can see that it has the properties of the Wheeler foam. Nevertheless, we have toexplain why observations measure a flat space structure. To this purpose, we have to recall that the foamyspacetime structure should be visible only at the Planck scale, while at greater scales it is likely that the flatstructure could be recovered by means of averages over the collective functional describing the semiclassicalfoam. Indeed if h is the spatial part of the flat metric, ordinarily we should obtaini j

C g C sh , 32² : Ž .i j i j

where g is the spatial part of the gravitational field. However in the foamy representation we should consider,i j

instead of the previous expectation value, the expectation value of the gravitational field calculated on wavefunctional representing the foam, i.e., to see that at large distances flat space is recovered we should obtain

C g C sh , 33² : Ž .foam i j foam i j

where C is a superposition of the single-wormhole wave functionalfoam

Nw

HC s C . 34Ž .Ýfoam iis1

This has to be attributed to the semiclassical approximation which render this system a non-interacting system.However, things can change when we will consider higher order corrections and the other terms of the actiondecomposition, i.e. the spin one and spin zero terms. Nevertheless, we can argue that only spin zero terms


Ž .associated with the conformal factor will be relevant, even if the part of the action which carries the physicalquantities is that discussed in this text, i.e., the spin two part of the action related to the gravitons.

Acknowledgements

I wish to thank R. Brout, M. Cavaglia, C. Kiefer, D. Hochberg, G. Immirzi, S. Liberati, P. Spindel and M.`Visser for useful comments and discussions.

References

w x Ž .1 J.A. Wheeler, Ann. Phys. 2 1957 604; Geometrodynamics, Academic Press, New York, 1962.w x Ž .2 L. Crane, L. Smolin, Nucl. Phys. B 1986 714.w x Ž .3 J.B. Hartle, G.T. Horowitz, Phys. Rev. D 24 1981 257.w x Ž .4 D.J. Gross, M.J. Perry, L.G. Yaffe, Phys. Rev. D 25 1982 330.w x5 R. Garattini, Probing foamy spacetime with variational methods, to appear in Int. J. Mod. Phys. A, Report gr-qcr98010045.w x Ž .6 B. Allen, Phys. Rev. D 30 1984 1153.

21 January 1999


Collinear factorization and splitting functions fornext-to-next-to-leading order QCD calculations 1

Stefano Catani a,2, Massimiliano Grazzini a,b

a Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerlandb Institute for Theoretical Physics, ETH-Honggerberg, CH-8093 Zurich, Switzerland 3¨

Received 20 October 1998Editor: R. Gatto

Abstract

We consider the singular behaviour of tree-level QCD amplitudes when the momenta of three partons becomesimultaneously parallel. We discuss the universal factorization formula that controls the singularities of the multipartonmatrix elements in this collinear limit and present the explicit expressions of the corresponding splitting functions. The

Ž . Ž 2.results fully include spin azimuthal correlations up to OO a . In the case of spin-averaged splitting functions we confirmS

similar results obtained by Campbell and Glover. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Ž .The universal properties of multiparton matrix elements in the infrared soft and collinear limit play aw xrelevant role in our capability to make reliable QCD predictions for hard-scattering processes 1,2 .

At leading order in the QCD coupling a , these properties are embodied in process-independent factoriza-Sw x w x Ž .tion formulae of tree-level 3–5 and one-loop 6–8,4 amplitudes that are at the basis of at least three

Ž .important tools in perturbative QCD. The leading-logarithmic LL parton showers, which are implemented inw xMonte Carlo event generators 1 to describe the exclusive structure of hadronic final states, are based on these

w x w xfactorization formulae supplemented with ‘jet calculus’ techniques 9 and colour-coherence properties 10,11 .Analytical techniques to perform all-order resummation of logarithmically enhanced contributions at next-to-

Ž . w xleading logarithmic NLL accuracy 12 rely on the factorization properties of soft and collinear emission. Morerecently, the leading-order factorization formulae have been fully exploited to set up completely general

1 This work was supported in part by the EU Fourth Framework Programme ‘‘Training and Mobility of Researchers’’, NetworkŽ .‘‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’’, contract FMRX–CT98–0194 DG 12 – MIHT .

2 On leave of absence from I.N.F.N., Sezione di Firenze, Florence, Italy.3 Address after 1st October 1998.


( )S. Catani, M. GrazzinirPhysics Letters B 446 1999 143–152144

w xalgorithms 7,13–15 to handle and cancel infrared singularities when combining tree-level and one-loopŽ .contributions in the evaluation of jet cross sections at the next-to-leading order NLO in perturbation theory.

Ž .Any further although challenging theoretical improvement of these tools requires the understanding ofinfrared-factorization properties at the next order in a .S

This topic has recently received considerable attention. The infrared singular behaviour of two-loop QCDw x w xamplitudes has been discussed in Ref. 16 . The collinear limit of one-loop amplitudes is known 17 . In the case

w x w xof tree-level amplitudes, the factorization structure in the double-soft 18 and triple-collinear 19 limits hasbeen studied.

w xIn this letter, we reconsider the triple-collinear limit examined by Campbell and Glover 19 . In particular, weŽ .extend their results by fully taking into account azimuthal spin correlations. Besides improving the general

understanding of the collinear behaviour of tree amplitudes, this extension is essential to apply some generalŽ .methods to perform exact fixed-order calculations at the next-to-next-to-leading order NNLO in perturbation

w xtheory. For instance, the subtraction method 14,15 works by regularizing the infrared singularities of thetree-level matrix element by identifying and subtracting a proper local counterterm. Thus, the knowledge of the

w xazimuthally aÕeraged collinear limit studied in Ref. 19 is not sufficient for this purpose.The outline of the paper is as follows. In Section 2, we define our notation and the collinear kinematics and,

Ž .after reviewing the known collinear-factorization formulae at OO a , we present their generalization at the nextSŽ 2 .perturbative order. Our explicit results for the collinear-splitting functions at OO a are given in Section 3,S

w xwhere the comparison with the azimuthally-averaged case considered in Ref. 19 is also discussed. Details onw xour calculation of the splitting functions are not discussed here and will be presented elsewhere 20 . Some final

remarks are left to Section 4.

2. Notation and kinematics

4 Ž .We consider a generic scattering process involving final-state QCD partons massless quarks and gluonsŽ ) 0 " .with momenta p , p , . . . Non-QCD partons g ,Z ,W , . . . , carrying a total momentum Q, are always1 2

understood. The corresponding tree-level matrix element is denoted by

MM c1 ,c2 , . . . ; s1 , s2 , . . . p , p , . . . , 1Ž . Ž .a ,a , . . . 1 21 2

� 4 � 4 � 4where c ,c , . . . , s ,s , . . . and a ,a , . . . are respectively colour, spin and flavour indices. The matrix1 2 1 2 1 2< Ž . < 2element squared summed over final-state colours and spins will be denoted by MM p , p , . . . . To thea ,a , . . . 1 21 2

purpose of the present paper, it is also useful to consider the sum over the spins of all the final-state partons butone. For instance, if the sum over the spin polarizations of the parton a is not carried out, we define the1

following ‘spin-polarization tensor’

TT s1 sX1 p , . . . ' MM c1 ,c2 , . . . ; s1 , s2 , . . . p , p , . . .Ž . Ž .Ý Ýa , . . . 1 a ,a , . . . 1 21 1 2

Xspins /s , s colours1 1

=X †c ,c , . . . ; s , s , . . .1 2 1 2MM p , p , . . . . 2Ž . Ž .a ,a , . . . 1 21 2

In the evaluation of the matrix element, we use dimensional regularization in ds4y2e space-timedimensions and consider two helicity states for massless quarks and dy2 helicity states for gluons. Thisdefines the usual dimensional-regularization scheme. Thus, the fermion spin indices are ss1,2 while to label

4 The case of incoming partons can be recovered by simply crossing the parton flavours and momenta.

( )S. Catani, M. GrazzinirPhysics Letters B 446 1999 143–152 145

the gluon spin it is convenient to use the corresponding Lorentz index ms1, . . . ,d. The d-dimensional averageof the matrix element over the polarizations of a parton a is obtained by means of the factors

1Xd 3Ž .ss2

Ž m n .for a fermion, and the gauge terms are proportional either to p or to p

1 1d p s yg qgauge terms 4Ž . Ž .Ž .mn mndy2 2 1yeŽ .

with

yg mnd p sdy2 , p m d p sd p pn s0 , 5Ž . Ž . Ž . Ž .mn mn mn

for a gluon with on-shell momentum p.Ž .The relevant collinear limit at OO a is approached when the momenta of two partons, say p and p ,S 1 2

become parallel. Usually, this limit is precisely defined as follows:

k 2 n m k 2 n mH Hm m m m m mp szp qk y , p s 1yz p yk y ,Ž .1 H 2 Hz 2 pPn 1yz 2 pPn

k 2H

s '2 p Pp sy , k ™0 . 6Ž .12 1 2 Hz 1yzŽ .Ž . Ž 2 . m mIn Eq. 6 the light-like p s0 vector p denotes the collinear direction, while n is an auxiliary light-like

Ž 2 . Ž .vector, which is necessary to specify the transverse component k k -0 k Ppsk Pns0 or, equiva-H H H HŽlently, how the collinear direction is approached. In the small-k limit i.e. neglecting terms that are lessH

2 . Ž . w xsingular than 1rk , the square of the matrix element in Eq. 1 fulfils the following factorization formula 1H

2 X X2 2 e ss ssˆ< <MM p , p , . . . , 4pm a TT p , . . . P z ,k ;e , 7Ž . Ž . Ž . Ž .a ,a , . . . 1 2 S a , . . . a a H1 2 1 2s12

ssX Ž .where m is the dimensional-regularization scale. The spin-polarization tensor TT p, . . . is obtained bya, . . .Ž .replacing the partons a and a on the right-hand side of Eq. 2 with a single parton denoted by a. This parton1 2

carries the quantum numbers of the pair a qa in the collinear limit. In other words, its momentum is p m and1 2Ž .its other quantum numbers flavour, colour are obtained according to the following rule: anythingqgluon

gives anything and quarkqantiquark gives gluon.ˆ Ž . w xThe kernel P in Eq. 7 is the d-dimensional Altarelli–Parisi splitting function 21 . It depends not onlya a1 2

on the momentum fraction z involved in the collinear splitting a™a qa , but also on the transverse1 2c, . . . ; s, . . . Ž .momentum k and on the helicity of the parton a in the matrix element MM p, . . . . More precisely,H a, . . .

ˆ XP is in general a matrix acting on the spin indices s,s of the parton a in the spin-polarization tensora a1 2ssX Ž .TT p, . . . . Because of these spin correlations, the spin-average square of the matrix elementa, . . .

c, . . . ; s, . . . Ž . Ž .MM p, . . . cannot be simply factorized on the right-hand side of Eq. 7 .a, . . .ˆThe explicit expressions of P , for the splitting processesa a1 2

a p ™a zpqk qOO k 2 qa 1yz pyk qOO k 2 , 8Ž . Ž . Ž .Ž . Ž .Ž . Ž .1 H H 2 H H

depend on the flavour of the partons a ,a and are given by1 2

21qzX Xss ssˆ ˆ XP z ,k ;e sP z ,k ;e sd C ye 1yz , 9Ž . Ž . Ž . Ž .q g H q g H ss F 1yz

21q 1yzŽ .X Xss ssˆ ˆ XP z ,k ;e sP z ,k ;e sd C ye z , 10Ž . Ž . Ž .g q H g q H ss F z


m nk kH Hmn mn mnˆ ˆP z ,k ;e sP z ,k ;e sT yg q4 z 1yz , 11Ž . Ž . Ž . Ž .qq H qq H R 2k H

m nz 1yz k kH Hmn mnP z ,k ;e s2C yg q y2 1ye z 1yz , 12Ž . Ž . Ž . Ž .g g H A 2ž /1yz z kH

Ž .where the SU N QCD colour factors arec

N 2 y1c 1C s , C sN , T s , 13Ž .F A c R 22 Nc

and the spin indices of the parent parton a have been denoted by s,sX if a is a fermion and m,n if a is a gluon.Ž Ž . Ž ..Note that when the parent parton is a fermion cf. Eqs. 9 and 10 the splitting function is proportional to

Ž .the unity matrix in the spin indices. Thus, in the factorization formula 7 , spin correlations are effective only inthe case of the collinear splitting of a gluon. Owing to the k -dependence of the gluon splitting functions inH

Ž . Ž .Eqs. 11 and 12 , these spin correlations produce a non-trivial azimuthal dependence with respect to thedirections of the other momenta in the factorized matrix element.

Ž . Ž .Eqs. 9 – 12 lead to the more familiar form of the d-dimensional splitting functions only after average overŽ .the polarizations of the parton a. The d-dimensional average is obtained by means of the factors in Eqs. 3 and

ˆ ˆŽ . ² :4 . Denoting by P the average of P over the polarizations of the parent parton a, we have:a a a a1 2 1 2

21qzˆ ˆ² : ² :P z ;e s P z ;e sC ye 1yz , 14Ž . Ž . Ž . Ž .q g q g F 1yz

21q 1yzŽ .ˆ ˆ² : ² :P z ;e s P z ;e sC ye z , 15Ž . Ž . Ž .g q g q F z

2 z 1yzŽ .ˆ ˆ² : ² :P z ;e s P z ;e sT 1y , 16Ž . Ž . Ž .qq qq R 1ye

z 1yzˆ² :P z ;e s2C q qz 1yz . 17Ž . Ž . Ž .g g A 1yz z

Ž 2 .In the rest of the paper we are interested in the collinear limit at OO a . In this case three parton momentaS

can simultaneously become parallel. Denoting these momenta by p , p and p , their most general parametriza-1 2 3

tion is

k 2 n mH im m mp sx p qk y , is1,2,3 , 18Ž .i i H i x 2 pPni

Ž . mwhere, as in Eq. 6 , the light-like vector p denotes the collinear direction and the auxiliary light-like vectorm Ž . Žn specifies how the collinear direction is approached k Ppsk Pns0 . Note that no other constraint e.g.H i H i

.Ý x s1 or Ý k s0 is imposed on the longitudinal and transverse variables x and k . Thus, we cani i i H i i H iŽ .easily consider any asymmetric collinear limit at once.

< Ž . < 2In the triple-collinear limit, the matrix element squared MM p , p , p , . . . has the singulara ,a ,a , . . . 1 2 31 2 3

< Ž . < 2 Ž X. X Ž Ž .2 .behaviour MM p , p , p , . . . ;1r ss , where s and s can be either two-particle s s p qpa ,a ,a , . . . 1 2 3 i j i j1 2 3

Ž Ž .2 . w xor three-particle s s p qp qp sub-energies. More precisely, it can be shown 19,20 that the matrix123 1 2 3Ž .element squared still fulfils a factorization formula analogous to Eq. 7 , namely

4 X X22 2 e ss ssˆ< <MM p , p , p , . . . , 4pm a TT p , . . . P . 19Ž . Ž . Ž .Ž .a ,a ,a , . . . 1 2 3 S a , . . . a a a21 2 3 1 2 3s123


Ž . ssX Ž .Likewise in Eq. 7 , the spin-polarization tensor TT p, . . . is obtained by replacing the partons a , a anda, . . . 1 2Ž .a with a single parent parton, whose flavour a is determined see Section 3 by flavour conservation in the3

splitting process a™a qa qa .1 2 3ˆ Ž .The three-parton splitting functions P generalize the Altarelli–Parisi splitting functions in Eq. 7 . Thea a a1 2 3

spin correlations produced by the collinear splitting are taken into account by the splitting functions in aŽ .universal way, i.e. independently of the specific matrix element on the right-hand side of Eq. 19 . Besides

ˆdepending on the spin of the parent parton, the functions P depend on the momenta p , p , p . However,a a a 1 2 31 2 3

due to their invariance under longitudinal boosts along the collinear direction, the splitting functions can dependin a non-trivial way only on the sub-energy ratios s rs and on the following longitudinal and transversei j 123

variables:xi

z s , 20Ž .i 3

xÝ jjs1

3xim m mk sk y k , 21Ž .Ýi H i H j3js1xÝ k

ks1

3 3 ˜which automatically satisfy the constraints Ý z s1 and Ý k s0. To simplify the explicit expressions ofis1 i is1 i

the splitting functions, we find it convenient to introduce also the variablesz s yz s z yzi jk j i k i j

t '2 q s . 22Ž .i j ,k i jz qz z qzi j i j

The results of our calculation are presented in the next section.

( 2 )3. Collinear splitting functions at OO a S

To evaluate the three-parton splitting functions, we use power-counting arguments and the universalw xfactorization properties of collinear singularities. The method 22 consists in directly computing process-inde-

pendent Feynman subgraphs in a physical gauge. Details on the method and on our calculation are given in Ref.w x20 . In the following we present the complete results for the spin-dependent splitting functions.

Ž .The list of non-vanishing splitting processes that we have to consider is as follows:X X X Xq™q qq qq , q™q qq qq , 23Ž .Ž .1 2 3 1 2 3

q™q qq qq , q™q qq qq , 24Ž .Ž .1 2 3 1 2 3

q™g qg qq , q™g qg qq , 25Ž .Ž .1 2 3 1 2 3

g™g qq qq , 26Ž .1 2 3

g™g qg qg . 27Ž .1 2 3X XThe superscripts in q ,q denote fermions with different flavour with respect to q,q. The splitting functions for

Ž . Ž .the processes in parenthesis in Eqs. 23 and 24 can be simply obtained by charge-conjugation invariance, i.e.ˆ X X ˆ X X ˆ ˆP sP and P sP . In summary, we have to compute five independent splitting functions.q q q q q q q q q q q q1 2 3 1 2 3 1 2 3 1 2 3

Ž Ž . Ž ..In the case of the splitting processes that involve a fermion as parent parton see Eqs. 23 – 25 , we findthat spin correlations are absent. We can thus write

ˆ ssXX X ssX ˆ X X² :P sd P , 28Ž .q q q q q q1 2 3 1 2 3

ˆ ssX ˆ ssX Ž .and likewise for P and P . This feature is completely analogous to the OO a case and follows fromq q q g g q S1 2 3 1 2 3

helicity conservation in the the quark–gluon vector coupling.


The spin-averaged splitting function for non-identical fermions in the final state is221 s t 4 z q z yz sŽ .123 12,3 3 1 2 12

X Xˆ² :P s C T y q q 1y2e z qz y . 29Ž . Ž .q q q F R 1 21 2 3 ž /2 s s s z qz s12 12 123 1 2 123

The analogous splitting function in the case of final-state fermions with identical flavour can be written inŽ .terms of that in Eq. 29 , as follows

Ž id .X Xˆ ˆ ˆ² : ² : ² :P s P q 2l3 q P , 30Ž . Ž .q q q q q q q q q1 2 3 1 2 3 1 2 3

where

21 2 s s 1qz23 123 1Ž id .ˆ² :P sC C y C 1ye ye qŽ .q q q F F A1 2 3 ž / ž /½2 s s 1yz12 12 2

22 z 1yz 2 zŽ .2 3 2 2y ye q1qz y ye 1yzŽ .1 3ž /1yz 1yz 1yz3 2 3

2 2s z 1qz 1yz123 1 1 2 2y ye 1q2 ye q 2l3 . 31Ž . Ž .ž / 5s s 2 1yz 1yz 1yzŽ . Ž .12 13 2 3 3

The splitting function of the remaining quark-decay subprocess can be decomposed according to the differentcolour coefficients:

ˆ 2 ˆŽab. ˆŽnab.² : ² : ² :P sC P qC C P , 32Ž .g g q F g g q F A g g q1 2 3 1 2 3 1 2 3

and the abelian and non-abelian contributions are32 2 2 2s 1qz z qz s z 1yz q 1yzŽ . Ž .123 3 1 2 123 3 1 2Žab.ˆ² :P s z ye ye 1qe qŽ .g g q 31 2 3 ½ 2 s s z z z z s z z13 23 1 2 1 2 13 1 2

1yz s2 232 2 2qe 1qz ye z qz z qz q 1ye ey 1ye q 1l2 ,Ž . Ž . Ž . Ž .Ž .3 1 1 2 2 5z z s1 2 13

33Ž .22 2t 1 e s 1yz 1ye q2 zŽ . Ž .12 ,3 123 3 3Žnab.ˆ² :P s 1ye q y qŽ .g g q 21 2 3 ½ ž /4 2 2 s s z4 s 12 13 212

22 2z 1ye q2 1yz s 1yz 1ye q2 zŽ . Ž . Ž . Ž .2 2 123 3 3q y z qe 1yeŽ .31yz 4 s s z z3 13 23 1 2

2 2s z 2y2 z qz yz 6y6 z qz z z y2 z yzŽ .Ž . Ž .123 1 1 1 2 2 2 3 1 2 2q 1ye q2eŽ .

2 s z 1yz z 1yzŽ . Ž .12 2 3 2 3

3 2s 1yz qz yz 2 1yz z yzŽ . Ž . Ž .123 2 3 2 2 2 3q 1ye ye yz qzŽ . 1 2ž /2 s z 1yz z 1yzŽ . Ž .13 2 3 2 3

3 2 2z 1yz q 1yz z qzŽ . Ž .3 1 2 1 2y qe 1yz ye q 1l2 . 34Ž . Ž . Ž .2 5ž /z z z z1 2 1 2


Ž Ž . Ž ..In the case of collinear decays of a gluon see Eqs. 26 and 27 , spin correlations are highly non-trivial.The colour-factor decomposition of the splitting function for the decay into a qq pair plus a gluon is

ˆmn ˆmn Žab. ˆmn Žnab.P sC T P qC T P , 35Ž .g q q F R g q q A R g q q1 2 3 1 2 3 1 2 3

where the abelian and non-abelian terms are given by

22 s s q 1ye s ys 4 sŽ . Ž .123 23 123 23 123mn Žab. mn m n m n m nˆ ˜ ˜ ˜ ˜ ˜ ˜P syg y2q q k k qk k y 1ye k k ,Ž .ž /g q q 3 2 2 3 1 11 2 3 s s s s12 13 12 13

36Ž .

m n2 2 2° ˜ ˜ ˜ ˜1 s t z z k k k k123 23,1 2 3 2 3 2 3mn Žnab. mn~P s g y16 y yg q q 21 2 3 ¢ ž / ž /4 s z 1yz z z z zs Ž .123 1 1 2 3 2 323

s123 mn m n m n m n mn˜ ˜ ˜ ˜ ˜ ˜q 2 s g y4 k k qk k y 1ye k k yg y 1y2eŽ . Ž .ž /123 2 3 3 2 1 1s s12 13

2s 1yz s 1yz q2 z s z 1y2 zŽ .123 3 123 1 1 123 2 1mnq2 q2 q y2 s g123s z 1yz s z 1yz s s z 1yzŽ . Ž . Ž .12 1 1 23 1 1 12 23 1 1

2 ¶z 2 z z yzŽ .2 2 3 1m n m n m n m n •˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜y16k k q8 1ye k k q4 k k qk k q 1yeŽ . Ž .ž /3 3 2 2 2 3 3 2 ž /ßz 1yz z 1yzŽ . Ž .1 1 1 1

q 2l3 . 37Ž . Ž .

In the case of gluon decay into three collinear gluons we find

m n2 2° ˜ ˜ ˜ ˜1ye z z k k k kŽ . 1 2 2 1 2 1mn 2 mn 2~P sC yg t q16 s y yg g g A 12,3 12321 2 3 ¢ ž / ž /z 1yz z z z z4 s Ž .3 3 2 1 2 112

23 s 1 2 1yz q4 z 1y2 z 1yzŽ . Ž .123 3 3 3 3mn mny 1ye g q g yŽ .4 s z 1yz z 1yzŽ .12 3 3 1 1

s 1ye 1y2 z 1y2 zŽ .123 3 2m n m n˜ ˜ ˜ ˜q 2 z k k qk k1 2 2 3 3ž /s s z 1yz z 1yzŽ . Ž .12 13 3 3 2 2

s 4 z z q2 z 1yz y1 1y2 z 1yzŽ . Ž .123 2 3 1 1 1 1mnq g yž /2 1ye 1yz 1yz z zŽ . Ž . Ž .2 3 2 3

¶2 z 1yzŽ .2 2m n m n •˜ ˜ ˜ ˜q k k qk k y3 q 5 permutations . 38Ž . Ž .ž /2 3 3 2 ž /ßz 1yzŽ .3 3


Ž . Ž .The splitting functions in Eqs. 36 – 38 can be averaged over the spin polarizations of the parent gluonŽ .according to Eq. 4 :

1mnˆ ˆ² :P ' d p P . 39Ž . Ž .a a a mn a a a1 2 3 1 2 32 1yeŽ .

Performing the average we obtain

1 1 s2 z q2 z z123 1 2 3Žab. 2ˆ² :P sy2y 1ye s q q2 1qz yŽ .g q q 23 11 2 3 ž /ž /s s s s 1ye12 13 12 13

s z qz s z qz123 1 2 123 1 3y 1q2 z qey2 y 1q2 z qey2 , 40Ž .1 1ž / ž /s 1ye s 1ye12 13

32 2 3t s 1yz yz 2 z 1yz y2 z zŽ . Ž .23 ,1 123 1 1 3 3 1 2Žnab.ˆ² :P s y q z yg q q 321 2 3 ½ 2 s s z 1yz 1ye z 1yz4 s Ž . Ž . Ž .13 23 1 1 1 123

s 1 2 z 1yzŽ .123 2 2q 1yz 1q yŽ .22 s z 1yz 1ye z 1yzŽ . Ž . Ž .13 1 1 1 1

23s 1qz z z yz y2 z z 1qz 1 eŽ . Ž .123 1 1 3 2 2 3 1q q y q

2 s z 1yz 1ye z 1yz 4 2Ž . Ž . Ž .23 1 1 1 1

s2 z q2 z z123 1 2 32y 1qz y q 2l3 , 41Ž . Ž .1ž / 52 s s 1ye12 13

1ye 3 s z z y1 z z y2 3 5Ž . 123 1 2 1 22 2ˆ² :P sC t q 1ye q 4 q q q zŽ .g g g A 12,3 321 2 3 ½ 4 s 1yz z 2 24 s 12 3 312

2 21yz 1yz s z z 1yz 1y2 z z 1q2 zŽ . Ž . Ž . Ž .Ž .3 3 123 1 2 2 3 1 1q q qz z y2q2 3z z 1yz s s z 1yz 2Ž . Ž .3 1 1 12 13 3 3

1q2 z 1qz 1y2 z 1yzŽ . Ž .1 1 1 1q q q 5 permutations . 42Ž . Ž .52 1yz 1yz 2 z zŽ . Ž .2 3 2 3

Ž 2 .The OO a -collinear behaviour of tree-level QCD matrix elements has been independently examined bySw xCampbell and Glover 19 . Their study differs in many respects from our analysis. Taken for granted the

Ž .universal factorization formula 19 , they compute the three-parton splitting functions by directly performing thecollinear limit of the explicit expressions of the g ) ™ four- and five-parton squared matrix elements.Moreover, they treat the colour structure in a different way and consider the collinear limit of the colour-ordered

w xsub-amplitudes 3 . Finally, they neglect spin correlations and present only the explicit expressions of thepolarization-averaged splitting functions.

By properly taking into account the differences in the colour treatment, we have compared our results withw x 5those of Ref. 19 and found complete agreement for the spin-averaged splitting functions. To be precise, the

5 w x Ž Ž . Ž ..Note that the published version of Ref. 19 contains two misprints in Eqs. 5.8 and 5.19 that have been corrected in the archiveversion hep-phr9710255 v3.


w xcolour-connected splitting functions P of Ref. 19 are related to our spin-averaged splitting functionsa a a ™ a1 2 3

as follows:

s2123 non - ident .X X X Xˆ² :P s T C P ,q q q R F q q q ™ q3 1 21 2 3 4

s2123 1Ž id . ident .ˆ² :P s C C y C P ,Ž .q q q F F A q q q ™ q21 2 3 3 1 24

s2123Žab.ˆ² :P s P ,g g q q g g ™ q˜ ˜1 2 3 3 1 24

s2 1123Žnab.ˆ² :P s P qP yP ,Ž .g g q q g g ™ q q g g ™ q q g g ™ q˜ ˜1 2 3 3 1 2 3 2 1 3 1 24 2

s2123Žab.ˆ² :P s P ,g q q q g q ™ g˜ ˜1 2 3 2 1 34

s2 1123Žnab.ˆ² :P s P qP yP ,ž /g q q g q q ™ g q q g ™ g q g q ™ g˜ ˜1 2 3 1 3 2 3 2 1 2 1 34 222s C123 Aˆ² :P s P q 5 permutations . 43Ž . Ž .g g g g g g ™ g1 2 3 1 2 3ž /4 2

Owing to the completely different methods used by the two groups, this agreement can be regarded as animportant cross-check of the calculations.

4. Summary

We have considered the three-parton collinear limit of tree-level QCD amplitudes. In this limit the singularŽ .behaviour of the matrix element squared is given by the universal factorization formula 19 and is controlled by

process-independent splitting functions, which are analogous to the Altarelli–Parisi splitting functions. InŽ 2 .Section 3 we have presented the explicit expressions of the splitting functions at OO a , taking fully intoS

account spin correlations.These splitting functions are one of the necessary ingredients needed to extend QCD predictions at higher

perturbative orders. In particular, they are relevant to perform analytic resummed calculations beyond NLLaccuracy and to set up general methods to compute jet cross sections at NNLO. The knowledge of the collinearsplitting functions, when combined with a consistent analysis of soft-gluon coherence, can also give prospects ofimproving the logarithmic accuracy of parton showers available at present for Monte Carlo event generators.

Acknowledgements

Ž .We thank Nigel Glover for discussions. One of us M.G. would like to thank the Fondazione ‘Angelo dellaRiccia’ and INFN for financial support.

References

w x1 R.K. Ellis, W.J. Stirling, B.R. Webber, QCD and collider physics, Cambridge University Press, Cambridge, 1996, and referencestherein.


w x Ž .2 S. Catani, hep-phr9712442, in: A. De Roeck, A. Wagner Eds. , Proc. of the XVIII International Symposium on Lepton-PhotonInteractions, LP97, World Scientific, Singapore, 1998, p. 147, and references therein.

w x Ž .3 M.L. Mangano, S.J. Parke, Phys. Rep. 200 1991 301, and references therein.w x Ž .4 S. Catani, M.H. Seymour, Phys. Lett. B 378 1996 287.w x Ž .5 D.A. Kosower, Phys. Rev. D 57 1998 5410.w x Ž .6 Z. Bern, L. Dixon, D.A. Kosower, Annu. Rev. Nucl. Part. Sci. 46 1996 109, and references therein.w x Ž .7 W.T. Giele, E.W.N. Glover, Phys. Rev. D 46 1992 1980.w x Ž .8 Z. Kunszt, A. Signer, Z. Trocsanyi, Nucl. Phys. B 420 1994 550.´ ´w x Ž .9 K. Konishi, A. Ukawa, G. Veneziano, Nucl. Phys. B 157 1979 45.

w x Ž .10 B.I. Ermolaev, V.S. Fadin, JETP Lett. 33 1981 269.w x Ž .11 A. Bassetto, M. Ciafaloni, G. Marchesini, Phys. Rep. 100 1983 201; Yu.L. Dokshitser, V.A. Khoze, A.H. Mueller, S.I. Troian, Basics

of Perturbative QCD, Editions Frontieres, Gif-sur-Yvette, 1991, and references therein.`w x Ž .12 G. Sterman, in: R. Raja, J. Yoh Eds. , Proc. 10th Topical Workshop on Proton-Antiproton Collider Physics, AIP Press, New York,

Ž .1996, p. 608; S. Catani, in: J. Tran Than Van Ed. , Proc. of the 32nd Rencontres de Moriond: QCD and High-Energy HadronicInteractions, Editions Frontieres, Paris, 1997, p. 331 and references therein.`

w x Ž .13 W.T. Giele, E.W.N. Glover, D.A. Kosower, Nucl. Phys. B 403 1993 633.w x Ž . Ž .14 Z. Kunszt, D.E. Soper, Phys. Rev. D 46 1992 192; S. Frixione, Z. Kunszt, A. Signer, Nucl. Phys. B 467 1996 399; Z. Nagy, Z.

Ž . Ž .Trocsanyi, Nucl. Phys. B 486 1997 189; S. Frixione, Nucl. Phys. B 507 1997 295.´ ´w x Ž . Ž Ž . .15 S. Catani, M.H. Seymour, Nucl. Phys. B 485 1997 291 E ibid. B 510 1998 503 .w x Ž .16 S. Catani, Phys. Lett. B 427 1998 161.w x Ž .17 Z. Bern, G. Chalmers, L. Dixon, D.A. Kosower, Phys. Rev. Lett. 72 1994 2134; Z. Bern, L. Dixon, D.C. Dunbar, D.A. Kosower,

Ž . Ž .Nucl. Phys. B 425 1994 217; Z. Bern, L. Dixon, D.A. Kosower, Nucl. Phys. B 437 1995 259; Z. Bern, G. Chalmers, Nucl. Phys. BŽ .447 1995 465.

w x Ž .18 F.A. Berends, W.T. Giele, Nucl. Phys. B 313 1989 595; S. Catani, in: Proceedings of the Workshop on New Techniques forCalculating Higher Order QCD Corrections, report ETH-THr93-01, Zurich, 1992.

w x Ž .19 J.M. Campbell, E.W.N. Glover, Nucl. Phys. B 527 1998 264.w x20 S. Catani, M. Grazzini, CERN preprint in preparation.w x Ž .21 G. Altarelli, G. Parisi, Nucl. Phys. B 126 1977 298.w x Ž .22 J. Kalinowski, K. Konishi, T.R. Taylor, Nucl. Phys. B 181 1981 221; J. Kalinowski, K. Konishi, P.N. Scharbach, T.R. Taylor, Nucl.

Ž . Ž .Phys. B 181 1981 253; J.F. Gunion, J. Kalinowski, L. Szymanowski, Phys. Rev. D 32 1985 2303.

21 January 1999


Gluino contribution to the 3-loop quark mass anomalousdimension in the minimal supersymmetric standard model

L. Clavelli, L.R. SurguladzeDepartment of Physics & Astronomy, UniÕersity of Alabama, Tuscaloosa, AL 35487, USA

Received 30 April 1998; revised 15 November 1998Editor: H. Georgi

Abstract

We deduce the gluino contribution to the three-loop QCD quark mass anomalous dimension function within the minimalŽ .supersymmetric Standard Model MSSM from its standard QCD expression. This work is a continuation of the program of

computation of MSSM renormalization group functions. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 11.30.Pb; 12.60.Jv; 12.38.Bx; 14.80.Ly

w xThe renormalization group method 1 is a power-ful tool for the study of many physically interestingquantities within the Standard Model and beyond.Although experimental measurements at the highestavailable energy are consistent with the standard

w xmodel 2 , the observed relationship of the strongcoupling constant at the Z and the weak angle aswell as the value of the m rm ratio vis-a-vis the`b t

top quark mass remain strong indications of a super-Ž . 16symmetric SUSY grand unification above 10 GeV

and a SUSY threshold for squarks and sleptons inthe 0.1 to 1 TeV region. To use the renormalizationgroup method to study the above and other quantitiesone needs to know the renormalization group func-tions, the b function and the quark mass anomalousdimension g . In our previous work we have calcu-m

lated the three-loop QCD b function with the gluinow xcontribution included 3 . In the present work we

deduce the three-loop quark mass anomalous dimen-sion function including the gluino contribution. We

use the known result for the standard QCD expres-sion of the three-loop quark mass anomalous dimen-

w xsion 4 . The two-loop contributions to the ratiom rm have been found to be 20% of the leadingb t

w xcontribution 6 . For this reason a full analysis of thethree-loop contribution would be useful.

The quark mass anomalous dimension is definedas usual

E lnm2ym sg a , 1Ž . Ž .m s2Em

The renormalization of a quark mass within thew xMS type framework 7,8 has the following form

` a aŽ .i sBm sZ msm 1q 2Ž .Ým i´i

where B indicates the ‘‘bare’’ mass. Within the MSframework, the anomalous dimension function is


( )L. ClaÕelli, L.R. SurguladzerPhysics Letters B 446 1999 153–157154

determined by the lowest order pole term in thequark mass renormalization constant. That is,

E a aŽ .1 sg a saŽ .m s s Ea s

2 3a a as s ssg qg qg y . . .1 2 3ž / ž /4p 4p 4p

3Ž .

The a coefficients are related via a renormaliza-i

tion group equation that serves as a powerful checkof the calculation.

Eg a yb a a aŽ . Ž . Ž .m s s i sž /Eas

Esya a a , 4Ž . Ž .s iq1 sEas

where the three-loop QCD b function including thegluino contribution and ignoring squarks is defined

w xas follows 3 .

Ea s2m sa b a , 5Ž . Ž .s s2Em

where2 3a a as s s

b a sb qb qb q . . . .Ž .s 1 2 3ž / ž /4p 4p 4p

6Ž .

with

ng11 4b sy C q N Tq C , 7Ž .1 A f A3 3 ž /2

ng34 202 2b sy C q N TC q C2 A f A A3 3 ž /2ng 2q4 N TC q C , 8Ž .f F Až /2

2857 205 14153 2 2b sy C yN T 2C y C C y CŽ .3 A f F F A A54 9 27

2 44 158 988 3y N T C q C q n CŽ . Ž .f F A g A9 27 27 ˜

224 22 1452 2 3yn N T C q C C y n C . 9Ž .Ž .g f A A F g A27 9 54˜ ˜

The MS renormalization of quark mass can beexpressed as the following multiplicative renormal-ization

y1Z sZ =Z 10Ž .m cc 2

Here Z is the quark propagator renormalization2

constant and Z renormalizes the quark propagatorcc

Ž . Ž .with Hc x c x dx operator insertion. The gluinocontributions to the above renormalization constantsare in one-to-one correspondence with quark loopgraphs and differ from them only by color andsymmetry factors. Our procedure, as used to deter-mine the gluino contributions to Z decay at four-loop

w xlevel 5 and to the b-function at three-loop level, isto decompose the known QCD results into contribu-tions from separate graphs. Then one can determinethe color factors that relate each graph with aninternal quark loop to the corresponding graph with agluino loop. In each case the gluino contribution isgiven by replacing the fundamental representationmatrices by the adjoint representation matrices andsupplying a symmetry factor of 1r2. For instance, asubgraph consisting of a simple quark loop has thecolor factor

N Tr T aT b sN Td abŽ .f f

while the color factor for a simple gluino loop hasŽ .the color and symmetry factor

n ng g˜ ˜a b abTr F F s C dŽ . A2 2

The relative factor 1r2 is due to the Majorana natureof the gluino. Here T a and F a are the gauge groupgenerators in the fundamental and adjoint representa-

Ž .tions respectively. For the gauge group SU N theysatisfy

Tr T aT b sTd ab , Tr F aF b sC d abŽ . Ž . A

Thus for graphs with a simple fermion loop sub-graph one obtains the gluino contribution by makingthe trivial substitution

ngN T™N Tq C 11Ž .f f A2

Here n s0 is the Standard Model limit and n s1g g˜ ˜corresponds to the minimal SUSY extension withone octet of gluinos. The full set of three-loop graphscontributing in the standard QCD Z and Z were2 cc

w xgiven in Ref. 4 . It is sufficient to focus one’sattention on graphs with one or more internal fermionloops which, at present order, appear with two, three,or four gluon attachments. The graphs with only twogluons attached to an internal fermion line are gener-

( )L. ClaÕelli, L.R. SurguladzerPhysics Letters B 446 1999 153–157 155

Fig. 1. Three-loop graphs giving nontrivial contributions to thequark mass anomalous dimension function with the gluino in-cluded. Wavy lines denote gluons and the solid loop correspondsto a quark or gluino.

alized to include gluinos as discussed above. Thegraphs with three gluons attached to the internalfermion line are generalized by the replacement

aw b c x aw b c xN Tr T T ,T ™N Tr T T ,TŽ . Ž .f f

n g a b cw xq Tr F F ,F 12Ž .Ž .2

It is easy to see that here also the gluino contribu-tions are trivially incorporated by the replacement of

Ž .Eq. 11 . Only the graphs with four gluon attach-ments depart from this rule. For the graph of Fig. 1a,the required replacement is

N Tr T aT cT cT b ™N Tr T aT cT cT bŽ . Ž .f f

n g a c c bq Tr F F F F 13Ž . Ž .2

or

C 2 n 27nA g g˜ ˜N ™N q sN q . 14Ž .f f f2TC 4F

The corresponding replacement for the graph of Fig.1b takes the form

N Tr T aT cT bT c ™N Tr T aT cT bT cŽ . Ž .f f

n g a c b cq Tr F F F F 15Ž . Ž .2

or

C 2 nA gN ™N q sN y27n . 16Ž .f f f g4T C yC r2Ž .F A

Resumming all graphs, we obtain the following re-sult for the three-loop quark mass anomalous dimen-sion function with gluino included.

g s3C 17Ž .1 F

ng3 97 10g sC C q C y TN q C 18Ž .2 F F A f A2 6 3 ž /2

129 1292g sC C y C C3 F F F A2 4

yC TN 46q48z 3 C yCŽ . Ž .Ž .F f A F

11413 5562q C y C TNA A f108 27

2ng140y TN q Cf A27 ž /2

1 1771 2yn C C C q C 19Ž .g F F A A2 54˜

The eigenvalues of the Casimir operators for theŽ . Ž .adjoint N s8 and the fundamental N s3 rep-A F

Ž .resentations of SU 3 arec

C s3, C s4r3, and Ts1r2. 20Ž .A F

We obtain the following values for the above pertur-bative coefficients of the g-function.

g s4 21Ž .1

202 20 20g s y N y n 22Ž .2 f g3 9 3 ˜

2216 160 140 2g s1249y q z 3 N y NŽ .Ž .3 f f27 3 81

3566 280 140 2yn q N y n 23Ž .Ž .g f g9 27 9˜ ˜

We see that the gluino gives a substantial contri-bution to the two-loop and especially to the three-looplevels. Indeed, the gluino contribution reduces thetwo-loop coefficient by about 10% and reduces thethree-loop contribution by about 50%. Such a largecontribution might ultimately be important in phe-nomenological applications.

The anomalous dimension of the quark mass alongwith the QCD b-function determines the running ofthe quark mass. Indeed, at the three-loop level, for

Ž w x.the running quark mass one has see, e.g., 9

m m f a mŽ . Ž .Ž .f 1 s 1s , 24Ž .

m m f a mŽ . Ž .Ž .f 2 s 2

( )L. ClaÕelli, L.R. SurguladzerPhysics Letters B 446 1999 153–157156

whereg1

yba m 1Ž .s

f a m s y2bŽ .Ž .s 1ž /4p

=g b g a mŽ .2 2 1 s

1y y 2ž /½ b 4pb1 1

21 g b g g2 2 1 3q y y2ž /2 b bb1 11

2b g b g b g2 2 3 1 2 1q q y2 2 3b b b1 1 1

=

2a mŽ .s

25Ž .ž / 54p

The above equation indicates that there will be asubstantial shift in the running of a quark mass dueto the gluino.

To verify this, we will need the following equa-tions. The running coupling is parametrized as fol-lows:

a m 1 b log L 1Ž .s 2sy y y3 2 5 34p b L b L b L1 1 1

= b 2 log2Lyb 2 log Lqb b yb 2Ž .2 2 3 1 2

qO Ly4 , 26Ž . Ž .2 2Ž .where Ls log m rL .MS

The general evolution equation for the runningŽ 3. w xcoupling to O a 9 has the forms

2Ž .Žn. ŽN . Na m a M a MŽ . Ž . Ž .s s ss y ž /4p 4p 4p

=M 2 m2

l2Ž .Nb log y logÝ1 32 2ž /m ml

3Ž .N 2a M MŽ .s Ž .Ny b log2 2ž /4p m

m2 M 2l38 Ž .Ny log q b logÝ 13 2 2žm ml

22ml2 50y log q Nyn 27Ž . Ž .Ý3 92 /ml

Ž .where the superscript n N indicates that the corre-Ž .sponding quantity is evaluated for numbers n N ofŽ .participating quark flavors. Conventionally, n N is

specified to be the number of quark flavors withŽ . Ž .mass Fm FM . However, the Eq. 27 is relevant

for any nFN and arbitrary m and M, regardless ofthe conventional specification of the number of quarkflavors. The log m rm terms are due to the ‘‘quarkl

threshold’’ crossing effects and the constant coeffi-cients 2r3sb Žky1.yb Žk ., 38r3sb Žky1.yb Žk .

1 1 2 2

represent the contributions of the quark loop in theb-function. The sum runs over Nyn quark flavorsŽ .e.g., lsb if ns4 and Ns5 . Note that m is thel

pole mass of the quark with flavor l. Quark massescan be estimated from QCD sum rules. For instance,

w xthe b quark pole mass is m s4.72 GeV 10 . InbŽ .Fig. 2 we show the evolution of m m from msb

4.72 GeV to M . We see that the gluino effect is aZ

few percent at M which could ultimately be impor-Z

tant for grand unification studies. If the gluino is notlight but is nevertheless below the squark mass, ourcurrent result will still have a region of relevance.Ultimately, of course, it will be desirable to have thefull SUSY three-loop effect including squark contri-butions.

Analyzing numerical results, we see that thethird-loop effect is a few MeV. Based on this, weconclude that the three-loop anomalous dimensionwith the gluino included is a good approximation forphenomenological applications and the error in the

Ž . Ž .perturbative evaluation of m m ym m canb b b Z

probably now be trusted to a few MeV, i.e. about0.1%.

Fig. 2. The gluino effect on the running of the b quark mass

( )L. ClaÕelli, L.R. SurguladzerPhysics Letters B 446 1999 153–157 157

An observable quantity R is invariant under therenormalization group transformations and obeys thehomogeneous renormalization group equation:

E E E2m qb a a yg a mŽ . Ž . Ýs s m s f2ž /Ea E mEm s ff

=R m ,a ,m s0 28Ž . Ž .s f

The quantity R may denote various cross sectionsand decay rates calculated using perturbation theory.In previous work we have calculated the gluinocontribution to the QCD b-function at the three-loop

w xlevel 3 and to the hadronic decay rate of the Zw xboson to the four-loop level 5 . Thus, the present

work completes the evaluation of gluino contribu-Ž 3.tions necessary for O a renormalization groups

analysis for the above quantity. These results canalso be applied to a renormalization group analysisof other quantities such as hadronic decay rates ofthe t-lepton and various Higgs bosons.

If the gluino lies above the squark, the currentcalculation provides the contribution from gluinosalone up to three-loop order. This is a gauge invari-ant subset of the full SUSY three-loop graphs and

Žleaves a vastly reduced number of graphs those with.one or more internal squark lines still to be calcu-

lated at this order. If the gluino lies lower in massthan the squarks, the current calculation provides the

full SUSY anomalous dimension of quark mass up toand including three-loop order in the region up to thesquark mass scale.

Acknowledgements

We thank Phil Coulter for useful discussions. Thiswork was supported by the US Department of En-ergy under grant no. DE-FG02-96ER-40967.

References

w x1 N.N. Bogolyubov, D.V. Shirkov, Introduction to the Theoryof Quantized Fields, Wiley, New York, 1980.

w x2 M. Demarteau, in: Proc. DPF 96 Meeting, Minneapolis,Ž .August 1996, to be published ; J. Hewett, ibid.

w x3 L.J. Clavelli, P.W. Coulter, L.R. Surguladze, Phys. Rev. DŽ .55 1997 4268.

w x4 O.V. Tarasov, Anomalous dimensions of quark masses inthree-loop approximation, Dubna Joint Institute for NuclearResearch Preprint No. JINR-P2-82-900, 1982.

w x Ž .5 L.J. Clavelli, L.R. Surguladze, Phys. Rev. Lett. 78 19971632.

w x Ž .6 D.V. Nanopoulos, D.A. Ross, Phys. Lett. 108B 1982 351.w x Ž .7 G. ’t Hooft, M. Veltman, Nucl. Phys. B 44 1972 189.w x Ž .8 G. ’t Hooft, Nucl. Phys. B 61 1973 455; W. Bardeen, A.

Ž .Buras, D. Duke, T. Muta, Phys. Rev. D 18 1978 3998.w x Ž .9 L.R. Surguladze, Phys. Lett. B 341 1994 60.

w x Ž .10 C.A. Dominguez, N. Paver, Phys. Lett. 293B 1992 197.

21 January 1999


The intercept of the BFKL pomeron from forward jets at HERA

J.G. Contreras a,1

a Departamento de Fısica Aplicada, CINVESTAV–IPN, Unidad Merida, A.P. 73 Cordemex, 97310 Merida, Yucatan, Mexico´ ´ ´ ´

Received 20 October 1998Editor: H. Georgi

Abstract

Recently the H1 and ZEUS collaborations have presented cross sections for DIS events with a forward jet. The BFKLformalism is able to produce an excellent fit to these data. The extracted intercept of the hard pomeron suggests that when allhigher order corrections are taken into account the cross section will still rise very rapidly as expected for low x dynamics.q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 13.60; 13.87

Keywords: BFKL; DIS; Forward Jets

1. Introduction

Using a physical gauge, deep inelastic scatteringprocesses can be represented within perturbativeQCD with ladder diagrams like the one shown inFig. 1. In this picture the interaction between anelectron and a proton is mediated through the ex-change of a virtual photon of four momentum squaredq q msyQ2, which couples to a quark–antiquarkm

box at the top of the parton ladder inside the proton.w x w xThe DGLAP 1 and BFKL 2 schemes select

different leading logarithmic regions of the phasespace to describe the partonic evolution along thisladder. The DGLAP approach takes into account the

2 Ž .leading terms in lnQ , but neglects those in ln 1rx .


In this case the square of the transverse momentumof the partons along the ladder are strongly order,

2 2 Ž .k 4k cf. Fig. 1 , whereas the longitudinal mo-i iy1

menta obey x -x . On the other hand, in thei iy1Ž .BFKL formalism the ln 1rx are resummed in the

region of k 2 fk 2 and x <x . Due to thei iy1 i iy1

inherent approximations involved the domain of ap-plicability of the DGLAP equations is medium tohigh x–Bjorken, but the BFKL approach shouldonly be used at small x–Bjorken.

The electron–proton collider HERA opened upthe study of DIS to new domains of small values ofx–Bjorken and still sizable virtuality of the photon.It was thus expected that experiments at HERAwould be able to measure the transition from theregion of applicability of DGLAP to that of BFKL.Among the different proposed observables to test theBFKL approach to DIS, the measurement of forwardjets is considered to be one of the most promising.


( )J.G. ContrerasrPhysics Letters B 446 1999 158–162 159

Fig. 1. Representation of a deep inelastic event using a physicalguage within perturbative QCD.

Higher order corrections to the BFKL formalismw xhave been recently calculated 3 . They turned out to

be sizable and pointed out the need of still higherorder corrections. This reduced the quantitative pre-dictive power of the leading logarithmic approxima-tion to BFKL calculations, but the qualitative behav-ior at small x may still remain the same. Taking thisinto account a fit to HERA data, which of courseresumes all orders, may shed light into the problemof the numerical stability of the BFKL kernel underhigher order corrections. In this spirit the intercept of

the BFKL pomeron is extracted from a fit to theforward jet data of the H1 and ZEUS collaborations.

2. Forward jets at HERA

w xBack in 1990 Mueller 6 proposed to look forDIS events with one jet – other than the jet originat-ing from the struck quark – fulfilling the following

Ž .characteristics cf. Fig. 1 :Ø x small. The selection of the smallest x–Bjorken

experimentally possible implies going away fromthe domain of validity of DGLAP and into aphase space area governed by BFKL.

Ø x large. This, along with the previous item,J

provides the phase space for parton evolution andhas the extra advantage to enable the use of

Ž .parton density functions PDF of the proton in aregion where they have already been measured,so no extrapolation is needed.

Ø k 2 fQ2. This requirement suppresses DGLAPJ

evolution without affecting the BFKL dynamics.It also provides a big scale at both ends of theladder, avoiding thus the dangerous infrared re-gions of phase space and increasing the solidityof the analytic predictions.The process so selected provides a well defined

topology for the experimentalist. One has to look fora DIS event with a jet at high rapidities and enforcethat the virtuality of the proton is of the same scaleas the transverse momentum squared of the jet. Asthe direction of the proton is termed by the HERAexperiments the forward direction, this kind of events

Table 1w xCross sections for the forward jet production measured by the H1 4 collaboration along with average values for the kinematic variables

w xobtained from the Ariadne MC 10 . Note that the quoted errors are an average of the slightly asymmetric cuts reported by the collaboration2 2w x ² : ² : w x ² : w x ² : w xs nb x Q GeV E GeV k GeVJ J

H1 k )3.5 GeV 342 " 55 0.00073 21.5 34.4 5.0J

224 " 32 0.0012 26.9 35.4 5.5138 " 25 0.0017 31.4 36.9 5.867 " 11 0.0024 38.1 38.1 6.332 " 5 0.0035 47.0 38.8 6.9

H1 k )5.0 GeV 132 " 20 0.0012 32.2 37.9 6.3J

96 " 20 0.0017 334.8 39.3 6.555 " 10 0.0024 40.1 39.4 6.728 " 6 0.0035 48.2 39.6 7.2

( )J.G. ContrerasrPhysics Letters B 446 1999 158–162160

are known as forward jets events. Recently both thew x w xH1 4 and the ZEUS 5 collaborations have pre-

sented cross sections for this observable. The H1collaboration selects DIS events requiring the energyŽ . Ž .E and polar angle u of the scattered electron toe e

be E )11 GeV and 1608-u -1738. ZEUS onlye e

requires E )10 GeV and the polar angle u ise e

restricted by the rest of the kinematic constrains.Both of them select events with y )0.1 to avoidBj

the jet from the struck quark going forward. In thejet side, selected with a cone algorithm of radius 1,the cuts used are for H1: k )3.5 GeV, x fJ J

E rE sE r820 GeV)0.035 and 78-u -208.J beam J J

ZEUS ask for jets with k )5.0 GeV, x )0.036J J

and 8.58-u . Both collaborations require that 0.5-J

k 2rQ2 -2.0. The luminosity used is 2.8 and 6.4J

pby1 for the H1 and ZEUS collaborations respec-tively. The H1 collaboration also reports a similarsearch, where all cuts were kept the same, exceptthat k )5.0 GeV was required. The measured crossJ

sections for different bins in x–Bjorken are pre-sented in the first column of Tables 1 and 2. Notethat both collaborations presented slightly asymmet-rical systematic errors. The errors presented in Ta-bles 1 and 2 are an average of the systematic errorsadded in quadrature to the statistical error of themeasurement. Note also that the measurements at thelowest x–Bjorken have not been included. This isbecause, due to the 0.5-k 2rQ2 -2.0 cut, the ex-J

periments ran out of phase space. Thus these pointsmix dynamic effects with phase space restrictionsand are not useful for the analysis presented here.

Table 2Cross sections for the forward jet production measured by the

w xZEUS 5 collaboration along with average values for the kine-w xmatic variables obtained from the Ariadne MC 10 . Note that the

quoted errors are an average of the slightly asymmetric cutsreported by the collaboration

2w x ² : ² : ² : ² :s nb x Q E kJ J2w x w x w xGeV GeV GeV

77.8 " 7.6 0.0019 50.7 39.9 7.634.4 " 3.6 0.0033 75.6 43.8 8.714.1 " 2.1 0.006 113.6 49.6 10.46.5 " 0.7 0.010 176.4 58.5 12.92.7 " 0.4 0.018 244.7 67.3 15.10.6 " 0.3 0.031 366.8 78.8 18.8

3. BFKL fits to the forward jet data

The cross section for DIS events containing aforward jet has been calculated at leading logarith-mic approximation within the BFKL formalism in

w xRefs. 7–9 . There the following form has beenfound:

4 2d s Q2 2 2k x sCa Q F x ,QŽ . Ž .)J J s J2 2 2dxdQ dx dk kJ J J

=exp a y1 ln x rxŽ .p J

1r2ln x rxŽ .J

1Ž .

Ž 2 .where a Q is the QCD coupling constant at thes

scale Q2 and F is a generic parton distribution in theproton given by

42 2 2F x ,Q sx G x ,Q q x q x ,QŽ . Ž . Ž .J J J J J9

2qx q x ,Q . 2Ž .Ž .J J

w xIn 7–9 an explicit form for the parameters Cand a can be found. Recently the NLO correctionsp

w xto the BFKL kernel have been presented 3 . Thesecorrections turn out to be very large implying thatNNLO calculations are needed. In particular thecorrections affected the intercept of the BFKLpomeron a , reducing the predictive power of thep

w x w xexplicit forms given in 7–9 at LO and in 3 atNLO. This means that the exact power of the leading

Ž .behavior of Eq. 1 ,

a y1px J, 3Ž .ž /x

is not numerically known from BFKL calculations.Nonetheless given that the first experimental data forthis process are available, it is tempting to test if the

Ž .form of Eq. 1 does indeed describe the measure-ment, and if so, which intercept of the BFKLpomeron is favored by the data.

To perform the fit there are some other ingredi-ents needed. Standard parton density functions are

Ž .required to evaluate F. Also as Eq. 1 is a fourdifferential expression, values for x, Q2, x and k 2

J J

( )J.G. ContrerasrPhysics Letters B 446 1999 158–162 161

Table 3Values obtained for a from a fit of the BFKL formalism to the data on forward jet production using different PDFsp

PDF H1 k )3.5 GeV H1 k )5.0 GeV ZEUSJ J

2 2 2x a x a x ap p p

GRV–LO 1.05 1.6 " 0.4 0.24 1.7 " 0.5 0.81 1.15 " 0.13GRV–HO 1.0 1.8 " 0.4 0.24 1.8 " 0.5 0.76 1.24 " 0.14CTEQ–4M 0.99 1.8 " 0.4 0.22 1.8 " 0.5 0.79 1.26 " 0.14MRS–R1 1.0 1.8 " 0.4 0.24 1.8 " 0.5 0.91 1.26 " 0.14

are needed in each x–Bjorken bin. Both collabora-tions, H1 and ZEUS, report that the Monte Carlo

w xgenerator Ariadne 10 describes very well, not onlythe x–Bjorken dependence, but all distributions in-volved in the analysis. So a similar analysis to thatreported by both collaborations has been performedat the hadron level of the Ariadne Monte Carlo. Ithas been checked that the cross section obtained withthis procedure agrees with both, the reported dataand also with the expectations from the AriadneMonte Carlo as given by the H1 and ZEUS collabo-rations. Using this Monte Carlo data set for forwardjet events the mean values of the variables x, Q2, x J

and k have been estimated for each measured pointJ

of both collaborations. The results are presented inTables 1 and 2 along with the measured data pointsand their errors.

Using the input of Tables 1 and 2, the PDFs givenw x w x w xby GRV-LO 11 , GRV-HO 12 , CTEQ-4M 13 and

w xMRS-R1 14 – the last three calculated in the MSscheme – and an a value consistent with each ofs

the different parton density functions, a 2 parameterfit was performed separately to the H1 and the ZEUS

Ž . 2points using formula 1 . The x and the value ofa obtained are shown in Table 3.p

4. Discussion

The results of the fits can be summarized asfollows:

1. All data points could be successfully fit to theŽ .form of Eq. 1 . This is quite encouraging because

that was the main motivation for this kind of mea-surements.

2. The fits are insensitive to the PDF used. Thisresult is also as expected. As a matter of fact this

was one of the main advantages of the forward jetproposal.

3. Different values of the exponent a are foundp

when using LO or NLO PDF. On the one hand,Ž .being the formula 1 a LO approximation to BFKL,

one is tempted to consider only the use of LO PDFs.On the other hand, the whole idea of performing thefits is to have a data driven estimation of the effectsof higher order corrections to the BFKL kernel. Theactual variation of the value of a is expected in thep

basis of it being proportional to a in LO. As thes

value of a decreases in going from LO to NLO, thes

value of a must compensate this trend and increasep

from one case to the other.4. The values of a found using H1 data arep

different to those found using ZEUS measurements.On the one hand ZEUS data is more precise having afactor of 3 more luminosity. This allowed the ZEUScollaboration to measure the cross section in a regiondefinitely dominated by DGLAP, having thus a cleantransition from the box diagram at the top of Fig. 1to the case when the box is complemented with a

Ž .ladder. This greatly constrains the fit to Eq. 1 . Onthe other hand H1 points reach smaller x, which isthe interesting region for BFKL studies, althoughstill with huge errors. One possible explanation forthe difference in the values obtained using the dataof both collaborations, could be again the a depen-s

dence of a . Note that the average Q2 is a lot biggerp

in the ZEUS measurement than in the H1 case. In theLO approximation this would naively produce a 15to 20% increase of a from H1 data with respect top

that from ZEUS cross sections.Ž .5. The formula 1 is at parton level, i.e. it does

not includes hadronization effects. As reported byboth collaborations these are uncertain. Differentmodels yield not only different normalization, but

Žmay also yield a x dependence see for example

( )J.G. ContrerasrPhysics Letters B 446 1999 158–162162

w x.Fig. 8 in Ref. 5 of the correction from hadron toparton level.

6. Using the dipole approach to BFKL, it hasbeen shown that the measurements of the structurefunction F can be described using an intercept of2

w xa s1.28 15 . Some care is necessary when com-p

paring this value with the one obtained here. Thequestion of infrared divergencies due to the random

w xwalk generated by the BFKL kernel 16 is quitesensitive for the F case, but it does not appear in2

the case of forward jets. Nevertheless is quite com-forting that two so different observables yield resultscompatible with the BFKL formalism.

7. The possibility to experimentally reach lower xat still sizable Q2 is very important. The fits pre-sented here, although they could not used the lowestx points due to phase space constrains, have shownthat the BFKL dynamics enforces a steep rise of thecross section for a process governed by a hardpomeron inspite the NLO corrections to the BFKLkernel. Reaching smaller x will allow to access thesaturation region of hot spots in the proton, whichwas one of the primary motivations of the forwardjet proposal. This goal may still be reachable atHERA.

5. Conclusion

A fit was performed of a BFKL prediction toforward jet production as measured by the H1 andZEUS collaborations. All data were consistent withthe assumption of using the BFKL formula for this

process. Difference in the intercept of the pomeronobtained with different sets of data, may be assignedto its a dependence. The fits support the idea of as

hard pomeron and point in the direction that when allhigher order corrections are taken into account theforward jet cross section will still be rising quiterapidly.

References

w x Ž .1 Yu.L. Dokshiter, Sov. Phys. JETP 46 1977 641; V.N.Ž .Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 1972 438,

Ž .675; G. Altarelli, G. Parisi, Nucl. Phys. 126 1977 297.w x2 E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45

Ž .1972 199; Y.Y. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys.Ž .28 1978 822.

w x Ž .3 V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429 1998 127.w x4 H1 Collaboration, C. Adolff et al. DESY preprint DESY

98-143, hep-exr9809028, 1998.w x5 ZEUS Collaboration, J. Breitweg et al., DESY preprint DESY

98-050, hep-exr9805016, 1998.w x Ž . Ž .6 A.H. Mueller, Nucl. Phys. B Proc. Suppl. 18C 1991 125.w x Ž .7 J. Bartels, A. De Roeck, M. Loewe, Z. Phys. C 54 1992

635.w x Ž .8 W.-K. Tang, Phys. Lett. B 278 1992 363.w x9 J. Kwiecinski, A.D. Martin, P.J. Sutton, Phys. Rev. D 46

Ž .1992 921.w x Ž .10 L. Lonnblad, Comp. Phys. Comm. 71 1992 15.¨w x Ž .11 M. Guck, E. Reya, A. Vogt, Z. Phys. C 53 1992 127.¨w x Ž .12 M. Guck, E. Reya, A. Vogt, Z. Phys. C 67 1995 433.¨w x Ž .13 H.L. Lai, Phys. Rev. D 55 1997 1280.w x14 A.D. Martin, R.G. Roberts, W.J. Stirling, Phys. Lett. B 387

Ž .1996 419.w x15 H. Navelet, R. Peschanski, Ch. Royon, S. Wallon, Phys. Lett.

Ž .B 385 1996 357.w x Ž .16 J. Bartels, H. Lotter, Phys. Lett. B 309 1993 400.

21 January 1999


Possible nonstandard effects in Zg events at LEP2

K.J. Abraham a, Bodo Lampe b

a Dept. of Physics & Astronomy, Iowa State UniÕersity, Ames, IA 50011, USAb Max Planck Institut fur Physik, Fohringer Ring 6, D-80805 Munchen, Germany¨ ¨ ¨

Received 9 October 1998; revised 24 November 1998Editor: R. Gatto

Abstract

We point out that the so-called ‘radiative return’ events eqey™Zg are suited to the study of nonstandard physics,Ž .particularly if the vector bosons are emitted into the central detector region. An effective vertex amplitude is constructed

which contains the most general gauge invariant eqeyZg interaction. The interference pattern between this and the StandardModel eqeyZg amplitude is calculated. Phenomenolgocial consequences of the results are examined. Low EnergyConstraints on the effective vertex are discussed as well. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

In the last decade a huge number of Z’s has beenproduced by the LEP1 experiment working on theZ-pole and the data obtained has been used for highprecision tests of the Standard Model and to estab-lish stringent bounds on physics beyond the StandardModel. Subsequently, the LEP2 experiment hasstarted data taking at higher energy with the primaryaim of determining the mass and selfinteractions ofthe W-bosons. Unfortunately, only a few thousandW-pairs will be available for this study, becausecross sections at LEP2 are generally much smallerthan at LEP1, and the bounds on new physics will be

w xcorrespondingly weak 1 .On the other hand, LEP2 is collecting a relatively

large number of events with a very hard photon andan on-shell Z in the final state which in the StandardModel are produced by Bremsstrahlung of the pho-ton from the eqey legs. From the experimentalpoint of view the production of Z’s together with a

hard prompt photon is a very clear and pronouncedsignature. Nevertheless, this class of events is usu-

w xally not considered very interesting 1 , because theyŽseem to lead back to LEP1 physics ‘radiative re-

.turn’ to on-shell Z production . However, it is ex-w x Ž .pected 2 that very roughly about 10000 Zg events

will be collected until the end of LEP2, with anangle u of the photon larger than 5 degrees withrespect to the eqey beam direction. There will beless of such events in the central region u)308 ofthe detector, but still about 1000 of them will beavailable to the analysis. The primary motivation forthe present study is whether at all there is a possibil-ity to use these events in a nonstandard physicsdiscussion.

In our approach we shall use an effective vertexfor the eqeyZg interaction which contains the Stan-dard Model part plus a small admixture of a newcontribution. The new contribution will be presentedin its most general form, i.e. as a sum of independentspinor and tensor structures. The approach comprises


( )K.J. Abraham, B. LamperPhysics Letters B 446 1999 163–169164

contributions from operators of arbitrary high dimen-sions. It can also be considered to effectively de-scribe the exchange of new heavy particles or someother exotic mechanism for g Z production like

q y ) w x q ye e ™Z ™Zg 3 . Point-like e e Zg vertices ofthe kind we consider can also be induced by exten-sions of the Standard Model such as the MSSM aswell as in models with composite fermions andgauge bosons. A study of the phenomenologicalconsequences of compositeness on g Z final states at

w xhadron colliders has been undertaken in Ref. 4 ; ourinvestigation of the same final state is howeverindependent of any given extension of the StandardModel.

The new contribution is supposed to be small, ofthe order of some small coupling constant d<1.Therefore, we will be mainly interested in interfer-ence terms between the Standard Model and the newvertices. More precisely, if s is the StandardSM

Model contribution to any differential cross sectionand ds is the interference contribution, we shallNEW

consider the ratio ds rs . To a first approxi-NEW SM

mation it is sufficient to use the lowest order Stan-dard Model result s with an on-shell Z in theLOSM

calculation of that ratio. We assume that the Stan-dard Model Radiative Corrections are sufficientlysmall so that only the interference amplitude be-tween the Standard Model Contribution and the Ra-diative Corrections are relevant. This interferenceamplitude can be recast in terms of form factors wewill introduce later on. New Physics can appear asform factors which do not arise in the StandardModel interference amplitude or as unexpected val-ues for form factors which do arise in the StandardModel interference amplitude.

2. Construction of the eHe IZg vertex

qŽ . yŽ . Ž . Ž .The process e p e p ™g k Z p is de-q y Z

picted as a Feynman diagram in Fig. 1a and in akinematic view in Fig. 1b. The polarization indicesof the photon and of the Z are denoted by a and m,respectively. We parametrize the momenta as being

s' Ž . Ž .p s 1,0,0,"1 , ksE 1,0,sinu ,cosu and p" g Z2

Ž 2 2 .sp qp yk. For an on-shell Z p sm oneq y Z Z

Fig. 1. The kinematics of the Zg process.

Žobtains a hard monochromatic photon of normal-.ized energy

E m2g Z

x ' s1y 1Ž .g ' ss r2

which is about x s0.75–0.8 at LEP2 energies andg

goes up to almost 0.97 at an eqey collider with's s500 GeV. As discussed above, an on-shell Z isa reasonable approximation to study nonstandardeffects. E being constant, the process’ real kinemat-g

ical variable is the production angle u of the photon.There are essentially 2 regions for photon detection,depending on the polar angle,Ø The region collinear to the beam with large

bremsstrahlung contributions is dominated by theStandard Model amplitude which has poles atŽ .2 2kyp sm ." e

Ø The central region of the detector, where theStandard Model cross section has its minimumvalue, so that one may be sensitive to nonstandardphysics.The amplitude for Zg production has the general

form G e Ze g where e Z is the polarization vectorma m a m

of the Z and e g is the polarization vector of thea

photon. The vertex G may be decomposed asma

G sG SM qdG 2Ž .ma ma ma

The first term in this expression is the StandardModel contribution to the vertex

ku ypuqSM 2G syie Q g g Õ qa gŽ .ma l a m l l 52 2½ kyp ymŽ .q e

ku ypuyyg Õ qa g g 3Ž . Ž .m l l 5 a2 2 5kyp ymŽ .y e

( )K.J. Abraham, B. LamperPhysics Letters B 446 1999 163–169 165

where

1a s f0.594l 4 s cW W

and

1 sWÕ sy q fy0.05l 4 s c cW W W

are the vector and axial couplings of the electron tothe Z. Note that the numerical value of Õ is small asl

compared to a . This will be important later on,l

when interference terms between G SM and dGma ma

will be discussed and terms of the order Õ d will bel

neglected as compared to terms of order a d . ThelŽ .second term in Eq. 2 has the general form

1 1dG s ie f kug g q f g k ykugŽ .ma 1 a m 2 a m am½ s s

1q f g p kp yp kpŽ .3 m q,a y y,a q2s

1q f p g kp yp kuŽ .4 y,m a q q,a2s

1q f p g kp yp kuŽ .5 q,m a q q,a2s

1q f p g kp yp kuŽ .6 y,m a y y,a2s

1q f p g kp yp ku 4Ž .Ž .7 q,m a y y,a2 5s

where f <1 are dimensionless coupling constantsi

whose strength cannot be predicted within our ap-proach. One factor of e has been introduced in the

Ž .definition of the vertex 4 for convenience. In writ-ing down this formula several requirements are takencare of:Ø The requirement of electromagnetic gauge invari-

Ž .ance is fulfilled by forming in Eq. 4 suitablecombinations such that k adG s0. under in-ma

finitesimal gauge transformations.Ž . Ž .Ø Eq. 4 is not explicitly SU 2 invariant. TheL

point is that in our approach the new interactionsare not necessarily tied to very high energies butcould in principle be related to energies below 1TeV. Therefore we have refrained from make Eq.Ž . Ž .4 explicitly SU 2 invariant. However, if de-L

Ž .sired, one can enforce global SU 2 by adding aL

similar correction to the engW vertex. This willenforce additional low-energy constraints on thenew vertex to be discussed in Section 4.It should be noted that our vertex is trivially

Ž .invariant under those local SU 2 gauge varia-L

tions which transform the Z into itself, becausethese are given by de m ;p m and thus one needsZ Z

to have p mdG s0. Although this condition isZ ma

Ž .not explicitly fulfilled by Eq. 4 , the situation isautomatically cured, because one can replace all

qpZ4-vectors qsp , p ,k by q y p and g2y q m Z,m ammZ

p pZ,m Z ,aby g y , and this replacement does notam2mZ

change the cross section, because terms ;pZ,mŽ .give O m when sandwiched between the leptone

spinors.Ø The ‘coupling constants’ f are really form factorsi

Ž .f s f x ,cosu . Powers of s have been intro-i i g

Ž .duced in Eq. 4 in such a way as to make the fi

dimensionless. The new contribution would havethe same overall energy dependence as the Stan-dard Model contribution, if one would assume thef to first approximation to be constant in energyi

x s1ym2rs. However, this would be a ratherg Z

unpleasant feature, because it would induce ef-Žfects already at the lowest energies see the dis-

.cussion of low energy constraints in Section 4 .Therefore, it is a good idea to assume that theform factors behave like some negative power of1yx in order to cut away the low energyg

constraints. Physically, such a behavior arises, forexample, if the new physics is induced at somescale L;1 TeV which forces the form factors tobehave like a power of srL2. Furthermore, suchbehaviour is also desirable for the purpose ofsatisfying constraints from unitarity at high ener-gies 1.

Ø The coupling constants f are of the general formi

f sÕ qa g and are really form factors f si i i 5 iŽ .f x ,cosu .i g

1 w xIn an earlier work 5 , the effects of a single form factor, alinear superposition of f . . . f was considered in the context of a4 7

constituent model for quarks and gauge bosons. Our analysis goesbeyond this single form factor and is not restricted to a fixedmodel.


'Ø Terms which give contributions of order m r se

when the interference with the Standard Model isŽ .formed, have not been included in Eq. 4 . Such

terms consist of a product of an even number of g

matrices.Ž .Ø The interactions in Eq. 4 conserve CP. Using

complex form factors one could also undertake asearch for CP violating interactions – in analogy

w xto what has been done in Ref. 6 concerning thebbg Z vertex.

3. Quantitative phenomenological consequences

Ž .With the vertex 4 at hand one can calculate thecross section dsrdcosu where u is the productionangle of the photon. dsrdcosu consists of a Stan-dard Model term ds rdcosu which is propor-SM

tional to Õ 2 q a2 and an interference terml l

ds rdcosu . These interference contributions be-NEW

tween G SM and dG are ;Õ Õ ya a . For conve-ma ma l i l i

nience the numerical analysis will be done only forthe form factors f but not for f . Due to the1,2,3,4 5,6,7

fact that the Standard Model coupling Õ almostl

vanishes it turns out that the Õ practically do noti

contribute to the interference term and that the ratioof ds rdcosu and ds rdcosu is proportionalNEW SM

to a ra . Explicitly one hasi l

ds rdcosuNEW

ds rdcosuSM

1 1s 2 2 2 2Õ qa 4 x q2y2 x rsin uy2Ž .l l g g

= Õ Õ ya a 2yx 1qcosuŽ . Ž .� Ž .l 1 l 1 g

q Õ Õ ya a x cosuŽ .l 2 l 2 g

q Õ Õ ya a 2 x y1Ž . Ž .l 3 l 3 g

q Õ Õ ya a x cosuy1 q PPP 5Ž . Ž . Ž .4l 4 l 4 g

where x s1ym2rs as before and the dots standg Z

for similar terms stemming from the other formfactors i)4. Note that ds rdcosu becomes verySM

large in the collinear regions cosuf"1 and staysroughly constant and small in the central regioncosuf0. This can be seen in Fig. 2 where the shapeof the Standard Model distribution is plotted. Fig. 3

Fig. 2. SM cross section as a function of the cosine of thephoton’s angle.

ds rNE W dcosu Ž .shows the ratio , cf. Eq. 5 , as a functionds rdcosuSM

of cosu for form factors with numerical valuesa s0.025, a s0.05, a s0.05 and a s0.05 at1 2 3 4'Ž .x s0.77 corresponding to s s190 GeV . It isg

nicely seen that the different form factors contributequite differently to the u distribution. The knots atcosus"1 are due to the fact that the new contribu-tions remain regular at this point whereas the Stan-dard Model cross section is large. Note that the resultFig. 3 is linear in the a , so that results for otheri

values may be easily determined by rescaling. Notefurther that the order of magnitude of the result isroughly the same as the magnitude of the a . Thei

values we choose for the form factors a are some-i

what arbitrary, as we do not wish to consider a fixedextension of the Standard Model. In the MSSM forexample, it may be expected that the induced valuesfor a are substantially smaller than those we con-i

sider. In strongly coupled models however, there areno computations from first principles available forthe values of a , experimental bounds will be ofi

importance in shedding light on the dynamics ofthese models.

The cross-section for Zg production with y0.6-cosu-0.6 is 4.2 pb, corresponding to about 2000events, assuming a luminosity of 500 pby1. Neglect-ing systematic errors, this points to a discovery limitcorresponding to a deviation of about 3–4 % fromthe Standard Model. From Fig. 3, the resulting dis-


Fig. 3. Ratio of nonstandard physics to SM as a function of thecosine of the photon’s angle.

covery bounds on the values of the anomalous formfactors a can be estimated to be ;0.1 2.i

This information is based solely on the g spec-trum. However, additional constraints on possibleanomalous couplings can be obtained from the deter-mination of the helicity of the outgoing Z. Thisrequires an analysis of the spectra of fermions fromZ decay. Even though one has in principle a differentfinal state, we work in the narrow width approxima-tion of the Z, thus avoiding complications fromadditional form factors and other processes, likeeq ey™ Z ) ™ mq my g . Experimentally, thisamounts to isolating photon energies to a narrowregion corresponding to on-shell Z production x fg

1ym2rs, as discussed in detail in the introduction.Z

It is assumed throughout that there are no non-standard contributions to the process Z™ ff. This isjustified, as such contributions are constrained fromLep-1 data to be very small. Therefore the complete

Ž .matrix-elements including Z decay contains termsŽ 2 2 .proportional either to Õ qa or to Õ a , wherem m m m

Õ and a are the SM vector and axial couplings ofm mŽ .the fermions leptons or quarks from Z decay.

Let us now discuss in detail some distributions ofZ decay products, assuming first that the Z decaysleptonically. As without Z-decay, all ratios

2 As in the case of anomalous triple gauge boson vertices theseexperimental limits are possibly larger than the magnitude of loopcorrections to the process. Loop corrections should therefore inprinciple be incorporated in the analysis.

ds rds behave like a ra to a good approxi-NEW SM i l

mation. The reason for that is mainly due to Õ f0m

for leptons. The Standard Model terms either go withŽ 2 2 .Ž 2 2 . 4 Ž .Ž .Õ qa Õ qa fa or with Õ a Õ a f0l l m m l l l m m

where one factor in the squares is due to productionof the Z and the other factor is due to the decay. The

Ž .Ž 2interference terms either go with Õ Õ ya a Õ qi l i l m2 . 3 Ž .a f a a or Õ a y a Õ Õ a f 0, so thatm i l i l i l m m

ds rds ; a ra as claimed. In practise,NEW SM i l

hadronic decays should be included as well, becausethey yield better statistics. In our framework, theycan be treated in a similar fashion like leptonicdecays.

In order to be as sensitive as possible to newphysics contributions, a sample of events with pho-tons in the central region of the detector say 0-

cosu-0.4 should be chosen. We have taken anŽ .asymmetric bin i.e. no event with cosu-0 because

some form factors yield contributions asymmetric incosu , as seen in Fig. 3. We parametrise the muonmomentum from Z decay in the lab frame by pf

s' Ž .s x 1,sinf sinu ,cosf sinu ,cosu . In Figs. 41 1 1 1 1 12

and 5 we show the deviation from the StandardModel predictions for x and c :scosu due to the1 1 1

anomalous form factors. The same values of cou-plings a s0.025, a s0.05, a s0.05 and a s1 2 3 4

0.05 and the same energy x s0.77 as in Fig. 3haveg

been chosen. Furthermore, we have averaged overthe bin 0-cosu-0.4 and in Fig. 4 in addition over0.3-x -0.7 and in Fig. 5 over y0.4-c -0.1 1

From Figs. 4 and 5 one sees that the decay lepton

Fig. 4. Ratio of nonstandard physics to SM as a function of thecosine of the muon’s angle.


Fig. 5. Ratio of nonstandard physics to SM as a function of themuon’s energy.

spectra contain useful additional information on pos-sible anomalous couplings.

4. Summary and discussion

In this letter possible new physics contributions tothe LEP2 process eqey™Zg have been analyzedby an effective vertex ansatz. Several nontrivial fea-tures of the new interactions have been derived. Onemay ask the question why we did not use the fash-ionable effective Lagrangian approach, in which newinteractions are expanded in powers of higher dimen-sional operators, in particular dimension 6, and addedto the Standard Model Lagrangian. Such operatorsare preferably chosen to respect the Standard Model

Ž . Ž .SU 2 =U 1 gauge symmetries. The complete setL Y

of these operators inducing the process eqey™Zg

is given bym mØ lD eD f D leD fm m

a aØ ls t efWmn mna aØ ilt g D lW ilg D lBm n mn m n mn

Ø ieg D B em n mn

where e and l denote the righthanded electron andleft handed lepton doublet respectively. f is the

² :Higgs doublet with vacuum expectation value f

Ž . a Ž . Ž .s 0,Õ and W and B are the SU 2 and U 1mn mn

field strengths. Note that in all the cases the eqeyZg

interaction is induced indirectly as a higher ordereffect either by the gauge field in a covariant deriva-tive D or by the nonlinear term in the nonabelianm

field strength. This implies that all these dimension 6

interactions induce eqeyg or eqeyZ couplingsmuch stronger than the eqeyg Z coupling. For all ofthem therefore exist much stronger constraints fromLEP1 than from LEP2 3. This is the reason why wehad to do without the effective Lagrangian approachto study new physics effects in eqey™Zg . In ourapproach we avoided the restriction to dimension 6,because the effective vertex in principle collectscontributions from operators of arbitrary high dimen-sion. For example, the first form factor f gets a1

contribution from a dimension 8 operator of the formm n sl t w xlg le D B B first studied in Ref. 7 . Further-mnst l

more, in strongly coupled theories, such as thosew xconsidered in 4 , there is no guarantee that operators

of a fixed dimension are dominant at a given energy.In such cases an approach which is not restricted tooperators of a fixed dimension, such as ours, isadvantageous.

Now we come to the question of constraints fromlower energies. As has already been discussed,bounds from really low energies can be avoided byassuming a suitable energy dependence of the formfactors of the form f ;s. In that case the onlyi

potentially important experiment to consider is LEP1with an energy only a factor 2 below LEP2. AtLEP1, events of the form Z™ ffg can be describedby our vertex. With an energy dependence f ;s thei

expected ratios ds rs are roughly a factor ofNEW SM

4 smaller than those considered in Section 3. Thismeans for a comparable study LEP1 would needabout 4=10000 events of the type Z™ ffg . LEP1has obtained a lot of photon events and analyzedthem in various studies. However, the photons ofinterest here are hard and noncollinear, and there areonly of the order of thousand such events in eachdetector at LEP1. They have been used by the L3

w xgroup 8 to derive constraints on ZZg and Zgg

couplings and could in principle be used to obtainlimits on our form factors as well.

Ž . Ž .If one considers our vertex 4 as part of a SU 2 L

symmetric system, one has in principle constraintsfrom lepton–neutrino scattering processes ln™Wg .However, apart from being done at rather low ener-gies, lepton–neutrino scattering has much too small

3 Actually, the first 3 operators above flip the helicity andŽ .therefore contribute only O m to all cross sections.e


cross sections to compete with Zg production atLEP2.

Another constraint could come from Tevatrondata pp™Zg X if one assumes in the spirit ofquark–lepton universality that light quarks have sim-ilar anomalous couplings to Zg as electrons. Thisprocess is well studied in conjuction with a parallelanalysis of pp™Wg X which constrains the the

)triple WWg gauge coupling via ud™W ™Zg .w xThe authors of ref 9 have given a exhaustive review

about the results from D0 and CDF. It turns out thatthere are less than 100 candidate events of the typeqq™Zg if one assumes that the Z decays leptoni-cally. This latter restriction is important because thebackground in the hadronic channel is too large. Thenumber of events is too low to compete with thestatistics of the ‘radiative return’ events at LEP2.The ability to test nonstandard physics is furtherreduced, because the photon transverse momenta arenot really large, typically Eg )10 GeV. Further-T

more, the form of the Eg distribution is not asT

sensitive to the structure of new physics as the cosu

distribution which was considered for LEP2.Finally we want to stress that the main aim of this

letter is to show that one can construct new physicspossibilities for the LEP2 process eqey™Zg inspite of various obstacles like gauge invariance,

dominance of low dimensional operators, LEP1 con-straints etc. Our main conclusion is therefore that itis worthwhile to analyze these events as precise aspossible and that one should not look at them just asboring background from the Standard Model.

Acknowledgements

We are indebted to Gunter Duckeck and Ron¨Settles, who have informed us about important ex-perimental aspects. Gunter has also provided the¨Standard Model Monte Carlo results. Arnd Leike hastriggered our analysis of low energy constraints.

References

w x1 Physics at LEP2, CERN Yellow Report 96–01.w x2 G. Duckeck, private communication.w x3 A. Barroso, F. Boudjema, J. Cole, N. Dombey, Z. Phys. C 28

Ž .1985 149.w x Ž .4 A. Suzuki, Int. J. Mod. Phys. A 2 1987 1333.w x Ž .5 Z. Ryzak, Nucl. Phys B 289 1987 301.w x Ž .6 B. Lampe, K.J. Abraham, Phys. Lett. B 326 1994 175.w x Ž .7 S.D. Drell, S.J. Parke, Phys. Rev. Lett. 53 1984 1993.w x8 M. Acciarri et al., L3 Collaboration, CERN–EPr98-096, 1998,

to appear in Phys. Lett. B.w x9 J. Ellison, J. Wudka, hep-phr9804322.

21 January 1999


The speed of light in confined QED vacuum:faster or slower than c?

M.V. Cougo-Pinto 1, C. Farina 2, F.C. Santos 3, A. Tort 4

Instituto de Fısica – UniÕersidade Federal do Rio de Janeiro, Caixa Postal 68528, CEP 21945-970, Rio de Janeiro, Brazil´

Received 18 June 1998; revised 24 November 1998Editor: M. Cvetic

Abstract

We consider the propagation of light in the QED vacuum between an unusual pair of parallel plates, namely: a perfectlyŽ . Ž .conducting one e™` and an infinitely permeable one m™` . For weak fields and in the soft photon approximation we

Žshow that the speed of light for propagation normal to the plates is smaller than its value in unbounded space in contrast tow Ž . Ž .the original Scharnhorst effect K. Scharnhorst, Phys. Lett. B 236 1990 354, G. Barton, Phys. Lett. B 237 1990 559, G.

Ž . x.Barton, K. Scharnhorst, J. Phys. A 26 1993 2037 . q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 11.10.-z; 03.70.qk; 12.20.-m; 12.20.Ds; 42.25.B

Keywords: Scharnhorst effect; QED vacuum

The QED vacuum may be regarded as the settingof a continuous creation and annihilation of virtualelectron-positron pairs, which during their fleetingexistence dictated by the Heisenberg principle canexchange virtual photons, which in their turn canpolarize again the vacuum and so on. Due to thesepolarization effects for many purposes the QED vac-uum behaves as a macroscopic medium imbued withmaterial properties. Its electromagnetic properties re-

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

spond to any change in the environment as forinstance the application of external electromagneticfields, the introduction of conducting parallel plates,

w xas in the original Casimir effect 4 , etc. Interestingprocesses that are not expected at the classical level,as for example, the scattering of light by light, oreven the scattering of light by an external electro-magnetic field can occur. In fact, light propagation ina non-trivial vacuum is one of the manifestations ofits material properties and has attracted the attentionof many physicists in the last years. Shifts in thespeed of light due to an external homogeneous mag-

w xnetic field were first considered in the early 50’s 5and received renewed attention since the early 70’s


( )M.V. Cougo-Pinto et al.rPhysics Letters B 446 1999 170–174 171

w x Ž w x .6,7 see 8 and references therein . For the case ofŽQED in curved spacetimes Schwarzschild, Robert-

.son-Walker and gravitational waves space-times , thew xpioneering work of Drummond and Hathrell 9

opened the possibility of superluminal photon propa-gation. They showed that, under certain circum-stances, the light cone for soft photons can lie in theusual spacelike region. However, this is not enoughto guarantee that superluminal propagation can beobserved, since the determination of the speed of thesignal propagation requires the knowledge of thecorrect dispersion relation including also the highfrequency limit. Regarding superluminal propagationin different gravitational backgrounds see the paper

w xby Daniels and Shore 10 and also the paper byw xShore 11 and references therein. Concerning the

influence of thermal effects on light propagation, aw xfirst calculation was made by Tarrach 12 and sub-

w xsequent discussions can also be found in Refs. 2,13 .Last, but not least, light velocity shifts induced bythe presence of two conducting parallel plates were

w xfirst computed by Scharnhorst 1 and rederived byw xBarton 2 . Both authors showed that the speed of

light between two perfectly conducting parallel platesis greater than its value in free space when thepropagation is perpendicular to the plates. All theprevious examples are valid only in the soft photonapproximation and for weak fields.

w xRecently, Latorre et al. 13 discovered that all theabove examples satisfy a unifying formula whichstates that the change in the velocity averaged overdirection and polarization is proportional to the vac-uum energy density r in the corresponding externalo

background. Denoting the change in the averaged² : w xvelocity by D Õ their formula reads 13 :

44a 2

² :D Õ sy r , 1Ž .o4135me

where a is the fine structure constant and m is thee

electron mass 5. More recently, Dittrich and Gies

5 For the case of a gravitational background a 2 must bereplaced by a Gm2 , where G is the Newton constant.e

w x14 developed a formalism in the framework ofQED effective actions that allows for the computa-tion of the velocity shifts for soft photons that goesbeyond the weak field approximation. They clarifymany aspects of the influence of a class of non-triv-ial vacua in the propagation of light and explain theorigin of the unifying formula. In particular, theyshow that in their approach it corresponds to take theweak field limit.

The purpose of this letter is to give one morew xexample of the Scharnhorst effect 1–3 , but with

boundary conditions different from those imposed bymetallic plates. The Scharnhorst effect is the onlyexample of the influence on the propagation of lightcaused by a non-trivial electromagnetic vacuum in-duced by spatial boundary conditions and henceanother explicit example of this kind may shed somelight on these issues.

In deriving their results, Scharnhorst and Bartonw x1–3 assumed that the metallic plates impose bound-ary conditions only on the radiation field, but do notaffect the fermion field. An immediate consequenceof this assumption is that the Scharnhorst effect is atwo-loop effect. Since the Scharnhorst effect is aperturbative effect one can obtain the first correc-tions to the permittivity e and permeability m of theQED vacuum just by computing the first relevantFeynman diagrams that contribute to the effectiveaction. This was precisely Scharnhorst’s approach,who employing a representation for the photon prop-agator obtained by Bordag, Robaschik and Wiec-

w xzorek 15 computed the corresponding change in thew xvelocity of light. On the other hand, Barton 2

reobtained Scharnhorst’s results in such a way thatthe connection of the effect to the Casimir energydensity becomes more transparent. Barton’s startingpoint was to consider a Lagrangian density for theelectromagnetic field that includes the correction

w xgiven by the so called Euler-Heisenberg 16 effec-tive Lagrangian density. In the soft photon approxi-

Ž .mation v<m and for field strengths well belowe

the critical field m2re, the first non-linear correc-e

tions to Maxwell equations are contained in thecontribution of fourth order in the fields:

1 e42 22 2D LLs E yB q7 EPB . 2Ž . Ž . Ž .½ 52 4360p me

( )M.V. Cougo-Pinto et al.rPhysics Letters B 446 1999 170–174172

Ž .The quartic terms in Eq. 2 in the effective La-grangian density will induce a vacuum polarizationand magnetization given by

E D LLŽ .2 2Ps s4 g E yB EŽ .

E E

q14 g EPB B ,Ž .E D LLŽ .

2 2Ms sy4 g E yB BŽ .E B

q14 g EPB E , 3Ž . Ž .2 Ž 3 2 2 4.where g:sa r 2 3 5p m .e

Let us now turn our attention to the coupling ofthe fermionic field to the confined quantized electro-

w xmagnetic field. According to Barton 2 , this can beaccomplished by the replacements E™E cl qE and

cl Ž .B™B qB in Eq. 3 , where E and B are now thequantized electromagnetic fields and E cl and B cl

describe the external fields of the propagating wave.Using the constitutive equations, linearizing in E cl

and B cl and taking the corresponding vacuum expec-w xtation values, it can be shown that 2

e sd q4px Že.s:d qDe sdi j i j i j i j i j i j

2 2² : ² :q16p g E yB d q2 E E0 0i j i j

² :q56p g B B ; 4Ž .0i j

and

m sd q4px Žm.s :d qDmi j i j i j i j i j

2 2² :sd q16p g y E yB d0i j i j

² : ² :q2 B B q56p g E E , 5Ž .0 0i j i j

² :where we made use of the fact that E B s0 due0i j

to time reversal invariance.A comment is in order here: the vacuum expecta-

tion values in the above equations are in fact theproperly regularized difference between their valuesin confined and unconfined space. Notice that theScharnhorst effect can be viewed as the response ofthe refractive index of the QED vacuum to thechange in the zero-point energy density of the quan-tized electromagnetic field imposed by boundaryconditions. Let us then compute the field operator

Ž . Ž .correlators involved in Eq. 4 and 5 between ourunusual pair of plates. These unusual boundaries

w xwere introduced by Boyer 17 in the context of

random electrodynamics and recently used in Refs.w x18–20 in a different approach. The setup consid-ered by Boyer is the simplest example where we cansee repulsive Casimir forces in action.

Let us assume that the perfectly conducting andthe perfectly permeable plates are perpendicular tothe OOZ-axis and located at zs0 and zsL, respec-tively. We then have z=Es0szPB on the planeˆ ˆzs0 and zPEs0sz=B on the plane zsL. Theˆ ˆelectromagnetic field can be quantized with theseboundary conditions and the relevant field operator

w xcorrelators are given by 21

² :E r ,t E r ,tŽ . Ž . 0i j

4p 2 7 15 Hs y yd qdŽ . i jž / ž /L 3p 8 120

qd G p zrL , 6Ž . Ž .i j

and

² :B r ,t B r ,tŽ . Ž . 0i j

4p 2 7 15 Hs y yd qdŽ . i jž / ž /L 3p 8 120

yd G p zrL , 7Ž . Ž .i j

Ž .where G j is defined as

1 d3 1G j sy csc j , 8Ž . Ž . Ž .3ž / ž /8 2dj

d I :sd d qd d and d H :sd d . Substitut-i j i x j x i y j y i j i z j zŽ . Ž . Ž . Ž .ing Eqs. 6 and 7 into Eqs. 4 and 5 , we obtain

4p 16 7 115 HDe sg y yd qdŽ . i ji j ž / ž / ž /L 3 8 120

q3d G p zrL , 9Ž . Ž .i j

and

4p 16 7 115 HDm sg y yd qdŽ . i ji j ž / ž / ž /L 3 8 120

y3d G p zrL . 10Ž . Ž .i j

( )M.V. Cougo-Pinto et al.rPhysics Letters B 446 1999 170–174 173

Although both De and Dm display a z-depen-i j i j

dence that leads to a divergence at the plates, thedivergent terms are equal and opposite so they will

Žcancel out in the expression for Dn recall that nŽ .Ž ..'s em so that Dnf 1r2 DeqDm . Due to the

divergences at the plates, a constant n is a goodapproximation only for distances to the plates greaterthan a few Compton wavelengths. This requires thatthe separation L between the plates should satisfyL41rm . Moreover, this condition also allows thee

WKB-type approximation on which relies the calcu-Ž Ž .lation of the refractive index see comment ii at the

w x. Ž .end of Ref. 2 . As a consequence of Eqs. 9 andŽ .10 , the speed of light for propagation parallel to theplates is the same as in free space. However, for

Žpropagation perpendicular to the plates OOZ-direc-.tion we have

1Õ f 1yDn s1y De qDm s1Ž . Ž .H H 11 222

7 11p 2 a 2

q y = - 1 . 11Ž .2 4 2 4ž /8 2 P3 P5 m LŽ .e

The y7r8 factor in this equation shows that thereplacement of one of the conducting plates by aperfectly impermeable one results in a change invelocity with opposite sign and with 7r8 of themagnitude of the change with two conducting plates.The factor y7r8 was expected on the light of Eq.Ž .1 because this is exactly the ratio between thezero-point energy density for two perfectly conduct-

w xing plates 4 and the zero-point energy density for aperfectly conducting plate and a perfectly permeable

w xone 17 . When light propagates in a direction mak-ing an angle u with the normal to the plates, wehave

7 11p 2 a 22Õ u f1y = cos u . 12Ž . Ž .2 4 2 48 2 P3 P5 m LŽ .e

Now we compute the average of this velocity indirection and polarization. Since in the present case

Ž .the obtained velocity 12 is independent of polariza-tion we obtain:

1 44a 2 7 p² :Õ s Õ u dVs1y = ,Ž .E 4 4ž /4p 8135m 240Le

13Ž .

where the quantity in the round brackets is thezero-point energy density for the considered set-up

w xas obtained for the first time by Boyer 17 . As wehad anticipated, this example also satisfies the gen-

² :eral relation between D Õ and r discussed inow x ² :Refs. 13,14 . The proportionality between D Õ

and r is quite reasonable, since for ordinary opticalo

materials the index of refraction is closely connectedto the density of matter of the material. In the case ofthe quantum vacuum, we may say that the zero-pointenergy density of the quantized fields plays the roleof the density of matter. We should emphasize herethat this simple heuristic argument relies on the factthat the change of velocity has been averaged. Alter-natively, one can understand this effect as follows:the external field describing the propagation of aplane wave interacts with the quantized electromag-netic field through the fermionic loops and hence,any change in the quantized fields can in principlemodify the wave propagation.

We showed in this letter by means of a concreteexample that the speed of light in a spatially con-strained QED vacuum can be not only greater than c,as in the original Scharnhorst effect, but also smallerthan c, according to the type of the imposed bound-ary conditions. However, some caution must be exer-cised: what has been computed is the phase velocitywhich coincides with the group velocity for smallfrequencies. In order to investigate if this kind of

Ž .effect despite its smallness can indeed be mea-sured, at least in principle, we need to construct avery narrow wave packet between the plates, whichin turn involves necessarily high frequencies and, asfar as we know, no one has determined the exact

Ž w xdispersion relation for this problem see Refs. 22,8and references therein for further comments on this

.subject .Regarding thermal effects, it seems that it does

not exist any temperature at which the ScharnhorstŽ .effect vanishes at least for this kind of plates , for

Žthe thermal contribution to Dn that one coming.from the black body radiation averages has the same

sign as the zero temperature one.We would like to point out that the analogue of

the Scharnhorst effect for scalar QED also exists.The results for this case are analogous to those foundin the original Scharnhorst effect, showing that thevacuum of scalar QED and the vacuum of the usual

( )M.V. Cougo-Pinto et al.rPhysics Letters B 446 1999 170–174174

QED behave very similarly when the photon field issubjected to boundary conditions dictated by the

w xpresence of material plates 21 .As a final remark, we would like to emphasize

that light propagation in the QED vacuum is alsoaffected if instead of imposing boundary conditionson the photon field we do it on the fermionic fieldw x23 . In this case, although the effect is suppressed

Žexponentially with the product m L this is true fore.the plane geometry , it shows up at one-loop level.

Since one-loop calculations are less involved thantwo-loop ones, maybe this is the appropriate scenariofor deeper investigations on the high frequency limit.

Acknowledgements

The authors would like to thank the referee forextremely valuable remarks and suggestions. Wealso thank K. Scharnhorst, P.A. Maia Neto and G.Matsas for enlightening conversations on the subject.C.F. and M.V.C.-P. would like to acknowledge CNPqŽ .The National Research Council of Brasil for partialfinancial support.

References

w x Ž .1 K. Scharnhorst, Phys. Lett. B 236 1990 354.w x Ž .2 G. Barton, Phys. Lett. B 237 1990 559.

w x Ž .3 G. Barton, K. Scharnhorst, J. Phys. A 26 1993 2037.w x Ž .4 H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51 1948 793.w x5 J.S. Toll, The Dispersion Relation for Light and Its Applica-

tion to Problems Involving Electron Pairs, Ph.D. thesis,Princeton, 1952.

w x6 Z. Bialynicka-Birula, I. Bialynicki-Birula, Phys. Rev. D 2Ž .1970 2341.

w x Ž . Ž .7 S.L. Adler, Ann. Phys. NY 67 1971 599.w x8 K. Scharnhorst, The velocities of light in modified QED

Ž Ž .vacua Talk delivered at the workshop ‘‘Superluminal ?.Velocities’’, Cologne, July 7–10, 1998 , hep-thr9810221.

w x Ž .9 I.T. Drummond, S.J. Hathrell, Phys. Rev. D 22 1980 343.w x Ž .10 R.D. Daniels, G.M. Shore, Nucl. Phys. B 425 1994 634.w x Ž .11 G.M. Shore, Nucl. Phys. B 460 1996 379.w x Ž .12 R. Tarrach, Phys. Lett. B 133 1983 259.w x Ž .13 J.I. Latorre, P. Pascual, R. Tarrach, Nucl. Phys. B 437 1995

60.w x Ž .14 W. Dittrich, H. Gies, Phys. Rev. D 58 1998 025004.w x Ž .15 M. Bordag, D. Robaschik, E. Wieczorek, Ann. Phys. NY

Ž .165 1985 192.w x Ž .16 W. Heisenberg, H. Euler, Z. Phys. 98 1936 714.w x Ž .17 T. Boyer, Phys. Rev. A 9 1974 2078.w x Ž .18 V. Hushwater, Am. J. Phys 65 1997 381.w x19 M.V. Cougo-Pinto, C. Farina, A. Tenorio, z-function method

for repulsive Casimir effect, to appear in the Braz. J. Phys.w x20 F.C. Santos, A. Tenorio, A. Tort, Zeta function method for

repulsive Casimir forces at finite temperature, hep-thr9807162.

w x21 M.V. Cougo-Pinto, C. Farina, F.C. Santos, A. Tort, QEDvacuum between two unusual pair of plates, hep-thr9811062.

w x Ž .22 P.W. Milonni, M.-L. Shih, Contemp. Phys. 33 1992 313.w x23 M.V. Cougo-Pinto, C. Farina, J. Rafelski, A. Tort, Phys.

Ž .Lett. B 434 1998 338.

21 January 1999


Superfield formulation of the phase space path integral

I.A. Batalin a, K. Bering b, P.H. Damgaard c

a LebedeÕ Physics Institute, 53 Leninisky Prospect, Moscow 117924, Russiab Institute for Fundamental Theory, Department of Physics, UniÕersity of Florida, GainesÕille, FL 32611, USA

c The Niels Bohr Institute, BlegdamsÕej 17, DK-2100 Copenhagen, Denmark

Received 5 November 1998Editor: P.V. Landshoff

Abstract

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first classconstraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the originalphase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended tothe case of second-class constraints, the correct path integral measure is again reproduced after integrating over thesuperpartners. These results suggest that the superfield formulation is of first-principle nature. q 1999 Elsevier Science B.V.All rights reserved.

1. Introduction

w xIn a recent paper 1 , we have shown that anarbitrary Hamiltonian quantum field theory can begiven a superfield formulation. Although the formal-

w xism of Ref. 1 and the constructions explained belowcan be formulated in operator language, we shallhere focus on the path integral formalism. The neededsuperspace is two-dimensional, consisting of time tand a new Grassmann-odd superpartner, which wedenote by u . All original phase space coordinates

AŽ .z t are then treated as zero-components of super0

phase space coordinates

z A t ,u sz A t qu z A t . 1Ž . Ž . Ž . Ž .0 1

AŽ .In particular, z t,u has the same statistics asAŽ .z t , which we denote by e . One essential ingredi-0 A

w xent of Ref. 1 was the introduction of a superspacederivative

d dD' qu , 2Ž .

du dt

which acts like a ‘‘square root’’ of the time deriva-tive:

d2D s . 3Ž .

dt

The superspace extends in an obvious manner to aŽ .d q 1 -dimensional superspace of coordinatesŽ m .x ,u when considered in the context of a Lorentzinvariant quantum field theory in d dimensions, butwe shall here restrict ourselves to the finite-dimen-sional case of 2 N phase space variables.

The purpose of the present paper is to demon-w xstrate that the superspace formalism developed in 1


( )I.A. Batalin et al.rPhysics Letters B 446 1999 175–178176

reaches one step deeper than could have been antici-pated. By considering here the extension to a phasespace with a non-constant symplectic metric, weshall show that the required superspace generaliza-

w xtion of the phase space path integral 2,3 leads, afterintegrating out the fermionic coordinate u , to pre-cisely the correct path integral measure. This is aquite non-trivial fact, completely independent of

Ž .whether we consider a system with first class con-straints or not. Moreover, when considered in thepresence of second-class constraints it turns out thatour formalism also here directly yields all requiredfactors in the path integral.

2. Symplectic structure

Ž . Ž .In addition to Eqs. 1 and 2 , the few ingredientswe need are as follows. Define a graded Poissonbracket by

§ ™A B� 4F ,G 'F E v E G , 4Ž .A B

Ž Ž .. Ž Ž ..for functions FsF z t,u , GsG z t,u . HereŽ .the non-degenerate symplectic metric,

v A B sv A B z t ,u s z A t ,u , z B t ,u , 5� 4Ž . Ž . Ž . Ž .Ž .AŽ .is allowed to depend on z t,u . We will in what

follows suppress some of the arguments to make theformulas more readable. For precise details we refer

w xto the original paper 1 . The symmetry propertiesare as follows:

e eA BB A A B A Bv sy y1 v , e v se qe ,Ž . Ž . A B

6Ž .

which ensures

Ž . Ž .e F e G� 4 � 4F ,G sy y1 G ,F . 7Ž . Ž .Similarly, the super Jacobi identity

Ž . Ž .e F e H� 4F ,G , H y1� 4 Ž .qcyclic perm. F ,G , H s0 , 8Ž . Ž .

is satisfied ife eA CA D BCv E v y1 qcyclic perm. A , B ,C s0 .Ž . Ž .D

9Ž .

As usual, we define an inverse symplectic metricv by v A Bv sd A. Its symmetry properties areA B BC C

quite different:Ž .Ž .e q1 e q1A Bv s y1 v , e v se qe .Ž . Ž .B A A B A B A B

10Ž .Crucial in this context is that the Jacobi identityturns into a closedness relation

Ž .e q1 eC BE v y1 qcyclic perm. A , B ,C s0 ,Ž . Ž .C A B

11Ž .which implies that locally we can represent v inA B

terms of a symplectic potential V :A

e e eA B Bv s E V y y1 E V y1 . 12Ž . Ž . Ž .Ž .A B A B B A

Our primary aim is not to elaborate on global issues.We shall for simplicity assume that the phase spaceis simply connected and that there exists a globallydefined symplectic potential.

3. Super Hamiltonian

Let there now be given a Grassmann-odd BRSTŽ Ž ..generator VsV z t,u and an Hamiltonian Hs

Ž Ž .. w xH z t,u with the properties 4

� 4 � 4V ,V s0 and H ,V s0 . 13Ž .We combine these two fundamental objects into aGrassmann-odd superfield Q:

Q z t ,u ,u 'V z t ,u qu H z t ,u .Ž . Ž . Ž .Ž . Ž . Ž .14Ž .

It is nilpotent in terms of the Poisson bracket, byŽ .virtue of Eq. 13 :

� 4Q,Q s0 . 15Ž .This nilpotency condition is preserved under supercanonical transformations

Q¨Q 'eadC Q , 16Ž .C

which infinitesimally are generated by the adjointaction

� 4adC' C , P . 17Ž .Here C is a superfield,

C z t ,u ,u sC z t ,u quC z t ,u ,Ž . Ž . Ž .Ž . Ž . Ž .0 1

18Ž .

( )I.A. Batalin et al.rPhysics Letters B 446 1999 175–178 177

which plays the role of a generalized gauge-fixingˆfermion. More precisely, C itself is Grassmann-eÕen,and it is the 1-component C which directly corre-1

sponds to the gauge-fixing fermion. Instead, thebosonic zero-component C is a generator of ordi-0

w xnary canonical transformations 1 .

4. The action

The classical equations of motion are taken to be

Dz A sy Q , z A . 19Ž .� 4C

w xAs was shown in Ref. 1 , these reduce to thestandard equations of motion in the original phase

AŽ .space variables z t . An action which yields these0

equations of motion is

t ef BA Bw xS z s dt du z v Dz y1 yQ 20Ž . Ž .H A B Cti

where

1y1Cv ' z E q2 v s v a z a da .Ž .Ž . HA B C A B A B0

21Ž .Note that we regain the well-known kinetic term in

1the case of a constant v for which v s v .A B A B A B2

We may rewrite the action as

t tf fAw xS z s dt du V Dz yQ y W z t ,0 ,Ž .Ž .H tA C iti

22Ž .where the boundary term is given by

AW z 'z V , 23Ž . Ž .A

and

1y1CV ' z E q1 V s V a z da . 24Ž . Ž .Ž . HA C A A0

Ž . Ž .From Eqs. 12 and 21 it follows thate e eA B B

v s E V y y1 E V y1 . 25Ž . Ž . Ž .Ž .A B A B B A

5. Partition function

Ž .We therefore take the action Eq. 20 , or equiva-Ž .lently Eq. 22 , as the correct candidate to be expo-

nentiated, and integrated over in the superfield pathintegral:

iw x w xZZs dz exp S z . 26Ž .H

"

Note that this path integral contains no additionalmeasure factors. This is not needed because the

w xmeasure dz remarkably transforms as a scalar un-der general coordinate transformations, due to thebalance between bosonic and fermionic degrees offreedom in the superfield formulation. In this case,on a curved phase space manifold, a crucial test ofthe present formalism is to see if we recover thecorrect path integral measure after integrating out thefermionic u-coordinate. The calculation is straight-forward, and results in SsS qS with0 1

t tf fAS s dt z V z yH z y W z t ,0Ž . Ž . Ž .Ž .˙H t0 0 A 0 C 0 iti

t f B1 A B AS s dt z v z z y1 yz E V z .Ž . Ž . Ž .H1 1 A B 0 1 1 A C 02ti

27Ž .Ž .Here H and V are defined according to Eq. 14C C

Ž .and Eq. 16 . After a gaussian integration over thesuperpartner z A, and use of the nilpotency relation1� 4V ,V s0, one arrives at the standard formC C

iw x w xZZs dz Pf v exp S z , 28Ž . Ž .H 0 0 0

"

where the Pfaffian of an arbitrary even supermatrixŽ . Ž Ž ..1r2is given by Pf M s Ber M .

6. Second class constraints

To test yet again how fundamental the presentsuperfield formulation is, let us now consider the

Ž Ž ..case of 2n second class constraints F sF z t,ua a

of Grassmann parity e . To impose such constraintsa

in the path integral, we introduce an auxiliary super-field

la u sla qula 29Ž . Ž .0 1

of Grassmann parity e q1, and consider the parti-a

tion function

i t f aw x w x w xZZs dz dl exp S z q dt du F l .H H až /" ti

30Ž .

( )I.A. Batalin et al.rPhysics Letters B 446 1999 175–178178

Note again the absence of any non-trivial measurefactors. Let us show that this superfield partitionfunction completely reproduces the standard versionof the partition function with second class con-straints. The crucial property of second-class con-straints is that the matrix

e ea bv s F ,F sy y1 v 31� 4 Ž . Ž .ab a b ba

is invertible. Let us denote the inverse matrixŽ .Ž .e q1 e q1a b baa bv s y1 v . 32Ž . Ž .

According to the standard Dirac procedure, the Pois-son bracket should be replaced by the Dirac bracket:

� 4 � 4 � 4 abF ,G s F ,G y F ,F v F ,G . 33� 4 Ž .D a b

Let us now trace the additional terms in the pathintegral due to the second-class constraints. We dothis as before by integrating over the fermioniccoordinate u . The result is as follows. First, the zerocomponent part S of the action picks up a delta0

function term that precisely enforces the second-classconstraints in the original phase space variables:

t f aS ™ S q dt l F z . 34Ž . Ž .H0 0 1 a 0ti

In the S -part of the action, integration over u effec-1Ž .tively just corresponds to replacing V z byC 0

V z yF z la .Ž . Ž .C 0 a 0 0

Therefore the gaussian integration over the super-partner z A, besides yielding the correct Pfaffian1Ž .Pf v as before, also produces a term

1 a bV z yF z l , V z yF z lŽ . Ž . Ž . Ž .� 4C 0 a 0 0 C 0 b 0 02

35Ž .

in the action by completing the square 1. If we nextperform also the gaussian integration over the zero

a Ž� 4.component l , we get Pf F ,F . The rest of the0w xaction conspires to yield 4

1 1 ab� 4 � 4V ,V y V ,F v F ,V� 4C C C a b C2 2

1 � 4s V ,V s0 . 36Ž .C C2 D

1 Without second-class constraints this term was just1 � 4V ,V , which in that case would vanish on account of theC C2

Ž .nilpotency condition Eq. 13 .

Therefore we quite remarkably arrive at just thew xstandard form of the partition function 5 :

iw x w xZZs dz Pf v exp S zŽ .H 0 0 0

"

= � 4d F Pf F ,F . 37Ž . Ž .Ž .

7. Conclusions

w xThe superfield formulation introduced in 1 thusin a very precise and non-trivial manner encodes allthe information required for Hamiltonian path inte-gral quantization for systems with or without anycombination of first and second class constraints, onan arbitrary curved phase space. In view of this, wepropose to consider our superfield formalism as afirst principle on which to base quantization. Anoperatorial formulation of precisely the same super-field formulation also exists, with or without firstand second class constraints, and with possibly non-constant symplectic v .A B

Acknowledgements

I.A.B. and K.B. would like to thank the NielsBohr Institute for the warm hospitality extended tothem there. The work of I.A.B. and P.H.D. is par-tially supported by grant INTAS-RFBR 95-0829.I.A.B. also acknowledges the funding by grants IN-TAS 96-0308, RFBR 96-01-00482 and RFBR 96-02-17314. The work of K.B. is supported by DoE grantDE-FG02-97ER41029 and Nordita.

References

w x1 I.A. Batalin, K. Bering, P.H. Damgaard, Nucl. Phys. B 515Ž .1998 455.

w x Ž .2 I.A. Batalin, E.S. Fradkin, Mod. Phys. Lett. A 4 1989 1001.w x Ž .3 I.A. Batalin, E.S. Fradkin, Nucl. Phys. B 326 1989 701.w x Ž .4 E.S. Fradkin, G.A. Vilkovisky, Phys. Lett. B 55 1975 224;

Ž .I.A. Batalin, G.A. Vilkovisky, Phys. Lett. B 69 1977 309;Ž .E.S. Fradkin, E.S. Fradkina, Phys. Lett. B 72 1978 343; I.A.

Ž .Batalin, E.S. Fradkin, Phys. Lett. B 122 1983 157.w x5 E.S. Fradkin, in: Proc. X Winter School of Theo. Phys.,

Karpacs, 1973, No. 207, p. 93; P. Senjanovic, Ann. Phys. 100Ž .1976 227.

21 January 1999


Observation of strong final-state effects in pq production in ppcollisions at 400 MeV 1

A. Betsch b, R. Bilger b, W. Brodowski b, H. Calen c, H. Clement b, J. Dyring c,´C. Ekstrom d, K. Fransson c, L. Gustafsson c, S. Haggstrom c, B. Hoistad c,¨ ¨ ¨ ¨

J. Johanson c, A. Johansson c, T. Johansson c, K. Kilian e, S. Kullander c, A. Kupsc f,´´G. Kurz a,2, P. Marciniewski f, B. Morosov g, A. Mortsell c, W. Oelert e, J. Patzold b,¨ ¨R.J.M.Y. Ruber c, M.G. Schepkin h, J. Stepaniak f, A. Sukhanov g, A. Turowiecki i,

G.J. Wagner b, Z. Wilhelmi i, J. Zabierowski j, A. Zernov g, J. Złomanczuk c´a ETH ZurichrPSI Villigen, Switzerland¨

b Physikalisches Institut, UniÕersitat Tubingen, 72076 Tubingen, Germany¨ ¨ ¨c Department of Radiation Sciences, UniÕersity of Uppsala, Sweden

d The SÕedberg Laboratory, Uppsala, Swedene Institut fur Kernphysik, Forschungszentrum Julich, Germany¨ ¨

f Sołtan Institute for Nuclear Studies, Warsaw, Polandg Joint Institute for Nuclear Research, Dubna, Russia

h Institute for Theoretical and Experimental Physics, Moscow, Russiai Institute of Experimental Physics, Warsaw UniÕersity, Poland

j Institute for Nuclear Studies, Lodz, Poland´

Received 19 October 1998; revised 27 November 1998Editor: L. Montanet

Abstract

Differential cross sections of the reactions pp™dpq and pp™pnpq have been measured at T s400 MeV byp

detecting the charged ejectiles in the angular range 48FQ F218. The deduced total cross sections agree well with thoseLab

published previously for neighbouring energies. The invariant mass spectra are observed to be strongly affected by D

production and NN final-state interaction. The data are well described by Monte Carlo simulations including both theseeffects. The ratio of pp™pnpq and pp™dpq cross sections also compares favourably to a recent theoretical predictionwhich suggests a dominance of np-production in the relative 3S -state. q 1999 Elsevier Science B.V. All rights reserved.1

PACS: 13.75.Cs; 13.60.Le; 21.30.Cb; 25.40.Qa

1 Ž . Ž .Supported by the BMBF 06 TU 886 , DFG Mu 705r3, Graduiertenkolleg , DAAD and NFR and INTAS-RFBR.2 Present address.

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01524-X

( )A. Betsch et al.rPhysics Letters B 446 1999 179–184180

The single pion production has received renewedinterest in recent years both from experimental andtheoretical points of view. Theoretical aspects cur-rently under discussion include possible heavy me-son exchange, the nature of the p NN vertex and the

Ž .role of final-state interactions FSI in these reac-tions. Experimentally the avail-ability of storage ringshas opened the possibility to measure single pionproduction with high statistics even close to thresh-

w xold 1–4 . These data show an energy dependence ofthe total cross section near threshold, which deviatessubstantially from phase space suggesting a stronginfluence of the NN FSI. In this Letter we show thatFSI effects can be explicitly observed and identified

Ž . Ž .in the invariant M and missing mass MM spectraof the reaction pp™pnpq. The energy of T s400p

MeV is already high enough for D-production to beobserved clearly in M q and M q, whereas the pnnp pp

FSI still strongly influences M even though thispn

energy is well above threshold.The measurements have been performed at the

CELSIUS storage ring at T s400 MeV using thep

WASArPROMICE detector setup including a hy-drogen cluster jet target. Details of the detector and

w xits performance are given in 5 . For the data pre-sented here only the forward detector has been uti-lized, which allows the determination of the four-momentum of charged particles in the angular rangeof 48FQ F218. It is composed of a tracker withLab

proportional counter straw chambers for an accuratedetermination of particle trajectories, followed bysegmented scintillator trigger and range hodoscopesfor dE and E determinations, respectively. Particleidentification has been made by use of the dE–Emethod. Fig. 1 shows a three-dimensional plot of thedE–E spectrum. Deuterons, protons and pions appearwell separated. For the pq identification the delayed

Žpulse technique observation of the delayed pulseq q .from m decay following the p decay has been

utilized in addition. Though protons and deuteronsappear to be well separated in Fig. 1, their separationis not perfect and the contamination of reconstructedpnpq events with dpq events, and vice versa, isnon-negligible. In order to get rid of this contamina-tion we have requested the dpq events to be planarŽ . qDFs1708–1908 and the pp events to be non-planar. This way mutual contaminations could bekept below 1%. The final number of good events

Ž .Fig. 1. Plot of the dE–E spectrum for two charged particles in the forward detector FD . The energy loss dE has been measured by theŽ . w xforward trigger hodoscope FTH , see Ref. 5 . Energies are given in units of GeV.

( )A. Betsch et al.rPhysics Letters B 446 1999 179–184 181

which passed all criteria has been about 105 for dpq

and 7=104 for pnpq. The integral luminosity hasbeen determined to better than 5% by the simultane-ous measurement of pp elastic scattering and its

w xcomparison to literature values 7 . Detector responseand efficiencies have been determined by Monte

Ž .Carlo MC simulations, utilizing the program pack-w xage GEANT 6 including the treatment of secondary

interactions in the detector.The data obtained for pp™dpq are shown in

Fig. 2. In the upper part the pq missing massMM q which gives a peak at the position of thep

deuteron mass is displayed together with the corre-sponding MC simulations. The good agreement

Fig. 2. Top: Spectrum of the pq missing mass MM q asp

obtained from identified dpq events together with the corre-Ž .sponding MC simulation shaded area of the detector response.

Bottom: Measured pq angular distribution for pp™ dpq inw xcomparison with the SAID phase shift prediction 7 .

demonstrates that the detector response is understoodto high precision. The measured pq angular distri-

Ž .bution in the center-of-mass c.m. system is dis-played in the lower part of Fig. 2 together with the

w xSAID phase shift prediction 7 . Note that the angu-lar distribution is symmetric about 908 due to thesymmetry in the entrance channel. The limited angu-lar range of the data results from the experimentalrequirements of 48FQ F218 for both d and pq.Lab

The experimental points are compatible with SAID,and we may use the angular dependence of the latterto extrapolate our data to 4p . This way we obtain a

Ž q.total cross section s pp™dp s0.78 mb, whichcompares quite favorably with the SAID value of0.82 mb. Whereas the statistical uncertainty is lessthan 1%, the systematic uncertainty is estimated tobe about 7% comprising uncertainties both from thedetermination of the luminosity and from the han-dling of deuteron breakup in the detector by the MCsimulation.

Results for the pp™pnpq reaction are com-prised in Figs. 3–5. For the angular range 48FQ Lab

F218 the invariant and missing mass spectra recon-structed from the measured ppq events are shownin Fig. 3 together with curves from MC simulations

Ž .assuming pure phase space dotted and includingqqŽ q. qŽ q.either D D excitation in the pp np system

Ž .dash-dotted or the s-wave FSI in the pn systemŽ .dashed . For the latter the effective range approxi-

w xmation of Migdal-Watson type as given in 8 hasbeen used. The only free parameter, R, accounts forthe size of the interaction region, from which the twonucleons emerge, and effectively subsumes also thepart of the reaction, where the two nucleons are notin relative s-waves. For a point-like vertex one finds

w xRf0.8 fm 8 . Here in the MC simulations R hasbeen adjusted for best reproduction of the data result-ing in Rs2.5 fm, if we assume the s-wave pnsystem to be in pure 3S states as previous work1

w xsuggests 9,10 . This assumption will also be corrob-orated by a comparison of the dpq and pnpq

channels below. The value for R is compatible withthose obtained in analyses of the NN FSI in the

w xdeuteron breakup on the proton at low energies 11 .The D excitation in the exit channel is assumed totake place between the pq and either the proton orthe neutron according to the isospin ratio of 9:1 withthe pq being in relative p-state in the p N system.


Fig. 3. Spectra for ppq invariant mass M q and missingpp

masses MM and MM q for pp™ pnpq reconstructed from thep p

ppq events detected for 48FQ F218. Corresponding MCLabŽ .simulations are shown assuming pure phase space dotted , includ-

Ž . Ž .ing either pn FSI dashed or D excitation dash-dotted and bothŽ .together shaded histograms . Note that due to the limited angular

range in Q the missing mass spectra are not structureless evenLab

in case of pure phase space. Notably the apparent structures inMM q and MM near 1.92 GeV and 1.1 GeV, respectively, are ap p

consequence of this angular limitation.

Ž . Ž .Fig. 4. Angular distributions of protons top and pions bottomfor pp™ pnpq reconstructed from the detected ppq events.

2Ž .They are plotted in dependence of cos Q and are compared toŽ .MC simulations assuming pure phase space dotted and includingŽ .both pn FSI and D excitation with bs0.1 dashed or bs0.4

Ž .solid . All simulations are normalized to an integral cross sectionof s s0.62 mb.

Note that parity conservation requires the pion alsoto be in a p-wave relative to 3S pn final state.1

The MC simulation including both FSI effectstogether is displayed in Fig. 3 by the shaded his-tograms which reproduce the data very well. Theinvariant mass spectrum M q and the missing masspp

Ž .qspectrum MM corresponding to M are peakedp np

Žqtowards large masses, whereas MM correspond-p

.ing to M is strongly peaked towards low massespnŽ .in contrast to phase space distributions dotted lines .

In all three spectra both FSI effects play a significantrole. However, the fact that the peaking towardslarge masses is much more pronounced in M qpp

than in MM , clearly exhibits a strong influence ofp

( )A. Betsch et al.rPhysics Letters B 446 1999 179–184 183

Ž . q qFig. 5. Ratio R Q of the pp™ pnp and pp™ dp differen-tial cross sections. The solid curve shows the prediction of Ref.w x9 .

the D production, since it is nine times more likelyin the ppq system than in the npq system. Thisobservation in M q and M q is strongly differentpp np

from the one at T s310 MeV, where both spectrapw xhave been observed to be of comparable shape 1 .

However, it is in agreement with the trend observedw xfor T F330 MeV 2 . From the partial-wave analy-p

w xsis of those data it was found 2 that the resonantp-wave contribution, though still small at these ener-gies, is steeply rising with incident energy. From thisanalysis the D contribution is expected to get domi-

Žnant at energies of T f400 MeV see Fig. 20 ofpw x.Ref. 2 . This is indeed what we observe in our data.

We note that the strong influence of the FSI on thepion energy distribution is already apparent in thedE–E plot in Fig. 1, where the pions of the pnpq

reaction are seen to be concentrated towards the peakof the dpq reaction.

In Fig. 4 the experimental pion and proton angu-lar distributions, corrected for detector efficiency andacceptance, are compared to MC simulations for thereaction pp™pnpq. As in pp™dpq the angulardistributions have to be symmetric about 908. Theproton angular distribution is essentially isotropicbeing affected only slightly by the D excitation. Thepion angular distribution, on the other hand, dependsstrongly on the D excitation. Conventionally thepion angular distributions in single pion production

Ž . 2are parametrized by s Q ;1r3qbcos Q , wherep p

b is the so-called anisotropy parameter. Previousanalyses yielded values for b up to 0.5 for np™

q w x 0 w xnnp 12 and 0.3 for pp™ppp 13 with the

maximum b being reached near T f550 MeV. Thep

b values are observed to decrease with decreasingT . At T s460 MeV, the lowest energy analyzed inp p

those studies, b gets as small as 0.1. On the otherw x qhand the IUCF measurements 1,2 of pp™pnp

show a strong rise of b already close to thresholdŽ .reaching bs0.23 6 at their highest energy of T sp

330 MeV. Whereas from the latter we would expectto find already a quite substantial value of b at 400MeV, the previous analyses would suggest rather asmall value. Hence we show in Fig. 4 two MCcalculations including pn FSI and D excitation, one

Ž .with bs0.1 dashed lines and one with bs0.4Ž .solid – in addition to the phase-space expectationŽ .dotted line . In the measured range of Q the datap

2Ž .clearly prefer the larger b value. From the cos Qp

plot of the pion angular distribution in Fig. 4 it isreadily seen that bs0.7 would be an upper limitcompatible with our data. For a more precise deter-mination of b larger pion angles are necessary,which are not covered by this measurement. Hencealso the determination of the total cross section fromour data is not independent of the assumption for b.Whereas for bs0.1 one would obtain a value ofŽ q. Ž .s pp™pnp s0.73 4 mb, our data favour val-

Ž . Ž .ues of 0.62 4 and 0.57 4 mb for bs0.4 and 0.7,respectively, where the assigned uncertainty is essen-tially due to that of the luminosity. The latter valuesare in good agreement with literature values at

Ž w x.neighbouring energies see 1 .The close relation between dpq channel and

pnpq channel as the breakup channel of the formerhas recently been pointed out by Boudard, Faldt and¨

w xWilkin 9 . Assuming that the final pn system is inthe 3S state, and that the pion production operator is1

Ž Ž .of short range, they derive a simple relation Eq. 6w x.of Ref. 9 for the ratio of differential cross sections

defined by

d2sqR Q s2p B pp™pnpŽ . Ž .d dV dQp

'ds p x xŽ .qr pp™dp sŽ .

dV p y1 1qxŽ .p

which should be independent of the pion scatteringangle. Here B denotes the deuteron binding energyd

and QsM ym the excitation energy in the pnpn d


Ž .system, xsQrB and p x is the pion c.m. mo-dŽ .mentum. Our experimental result for R Q is plotted

w xin Fig. 5 together with the prediction of Ref. 9 .Though there is good qualitative agreement, the dataexhibit a significantly steeper slope than the predic-tion. In particular, the data are higher at low Q,

w xwhere the approximations made in Ref. 9 should beŽ .valid best. Experimentally the determination of R Q

from the simultaneous measurement of both reac-tions is expected to be particularly reliable, sinceuncertainties in the determination of the luminosityand detector response cancel to large degree in theratio, the only major source of possible error beingthe treatment of the deuteron breakup in the detector.

Ž q.However, since our result for s pp™dp agreeswell with the literature values at neighbouring ener-gies, we do not see a significant problem there

w xeither. A closer inspection of Fig. 1 in Ref. 9Ž .indicates that experimentally R Q is not fully angle

w xindependent. The TRIUMF data 10 plotted thereexhibit a systematic trend around the maximum ofŽ . Ž .R Q : the data taken at small angles Q s468,568Lab

lie systematically above the theoretical curve,Ž .whereas the ones taken at larger angles 738–888 lie

significantly below. Our result for the forward angu-lar range fits very well into this trend in the TRI-

Ž .UMF data. Since in the calculation of R Q only theisoscalar 3S channel of the pn system is considered,1

the surplus of approximately 10% in our data atsmall Q may be associated with contributions fromother partial wave channels, notably the 1S channel0

of the pn system. This conclusion conforms quitenicely with the expectations from the partial-wave

w xanalysis at lower energies 2 . The observed discrep-ancy between calculation and data at higher Q maybe associated with effects from higher partial wavesas well as with kinematic approximations made in

Ž .the derivation of the theoretical expression for R Qw x9 .

Summarizing we observe in the exclusive mea-surement of pp™pnpq at T s400 MeV strongp

FSI effects in the invariant and missing mass spectrawhich are identified as being due to pn FSI andp ND excitation. The dominance of the latter is inagreement with expectations from partial-wave anal-yses close to threshold. The influence of the formeris in agreement with expectations for the Migdal-Watson effect in the pn system, if s-waves are

Ž .dominating there. The observed ratio R Q for theexit channels pnpq and dpq is compared with the

w xprediction of Ref. 9 relating the channel of thebound system with its breakup channel. Although thegeneral agreement is good, there are significant dif-ferences which call for a more refined theoreticaltreatment of both channels.

Acknowledgements

We gratefully acknowledge valuable discussionswith Colin Wilkin. We are also grateful to the per-sonnel at the The Svedberg Laboratory for their helpduring the course of this work.

References

w x Ž .1 J.G. Hardie, Phys. Rev. C 56 1997 20.w x Ž .2 R.W. Flammang, Phys. Rev. C 58 1998 916.w x Ž .3 A. Bondar, Phys. Lett. B 356 1995 8.w x Ž .4 H.O. Meyer et al., Phys. Rev. Lett. 65 1990 2846; Nucl.

Ž .Phys. A 539 1992 633.w x Ž .5 H. Calen, Nucl. Instr. Meth. A 379 1996 57.w x6 Program package GEANT, CERN library.w x7 Program package SAID; Chang Heon Oh, R. Arndt, I.

Ž .Strakovsky, R. Workman, Phys. Rev. C 56 1997 635.w x8 M. Schepkin, O. Zaboronski, H. Clement, Z. Phys. A 345

Ž .1993 407.w x Ž .9 A. Boudard, G. Faldt, C. Wilkin, Phys. Lett. B 389 1996¨

440.w x Ž .10 W.R. Falk, Phys. Rev. C 32 1985 1972.w x Ž .11 H. Bruckmann, Phys. Lett. B 30 1969 460.¨w x Ž .12 A. Bannwarth, Nucl. Phys. A 567 1994 761.w x Ž .13 G. Rappenecker, Nucl. Phys. A 590 1995 763.

28 January 1999


Solution of the Schrodinger equation including two-, three- and¨four-body correlations – Bose systems

Nir Barnea 1

ECT ) , European Center for Theoretical Studies in Nuclear Physics and Related Areas, Strada delle Tabarelle 286, 1-38050 Villazzano( )Trento , Italy

Received 20 August 1998; revised 24 November 1998Editor: J.-P. Blaizot

Abstract

Two-, three- and four-body correlations are introduced in the wave function of a many-body system by decomposing theSchrodinger equation into a set of differential equations which contain the appropriate correlations. q 1999 Published by¨Elsevier Science B.V. All rights reserved.

PACS: 31.15.Ja; 05.30.Jp; 24.10.Cn

In this paper we propose a method for solving themany-body Schrodinger equation by expanding the¨wave-function in terms of two-, three- and four-bodycorrelations. Our procedure is a generalization of thepotential harmonics method developed by Fabre de

w xla Ripelle 1 to account for the two-body correla-tions. The basic notion in this procedure is to expandthe Schrodinger wave-function in terms of few-body¨correlations

Cs c Ž2. q c Ž3. q c Ž4. q . . . ,Ý Ý Ýi i i i i i i i i1 2 1 2 3 1 2 3 4i i i i i i i i i1 2 1 2 3 1 2 3 4

1Ž .

where the partial amplitudes c Žn. depend in general1 2on the collective variable rs Ý r and on the( i) j i jN

internal coordinates of the n-body system. Assumingthat for saturated systems the contribution of the


high order amplitudes is negligible we can truncatethis expansion. In this paper we shall disregardfive-body and higher order correlations. Our aim isto derive the differential equations for the n-bodycorrelation function. It is shown that this goal can beachieved by decomposing the Schrodinger equation¨into a set of equivalent equations and projecting eachequation on the sub-space defined by r and theinternal coordinates of the subsystem. Expanding theamplitudes using the hyperspherical harmonics, theprojection procedure is carried out through the hy-perspherical harmonic transformation coefficients.These coefficients are known analytically for the

w xtwo-body correlations 2 , and for higher order corre-lations they can be calculated numerically using a

w xprocedure recently proposed by Viviani 3 .Before proceeding to the derivation of the differ-

ential equations let us briefly review the hyperspheri-cal harmonics functions. To introduce the hyper-spherical coordinates we start by removing the center


( )N. BarnearPhysics Letters B 446 1999 185–190186

of mass motion and describe the internal dynamicsof the system by the normalized reversed orderNy1 Jacobi coordinates

Ny1 1h s r y r qr q . . . qrŽ .(1 1 2 3 Nž /N Ny1

PPP

2 1h s r y r qr( Ž .Ž .Ny2 Ny2 Ny1 N3 2

1h s r yr . 2( Ž . Ž .Ny1 Ny1 N2

The Jacobi coordinates are then transformed to thehyperradial coordinate

12 2 2 2(rs h qh q . . . qh s r 3Ž .Ý1 2 Ny1 i j(N i)j

and a set of 3Ny4 angular coordinates V 'Ny1� 4h , . . . , h ,a , . . . ,a . The hyper angles aˆ ˆ1 Ny1 2 Ny1 j

are defined by

hjsina s . 4Ž .j 2 2h q . . . qh( 1 j

The Laplace operator associated with the Jacobicoordinates can be written as a sum of two terms, anhyperradial term and an hyperspherical term, some-times called the generalized angular momentum op-erator as it can be regarded as a generalization of theusual angular momentum operator. The hyperspheri-cal harmonics, HH, functions YY are eigenfunc-w K xtions of this hyper-angular operator. The explicit

w xexpression for the HH functions is given by 2 :

² < :YY s ll m ll m L MÝw K x 1 2 2 21 2m , . . . ,m1 Ny1

=² < :L M ll m L M = PPP2 2 3 3 33

² < := L M ll m L MNy2 Ny2 Ny1 Ny1 Ny1Ny1

=Ny1 Ny1

Y h NN K ; ll KˆŽ . Ž .Ł Łll m j j j jy1jjjjs1 js2

=

1 3 jy5ll q , K qjy 1jž /ll K 2 2j jy1sina cosa PŽ .Ž .j j n j

= cos2a , 5Ž .Ž .j

where Y are the spherical harmonic functions,ll m

P Ž a , b . are the Jacobi polynom ials andnŽ . w xNN K ; ll K are normalization constants 4 . Thej j jy1j

w xsymbol K stands for the set of quantum numbersll , . . . , ll , L , . . . , L , n , . . . ,n and M .2 Ny1 2 Ny1 Ny11 Ny1

LsL is the total angular momentum of the stateNy1

and MsM is the magnetic projection. TheNy1

quantum numbers K are given by K s2n qKj j j jy1

q ll , where n '0, K s ll and K'K , the1 1 Ny1j 1

n ’s are non-negative integers. By construction,j

r K YY is an harmonic polynomial of degree K. Thew K xHH functions YY are eigenfunctions of the gener-w K xalized angular momentum operator with eigenvaluesŽ .K Kq3Ny5 . In what follows we shall use the

Ž . Ž .terms potential basis PB triplet basis TB andŽ .quartet basis QB to describe HH wave functions

that depend only on two, three and four particlesrespectively. With the proper choice of Jacobi coor-dinates these are HH wave functions that depend onthe last, last two or last three Jacobi coordinates.

Following now the derivation of the potentialŽ . w xbasis PB equations 1 the N-body wave function

Ž .C h , . . . ,h is written as a sum of two-body1 Ny1

amplitudes

C h , . . . ,h sC r ,V s C ,Ž . Ž . Ý1 Ny1 Ny1 i i1 2i -i1 2

6Ž .

and the Schrodinger equation is decomposed into an¨equivalent set of Faddeev like equations

N Ny1Ž .Tq V r yE CŽ .0 i i1 22

sy V yV r C . 7Ž . Ž .Ž .i i 01 2

The collective hyperradial part of the potential,Ž .V r , is an average of the two-body interaction0

over the hyper angles. The PB approximation con-sists of writing C asi i1 2

C sC r , z ; z sr rr . 8Ž . Ž .i i i i i i i i i i1 2 1 2 1 2 1 2 1 2

( )N. BarnearPhysics Letters B 446 1999 185–190 187

Ž .As a result the lhs of Eq. 7 will depend only onr, z and one must find a projection operator Pi i i i1 2 1 2

that will project C into that space. This amounts to

N Ny1Ž .rTq V r yE C syV P C ,Ž .0 i i i i i i1 2 1 2 1 22

9Ž .

where for the sake of brevity we defined V r 'Vi i i i1 2 1 2

Ž . Ž .yV r . In writing Eq. 9 we used the fact that the0

lhs is invariant to the action of the projection opera-tor and assumed that the potential V commutesi i1 2

with the projection operator P . For identical parti-i i1 2

cles it is sufficient to consider the equation for theamplitude C . It is clear that C can beN, Ny1 N, Ny1

expanded into the hyperspherical series

C r , z sC r ,h rrŽ . Ž .N , Ny1 N , Ny1 N , Ny1 Ny1

s R r YY V . 10Ž . Ž . Ž .Ý w K x w K x Ny1w xK gPB

Ž .The sum over the HH functions in 10 is restrictedto HH functions that are independent of the firstNy2 Jacobi coordinates h , . . . h . By inspec-1 Ny2

Ž . Ž . w xtion of Eqs. 4 and 5 it is evident that K gPB ifK s K s . . . s K s 0 and ll s ll s . . . s2 3 Ny2 1 2

ll s0. The projection operator P is realizedN, Ny1Ny1

by

< :² <P s YY YY . 11Ž .ÝN , Ny1 w K x w K xw xK gPB

Ž . Ž .Substituting Eqs. 10 and 11 into the Faddeev Eq.Ž .9 and using the permutation symmetry we get thefollowing equation:

1T r q N Ny1 V r yE R rŽ . Ž . Ž . Ž .K 0 w K x2

² < r < :Xsy YY V YYÝ w K x N , Ny1 w K xX XXw x w xK , K gPB

= N , Ny2X XX X XXd q2 Ny2 FŽ .w K x ,w K x w K x w K x

1 Ny2, Ny3X XXq Ny2 Ny3 FŽ . Ž . w K x w K x2

=R XX r , 12Ž . Ž .w K x

where

21 d 3Ny4 dT r sy qŽ .K 22 r drdr

K Kq3Ny5Ž .y . 13Ž .2r

Ž .Eq. 12 is the main equation of the PB approxima-w xtion and it can be either solved directly 5 or trans-

w xformed into an integral equation 6 . The matrixN , Ny 2 ² < :X XX X XXelements F s YY YY andN , Ny 2w K x w K x w K x w K x

Ny2, Ny3 ² < :X XX X XXF s YY YY are respectivelyNy2, Ny3w K x w K x w K x w K xthe matrix elements between the HH constructed

Ž .with the Jacobi coordinates defined in Eq. 2 andHH that are constructed with a set of coordinates

1 Ž .such that h s r y r or h(Ny 1 Ny 2 N Ny 12

1 Ž .s r yr . These are basically the HH( Ny3 Ny22

transformation matrix elements from one set of Ja-cobi coordinates to another set. It should be notedthat these coefficients are diagonal with respect tothe quantum numbers K , L and M.

Ž .Generalizing Eq. 9 for three- and four-bodyŽ .correlations or equivalently Generalizing Eq. 12 to

Ž .include the HH triplet basis TB and quartet basisŽ .QB we shall take the following stepsØ Write the N-body wave function as a sum of

three- or four-body amplitudes. Generalize theŽ .Faddeev equation, Eq. 7 , and obtain a set of

differential equations for the amplitudes.Ø Project the equations to the sub-space of r and

the internal three-body or four-body degrees offreedom.

Ø Using the permutation symmetry reduce the am-plitudes equations into a single equation.

It should be noted that the amplitude equations arean exact decomposition of the Schrodinger equation.¨The approximation starts when we project the equa-tions on a limited sub-space. Carrying out the aboveprocedure we should demand that the sum of theamplitude equations must reproduce the Schrodinger¨equation and that the equations do not possess anyspurious bound states, i.e. solutions with E-0 andCs0. We should also try to establish a clear con-nection between the two-body approximation, thethree-body approximation and the four-body approx-imation.


The natural way of generalizing the Faddeev de-Ž .composition, Eq. 7 , is by expanding the two-body

w xamplitudes in terms of three-body amplitudes 7

C s C . 14Ž .Ýi i i i , i1 2 1 2 3i /i , i3 1 2

The TB approximation then leads to

C sC r , z , z ;Ž .i i , i i i , i i i i i , i1 2 3 1 2 3 1 2 1 2 3

r qri i1 2z s r y rr , 15Ž .i i , i iž /1 2 3 3 2

and the corresponding equation for C isi i , i1 2 3

N Ny1Ž .rTq V r qV yE CŽ .0 i i i i , i1 2 1 2 32

rsyV P C qCi i i i , i i i , i i i , i1 2 1 2 3 1 3 2 3 2 1

1q C qC q C .Ž .Ý i i , i i i , i i i21 4 3 4 2 3 3 4i /i , i , i4 1 2 3

16Ž .

P is the projection operator on the sub-space ofi i , i1 2 3

functions of r, z , z . It should be noted that thei i i i , i1 2 1 2 3

particles i i play a different rule in C then the1 2 i i , i1 2 3

particle i . C should be symmetric with regards3 i i , i1 2 3

Ž .to the permutation i ,i but not necessarily with1 2Ž . Ž .regard to i ,i or i ,i . The total symmetry of the1 3 2 3

wave function is ensured by the summation. Omit-Ž .ting the projection operator and summing Eq. 16 on

Ž .i one can reproduce the Faddeev Eq. 7 . More then3

that, it can be seen that if C does not depend oni i , i1 2 3

Ž .z we reproduce Eq. 9 .i i , i1 2 3

As in the two-body case, we shall consider theequation for the amplitude C . The ampli-N Ny1, Ny2

Ž .tude C , Eq. 15 , depends only on the lastN Ny1, Ny2Ž .two Jacobi coordinates, Eq. 2 , therefore it can be

expanded into the hyperspherical series that containsw xonly the TB functions. An HH function K gTB if

K s ll s0 for i)2, and ll is even. TheNy i Nyi Ny1

projection operator P is realized byN Ny1, Ny2< :² <P s Ý YY YY . SubstitutingN Ny1, Ny2 w K xg TB w K x w K x

Ž .C , P in Eq. 16 and summingN Ny1, Ny2 N Ny1, Ny2

equivalent terms we finally get1T r q N Ny1 V r yE R rŽ . Ž . Ž . Ž .K 0 w K x2

r² < < :X X XXsy YY V YY dÝ w x w xw K x N , Ny1 w K x K , KX XXw x w xK , K gTB

q2 F N Ny2, Ny1X XXw x w xK K

q Ny3 F Ny2 Ny3, NX XXŽ . w x w xK K

q2 Ny3 F N Ny3, Ny2X XXŽ . w x w xK K

1 Ny2 Ny3, Ny4X XXq Ny3 Ny4 FŽ . Ž . w x w xK K2

=R XX r 17Ž . Ž .w xK

As for the PB, the matrix elements F X XXi1 i2 , i3 sw K x w K x

² < :X XXYY YY are matrix elements between thei i , iw K x w K x 1 2 3

HH constructed with the Jacobi coordinates definedŽ .in Eq. 2 and HH that are constructed with a set of

1 Ž .coordinates such that h s r y r and(Ny1 i i2 2 1

2 1Ž Ž ..h s r y r qr . These are basically(Ny2 i i i3 23 1 2

the HH transformation matrix elements from one setof Jacobi coordinates to another set. The method forcalculating these matrix elements was recently pre-

w xsented by Viviani 3 .Proceeding in the same way we may derive the

four-body, QB, equations. As for the three-bodyŽ .case, Eq. 14 , we shall present the four-body corre-

lations by expanding the three-body amplitudes intofour-body amplitudes,

C s C , 18Ž .Ýi i , i i i , i , i1 2 3 1 2 3 4i /i , i , i4 1 2 3

which we shall look for at the level of the QBŽapproximation, C sC r, z , z ,i i , i , i i i , i , i i i i i , i1 2 3 4 1 2 3 4 1 2 1 2 3

1. Ž Ž ..z , z s r y r qr qr rr. Thei i i , i i i i , i i i i i31 2 3 4 1 2 3 4 4 1 2 3

corresponding equation for C isi i , i , i1 2 3 4

N Ny1Ž .rTq V r qV yE CŽ .0 i i i i , i , i1 2 1 2 3 42

rsyV P C qCi i i i , i , i i i , i , i i i , i , i1 2 1 2 3 4 1 3 2 4 3 2 1 4

1qC qC q C qCŽ .i i , i , i i i , i , i i i , i i i , i21 4 3 2 4 2 3 1 3 4 1 3 4 2

q C qCŽÝ i i , i , i i i , i , i1 5 3 4 5 2 3 4i /i , i , i , i5 1 2 3 4

1q C . 19Ž ..i i , i2 3 5 4

( )N. BarnearPhysics Letters B 446 1999 185–190 189

As in the two- and three-body equations P isi i , i , i1 2 3 4

the projection operator on the sub-space of functionsof r, z , z z . Disregarding the projectioni i i i , i i i i , i1 2 1 2 3 1 2 3 4

Ž .operators, summing Eq. 19 on i one can repro-4Ž .duce the three-body Eq. 16 . More than that, it can

Ž .also be seen that we reproduce Eq. 16 if Ci i , i , i1 2 3 4

do not depend on z . Expressing the amplitudei i i , i1 2 3 4

C and the projection operator P ini i , i , i i i , i , i1 2 3 4 1 2 3 4

terms of the HH functions, and using symmetryarguments we get to the four-body equivalence of

Ž . Ž .Eqs. 12 and 17 ,1T r q N Ny1 V r yE R rŽ . Ž . Ž . Ž .K 0 w K x2

r² < < :X X XXsy YY V YY dÝ w K x N , Ny1 w K x w K x ,w K xX XXw x w xK , K gQB

qF Ny2 Ny3, N , Ny1X XXw K x w K x

q2 F N Ny2, Ny1, Ny3X XXw K x w K x

q2 F N Ny3, Ny2, Ny1X XXw K x w K x

q Ny4 F Ny2 Ny3, N , Ny4X XXŽ . w K x w K x

q Ny4 F Ny2 Ny4, Ny3, NX XXŽ . w K x w K x

q2 Ny4 F N Ny4, Ny2, Ny3X XXŽ . w K x w K x

1q Ny4 Ny5Ž . Ž .2

Ny2 Ny4, Ny3, Ny5X XX XX=F R r 20Ž . Ž .w K x w K x w K x

w xAn HH function K gQB if K s ll s0 forNy i Nyi

i)3, and ll is even. As for the PB and the TB,Ny1i1 i2 , i3, i4 ² < :X XX X XXthe matrix elements F s YY YY i i , i , iw K x w K x w K x w K x 1 2 3 4

are matrix elements between the HH constructedŽ .with the Jacobi coordinates defined in Eq. 2 and

HH that are constructed with a set of coordinates1 2Ž . Žsuch that h s r y r , h s r( (Ny1 i i Ny2 i2 32 1 3

1 3 1Ž .. Ž Žy r qr , and h s r y r qr q(i i Ny3 i i i2 4 31 2 4 1 2

..r . For the QB these matrix elements can bei3

calculated by a strait forward generalization of thew xprocedure of Ref. 3 from the TB to the QB.

Ž .The QB equation, Eq. 20 , provides a clear andtransparent connection between the two-, three- andfour-body correlations as it provides a systematicway to improve the PB approximation by includingTB and QB functions. This can be seen by limiting

Ž .the summation in 20 to the TB functions. As aresult the forth index in the transformation matrixelements become irrelevant and by collecting to-

Ž .gether equal terms one can reproduce Eq. 17 . Byfurther restricting the basis functions to the PB we

Ž .can recover the two-body equation, Eq. 12 .In order to get some perspective for the proposed

method, let us consider perturbation expansion forthe Hamiltonian

N Ny1Ž .HsH qlH s Tq V rŽ .0 1 02

ql V yV r . 21Ž . Ž .Ž .Ý i i 01 2i i1 2

The unperturbed Hamiltonian H is diagonal with0

respect to the HH functions. Therefore, the bosonicŽ .ground state wave function is given by f sf r ,0 0

and the corresponding expansion for the wave func-tion must take the following form:

Csf r ql f r , zŽ . Ž .Ý0 i i i i1 2 1 2i i1 2

2ql f r , z , zŽ .Ý i i i i i i , i1 2 3 1 2 2 3i i i1 2 3

q f r , z , z q . . . . 22Ž . Ž .Ý i i i i i i i , i1 2 3 4 1 2 3 4i i i i1 2 3 4

It is evident that the PB approximation correspondsto 1st order perturbation expansion. The TB approxi-mation includes only the first 2nd order term and theQB approximation includes all the 2nd order term.From this expansion we may conclude that in orderto take into account the full contribution of the 2ndorder perturbation expansion one must include up tofour-body correlations in the expansion of theSchrodinger wave function. The convergence of this¨expansion depends on the value of l. From the work

w xof Brizzi et al. 6 we see that in comparison withother many-body methods the PB approximationgives a rather good approximation to the bindingenergy of light nuclei and may conclude that forsuch systems it might be sufficient to consider 2ndorder corrections in l.

In conclusion, we have developed a systematicway to include two-, three- and four-body correla-tions in the bosonic N-body wave function. Thebasic ingredient of the method is the decompositionof the Schrodinger equation into an equivalent set of¨


amplitude equations which we project on the propersub-space. The HH basis functions are then used torealize the projection operators and to expand theamplitudes. In practice, once the transformation co-efficients are known, the solution of the TB or theQB equations is equivalent to the solution of a three-or four-body problem. It is also possible to general-ize the above formalism to include higher-order cor-relations, however we hope that in view of therelative success of the application of the PB, it wouldbe sufficient to include only three- and four-bodycorrelations.

Turning now to the more complicated nuclear, orin general Fermi, systems, it should be noted that theformalism developed in this paper is still valid givena different realization of the basis functions and theprojection operators. In fact the PB approximation

Ž w xwas already applied long ago see 6 and refs..therein to the study of real nuclei. The application

of the TB and QB approximations to the nuclearproblem is more involved but still doable.

Acknowledgements

The author wishes to thank M. Viviani, V.B.Mandelzweig, S.L. Yakovlev and R. Krivec for use-ful discussions and advice during the preparation ofthis work.

References

w x Ž .1 M. Fabre de la Ripelle, Phys. Lett. 135 1984 5.w x Ž . Ž .2 M. Fabre de la Ripelle, Ann. Phys. NY 147 1983 281.w x3 M. Viviani, Few-Body systems, accepted for publication.w x Ž . w4 V.D. Efros, Yad. Fiz. 15 1972 226 Sov. J. Nucl. Phys. 15

Ž . x1972 128 .w x5 M. Fabre de la Ripelle, Y.J. Jee, A.D. Klemm, S.Y. Larsen,

Ž .Ann. Phys. 212 1991 195.w x6 R. Brizzi, M. Fabre de la Ripelle, M. Lassaut, Nucl. Phys. A

Ž .596 1996 199.w x Ž .7 S.P. Merkuriev, S.L. Yakovlev, Teor. Mat. Fiz. 56 1984 60

w Ž . xTheor. Math. Phys. 56 1984 673

28 January 1999


Unlike particle correlations and the strange quark matterdistillation process

D. Ardouin a,1, S. Soff a, C. Spieles a, S.A. Bass a, H. Stocker a, D. Gourio b,¨S. Schramm b, C. Greiner c, R. Lednicky d,2, V.L. Lyuboshitz e,3, J.-P. Coffin f,

C. Kuhn f

a Institut fur Theoretische Physik, J.W. Goethe-UniÕersitat, Postfach 11 19 32, D-60054 Frankfurt am Main, Germany 4¨ ¨b GSI Darmstadt, Postfach 11 05 52, D-64220 Darmstadt, Germany

c Institut fur Theoretische Physik, J. Liebig-UniÕersitat, Heinrich-Buff-Ring 16, D-35392 Gießen, Germany¨ ¨d Institute of Physics of the Academy of Sciences of the Czech Republic, Na SloÕance 2, 18040 Prague 8, Czech Republic

e JINR Dubna, 141980, Moscow, Russiaf CRN Strasbourg, UniÕersite L. Pasteur, Strasbourg, France´

Received 7 November 1998; revised 28 November 1998Editor: J.-P. Blaizot

Abstract

We present a new technique for observing the strange quark matter distillation process based on unlike particlecorrelations. A simulation is presented based on the scenario of a two-phase thermodynamical evolution model. q 1999Published by Elsevier Science B.V. All rights reserved.

PACS: 25.75.q r; 12.38.Mh; 12.39.Ba; 12.39.Mh

1. Motivation

The possibility to create strangelets or droplets ofmetastable cold strange quark matter in ultra-relativ-istic collisions has been proposed and studied by

w xseveral authors 1–7 . The existence of such exotic

1 On leave from University of Nantes, U.M.R. Subatech.2 Supported by GA AV CR, Grant No. A1010601 and GA CR,

Grant No. 202r98r1283.3 Supported by RFFI, Grant No.97-02-16699.4 Supported by GSI, BMBF, DFG and Buchmann Fellowship.

states, as well as metastable exotic multi-strangeŽ .baryonic objects MEMO’s , has fundamental impor-

tance in cosmological models and in the underlyingdescription of strong interactions. Several experi-ments are being carried out at the Brookhaven AGSŽ . Ž .E864, E878 and the CERN SPS NA52 , which

Žlook for strangelet production searching for small. w xZrA ratios 8–10 .

w xAmong the different theoretical approaches 1–7 ,a mechanism of separation of strangeness from anti-

Ž .strangeness distillation process has been proposedw x4 during hadronization of a system at finite baryondensities. This scenario, which assumes a first order


( )D. Ardouin et al.rPhysics Letters B 446 1999 191–196192

phase transition, predicts a relative time delay be-tween the production of strange and anti-strangeparticles. In the present paper, we propose to searchfor this delay with the help of the novel correlation

w xmethod proposed in 11,12 . It exploits the sensitivityof the directional dependence of the correlation func-tion of two non-identical particles to the time orderof their emission. This new technique can allow forthe study of the transient strange quark matter stateeven if it decays on strong interaction time scalesw x13–15 .

A different method, exploiting K 0K 0 correla-s s

tions, also allowing for a study of the relative delaysŽ .but not their sign in K and K emission was

w xproposed in 16 . It is based on the fact that theK 0K 0 system, due to the positive CP parity of K 0,ss s

0 0triggers out the symmetric combination of the K Kstates, thus leading to the familiar Bose–Einstein

w xcorrelation pattern 17 . This pattern, contrary to thecase of non interacting identical bosons, is howeversubstantially modified by the effect of the strong

w xfinal state interactions 18 .w xHere we will use the general method of 11,12 to

study the impact of the time delays on the correla-tions in KqKy pairs. The choice of this system,similar to the case of neutral kaons, apart from thepossibility to determine the signs of the time delays,has the advantage of weaker distortions caused by

Žthe resonance production more than 50% of kaonsw x.are predicted to originate from direct emission 19 ,

as compared to pions, while the experimental feasi-bility is better than in the case of neutral kaons.

2. The mixed phase thermodynamical approach

The dynamical evolution of the mixed phase con-sisting of a quark gluon plasma and hadronic gaswill be described in a two-phase model which takesinto account equilibrium as well as non-equilibrium

w xfeatures 20,21 .Within this model, two main assumptions are

made. Firstly, the QGP is surrounded by a layer ofŽhadron gas and equilibrium described by Gibbs

.conditions is maintained during the evolution. Sec-ondly, non-equilibrium evaporation is incorporatedby a time dependent emission of hadrons from the

surface of the hadronic fireball. Within this model, itis possible to follow the evolution of mass, entropyand strangeness fraction f of the system and extracts

relative yields of particles which compare reasonablyw xwell with experimental data 19 .

One specific feature of these calculations is theprediction that the system quickly enters thestrangeness sector leaving the usual m yT plane ofq

the phase diagram. While the net-baryon number AB

decreases during the hadronization process, the squark chemical potential m increases from m ss s

Ž ini0 MeV for f s0 if we start with a pure u,d quarks.phase to several tens of MeV. Consequently, the

Ž .strangeness fraction f s N yN rA increases.s s s tot

Fig. 1. Time evolution of baryon number A, strangeness fractionq y Žf , temperature T and yields of K and K for set 1 lefts

. Ž . Ž . Žcolumn , set 2 central column and set 3 right column see.text .

( )D. Ardouin et al.rPhysics Letters B 446 1999 191–196 193

w xThis is the so-called ‘‘distillation process’’ 4 whichcan result in the formation of stable or metastableblobs of strange matter in the case of low bag model

1r4 Žconstants B -180 MeV. In this case, a cooling.of the system is predicted. Strange and anti-strange

quarks, produced in equal amount in the hot plasma,do not hadronize at the same time. Because of the

Ž . Žexcess of massless u and d quarks as compared to.their anti-quarks in the case of a baryon rich plasma,

q 0 Žthe hadronization of K and K containing sy.quarks will be favored preferentially to K and K0

Ž .containing s quarks . The evaporation from the sur-face of the hadron gas, which is rich of anti-strange-ness, carries away Kq, K 0, . . . thus charging up theremaining mixed system with net positivestrangeness. Fig. 1 shows, for two rather different

w xbag constants, the time evolution 19 of the baryonnumber A, the strangeness fraction f and the tem-s

perature T for initial entropy SrA s10,45 andini

f s0. Another prediction of interest is the hadronics

freeze-out time. Strange and antistrange hadrons arenot emitted at the same time since the hadron densi-ties in the outer layer are dictated by the stronglytime dependent chemical potentials and the time

w xdependent temperature 20,19 . One should also no-tice that the difference between KqN and KyNcross sections in a baryon rich gas will slow down

Ž .the diffusion of negative strangeness s , thus leadingw xto an additional creation of time differences 20,19 .

In the present work, this time separation charac-terizes the transient existence of the plasma. In thefollowing, we will show that it can be used to searchfor the distillation process by using correlation tech-

w xniques of unlike particle pairs 22,23,18 .

3. Unequal particle correlations: time-orderingsensitivity

Particle correlations at small relative momentumare mainly driven by strong and Coulomb final state

Ž .interactions in the case of unlike particle pairs .At high bombarding energies, this effect has been

often treated with a well known size-independentCoulomb correction factor. Contrary, at low ener-gies, the Coulomb effect becomes the main tool forthe study of this evolution due to larger time se-quence of emission.

Unlike particle correlations have been used exper-imentally and theoretically for more than 15 yearsw x18,22 , particularly for the study of heavy ion colli-

w xsion mechanisms around the Fermi energy 22–25 .Particularly, evidence for three-body Coulomb ef-fects on two-particle correlations was observed inw x24 and taken into account to describe a variety oftwo-particle correlation patterns using classical tra-

w xjectory calculations 25 or a complete three-bodyw xquantum approach 11,26 in the adiabatic limit. The

sensitivity of unlike particle correlations to the orderof particle emission was pointed out and studied

w xusing both quantum 11,12 and classical trajectoryw x27 approaches. In the framework of the former oneand under the assumption of sufficiently small den-sity in phase space, the two–particle correlationfunction at a given relative c.m.s. momentum qs2 kis determined by the modulus squared of the two–particle amplitude averaged over the relative c.m.s.coordinates r ) of the emission points. The sensitiv-ity of the correlation function to the time delays, orgenerally to the space-time asymmetries in particleproduction, appears due to the dependence of thetwo-particle amplitude on the scalar product kr ). Inparticular, in the limit of large relative emission

< <times ts t y t , Õ t 4r, the Lorentz transforma-1 2

tion from the source rest frame to the two-particleŽc.m.s Õ and g are the pair velocity and Lorentz.factor :

r ) sg r yÕt , r ) sr , 1Ž . Ž .L L T T

indicates that the vector r ) fyg zt is only slightlymodified by averaging over the spatial location ofthe emission points in the rest frame of the source.So the vector r ) is nearly parallel or antiparallel tothe pair velocity z, depending on the sign of thetime difference t. The dependence of the amplitudeon kPr ) is thus transformed into the dependence ofthe correlation function on ykPzg t. Therefore, thesensitivity to the sign of the time difference t is dueto the odd part, in kPz, of the correlation function.The mean relative emission time, including its sign,can be determined by comparing the correlationfunctions Rq and Ry corresponding to kPz)0

w xand -0 12 . Noting that for particles of equalmasses the sign of kPz coincides with the sign ofthe velocity difference Õ yÕ , we can see the sim-1 2

ple classical meaning of the above selection. It corre-


sponds to the intuitive expectation of different parti-cle interaction in the case when the faster particle isemitted earlier as compared to the case of its lateremission.

For charged particles characterized by a large< < ² ): < < ŽBohr radius, a 4 r , Re f f is the amplitude

. Ž q. Ždue to the strong interaction , the ratio 1qR r 1y.qR at very small q takes on a simple analytical

w xform 28 :

1qRq r 1qRyŽ . Ž .² ): ² :f1q2 r ra™1y2 g Õt ra, 2Ž .L

< <where the arrow indicates the limit Õ t 4r. Thus,q y Ž .for K K -system asy111 fm , each fm in the

² ): ² :asymmetry r or g Õt transforms in a 2%LŽ q. Ž y.change of the 1qR r 1qR ratio at q™0.

The correlation functions Rq and Ry as predictedw xin 11,12 well agree with the proton–deuteron corre-

lation functions measured in low energy nuclearw xcollisions studied at GANIL 29,14,30,31 . As pro-

w xposed in 12 , this technique can be extended for astudy of the emission time differences between anykinds of interacting unlike particles.

In preliminary studies using simple event genera-w x q ytors, it has been demonstrated 13–15 that K K

pairs with mean emission time differences as smallas "5 fmrc can give rise to observable differencesbetween Rq and Ry correlation functions. It hasbeen also found that most of the effect at q-

10 MeVrc originates from the Coulomb interactionbetween the two kaons and that, in full correspon-

Ž . Ž .dence with Eq. 2 , the later on the average emis-y Ž q. Žsion of K ’s is associated with the 1qR r 1q

y.R ratio less than unity.In this paper, we will use this method in conjunc-

Žtion with the dynamical two phase description see.previous section in order to quantitatively demon-

strate its sensitivity to the predicted delays betweenthe Kq and Ky emission related to the productionof a transient strange quark matter state.

w xThree sets of parameters 15,19 have been cho-sen:

Žset 1 & 2 representing AuqAu at AGS energiesw x.19 : initial mass As394, entropy per baryon SrAs10, initial net strangeness f s0;s

1r4 Žset 1: B s 160 MeV, strangelets albeit. w xmetastable are formed 19 .

set 2: B1r4 s235 MeV, strangelets are not dis-w x Ž .tilled 19 . They are not metastable.

Ž w x.set 3 representing SqAu at SPS energies 19 :bag model constant B1r4 s235 MeV, initial massAs100, entropy per baryon SrAs45, initial netstrangeness f s0.s

Ž 1r4In the case of the low bag constant B s.160 MeV – set 1 – a cooling of the system is

predicted leading to rather long kaon emission times.q ŽIn the early stage mainly K ’s are emitted see Fig.

.1, l.h.s. column so that a cold strangelet emerges inŽ . Ž .a few tens fmrc. According to Eqs. 1 and 2 and

the corresponding discussion, the later emission ofKy ’s should lead to the correlation function ratioŽ q. Ž y.1qR r 1qR less than unity provided that the

² ): q yasymmetry r in K K c.m.s. is dominated byL² :the time asymmetry term Õt . As seen from Table 1

and Fig. 2 this is indeed the case: the spatial asym-² : Žmetry term r contributes by less than 10% in theL

.same direction ; the predicted value of the deviationof the correlation function ratio from unity of y26%at q™0 is in good agreement with the interceptvalue of the calculated ratio curve with the ordinateaxis. Also, the deviation from unity remains impor-tant and statistically meaningful up to the regionqf30 MeVrc. Thus, in spite of experimental diffi-culties to measure correlations down to a fewMeVrc, the signal proposed for consideration isaccessible to usual correlation measurements. Sincefor various two-particle systems produced in themid-rapidity region, the ordinary dynamical mecha-nisms lead to the intercept values deviating from

w xunity by less than 10% 28 , we can consider theeventual observation of a large negative asymmetryin the KqKy correlation function ratio as a signal ofthe strangelet formation.

Ž 1r4In the case of the high bag constant B s.235 MeV – sets 2 and 3 – the system heats up

w xslightly. This results in a fast hadronization 20 and,

Table 1Mean values of the relative space-time coordinates for pairs withrelative momenta qF50MeVrc calculated within the mixed

Ž .phase thermodynamical model see text) )² : Ž . ² : Ž . ² : Ž . ² : Ž .Set Õt fm r fm r fm r fmL L

1 y10.9 1.1 14.3 17.52 y1.4 0.5 2.6 5.53 y0.7 0.2 1.3 6.3

( )D. Ardouin et al.rPhysics Letters B 446 1999 191–196 195

Ž q . Ž y .Fig. 2. Ratios of correlation functions 1q R r 1q R forthree parameter sets in a mixed-phase thermodynamical modelŽ .see text . The strange quark matter distillation process predicted

Ž .in the case of a low bag constant upper curve results in ameasurable deviation from unity in the range q-30MeVrc.

subsequently, in a small difference between the Kq

y w x Žand K emission times 19 see the middle and.r.h.s. columns in Fig. 1 . Similar to the previous case

the Kq ’s are emitted earlier than Ky ’s. Also the² ):asymmetry r is dominated by the time asymme-L

² : ² :try term Õt , the spatial one, r , contributing byLŽabout 25% in the same direction see Table 1, rows 2

.and 3 . However, the asymmetry effect is now muchŽ .weaker Fig. 2 : the predicted intercepts with the

ordinate axis of the correlation function ratio deviatefrom unity by y4.7% and y2.3% for sets 2 and 3,respectively.

Clearly, in this case, such a relatively weak asym-metry effect cannot be considered as a signal of thestrangeness distillation without detailed studies of

Žthe impact of the ordinary mechanisms rescattering.effects, for example .

Regarding the width of the effect seen in theŽ .correlation function ratios Fig. 2 , it decreases with

² ): Ž .the increasing r see Table 1 in correspondencewith the narrowing of the correlation effect itself.

Ž² ): .Thus, for set 1 r f18 fm the effect rapidlyvanishes with increasing q and extends up to about

Ž² ):30 MeVrc only. For sets 2 and 3 r f6 fm in.both cases the deviation from unity is softer in the

same range of q values and extends up to about60 MeVrc.

4. Conclusions

We have presented a novel method which can beapplied to characterize the possible existence of astrange quark matter distillation process in ultra-rela-tivistic heavy-ion collisions. The method is based on

w xthe predictions 18,11–14 for unequal particle corre-lations and exploits the predicted properties of the

w xtransient, strange quark matter state 4,20,21,19 ,even if it decays on strong interaction time scales.Using the description of this strangeness distillationprocess by a dynamical evolution model for themixed QGP-hadronic phase, we have quantitativelydemonstrated the sensitivity to the bag constant cor-related to the stability of the quark matter possiblyencountered in available experimental situations.

For the lower bag constant case a strong andsharp negative asymmetry effect is predicted in thecorrelation function ratio, thus offering a clear signalof the predicted strangeness distillation process pro-vided the experimental resolution and statistics aresufficient to measure the ratio down to about10 MeVrc. Weaker though wider signals are pre-dicted for the higher bag constant hypothesis. In thiscase the competition with other collision mecha-nisms generating the asymmetry is possible. Thesemechanisms, including the expansion of the hadronic

w xsystem, are currently under investigation 32,33 .Finally, we think that the unlike particle correla-

tion technique may offer a very valuable tool todisentangle between different and presently debatedscenarios for the phase transition in QCD matter atfinite net baryon density.


Acknowledgements

The authors would like to thank B. Erazmus, L.Martin, D. Nouais, C. Roy, and A.Dumitru for usefuldiscussions andror for providing numerical pro-grams. D.A. is pleased to thank the Institut fur¨Theoretische Physik at the University of Frankfurtfor an invitation and kind hospitality.

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w x Ž .15 S. Soff et al., J. Phys. G, Nucl. Part. Phys. 23 1997 2095;Special Issue of Int. Symposium on Strangeness in QuarkMatter, Santorini, Greece, 1997.

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w x Ž .20 C. Greiner, H. Stocker, Phys. Rev. D 44 1991 3517.¨w x21 C. Spieles, L. Gerland, H. Stocker, C. Greiner, C. Kuhn, J.P.¨

Ž .Coffin, Phys. Rev. Lett. 76 1996 1776.w x22 For a review see e.g., D.H. Boal, C.K. Gelbke, B.K. Jen-

Ž .nings, Rev. Mod. Phy. 62 1990 553, and references therein.w x Ž .23 J. Pochodzalla, Phys. Rev. C 35 1987 1695.w x Ž .24 J. Pochodzalla et al., Phys. Lett. B 161 1985 256; B 174

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lished.w x Ž .33 S. Soff et al., 1997 to be published.

28 January 1999


Direct observation of the inversion of flow

R.C. Lemmon a, M.B. Tsang a, W. Trautmann b, R.J. Charity c, J.F. Dempsey c,J. Dinius a, W. Dunnweber d, S. Gaff a, C.K. Gelbke a, M.J. Huang a, G.J. Kunde a,¨

W.G. Lynch a, L. Manduci a, R. Popescu a, R. Ronningen a, L.G. Sobotka c,L. Weathers a, D. White a

a NSCL and Department of Physics and Astronomy, Michigan State UniÕersity, East Lansing, MI 48824, USAb Gesellschaft fur Schwerionenforschung, D-64291 Darmstadt, Germany¨

c Department of Chemistry, Washington UniÕersity, St. Louis, MO 63130, USAd Sektion Physik, UniÕersitat Munchen, D-85748 Garching, Germany¨ ¨

Received 6 July 1998; revised 4 December 1998Editor: J.P. Schiffer

Abstract

The sign of the mean transverse momentum of non-equilibrium light charged particles was determined from the circularpolarization of coincident g-rays emitted from residual nuclei for 14N-induced reactions on 154Sm at incident energies,ErAs35, 100 and 155 MeV. Results of the emitted a-particles at mid-central collisions show the predicted transition frommean-field dominated dynamics at low energies to nucleon-nucleon collision dominated dynamics at high energies. q 1999Published by Elsevier Science B.V. All rights reserved.

Transverse momentum transfer in heavy-ion colli-sions is an important observable that reflects thebalance between the mean field and collisional dy-

w xnamics 1–4 . This balance evolves with both impactw xparameter and bombarding energy 5 . At low ener-

Ž .gies ErAs10 MeV , the mean field is attractiveand two-body collisions are suppressed by the Pauliexclusion principle. There, the sense of rotation ofthe intermediate reaction complex has been deter-mined via measurements of the circular polarization

w xof g-rays emitted by the excited target residue 6,7 .Coincidence measurements between these g-rays andthe reaction products demonstrated the existence of

w xorbiting trajectories at the Coulomb barrier 6 and

predominantly negative deflection angles for non-w xequilibrium light charged particles and fragments 7 .

Ž .At higher energies ErA)100 MeV , two-bodycollisions are frequent, and eventually the repulsivenature of the nucleon-nucleon collisions dominates.Both the attractive and repulsive collective motionsare studied in global transverse momentum analysesw x8 in which the transverse momentum of each frag-ment is projected onto an estimated reaction planefor the event. The dominant correlation between thefragmentsX transverse momenta is caused by collec-tive motion, the signal of which can be deduced byaveraging the transverse momentum over manyevents. Directed collective flow can then be defined


( )R.C. Lemmon et al.rPhysics Letters B 446 1999 197–202198

by the slope of the average in-plane transverse mo-mentum versus rapidity around the mid-rapidity re-gion. In such global transverse momentum analyses,however, information about the overall sign of thedirected momentum is lost even though transportmodel calculations require sidewards collective flowon the same side as the projectile for the successfulinterpretation of flow. Until now, the sign of the

w xtransverse flow has not been determined 1–4 thoughindirect conclusions may be drawn from the recent

w xmeasurements of pion flow 9 .The transition from the attractive mean-field dom-

inated dynamics of low-energy reactions to the re-pulsive nucleon-nucleon collision dominated dynam-ics of high-energy reactions is expected to occur atincident velocities comparable to the Fermi velocity

w xwhere Pauli blocking becomes less effective 10 .The most direct way to observe the transition fromattractive to repulsive interactions is to measure thechange in sign of the transverse momentum vector.Unfortunately, such experiments are rare and verydifficult technically. Instead, most studies measuredthe ‘‘balance energy’’, the energy where the attrac-tive and repulsive interactions supposedly balanceand the flow should vanish. These measurementsrely on deducing the minimum in the transverse flowvalues. The ‘‘disappearance’’ of flow has been esti-mated to be around 100 MeV per nucleon bombard-

w xing energies for light symmetric systems 11 andw xdepends on impact parameters 5,12 .

In this article, we describe an experiment de-signed to determine the mean sign of the scatteringangles of the emitted particles for the 14 N q 154Smreaction at ErAs35, 100 and 155 MeV. We usethe technique of measuring the circular polarizationof g-rays emitted in coincidence with the lightcharged particles. In order to observe the change ofsign, an impact parameter filter is required. Thisfilter must be transparent to the g-rays and compactenough to fit between the two polarimeters whichdetermine the circular polarization of the emittedg-rays.

The measurements were performed at the Na-tional Superconducting Cyclotron Laboratory atMichigan State University. The accelerator providedbeams of 35 A MeV, 100 A MeV and 155 A MeV14 N which impinged on an isotopically enriched

154 Ž .target of Sm 98.7% of areal density 3.15

mgrcm2. Evaporation residue, fission and total fu-sion cross-sections were previously measured for thisreaction over the same range of incident energies,and significant residue cross-sections were found to

w xexist even at the highest energies 13 .In the polarization experiment, a doubly symmet-

ric arrangement of the experimental apparatus wasused. A schematic of the experimental set up isshown in Fig. 1. Two DEyE telescopes were posi-tioned at fs08 and fs1808 around the beam axisand subtended approximately 108FuF358, where u

and f are the polar and azimuthal angles respec-tively. These were used to detect and identify chargedparticles. The DE detector consisted of a 5 cm = 5cm 16-strip Si detector, 300 mm thick, and waspositioned 135 mm from the target position. The E

Ž .detector consisted of nine tapered CsI Tl detectors,7 cm long. These were arranged in a square 3=3geometry and placed immediately behind the Si de-tector. A compact cylindrical multiplicity filter, theMinitube, consisting of 58 scintillating fibers, was

Fig. 1. Cross-sectional view of the detector setup inside thescattering chamber. Two Si-CsI detector arrays were placed sym-metrically around the beam axis in forward direction, coveringpolar angles 108Fu F358. The multiplicity of charged particleswas measured with the cylindrical Minitube, mounted symmetri-cally around the beam axis in the gap between the two polarime-

Ž .ters not shown . The inner circle indicates the outer diameter ofthe polarimeters mounted perpendicular to the plane defined bythe beam and the two charged particle detector telescopes.

( )R.C. Lemmon et al.rPhysics Letters B 446 1999 197–202 199

placed coaxially around the beam axis in the gapbetween the two polarimeters. The obtained informa-tion on the multiplicity of light charged particles wasused to derive the reduced impact parameter, brb ,max

w xduring the collision 14 .The circular polarization of g-rays emitted per-

pendicular to the reaction plane defined by the beamaxis and the coincident light charged particles, wasmeasured with two forward-scattering polarimetersw x15 . These were positioned at us908, fs908 and

w xus908, fs2708. The sign convention adopted 6,7to define the polarizations with respect to the quanti-

< <zation axis n, is given by nsp =p r p =p wherei f i f

p and p are the momentum vectors of the beami f

and the detected particle, respectively. The measure-ment relies on the assumption that the angular mo-mentum transferred to the target residue will bepreferentially oriented in the direction expected for afriction-like mechanism between the surfaces of the

w xcolliding nuclei 6 . Thus positive circular g-ray po-larizations correspond to a photon spin parallel to n,and a deflection of the emitted particle to negativeangles by the nuclear mean field. Negative circularg-ray polarizations correspond to a photon spin anti-parallel to n and a deflection of the emitted particleto positive angles, caused by repulsive effects ofnucleon-nucleon collisions. Zero polarization valuesmay indicate that either the attractive and repulsiveinteractions balance each other or that the detectedlight-charged particles, in the absence of a collectivevelocity component, are emitted at random azimuthaldirection.

Experimentally, the count-rate asymmetry, P A,g

is measured. For the doubly symmetric detector sys-tem, the count rate asymmetry can be expressed asw x15

2N N 1qP A11 22 g

s 1Ž .ž /N N 1yP A12 21 g

where N are the coincidence count rates of particlei j

detector i and polarimeter j. The analyzing power Acorresponds to the sensitivity of the overall polarime-ter setup to the circular polarization of the emittedg-rays. The direction of the polarimeter magneticfield was reversed every hour during the experimentin order to detect and cancel out spurious count rateasymmetries.

Measurements of A were made using g-raysources and compared to theoretical simulations us-ing Monte Carlo techniques, which lead to a value of

w xAf1.5% 15 . The value of the effective analyzingpower is considerably reduced by neutrons emittedin the reaction. The neutrons interact with the detec-

Ž X .tor and produce g-rays via n,ng reactions which inturn interact with the detector as well. A comprehen-sive experimental study of the effects of the neutronmultiplicity on the analyzing power was thereforemade and values of A were estimated for the ener-gies measured and impact parameter gates used inthe subsequent analysis. These values were found tovary between 0.85% and 0.95%, similar to valuesobtained previously for the same reaction at ErAs

w x35 MeV 7 .Ž .Assuming that N fN fN fN in Eq. 111 12 21 22

and Af0.9%, the uncertainties in P can be de-g

rived.

1d P s110 2Ž .Ž . (g N

where NsN qN qN qN is the total num-11 12 21 22

ber of coincidence counts. Thus a measurement witha statistical accuracy of 5% will require a minimumof 5 million counts. Even if one is to determine P tog

within the statistical uncertainty of 30%, 150,000coincidence particles will still be necessary. Thus,the present experiment is very challenging both tech-nically and in terms of statistics.

Angular correlation measurements have demon-strated that charged particles are preferentially emit-

w xted in the reaction plane 16 . Individual chargedparticles are therefore useful for an approximatedetermination of the reaction plane. Note also thatfor the observation of the polarization, a much poorerdetermination of the reaction plane will be sufficientthan needed for the observation of the alignmentapparent in an azimuthal angular distribution. Since

w xthe magnitude of polarization values increases 7while the dispersion of the reaction plane decreases

w xwith fragment mass 16 , polarization values areexpected to be the largest in magnitude, and thechange in sign the strongest, for a particles ascompared to protons, deuterons and tritons. Thisfollows from the fact that the random momenta ofthe constituent nucleons, relative to the collective


Fig. 2. The circular polarization of coincident g-rays emitted fromresidual nuclei for 14 N-induced reactions on 154Sm as a function

Ž .of incident energy for mid-central solid points and peripheralŽ .collisions open points for a particles.

momenta, increasingly cancel. The measured circularpolarization of g-rays in coincidence with a parti-cles is shown in Fig. 2 and listed in Table 1 as afunction of incident energy. The quoted errors corre-spond to the statistical uncertainty of P A and dog

not include the uncertainty of A.Three impact parameter bins have been chosen,

Ž .corresponding to central brb -0.2 , mid-centralmaxŽ . Ž .0.2 F brb F0.6 and peripheral brb )0.6max max

collisions. The impact parameter bins were chosen tomaximize the magnitude of the circular polarizationof g-rays with maximum statistics to reduce the

uncertainties. Finer impact parameter bins with rea-sonable statistical uncertainties could not be im-posed. To maximize pre-equilibrium emissions andminimize the contributions from evaporation, an an-gular gate of 258FuF358 and a threshold energy of5 MeV per nucleon are imposed on the emittedparticles.

At all incident energies the central impact parame-ter bin has a polarization that is statistically consis-tent with zero. Due to lack of statistics, these datahave very large error bars and are not plotted here.Rather they are listed in Table 1 together with all thepolarization data points measured in the present work.

ŽFor peripheral collisions open points in Fig. 2 and.Table 1 , the polarization is positive at 35 A MeV

and very close to zero at the two higher energies of100 A and 150 A MeV. Similar trends are observedfor the measured P for particle mass AF3. Theg

observation is consistent with previous studies wherethe maximum flow has been observed at mid-central

w xcollisions 17 .Ž .For mid-central collisions closed points , the most

striking feature is the change in sign of the a-particleassociated polarization from positive values at 35 AMeV to negative values at 100 A MeV and 155 AMeV. This change in sign of the a-particle associ-ated polarization provides a direct observation of thechange from attractive mean field dominated dynam-

Ž .ics at low energies P )0 to repulsive nucleon-g

nucleon collision dominated dynamics at the higherŽ .energies P -0 . The change of sign occurs aroundg

ErAs70 MeV. The previous study of the balanceenergy for symmetric systems ranging from CqC to

w xKrqNb reactions 11 , suggests the same value of

Table 1Circular polarization of coincident g-rays emitted from residual nuclei for the reaction 14 Nq154Sm

3Ž . Ž . Ž . Ž . Ž . Ž .ErA MeV brb P p P d P t P He P amax g g g g g

35 0.1"0.1 y0.09 "0.29 0.11 "0.40 4.04 "3.79 y0.35 "1.02 y0.34 "0.410.4"0.2 0.046"0.083 0.20 "0.12 0.40 "0.26 0.47 "0.29 0.29 "0.120.8"0.2 0.11 "0.03 0.15 "0.05 0.13 "0.06 0.15 "0.10 0.14 "0.04

100 0.1"0.1 y0.015"0.088 y0.12 "0.12 0.12 "0.15 0.086"0.276 0.11 "0.150.4"0.2 0.025"0.038 y0.002"0.052 0.050"0.068 0.16 "0.12 y0.17 "0.060.8"0.2 0.019"0.022 y0.010"0.031 y0.003"0.043 0.063"0.067 0.079"0.043

155 0.1"0.1 0.124"0.428 y0.19 "0.65 y0.531"0.80 y0.61 "1.50 y0.139"0.530.4"0.2 0.062"0.092 y0.078"0.142 0.038"0.181 0.19 "0.32 y0.29 "0.140.8"0.2 y0.023"0.021 y0.034"0.035 y0.012"0.048 0.036"0.078 0.011"0.054

( )R.C. Lemmon et al.rPhysics Letters B 446 1999 197–202 201

about ErAs70 MeV for mass 170. It remains to beunderstood, however, why the balance energy for anasymmetric system of 14 Nq154Sm should follow themass dependence established for symmetric systems.

Fig. 3 shows the measured g-ray polarizations incoincidence with protons, deuterons and tritons forthe mid-central collisions. The magnitude of thepolarizations are smaller than the P associated withg

the emitted a particles. This may be partly related tothe fact that the statistics are much poorer fordeuterons and tritons and that the reaction planes areless well defined.

To provide consistency checks between the mea-sured circular polarizations associated with p, d, tand a-particles, we adopt the coalescence model. Inthis model, the cross section of a composite particlewith mass A is proportional to the Ath power of thenucleon cross-sections

As u A s u 3Ž . Ž . Ž .A As1

Assuming the light particles are emitted in theŽ .reaction plane, the g-ray polarization P A associ-g

ated with a fragment with mass A can be related to

Fig. 3. The circular polarization of coincident g-rays emitted fromresidual nuclei for 14 N-induced reactions on 154Sm as a function

Ž .of incident energy for mid-central solid points collisions for p, d,t particles. The hatched areas indicate the predicted polarizationsextracted by scaling the alpha circular polarization according to

w xthe coalescence model 7 .

the difference between positive- and negative-anglecross sections.

s yu ys quŽ . Ž .A AP A ,u A 4Ž . Ž .g

s yu qs quŽ . Ž .A A

Ž . Ž . ² :If we define Ds ss yu ys qu and s sA A A Aw Ž . Ž .xs yu qs qu r2A A

P A ,uŽ .g

A A² : ² :s qDs r2 y s yDs r2Ž . Ž .1 1 1 1A 5Ž .A A² : ² :s qDs r2 q s yDs r2Ž . Ž .1 1 1 1

P A fA P 1 6Ž . Ž . Ž .g g

Ž .for small Ds , and P 1 is the proton g-ray polar-1 g

ization.Ž .Using Eq. 6 , we can compute the expected

values of P for p, d, t from the measured Pg g

associated with the a particles as indicated by thehatched areas in Fig. 3. The magnitude of the mea-sured polarizations are consistently smaller than thescaled P . For tritons, the expected change of signg

for the polarization is not observed. However, muchmore statistics will be needed in order to study themass dependence on the sign change of the circularpolarization associated with emitted light particles.In previous balance energy measurements on sym-

w xmetric systems 11 , the balance energy was assumedto be independent of the mass of the emitted parti-cles, however the measurements were not definitivedue to lack of statistics especially for deuterons andtritons as in the present study.

In summary, we have measured the circular polar-ization of coincident g-rays emitted from residualnuclei for 14 N-induced reactions on 154Sm at ErAs35, 100 and 155 MeV for mid-central and peripheralimpact parameters. We deduced the sign of the meantransverse momentum of non-equilibrium lightcharged particles. The circular g-ray polarization as-sociated with a particles emitted in mid-centralcollisions changes sign from positive at 35 A MeV tonegative at 100 A MeV and 155 A MeV, demonstrat-ing directly the predicted transition from nuclearmean-field dominated dynamics at low energies tonucleon-nucleon collision dominated dynamics athigh energies. The change of sign was not observedfor a-associated polarization in peripheral or centralcollisions. Due to lack of statistics, the mass depen-


dence of the polarization is not well determined forparticles lighter than alphas at high incident energy.

Acknowledgements

The authors would like to thank H.J. Maier formaking the isotopically enriched 154Sm targets. Thiswork is supported by the National Science Founda-tion under Grants No. PHY-95-28844.

References

w x1 H.H. Gutbrod, A.M. Poskanzer, H.G. Ritter, Rep. Prog.Ž .Phys. 52 1989 1267, and references therein.

w x Ž .2 M.J. Huang, Phys. Rev. Lett. 77 1996 3739.w x Ž .3 P. Danielewicz, Phys. Rev. C 51 1995 716.w x Ž .4 C.A. Ogilvie, Phys. Rev. C 40 1989 2592.

w x5 G.F. Bertsch, W.G. Lynch, M.B. Tsang, Phys. Lett. B 189Ž .1987 384.

w x Ž .6 W. Trautmann et al., Phys. Rev. Lett. 53 1984 1630; W.Ž .Trautmann et al., Nucl. Phys. A 422 1984 418; W.

Ž .Dunnweber, K.M. Hartmann, Phys. Lett. B 80 1978 23.¨w x Ž .7 M.B. Tsang et al., Phys. Rev. Lett. 57 1986 559; M.B.

Ž .Tsang et al., Phys. Rev. Lett. 60 1988 1479.w x Ž .8 P. Danielewicz, G. Odyniec, Phys. Lett. B 157 1985 146.w x Ž .9 J.C. Kintner, Phys. Rev. Lett. 78 1997 4165.

w x Ž .10 J.J. Molitoris, H. Stocker, Phys. Lett. B 162 1985 47.¨w x Ž .11 G.D. Westfall et al., Phys. Rev. Lett. 71 1993 1986, and

Ž .refs. therein; J.P. Sullivan et al., Phys. Lett. B 249 1990 8;Ž .V. de La Mota et al., Phys. Rev. C 46 1992 677; D.

Ž .Klakow, G. Welke, W. Bauer, Phys. Rev. C 48 1993 1982.w x12 S. Soff, S.A. Bass, Ch. Hartnack, H. Stocker, W. Greiner,¨

Ž .Phys. Rev. C 51 1995 3320.w x Ž .13 A.A. Sonzogni, Phys. Rev. C 53 1996 243.w x Ž .14 C. Cavata et al., Phys. Rev. C 42 1990 1760; Y. Lou et al.,

Ž .Nucl. Phys. A 604 1996 219.w x Ž .15 W. Trautmann, Nucl. Instrum. Methods 184 1981 449.w x Ž .16 M.B. Tsang, Phys. Rev. C 47 1992 2717, and the refer-

ences therein.w x Ž .17 R. Pak, Phys. Rev. Lett. 78 1997 1022.

28 January 1999


Viscosity of scalar fields from classical theory

A. Jakovac 1´Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany

Received 20 August 1998; revised 20 November 1998Editor: P.V. Landshoff

Abstract

We show how the resummation for time dependent quantities at high temperature can be performed with an effectiveclassical theory. As an application we demonstrate that the leading term in the shear viscosity, which is related to the r 2 2F F

spectral function can be calculated classically, either using classical linear response theory or from the classical F 2

correlation function. The classical result depends explicitly on the cutoff, and the choice L;T reproduces the knownquantum result. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

High temperature quantum field theories areknown to suffer from a number of IR problemswhich make the direct application of perturbationtheory unreliable. Part of these problems can becured by integrating out the hard thermal modes. Forthe static quantities this yields dimensional reductionw x1 , for the nonstatic ones, depending on the way we

w xdo it, we arrive at the HTL effective action 2 or anw xeffective classical field theory 3–7 . In particular it

was shown that in the F 4 theory the self-energy canbe calculated classically, and also the first quantumcorrection can be reproduced from the effective the-

w xory 3,6 .The transport coefficients can be calculated from

the microscopic theory using linear response theoryw x8 . The shear viscosity gets the dominant contribu-


2Ž .tion from the spectral function of the operator F xw x9

r 2 2 p , pŽ .F F 0hs lim

pp , p™00 0

2 2Disc F x ,F 0² :Ž . Ž .s lim . 1Ž .

pp , p™00 0

ŽOne can use a quantum theory with effective re-.summed spectral functions to compute this quantity

w x9–11 . The goal of this paper is to show that thesame results can be obtained from the classical the-ory as well, which provides a simple calculationalpossibility as well as a feasible numerical frame-work.

ŽIn the followings we shortly recall cf. Refs.w x.4,5,7 and generalize, how one can use the classicaltheory to perform resummation for time dependentquantities. We will then apply the formalism to theviscosity.


( )A. JakoÕacrPhysics Letters B 446 1999 203–208´204

2. Resummation with classical theory

Dimensional reduction, which yields an extremelypowerful method to compute static quantities at hightemperatures uses a very general, renormalization

Ž .group RG inspired technique to get rid of IRdivergencies: it identifies the most IR sensitive de-grees of freedom, separates them and integrates overthe remaining ones. In this form it can be generalizedto develop a resummation method also for the non-static quantities. To this end we have first to find theIR sensitive degrees of freedoms, with other wordsthe source of the IR divergencies of the Feynmandiagrams. The diagrams are generated by the gener-ating functional; for the F 4 theory at finite tempera-

w xture it reads 12

dw xZ j sexp yi HH I ž /ž /id j tŽ .c k

=1

X Xexp y j t G k ,t ,t j t ,Ž . Ž . Ž .H yk kž /2 c

2Ž .

where

1yi v t i v tk kG k ,t s e Q t qe Q ytŽ . Ž . Ž .Ž .c c2vk

n vŽ .kq cosv t 3Ž .k

vk

Ž . Ž bv .y1is the propagator, n v s e y1 the Bose-ŽEinstein distribution, c is a real time contour eg. the

. 2 2 2Keldysh contour and v sk qm . In the IR< <regime where k <T the vacuum part behaves as

2 Ž .;1rv, the matter part is ;Trv , because n v sTrv q OO 1 . For massless fields the latter isŽ .quadratically, the former just linearly diverges in theIR. Let us denote by G the IR sensitive part of theIR

propagator; it is defined asymptotically, its generalform reads

TG k ,t s 1qOO bv cosv t . 4Ž . Ž . Ž .Ž .IR k2vk

Let us moreover denote the IR regularized propaga-˜tor by GsGyG .IR

The dimensional reduction technique suggests thatwe should rearrange the perturbation theory and

postpone the calculation with the most singular prop-agator. We can try to represent the IR propagator bya Gaussian path integral, like in the RG, and theninterchange the order of the operations. Symboli-cally, if

1 1)df exp y f fq ijf ;exp y jG jH IRž /ž /2G 2IR

5Ž .

then the effective theory is given by

1)w xZ j s df exp y f fH ž /2G IR

=d

exp y HH I ž /ž /id j

=1

˜exp y jGjq ijf . 6Ž .ž /2

The perturbation theory generated by the interactionsis now IR finite, and provides an action for itsbackground field which will be the kernel of thesubsequent path integral. In the standard RG we use

˜Ž .the formula to lower the cutoff by considering G kŽ < < . Ž .sQ L) k )LyDL G k , but in this way we

can introduce completely new type of degrees offreedom.

In our case the very special form of the timedependence of the IR propagator allows a 3D repre-sentation. In fact we can write

1˜ w xexp y jG j s DDf DDp exp yH f ,pH HIR 0žž /2 c

˜ w xqi jF f ,p , 7Ž .H /c

where

d3k 12˜ w xH f ,p s p p qv f fŽ .H0 yk k k yk k3 2 K2pŽ . k

d3k hkF f ,p s z f q p , 8Ž . Ž .H k k k3 ž /v2pŽ . k

and, in leading order

K sT , z t scosv t , h t ssinv t . 9Ž . Ž . Ž .k k k k k

Since the IR propagator is defined only in the lead-Ž Ž ..ing order cf. 4 , we can freely choose the sublead-

( )A. JakoÕacrPhysics Letters B 446 1999 203–208´ 205

ing parts of K , z and h. The different choices of thesubleading parts lead to different 3D theories, corre-sponding to the different prescriptions of the papersw x4,5,7 . We will use here

K sTQ LyvŽ .k k

z t ,h tŽ . Ž .Ž .k k

cos v t ,sin v t , if tgCŽ . Ž .Ž .k k 1,2s 10Ž .½ 1,0 , if tgC ,Ž . 3

with the cutoff L;T which describes the validityw xrange of the classical approximation 7 . The cutoff

is useful also to suppress the spatial nonlocalities inw xthe effective action 13 , and to ensure the quantum

w xdecoherence 7 .To simplify notations we can introduce the effec-

tive generating functional

1 d 1˜ ˜w xZ j s exp yi H exp y jGjH HHI ž /ž /ž /NN id j 2c c c

= ˜ w xexp i jF f ,p , 11Ž .Hž /c

˜w xwhere the normalizing factor NN assures Z 0 s1and thus cancels vacuum diagrams. It is backgrounddependent, and we will include it into the Hamilto-nian

˜yH wf ,p x ˜w x w xZ j s DDf DDp e Z j , 12Ž .H˜ ˜where HsH q ln NN. Because of the special form0

of the background field on the Matsubara contourŽ . w x10 the effective Hamiltonian can be computed 5

b d3k˜ w xHsG f q p p , 13Ž .Hdim .red yk k32 2pŽ .

w xwhere G f is the effective action of the di-dim.red

mensional reduction.It has to be emphasized that with any choice of

the IR propagator we get a theory which is equiva-lent with the original one because in the constructionwe have never used any ad hoc assumption. There-fore any computation performed in the effective orthe original theory yields the same result. In particu-lar, since the original theory does not know about theauxiliary cutoff L, the cutoff dependence of theeffective theory will cancel in the final results.

The background field, beeing free, can be easilygenerated by the initial conditions

˜ X ˜ ˜ XiF t sy dt J t G t ,t , 14Ž . Ž . Ž . Ž .Hc

where

˜ ˜ ˜J t s EF t qF t E d ty t 15Ž . Ž . Ž . Ž . Ž .Ž .0 0 t c 0

is the current localized at the initial time. This yields

1 1 1˜ ˜ ˜ ˜ ˜y jGjq i jFsy JGJq JGJ , 16Ž .H H H H

2 2 2c c c c

˜where Js jqJ. The last term is current-indepen-dent, so it is canceled by the normalization. What

˜ ˜ ˜w x w xremains is that Z j,F sZ J,0 , the backgrounddependence can be absorbed into a redefined current.

We can also perform a Legendre transformationwith respect to this current. The resulting effective

˜ w xaction G w is background independent, it can becalculated using the ordinary real time perturbation

˜theory with the propagator G. The 1PI vertex func-˜tions can be obtained by differentiating G with

respect to w, and take it at the physical point wphys

which corresponds js0

˜ w xdG wŽn.G x , . . . x s .Ž .1 n

dw x . . . dw xŽ . Ž .1 n wsw phys

17Ž .˜ ˜js0 means JsJ. The Dirac deltas in J at the

initial time set the initial conditions for the timeevolution. Therefore

˜ w t sfŽ .dG phys 0s0, 18Ž .½ E w t sp .dw Ž .t phys 0phys

That is, while the averaging over the initial condi-tions corresponds to the local effective action, thetime evolution is governed by the time dependenteffective action. The difference comes from the

w xtime-nonlocal loops 7 , a genuine quantum effect.

3. Shear viscosity in scalar field theories

As an application we can calculate the shearviscosity in F 4 theory. First we recall the quantum

w xresult 9,10 , then the different classical approaches.


3.1. The quantum result

The calculation is not too involved even in thequantum case. The retarded Green functions have tobe computed by the rule

G R x sG11 x yG12 x . 19Ž . Ž . Ž . Ž .A B A B A B

In the Feynman diagrams we have to use the propa-gators

G11 k s iG R k q iG12 k ,Ž . Ž . Ž .iG12 k sn k r k ,Ž . Ž . Ž .0

iG22 k syiG R k q iG21 k ,Ž . Ž . Ž .iG21 k s 1qn k r k , 20Ž . Ž . Ž . Ž .Ž .0

RŽ . Ž . Ž .where iG k,t sQ t r k,t . We have performedhere a self-consistent resummation and used interact-

Žing spectral function instead of the free one cf.w x. Ž .9,10 . This yields including the symmetry factor 2

d3kR

2 2iG p ,t s2 Q tŽ . Ž .HF F 32pŽ .= r k ,t r pyk ,tŽ . Ž .

-q2 r k ,t iG k ,t . 21Ž . Ž . Ž .Performing a Fourier transformation in time

d3k dv dvX

R2 2G p s2Ž . HF F 3 2p 2p2pŽ .

=r k ,v r pyk ,vXŽ . Ž .

Xp yvyv q i´0

= 1qn v qn vX . 22Ž . Ž . Ž .Ž .

The discontinuity can be simply obtained usingŽ .y1 Ž .Disc xq i´ sy2p id x

d4k2 2r p s2 r k r pyk 1qn kŽ . Ž . Ž . Ž .ŽHF F 042pŽ .

qn p yk . 23Ž . Ž ..0 0

Using the identity

1qn v qn vX s 1yeyb Žvqv

X . 1qn vŽ . Ž . Ž . Ž .Ž .= 1qn v

X 24Ž . Ž .Ž .w xwe get back the previous results 9,10 .

Fig. 1. A typical classical contribution to the effective operator.

3.2. The classical approach

As it is proven before, the same quantum resultcan be obtained using a two-step method, where inthe first step we compute the effective operator only,with the IR stable propagator and in the backgroundof the IR fields. A typical diagram contributing to

Ž .the classical ie. without quantum loops part for theretarded Green’s function is shown on Fig. 1. Informula

G R2 2 x s4F x G R x ,0;F F 0 , 25Ž . Ž . Ž . Ž . Ž .F F ,e f f FF

R Ž .where G x,0;F is the retarded Green’s functionFF

Ž w x.cf. 6,7 .The same result can be obtained from the classical

linear response theory. To see it let us add a currentterm to an action

S j sSq jf F , 26Ž . Ž . Ž .Hwhere f is an arbitrary function. We want to exam-

Ž .ine the linear response of the function g F to thischange

d g F j, xŽ .Ž .X XRG xyx s sg F 0, xŽ . Ž .Ž .Xf g

d j xŽ . js0

=G R xyxX . 27Ž . Ž .fF

We can get G R from the modified equations offF

motion

dS j dSŽ .X0s s q j x f F x 28Ž . Ž . Ž .Ž .

dF x dF xŽ . Ž .by differentiating with respect to j:

dSXRG x yyŽ .H X fF

dF x dF xŽ . Ž .syf X

F y d xyy . 29Ž . Ž . Ž .Ž .

( )A. JakoÕacrPhysics Letters B 446 1999 203–208´ 207

Its solution is simply

G R xyy sG R xyy f XF y , 30Ž . Ž . Ž . Ž .Ž .fF FF

so finally

G R xyxX sgXF x G R xyxX f X

F xX .Ž . Ž . Ž . Ž .Ž . Ž .f g FF

31Ž .2 Ž .For fsgsF we reproduce 25 .

After the 3D integration the background lines areclosed with

4² :F k F q s 2p d kqq iG k . 32Ž . Ž . Ž . Ž . Ž . Ž .3 D

The background fields, however, also follow a non-trivial, nonlinear time evolution, and the completecalculation cannot be performed in its generality. Wewill make an approximation similar to the one in thequantum theory: we join all the background lines on

Žone propagator i.e. sum up the the self-energy dia-.grams and work further with these effective propa-

gators. This yields the spectral representations

dv r k ,vŽ .G k s ,Ž . HR 2p k yvq i´0

TiG k s r k , 33Ž . Ž . Ž .3 D k0

where we should use the complete spectral functionsinstead of the free ones. The retarded Green’s func-tion reads

d4kR

2 2iG p s4 iG pyk iG k ,Ž . Ž . Ž .HF F R 3 D42pŽ .34Ž .

Žwhich yields the discontinuity note the symmetry.k lp yk0 0 0

d4k2 2r p s2T r pyk r kŽ . Ž . Ž .HF F 42pŽ .

=1 1

q . 35Ž .ž /k p yk0 0 0

Ž .There is another use of the approximation 33 ,because it provides a simple way to extract the

Ž w x .spectral function cf. 10 for the quantum case

r p sb p iG p . 36Ž . Ž . Ž .A B 0 A B ,3 D

In our case we have to compute the classical expec-tation value

² 2 2 :F x F 0 s2 iG x iG x 37Ž . Ž . Ž . Ž . Ž .3 D 3 Dclass

and, after Fourier transformation

d4k T T2 2r p s2b p r kŽ . Ž .HF F 0 4 k p yk2pŽ . 0 0 0

=r pyk 38Ž . Ž .Ž .which is indeed identical with 35 .

Let us finally compute the viscosity from theclassical theory. We approximate the spectral func-tion by

4k g0 kr k f , 39Ž . Ž .22 2 2 2k yv q16k gŽ .0 k 0 k

which yields in the narrow width approximationw x9,10

pd k 2 yv 2Ž .0 k2r k f . 40Ž . Ž .

v gk k

To be consistent we have to use the classical valueof the damping rate. Its parametric form has been

w xgiven by 6 , it is equivalent to a high temperatureapproximation of the parametric quantum resultw x11,14,15 . Therefore we can extract the on shellclassical damping rate from the corresponding quan-

w xtum expression 14 performing a high temperatureexpansion

2 2 2l T 6 dq v< <k kg s 1y L 1y ,Hk 22 2< < ž /1536pv kp v0k q

41Ž .z Ž .where L syH ln 1y t rt dt is the Spence func-2 0

tion. After interchanging the order of integrations wecan write

dq v 2 ln 1q t2 2 Ž .< <k k k rmTL 1y sy dtH H2 2< < 'ž /k v t 1q t0 0q

=

2tmT1y . 42Ž .( 2k

< < < <The integral vanishes for k s0; for large k wefind

< <m kTg v s4g v 1qOO ln . 43Ž .< <k 4mk k 0 0T ž /ž /< <k mT


Ž . Ž . Ž .From 1 , 38 and 40 we then obtain24

2 2r p d k r kŽ . Ž .F Fhs lim s2TH 4 2p kp™0 2pŽ .0 0

d3k 1fTH 3 4g v2pŽ . k k

768 k 2 g vL 0 0s dk . 44Ž .H2 3r22 2 g vl Tp 0 k kk qmŽ .T

Since g v is bounded, h will have a logarithmick k

divergence, its coefficient can be extracted from theŽ Ž ..large momentum behaviour of g v cf. 43k k

192 Lhs ln qconst. . 45Ž .2 ml Tp T

With L;T we get back the leading term of thew xquantum result 10 . Using the complete expression

Ž .41 we find consts2.1552.This divergence cannot be canceled by any local

counterterm in the Lagrangian. It is the consequenceof having a composite operator which needs renor-malization even in the classical case. The completeresult, of course is independent on this auxiliarycutoff, the UV integration should carry the appropri-ate counterterms.

4. Conclusion and outlook

We have summarized, how an effective classicaltheory can solve the resummation problems in hightemperature quantum field theories. For static quanti-ties this classical theory is equivalent to dimensionalreduction, the time evolution is governed by theeffective quantum action.

With this effective classical theory we could re-produce in the F 4 model the quantum result for theshear viscosity. It could be computed either by using

Ž .the definition from the retarded Green’s function ordirectly from the 3D expectation value of² 2 2 :F x F 0 . The usual classical theory givesŽ . Ž . classa cutoff dependent result even after a proper renor-malization. This cutoff dependence has to vanish ifwe calculate also the UV contributions to the effec-tive operator.

The viscosity in this form is not complete, asw x Žshown in 11 . The correction terms ladder dia-

.grams can be of the same order as the leading one.Summing them up is far from beeing trivial. Since,however, the important 1rl2 behaviour of the vis-cosity is essentially classical, one can try to performthe summation of the ladder diagrams in this ap-proach, which may be easier than in the quantumtheory. This is a task for future studies.

The classical theory, on the other hand, can besimulated on computers. Since the viscosity is di-rectly proportional to the F 2 two-point function, itis a relatively simply accessible quantity. The simu-lations have to be performed with a finite cutoff ofthe order of the temperature, or with the properrenormalization factor stemming from the UV inte-gration.

Acknowledgements

I would like to thank U. Heinz and E. Wang forcalling my attention to this problem and for discus-sions, and D. Bodeker, W. Buchmuller, Z. Fodor, A.¨ ¨Patkos and P. Petreczky for discussions. This work´has been partially supported by the Hungarian NFSunder contract OTKA-T22929.

References

w x1 For discussion and references see V.A. Rubakov, M.E. Sha-Ž .poshnikov, Usp. Fiz. Nauk. 166 1996 493.

w x Ž .2 E. Braaten, R.D. Pisarski, Nucl. Phys. B 337 1990 569; E.Ž .Braaten, R.D. Pisarski, Nucl. Phys. B 339 1990 310.

w x Ž .3 G. Aarts, J. Smit, Phys. Lett. B 393 1997 395.w x Ž .4 G. Aarts, J. Smit, Nucl. Phys. B 511 1998 451.w x5 B.J. Nauta, Ch.G. van Weert hep-phr9709401.w x Ž .6 W. Buchmuller, A. Jakovac, Phys. Lett. B 407 1997 39.¨ ´w x Ž .7 W. Buchmuller, A. Jakovac, Nucl. Phys. B 521 1998 219.¨ ´w x Ž .8 A. Hosoya, M.-A. Sakagami, M. Takao, Ann. Phys. NY

Ž .154 1984 229.w x Ž .9 S. Jeon, Phys. Rev. D 47 1993 4586.

w x Ž .10 E. Wang, U. Heinz, X. Wang, Phys. Rev. D 53 1996 5978.w x Ž .11 S. Jeon, Phys. Rev. D 52 1995 3591; S. Jeon, L.G. Yaffe,

Ž .Phys. Rev. D 53 1996 5799.w x Ž .12 N.P. Landsmann, Ch.G. van Weert, Phys. Rep. 145 1987

141.w x Ž .13 A. Jakovac, A. Patkos, Nucl. Phys. B 494 1997 54.´ ´w x Ž .14 E. Wang, U. Heinz, Phys. Rev. D 53 1996 899.w x Ž .15 H.A. Weldon, Phys. Rev. D 28 1983 2007.

28 January 1999


Evidence for discrete chiral symmetry breaking in Ns1supersymmetric Yang-Mills theory

DESY-Munster Collaboration¨

R. Kirchner a, I. Montvay a, J. Westphalen a, S. Luckmann b, K. Spanderen b

a Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germanyb Institut fur Theoretische Physik I, UniÕersitat Munster, Wilhelm-Klemm-Str. 9, D-48149 Munster, Germany¨ ¨ ¨ ¨


Abstract

Ž .In a numerical Monte Carlo simulation of SU 2 Yang-Mills theory with dynamical gauginos we find evidence for twodegenerate ground states at the supersymmetry point corresponding to zero gaugino mass. This is consistent with theexpected pattern of spontaneous discrete chiral symmetry breaking Z ™Z caused by gaugino condensation. q 19994 2

Published by Elsevier Science B.V. All rights reserved.

1. Introduction

The basic assumption about the non-perturbativeŽ .dynamics of supersymmetric Yang-Mills SYM the-

ory is that there is confinement and spontaneousw x Žchiral symmetry breaking, similar to QCD 1 . For a

w x .more recent introduction and review see also 2 . Inthe past years there has been great progress in theunderstanding of the non-perturbative properties ofsupersymmetric gauge theories, in particular follow-

w xing the seminal papers of Seiberg and Witten 3 . Incase of Ns1 SYM theory the non-perturbativeresults are not rigorous but fit into a self-consistentplausible picture of low energy dynamics of super-

Ž . w xsymmetric QCD SQCD 4 . The features of the lowenergy dynamics, like symmetries and bound statespectra, are formulated in terms of low energy effec-

w xtive actions 5,6 . Lattice Monte Carlo simulations

may contribute by directly testing some of thesepredictions.

The expected pattern of spontaneous chiral sym-metry breaking in SYM theories is quite interesting:

Ž .considering for definiteness the gauge group SU N ,c

the expected symmetry breaking is Z ™Z . This2 N 2c

is because the global chiral symmetry of the gauginoŽ .a Majorana fermion in the adjoint representation isanomalous. The symmetry transformations are

yi wg yiwg5 5C ™e C , C ™C e , 1Ž .x x x x

where the Dirac-Majorana fields are used whichsatisfy, with the charge-conjugation Dirac matrix C,

TC sCC , C sC C . 2Ž .x x x x

Ž .The group of symmetry transformations in 1 coin-cide with the R-symmetry and hence will be called


( )R. Kirchner et al.rPhysics Letters B 446 1999 209–215210

Ž .U 1 . The transformation is equivalent to the trans-R

formation of the gaugino mass m and a shift of theg

u-parameter:

m ™m ey2 iwg 5 , u™uy2 N w . 3Ž .g g c˜ ˜

Since u is periodic with period 2p , in the supersym-Ž .metric case with m s0 the U 1 symmetry isg R˜

unbroken if

kpwsw ' ks0,1, . . . ,2 N y1 . 4Ž . Ž .k cNc

Gaugino condensation means a non-zero vacuumexpectation value

a a² : ² :C C s l l ql l /0 . 5Ž .x x x x a x x a

ŽHere, besides the Dirac-Majorana field, the Weyl-a aMajorana field components l and l are alsox x

.introduced. The gaugino condensate is transformedŽ .under U 1 according toR

y2 iwg 5² : ² :C C ™ C e C . 6Ž .x x x x

If such a condensate is produced by the dynamicsthen it breaks the Z symmetry to Z : the expected2 N 2c

spontaneous chiral symmetry breaking is Z ™Z .2 N 2c

This implies the existence of N discrete degeneratecŽ .ground vacuum states with different orientations of

Ž . Ž .the gaugino condensate according to 4 , 6 .Ž .A non-zero gaugino mass m /0 breaks theg

supersymmetry softly. As a function of the gauginomass the degeneracy of the N ground states isc

resolved. At m s0 the lowest ground state isg

changing. This gives rise to a characteristic patternof first order phase transitions.

Ž .In the special case of SU 2 gauge group, whichwill be considered in this paper, we have Z ™Z4 2

and in the two vacua the gaugino condensate hasopposite signs. At m s0 the lowest ground statesg

are exchanged and a first order phase transitionoccurs. In this letter we report on a large scalenumerical Monte Carlo simulation with the aim tofind numerical evidence for the existence of thisphase transition.

2. Lattice formulation

The definition of an Euclidean path integral forw xMajorana fermions 7 may be obtained by starting

w xfrom the well known Wilson formulation 8 of aDirac fermion in the adjoint representation. If theGrassmanian fermion fields in the adjoint representa-

r rtion are denoted by c and c , with r being thex x

adjoint representation index, then the fermionic partof the lattice action can be written as

Õ uS s c Q c . 7Ž .Ýf y y Õ , x u xxu , yÕ

Here the fermion matrix Q is defined by

w xQ 'Q U 'd dy Õ , x u y Õ , x u y x Õu

4

yK d 1qg VŽ .Ý y , xqm m Õu , xmˆms1

Tqd 1yg V . 8Ž .Ž .yqm , x m Õu , ymˆ

K is the hopping parameter and the matrix for thegauge-field link in the adjoint representation is de-fined as

w x † )V 'V U '2Tr U T U T sVŽ .r s , xm r s , xm xm r xm s r s , xm

sVy1T . 9Ž .r s , xm

1The generators T ' l satisfy the usual normaliza-r r21Ž . Ž .tion Tr l l s d . In case of SU 2 we haver s r s2

1T ' t with the isospin Pauli-matrices t . Startingr r r2

from the Dirac fermion fields one can introduce twoŽ1,2. Ž .Dirac-Majorana fields C satisfying 2 :

1 iŽ1. T Ž2. TC ' cqCc , C ' ycqCcŽ . Ž .' '2 2

10Ž .

and S can be rewritten asf

21Ž j.Õ Ž j.uS s C Q C . 11Ž .Ý Ýf y y Õ , x u x2 xu , yÕjs1

Using this, the fermionic path integral for Diracfermions becomes

yS yc Qcfdc dc e s dc dc e sdetQH H2

Ž j. Ž j.Ž j. yC QC r2w xs dC e .ŁHjs1

12Ž .

( )R. Kirchner et al.rPhysics Letters B 446 1999 209–215 211

For Majorana fields the path integral involves onlyw Ž j.xdC , either with js1 or js2 hence, omitting

Ž .the index j , we have

TyC QC r2 yC CQC r2w x w xdC e s dC eH HsPf CQ sPf M . 13Ž . Ž . Ž .

Here the Pfaffian of the antisymmetric matrix M'

CQ is introduced. The Pfaffian can be defined for ageneral complex antisymmetric matrix M syMab ba

Ž .with an even number of dimensions 1Fa ,bF2 Nby a Grassmann integral as

w x yfa Mab fb r2Pf M ' df eŽ . H1

s e M . . . M . 14Ž .a b . . . a b a b a bN 1 1 N N 1 1 N NN !2w xHere, of course, df 'df . . . df , and e is the2 N 1

totally antisymmetric unit tensor. It can be easilyshown that

2Pf M sdet M . 15Ž . Ž .One way to prove this is to use det MsdetCQs

Ž . Ž .detQ and Eqs. 12 and 13 . Besides the partitionŽ .function in 12 , expectation values for Majorana

w xfermions can also be similarly defined 9,10 .w xIt is easy to show 11 that the adjoint fermion

matrix Q has doubly degenerate real eigenvalues,Ž .therefore detQ is positive and Pf M is real. Omit-

Ž .ting the sign of Pf M one obtains the effectivew xgauge field action 12 :

1 1 w xS sb 1y Tr U y logdetQ U , 16Ž .Ž .ÝC V pl2 2pl

with the bare gauge coupling given by b'2 N rg 2.c1The factor in front of logdetQ tells that we2

1effectively have a flavour number N s of adjointf 2

fermions. The omitted sign of the Pfaffian can betaken into account in the expectation values:

² :A sign Pf MŽ . C V² :A s . 17Ž .² :sign Pf MŽ . C V

This sign problem is very similar to the one in QCDwith an odd number of quark flavours.

The value of the Pfaffian, hence its sign, can benumerically determined by calculating an appropriate

w xdeterminant 13 . It turns out that in updating se-quences with dynamical gauginos configurations with

positive Pfaffian dominate. This is shown by explicitevaluation on 43 P8 lattices. It is plausible that thesign changes, as a function of the valence hoppingparameter, typically occur at higher values than the

w xvalue of K in the dynamical updating 13 . There-fore, in the present work, we consider the effective

Ž .gauge action in 16 and neglect the sign of thePfaffian. To take into account the sign is possible butnumerically demanding, therefore we postpone it forfuture studies.

Since the Monte Carlo calculations are done onfinite lattices, one has to specify boundary condi-tions. In the three spatial directions we take periodicboundary conditions both for the gauge field and thegaugino. This implies that in the Hilbert space ofstates the supersymmetry is not broken by theboundary conditions. In the time direction we takeperiodic boundary conditions for bosons and an-tiperiodic ones for fermions, which is obtained if one

Žwrites traces in terms of Grassmann integrals. Theminus sign for fermions is the usual one associated

.with closed fermion loops. Of course, boundaryconditions do not influence the physical results inlarge volumes. For instance, we explicitly checkedthat the distribution of the gaugino condensate is noteffected if in the time direction periodicity is as-

Ž .sumed for the fermions, too see below . Anotherinteresting possibility would be to consider twisted

w xboundary conditions 14 which are useful in theoret-ical considerations about supersymmetry breakingw x15 .

3. Monte Carlo simulation

The expected first order phase transition at zerogaugino mass should show up as a jump in the

Ž .expectation value of the gaugino condensate 5 . Therenormalized gaugino mass is obtained from thehopping parameter K as

Z am 1 1Ž .mm s y 'Z am m . 18Ž . Ž .R g m 0 g˜ ˜2 a K K0

Here a denotes the lattice spacing, m is the renor-Ž .malization scale and K sK b gives the b-de-0 0

pendent position of the phase transition, which isexpected to approach K s1r8 in the continuum0


limit b™`. The bare gaugino mass m is defined,0 g

as usual, by omitting the multiplicative renormaliza-tion factor Z . The renormalized gaugino condensatem

is also obtained by additive and multiplicative renor-malizations:

² : ² :C C sZ am C C yb am . 19Ž . Ž . Ž .RŽ m .x x x x 0

The renormalization factors Z and Z are expectedmŽ .to be of order OO 1 . The presence of the additive

Ž .shift in the gaugino condensate b am implies that0

the value of its jump at m s0 is easier availableR g

than the value itself.A first order phase transition should show up on

small to moderately large lattices as metastabilityexpressed by a two-peak structure in the distributionof some order parameter, in our case the value ofthe gaugino condensate. By tuning the bare parame-ters in the action, in our case the hopping parameterK for fixed gauge coupling b , one can achieve that

Ž .the two peaks are equal in height or area . This isthe definition of the phase transition point in finitevolumes. By increasing the volume the tunnelingbetween the two ground states becomes less and lessprobable and at some point practically impossible.

In our simulations, besides the distribution of thegaugino condensate, we also studied other quantitiesas the string tension or the masses of the lightestbound states. The first results have been published

w xrecently 13,16 together with a first hint for theexistence of a phase transition from a simulation atŽ .bs2.3, Ks0.195 . In the present paper we keepthe gauge coupling at bs2.3 and exploit the regionaround Ks0.195.

The Monte Carlo simulations are done by a two-w xstep variant of the multi-bosonic algorithm 17 pro-

w xposed in 9 . We use polynomial approximations

w xdiscussed in detail in 18 and correction procedureswhich are adapting some known methods from the

1w xliterature 19,20 to the present situation with N sf 2

flavours. Our experience with this algorithm hasbeen described already in previous publicationsw x21,13,16 and will be discussed in detail in a forth-

w xcoming paper 22 .The parameters of the numerical simulations on

63 P12 lattice at bs2.3 are summarized in Table 1.The run with an asterisk had periodic boundaryconditions for the gaugino in the time direction, therest antiperiodic. K is the hopping parameter andw xe ,l is the interval of approximation for the firstthree polynomials of orders n , respectively. The1,2,3

w xfourth polynomial of order n is defined on 0,l . In4

the eighth column the number of performed updatingcycles is given. The ninth column contains the accep-tance rate in the noisy correction step A , the tenthnc

column gives the exponential autocorrelation lengthfor plaquettes t observed in the range of aboutplaq

100 updating steps. The integrated autocorrelation isroughly a factor four higher, with large errors: forinstance at Ks0.1925 t int ,900"300. The lastplaq

column contains the value of the autocorrelationfunction of the gaugino condensate at a distance 240,where the measurements were performed.

The order parameter of the supersymmetry phasetransition at zero gaugino mass is the value of thegaugino condensate

1r' C C . 20Ž .Ž .Ý x x

V x

The normalization is provided by the number oflattice points V . We determined the value of r on a

Table 1Parameters of the numerical simulations on 63 P12 lattice at bs2.3

Ž240.K e l n n n n Updates A t C1 2 3 4 nc plaq r

Ž . Ž .0.19 0.0005 3.6 20 112 150 400 1487360 0.888 214 9 0.136 42Ž . Ž .0.1925 0.0001 3.7 22 132 180 400 3655680 0.889 220 7 0.220 36Ž . Ž .0.195 0.00001 3.7 24 200 300 400 460800 0.892 256 15 0.063 38

)0.195 0.00003 3.7 22 66 102 400 1224000 0.823 – –Ž . Ž .0.196 0.00001 3.7 24 200 300 400 952320 0.889 321 26 0.180 32Ž . Ž .0.1975 0.000001 3.8 30 300 400 500 506880 0.926 295 17 0.367 31Ž . Ž .0.2 0.000001 3.9 30 300 400 500 599040 0.925 317 16 0.424 26


Fig. 1. The probability distributions of the gaugino condensate for different hopping parameters at bs2.3 on a 63 P12 lattice. The dashedlines show the Gaussian components.


Table 2Fit parameters of the order parameter distributions corresponding to the runs in Table 1. The statistical errors in last digits are given inparentheses

2K p m m s x rd.o.f.1 1 2

Ž . Ž .0.19 1.0 11.0023 26 – 0.0423 16 27.9r20Ž . Ž .0.1925 1.0 10.8807 30 – 0.0524 17 25.9r20

Ž . Ž . Ž . Ž .0.195 0.89 7 10.762 30 10.608 30 0.066 7 16.5r18) Ž . Ž . Ž . Ž .0.195 0.83 6 10.78 3 10.60 3 0.055 7 16.3r18

Ž . Ž . Ž . Ž .0.196 0.35 7 10.722 11 10.588 11 0.073 3 5.7r18Ž . Ž . Ž . Ž .0.1975 0.26 5 10.626 17 10.484 17 0.056 4 19.5r18

Ž . Ž .0.2 0.0 – 10.3363 37 0.0562 18 21.4r20

gauge configuration by stochastic estimatorsNh1

y1h Q h 21Ž .Ž .Ý Ý y , i y x x , iNh x yis1

on normalized Gaussian random vectors h . In prac-x , i

tice N s25 works fine. Outside the phase transitionh

region the observed distribution of r can be fittedwell by a single Gaussian, but in the transition regiona reasonably good fit can only be obtained with two

Ž .Gaussians see Fig. 1 . The fit parameters of theŽ .distributions is1 or is1,2

2p rymŽ .i iexp y 22Ž .2½ 5' 2ss 2p ii

and the x 2 values per degrees of freedom are givenin Table 2. The normalization is such that p qp s1 2

1. Exact supersymmetry would imply that the widthsof the two Gaussians are equal. This relation isbroken by the lattice regularization and by the non-zero gaugino mass away from the phase transitionpoint. In order to keep the number of fit parameterssmall we neglect this small symmetry breaking and

Table 3Fit parameters of the plaquette distributions on 63P12 lattice atb s2.3 for different hopping parameters. The statistical errors inlast digits are given in parentheses

2K p m s x rd.o.f.1 1 1

Ž . Ž . Ž .0.19 0.974 25 0.63165 8 0.00425 13 0.89r47Ž . Ž . Ž .0.1925 1.014 27 0.63511 8 0.00461 15 0.74r47Ž . Ž . Ž .0.195 0.997 59 0.63811 19 0.00481 35 2.58r47Ž . Ž . Ž .0.196 1.059 63 0.64182 22 0.00518 36 1.77r47Ž . Ž . Ž .0.1975 0.987 54 0.64452 18 0.00444 30 2.26r47Ž . Ž . Ž .0.2 1.018 44 0.64846 13 0.00424 22 2.00r47

the fits are done under the assumption s ss 's .1 2

The statistical errors of the fit parameters are deter-mined by jack-knifing 64 statistically independentparallel runs.

As Fig. 1 and Table 2 show, in the region 0.195FKF0.1975 the distribution of the gaugino con-densate can only be fitted well by two Gaussians.Comparing the two runs at Ks0.195 with antiperi-odic, respectively, periodic boundary conditions inthe time direction, one can see that the differentboundary conditions do not have a sizable effect onthe distributions, as remarked before. For increasing

Ž .K decreasing bare gaugino mass the weights shiftfrom the Gaussian at larger r to the one with smallerr, as expected. The two Gaussians represent thecontributions of the two phases on this lattice. Theposition of the phase transition on the 63 P12 latticeis at K s0.1955"0.0005. The jump of the order0

parameter is Dr'm ym ,0.15.1 2

The two-phase structure can also be searched forin pure gauge field variables as the plaquette orlonger Wilson loops. It turns out that the distribu-tions of Wilson loops are rather insensitive. They canbe well described by single Gaussians with almostconstant variance in the whole range 0.19FKF0.2Ž .see, for instance, Table 3 . This speaks against the

w xappearance of a third chirally symmetric phase 23 ,w xwhich has been suggested in 24 .

4. Summary and discussion

The observed dependence of the distribution ofthe gaugino condensate on the gaugino mass mg

near m s0 is consistent with a typical behaviourg


characteristic of a first order phase transition be-Ž .tween two phases see Fig. 1 and Table 2 . Our

Ž 3 3 .lattice volume L PTs6 P12 is, however, still notvery large in physical units, therefore the expectedtwo-peak structure is not yet well developed. Forinstance, at Ks0.1925 we have LM 0q

,3.6, withg g0q w xthe smallest glueball mass M 13,22 . In fact, ag g

behaviour corresponding to a true first order phasetransition can only be established in a detailed studyof the volume dependence, which we postpone forfuture work. Therefore, the present observations arealso consistent with a rapid cross-over at finite latticespacings, approaching to a first order phase transitionin the continuum limit b™`. On our 63 P12 lattice

Ž .for bs2.3 the phase transition or cross-over is atK s0.1955"0.0005. The jump of the gaugino0

condensate in lattice units is Dr,0.15.A rather positive aspect of our Monte Carlo simu-

lations is the ability of the two-step multi-bosonicw xalgorithm 9 to cope with the difficult situation at

small dynamical fermion mass in the environment ofmetastability of phases.

In the numerical simulations we considered up tonow only the unrenormalized gaugino mass andgaugino condensate. The transformation to the corre-sponding renormalized quantities defined in Eqs.Ž . Ž .18 and 19 will, however, not change the qualita-tive behaviour, because the multiplicative renormal-

Ž .ization constants are expected to be of OO 1 . Onehas to note that in the exploited range the baregaugino masses m are small compared to the0 g

lightest bound state masses. With K s0.1955 at0

Ks0.1925 we have m rM 0q,0.07. Similarly to0 g g g˜

QCD, it is expected that the mass gap in the spec-trum is of the same order of magnitude as the scaleparameter for the asymptotically free coupling L. Asthe preliminary results on the bound state masses

w xshow 13,22 , at Ks0.1925 we already have anapproximate degeneracy of the states which are ex-pected to form the lowest chiral supermultiplet.

Besides the volume dependence, another interest-ing question is the development of the phase transi-tion signal towards the continuum limit at bs`. In

Ž Ž .fact, the arguments in the introduction at Eqs. 1 –Ž ..6 for the spontaneous chiral symmetry breakingZ ™Z refer to the continuum limit. The present2 N 2c

numerical evidence shows that the discrete chiralsymmetry breaking is manifested at non-zero lattice

spacing in feasible numerical simulations and can beinvestigated by well established methods.

Acknowledgements

It is a pleasure to thank Gernot Munster for¨helpful discussions. The numerical simulations pre-sented here have been performed on the CRAY-T3E-512 computer at HLRZ Julich. We thank HLRZ and¨the staff at ZAM for their kind support.

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w x22 R. Kirchner, S. Luckmann, I. Montvay, K. Spanderen, J.Westphalen, in preparation.

w x23 N. Evans, S.D.H. Hsu, M. Schwetz, hep-thr9707260.w x Ž .24 A. Kovner, M. Shifman, Phys. Rev. D 56 1997 2396.

28 January 1999


ž /Non-holomorphic effective potential in Ns4 SU n SYM

E.I. Buchbinder a, I.L. Buchbinder b, S.M. Kuzenko c,1

a Department of Physics, Tomsk State UniÕersity, Tomsk 634050, Russiab Department of Theoretical Physics, Tomsk State Pedagogical UniÕersity, Tomsk 634041, Russia

c Sektion Physik, UniÕersitat Munchen, Theresienstr. 37, D-80333 Munchen, Germany¨ ¨ ¨


Abstract

Ž .We compute the one-loop non-holomorphic effective potential for the Ns4 SU n supersymmetric Yang-Mills theoryŽ .ny1with the gauge symmetry broken down to the maximal torus U 1 . Our approach remains powerful for arbitrary gauge

groups and is based on the use of Ns2 harmonic superspace formulation for general Ns2 Yang-Mills theories along withthe superfield background field method. q 1999 Elsevier Science B.V. All rights reserved.

Extended supersymmetry imposes strong restrictions on the structure of quantum field theories. One of themost prominent examples where extended supersymmetry has played a substantial role is the exact solution for

Ž .non-perturbative low-energy effective action in the Ns2 SU 2 supersymmetric Yang-Mills theory given byw x w xSeiberg and Witten 1 . Their construction was generalized to arbitrary gauge groups in Ref. 2 . Another

intriguing example comes from the Ns4 Yang-Mills theory where the powerful symmetry properties allow oneŽ w x .to exactly compute some Green’s functions see Ref. 3 and references therein .

In the background field formulation, the effective action of Ns2, Ds4 super Yang-Mills theories is amanifestly gauge invariant and supersymmetric functional of the covariantly chiral strength W and its conjugate

w xW 4 . In the Coulomb branch the effective action is in general reads

4 4 4 8w xG W ,W s Im d xd u FF W q d xd u HH W ,W q . . . 1Ž . Ž . Ž .H Hwhere the first term is integrated over the chiral subspace of Ns2 superspace while the second term is

M m a iŽ .integrated over the full Ns2 superspace parametrized by z ' x ,u ,u . The ellipsis denotes all termsi a

Ž .involving derivatives of the strengths. The holomorphic potential FF W dominates at low energies and presentsŽ .itself the main object of the Seiberg-Witten theory. The non-holomorphic potential HH W,W constitutes the

Ž .next-to-leading finite quantum correction. For finite Ns2 Yang-Mills theories with matter, FF W coincides

1 On leave from: Department of Physics, Tomsk State University, Tomsk 634050, Russia. Supported in part by Deutsche Forschungsge-meinschaft.


( )E.I. Buchbinder et al.rPhysics Letters B 446 1999 216–223 217

Ž .with the classical gauge action, and hence HH W,W is the dominant quantum correction. An importantŽrepresentative of such superconformal models is the Ns4 Yang-Mills theory the first finite quantum field

w x.theory found 5 in which the Ns2 matter is realized by a single hypermultiplet in the adjoint representation.w xIn a recent paper 6 Dine and Seiberg showed that the requirement of scale and chiral invariance severely

Ž . Ž .restricts the possible structure of HH W,W in the Ns4 Yang-Mills theory. For the Ns4 SU 2 theory brokenŽ .down to U 1 , they found out the only admissible form for the one-loop non-holomorphic potential:

2 2W WHH W ,W sc ln ln 2Ž . Ž .2 2L L

Ž .with some numerical coefficient c and some scale L. The corresponding action given by the second term in 1Ž .turns out to be independent on L. Moreover, Dine and Seiberg argued that HH W,W gets neither higher-loop

w xperturbative nor instanton corrections, which was confirmed by instantons calculations 7 and two-loopw xsupergraph analysis 8 . The problem of explicit calculation of the coefficient c has been recently solved in

w x Ž .y2Refs. 9–11 on the base of different techniques, the final result being cs 8p . This value for c was givenw xin Ref. 9 to be the result of calculations based on the use of Ns1 superspace formulation for the Ns4

w xYang-Mills theory. Gonzalez-Rey and Rocek 10 computed, in the framework of Ns2 projective superspaceˇapproach, a special sector of the hypermultiplet low-energy action and then gave some grounds that the

Ž . w xnon-holomorphic effective potential HH W,W should have the same functional form. Finally, in our paper 11we directly analysed, in the framework of the Ns2 harmonic superspace approach, the effective actioncorresponding to the Ns2 gauge multiplet of the full Ns4 Yang-Mills theory.

w x Ž .In the present paper we extend the results of our work 11 to the case of Ns4 SU n Yang-Mills theoryny1Ž . Ž .with the gauge group broken down to U 1 . Our method of computing HH W,W is equally powerful for

Ž .arbitrary semi-simple gauge groups and naturally leads to a nice algebraic structure encoded in HH W,W .Ž .It is interesting to note that HH W,W is in general unambiguously defined when W lies along the flat

directions of the Ns2 Yang-Mills potential

w xW ,W s0 . 3Ž .w xOtherwise, the following identity 4

i j i j w xDD , DD Ws2i ´ ´ W ,W 4Ž .� 4a b a b

Ž .implies that some higher derivative terms, which are denoted by the dots in 1 , can also contribute toŽ . Ž .HH W,W . Such problems do not appear when Eq. 3 holds.As is well known, the most powerful approach to investigate quantum supersymmetric field theories is to

make use of an unconstrained superfield formulation. Unfortunately, such a manifestly supersymmetricformulation for the Ns4 Yang-Mills theory is not known. For our present purpose, however, it is sufficient torealize the Ns4 Yang-Mills theory as a theory of Ns2 unconstrained superfields. The Ns2 harmonic

w xsuperspace 12 is the only manifestly supersymmetric formalism developed to describe general Ns2Ž .Yang-Mills theories in terms of unconstrained analytic superfields. This approach has been successfully

w xapplied for investigating effective action in various Ns2 supersymmetric models in recent papers 13–15,8,11 .From the point of view of Ns2 supersymmetry, the Ns4 Yang-Mills theory describes coupling of the

Ns2 vector multiplet to the hypermultiplet in the adjoint representation. In the harmonic superspace approach,the vector multiplet is realized by an unconstrained analytic gauge superfield Vqq. As concerns the hypermulti-

Ž .plet, it can be described either by a real unconstrained analytic superfield v v-hypermultiplet or by a complexq qŽ .unconstrained analytic superfield q and its conjugate q q-hypermultiplet . In the v-hypermultiplet realiza-˘

tion, the classical action of Ns4 Yang-Mills theory reads

1 1qq 4 4 2 Žy4. qq qqw xS V ,v s tr d xd u W y tr dz = v= v 5Ž .H H2 22 g 2 g

( )E.I. Buchbinder et al.rPhysics Letters B 446 1999 216–223218

where the second term describes the v-hypermultiplet action and is integrated over the analytic subspace ofŽ w x . Ž .harmonic superspace see Refs. 12,15,11 for more details and notation . The first term in 5 is the pure Ns2

qq w xYang-Mills action. The explicit expression for the strength W via the prepotential V is given in Ref. 16 .Ž . Ž .The theory with action 5 is manifestly Ns2 supersymmetric. However, the action 5 turns out to be invariant

w xunder two hidden supersymmetric transformations 12

qq q a i q i qad V su e u qe u v ,ž /i a a

221 y q i y q i ydvsy u D e u W q D e u W . 6Ž . Ž . Ž .Ž . Ž .½ 5i l l8

w xHere W denotes the strength in the l-frame 12,15 . In the q-hypermultiplet realization, the Ns4 Yang-Millsl

theory is given by the action

1 1qq q q 4 4 2 Žy4. qi qq qS V ,q ,q s tr d xd u W y tr dz q = q 7Ž .˘ H H i2 22 g 2 g

where

qq s qq,qq , qqi s´ i jqq s qq,yqq . 8Ž . Ž .Ž .˘ ˘i j

This model is manifestly Ns2 supersymmetric. It also possesses two hidden supersymmetries

qq a i q i qa q˙d V s e u qe u q ,ž /a a i˙

221qi q i y q i yd q sy D e u W q D e u W . 9Ž . Ž . Ž .Ž . Ž .½ 5l l4

To provide manifest gauge invariance and supersymmetry at the quantum level, we study the effective actionŽ . Ž . w xfor the classically equivalent theories 5 and 7 within the Ns2 superfield background field method 15,8 . In

w xaccordance with 15,11,13 , the one-loop effective action in both realizations is given by

i iL LŽ1. qqw xG V s Tr lnIy Tr lnI 10Ž .Ž2,2. Ž4 ,0.2 2L

w xwhere I is the analytic d’Alambertian introduced in Ref. 15

i i iL m qa y q ya qa q yy˙IsDD DD q DD W DD q DD W DD y DD DD W DDŽ . Ž .ž /m a a a˙2 2 4

i1qa y � 4q DD , DD Wq W ,W . 11Ž .a 28

L Lw xThe formal definitions of the Tr lnI and Tr lnI are given in Ref. 11 .Ž2,2. Ž4,0.

Ž .For computing HH W,W it is sufficient in fact to consider a special background

DDa Ž i DD j.Ws0 . 12Ž .a

Ž1. w xThen, one can get the following path integral representation for G 11

i LŽ .qq y4 qq qqDDFF exp y tr dz FF I FFH H½ 52Ž .1exp i G s . 13Ž .Ž . iŽ .qq y4 qq qqDDFF exp y tr dz FF FFH H½ 52


qqŽ . qqŽ . i jŽ . q qThe superfield FF z,u belonging to the adjoint representation looks like FF z,u sFF z u u , withi j

FF i j sFF ji satisfying the constraints

Ž i jk . Ž i jk . i jDD FF sDD FF s0 , FF sFF . 14Ž .a a i j˙

L i jThe operator I acts on FF as follows

i iL 1i j m i j a Ž i j.k Ž i a j.k˙� 4I FF s DD DD q W ,W FF q DD W DD FF q DD W DD FF . 15Ž .Ž .m a < k < a < k <2 ˙3 3

Ž .Representation 13 involves path integrals over constrained Ns2 superfields. Our aim now is to transformthese path integrals to those over unconstrained Ns1 superfields. We introduce Ns1 Grassmann coordinates

a a a 1 1 a a˙ ˙Ž .u ,u by the rule u su , u su , the corresponding gauge covariant derivatives DD sDD , DD sDD anda 1 a a a a 1˙ ˙ ˙Ž M . <then define the Ns1 projection of an arbitrary Ns2 superfield f z by the standard rule f s

m a i2Ž . <f x ,u ,u . As is well known, from the Ns2 Yang-Mills strength W one obtains two Ns1u su s0i a 2

< 2 < i jcovariantly chiral superfields FsW and 2iW sDD W . The Ns1 projections of FF reada a

22 11 12< < <CsFF , CsFF , FsFsy2 iFF 16Ž .

and satisfy the constraints

1 2 w xDD Cs0 , y DD Fq F ,C s0 . 17Ž .a 4˙

Therefore, C is a covariantly chiral Ns1 superfield while the real superfield F is subject to a modified linearconstraint.

Ž .Until this point, the Ns2 Yang-Mills strength was constrained only by Eq. 12 . Now, we specify W toŽ .belong to the Cartan subalgebra and, hence, to satisfy Eq. 3 . Moreover, we require the Ns1 components of

Ž .W to be covariantly constant,

DD Fs0 , DD W s0 . 18Ž .a a b

Ž .Such a background is still sufficient for calculating HH W,W , since the identity

4E HH F ,FŽ .4 8 8 a ad xd u HH W ,W s d z W W W W qderivatives 19Ž . Ž .H H a a 2 2EF EF

8Ž .along with the requirement of scale and chiral invariance allow us to uniquely restore HH W,W . Here d zL

denotes the full Ns1 superspace measure. For the background chosen the operator I does not mix thesuperfields C , C and F

Li j i j< <D FF s I FF 20Ž .Ž . Ž .

where

1m a a � 4DsDD DD yW DD qW DD q F ,F . 21Ž .m a a 2˙


Ž .Expressing the Ns2 integration variables in 13 via their Ns1 projections, we obtain the followingŽ1. Ž . 2representation for G in terms of path integrals over still constrained Ns1 superfields

18DDC DDC DDFexp itr d z yCDCq FDFŽ .H H 2½ 5Ž .1exp i G s . 22Ž .Ž .

18 2DDC DDC DDFexp itr d z yCCq FŽ .H H 2½ 5Ž .Our next step is to evaluate the right hand side of Eq. 22 .

Until now the gauge group was completely arbitrary. Let us specialize our consideration to the case ofŽ . Ž . w xSU n . To start with we make a quick tour through the corresponding Lie algebra su n 17 consisting of

� 4 Ž . 3hermitian traceless matrices. We introduce the Weyl basis e of su nk l

e sd d , k ,l , p ,qs1,2, . . . ,n 23Ž . Ž .p qk l k p lq

Ž .Then an arbitrary element agsu n looks liken n

k k l k l lk kas a e q a e , a sa , a s0 24Ž .Ý Ý Ýk k k lks1 k/l ks1

with ai being real. The elements r of the Cartan subalgebra aren n

k 1 2 n irs r e sdiag r ,r , . . . ,r , r s0 . 25Ž . Ž .Ý Ýk kks1 is1

For any elements of the Weyl basis we have

tr e e s2ntr e e s2n d d . 26Ž . Ž . Ž .p q k l F p q k l p l qk

Here ‘tr ’ denotes the trace in the fundamental representation. From here one gets important consequencesF

tr e e s2n ; tr e e s0 , p/ l , q/k . 27Ž . Ž . Ž .k l lk p q k l

Given an element r of the Cartan subalgebra, one finds

w x k lr ,e s r yr e 28Ž . Ž .k l k l

Ž k l. Ž .with the eigenvalues r yr defining the roots of su n .Ž .For the gauge group chosen, the strengths W and W lie in the Cartan subalgebra of su n

n1 2 n kWsdiag W ,W , . . . ,W , W s0 . 29Ž . Ž .Ý

ks1

Ž .Since we are interested in the situation when the gauge group SU n is broken down to the maximal torusŽ .ny1 k l kU 1 , we should have W yW /0 for k/ l. In the opposite case, when several eigenvalues W coincide,

Ž . <some nonabelian group HgSU n remains unbroken. Introducing the Ns1 projections FsW and W sai 2 k l k l<y DD W associated with W, we obtain the Ns1 superfield roots F yF and W yW . The abovea a a2

restrictions on W k are equivalent to F k yF l /0 for k/ l.Ž .Let us return to Eq. 22 . Since the strengths F and W belong to the Cartan subalgebra, the components ofa

the quantum superfields C , C , F which lie in the Cartan subalgebra do not interact with the background field

2 Ž . Ž . Ž1.It is worth pointing out that we deduce Eq. 22 from the representation 13 for G which is manifestly Ns2 supersymmetric andŽ .invariant with respect to the automorphism SU 2 symmetry. That is why it is in our power to make use of any useful technique in order toR

compute special contributions to G Ž1., in particular, to reduce G Ž1. to Ns1superfields. This is completely different to the case when theNs2 or Ns4 theories are formulated from the very beginning in Ns1 superfields, when only Ns1 supersymmetry is realized off-shell;in such a case the effective action possesses Ns1 supersymmetry only. By construction, our approach is manifestly Ns2 supersymmetric,

w xin spite of the comments given in Ref. 19 .3 Ž .From now on, small Latin letters are used for SU n indices.


and therefore they completely decouple. On the other hand, the components of C and C out of the CartanŽ .subalgebra are expressed via F and F with the aid of constraints 17

2 k l 2 k lDD F DD Fk l k lC s , C s , 30Ž .k l k l4 F yFŽ . 4 F yFŽ .

and these expressions are nonsingular in the case under consideration. As a result, we can transform the rightŽ .hand side of Eq. 22 to path integrals over unconstrained superfields

k l k l k l lkV 'F , V 'F , k- l . 31Ž .Ž . Ž . Ž .Taking into account Eqs. 27 and 30 , we can transform the integral in the denominator of Eq. 22 as

follows

18 2 8 k l k ltr d z yCCq F s2n d z V B V 32Ž .Ž . ÝH H k l2k-l

where2 2� 4DD , DD

1B s q1 . 33Ž .k l 16 2k l< <F yF

Ž .It is worth pointing out that the sum in 32 is taken over half the roots and we can choose the positive roots toŽ .contribute to 32 . As a result

18 2DDC DDC DDFexp i tr d z yCCq FŽ .H H 2½ 5k l k l 8 k l k ls DDV DDV exp 2n i d z V B VÝH H k l½ 5

k-l

s Dety1 B . 34Ž . Ž .Ł k lk-l

Ž . Ž . k lNext we turn to the nominator in 22 . First of all we find the action of D 21 on the superfields F . The resultreads

D F k le s D F k l e no sum 35Ž . Ž .Ž . Ž .k l k l k l

where2m k a la k l a k l˙ < <D sDD DD y W yW DD q W yW DD q F yF . 36Ž . Ž .ž /k l m a a a˙ ˙

Ž .Using 30 and fulfilling straightforward calculations we get

18 8 k l k ltr d z FDFyCDC s2n d z V B D V 37Ž .Ž . ÝH H k l k l2k-l

Ž .where B is given by Eq. 33 . From here we obtaink l

18DDC DDC DDFexp i tr d z yCDCq FDFŽ .H H 2½ 5k l k l 8 k l k ls DDV DDV exp 2n i d z V B D VÝH H k l k l½ 5

k-l

s Dety1 B Dety1 D . 38Ž . Ž . Ž .Ł k l k lk-l

The result is

ei G Ž1.s Dety1 D . 39Ž . Ž .Ł k l

k-l


It is seen that the one-loop correction G Ž1. to effective action is determined by the functional determinant of theŽ . Ž .operator 36 on the space of unconstrained Ns1 superfields under the Feynman boundary conditions. Eq. 39

can be rewritten as follows

G Ž1.s G , G s i Tr ln D . 40Ž .Ý k l k l k lk-l

Ž . Ž . Ž . w xThe SU n -operator d 36 has the same structure as the SU 2 -operator D introduced in Ref. 11 .k lw xTherefore we can apply the technique developed in Ref. 11 and obtain

a k l k l k l a k l˙1 W W W Wa a8G s d z , 41Ž .Hk l 2 22k l k l4pŽ . F FŽ . Ž .where

F k l sF k yF l , W k l sW k yW l . 42Ž .a a a

Ž . Ž . Ž .Eqs. 40 – 42 define the non-holomorphic effective potential HH W,W of the Ns4 Yang-Mills theory interms of the Ns1 projections of W and W.

Ž . Ž . Ž . Ž .From Eqs. 19 and 40 – 42 one can easily restore HH W,W :

Ž1. 4 8G s d xd u HH W ,WŽ .H2 2k l k l1 W yW W yW

HH W ,W s ln ln 43Ž . Ž .Ý2 ž /ž /L L8pŽ . k-l

k Ž . k l Ž .where the strengths W are chosen as in 29 , with W yW /0 for k/ l. Eq. 43 is our final result. SimilarŽ . w xto the holomorphic effective potential FF W 2 , the non-holomorphic effective potential is constructed in terms

Ž . Ž1.of the roots of SU n and obviously invariant under the Weyl group. Some bosonic contributions to G werew xdiscussed in Ref. 18 .

Ž .It is necessary to point out that our method to compute the non-holomorphic effective potential HH W,W isŽ .general and perfectly works for arbitrary semi-simple gauge groups, for instance, SO n . The starting point is

Ž .representation 13 . Then, one has to specify the Cartan subalgebra and Weyl basis for the gauge group in fieldand, finally, it remains to repeat the technical steps described. Given a semi-simple rank-r gauge group G, we

ˆ� 4introduce its Weyl basis h ,e ,e , where the elements h span the Cartan subalgebra, is1, . . . ,r, and "aˆ î qa ya iˆ ˆŽ . Ž . rare the positive negative roots. When the gauge group is broken down to its maximal torus U 1 , the Ns2

w xstrength looks like WsÝW h , W,e sW e , with all W being non-vanishing. The non-holomorphicˆ î i qa qa qa qaˆ ˆ ˆ êffective potential reads

2 21 W Wqa qaˆ ˆHH W ,W s ln ln 44Ž . Ž .Ý2 ž /ž /L L8pŽ . pos. roots

w xand this is similar to the structure of perturbative holomorphic effective potential 2 .w xWhen this work was completed, there appeared recent papers 19,20 where similar results were obtained by

different methods.

Acknowledgements

The authors are grateful to E.A. Ivanov, B.A. Ovrut and S. Theisen for valuable discussions. We are gratefulw xto A.A. Tseytlin for bringing Ref. 18 to our attention. We acknowledge a partial support from INTAS grant,

INTAS-96-0308. I.L.B. and S.M.K. are grateful to RFBR grant, project No 96-02-16017 and RFBR-DFG grant,project No 96-02-00180 for partial support.


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28 January 1999


Stabilized NMSSM without domain walls

C. Panagiotakopoulos a, K. Tamvakis b

a Physics DiÕision, School of Technology, Aristotle UniÕersity of Thessaloniki, 54006 Thessaloniki, Greeceb Physics Department, UniÕersity of Ioannina, 45110 Ioannina, Greece

Received 1 October 1998Editor: L. Alvarez-Gaume

Abstract

Ž .We reconsider the Next to Minimal Supersymmetric Standard Model NMSSM as a natural solution to the m-problemand show that both the stability and the cosmological domain wall problems are eliminated if we impose a ZZ R-symmetry2

on the non-renormalizable operators. q 1999 Elsevier Science B.V. All rights reserved.

The Ns1 supersymmetric extension of the Stan-dard Model provides a well defined framework for

w xthe study of new physics beyond it 1 . The lowenergy data support the unification of gauge cou-plings in the supersymmetric case in contrast to thestandard case. The Minimal Supersymmetric exten-

Ž .sion of the Standard Model MSSM is defined bypromoting each standard field into a superfield, dou-bling the Higgs fields and imposing R-parity conser-vation. The most viable scenario for the breaking ofsupersymmetry at some low scale m , no larger thans

;1 TeV, is the one based on spontaneously brokensupergravity. Although this scenario does not employpurely gravitational forces but could require the ap-pearance of gaugino condensates in some hiddensector, it is usually referred to as gravitationallyinduced supersymmetry breaking. The resulting bro-ken theory, independently of the details of the under-lying high energy theory, contains a number of soft

Ž .supersymmetry susy breaking terms proportional topowers of the scale m . Probably the most attractives

feature of the MSSM is that it realizes a version of

‘‘dimensional transmutation’’ where radiative correc-tions generate a new scale, namely the electroweakbreaking scale M . This is a highly desirable, butW

also non-trivial, property that is equivalent to deriv-ing M from the supersymmetry breaking scale asW

opposed to putting it by hand as an extra arbitraryparameter. Unfortunately, a realistic utilization of

w xradiative symmetry breaking 2 in MSSM requiresthe presence of the so called m-term coupling di-rectly the Higgs fields H and H , namely mH H ,1 2 1 2

with values of the theoretically arbitrary parameter m

close to m or M . This nullifies all merits ofs W

radiative symmetry breaking since it reintroduces anextra arbitrary scale from the back door. Of course,there exist explanations for the values of the m-term,

w xalas, all in extended settings 3 .At first glance, the most natural solution to the

m-problem would be to introduce a massless gaugesinglet field S, coupled to the Higgs fields as

Ž .lSH H , whose vacuum expectation value vev1 2

would turn out to be of the order of the other scalesfloating around, namely m and M . This leads tos W


( )C. Panagiotakopoulos, K. TamÕakisrPhysics Letters B 446 1999 224–227 225

the simplest extension of the MSSM the so calledw x‘‘Next to Minimal’’ SSM or NMSSM 4 with a

Ž .cubic renormalizable superpotentialk

3 Žu. cWW slSH H q S qY QU Hren 1 2 13

qY Žd .QDcH qY Že.LEcH . 1Ž .2 2

Unfortunately, the above scenario runs into diffi-culties. As can be readily seen the NMSSM at therenormalizable level possesses a discrete non-anoma-lous ZZ global symmetry under which all super-3

fields are multiplied by e2p i r3. The discrete symme-try is broken during the phase transition associatedwith the electroweak symmetry breaking in the earlyuniverse and cosmologically dangerous domain wallsare produced. These walls would be harmless pro-vided they disappear effectively before nucleosyn-thesis which, roughly, requires the presence in theeffective potential of ZZ -breaking terms of magni-3

tude4 y12 4d VRO 1 MeV ;10 GeV .Ž .

Such an estimate is not very different from the morew xelaborate one 5

d VR10y7Õ3M 2 rM ,W P

where Õ is the scale of spontaneous breaking of thediscrete symmetry and M ,1.2=1019 GeV is theP

Planck mass. The above magnitude of ZZ -breaking3

seems to correspond to the presence in the superpo-tential or in the Kahler potential of ZZ -breaking¨ 3

operators suppressed by one inverse power of thePlanck mass. However, these ZZ - breaking non-re-3

normalizable terms involving the singlet S werew xshown 5 to induce quadratically divergent correc-

tions 1 which give rise to quadratically divergentw xtadpoles for the singlet 6 . Their generic form, cut-off

at M , isP

j m2 M SqS) , 2Ž . Ž .s P

where m is the scale of supersymmetry breaking ins

the visible sector. The value of j depends on the

1 These non-renormalizable terms appear either as D-terms inthe Kahler potential or as F-terms in the superpotential. The¨natural setting for these interactions is Ns1 Supergravity sponta-neously broken by a set of hidden sector fields.

Žloop order of the associated graph two or three in.this case which, in turn, depends on the particular

non-renormalizable term that gives rise to the tad-pole. Such terms lead to a vev for the light singlet Smuch larger than the electroweak scale. Thus, itseems that the non-renormalizable terms that are ableto make the walls disappear before nucleosynthesisare the ones that destabilize the hierarchy.

The purpose of the present article is to address thetwo problems of domain walls and destabilizationthat arise in the NMSSM and show that, despite theimpass that the previous arguments seem to indicate,there is a simple way out rendering the model aviable solution to the m-problem. The crucial obser-vation is that due to the divergent tadpoles a ZZ -3

breaking operator could have a much larger effect onthe vacuum than its dimension naively indicates.Thus, it is conceivable that non-renormalizable termssuppressed by more than one inverse powers of MP

are able to generate linear terms in the effectivepotential which are strong enough to eliminate thedomain wall problem although, at the same time,they are too weak to upset the gauge hierarchy.Clearly, it would be very helpful to obtain a betterunderstanding of both the symmetries that could beimposed on the model and the magnitude of destabi-lization that the various non-renormalizable opera-tors generate.

The renormalizable part of the NMSSM superpo-Ž .tential 1 possesses the following global symme-

tries:1 1 1c c cU 1 :Q , U y , D y , L 0 , E 0 ,Ž . Ž . Ž .Ž . Ž . Ž .B 3 3 3

H 0 , H 0 , S 0Ž . Ž . Ž .1 2

U 1 : Q 0 , U c 0 , Dc 0 , L 1 , Ec y1 ,Ž . Ž . Ž . Ž . Ž . Ž .L

H 0 , H 0 , S 0Ž . Ž . Ž .1 2

U 1 : Q 1 , U c 1 , Dc 1 , L 1 , Ec 1 ,Ž . Ž . Ž . Ž . Ž . Ž .R

H 1 , H 1 , S 1Ž . Ž . Ž .1 2

Žwhere in parenthesis is given the charge of the.superfield under the corresponding symmetry . The

Ž .last U 1 is an anomalous R-symmetry under whichthe renormalizable superpotential WW has charge 3.ren

The soft trilinear susy-breaking terms break the con-Ž .tinuous R-symmetry U 1 down to its ZZ subgroupR 3

that we mentioned earlier which, however, is not aR-symmetry. We see that the renormalizable part of

( )C. Panagiotakopoulos, K. TamÕakisrPhysics Letters B 446 1999 224–227226

the model possesses a genuinely discrete symmetrywhose spontaneous breakdown produces domainwalls.

Of cource, one does not have to impose all theabove continuous symmetries in order to obtain therenormalizable superpotential WW of the NMSSM.ren

The same WW can be obtained if we impose aren

discrete symmetry. There are various choices amongwhich it is useful to consider two interesting possi-bilities:

. M P M Pa ZZ =ZZ . The matter parity ZZ is gener-2 3 2

ated by

ZZ M P : Q,U c , Dc , L, Ec ™y Q,U c , Dc , L, Ec ,Ž . Ž .2

H , H ,S ™ H , H ,SŽ . Ž .1 2 1 2

and the ZZ symmetry by3

ZZ : Q,U c , Dc , L, Ec , H , H ,SŽ .3 1 2

™e2p i r3 Q,U c , Dc , L, Ec , H , H ,S .Ž .1 2

Ž .Note that ZZ ;U 1 , as already mentioned. Both3 RM P Ž .ZZ and ZZ are not R-symmetries WW™WW .2 3. M P ŽR. M Pb ZZ =ZZ . The matter parity ZZ genera-2 4 2

tor is defined as in the previous case. The ZZ4ŽR. Ž .R-symmetry ZZ ;U 1 generator is defined by4 R

ZZ ŽR. : Q,U c , Dc , L, Ec , H , H ,SŽ .4 1 2

™ i Q,U c , Dc , L, Ec , H , H ,S , WW™yiWW .Ž .1 2

Although it makes no difference which of theabove symmetries are imposed on the renormalizablesuperpotential, we should make sure that the ZZ3

symmetry, or any other symmetry containing it, isnot a symmetry of the non-renormalizable operators.If ZZ invariance is imposed on the complete theory3

the domain walls will not disappear. In contrast, theZZ ŽR. symmetry can be imposed on the non-renormal-4

izable operators and no domain walls associated withits breaking will form because the soft susy-breakingterms break ZZ ŽR. completely.4

Let us now move to the other important issue thathas to be addressed in the presence of the gaugesinglet superfield S, namely the destabilization of theelectroweak scale due to quadratically divergent tad-pole diagrams involving non-renormalizable opera-tors which generate in the effective action linear

Ž .terms of the type 2 . As mentioned, such terms leadto a vev for the light singlet which, in general, is

w xmuch larger than the electroweak scale. Abel 7 has

shown that the potentially harmful non-renormaliz-able terms are either eÕen superpotential terms orodd Kahler potential ones. Such terms are easily¨avoided if we impose on the non-renormalizableoperators a ZZ R-symmetry ZZ ŽR. under which the2 2

superpotential as well as all superfields flip sign.Ž .This symmetry is a subgroup of both U 1 andR

ZZ ŽR.. Therefore, one could impose on all operators4

the symmetry ZZ M P =ZZ ŽR. which ensures the form2 4Ž .1 of the renormalizable superpotential WW of theren

NMSSM and solves the stability problem of themodel at the same time. Alternatively, we couldimpose the full ZZ M P =ZZ ŽR. symmetry group on the2 4

renormalizable superpotential and solve the stabilityproblem by imposing on the non-renormalizable op-erators its ZZ ŽR. subgroup only.2

Notice that the non-renormalizable terms allowedby ZZ ŽR. or ZZ ŽR., although not harmful to the gauge2 4

hierarchy, are still able to solve the ZZ -domain wall3

problem since they generate in the effective actionthrough n-loop tadpole diagrams linear terms of theform

yn2 3 )d V; 16p m SqS .Ž . Ž .s

These terms are small to upset the gauge hierarchybut large enough to break the ZZ symmetry and3

eliminate the domain wall problem. For example, thepresence of the term S7rM 4 in the superpotential,P

allowed by both symmetries ZZ ŽR. and ZZ ŽR., is able2 4

to generate at four loops such a harmless linear term,w xas shown by Abel 7 .

Combining all the above we see that by adoptingŽ .the renormalizable superpotential 1 of the NMSSM

and imposing on the non-renormalizable operatorsjust a ZZ R-symmetry ZZ ŽR. we are able to solve2 2

both the cosmological and the stability problems ofthe model 2. Thus, NMSSM can be finally regardedas a solution to the m-problem of the MSSM.

Acknowledgements

C.P. was supported in part by CERN as a Corre-sponding Associate and by the TMR network ‘‘Be-

2 ŽR. Ž ŽR..Incidentally notice that ZZ or ZZ eliminates all dimen-2 4

sion five operators leading to fast proton decay.

( )C. Panagiotakopoulos, K. TamÕakisrPhysics Letters B 446 1999 224–227 227

yond the Standard Model’’. C.P. wishes to thank S.Abel for a useful discussion. K.T. wishes to thankthe TMR network ‘‘Beyond the Standard Model’’and the Greek Ministry of Science and Technology,through its PENED and research collaboration withCERN programs, for travelling support. He alsowishes to thank the CERN Theory Division for itshospitality during the summer of 1998.

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28 January 1999


Model-independent analysis of soft masses in heterotic stringž /models with anomalous U 1 symmetry

Yoshiharu Kawamura 1

Department of Physics, Shinshu UniÕersity, Matsumoto 390-0802, Japan

Received 16 November 1998; revised 7 December 1998Editor: M. Cvetic

Abstract

Ž .We study the magnitudes of soft masses in heterotic string models with anomalous U 1 symmetry model-independently.In most cases, D-term contribution to soft scalar masses is expected to be comparable to or dominant over other

Ž .contributions provided that supersymmetry breaking is mediated by the gravitational interaction andror an anomalous U 1symmetry and the magnitude of vacuum energy is not more than of order m2 M 2. q 1999 Published by Elsevier Science3r2

B.V. All rights reserved.

PACS: 04.65.qe; 11.25.Mj; 12.60.Jv

Ž .Keywords: Superstring theory; Anomalous U 1 symmetry; Soft SUSY breaking parameters

1. Introduction

Superstring theories are powerful candidates for the unification theory of all forces including gravity. TheŽ . Ž .supergravity theory SUGRA is effectively constructed from 4-dimensional 4D string model using several

w x Ž .methods 1–3 . The structure of SUGRA is constrained by gauge symmetries including an anomalous U 1Ž Ž . . w x w xsymmetry U 1 4 and stringy symmetries such as duality 5 .A

4D string models have several open questions and two of them are pointed out here. The first one is what theŽ .origin of supersymmetry SUSY breaking is. Although interesting scenarios such as SUSY breaking mecha-

w x w xnism due to gaugino condensation 6 and Scherk-Schwarz mechanism 7 have been proposed, realistic one hasŽ .not been identified yet. The second one is how the vacuum expectation value VEV of dilaton field S is

stabilized. It is difficult to realize the stabilization with a realistic VEV of S using a Kahler potential at the tree¨w xlevel alone without any conspiracy among several terms which appear in the superpotential 8 . A Kahler¨

potential generally receives radiative corrections as well as non-perturbative ones. Such corrections may bew xsizable for the part related to S 9,10 . It is important to solve these enigmas in order not only to understand the

structure of more fundamental theory at a high energy scale but also to know the complete SUSY particle



( )Y. KawamurarPhysics Letters B 446 1999 228–237 229

spectrum at the weak scale, but it is not an easy task because of ignorance of the explicit forms of fullycorrected total Kahler potential. At present, it would be meaningful to get any information on SUSY particle¨spectrum model-independently 2.

In this paper, we study the magnitudes of soft SUSY breaking parameters in heterotic string models withŽ .U 1 and derive model-independent predictions for them without specifying SUSY breaking mechanism andA

w xthe dilaton VEV fixing mechanism. The idea is based on that in the work by Ref. 12 . The soft SUSY breakingterms have been derived from ‘‘standard string model’’ and analyzed under the assumption that SUSY is brokenby F-term condensations of the dilaton field andror moduli fields M i. We relax this assumption such that

i Ž .SUSY is broken by F-term condensation of S, M andror matter fields with non-vanishing U 1 charge sinceAŽ . w xthe scenario based on U 1 as a mediator of SUSY breaking is also possible 13,14 . In particular, we make aA

comparison of magnitudes between D-term contribution to scalar masses and F-term ones and a comparison ofmagnitudes among scalar masses, gaugino masses and A-parameters. The features of our analysis are asfollows. The study is carried out in the framework of SUGRA model-independently 3, i.e., we do not specifySUSY breaking mechanism, extra matter contents, the structure of superpotential and the form of Kahler¨potential related to S. We treat all fields including S and M i as dynamical fields.

The paper is organized as follows. In the next section, we explain the general structure of SUGRA brieflywith some basic assumptions of SUSY breaking. We study the magnitudes of soft SUSY breaking parameters in

Ž .heterotic string models with U 1 model-independently in Section 3. Section 4 is devoted to conclusions andA

some comments.

2. General structure of SUGRA

w xWe begin by reviewing the scalar potential in SUGRA 18,19 . It is specified by two functions, the totalŽ . Ž .Kahler potential G f,f and the gauge kinetic function f f with a , b being indices of the adjoint¨ ab

Ž . Ž .representation of the gauge group. The former is a sum of the Kahler potential K f,f and the logarithm of¨Ž .the superpotential W f

22 3< <G f ,f sK f ,f qM ln W f rM , 1Ž . Ž .Ž . Ž .'where MsM r 8p with M being the Planck mass, and is referred to as the gravitational scale. We havePl Pl

Idenoted scalar fields in the chiral multiplets by f and their complex conjugate by f . The scalar potential isJ

given by2 I 12 G r M y1 J 2 y1 a bˆ ˆVsM e G G G y3M q Re f D D , 2Ž . Ž .Ž . abž /JI 2

whereIa a a JD sG T f s fT G . 3Ž . Ž .Ž .I J

I J aHere G sE GrEf , G sE GrEf etc., and T are gauge transformation generators. Also in the above,I JŽ y1 . Ž y1 . I IRe f and G are the inverse matrices of Re f and G , respectively, and a summation overab J a b J

Ž .a , . . . and I, . . . is understood. The last equality in Eq. 3 comes from the gauge invariance of the total Kahler¨potential. The F-auxiliary fields of the chiral multiplets are given by

2 II G r2 M y1 JF sMe G G . 4Ž . Ž .J

2 w xThe stability of S and soft SUSY breaking parameters are discussed in the dilaton SUSY breaking scenario in Ref. 11 .3 w xThe model-dependent analyses are carried out in Ref. 13–17 .

( )Y. KawamurarPhysics Letters B 446 1999 228–237230

The D-auxiliary fields of the vector multiplets are given bya y1 ˆ bD s Re f D . 5Ž .Ž . ab

Using F I and Da, the scalar potential is rewritten down by

VsV qV , V 'F K I F J y3M 4eG r M 2, 6Ž .F D F I J

1 a bV ' Re f D D . 7Ž .D a b2

Let us next summarize our assumptions on SUSY breaking. The gravitino mass m is given by3r2

² G r2 M 2:m s Me 8Ž .3r2

² :where PPP denotes the VEV. As a phase convention, it is taken to be real. We identify the gravitino masswith the weak scale in most cases. It is assumed that SUSY is spontaneously broken by some F-term

Ž² : .condensations F /0 for singlet fields under the standard model gauge group andror some D-termŽ² : . I acondensations D /0 for broken gauge symmetries. We require that the VEVs of F and D should satisfy

1r2I J² :F K F FO m M , 9Ž .Ž .Ž .I J 3r2

² a:D FO m M 10Ž .Ž .3r2

Ž . Ž . ² : 2 2for each pair I, J in Eq. 9 . Note that we allow the non-zero vacuum energy V of order m M at this3r2

level, which could be canceled by quantum corrections.In order to discuss the magnitudes of several quantities, it is necessary to see consequences of the stationary

² I: Ž .condition E VrEf s0. From Eq. 2 , we findV X X2 2F 1I 2 G r M G r2 M J I J a b a a J

X XE VrEf sG qM e qMe G F yF G F y Re f D D qD fT G .Ž . Ž .I I J I J I a b I2 J2 , Iž /M11Ž .

Taking its VEV and using the stationary condition, we derive the formula² :V X XFJ 2 I J² : ² : ² : ² : ² : ² :X Xm G F sy G qm q F G F3r2 I J I 3r2 I J I2ž /M

1 a b a a J² : ² : ² : ² : ² : ² :q Re f D D y D fT G . 12Ž .Ž . Ž .ab I2 J, I

Ž² : ² : ² : 2We can estimate the magnitude of SUSY mass parameter m ' m G q G G rM yI J 3r2 I J I J² Ž y1 . I X J X:. Ž . Ž a . I Ž .X XG G G using Eq. 12 . By multiplying T f to Eq. 11 , a heavy-real direction is projected on.I J I J

Using the identities derived from the gauge invariance of the total Kahler potential¨J Ja a J aG T f qG T yK fT s0, 13Ž . Ž . Ž .Ž .II J J I J

X X X JJ JJ a J a J aX X XK T f qK T y G fT s0, 14Ž . Ž . Ž .Ž .II J J J I

we obtainE V V I2FI I1a 2 G r M a J a a b gˆ ˆT f s q2 M e D yF F D y Re f T f D DŽ . Ž .Ž . Ž .JI bg2I 2 , Iž /Ef M

Ib J a bq fT G T f D . 15Ž . Ž .Ž . IJ

Taking its VEV and using the stationary condition, we derive the formulaab2 ² :M VŽ . IV F 2 b J aˆ² : ² : ² : ² : ² :q q2m Re f D s F F DŽ . J3r2 a b I2ž /½ 52 g g Ma b

I1 a b g² : ² : ² : ² :q Re f T f D D , 16Ž . Ž .Ž .bg2 , I


2 a b b J a IŽ . ²Ž . Ž . :where M s2 g g fT K T f is the mass matrix of the gauge bosons and g and g are theV a b J I a b

Ž . ² b:gauge coupling constants. Using Eq. 16 , we can estimate the magnitude of D-term condensations D .Using the scalar potential and gauge kinetic terms, we can obtain formulae of soft SUSY breaking scalar

Ž 2 . J w xmasses m , soft SUSY breaking gaugino masses M and A-parameters A 20,21 ,I a I JK

J J J2 2 2m s m q m , 17Ž . Ž .Ž . Ž .I F DI I

² :V X X XJ J ’’F2 2 J I I ’’ y1 J J J J² : ² : ² : ² :X X Xm ' m q K q F K K K yK F q PPP , 18Ž . Ž .Ž . ž /I ’’F 3r2 I I I J ’’ I I JI 2ž /MJ2 a a Jˆ ˆ² : ² :m ' q D K , 19Ž .Ž . ÝD I II

a

y1I² : ² : ² :M s F Re f f , 20Ž . Ž .a a a , I

² :XKX X J ’’II J y1X² : ² : ² : ² : ² : ² :X XA s F f q f y K K f , 21Ž . Ž .JI JK I JK , I I JK Ž II J ’’ JK .2ž /M

where the index a runs over broken gauge generators, Re f 'Re f and f ’s are Yukawa couplings someˆ a a a I JKŽ . Ž .of which are moduli-dependent. The I PPP JK in Eq. 21 stands for a cyclic permutation among I, J and K.

Ž 2 . J Ž 2 . JThe ellipsis in m stands for extra F-term contributions and so forth. The m is a D-term contribution toF I D I

scalar masses.

( )3. Heterotic string model with anomalous U 1

Effective SUGRA is derived from 4D string models taking a field theory limit. In this section, we study softŽ . 4SUSY breaking parameters in SUGRA from heterotic string model with U 1 . Let us explain our startingA

Ž . Xpoint and assumptions first. The gauge group GsG =U 1 originates from the breakdown of E =ESM A 8 8Ž . Ž . Ž . Ž .gauge group. Here G is a standard model gauge group SU 3 =SU 2 =U 1 and U 1 is an anomalousSM C L Y A

Ž . w xU 1 symmetry. The anomaly is canceled by the Green-Schwarz mechanism 24 . Chiral multiplets are classifiedinto two categories. One is a set of G singlet fields which the dilaton field S, the moduli fields M i and someSM

of matter fields f m belong to. The other one is a set of G non-singlet fields f k. We denote two types ofSMl � m k4matter multiplet as f s f ,f .

A Ž . Ž .The dilaton field S transforms as S™Sy id Mu x under U 1 with a space-time dependent parameterGS AŽ . A Ž .u x . Here d is so-called Green-Schwarz coefficient of U 1 and is given byGS A

1 1A A Ad s Tr Q s q , 22Ž .ÝGS l2 296p 96p

l

A Ž . A Ž . l Ž .where Q is a U 1 charge operator, q is a U 1 charge of f and the Kac-Moody level of U 1 isA l A A< A A < Ž y1 . Ž y2 . w xrescaled as k s1. We find d rq sO 10 ;O 10 in explicit models 25,26 .A GS m

Ž .The requirement of U 1 gauge invariance yields the form of Kahler potential K as,¨A

AA i i q V lm AKsK SqSqd V , M , M ,f e ,f 23Ž .ž /GS A m

up to the dependence on G vector multiplets. We assume that derivatives of the Kahler potential K with¨SM

respect to fields including moduli fields or matter fields are at most of order unity in the units where M is taken

4 Based on the assumption that SUSY is broken by F-components of S andror a moduli field, properties of soft SUSY breaking scalarw xmasses have been studied in Ref. 22,23 .


to be unity. However we do not specify the magnitude of derivatives of K by S alone. The VEVs of S and M i

are supposed to be fixed non-vanishing values by some non-perturbative effects. It is expected that thestabilization of S is due to the physics at the gravitational scale M or at the lower scale than M. Moreover we

Ž . ² :assume that the VEV is much bigger than the weak scale, i.e., O m < K . The non-trivial transformation3r2 SŽ . Ž .property of S under U 1 implies that U 1 is broken down at some high energy scale M .A A I

Hereafter we consider only the case with overall modulus field T for simplicity. It is straightforward to applyour method to more complicated situations with multi-moduli fields. The Kahler potential is, in general, written¨by

ŽS . A ŽT . ŽS ,T . m AKsK SqSqd V qK TqT qK q s SqSqd VŽ .Ž . Ž .ŽÝGS A l GS Al,m

m mŽS ,T . lqt TqT qu f f q PPP , 24Ž . Ž ..l l m

ŽS,T . mŽS,T . ² ŽS,T .: ² m: ² mŽS,T .:where K and u are mixing terms between S and T. The magnitudes of K , s and ul l l

Ž 2 . Ž . Ž . Ž .are assumed to be O e M , O e and e where e ’s ns1,2,3 are model-dependent parameters whose1 2 3 n

orders are expected not to be more than one 5. We estimate the VEV of derivatives of K in the form including² m : Ž . Ž . le . For example, K FO e rM ps2,3 . Our consideration is applicable to models in which some of fn lS p

are composite fields made of original matter multiplets in string models if the Kahler potential meets the above¨ˆ AŽ .requirements. Using the Kahler potential 24 , D is given by¨

lA A m AD syK d Mq K f q f q PPP . 25Ž .Ž .ÝS GS l m

l,m

Ž . <² m: < ŽŽ² : A A.1r2 .The breaking scale of U 1 defined by M ' f is estimated as M sO K d Mrq from theA I I S GS m² A: Ž .requirement D FO m M . We require that M should be equal to or be less than M, and then we find3r2 I

² : Ž A A .that the VEV of K has an upper bound such as K FO q Mrd .S S m GSŽ . Ž 2 . AThe U 1 gauge boson mass squared M is given byA V

A 22 2 S A A A n m² : ² : ² : ² :M s2 g K d M q q q K f f , 26Ž .Ž . Ž . ÝV A S GS m n m n½ 5m ,n

Ž . Ž 2 . A 2where g is a U 1 gauge coupling constant. The magnitude of M rg is estimated asA A V AŽ Ž² S:Ž A .2 . Ž A2 2 .. Ž 2 . A 2 Ž A2 2 .Max O K d M ,O q M . We assume that the magnitude of M rg is O q M . It leads toS GS m I V A m I

² S: ŽŽ A A .2 .the inequality K FO q M rd M .S m I GSw xThe formula of soft SUSY breaking scalar masses on G non-singlet fields is given by 23SM

² :Vk F2 2 k I J k J k² : ² : ² : ² : ² :m s m q K q F F R q X , 27Ž . Ž .Ž .l 3r2 l J I l I l2ž /MX J X

J k I y1 k J J kX² : ² :XR ' K K K yK , 28Ž . Ž .ž /II l I l J I l

y1 JAJ k A 2 A kˆ² : ² : ² :X 'q M D K . 29Ž .Ž .Ž . Iž /I l k V l

Here we neglect extra F-term contributions and so forth since they are model-dependent. The neglect of extraF-term contributions is justified if Yukawa couplings between heavy and light fields are small enough and the

Ž .R-parity violation is also tiny enough. We have used Eq. 16 to derive the part related to D-term contribution.

5 m lThe existence of s f f term in K and its contribution to soft scalar masses are discussed in 4D effective theory derived through thel m

w xstandard embedding from heterotic M-theory 27 .


Table 1² J k: ² J k:The magnitudes of R and XIl I l

J k J kŽ . ² : ² :I, J R XIl I l

2 S 2Ž . Ž . Ž Ž² : ² :. Ž ..S,S O e rM Max O K r K ,O e rMp SS S p2 2Ž . Ž . Ž Ž Ž² : .. Ž ..T ,T O 1rM Max O e r K M ,O 1rM1 S2 2Ž . Ž . Ž .m,m O 1rM O 1rMI

2 2Ž . Ž . Ž Ž Ž² : .. Ž ..S,T O e rM Max O e r K M ,O e rMp 1 S 33Ž . Ž . Ž Ž Ž² : .. Ž Ž ..S,m O e M rM Max O e r K M ,O e r MMp I p S p I

3Ž . Ž . Ž Ž Ž² : .. Ž Ž ..T ,m O M rM Max O e r K M ,O 1r MMI 3 S I

Ž . Ž 2 . A 2 2Note that the last term in r.h.s. of Eq. 16 is negligible when M rg is much bigger than m . Using theV A 3r2² J k: ² J k:above mass formula, the magnitudes of R and X are estimated and given in Table 1. Here we assumeI l I l

A A Ž .q rq sO 1 .k mŽ 2 .kNow we obtain the following generic features on m .l

Ž . ² J k: ² J k:1 The order of magnitude of X is equal to or bigger than that of R except for an off-diagonal partI l I lŽ . Ž .I, J s S,T . Hence the magnitude of D-term contribution is comparable to or bigger than that of F-term

Ž 2 ² : 2 .² k:contribution except for the uniÕersal part m q V rM K .3r2 F lŽ . ² : Ž .2 In case where the magnitude of F is bigger than O m M and M)M , we get the inequalitym 3r2 I I2 2 ˆ A 2Ž . Ž . ² : Ž .m )O m since the magnitude of D is bigger than O m .D k 3r2 3r2Ž . ŽŽ 2 . . ŽŽ 2 . .3 In order to get the inequality O m )O m , the following conditions must be satisfiedF k D k

simultaneously,

m M3r2² : ² : ² :F , F <O m M , F sOŽ .T m 3r2 S 1r2Sž /² :KS

2² S :M K eSS p-O 1 , -O 1 ps2,3 30Ž . Ž . Ž . Ž .S S² : ² : ² :K K KS S S

ˆ A² : ² : Ž .unless an accidental cancellation among terms in D happens. To fulfill the condition F <O m M ,T ,m 3r2² : ² 2 :a cancellation among various terms including K and M W rW is required. Note that the magnitudes ofI I

² : ² : Ž . Ž .K and K are estimated as O M and O M , respectively.T m I

The gauge kinetic function is given by

S TŽm. lf sk d qe d q f f 31Ž .Ž .ab a a b a a b a bM M

w xwhere k ’s are Kac-Moody levels and e is a model-dependent parameter 28 . The gauge coupling constantsa ay2 ² :g ’s are related to the real part of gauge kinetic functions such that g s Re f . The magnitudes ofa a a a

gaugino masses and A-parameters in MSSM particles are estimated using the formulae

² I: ² :M s F h , 32Ž .a a I

² : ² :y1² :h ' Re f f 33Ž .a I a a , I

² I: ² :X XA s F a , 34Ž .k l l k l l I

² :K X JI I y1X² : ² : ² : ² : ² : ² :X X X Xa ' f q f y K K f . 35Ž . Ž .Ik l l I k l l , I k l l Žk I J l l .2M


Table 2² : ² X :The magnitudes of h and aa I k l l I

X² : ² :I h aa I k l l I

2Ž . Ž Ž² : . Ž ..S O 1rM Max O K rM ,O e rMS pŽ . Ž .T O e rM O 1rMa

2 2Ž . Ž .m O M rM O M rMI I

y2 Ž .The result is given in Table 2. Here we assume that g sO 1 .a

²Ž S.1r2 : ² :In case that SUSY is broken by the mixture of S, T and matter F-components such that K F , F ,S S T² : Ž .F sO m M , we get the following relations among soft SUSY breaking parametersm 3r2

M 222 2 2 2

Xm G m sO m G A sO m , 36Ž . Ž . Ž .Ž . Ž .k D 3r2 k l l 3r2k 2ž /MI

M 2I2 y12 S 2² :M sO m PMax O K ,O e ,O . 37Ž . Ž .Ž .Ž .Ž .a 3r2 S a 2ž /ž /M

Finally we discuss the three special cases of SUSY breaking scenario.1. In the dilaton dominant SUSY breaking scenario

1r2S² : ² : ² :K F sO m M 4 F , F , 38Ž .Ž .Ž .S S 3r2 T m

the magnitudes of soft SUSY breaking parameters are estimated as2² S :M K eSS p2 2m sO m PMax O 1 ,O ,O ,Ž . Ž .Ž .k 3r2 S Sž /ž /² : ² : ² :ž /K K KS S S

² :m K3r2 SXM sO , A sO m PMax O ,O e .Ž . Ž .a k l l 3r2 p1r2S ž /ž / ž /M² :KS

ŽŽ 2 . . ŽŽ .2 .XHence we have a relation such that O m GO A .k k l l² S: Ž 2² S :. Ž² :.Gauginos can be heavier than scalar fields if K is small enough and O M K -O K . In thisS SS S

Ž .case, dangerous flavor changing neutral current FCNC effects from squark mass non-degeneracy areavoided because the radiative correction due to gauginos dominates in scalar masses at the weak scale. In

w xRef. 17 , it is shown that gauginos are much lighter than scalar fields from the requirement of the conditionw xof vanishing vacuum energy in the SUGRA version of model proposed in Ref. 14 . In appendix, we study it

² : ² S: ² S :by deriving the relations among the magnitudes of K , K and K under some assumptions.S S SS

2. In the moduli dominant SUSY breaking scenario1r2S² : ² : ² :F sO m M 4 K F , F , 39Ž .Ž . Ž .T 3r2 S S m

the magnitudes of soft SUSY breaking parameters are estimated as

e M12 2m sO m PMax O 1 ,O ,Ž . Ž .Ž .k 3r2 ž /ž /² :KS

M sO e m , A sO m .Ž . Ž .a a 3r2 k lm 3r2

ŽŽ 2 . . ŽŽ .2 . ŽŽ .2 .XHence we have a relation such that O m GO A GO M . The magnitude of m is estimatedk k l l a T TŽ .as m sO m .T T 3r2

3. In the matter dominant SUSY breaking scenario1r2S² : ² : ² :F sO m M 4 K F , F , 40Ž .Ž . Ž .m 3r2 S S T


the magnitudes of soft SUSY breaking parameters are estimated as

M 2 MI2 2Xm sO m , M , A sO m .Ž . k 3r2 a k l l 3r22 ž /ž / MMI

Ž 2 . ŽŽ .2 . ŽŽ .2 .XThe relation m 4O M sO A is derived when M4M . The magnitude of m is estimatedk a k l l I m nŽ . w xas m sO m MrM . This value is consistent with that in Ref. 13 .m n 3r2 I

4. Conclusions

We have studied the magnitudes of soft SUSY breaking parameters in heterotic string models withŽ . XG =U 1 , which originates from the breakdown of E =E , and derive model-independent predictions forSM A 8 8

them without specifying SUSY breaking mechanism and the dilaton VEV fixing mechanism. In particular, wehave made a comparison of magnitudes between D-term contribution to scalar masses and F-term ones and acomparison of magnitudes among scalar masses, gaugino masses and A-parameters under the condition thatŽ . ² : Ž A A . Ž 2 . A 2 Ž A2 2 . ² : Ž 2 2 .O m < K FO q Mrd , M rg sO q M and V FO m M . The order of magnitude3r2 S m GS V A m I 3r2

Ž .of D-term contribution of U 1 to scalar masses is comparable to or bigger than that of F-term contributionA² I: ² : ² J k: Ž 2 ² : 2 .² k:F F R except for the universal part m q V rM K . If the magnitude of F-term conden-J I l 3r2 F l

² : Ž . Ž 2 .sation of matter fields F is bigger than O m M , the magnitude of D-term contribution m is biggerm 3r2 I D kŽ 2 . ŽŽ 2 . . ŽŽ 2 . .than O m . In general, it is difficult to realize the inequality O m -O m unless conditions such as3r2 D k F k

Ž .Eq. 30 are fulfilled. We have also discussed relations among soft SUSY breaking parameters in three specialscenarios on SUSY breaking, i.e., dilaton dominant SUSY breaking scenario, moduli dominant SUSY breakingscenario and matter dominant SUSY breaking scenario.

The D-term contribution to scalar masses with different broken charges as well as the F-term contributionfrom the difference among modular weights can destroy universality among scalar masses. The non-degeneracyamong squark masses of first and second families endangers the discussion of the suppression of FCNC process.On the other hand, the difference among broken charges is crucial for the scenario of fermion mass hierarchy

w xgeneration 29 . It seems to be difficult to make two discussions compatible. There are several way outs. TheŽ .first one is to construct a model that the fermion mass hierarchy is generated due to non-anomalous U 1

Ž .symmetries. In the model, D-term contributions of non-anomalous U 1 symmetries vanish in the dilatondominant SUSY breaking case and it is supposed that anomalies from contributions of the MSSM matter fieldsare canceled out by an addition of extra matter fields. The second one is to use ‘‘stringy’’ symmetries for

w xfermion mass generation in the situation with degenerate soft scalar masses 30 .w xFinally we give a comment on moduli problem 31 . If the masses of dilaton or moduli fields are of order of

the weak scale, the standard nucleosynthesis should be modified because of a huge amount of entropy²Ž S.1r2 : Ž .production. The dilaton field does not cause dangerous contributions in the case with K F sO m MS S 3r2

² S: 6 Ž 2 . Ž 2 ² S:2 .if the magnitude of K is small enough , because the magnitudes of m are given by O m r K .S F S 3r2 S

Acknowledgements

The author is grateful to T. Kobayashi, H. Nakano, H.P. Nilles and M. Yamaguchi for useful discussions.

6 w xThis possibility has been pointed out in the last reference in 10 .


Appendix A. On derivatives of Kahler potential related to dilaton¨

² : ² S: ² S :We discuss the relations among K , K and K using SUSY breaking conditions and the stationaryS S SS

conditions of scalar potential. We list our assumptions first.²Ž S.1r2 : ² :1. SUSY is broken by the mixture of S, T and matter F-components such that K F , F ,S S T

² : Ž .F sO m M .m 3r2² : Ž . Ž A A .2. The magnitude of K is much bigger than O m and it is comparable to or smaller than O q Mrd .S 3r2 m GS

<² m: < Ž .The latter is equivalent to the condition that the magnitude of M ' f is at most O M .I² S:1r2 ² T: ² m:3. The magnitude of K is much bigger than those of K and K , and it is comparable to or smallerS S S

Ž A A . Ž 2 . A 2than O q M rd M . The latter is equivalent to the condition that the magnitude of M rg ism I GS V AŽ A2 2 .O q M .m I

A A Ž . Ž .4. The magnitude of d rq is O 1r10 ;O 1r100 .GS m² : 2² : ² :5. No cancellation happens among terms in K and M W r W .S S

Under these assumptions, the following relation is derived

² : ² : ² :G K WS S S1r2S² :K sO sO qM A.1Ž .S ž / ž /² :M M W

Ž . ² : 2² : ² : ² S:1r2 Ž² : .by the use of the definition 1 . If K is bigger than M W r W , we find that K sO K rMS S S SŽ A A .FO q rd .m GS

² : ² S: ² S : Ž .Further we can get the following relation among K , K and K from the stationary conditions 12S S SSŽ .and 16 ,

² S : ² S: ² S:1r2K K KSS S SsMax O ,O . A.2Ž .S ž /ž /² :² : ž /K MK SS

Let us consider a typical case with non-perturbative superpotential derived from SUSY breaking scenario bygaugino condensations. The non-perturbative superpotential W is, generally, given bynon

yb SilW s a f ,T exp , A.3Ž .Ž .Ýnon i Až /d MGSi

l Ž A.where a ’s are some functions of f and T , and b ’s are model-dependent parameters of O q . Using thei i mŽ . Ž Ž .. ² S:1r2 Ž ² : ² :. Ž A A .second assumption, Eqs. A.1 and Eq. A.3 , we get the relation K sO M W r W sO q rdS S m GS

Ž² :. Ž² :.if O W sO W . This relation leads to M sM from the third assumption and the relation such thatnon I² S: ² : Ž A A . ² S : ² S: Ž A A . Ž .M K r K sO q rd . We obtain the relation M K r K sO q rd using Eq. A.2 . Hence itS S m GS SS S m GS

is shown that the magnitude of gaugino masses is comparable to or smaller than that of scalar masses. It isnecessary to relax some of assumptions in order to have the SUSY spectrum at M such that gauginos are muchI

heavier than scalar fields.

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28 January 1999


Cosmological consequences of superconducting string networks

Konstantinos Dimopoulos 1, Anne-Christine Davis 2

Department of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge, CB3 9EW, UK

Received 8 June 1998; revised 2 December 1998Editor: P.V. Landshoff

Abstract

We consider the cosmological consequences of a network of superconducting cosmic strings. For strong enough currentthe period of friction domination never ends. Instead a plasma scaling solution is reached. We demonstrate that this givesrise to a very different cosmology than the usual horizon scaling solution. In particular the string network gives rise to adistinct imprint on the microwave sky, giving non-Gaussian features on much smaller angular scales. It also gives rise to afilament structure in string wakes. Because of the presence of the string magnetocylinder, the string magnetic field cannotcreate a primordial magnetic field. Similarly, it evades nucleosynthesis constraints. We also show that strings formed at thesupersymmetry breaking scale can create the required baryon asymmetry of the universe. q 1999 Published by ElsevierScience B.V. All rights reserved.

1. Introduction

The microphysics of cosmic strings has receivedw xconsiderable attention. In particular, Witten 1

showed that cosmic strings become superconductingas a result of boson condensates or fermion zeromodes in the string core. Such strings are capable ofcarrying a sizable current, with the maximum currentbeing about 1020A for a grand unified scale string.Inevitably, such currents have cosmological and as-

w xtrophysical 2 consequences. The consequences forw xemission of synchrotron radiation 3 and for high

w xenergy g-rays 4 have been explored. However, allthese studies have assumed that the evolution of anetwork of superconducting strings is similar to thatof ordinary strings.

1 E-mail: [email protected] E-mail: [email protected]

w xEarly studies using both analytic 5 and numeri-w xcal techniques 6 showed that the string evolution

was indeed similar to that of ordinary cosmic strings.However, these studies neglected the very early timeswhen the string is interacting strongly with the sur-

w xrounding plasma. As shown recently in 7 , for strongenough currents, this friction dominated period maynever end. In this case the network reaches theso-called plasma-scaling solution, where the densityof strings may be much larger than that of the usualhorizon-scaling string networks. In this case thestrings are highly tangled and move rather slowly.Also considering the attractive gravitational fields

w xgenerated by superconducting strings 8 , it is evidentthat, compared with ordinary cosmic strings, a net-work of superconducting strings could have verydifferent cosmological implications on the matterand radiation content of the early universe.

In this letter we briefly explore some of the mostimportant cosmological consequences of a plasma-


( )K. Dimopoulos, A.-C. DaÕisrPhysics Letters B 446 1999 238–246 239

scaling superconducting string network. Assuming aconstant string current, we first discuss the character-istics of the plasma-scaling solution. We then de-scribe the string spacetime and its implications onthe surrounding particles and radiation. Afterwards,we explore the implications of the network regardingthe Cosmic Microwave Background RadiationŽ .CMBR anisotropies and the formation of the Large

Ž .Scale Structure LSS of the Universe. We alsodiscuss any effects a plasma-scaling string network

Ž .may have on Big Bang Nucleosynthesis BBN andwhether it can generate a Primordial Magnetic FieldŽ .PMF sufficient enough to account for the currentlyobserved magnetic fields of the galaxies.

Finally, we apply our results to a recently pro-posed baryogenesis mechanism with superconduct-

w xing cosmic strings 9 . This mechanism uses stringsformed at the supersymmetry breaking scale. Withour plasma scaling solution this mechanism producesa sufficiently strong baryon asymmetry to account

w xfor nucleosynthesis. This improves the result of 9 ,where vorton domination was needed to obtainenough baryon asymmetry. In what follows, unless

Ž .stated otherwise, we use natural units "scs1 .

2. Friction and plasma-scaling

After the formation of the string network, curvesand wiggles on the strings tend to untangle due tostring tension, which results in oscillations of thecurved string segments on scales smaller than thecausal horizon and larger than their curvature radii.Friction dissipates the energy of these oscillations

w xand leads to their gradual damping 10 . Thus, thestrings become smooth on larger and larger scaleswith their curvature radius R growing accordingly.

Friction on a cosmic string is caused by theinteraction of the string fields with the plasma parti-

w xcles. As shown in 7 , for string currents J largerthan a critical value J , the friction force is deter-c

mined by the string magnetic field generated by thecurrent. This critical current is estimated as,

'J ;J LGm 1Ž .c max

where m is the string mass per unit length, Gsmy2P

Žis Newton’s gravitational constant with m s1.22P19 .=10 GeV being the Planck mass and Jmax

'; mrL is the maximum acceptable string current,over which the string loses its superconducting prop-

w x Ž .erties 1 , with L, ln LR being the self-inductancey1 w x 3of a string of radius L 8 .

For JGJ friction prevents the network fromcw xreaching horizon-scaling 7 . Instead the strings are

found to move with a more or less constant terminalvelocity,

1r2J GmcÕ; ; <1 2Ž .(

2'J GJ

In this case the string network does satisfy aŽ .scaling solution so-called plasma-scaling , which,

however, may differ substantially from the usualhorizon-scaling of ordinary strings. Indeed, aplasma-scaling network consists of slowly moving,highly tangled strings, with curvature radius andinter-string distance much smaller than the horizon,since R;Õt<Hy1, where H; ty1 is the Hubbleparameter 4. Still, although denser, a plasma-scalingnetwork is not in danger of dominating the overallenergy density of the universe because, r rrs

2 2'; GJ <1, where r ;mrR is the energy den-s

sity of the strings and r;1rGt 2 is the energydensity of the universe with t being the cosmic time.

It should be noted here that J is the value of thelocal, coherent current on the string, which generatesthe string magnetocylinder and determines the fric-tional cross-section between the string and theplasma. J should be thought as a free parameter,which, however, is expected to assume large valuesw x1 either directly at current generation or throughsubsequent interactions with weak primordial mag-

w xnetic fields 11 . On large scales the orientation ofthe string current is expected to be the stochastic, or

w xKibble current 12 . The string current is dynamicallyconserved, which, in view of the growth in thecurvature radius during string network evolution andcausal interaction of different current patches on the

3 w xIn 7 , the factors of L were omitted for simplicity. However,it is important to maintain them for quantitative calculations.Indeed, it can be easily shown that, for strings formed at the

Ž .breaking of the Grand Unified Theory GUT-strings L;100.4 Note that in the one-scale model the curvature radius and the

inter-string distance of the string network are of similar magni-tude.

( )K. Dimopoulos, A.-C. DaÕisrPhysics Letters B 446 1999 238–246240

string, results in the local value J remaining approx-w ximately constant 7 . We note that superconducting

strings in the friction dominated era have also beenw xconsidered in 13 . However, the much smaller rms

current was used rather than the local, coherentcurrent. As a consequence the effect of the magneto-cylinder, and thus the string interaction with the

w xplasma, was not included. This resulted in Ref. 13concluding that frictional effects would be small in

w xcontrast to our results 7 .

3. The string gravitational field

The exact metric of the spacetime around a cur-rent carrying string was first calculated by Moss and

w xPoletti 14 . The implications of this spacetime ontest particles and light rays was also investigatedw x w x15 , reaching similar conclusions as Ref. 14 . How-

w xever, it was Linet 16 who first attempted to explorethe gravitational properties of a superconductingstring system in a more realistic way, by consideringonly first order terms in G. Linet demonstrated thatthis was fully consistent with the original, exactsolution. A more thorough study of the linearised

w xspacetime of a superconducting string 17 arrived atsimilar conclusions. Finally, these results were ex-

w xtended 18 by also considering higher order terms inthe gauge coupling. However, in all the above workthe importance of self-inductance effects on the stringspacetime has not been fully appreciated. Taking

w xthese into account 8 has shown that the perpendicu-lar to the string geometry is described by,

2 2 2 2 2ds s 1q2F ydt qdr q 1ydrp r duŽ . Ž .H

3Ž .

Ž 2 . Ž .where F;L GJ ln Lr is the attractive gravita-tional potential and d is the deficit angle estimated

w xas 8 ,

d,8p G mqLJ 2 4Ž .Ž .Therefore, the existence of a current on the string

generates an attractive gravitational field. This fieldalong with the conical form of the spacetime affectsthe surrounding particles while the string moves in

w xthe plasma. In 8 it is shown that the velocity boost

felt by the particles towards the perpendicular direc-tion to the string motion is,

12us8p GmÕgq8p GJ L Õgq 5Ž .ž /Õg

y1 2'where g s 1yÕ is the Lorentz factor. In theabove the first two terms are due to the deficit anglewhereas the last term is due to the attractive gravita-tional field. The gravitational field dominates forJGJ where 5,G

1r6J ;J LGm 6Ž . Ž .G max

The effect of the existence of strong string cur-rents on the string spacetime morphology and on thecharacteristics of the string network scaling solutionis expected to reflect itself on the numerous cosmo-logical implications of strings, in particular theCMBR anisotropies and the formation of the LSS ofthe universe.

4. Anisotropies on the microwave sky

Ž .The root mean square rms CMBR temperatureanisotropies generated by cosmic strings may be

w xestimated as 19 ,

DT DT', N 7Ž .ž / ž /T Trms S

Ž .y2where N; HR is the number of strings inside aŽ w x.horizon volume see 7 and

DT,d Õg 8Ž .ž /T S

w xis the anisotropy generated by a single string 20 .Ž .Thus, from 4 and the above, the rms anisotropy

generated by a network of superconducting strings isw x7 ,

DT2,d,8p G mqLJ 9Ž .Ž .ž /T rms

The above suggests that the rms effect of thestring spacetime on radiation does not depend on the

5 It can be easily verified that, for realistic values of theparameters, J ) J .G c


gravitational field of the strings. This is to be ex-Ž .pected since, for radiation, ds s0 and 3 suggestsH

Ž .that the prefactor 1q2F cannot influence theshape of the null geodesics. Moreover, because LJ 2

F m the magnitude of the rms temperatureanisotropies is little affected by the string current.However, in terms of the stochastic nature of theanisotropy distribution, a plasma-scaling string net-work may produce a distinct imprint on the mi-crowave sky, due to the larger number of strings perhorizon. Since the string network is denser onepossible effect is to shift the position of the Dopplerpeak to smaller values of l.

Indeed, the distribution of CMBR temperatureanisotropies generated by a horizon-scaling networkof ordinary strings is expected to be non-Gaussian

Ž .over angular scales smaller than Dq ;18, which0

corresponds to the angular scale of the horizon at thew xtime of last scattering 21 . However, as the inter-

string distance is much smaller in a plasma-scalingstring network, one would expect to discover non-Gaussian signatures only on angular scales smallerthan,

RDq; Dq ;Õ 8 <18 10Ž . Ž . Ž .0y1H

Ž .For GUT-strings the rms anisotropy is DTrT rms

,8p Gm;10y5, in good agreement with the obser-vations. In this case, it is easy to see that, formaximum string current, the Gaussianity of the dis-tribution appears over angular scales less than 0.18

as,

1r4Õ J ; LGm 11Ž . Ž . Ž .max

However, in order to ascertain the full stringpredictions for the CMBR anisotropies large com-puter simulations are necessary. Whilst early simula-tions produced disappointing results suggesting the

w xlack of a Doppler peak 23 the most recent simula-tion suggests that local cosmic strings can account

w xfor the observed CMBR 24 . It is likely that ourdenser network will have more power on smallscales since its scaling distance is smaller. A fullscale numerical simulation is required to compute

the exact scale of the peak in the power spectrum.This is the subject of a future investigation.

5. Large scale structure overdensities

The angular deficit of the string spacetime and theattractive gravitational field generate two overlap-ping streams of matter behind a moving string. Thisis because of the relative boost, u, felt by the plasmaparticles towards the string trail. Thus, the matteroverdensity generated by a moving string may beestimated as, drsbr, where 0-bF1 is deter-mined by the fraction of the matter streams thatremain inside the string wake, rather than dissipatinginto the inter-string space. The b factor is stronglyrelated to the nature of the dark matter of the uni-

Ž .verse. For baryonic or Cold Dark Matter CDMb,1 as almost all the overdensity is contained

Ž .inside the string wake. For Hot Dark Matter HDMthough, b can be substantially smaller as an impor-tant fraction of the overdensity diffuses away due to

w xfree streaming effects 22 .Ž .The length of a string wake is l t ;Õt and its

Ž . Ž .thickness is d t ;ut, where u is given by 5 .Thus, the linear mass overdensity of the wake is

Ž . 2dms dr dl,bruÕt . Therefore, the total overden-sity of a string wake is,

dr 1 dm u, ;b 12Ž .2ž /r r ÕR

For currents smaller than J the boost u isGŽ .determined by the deficit angle terms in 5 . In this

case it easy to see that,

dr,bd 13Ž .ž /r J-JG

where we have taken g,1 since the coherent mo-tion of the strings is never expected to be ultrarela-

w xtivistic 7 .The above estimate is not very different from the

case of ordinary strings, which again is due to thefact that the deficit angle is largely insensitive to thestring current. However, when the gravitational fieldbecomes important, the situation is drasticallychanged.


For very strong currents the gravitational attrac-Ž .tion term dominates in 5 . In this case, using also

Ž .2 one finds,

3r22dr GJŽ .,b 8p L 14Ž . Ž .ž /r GmJGJG

For maximum current the above becomes,

dr 8p ',b Gm 15Ž .ž / ž /'r LJsJmax

Observations of the galaxy correlation function ofŽ . y5 w xthe LSS suggest that drrr ;10 22 . There-obs

fore, for GUT-strings with weak currents there isreasonable agreement for CDM models with b,1.

ŽHowever, for strong currents HDM or MDM Mixed.Dark Matter models are preferable. Indeed, for max-Ž .imum current 15 suggests that bF0.1. In general,

from the comparison with observation it can beshown that, for GUT-strings with JGJ one re-G

quires,

JFby1r3J 16Ž .G

The above constraints may be somewhat strength-ened if the distribution of dark matter is smootherthan the distribution of the galaxies. Indeed, it isbelieved that the observed galactic distribution, whichis used in order to estimate the density perturbationsin the universe today, represents only the peaks inthe actual density distribution of the dark matter. Itis, thus, believed that the overall density perturbation

Ž .of the universe relates to that observed as, drrr obsŽ .sb drrr , where bG1 is the so-called bias factor

w x22 . From the above it is evident that this factor maybe included in b and so the form of our resultsremains unaffected. Also, since b is expected to beof order unity, the quantitative estimates remainreliable.

Apart from the magnitude of the overdensities aplasma-scaling string network may generate LSSmorphologically different from the one due to ordi-nary strings. Indeed, the slow moving strings of afriction dominated network would produce filaments,rather than thin wakes. It is possible that thesefilaments are thickened by gravitational effects. Thedistribution of these filaments would be denser dueto the smaller inter-string distance of the network.This is rather unfortunate as the spectrum of density

distributions would lose power on large scales, aproblem already present for ordinary strings. Thus,one could argue that plasma-scaling superconductingstring networks are not sufficient to explain theoverall LSS, and some other density perturbationmechanism is required to seed the structure on verylarge scales. If such a mechanism exists then thesmaller filamentary structure generated by the stringscould be swept inside the ‘pancakes’ of the larger,horizon-sized density perturbations. Such structures,i.e. embedded filaments on large walls are indeed

w xobserved 25 . However, a full numerical simulationis required to investigate this and is the subject offuture investigation. If the numerical simulationsconfirm these tentative conclusions then possiblemechanism to generate both cosmic strings and largescale density perturbations could be Hybrid Inflationw x26 .

6. Nucleosynthesis and galactic magnetic fields

w xIn an early work of Butler and Malaney 5 it wassuggested that the existence of a network of electri-cally charged current carrying strings may seriously

Ž .disturb Big Bang Nucleosynthesis BBN . Their ar-gument was based on the fact that a current carryingstring generates a Biot-Savart magnetic field whichresults in the creation of a magnetocylinder around

w xthe string core 3,22,7 . This magnetocylinder isimpenetrable to charged plasma particles but not to

w xsingle neutrons. Thus, in 5 it was argued that insidethe trail of the moving strings an overdensity ofneutrons would be generated that may affect the rateof BBN’s reactions and the abundance of the result-ing elements.

However, it can be easily shown that this is notactually the case. Firstly, the charged plasma parti-cles that are pushed away on the border of themagnetocylinder follow the magnetic field lines in asimilar way that the solar wind is directed towardsthe Earth’s magnetic poles by the Earth’s Magneto-sphere. Thus, the orbits of the charged plasma parti-cles trace the surface of the magnetocylinder and,therefore, are expected to be sucked back into thetrail of the string after the string has passed. More-over, not only does the charged plasma close behind


the string magnetocylinder but some of it may evenw xpenetrate it from the back as discussed in 22 .

Another argument against the disastrous implica-w xtions of 5 is due to purely geometrical facts. The

dimensions of the string magnetocylinder are deter-mined by the pressure balance between the plasma

w xand the string magnetic field as 3,7 ,

Jr ; 17Ž .s 'r Õ

where Õ is the string velocity. The plasma-scalingsolution suggests that, inside a volume ;R3 onewould expect only about one string segment of length;R. This segment is expected to sweep, whilemoving, a volume DV;r =R=ÕDt. Thus, usings

R;Õt, the fraction of volume traced by string mag-netocylinders per Hubble time t is,

DV r J 2s y1 y2; ; FL ;10 18Ž .3 R mR

Therefore, since the duration Dt of BBN is noBBNw xmore than about a hundred Hubble times 25 , an

arbitrary point in space may be swept by a stringmagnetocylinder at most once or twice. Such anencounter would last about Dt;r rÕ;Ly1 t<s

Dt Thus, one would expect that any effect thatBBN

such an event may have on BBN’s processes wouldbe insignificant.

It would be misleading to believe that BBN’sprocesses could be disturbed by the long-rangeBiot-Savart string magnetic field. Indeed, the field is,in fact, not expected to extend beyond the border ofthe string magnetocylinder because the chargedplasma particles, while travelling on the magneto-cylinder’s border, generate a surface current of oppo-

w xsite orientation to the string current 7 . As a result,the string magnetic field is cancelled outside themagnetocylinder.

For the same reason the string magnetic fieldcannot be involved in any astrophysical processes,since, being contained inside the magnetocylindersof the strings, it never really comes into contact withthe cosmic plasma. Thus, such a field cannot freezeinto the plasma and be in any way directly responsi-ble for seeding the galactic magnetic fields. How-

w xever, as shown in 8 , superconducting cosmic stringsmay efficiently generate a primordial magnetic fieldindirectly, through dynamical friction. Such a fieldmay be strong and coherent enough to easily triggerthe galactic dynamo and generate the observed galac-tic magnetic fields. Also, it can be shown that the theplasma vorticity generated by the string motion andgravitational pull may be contribute to the fragmen-tation process of galaxy formation as the scale of thespinning plasma volumes compares to the proto-

w xgalactic scale before gravitational collapse 8 .

7. Baryogenesis

w xIn a recent work Brandenberger and Riotto 9have suggested a new mechanism for explaining thebaryon asymmetry in the Universe, involving super-conducting cosmic strings. Their model considers acosmic string network formed at the breaking ofsupersymmetry at temperatures of order 102 TeV.Charged sleptons and squarks condense in the stringcore, resulting in bosonic superconductivity. CP-violating interactions during the string network for-mation period may result in the confinement of anon-zero net baryon number inside the string core,which would be preserved due to dynamical andtopological current conservation until after the elec-troweak phase transition, when it could be releasedthrough string loop decay without being erased bysphaleron processes. In their treatment Branden-berger and Riotto have shown that the confinedbaryon number is released during the friction domi-nation period of the string network. However, theydid not take into account the effects of excessivefriction on the string evolution due to the existenceof a magnetocylinder around the string core.

As discussed in previous sections, for large cur-rents a network of superconducting strings remainsalways friction dominated. Thus, such a friction-scal-ing string network would be more tangled and denser,which would imply that the captured baryon numberdensity may be larger than the original considera-tions of Brandenberger and Riotto. Following the

w xreasoning of 9 we calculate the baryon numberdensity generated by a network of superconductingstrings carrying maximum current J;102 TeV.


Using the one-scale model, the number density ofw xloops created per unit time is given by 22 ,

dn dR Õy4sn R ; 19Ž .4dt dt R

where n is a numerical factor of order unity. Each ofthese loops contains a net baryon number trappedinside the strings at the time of their formation t . If0'Q is the baryon charge per unit length, and Q; m

w x9 then the charge per correlation length at thenetwork formation is, Q s QR , where R ;0 0 0

y1Ž . w x'l m is the initial correlation length 27 with l

being the self-coupling of the string vortex field. Thebaryon charge on larger string segments can beestimated using a random walk. Thus, when a loop

Ž .of radius R t is formed one would expect it tocontain baryon charge of order

1r2

R tŽ .Q t ; Q 20Ž . Ž .R 0a tŽ .

R0ž /a tŽ .0

Ž . 1r2where a t A t is the scale factor of the Universe0

and we have included the conformal stretching of thestrings. When the loop decays a fraction eF1 of thecaptured baryon charge is released as a net baryonnumber, Dn se Q , where e is determined by theB R

w xrates of CP-violating processes 9 .The total baryon number density generated by

w xloop decay is easily estimated as 9 ,

X 3r2dn tt X X Xn t s dte Q t t 21Ž . Ž . Ž . Ž .HB R ž /dt tti

where the final factor is due to cosmological redshiftand t is the earliest time, when loops that contributei

Ž .to the baryon number density are formed. Using 19Ž . Ž .and 20 Eq. 21 gives,

5r43r2t t0 05 y3n , en Q R 22Ž .B 0 04 ž / ž /t ti

This gives,

5r2n TB i3 y1;en Q l g 23Ž .0 ) ž /s T0

where s is the entropy density of the Universe,g ;102 is the number of degrees of freedom and

)

Ž .T sT t .i i

In order to evaluate the above one needs to decideon the choice of T . If the loops decay promptly theyi

do so in less than a Hubble time due to the efficientw xradiation emission 9 . In this case, at earlier times

than the electroweak phase transition, any baryonnumber released is expected to be ‘washed-out’ bysphaleron processes. Thus, only loops formed laterthan the time t of the transition may contribute toew

the net baryon number density. Therefore, T ,Ti ewŽ . 2 Ž .'T t ;10 GeV and Eq. 23 becomes,ew

5r2n TB ew3 y1;en Q l g 24Ž .0 ) ž /s T0

The above result differs substantially from the find-Ž Ž . w x.ings of Brandenberger and Riotto Eq. 30 in 9 ,

Ž .y2 6 w xby a factor T rT ;10 . Thus, the result of 9ew 0

underestimates the generated baryon asymmetry by aŽ .'million times! Taking Q; m i.e. Q ;1 it can0

be easily seen that the desired asymmetry n rs;B

10y10 can be achieved with rather natural values ofthe parameters: e;10y1 and l,n;1.

However, if instead of collapsing the string loopsform stable vortons, i.e. string rings stabilised by the

w xangular momentum of the current carriers 28 , theabove situation is modified. Vortons may releasetheir baryon number if they ever become unstable

w xand decay 29 . In this case, vortons manage topreserve their baryon number throughout the periodprior to the electroweak transition. Thus, the result-ing net baryon number density may receive contribu-tions even from the time of network formation, i.e.

Ž .T ,T . Consequently, 23 would give,i 0

nB 3 y1;en Q l g 25Ž .0 )s

Ž . w xwhich is identical with Eq. 31 of 9 . This is notŽ .surprising since, in this case, the integral of 21 is

dominated by the initial contribution at the time t of0

formation of the string network, so that the subse-quent frictional evolution of the strings does notaffect the results.


8. Conclusions

In conclusion, we have investigated the cosmolog-ical and astrophysical consequences of a plasma-scaling, charged-current carrying, open string net-work.

We have shown that such a network would gener-ate large scale structure with very different featuresthan the one produced by a horizon scaling network.Indeed, the slow moving strings would create fila-ments instead of thin wakes, whose separation dis-tances would be much smaller than the horizon. Thiscompounds the existing problem of structure forma-tion with ordinary strings, due to the lack of poweron large scales. One way to overcome this is byconsidering hybrid models, which incorporate infla-

Ž w x.tion with cosmic strings see for example 26 . Insuch models the large scale fluctuations could begenerated by inflation and the string-produced fila-ments swept into the horizon-sized ‘pancake’ struc-tures. As we have shown, the magnitude of suchfilamentary overdensities depends on the type ofdark matter assumed, which gives upper bounds onthe parameters, and may provide a link between thebias factor and the string current.

We have also found that the imprint of the stringson the microwave sky would be Gaussian on smallerangular scales than the horizon scale at decoupling.The scale of the non-Gaussian features depends onthe string current and is related to the terminalvelocity of the friction dominated strings.

Furthermore, we discussed possible effects of aplasma-scaling network on nucleosynthesis andshowed that the latter is not seriously disturbed evenfor maximum string currents. We briefly consideredthe possibility of direct generation of primordialmagnetic fields by the string magnetic fields. Sincesuch fields are shielded by the string magnetocylin-der they are unable to freeze into the cosmic plasmaand have any astrophysical effect.

We have shown that, regardless of the existenceof stable vortons, superconducting cosmic stringsthat are formed at the breaking of supersymmetry areable to generate the observed baryon asymmetry inthe Universe. We showed that, in contrast to the

w xclaim in 9 , a friction-scaling string network cancreate the required baryon number density evenwithout the production of stable vortons, for rather

natural values of the model parameters. This is dueto the fact that a friction-scaling network is muchdenser that a horizon-scaling one, producing substan-tially more string loops, whose decay eject sufficientbaryon charge when decaying to account for theobserved anisotropy.

In overall, the plasma-scaling solution of electri-cally charged current carrying superconductingstrings may result in a modified cosmic string cos-mology. Comparing this scenario with observationscould provide insight into the microphysics of stringsand the effect of cosmic string superconductivity.

Acknowledgements

We would to thank Nathalie Deruelle for discus-sions. This work was supported in part by PPARC.

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28 January 1999


Trace anomalies and the string-inspired definition ofquantum-mechanical path integrals in curved space

K. Schalm 1, P. van Nieuwenhuizen 2

Institute for Theoretical Physics, State UniÕersity of New York, Stony Brook, NY 11794-3840, USA

Received 16 November 1998Editor: L. Alvarez-Gaume

Abstract

We consider quantum-mechanical path integrals for non-linear sigma models on a circle defined by the string-inspiredmethod of Strassler, where one considers periodic quantum fluctuations about a center-of-mass coordinate. In this approachone finds incorrect answers for the local trace anomalies of the corresponding n-dimensional field theories in curved space.The quantum field theory approach to the quantum-mechanical path-integral, where fluctuations are not periodic but vanishat the endpoints, yields the correct answers. We explain these results by a detailed analysis of general coordinate invariancein both methods. Both approaches can be derived from the same operator expression and the integrated trace anomalies inboth schemes agree. In the string-inspired method the integrands are not invariant under general coordinate transformationsand one is therefore not permitted to use Riemann normal coordinates. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 04.62.qv

1. Introduction

Ž .One-dimensional quantum-mechanical path inte-Ž .grals in curved space non-linear sigma models have

w x w xbeen used to compute the chiral 1 and trace 2anomalies of n-dimensional quantum field theoriescoupled to external gravitation and Yang-Mills fields.In addition, quantum-mechanical path integrals inflat space have proven useful to calculate one- and

w xhigher-loop Green’s functions for field theories 3,4 .In both cases one considers the evaluation of thepartition function of the one-dimensional theory on a

w xfinite time interval yb ,b .1 2

1 E-mail address: [email protected] E-mail address: [email protected]

For the evaluation of anomalies the quantum-mechanical path integral has been defined, in anal-

Ž .ogy to quantum field theory QFT , by time slicingand requiring that all paths attain the same values at

Žyb and b for bosonic point particles or opposite1 2. w x 3values for fermionic point particles 5 . One de-

Ž . iŽ .composes the paths ‘‘fields’’ x t into a back-ground solution of the free theory satisfying theboundary conditions at yb and b and quantum1 2

deviations vanishing at these points. CompactifyingŽthe interval to a circle as seems natural for evaluat-

3 In the case of chiral anomalies, the fermionic point particlesattain the same values at y b , and b .1 2


( )K. Schalm, P. Õan NieuwenhuizenrPhysics Letters B 446 1999 247–255248

.ing the trace, which yields the partition function ,such paths will in general possess a kink-discontinu-ity because their derivatives need not be continuousat the point where the endpoints come together. Thekinetic operator in the space of functions whichvanish at the endpoints has no zero modes, and isreadily inverted. To compute the partition functionone finally integrates over all possible boundaryvalues.

On the other hand, for the evaluation of Green’sfunctions in flat space, a string-inspired approach to

w xthe path integral has been used 3,4 . Here the peri-odicity of the partition function is reflected in thefact that bosonic paths are decomposed into periodicfluctuations about a constant center-of-mass modeŽand fermionic paths are expanded into antiperiodic

.fluctuations, see footnote 1 . The propagator forthese deviations is constructed by inverting the ki-netic operator in the space of periodic functionsorthogonal to the constant modes, which are zeromodes of the one-dimensional Laplacian, and only atthe very last does one integrate over the center-of-mass coordinate.

In this letter we consider quantum-mechanicalpath integrals in curved space using the string-in-spired definition of path integrals; we will use thecalculation of trace anomalies to illustrate our points.Both definitions can be derived from the same opera-tor expression by time-slicing. At first sight onewould thus expect that this method should give thesame results as the QFT approach. In fact it has been

Ž .observed that for linear sigma models flat-space thetarget space Green’s functions in both methods differ

w xby total derivatives 4 and we have earlier shownwhy chiral anomalies are independent of the ap-

w xproach one uses 6 . We find that the results for localtrace anomalies differ, due to a subtlety having to do

Žwith general coordinate invariance Einstein invari-.ance in curved space. Namely, in the QFT approach

b ˆ² < Ž .the Riemann summands g x x exp y H( Ž .0 0 "

< : n Ž .x d x of the discretized partition function0 0b ˆŽ . Ž .Z b s Tr exp y H are Einstein invariant,"

whereas in the string-inspired approach the corre-Žsponding expressions are not Einstein invariant even

ˆ îf H is Einstein invariant, i.e., even if H commuteswith the generator of general coordinate transforma-

.tions . Consequently, one cannot simplify the calcu-lations in the string-inspired approach by using nor-

mal coordinates. Only the integrated expressions inthis approach are general coordinate invariant. In theQFT approach, on the other hand, one obtains di-rectly the correct local trace anomaly, and the invari-ance of the summands allows normal coordinates tobe used. Since both approaches are derived from thesame expression, this indicates that again the inte-grands computed with either method should differ bya total derivative. We present explicit two-loop andthree-loop calculations which support our arguments,and our conclusion is that the easiest way to computequantum-mechanical amplitudes in a curÕed space-time background is to follow the QFT approachrather than the string-inspired method.

2. Two approaches to the path integral

The basic object to compute is the partition func-²tion: the trace of the transition element z

b iˆ ˆ< Ž . < : Ž .exp y H y . Here H x , p is an arbitrary butˆ î"

fixed Hamiltonian operator with is1, . . . ,n. Forˆdefiniteness we shall choose an Einstein invariant H,

ˆwhich means that H commutes with the generatorˆ iŽ Ž ..G j x for general coordinate transformations withˆ

iŽ . w xparameters j x 7 . The Hamiltonian we considerîs given by

1 y1r4 i j 1r2 y1r4Hs g x p g x g x p g x ;Ž . Ž . Ž . Ž .ˆ ˆ ˆ ˆ ˆ î j2

g x sdet g x . 2.1Ž . Ž . Ž .ˆ î j

In the transition element one inserts Ny1 completesets of x-eigenstates and N-sets of p-eigenstates,ˆ ˆrewrites the Hamiltonian in Weyl-ordered form 4,replaces the x and p-operators by their c-numberˆ ˆ

4 w xIt has been shown in 9 that also in curved space one mayreplace the Weyl-ordered exponential by the exponential of theWeyl-ordered Hamiltonian, because the difference vanishes for

² Ž . Ž .:N™`. This is due to the fact that the propagators p t p ti 1 j 2j ˆ² Ž . Ž .:and p t q t are not singular in e ' brN. With H ini 1 2

Weyl-ordered form one may use the midpoint rule to evaluatearbitrary functions of the operators x and p in terms of theˆ ˆ

Ž .eigenvalues x and p. Weyl-ordering of 2.1 leads to well-knownextra noncovariant terms of order "

2 which are crucial for thegeneral coordinate invariance of the transition element.

( )K. Schalm, P. Õan NieuwenhuizenrPhysics Letters B 446 1999 247–255 249

eigenvalues and integrates over all p’s. The factorsŽ Ž ..det g x qx r2 produced by the integrationi j k ky1

over the p’s are exponentiated by anticommutingLee-Yang ghosts bi, ci and a commuting Lee-Yang

i w xghost a 2 . Since we are interested in taking traces,we set zsy.

From this point on the QFT approach and thestring-inspired approach go separate ways. In theQFT approach one expands the z i sx i , x i , . . . ,0 1

x i , x i sy i into a constant background part x i sNy1 N 0

z i sy i satisfying the free field equations of the free1 1 1i i jŽ . Ž . Žfield part of the action g x x yx x yi j 0 k ky1 k2 e e

1j i j. Ž . Ž .x ; g x x x where esbrN , and quan-˙ ˙ky1 i j 02

tum deviations qi which vanish at the endpointsk

ks0 and ksN. The q i are expanded into eigen-k

functions that satisfy these boundary conditions

Ny1 2 kmpi iq s r sin ; ks1, . . . , Ny1 ;(Ýk m ž /N Nms1

is1, . . . ,n . 2.2Ž .1 i i i iŽ . Ž .By coupling q qq and q yq re to ex-k ky1 k ky12

ternal sources, one obtains the discretized propaga-tors in closed form, and by taking the limit N™`

w x Žone reads off the correct Feynman rules 5 the ruleshow to evaluate integrals over products of distribu-

Ž . Ž .tions u syt and d syt as they appear in. Ž .Feynman graphs . The upshot is that d syt is to

be considered a Kronecker delta function and thatŽ . Žu 0 s1r2, so that for example we define tsbt

.for convenience

0 0ds dtd syt u syt u sytŽ . Ž . Ž .H H

y1 y1

0 01s s ds dtd syt u sytŽ . Ž .H H4y1 y1

=u tys . 2.3Ž . Ž .w xAfter a careful detailed analysis 5 one finds that

bˆ² < < :Tr z exp y H yž /"

1n yS r "int² :s d x g x e ,( Ž .H x0 0 0nr22pb "Ž .

2.4Ž .

where 5

1 1 0y S sy dt g x qqŽ .Hint i j 0

" 2b " y1

i j i j i jyg x q q qb c qa aŽ . ˙ ˙Ž .i j 0

b " 0y dt R x qqŽ .H 08 y1

i j n mqg G G x qq . 2.5Ž . Ž .im jn 0

n Ž .The integral Hd x g x on the r.h.s. of 2.4( Ž .0 0

comes from taking the trace and the factorŽ .yn r22pb " coincides with the Feynman measure.The terms with R and GG are created by Weyl

Ž .ordering Eq. 2.1 . The continuum limits of thepropagators to be used for the perturbative evalua-

Ž .tion of the last factor in Eq. 2.4 are

² i j : i j QFTq s q t syb "g x D s ,t ,Ž . Ž . Ž . Ž .x 00

² i j : i jb s c t sy2b "g x d sytŽ . Ž . Ž . Ž .x 00

² i j :sy2 a s a t 2.6Ž . Ž . Ž .x 0

with

DQFT s ,t ss tq1 u sytŽ . Ž . Ž .qt sq1 u tys . 2.7Ž . Ž . Ž .

² iŽ . jŽ .: v v Ž . ŽNote that q s q t ; D s ,t s1yd sy˙ ˙ QFT.t is singular for s™t but that this singularity

² i j: ² i j: ² i j:cancels in the sum q q q b c q a a . At˙ ˙higher loops the ai,bi,ci ghosts are crucial to remove

Ž .all ultraviolet syt™0 singularities and as a re-Žsult all integrals are finite as they should be in

quantum mechanics, despite the double derivative1 i jŽ . .interactions in g x x x . Since we work on the˙ ˙i j2

w x Ž w x .finite interval yb ,0 or y1,0 for ts trb thereare no infrared singularities.

The string-inspired approach to the partition func-tion starts from the same discretized expression ofthe transition element as an integral over the com-plete sets dn x , . . . ,dn x , but one then includes1 Ny1

the trace integration over x i sx i and treats all0 N

5 Our conventions for the Riemann and Ricci tensor are thatl l l m Ž . jR sE G q G G y il j and R s R .i jk i jk i m jk i k i jk


points x i , x i , . . . , x i on equal footing. One thus0 1 Ny1

considers the same object

bnˆ² < < :Tr z exp y H y s d x g x( Ž .H 0 0ž /"

=b

ˆ² < < :x exp y H x ,0 0ž /"

2.8Ž .

but now one expands the n=N integration variablesx i into periodic functions which are eigenfunctionsk

of the free action

Nr2 2 2kppi i ,cx s cos rÝk pž /' NNps0

Nr2y1 2 2kppi , sq sin r ;Ý pž /' NNps1

ks1, . . . , N . 2.9Ž .

Ž .We take N to be even. In the discretized expres-Ž .sion for 2.8 , only factors of the metric g ati j

Ž .midpoints occur, and no explicit g x survive. Wei j 0

decompose x i into a center of mass part x i sk ci,c i i'Ž .2r N r and fluctuations q around x . Cou-0 k c

pling the q i to the same external sources as before,kŽ .but substituting then 2.9 one finds after consider-

able tedious but straightforward algebra the propaga-tors for the string-inspired method in closed dis-

w xcretized form 6 . They contain discretized theta- anddelta-functions with the same properties as before; in

Ž .particular 2.3 holds again. The continuum propaga-Ž .tors are now given by 2.6 with

1SID s ,t s syt e sytŽ . Ž . Ž .2

21 1y syt y . 2.10Ž . Ž .2 12

and x replaced by x . As expected, this propagator0 c2 SIŽ .is translation invariant, and satisfies E D syt ss

Ž .d syt y1 which is the Dirac delta-function inŽ .the space of functions q t orthogonal to the con-

1 0 SIŽ .stant. The factor y ensures that H D s ,t dty112

² Ž . 0 Ž .s0, which should be satisfied as q s H q ty1:dt s0.

Ž .The expression for the trace in 2.8 is thenŽ .formally the same as obtained from 2.4 , namely

bn ˆ² < < :d x g x x exp y H x( Ž .H 0 0 0 0ž /"

1n yS r "int² :s d x g x e ,( Ž .H xc c cnr22pb "Ž .

2.11Ž .where the right-hand-side is now evaluated using

SIŽ . Ž .D s ,t . In particular S is the same as 2.5 butint

with x i replaced by x i . However, the integrand0 c1² Ž .:exp y S cannot be written as a matrix ele-xint" c

ment with well-defined position bras and kets; ratherŽ .it is a function which can be computed using 2.10

Ž .and 2.3 . In particular the x in g x is not a( Ž .c c

.true position like the x in g x , which is an( Ž0 0

eigenvalue of the x-operator. It is this fact which isˆof crucial importance for the issues of general co-variance. Starting with an Einstein invariant Hamil-tonian our final answer for the partition function willbe general coordinate invariant. In the QFT case the

b ˆ² < Ž . < :integrands z exp y H y and the tracing mea-"

nsure Hd x g x are separately Einstein invariant( Ž .0 0

by construction. In the string-inspired case there isno reason why the integrands on the right-hand-side

Ž .of 2.11 should be Einstein invariant, and explicitcalculations show that they are not Einstein invari-ant. Only the integrated expression for the partition

Žfunction is guaranteed to be covariant and is thesame in both approaches since in both cases one

.integrates over all n=N variables .We have now defined the path integral according

to the QFT method and the string-inspired method.Both have the same non-covariant order "

2 countert-erm needed for the Einstein invariance of the finalresults, and both have the same vertices because bothare based on time-slicing. The difference rests sole-ly in the order of the integration over dn x ,dn x ,1 2

. . . ,dn x ,dn x sdn x : in the QFT approach weNy1 N 0

first integrate over intermediate x i , . . . , x i using1 Ny1nŽ .2.7 and then integrate over g x d x , whereas( Ž .0 0

in the string-inspired method one first integrates overthe fluctuations about the center-of-mass x i sc

i nŽ .1rNÝ x using 2.10 , and then over g x d x .( Ž .k c c

Note that the number of eigenfunctions into whichquantum fluctuations are expanded is the same in

Žboth cases sines and cosines of the double angle vs.


.only sines of the single angle . However, if we viewthe integration region in both cases as a circle, wehave a kink in the paths in the first case but not inthe second case.

3. Comparison of approaches: trace anomalies inns2 and ns4 dimensions

The anomalies of n-dimensional quantum field-theories can be written as

bÂns lim Tr JJexp y RR , 3.1Ž .ž /"b™0

Ž . Ž .where JJ is the Jacobian Edf x rEf y of theŽ .fields f x transforming under the symmetry with

ˆŽ .df x and RR is a regulator. For consistent anoma-ˆ w xlies the form of RR is unique 10 and for local trace

Ž .Weyl anomalies for real scalar fields one finds thatˆ ˆ Ž .RR is equal to the Hamiltonian H in 2.1 with an

improvement term. The QFT approach then yieldsw x2

b1 Ân s lim Tr 2yn s x exp y H 3.2Ž . Ž . Ž .ˆW 4 ž /"b™0

Ž .where s x is the Weyl parameter, normalized toŽ . Ž .d g ss x g x andW i j i j

1 y1r4 1r2 i j y1r4Hs g x p g x g x p g xŽ . Ž . Ž . Ž .ˆ ˆ ˆ ˆ ˆ î j2

1 2y " j R x . 3.3Ž . Ž .ˆ2

Ž .The term with R x is the improvement term andˆ1 Ž . Ž . Ž .js ny2 r ny1 . The trace in 3.2 can be4

written as a path integral. For the QFT approach onefinds then that the local trace anomaly is propor-

1n Ž .² Ž .:tional to Hd x g x s x exp y S .( Ž . x0 0 0 int" 0

Naively, one might expect a similar expressionfor the trace anomaly in the string-inspired method,

1n Ž .² Žproportional to Hd x g x s x exp y( Ž .c c c "

.:S . One should, however, take care in definingxint c

the expectation value of a local operator in thestring-inspired approach. The naive equivalence withthe QFT approach by taking the local operator at thepoint x is incorrect since at a discretized intermedi-c

ate stageN1 1

s x exp y SŽ .ÝH k ž /N "ks1

N1 1/ s x s x exp y S 3.4Ž .ÝH c k ž /ž /N "ks1

n N iwhere H denotes Ł Ł H g x dx and S is( Ž .is1 ks1 k k

the discretized action. Using cyclicity of the traceŽ . Ž .3.2 , leads to the left-hand-side of 3.4 , not to theright-hand-side. Unambiguous is the global trace

Ž .anomaly where s x is a constant. In that case theânomaly is just proportional to the partition functionand this is the case we shall consider.

ŽFor ns2 we need all connected and discon-. 6nected two-loop graphs on the worldline . If one

uses normal coordinates, only graphs with the topol-ogy of the number 8 and the counterterm contribute.The local anomaly is then proportional to

² yS int r ":e x , ns20

b " 0v vsy R x dt D t ,t D t ,tŽ . Ž . Ž .�H

6 y1

b "v vqd 0 y D t ,t D t ,t y R x .Ž . Ž . Ž . Ž .4

83.5Ž .

Ž .The singularities with d 0 are well-defined in thediscretized expressions and cancel in both ap-proaches. Following the QFT approach, and substi-

Ž .tuting 2.7 in the above, one obtains the correct1answer y R. In the string-inspired method, substi-12

Ž .tuting 2.10 , the second term vanishes due to thev 1SIŽ .fact that D t ,t s0 and the final result is y R,9

which is incorrect.The results for the calculation in general coordi-

nates are summarized in Fig. 1, where we intro-duce the notation E 2 g sg i jg k lE E g , E iE jg sk l i j i j

g ik g jlE E g , E gsg i jE g , g sg inE g , g mk l i j m m i j m i m n

m n Ž .2 i p jq k rsg g , E g sg g g E g E g and simi-n i jk i jk p qr

larly for E g E g . For example in the first line ini jk j i k

Fig. 1 we find the contribution due to expanding1 11 k l i j i j i jw Ž . � 4exp y H q qE E g x q q qb c qa a˙ ˙y k l i j 02b " 2

x ² i jdt to first order and taking the contractions q q q˙ ˙i j i j: ² k l:b c qa a and q q . In the QFT approach one

b " 12 Ž .finds y E g y while in the string approach4 6b " 12 Ž .one obtains y E g y . In the last line one4 12

finds the contribution from the order "2 terms in the

action which were produced by Weyl ordering.

6 nŽ .The b independent terms in 2.4 come from 1q loops2

Žone-loop graphs are independent of b , two-loop graphs are.proportional to b , etc. .


Fig. 1. The results for the trace anomaly for a real scalar field in ns2 dimensions. Dotted lines denote ghosts. For each graph, the result inŽ .either the QFT or string-inspired SI approach is obtained by taking the product of the tensor structure and the number in the appropriate

v Ž .column. The tensor structure is the same in both cases as the vertices are the same. Because D t ,t s0 in the string-inspired approach,SI

there are fewer nonvanishing contributions than in the QFT approach. However, the sum of all contributions is general coordinate invariantin the QFT but not in the SI approach.

²Adding all contributions, we find for exp1Ž .: Žy S proportional to the local Weyl anomalyxint" 0

.of a real scalar field in ns2 dimensions in theQFT approach an Einstein invariant result, which ofcourse equals that of the normal coordinate calcula-tion.

1y S int² :e " x , ns20

1 1 1 12 i js y E g y q y E E gŽ . Ž .Ž . Ž .i j4 6 2 12

1 1mq y E gE g yŽ . Ž .m8 12

1 1mq y E ggŽ . Ž .m2 12

1 1mq y g g yŽ . Ž .m2 12

21 1q y E gŽ . Ž .ž /i jk4 4

1 1q y E g E g yŽ .Ž .i jk j i k2 6

1 1 1i j l kq y Ry g G G sy R . 3.6Ž .Ž .i k jl8 8 12

ŽWe have written each term as a product of a tensorstructure times the result of integrations over expres-

Ž .sions depending on D s ,t . Using normal coordi-nates one need only evaluate the first, second and

.last graph in Fig. 1. However, in the string-inspiredapproach we find a noncovariant result

1y S int² :e " x , ns2c

21 1 1 12s y E g y q y E gŽ .Ž . Ž .Ž . ž /i jk4 12 4 6

1 1q y E g E g yŽ .Ž .i jk j i k2 12

1 1 i j l kq y Ry g G G . 3.7Ž .Ž .i k jl8 8

Hence the integrands in both methods differ. Yet theŽ . Ž .integrated expressions 3.6 and 3.7 should be the

same, as we explained in the previous section. Bynintegrating these expressions with Hd x g x and( Ž .0 0

nHd x g x , respectively, and using partial inte-( Ž .c c

gration we may replace terms with double deriva-tives of the metric by products of terms with single


derivatives. This yields under the integral sign the‘‘equalities’’

212 m m' 'g E gm g y E gE gq E g qg E g ,Ž .ž /m i jk m2

1i j m m' 'g E E g m g y E gg qE g E g qg g .Ž .i j m i jk j i k m2

3.8Ž .

One then indeed finds that the integrated Weylanomalies agree

yS r "int² :dx g x e( Ž .H x , ns20 0 0

yS r "int² :s dx g x e . 3.9( Ž . Ž .H x , ns2c c c

The case ns2 is rather special as the answer isŽns2.'proportional to g R which is itself a total

derivative. Let us therefore consider the traceanomaly for a real scalar field in ns4 dimensionsand compute it using normal coordinates. This willexplicitly demonstrate that one gets incorrect resultsfrom the string-inspired method. As we explainedbefore, this is due to the fact that one can onlychoose normal coordinates at one point, but since the

Ž .integrands functions of x are not Einstein scalars,c

one cannot use normal coordinates at each point x .cŽ .In ns4 the improvement term in 3.3 is nonvan-

"2 Ž .ishing and effectively converts the term R in 2.58

"2

into R. The action yields the following vertices in24

normal coordinates

1 1 0 1 k ly S sy dt y R q qHint i k l j3" 2b " y1

1 1m k ly D R q q q y D D R�m ik l j m n ik l j6 20

2 t m n k ly R R q q q q4i k tm jl n45

i j i j i jq . . . q q qb c qa a˙ ˙Ž .0 1yb " dt R x qqŽ .ŽH 024

y1

1 i j l kq g G G x qq . 3.10Ž . Ž ..i k jl 08

ŽThe b-independent term which yields the trace.anomaly now arises from three-loop diagrams. Since

there are no three-point vertices in normal coordi-nates, the five-point vertices do not contribute. Thefour- and six-point vertices yield the graphs in Fig.2. In the one-but-last line of Fig. 2 one finds the

12 Ž .contribution from the order " vertex R x qq q0241 i j l kŽ .g G G x qq due to expanding it to secondi k jl 08

order in q i and contracting the two q i’s to an equaltime loop. Note that it would have been incorrect toomit the GG term altogether by arguing that it willnot contribute in normal coordinates. In the last lineone finds the contribution from the disconnected

Fig. 2. The results for the trace anomaly for a real scalar field in ns4 dimensions, calculated in Riemann normal coordinates. Only theŽ .2 Ž .topology of the graphs is indicated and all terms should be multiplied by an overall factor b " r72 to obtain the results in 3.11 and

Ž . v Ž .3.12 . In the string-inspired approach the fact that D t ,t s0 again leads to great simplifications but in this approach the integrands areSI

not Einstein invariant and the use of normal coordinates is therefore illegal.


graphs; they are proportional to the square of thens2 result, but with the improvement term theircontribution vanishes in the QFT approach. This iscoincidental as for the trace anomaly in ns6 di-mensions the contribution from disconnected graphs

Ždoes not cancel. It is given by the products of lower1Ždimensional trace anomalies, namely y R12

1 1 1 1 1 12 2 2. Ž . Žq R R yR y D R q y Rqm n p q m n10 720 5 6 12 101 13 2.R where the terms R and y D R come from10 5

.the improvement term.Adding all contributions, we obtain from the QFT

approach1

y S int² :e " x , ns40

2b "Ž .

1 3 12 2 2s y R q D Rq RŽ . Žm n m n p q6 20 10721 1 12 2 2q R q y R q y D R. Ž . Žm n m n p q15 4 4

2b "Ž .

1 2 2 2 2q R s R yR yD R.m n p q m n p q m n4 7203.11Ž .

w xwhich is the correct result 8 . In the string inspiredŽ .result assuming incorrectly that one may use nor-

mal coordinates in the integrands, we find instead1

y S int² :e " x , ns4c

2b "Ž .

7 12 2s y R q D RŽ . Žm n180 40721 1 12 2 2q R q R q y R. Ž .m n p q m n m n p q60 90 60

1 1 12 2 2q y D Rq R q RŽ .m n p q8 8 36

2b "Ž .

1 1 1 12 2 2 2s R y R y D Rq R .m n p q m n8 36 10 36723.12Ž .

The D2R terms come out the same, for which wehave no explanation at hand, but the other terms aredifferent. This demonstrates that using normal coor-dinates in the string-inspired approach yields incor-rect results for the local trace anomaly.

4. Conclusion

We have considered path integrals of quantum-mechanical non-linear sigma models. The operatorexpression for the partition function can be evaluated

Žin two different ways: the QFT approach where.paths vanish at the endpoints and the string-inspired

Ž .approach where paths are manifestly periodic . Atthe discretized level, where all expressions are well-defined, the difference is that in the QFT approachone first integrates over the intermediate points q sk

x yx for ks1, . . . , N and then over the endpointk 0

x sx , while in the string-inspired approach one0 N

first integrates over the deviations q sx yx andk k c1then over the center-of-mass x s Ý x . As thec kN

starting point is the same expression for both pathintegrals they should yield the same answers.

As a test we considered local trace anomalies,b ˆŽ . Ž .which are proportional to Tr s x exp y RR withˆ "

ˆ ˆRR a regulator equal to the Hamiltonian H with animprovement term. This yields the correct local traceanomalies using the QFT approach. The string-in-spired approach to path integrals does not reproduce

nthe correct local trace anomaly from Hd x g x( Ž .c c1Ž .² Ž .:s x exp y S . We understood this by go-xc int" c

Ž .ing back to the discretized level in which case s xc1 1Ž . Ž .equals s Ý x and not Ýs x . Interpretingk kN N

1n Ž .² Ž .:Hd x g x s x exp y S as a regular-( Ž . xc c c int" c

Ž .ization scheme for Tr s x , our result shows thatˆŽdifferent regularization schemes for example the.QFT approach and the SI approach lead to different

local trace anomalies. Since trace anomalies are nottopological, this result could have been expected.

Ž .The integrated or global anomaly, for which s xis constant, is proportional to the partition function.The answers for this case are the same, as theyshould, but a second subtlety was discovered in theprocess. Namely, the QFT approach has the addi-

²tional benefit that the transition element x0b ˆ< Ž . < :exp y H x is itself general covariant and one0"

may use normal coordinates in evaluating Feynmandiagrams. The string-inspired approach uses thecyclic symmetry of the partition function, whichcauses many diagrams to vanish trivially but the

1² Ž .:integrand exp y S is not covariant and thexint" c

benefits of cyclicity are offset by the fact that one isnot allowed to use normal coordinates.

Our final conclusions are twofold. First, one can-not use the string-inspired path integral to constructregulators for local trace anomalies which keep Ein-stein invariance at the quantum level. Second, onecan use the string-inspired method to evaluate parti-tion functions, but one cannot use normal coordi-


nates and the integrands differ from those in the QFTapproach by total derivatives.

Acknowledgements

The authors thank the organizers of the 1998summer workshop on String Theory and Black Holesat Amsterdam for their hospitality and discussions.

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28 January 1999


ž /2Neutrino masses and mixings from an anomaly free SMG=U 1model

C.D. Froggatt a,1, M. Gibson a,2, H.B. Nielsen b,3

a Department of Physics and Astronomy, UniÕersity of Glasgow, Glasgow G12 8QQ, UKb The Niels Bohr Institute, BlegdamsÕej 17-21, DK-2100 Copenhagen Ø, Denmark

Received 12 November 1998; revised 29 November 1998Editor: P.V. Landshoff

Abstract

A natural solution to the fermion mass hierarchy problem suggests the existence of a partially conserved chiral symmetry.We show that this can lead to a reasonably natural solution to the solar and atmospheric neutrino problems withoutfine-tuning or the addition of new low energy fermions. The atmospheric neutrino atmospheric neutrino anomaly is given bylarge mixing between n and n , with Dm2 ;10y3 eV2, and the solar neutrino deficit is due to nearly maximal electronm t atm

neutrino vacuum oscillations. We present an explicit model for the neutrino masses which is an anomaly free Abelianextension of the standard model that also yields a realistic charged fermion spectrum. q 1999 Elsevier Science B.V. Allrights reserved.

PACS: 14.60.Pq; 12.60.y i

1. Introduction

The observed hierarchy of charged fermion masses and quark mixing angles strongly suggests the existencew x Ž . w xof an approximate chiral flavour symmetry 1 beyond the standard model SM . In a previous paper 2 we

discussed the implications of such a symmetry for neutrino masses and mixings. We showed that the mostnatural scenario would correspond to nearly maximal mixing between n and n being responsible for both thee m

solar and atmospheric neutrino problems. However, the recent data on the atmospheric neutrino zenith anglew xdependence from Super-Kamiokande 3 indicate that this solution no longer gives an acceptable fit to the

atmospheric neutrino data. In this paper we show that approximately conserved chiral symmetries can still leadto a reasonably natural solution to the solar and atmospheric neutrino problems, if we relax the assumptions we

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]


( )C.D. Froggatt et al.rPhysics Letters B 446 1999 256–266 257

w xmade in 2 . We shall also present an explicit model for the neutrino masses and mixings, in which the chiralflavour symmetry comes from an Abelian extension of the standard model gauge group.

Previously we made two assumptions for the models with approximately conserved chiral symmetries:i. The low energy fermion spectrum of the model is the same as in the standard model – in particular we have

only three left-handed neutrinos.ii. The chiral symmetries lead to elements of the effective light neutrino mass matrix M which are of differentn

orders of magnitude, apart from those elements which are equal due to the symmetry M sM T.n n

w xAs we discussed in our earlier paper 2 , the only natural solution to the solar and atmospheric neutrinoproblems with these assumptions is if we have nearly maximal n yn mixing, and small mixing with n . Thise m t

no longer gives a good description of the atmospheric neutrino data. We cannot obtain any other types ofŽ . Ž .solution as a direct consequence of the assumptions i and ii .

Ž .Assumption i implies that we must have a 3=3 symmetric effective Majorana-like neutrino mass matrix.With a hierarchy between the elements of a symmetric mass matrix there are essentially two different forms forthe matrix depending on whether or not the diagonal elements dominate all of the eigenvalues. The first caseleads to small mixing between all three neutrinos, and this is unsuitable for a solution to the atmosphericneutrino problem. The second case gives large mixing between two nearly degenerate neutrinos, and small

Ž .mixing with the third non-degenerate neutrino. Since we have only three neutrinos we have two independentŽ 2 . 2mass-squared differences Dm for the neutrinos. The smaller of these Dm s determines the wavelength ofi j

oscillation for the two largely mixed neutrinos, which we must take to be n and n with Dm2 ;10y3 eV 2m t 23

Ž 2 2 y3 2 .and consequently the other mass-squared differences Dm ;Dm )10 eV if we wish to explain all of12 13

the data on the atmospheric neutrino problem. However, we cannot also explain the solar neutrino problem,since the electron neutrino is then only slightly mixed and the small angle MSW solution requires Dm2 ;

10y5 eV 2. Hence we see that it is necessary to relax our assumptions.The first assumption was made because of the desire for minimality in our theory. We do not wish to

introduce extra low energy fermions unless it is absolutely necessary, and consequently we will retainŽ . Ž .assumption i in this paper. The second assumption is often satisfied in models with chiral gauged symmetry

breaking; however, it is not uncommon to find two order of magnitude equal elements in the mass matrices.Ž Ž . w x.Indeed in the explicit model based on the anti-grand unified model AGUT , 4,5 from our previous paper we

Ž . Ž .found that the 1, 1 and 2, 2 elements of the neutrino mass matrix were approximately equal, although in thatcase this did not have any effect on the phenomenology. Hence, in this paper we shall relax the secondassumption and consider the case where there are two order of magnitude equal elements in our mass matrixŽ .other than those elements which are exactly equal due to the symmetry of the mass matrix . We do not expectthese elements to be exactly equal, since that would generally require fine-tuning which we are careful to avoid.

In the next section we discuss the structure of the neutrino mass matrix we would expect to have for naturalŽmodels of this type, and the phenomenology of the neutrino oscillations. We will show that with no

. Žfine-tuning we would typically obtain nearly maximal n vacuum oscillations with a linear combination ofe.n yn for the solar neutrinos, and large n yn oscillations for atmospheric neutrinos. We would expect tom t m t

w xsee nothing at LSND, much of the parameter space for which has already been ruled out by Karmen 6 , andw xBugey 7 .

w xWhilst there are numerous examples of models 4,5,8–10 which explain the fermion spectrum using globalŽ . Ž .U 1 symmetries, or which cancel gauged U 1 anomalies using the supersymmetric Green-Schwarz mecha-

w xnism, it seems to have become a common belief 11,12 that it is not possible to construct an anomaly freegauged Abelian extension of the SM which yields a realistic fermion mass spectrum. We present here an explicit

Ž .2 Ž . Žanomaly free model with gauge group SMG=U 1 where SMG is the SM gauge group , which with a.non-minimal Higgs field spectrum fits the charged fermion mass spectrum and yields solutions to the solar and

atmospheric neutrino problems. The charged fermion mass spectrum in this model is identical to that predictedby the AGUT model. However, the neutrino mass spectrum is considerably different from that given by theAGUT, and we show in Section 4 that it can yield neutrino masses of the form suggested in Section 2. In order

( )C.D. Froggatt et al.rPhysics Letters B 446 1999 256–266258

Ž .to obtain the required neutrino spectrum, it is necessary to introduce an SU 2 triplet Higgs field with a suitablevacuum expectation value. We also discuss some difficulty in naturally obtaining such a vacuum expectationvalue for this Higgs field from the scalar potential.

2. Neutrino phenomenology

In this section we shall examine the possible structures of the effective 3=3 light neutrino mass matrix,which can arise in models with approximately conserved chiral symmetries in a reasonably natural way. In thefollowing discussion we shall use the convention that

2 < 2 2 <Dm s m ym , 1Ž .i j n ni j

Dm2 -Dm2 , 2Ž .12 23

where n is the ith neutrino mass eigenstate. We then require Dm2 ;10y3 eV 2 and large n yn mixing fori 23 m t

the atmospheric neutrinos. We can have several types of solution to the solar neutrino problem, such as the wellknown MSW and ‘just-so’ solutions to the solar neutrino problem with

Dm2 ;10y5 , 10y10 eV 2 3Ž .solar

respectively. There is also some variation in the solar neutrino fluxes predicted by different solar models andthis theoretical uncertainty means that it is also possible to have an ‘energy-independent’ vacuum oscillation

w x 2solution to the solar neutrino problem 13 . By ‘energy-independent’ we mean that Dm is sufficiently largesolar

that many oscillation lengths lie between the sun and the earth, and what we observe is the averaged fluxsuppression which is the same for solar neutrinos of all energies. Hence we can have

10y10 QDm2 sDm2 Q10y4 eV 2 , 4Ž .12 solar

where the upper limit comes from the constraint that electron neutrino mixing does not make a largecontribution to the atmospheric neutrinos. This type of solution does not agree well with the solar neutrino data

Žif we take both the experimental and theoretical solar neutrino rates at face value. The Bahcall-PinsonneaultŽ . w x .BP98 model 14 rules out this possibility at 99% C.L. However, we note that there is still some freedomallowed in the choice of solar model.

w xThe analysis of 13 examines the possibility of having an energy-independent solution if the true solar modellies somewhere within the range of currently allowed solar models. Taking the energy-independent flux

Ž .suppression F as a free parameter they find

Fs0.50"0.06 5Ž .2 Ž .with a minimum x of 8. If Fs0.5 is not a free parameter as in our model below then this corresponds to a

confidence level of 5%. Even if the BP98 solar model is correct, the requirement for an energy-dependentŽ .solution to the solar neutrino problem rests essentially on only one experiment the Chlorine experiment. Given

the possibility of unknown systematic errors we would prefer to avoid relying too strongly on the result of anysingle experiment. Hence, whilst the MSW and ‘just-so’ solutions to the solar neutrino problem are empirically

Žfavoured we still consider the simpler energy-independent solution with maximal mixing between two.neutrinos to be a viable solution. The amount of mixing will be large for the vacuum oscillation solutions, and

may be either large or small for the MSW solutions.ŽAs we saw in our previous paper if we have a completely hierarchical mass matrix with all independent

.elements of different orders of magnitude , the only solution to the solar and atmospheric neutrino problems isto have nearly maximal n yn mixing responsible for both, which seems to be no longer compatible with thee m

atmospheric neutrino data. Hence we shall now look at the possible mass matrices with order of magnitudedegeneracies between the elements. One possibility would be to have an order of magnitude degeneracy in the


charged lepton mass matrix, leading to large mixing coming from the charged sector. It has been shownw xelsewhere in the literature 15–17 that this can yield an acceptable phenomenology, and we do not consider it

further here. So we now consider order of magnitude equal elements in the neutrino mass matrix. There areessentially three types of matrix which could potentially yield an acceptable phenomenology with a smallnumber of approximately equal elements,

I II IIIA = = = = = = A B 6Ž .= = A = A B A = =ž / ž / ž /= A = = B C B = =

where = denotes small elements and in each case A;B;C. We shall call these textures I, II and IIIrespectively.

From the form of texture I we see that this texture would require the imposition of an exact flavour symmetryŽ . Ž .relating M to M , for which we have no good reason. Hence we will not use texture I. In order to haven 11 n 23

a good phenomenology, type II would require AC;B2, which is not unlikely to occur by chance. However, italso requires three order of magnitude equal elements in the neutrino mass matrix, which we do not considerlikely in most models with approximately conserved chiral symmetries. Nevertheless, it has been obtained in a

w xsupersymmetric extension of the standard model with approximately conserved gauged chiral symmetries 10 .Type III has only two approximately equal elements and, as we shall see in Section 4, can occur reasonably

w xnaturally in a specific model. In fact type III has previously been considered in the literature in 18 , where thestructure of the mass matrix is assumed to be due to a global L yL yL symmetry. The fine-tuned casee m t

w xwhere BsA corresponds to the popular ‘bi-maximal mixing’ solution to the neutrino problems 19,20 . All ofŽ . w x w xthe textures I, II, and III examined here have previously been discussed in 21 by three of the authors of 18 .

However, they claim there that flavour symmetries which lead to textures II and III also yield large mixing fromthe charged lepton mass matrix. We do not find this to be the case here.

The mass matrix texture of type III has the eigenvalues:

2 2'" A qB ,0 7Ž .and can be diagonalised by the mixing matrix:

1 1y 0' '2 21 0 0

U ; 81 1 Ž .0 cosu ysinun ž / 00 sinu cosu ' '2 2� 00 0 1

1 1° ¶y 0' '2 2

1 1cosu cosu ysinus 9Ž .' '2 2

1 1sinu sinu cosu¢ ß' '2 2

where

Btanus . 10Ž .

A


Ž .From the first row of Eq. 9 we can see that n is maximally mixed between n and n , so that its mixing doese 1 2w xnot contribute to the atmospheric neutrino anomaly, and there will be no effect observable at Chooz 22 since

we take Dm2 -10y4 eV 2. The atmospheric neutrino anomaly will be entirely due to large n yn mixing and,12 m t2 Ž .in order that the mixing be large enough, we need sin 2uR0.7 95% C.L which requires

B0.56Q Q1.8. 11Ž .

A

So although A and B must be order of magnitude degenerate, it is not necessary to do any fine tuning. Thesolar neutrino problem is explained by vacuum oscillations, although whether it is an ‘energy-independent’ or a‘just-so’ solution will depend on the small elements which we have neglected. It is not entirely clear which ofthese types of solution will be more likely to occur in models with chiral symmetry breaking. We note howeverthat the elements of M which contribute to the Dm2 have to be about 8 orders of magnitude smaller than then 12

large elements A and B for the ‘just-so’ solution. The solar neutrino problem cannot be explained in this modelby an MSW type solution, since the mixing of the electron neutrino is too large for this type of solution.

( )23. Constructing an anomaly free SMG=U 1 model

We now introduce an anomaly free Abelian extension of the SM which we shall use in the next section toobtain a neutrino mass spectrum of the form we have just discussed. This extension has the gauge group

SMG=U 1 =U 1 12Ž . Ž . Ž .f 1 f 2

Ž . Ž .and we have only the standard model fermion spectrum at low energies. We shall break U 1 and U 1 withf 1 f 2

a non-minimal set of three Higgs fields, which are required to give a realistic charged fermion spectrum andŽ . Ž .which leave the SMG unbroken. The SMG will be broken down to SU 3 =U 1 by the usual Weinberg-Salam

Ž . Ž .Higgs field, although this will now also carry charges under U 1 and U 1 . We shall also introduce af 1 f 2

further Higgs field to generate a realistic spectrum of neutrino masses in the next section.Ž . Ž .The fermions will each have different charges under the chiral symmetries U 1 and U 1 , which willf 1 f 2

prevent most of them from acquiring masses by a direct Yukawa coupling with the Weinberg-Salam Higgs field.Ž . Ž .However, after the spontaneous breaking of U 1 and U 1 at some high mass scale M , the chargedf 1 f 2 F

fermions will all acquire effective mass terms in the low-energy effective theory via diagrams such as Fig. 1.Ž .The intermediate states are taken to be vector-like fermions of mass MsO M , and we assume that theF

Ž .fundamental couplings are O 1 . Fig. 1 then gives an effective mass to the bottom quark,

² : ² :2W u² :m ; f , 13Ž .b WS 2M MF F

² : ² :where W , u are the vacuum expectation values of Higgs fields W and u used to spontaneously break theŽ .2SMG=U 1 down to the standard model. The other charged fermions acquire their mass via similar diagrams.

Fig. 1. Feynman diagram for bottom quark mass in the full theory. The crosses indicate the couplings of the Higgs fields to the vacuum.


Table 1An anomaly free choice of Abelian charges for the fermion fields

Ž .1st. gen. c t c s t b m t m tL L R R R R L L R R

Q 0 0 1 4 0 0 y2 0 y3 0 y6f 1

Q 0 y1 0 y1 1 y3 1 3 0 5 1f 2

As we discussed earlier we do not wish to extend the low-energy fermion spectrum for reasons ofminimality, so we have the usual SM fermion spectrum with their usual representations under SMG. The

Ž . Ž .fermion charges under U 1 and U 1 are then severely constrained by the requirement that all the anomaliesf 1 f 2Ž . Ž . Ž .involving them cancel. If we denote the charges of the fermions under U 1 and U 1 by Q u suf 1 f 2 f i L Li

Ž .is1,2 etc., then the anomaly constraints are given by:2Tr SU 3 U 1 s2 u qc q t y u qd qs qc q t qb s0,Ž . Ž . Ž . Ž .f i Li Li Li R i R i R i R i R i R i

2Tr SU 2 U 1 s3 u qc q t qe qm qt s0,Ž . Ž . Ž .f i L i Li Li Li Li Li

2Tr U 1 U 1 su qc q t y8 u qc q t y2 d qs qb q3 e qm qtŽ . Ž . Ž . Ž . Ž .f iY Li Li Li R i R i R i R i R i R i Li Li Li

y6 e qm qt s0,Ž .R i R i R i

2 2 2 2 2 2 2 2 2 2 2 2 2Tr U 1 U 1 su qc q t y2 u qc q t qd qs qb y e qm qtŽ . Ž . Ž . Ž .f iY Li Li Li R i R i R i R i R i R i Li Li Li

qe2 qm2 qt 2 s0,R i R i R i

Tr U 1 U 1 U 1 s6 u u u qc c c q t t t y3 d d d qs s sŽ . Ž . Ž . Ž . Žf i f j f k Li L j Lk Li L j Lk Li L j Lk R i R j R k R i R j R k

qb b b qu u u qc c c q t t t q2 e e eŽ.R i R j R k R i R j R k R i R j R k R i R j R k Li L j Lk

qm m m qt t t y e e e qm m m qt t t s0,Ž ..Li L j Lk Li L j Lk R i R j R k R i R j R k R i R j R k

2Tr graviton U 1 s6 u qc q t y3 u qd qs qc q t qb q2 e qm qtŽ . Ž . Ž . Ž . Ž .f i L i Li Li R i R i R i R i R i R i Li Li Li

y e qm qt s0. 14Ž . Ž .R i R i R i

Ž .A possible choice of charges which is based on the AGUT Abelian charges satisfying these constraints isgiven in Table 1 and, as we shall see, a realistic charged fermion mass spectrum can be obtained for thesecharges by making a suitable choice of Higgs fields. The set of charges in Table 1 is not the only one which is

Ž .anomaly free. For example, the AGUT has four U 1 s, with linearly independent sets of charges which satisfyŽ . w x Ž .the anomaly constraints of Eq. 14 . In the AGUT 5 one of these U 1 s is broken before the others at the

Ž .Planck scale, leaving three unbroken U 1 generators. In this paper we choose the fermion charges to be a linearŽcombination of the charges under these unbroken generators. Our choice of charges is given by Q sy qy qY 1 2

w x.y ,Q s3 y and Q sy3 y qQ where y and Q are the AGUT fermion charges of Ref. 5 . We could3 f 1 3 f 2 2 f 1,2,3 fŽ .alternatively have chosen to use the charges under the broken U 1 for Q or Q ; however, we are unaware off 1 f 2

Ž Ž . . Ž .any choice of charges with only two non-standard model U 1 s involving this broken U 1 which yields arealistic charged fermion spectrum.

Table 2Higgs field charges which have been chosen to give a realistic charged fermion spectrum, and the vacuum expectation values for the chiralsymmetry breaking Higgs fields in units of the fundamental scale MF

yr2 Q Q Vacuum expectation valuef 1 f 2

1f y1 y3WS 2

5W 0 3 0.15831 1

u 0 0.2662 6

j 0 0 1 0.099


The Weinberg-Salam Higgs field, f , charges are chosen so that the top quark obtains its mass directlyWS

from its Yukawa coupling with f , and M is thus unsuppressed. The other fermions cannot couple directly toWS t

f since such couplings are protected by the chiral symmetries. Hence we introduce three other Higgs fieldsWSŽ . Ž .W, j and u to break the U 1 and U 1 with charges and vacuum expectation values chosen to give af 1 f 2

realistic fermion spectrum. The charges and vacuum expectation values of the Higgs fields are given in Table 2.We take the Higgs fields at the fundamental scale to be singlets under the standard model symmetries. Thecharged fermion effective SM Yukawa matrices are then given by

² :² :4² :2 ² :2² :2² : ² :² :4² :W u j W u j W u j

4 3 2 2 4² :² : ² : ² : ² : ² :² :H ; , 15Ž .W u j W u W uU � 03 2² : ² :² :j W u 1

² :² :4² :2 ² :² :4² : ² :6² :W u j W u j u j

4 4 6² :² : ² : ² :² : ² :H ; , 16Ž .W u j W u uD � 02 8 2 8 2² : ² : ² : ² : ² : ² :² :W u j W u W u

² :² :4² :2 ² :² :4² :3 ² :² :8² :W u j W u j W u j

4 5 4 8 2² :² : ² : ² :² : ² :² : ² :H ; , 17Ž .W u j W u W u jE � 010 3 2 8 2² :² : ² : ² : ² : ² :² :W u j W u W u

² : ² : ² :where the Higgs field vacuum expectation values W , j and u are in units of the fundamental scale, M .F

These mass matrices yield exactly the same masses and mixings at the fundamental scale as we obtained in thew x 2AGUT model in previous papers 5 , as can be seen by substituting the Higgs field combination u in this paper

Žby the Higgs field T in the AGUT, and relabelling the c and t fields. This is because after this trivialR R.relabelling of fermion fields the charges on the fermion fields are the same as a linear combination of the

Ž .remaining Abelian fermion charges in the AGUT after one of the AGUT U 1 ’s is spontaneously broken. TheŽ .2choice of Higgs fields in the SMG=U 1 model is however different and, whilst this leads to the sameŽ w x.charged fermion spectrum as in the AGUT see Table 3 for the best fit spectrum from 5 , it does not yield the

Žsame neutrino spectrum. The AGUT cannot produce the same neutrino mass matrix structure without.increasing the number of Higgs fields , since it is not possible to choose a consistent set of non-Abelian

Table 3Best fit to conventional experimental data. All masses are running masses at 1 GeV except the top quark mass which is the pole mass

Fitted Experimental

m 3.6 MeV 4 MeVu

m 7.0 MeV 9 MeVd

m 0.87 MeV 0.5 MeVe

m 1.02 GeV 1.4 GeVc

m 400 MeV 200 MeVs

m 88 MeV 105 MeVm

M 192 GeV 180 GeVt

m 8.3 GeV 6.3 GeVb

m 1.27 GeV 1.78 GeVt

V 0.18 0.22u s

V 0.018 0.041cb

V 0.0039 0.0035u b


Ž .2representations for the Higgs fields. We shall see however that, within the SMG=U 1 model, we can obtainan acceptable neutrino spectrum.

4. Neutrino masses and mixings from an explicit model

Neutrino masses can be generated in this model by the Weinberg-Salam Higgs field, via a see-saw likemechanism, giving a dominant off-diagonal element in the neutrino mass matrix,

² :2² :8² :4 ² :2² :8² : ² :2² :² :3W u j W u j W u j2f WS 2 8 10 2² : ² : ² : ² :² : ² : ² :M ; . 18Ž .W u j W u W un MF � 02 3 2 2 2 2² : ² :² : ² : ² : ² : ² : ² :W u j W u W u j

This yields nearly maximal n yn mixing between a nearly degenerate pair of neutrinos. As we discussedm t

earlier this does not lead to an acceptable phenomenology, and hence we require a different mechanism togenerate the dominant contribution to the neutrino masses and mixings. We do this here by introducing an

Ž . Ž . Ž .SU 2 triplet Higgs field D. The charges on this Higgs field are then chosen so that the 1, 2 and 1, 3elements of M are suppressed by equal amounts, givingn

y3 3,Q ,Q s 1, ,y . 19Ž .Ž .f 1 f 2 2 2ž /2

The neutrino mass matrix,

² :2 ² : ² :j j j

3 4 20² :² : ² : ² : ² :M ; D u , 20Ž .j j jn � 02 6² : ² : ² :j j u

is then generated by diagrams such as Fig. 2. We have ignored CP violating phases here, and there are unknownŽ .O 1 factors in front of each of the mass matrix elements.This mass matrix gives

Dm212 ² :; j 21Ž .2Dm23

which is not small enough for the ‘just-so’ or MSW solutions to the solar neutrino problem if we take

Dm2 ;10y3 eV 2 22Ž .23

for the atmospheric neutrino problem. Hence we shall use the ‘energy-independent’ vacuum oscillation solutionŽ .to the solar neutrino problem. The mixing from this mass matrix is similar to that given by Eq. 9 , although the

² 0: 3 2elements of order D u j in the mass matrix can have some effect on the mixing leading to some smallŽ .deviations from the form of Eq. 9 . The electron neutrino mixing remains very close to maximal regardless of

Ž .2Fig. 2. Example Feynman diagram for neutrino mass in the SMG=U 1 model.


Ž .the O 1 factors in the mass matrix, and makes almost no contribution to the atmospheric neutrino mixing.Ž . Ž .Depending on the O 1 factors the muon and tau neutrino mixing can differ slightly from that given by Eq. 9 ,

Ž .although if Eq. 11 is satisfied then the mixing between them remains large enough to solve the atmosphericneutrino problem.

² :Hence if we take D ;12 eV to give suitable masses for the atmospheric neutrino problem then we have

Dm2 ;10y4 eV 2 , sin2 2u ;1 23Ž .12 12

Dm2 ;10y3 eV 2 , sin2 2u s0.7–1.0 24Ž .23 23

for the solar and atmospheric neutrinos respectively. This means we will have an electron neutrino fluxsuppression of 1r2 for all of the solar neutrinos, and the atmospheric neutrino problem will be due to largen yn mixing. The neutrino masses are too small to make a significant contribution to dark matter, or to them t

w xanomaly observed at LSND 23 . Hence we predict that the LSND result will prove to be unfounded. TheŽ . Ž .amplitude of neutrinoless double beta decay is proportional to M , which we predict to be M ;2=n ee n ee

y3 Ž . w x10 eV, which is much less than the current limit of M F0.45 eV 24 and the sensitivities of current orn ee

planned experiments.² 0:In obtaining the spectrum of neutrino masses we have simply chosen D to have the required value for the

² 0:atmospheric neutrinos. However, there is some unnaturalness in obtaining a suitable value for D from thescalar potential. If we write down the low energy effective scalar potential we have

m22 2 2X XX† † †2 2 † †² :² : ² :V f ,D ;l f f ql D D ql M f D W j u yhM D Dy f fŽ . Ž .Ž .WS WS WS F WS F WS WS½ 5l

25Ž .X XX Ž .where we would typically expect l ,l ,hsO 1 . However, this leads to a vacuum expectation value for D of

² 2 :f WS 20² : ² :² : ² :D ; W j u . 26Ž .MF

² :Whilst we can choose M to give the required vacuum expectation value for D we then find that, since D isF² :2much less than the see-saw scale f rM , the neutrino mass matrix is dominated by the see-saw typeWS F

diagrams which as we noted earlier, do not yield an acceptable phenomenology. Hence, in order to avoid thisproblem, we would require a f†2 D coupling which is for some unknown reason much larger than expected. OfWS

course the scalar potential is in any case not well understood, since the lightness of the Weinberg-Salam Higgsfield is also something of a mystery.

It should be noted that, whilst in this case we have some difficulty in obtaining a suitable vacuum expectationvalue for the triplet Higgs field, this will not necessarily be the case for other models which use this mechanismfor generating the neutrino masses. If the see-saw neutrino masses are sufficiently suppressed by the symmetrybreaking parameters, then the masses coming from the triplet Higgs field will dominate and there will be noproblem.

5. Conclusions

Ž .We have shown that models with only the 3 standard model neutrinos in the low energy spectrum , andchiral symmetry breaking can explain the solar and atmospheric neutrino problems including the Super-Kamio-

Ž .kande zenith angle distribution. This can occur if the chiral symmetry does not lead to independent elements inŽ .M which are all of different orders of magnitude as we assumed in a previous paper . The atmosphericn

Ž .neutrino problem is explained by large n yn mixing, and for the mass matrix structure we examined them t


solar neutrino deficit is due to nearly maximal electron neutrino vacuum oscillations, which can be either‘just-so’ or ‘energy-independent’. We presented an explicit model, which is an anomaly free Abelian extensionof the SM, yielding this type of phenomenology, although there are unresolved problems in the scalar potential.This model is an extension of a model which gives a realistic 3 parameter fit to the charged fermion masses andmixings. It gives an ‘energy-independent’ solar neutrino suppression of 1r2, with Dm2 ;10y4 eV 2. We alsosolar

predict that the signal at LSND will not be confirmed by other experiments, and that the neutrinos will not makea significant contribution to hot dark matter.

w x w xThe prospects for examining this scenario are good. Experiments such as SNO 25 , Borexino 26 andw xKamLand 27 should provide us with more information on the solar neutrino spectrum. Super-Kamiokande will

also provide data on the day-night asymmetry and seasonal variations which will be important in determiningw x w xthe type of solution to the solar neutrino problem. Long baseline experiments such as K2K 28 and MINOS 29

should enable us to confirm the nature of the atmospheric neutrino oscillations with a better understood neutrinosource, and should tell us whether the n oscillations are to n or a sterile neutrino. The LSND result will alsom t

be further tested by Karmen at 95% C.L., and definitely by MiniBoone; neither of which we would expect tofind evidence of oscillations. In conclusion, we predict the atmospheric neutrino problem to be due to largen yn oscillations with Dm2 ;10y3 eV 2, and the solar neutrino deficit to be due to electron neutrino vacuumm t

oscillations of either the ‘just-so’ or ‘energy-independent’ type. This scenario should be confirmed or denied bya number of experiments in the near future.

Acknowledgements

H.B.N. and C.F. acknowledge funding from INTAS 93-3316-ext, and the EU grant HMC 94-0621. M.G. isgrateful for a PPARC studentship. We would also like to thank M. Jezabek for useful discussions.

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w x Ž .10 N. Irges, S. Lavignac, P. Ramond, Phys. Rev. D 58 1998 035003.w x Ž .11 J. Bijnens, C. Wetterich, Nucl. Phys. B 283 1987 237.w x Ž .12 P. Binetruy, S. Lavignac, P. Ramond, Nucl. Phys. B 477 1996 353.´w x13 G. Conforto, C. Grimani, F. Martelli, F. Vetrano, talk presented at Neutrino’98, Takayama, Japan, hep-phr9807306.w x14 J.N. Bahcall, P.I. Krastev, A.Yu. Smirnov, hep-phr9807216.w x15 Y. Grossman, Y. Nir, Y. Shadmi, hep-phr9808355.w x Ž .16 G.K. Leontaris, S. Lola, G.G. Ross, Nucl. Phys. B 454 1995 25.w x17 J. Pati, based on talk presented at Neutrino’98, hep-phr9807315.w x18 R. Barbieri, L.J. Hall, D. Smith, A. Strumia, N. Weiner, hep-phr9807235.w x19 V. Barger, S. Pakvasa, T.J. Weiler, K. Whisnant, hep-phr9806387.w x20 M. Jezabek, Y. Sumino, hep-phr9807310.w x21 R. Barbieri, L.J. Hall, A. Strumia, hep-phr9808333.w x Ž .22 CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B 420 1998 397.w x Ž . Ž .23 LSND Collaboration, C. Athanassopoulos et al., Phys. Rev. Lett. 81 1998 1774; 77 1996 3082.


w x24 H.V. Klapdor-Kleingrothaus, L. Baudis, J. Hellig, M. Hirsch, S. Kolb, H. Pas, Y. Ramachers, hep-phr9712381.¨w x Ž . Ž .25 SNO Collaboration, A.B. McDonald, Nucl. Phys. B Proc. Suppl. 48 1996 357.w x Ž .26 BOREXINO Collaboration, C. Arpesella et al., INFN Borexino proposal, vols. IrII, G. Bellini, R. Ragahaven et al. Eds. , Univ.

Milan, 1992.w x27 A. Suzuki, Talk presented at Neutrino’98, Takayama, Japan.w x28 Y. Oyama, hep-exr9803014.w x29 MINOS Collaboration, Neutrino Oscillation Physics at Fermilab: The NuMI-MINOS Project, NuMI-L-375, May 1998.

28 January 1999


Asymptotic Pade-approximant predictions for´renormalization-group functions of massive f 4 scalar field theory

F. Chishtie a, V. Elias a,1, T.G. Steele b,2

a Department of Applied Mathematics, The UniÕersity of Western Ontario, London, Ont., N6A 5B7, Canadab Department of Physics and Engineering Physics and Saskatchewan Accelerator Laboratory, UniÕersity of Saskatchewan, Saskatoon,

Sask., S7N 5C6, Canada

Received 25 September 1998; revised 12 November 1998Editor: M. Cvetic

Abstract

Within the context of massive N-component f 4 scalar field theory, we use asymptotic Pade-approximant methods to´estimate from prior orders of perturbation theory the five-loop contributions to the coupling-constant b-function b , theg

anomalous mass dimension g , the vacuum-energy b-function b , and the anomalous dimension g of the scalar fieldm Õ 2

propagator. These estimates are then compared with explicit calculations of the five-loop contributions to b , g , b , andg m Õ

are seen to be respectively within 5%, 18%, and 27% of their true values for N between 1 and 5. We then extend asymptoticPade-approximant methods to predict the presently unknown six-loop contributions to b , g , and b . These predictions, as´ g m Õ

well as the six-loop prediction for g , provide a test of asymptotic Pade-approximant methods against future calculations.´2

q 1999 Published by Elsevier Science B.V. All rights reserved.

w xA substantial body of work 1–5 already exists inwhich Pade-approximant methods are specifically´utilized to predict higher order corrections to pertur-

w xbation theory, in addition to other applications 6 ofPade techniques to quantum field theory. In the´present letter, we utilize such Pade methods to pre-´dict the leading-order unknown contributions to the

Ž .coupling-constant b-function b g , the anomalousgŽ .mass dimension g g , the anomalous scalar-fieldm

Ž .propagator dimension g g , and the vacuum-energy2


Ž . 4b-function b g for a massive f N-componentÕ

scalar field theory based on the Lagrangian 3

N1 1a a 2 a aLLs E f E f q m f fÝ m m2 2

as1

2 416p m h2a aq g f f q 1Ž . Ž .2ž /4! 4p gŽ .w < xSpecifically, we utilize the N 1 Pade-approximant´

w x Ž w xalgorithm of 2 see 3 for a subsequent pedagogical

3 Ž .The final constant term in 1 is of relevance only for theŽ . w xcalculation of b g 7 , and has not been incorporated in calcula-Õ

w x Ž . Ž . Ž .tions 8 of b g , g g , and g g .g m 2


( )F. Chishtie et al.rPhysics Letters B 446 1999 267–271268

.treatment , based upon asymptotic Pade-approximant´w xmethodology first delineated in 5 , in order to com-

Ž .pare predictions of five-loop contributions to b g ,gŽ . Ž . w xg g , and b g to their now-known 8,9 calcu-m Õ

lated values. These predictions are seen to beŽ .startlingly successful for b g , and for up to fiveg

scalar field components, are generally within 20% ofŽ .the correct value for the five-loop term in g g andm

Ž .b g . We then adapt asymptotic Pade methods to´Õ

predict the presently unknown six-loop contributionsŽ . Ž . Ž . Ž .to g g , b g , g g , and b g , with the hope2 g m Õ

that these predictions may be tested against six-loopcalculations in the not too distant future.

The asymptotic Pade approximant methods we´employ are directed toward perturbative field-theo-retical series of the form

`

kF x sF x 1q R x 2Ž . Ž . Ž .Ý0 kks1

where x typically denotes a field-theoretical cou-pling constant. Unknown coefficients R areNqMq1

w < xestimated via N M Pade approximants whose´Maclaurin expansions reproduce known coefficientsR – R :1 NqM

1qa xqa x 2 q . . . qa x N1 2 N

F x ,F xŽ . Ž .0 2 M1qb xqb x q . . . qb x1 2 M

2sF x 1qR xqR x q . . .Ž .0 1 2

NqM Pade NqMq1qR x qR xNqM NqMq1

NqMq2qOO x 3Ž . Ž .Within a quantum field-theoretical context in whichcoefficients R are expected to diverge like k!C kkg

kw x w x10 , it has been argued 2,4,5 that the relative error

w < xin using an N M Pade approximant to estimate the´coefficient R satisfies the asymptotic errorNqMq1

formula

PadeR yRNqMq1 NqMq1d 'NqMq1 RNqMq1

M ! AM

,y 4Ž .MNqMqaMqbŽ .� 4where A, a, b are constants to be determined.

w < x ŽWithin the context of N 1 approximants i.e. Ms. 21 , the Pade prediction for R is R rR , and´ Nq2 Nq1 N

Ž .the error formula 4 simplifies to

R2 yR R ANq1 N Nq2d s syNq2 R R Nq1q aqbŽ .N Nq2

5Ž .

Ž . Ž .Noting that R '1 in 2 , one easily sees from 50w xthat 2,3

d d d y2d2 3 2 3As , aqb s 6Ž . Ž .

d yd d yd3 2 3 2

Ž . w < xWe further note from 5 that the 2 1 Pade predic-´tion for R can be corrected by the asymptotic error4

w xformula to yield the following result 3 :

2 2R rR R 3q aqbŽ .3 2 3R s s4 1qd R 3yAq aqbŽ .4 2

2 3 3R R qR R R y2 R R3 2 1 2 3 1 3s 7Ž .3 3 2 2R 2 R yR R yR R2 2 1 3 1 2

Ž . Ž .The final expression in 7 is obtained via 6 , withd s R2 yR rR , d s R2rR yR rR .Ž . Ž .2 1 2 2 3 2 1 3 3

Table 1Ž . Ž . w xThe coefficients R –R in the coupling-constant b-function b g , as defined in 8 and as calculated in Ref. 8 for an N-component1 4 g

massive f 4 scalar field theory. The quantity RAPAP is the asymptotic Pade-approximant estimate of the five-loop contribution R , and E is´4 4Ž . APAPthe relative error defined in 9 between this estimate and R ’s true value. The quantity R is the asymptotic Pade estimate of the´4 5

Ž .six-loop contribution to b gg

APAP APAPN R R R R R E R1 2 3 4 4 5

171 y 10.84989 y90.53526 949.5228 947.8 0.002 y11744.29

2 y2 11.98433 y105.1544 1153.352 1165.0 0.010 y14814.4233 y 12.99584 y119.3579 1363.260 1394.1 0.023 y18116.211134 y 13.91515 y133.2477 1579.416 1633.9 0.035 y21659.16295 y 14.76355 y146.8943 1801.925 1883.4 0.045 y25451.913

( )F. Chishtie et al.rPhysics Letters B 446 1999 267–271 269

Table 2Ž .The coefficients R – R in the anomalous mass dimension g g ,1 4 m

Ž . w x APAPas defined in 10 and as calculated in 8 . The quantity R is4

the asymptotic Pade-approximant estimate of the five-loop term´w Ž .xR , E is the relative error in this five-loop term Eq. 9 , and4

RAPAP is the asymptotic Pade-approximant estimate of the six-loop´5Ž .contribution to g gm


5 71 y y19.956305 150.7555 135.07 0.104 y14786 25 472 y y23.90300 191.8840 168.37 0.123 y20556 125 133 y y27.98248 236.9432 203.06 0.143 y28826 35 194 y y32.19475 285.9396 239.18 0.164 y47386 45 315 y y36.53981 338.8796 276.80 0.183 q14466 6

We first test this asymptotic Pade-approximant´Ž .prediction APAP for R against the known values4

of the coefficients R . . . R in the coupling-constant1 4Ž .b-function for the scalar field theory 1 :

2 `g Nq8Ž .kb g s 1q R g 8Ž . Ž .Ýg k6 ks1

The coefficients R , R , R and R have been1 2 3 4w xcalculated explicitly 8 , and are tabulated in Table 1.

Ž .The APAP prediction for R , as obtained via 74Ž .from prior coefficients R . . . R is also tabulated1 3

as RAPAP in Table 1. The relative error4

APAPR yR4 4E' 9Ž .

R4

Ž .is seen to be astonishingly small 0.2% for single-Ž .component Ns1 scalar field theory, and remains

less than 5% for NF5 4.We have repeated the above procedure for coeffi-

w xcients R –R , as extracted from Ref. 8 , in the1 4Ž .anomalous mass dimension g g :m

`g Nq2Ž .kg g s 1q R g 10Ž . Ž .Ým k6 ks1

These values are tabulated in Table 2 for Ns� 41,2,3,4,5 . The APAP prediction for R , as obtained4

Ž .via 7 from R , R , R , is within 10.4% of the true1 2 3

4 The coefficients R for b have also been estimated quite4 gw xaccurately by a somewhat different procedure in 5 involving

explicit use of the N 4 dependence of R , as well as an overall fit4

of R ’s N-dependence within the context of a simplified version4Ž .of the error formula 7 .

Ž .value for a single component scalar field Ns1 ,and becomes progressively less accurate as the num-ber of scalar field components increases. Neverthe-less, even the five-component APAP estimate of R4

differs from the true value by only 18.3%, remark-able accuracy for a five-loop estimate that is ob-tained entirely from lower-order perturbative contri-butions. It should be noted, however, that even greateraccuracy has been exhibited in the successful APAPprediction of QCD’s four-loop anomalous mass di-

w xmension, as described in Ref. 2 .We suspect that our somewhat diminished accu-

racy in the estimate of the five-loop contribution ofg , as compared to that for b , may be a conse-m g

quence of the value for R remaining static at y5r61

while R , R , and R are all seen to increase in2 3 4

magnitude with the number of scalar-field compo-nents N. Indeed, a similar situation arises for the

Ž .vacuum-energy b-function b g , as defined andÕ

w x w xcalculated in 7 and 9 for the N-component mas-Ž .sive scalar-field theory 1 :

Ng Nq22b g s 1q g y0.032639868 Nq2Ž . Ž .Õ 4 24

= Nq8 g 3 q4 0.5781728935Ž . Žq0.4184309753Nq0.06839310856N 2

3 4q0.001860422132 N g q . . ..`Ng

k' 1q R g 11Ž .Ý k4 ks1

In the above expression, R is clearly zero for all N.1

Consequently, APAP estimates for R obtained via4

Table 3Ž .The coefficients R – R in the vacuum-energy b-function b g ,1 4 Õ

Ž .as defined in 11 . Values of R , R , and R are extracted from1 2 3w x w xRef. 7 ; the five-loop contribution R is extracted from Ref. 9 .4

The quantity RAPAP is the asymptotic Pade-approximant estimate´4

of the five-loop term R , E is the relative error in this five-loop4w Ž .x APAPterm Eq. 9 , and R is the asymptotic Pade-approximant´5

Ž .estimate of the six-loop contribution to b gÕ


11 0 y0.8812764 4.267430 3.1066 0.272 y16.83812 0 y1.305595 6.813963 5.1137 0.250 y28.43653 0 y1.795193 9.996941 7.7345 0.226 y43.7124

14 0 y2.3500705 13.861014 11.0457 0.203 y63.04475 0 y2.9702280 18.450833 15.1239 0.180 y86.8524

( )F. Chishtie et al.rPhysics Letters B 446 1999 267–271270

Table 4The coefficients R , R , R in the in the anomalous dimension1 2 3

Ž .g of the scalar field propagator, as defined in 12 and extracted2w x APAPfrom Ref. 8 . The quantity R is the asymptotic Pade-ap-´4

proximant estimate of the six-loop termAPAPN R R R R1 2 3 4

3 651 y y23.10702 134.74 165 552 y y28.33255 186.16 1211 7253 y y33.762556 252.012 144

654 y1 y39.37539 337.21213 2755 y y45.149425 448.912 48

Ž . 2 Ž .7 are simply R r 2 R . Surprisingly, such an esti-3 2

mate does substantially better than R2rR , the naive3 2w < xestimate of R from the 2 1 -approximant, as is4

evident from Table 3. We note that RAPAP actually4

improves in accuracy with increasing N; the relativeerror in RAPAP decreases from 27% to 18% as N4

increases from 1 to 5.The six-loop contribution to the scalar-field

anomalous dimension g has not, to our knowledge,2

been calculated. Only contributions up to five-loopw xorder are listed in Ref. 8 . These results, when

expressed in the form2 `g Nq2Ž .

kg g s 1q R g 12Ž . Ž .Ý2 k36 ks1

Ž .are tabulated in Table 4. The algorithm 7 is used topredict R , the presently unknown six-loop contribu-4

tion. A genuine test of the asymptotic error formulaeŽ .4 would be to compare the final column of Table 4with the results of a six-loop calculation.

It is perhaps of even greater value to see if thesuccess already evident in APAP predictions of

Ž . Ž . Ž .five-loop terms for b g , g g , and b g carryg m Õ

over to the next order of perturbation theory, byextending the APAP procedure based on the error

Ž .formula 4 to estimate six-loop order contributionsŽ .R from lower-order terms in perturbation theory.5

Ž .We note from 5 that2 2R R 4q aqbŽ .4 4

R s s 13Ž .5 R 1qd R 4yAq aqbŽ . Ž .3 5 3

Ž .We seek to improve upon 6 by incorporatingŽ .knowledge of R , hence of d , into A and aqb .4 4

Since this applies the asymptotic error formula at a

Ž .larger value of N, the results of the constants in 4should be more accurate. The equations

A Ad sy , d sy 14Ž .3 42q aqb 3q aqbŽ . Ž .have solutions

d d 2d y3d3 4 3 4As ; aqb s 15Ž . Ž .

d yd d yd4 3 4 3

Ž . 2Noting from 5 that d s R yR R r R R ,Ž .Ž .4 3 2 4 2 4

we obtain the asymptotic error formula prediction

2 3 3R R R y2 R R qR R R R4 1 3 4 2 1 2 3 4R s 16Ž .5 3 3 2 2R 2 R R yR R yR R3 1 3 4 2 2 3

The final column of Table 1 lists this APAP predic-� 4tion of the six-loop term R for Ns 1,2,3,4,5 In5

view of the accuracy already noted in the APAPŽ .prediction of the five-loop contribution to b g , ag

comparison of this column’s entries to an explicitcalculation of the six-loop contributions to the b-function would be of evident value.

For completeness, we have also added to Tables 2and 3 predictions of the six-loop contributions to

Ž . Ž . Ž .g g and b g , as obtained from 16 . Kasteningm Õ

w xhas already noted 9 that the six-loop contribution toŽ .b g is perhaps the easiest six-loop quantity inÕ

scalar field theory to calculate. Perhaps a directcomparison of such a calculation to the final columnof Table 3 will be possible in the near future.

Acknowledgements

We are grateful for support from the NaturalSciences and Engineering Research Council ofCanada.

References

w x1 J. Ellis, E. Gardi, M. Karliner, M.A. Samuel, Phys. Lett. 366Ž . Ž .1996 268; Phys. Rev. D 54 1996 6986.

w x2 J. Ellis, I. Jack, D.R.T. Jones, M. Karliner, M.A. Samuel,Ž .Phys. Rev. D 57 1998 2665.

w x3 V. Elias, T.G. Steele, F. Chishtie, R. Migneron, K. Sprague,Ž .Phys. Rev. D to appear: hep-phr9806324 .

w x4 M.A. Samuel, J. Ellis, M. Karliner, Phys. Rev. Lett. 74Ž .1995 4380.

( )F. Chishtie et al.rPhysics Letters B 446 1999 267–271 271

w x Ž .5 M.A. Samuel, J. Ellis, M. Karliner, Phys. Lett. B 400 1997176.

w x Ž .6 J.-F. Yang, Comm. Theor. Phys. 22 1994 207; I.T. Drum-mond, R.R. Horgan, P.V. Landshoff, A. Rebhan, Nucl. Phys.

Ž .B 524 1998 579.w x Ž .7 B. Kastening, Phys. Rev. D 54 1996 3965.w x8 H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin,

Ž . Ž .S.A. Larin, Phys. Lett. B 272 1991 39; B 319 1993 545Ž .E .

w x Ž .9 B. Kastening, Phys. Rev. D 57 1998 3567; see also S.A.Larin, M. Moenningmann, M. Stroesser, V. Dohm, cond-matr9805028.

w x Ž .10 A.I. Vainshtein, V.I. Zakharov, Phys. Rev. Lett. 73 19941207.

28 January 1999


Classical kinetic theory of Landau damping for self-interactingscalar fields in the broken phase

A. Patkos 1, Zs. Szep 2´ ´Department of Atomic Physics, EotÕos UniÕersity, Budapest, Hungary¨ ¨

Received 13 October 1998; revised 3 December 1998Editor: R. Gatto

Abstract

The classical kinetic theory of one-component self-interacting scalar fields is formulated in the broken symmetry phaseand applied to the phenomenon of Landau damping. The domain of validity of the classical approach is found by comparingwith the result of a 1-loop quantum calculation. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

< <For high enough temperature leading non-equilibrium transport effects of large wave number k 4Tfluctuations can be reproduced by a kinetic theory of the corresponding response functions. This approach has

w xbeen applied to the features of the QCD plasma 1 and proved successful in reproducing the contribution of< < w xhard loops to the Green functions of low k <T modes in Abelian and non-Abelian gauge theories 2,3 .

These investigations have assumed tacitly that no background fields are present, all symmetries are restored.This assumption is certainly not valid for all field theoretical models, since in some scalar models with specificinternal symmetries one finds arguments in favor of non-restoration of the symmetry at arbitrary high

w xtemperature 4,5 . Also certain approaches to non-Abelian gaugeqHiggs systems provide evidence for non-zerow xscalar expectation values even in the high-temperature phase 6 . Therefore, there exist intuitive hints for

possible relevance of the classical kinetic considerations in a non-zero scalar background even at hightemperature.

The classical kinetic theory for self-interacting scalar fields has been derived first by Danielewicz andw x w xMrowczynski 7 and its features are still being discussed 8 . These papers deal with scalar theories of´

non-negative squared mass parameter. Therefore their results are useful for the calculation of transportŽ . w xcharacteristics plasmon frequency, damping rates in the restored symmetry phase. In Ref. 8 an effective

action is derived, which accounts for the loop contribution of high-k fluctuations to the Green functions of the



( )A. Patkos, Zs. SzeprPhysics Letters B 446 1999 272–277´ ´ 273

low-k modes. As a parallel evolution one may note that a classical cut-off field theoretic approach to timew xdependent phenomena in the symmetric phase has been developed in Refs. 9–12 . Quantitative results relevant

for quantum systems were obtained by matching the classical theory to the parameters of the quantum theory.This effective theory reproduces, for instance, correctly the on-shell damping rate in the high temperature phase.

The present note rederives in a fully relativistic Lagrangian formalism the kinetic theory of scalar fields forŽ .the case of non-zero background field generated by a negative mass squared parameter . It starts by proposing a

Lagrangian for the effective particles, which accounts for the effect of the high-k w modes on the low-kfluctuations. A detailed motivation for this Lagrangian is provided by recalling results of earlier investigations.An advantage of this proposition is that it leads to the induced source density of the low-k fluctuations directly,without any reference to the quantum theory. Evidence for the correctness of the effective lagrangian can bepresented by comparing its consequences with the results of the corresponding quantum calculations. As a firsttest we compute a physically meaningful quantity, the Landau damping coefficient for the off-shell scalarfluctuations. This effect is present only in the broken symmetry phase of the scalar theory. Its classical value is

w xcompared with the result of a 1-loop quantum calculation 13 , establishing in this way the domain of validity ofthe proposed classical treatment.

2. Kinetic theory of the high-k modes in a non-zero background

The effective gas of high-frequency fluctuations is out of thermal equilibrium if an inhomogeneous lowfrequency background fluctuation is present. This state of the gas induces a source term into the wave equationof the low-k modes. A unified description of the two effects can be attempted if in addition to the action Scl

describing the low-frequency dynamics an appropriate Lagrangian can be introduced for a gas particle coupledŽ .to the low-frequency field along its trajectory j t :m

w xS sS w qDS, DSs dt L w j t . 1Ž . Ž .Ž .Ž .Heff cl particle

Ž .Variation of this additional piece of action with respect to the field variable w x should yield the inducedsource term to the wave equation of the low frequency modes when averaged over the statistical distribution ofthe gas particles. The distribution can be derived from the solution of a Boltzmann equation describing the gas

Ž .in the background field w. Variation of 1 with respect to the particle trajectory provides the expression of theforce exerted on the particle by the external field w. This information is used in the collisionless Boltzmann-equation.

The effective particle Lagrangian for the one-component selfinteracting scalar fields described by the theory:

1 1 l2 2 2 4w xL w s E w y m w y w 2Ž .Ž .m2 2 24

can be guessed intuitively by making use of the equation of motion for the real time Green function:) Ž . ² Ž . Ž .: w xD x, y s w x w y . Its Wigner-transform in the collisionless case fulfills 7

E 1 E m2 X EŽ .)p q D X , p s0, 3Ž . Ž .mž /E X 2 E X E pm m m

2Ž . 2 Ž . 2Ž . Ž .where m X sm q lr2 w X , with w representing the background at Xs xqy r2. This equationsuggests the relation

1 2mF s E m X , 4Ž . Ž .m m2

( )A. Patkos, Zs. SzeprPhysics Letters B 446 1999 272–277´ ´274

which is equivalent to

L sym w x . 5Ž . Ž .particle

On the basis of this argument we propose for the classical effective action of an effective particlerepresentation of the high-k modes of the one-component, self-interacting scalar field

l 22 2DSsy M w ,w j t dt , M w ,w x sm q wqw x , 6Ž . Ž . Ž . Ž .Ž . Ž .H loc loc 2

2where m is of negative sign, w denotes the spontaneously generated non-zero average background field, andŽ .w x is the amplitude of the low-k fluctuation field at x in the broken phase.

ŽWe propose to write the Boltzmann-equation for the effective particle with an x-independent mass m seeeff. w xbelow , which leads to a slight deviation from the kinetic equation derived by 7 . Its advantage is that the

momentum variable of the one-particle distribution is defined through an x-independent relation. This definitionŽ 2 .gives slightly different OO l thermal mass to the low-frequency waves, but the damping coefficient remains

unchanged. The necessary input into the derivation of the Boltzmann-equation is the equation for the kineticmomentum:

dj l2 2 2p sm , m w sm q w . 7Ž . Ž .m eff effdt 2

Ž .The relevant equation can be found from the canonical Euler-Lagrange equation:

d dM w ,w xŽ .loc 1 2˙M w ,w x j s E V w ,w x , V w ,w x sl wwq w , xsj t .Ž . Ž . Ž . Ž .Ž .ž /loc m m 2dt dV w ,w xŽ .8Ž .

From this equation, using explicitly the definition of M , one can express the proper-time derivative of theloc

kinetic momentum.

2dp m pm eff m w xm s E V w ,w x y pPE V w ,w . 9Ž . Ž . Ž .eff m2 2dt 2 M mloc eff

Ž .The second term on the right hand side is missing from 4 .Ž .With help of 9 one writes the collisionless Boltzmann-equation for the gas of these particles:

2m w p E f x , pŽ . Ž .eff m w xpPE f x , p q E V w ,w x y pPE V w ,w s0. 10Ž . Ž . Ž . Ž . Ž .m2 2 E p2 M w ,w mŽ . mloc eff

Its perturbative solution in the weak coupling limit l<1 is searched for in the form:

1f x , p s f p ql f w x , p , f p s . 11Ž . Ž . Ž . Ž . Ž .Ž .0 1 0 b p0e y1


Ž Ž . .One may emphasize that the small parameter allowing iterative solution of Eq. 10 is hidden in V . The formalw xsolution is easily found in full structural agreement with the result of 8 :

1 1 p 1 dfm 02f w x , p sy wqw x E w x y ww x q w x . 12Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž .1 m 2 ž /2 pPE 2 dpmŽ . meff

3. Source density induced by field fluctuations

Ž .The variation of 6 with respect to w provides the induced source term for the low-k fluctuations. For anumber of particles moving on definite trajectories one has

1 1 dV w ,w xŽ .Ž4.j x s dt d xyj t . 13Ž . Ž . Ž .Ž .ÝH iž /w x2 M w ,w dwloci

Here the index i refers to the trajectory of the i-th particle. Statistical average over the full momentum spaceand in a small volume around the point x introduces the one-particle distribution

d3pŽ4.² :m f x , p s dt d xyj t 14Ž . Ž . Ž .Ž .ÝH Heff i32p pŽ . i0

and produces the following ‘‘macroscopic’’ source density:3 w x1 d p m w dV w ,wŽ .eff 2 2 2j x s f x , p , p sm qp . 15Ž . Ž . Ž .Hav 0 eff3 w x2 M w ,w dw2p pŽ . loc0

Ž .The induced linear response to w x is found by retaining terms with linear functional dependence of j onav

w. The contribution of the equilibrium distribution f is easily found:0

3 2l d p lwŽ0.j x s w x f p 1y . 16Ž . Ž . Ž . Ž .Hav 0 03 2ž /2 2m2p pŽ . eff0

Ž 2 . Ž .The OO l contribution is the result of the expansion of M in the denominator of 15 . This contribution toloc

the thermal mass in the high-T limit gives the correct limiting value for the one-component scalar theory.The terms induced by f give rise to the following expression linear in w:1

2 3 2 2 3l w d p l w d p 1 df0Ž1.j x s f w x , p ;y E w xŽ . Ž . Ž .Ž .H Hav 1 03 32 4 pPE dpŽ .2p p 2p pŽ . Ž . 00 0

2 2`l w w x 1Ž .

q dp f p . 17Ž . Ž .H 0 0 02 2 28p m (p ymeff 0 eff

Ž 2 .The damping effect arises from the first term, while from the second another OO l contribution is obtained tothe thermal mass.

The imaginary part of the linear response can be evaluated by the use of the principal value theorem, whenŽ .y1 Ž .the e-prescription of Landau is applied to the pPE operator in the first term on the right hand side of 17 .

For an explicit expression one may assume that w represents an off-mass-shell fluctuation characterised by theŽ . Ž . Ž .4-vector: v,k , that is w x sw kPx . The evaluation of the integral is a not too difficult exercise. One is led

to the following expression for the imaginary part of the linear response function:2 2l w 1 v

2 2< <Im Ss P Q k yv . 18Ž .Ž .22 < << <'16p kb m f f r 1yv r kee y1

( )A. Patkos, Zs. SzeprPhysics Letters B 446 1999 272–277´ ´276

4. Discussion

In this note we have shown that in the broken phase of scalar theories Landau-type damping phenomenonoccurs according to an effective classical kinetic theory. The clue to its existence is the presence of the ;ww

term in the fluctuating part of the local mass, which is responsible for the emergence of a linear source-ampli-tude relation with non-zero imaginary part. On the other hand the imaginary part of the self-energy has been

4 w xcomputed earlier in the w -theory using its real time quantum formulation 13 , where ‘‘a term reminiscent ofLandau damping’’ was found:

q2 2 ybv 2k < <l w T 1ye v k 4M2 2 "< <Im S s ln Q k yv , v s " 1y . 19Ž .Ž .y (Landau k 2ybv 2k< <16p k 2 2 < <1ye v y k

Here M is the temperature dependent mass parameter in the finite temperature quantum theory. With explicit< <calculation one may check that in the limit bv,b k <1 the expression of the damping rate calculated from

Ž . Ž .19 goes over into 18 , under the assumption that m of the classical theory is identified with M. In this wayeff

the classical theory provides a convincing argument clearly demonstrating that Landau damping is the correctinterpretation of the result derived from quantum theory.

The extension of our discussion to the n-component scalar fields in the broken phase forces us to accept theŽ .choice 4 , since for the Goldstone fields m s0. Then one can see by analysing the equations of motion of theeff

Ž 2 .Green-functions composed from the different field components, that with accuracy OO l one-particle distribu-tions for each type of particles have their separate kinetic equation, decoupled from each other. Landau damping

Ž .of the heavy ‘‘Higgs’’ mode can be described in full analogy with the treatment of the present paper, writingŽ .for each particle a mechanical action of the form 6 with appropriate local mass expressions. Damping of the

Goldstone modes is the result of the non-zero correlation between these and the ‘‘Higgs’’ modes. One canconstruct an additional non-local, velocity dependent mechanical action for the heavy particles describing this

w xeffect 17 .If one completes the kinetic theory of the pure non-Abelian gauge fields by appropriate scalar fields, an

important generalisation of the present discussion can be made to the damping rates of gaugeqHiggs systems,w x w xwhere also the interpretation of lattice simulations 14 represents a non-trivial challenge 15 . Some results

concerning the hard thermal loop effects on thermal masses of the gauge bosons have been obtained recently inw x w x16 . An approach based on the coupled kinetic system is presently under study 17 .

Acknowledgements

The authors acknowledge informative discussions on the subject of the paper with A. Jakovac and P.´Petreczky. This work has been supported by the research contract OTKA-T22929.

References

w x Ž . Ž . Ž .1 U. Heinz, Phys. Rev. Lett. 51 1983 351; Ann. Phys. NY 168 1986 148.w x Ž .2 P.F. Kelly, Q. Liu, C. Lucchesi, C. Manuel, Phys. Rev. D 50 1994 4209.w x Ž .3 J.-P. Blaizot, E. Iancu, Nucl. Phys. B 421 1994 565.w x Ž .4 S. Weinberg, Phys. Rev. D 9 1974 3357.w x Ž .5 P. Solomonsson, B.-S. Skagerstam, Phys. Lett. B 155 1985 100.w x Ž .6 W. Buchmuller, O. Philipsen, Nucl. Phys. B 443 1995 47.¨w x Ž .7 S. Mrowczynski, P. Danielewicz, Nucl. Phys. B 342 1990 345.´w x Ž .8 F. Brandt, J. Frenkel, A. Guerra, Int. J. Mod. Phys. A 13 1998 4281.


w x Ž .9 G. Aarts, J. Smit, Phys. Lett. B 393 1997 395.w x Ž .10 G. Aarts, J. Smit, Nucl. Phys. B 511 1998 451.w x Ž .11 W. Buchmuller, A. Jakovac, Phys. Lett. B 407 1997 39.¨ ´w x Ž .12 W. Buchmuller, A. Jakovac, Nucl. Phys. B 521 1998 219.¨ ´w x Ž .13 D. Boyanovsky, I.D. Lawrie, D.-S. Lee, Phys. Rev. D 54 1996 4013.w x Ž .14 W.H. Tang, J. Smit, Nucl. Phys. B 510 1998 401.w x Ž .15 D. Bodeker, M. Laine, Phys. Lett. B 416 1998 169.¨w x Ž .16 C. Manuel, Phys. Rev. D 58 1998 016001.w x17 A. Jakovac, A. Patkos, P. Petreczky, Zs. Szep, work in progress.´ ´ ´

28 January 1999


Sudakov effects in electroweak corrections

P. Ciafaloni a, D. Comelli b

a INFN sezione di Lecce, Õia Arnesano, 73100 Lecce, Italyb INFN sezione di Ferrara, Õia Paradiso 12, 44100 Ferrara, Italy

Received 18 September 1998; revised 3 November 1998Editor: R. Gatto

Abstract

In perturbation theory the infrared structure of the electroweak interactions produces large corrections proportional to2Ž 2.double logarithms log srm , similar to Sudakov logarithms in QED, when the scale s is much larger than the typical mass

m of the particles running in the loops. These energy growing corrections can be particularly relevant for the planned Nextq yLinear Colliders. We study these effects in the Standard Model for the process e e ™ f f and we compare them with

similar corrections coming from SUSY loops. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. IR divergences: qualitative discussion

Ž .Infrared IR divergences arise in perturbativecalculations from regions of integration over themomentum k where k is small compared to thetypical scales of the process. This is a well known

w xfact in QED for instance 1 where the problem of anunphysical divergence is solved by giving the photona fictitious mass which acts a a cutoff for the IR

Ž .divergent integral. When real bremsstrahlung andvirtual contributions are summed, the dependence on

w xthis mass cancels and the final result is finite 1 . TheŽ .double logarithms coming from these contributionsare large and, growing with the scale, can spoilperturbation theory and need to be resumed. They

w xare usually called Sudakov double logarithms 2 . Inthe case of electroweak corrections, similar loga-rithms arise when the typical scale of the processconsidered is much larger than the mass of the

Ž .particles running in the loops, typically the W Zw xmass 3–5 . The expansion parameter results then

a s2log ,2 24sin u p Mw W

'which is already 10% for for energies s of theorder of 1 TeV. This kind of corrections becomestherefore particularly relevant for next generation of

Ž w x.linear colliders NLC 6 . In the case of correctionsŽ .coming from loops with W Z s, there is no equiva-

lent of ‘‘bremsstrahlung’’ like in QED or QCD: theŽ .W Z , unlike the photon, has a definite nonzero

mass and is experimentally detected like a separateŽ .particle. In this way the full dependence on the W Z

mass is retained in the corrections. Other singulari-ties arise in perturbation theory, namely those com-

Ž .ing from the ultraviolet UV region. These diver-gences can be treated with the usual renormalizationprocedure and can be resummed through RGE equa-

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01541-X

( )P. Ciafaloni, D. ComellirPhysics Letters B 446 1999 278–284 279

tions. However they produce single logs and weexpect them to be asymptotically subdominant withrespect to the double logs of IR origin.

q yWe consider here the process e e ™ ff in thelimit of massless external fermions. Our notation is

Ž . ythat p p is the momentum of the incoming e1 2Ž q. Ž .e and p p is the momentum of the outgoing3 4Ž .f f . Furthermore, we define the Mandelstam vari-

Ž .2 Ž .2ables: s s p q p s 2 p p ,t s p y p s1 2 1 2 1 3s s2Ž . Ž . Ž .y 1ycosu ,us p yp sy 1qcosu . In1 42 2

the following we consider only the dominant doublelogs corrections of IR and collinear origin comingfrom one loop perturbation theory and we neglect

Ž .systematically single logs IR, collinear or UV and‘‘finite’’ contributions that do not grow with energy.We discuss the kind of diagrams where we expectthese corrections to be present 1, and evaluate themin the asymptotic regime s4M 2.w

2. Sudakov logarithms in the vertices

We will consider first as an example, to have agrasp over the effect of the IR double logs, the‘‘SM-like case’’ in which a ‘‘W boson’’ havingmass M and coupling with fermions like the photonis exchanged. We take the Born QED amplitude asthe reference tree level amplitude. Then we denotethe tree level photon exchange amplitude with MM s i01 2 Ž . Ž . Ž . Ž .e Õ p g u p u p g Õ p and the tree levele 1 m e 2 f 3 m f 4s

Ž . Ž .photon vertex with VV syieÕ p g u p ; e is0 e 1 m e 2

the electron charge.Let us first consider IR divergences coming from

vertex corrections. Since we work in the limit ofmassless fermions, there is no coupling to the Higgssector. Moreover, by power counting arguments, it iseasy to see that the vertex correction where thetrilinear gauge boson coupling appears is not IRdivergent. The only potentially IR divergent diagramis then the one of Fig. 1, where a gauge boson isexchanged in the t-channel. It is convenient to choosethe momentum of integration k to be the one of the

1 Only vertex and box corrections will be analyzed, sincevacuum polarization corrections give only single logs, both ofultraviolet and infrared origin.

Ž . Ž .Fig. 1. Vertex diagram in SM left and SUSY right generating a2Ž 2 .log srM . p and p are ingoing.1 2

exchanged particle, the boson in this case. Then, bysimple power counting arguments it is easy to seethat the IR divergence can only be produced byregions of integration where kf0. The only poten-tially IR divergent integral is then the scalar integral,

w xusually called C in the literature 7 . Any other0

integral with k ,k k in the numerator cannot, againm m n

by power counting, be IR divergent. To understandthe origin of the divergences, let us consider thediagram of Fig. 1 with all the masses set to zero. Forkf0 the leading term of the vertex amplitude isgiven by:

a d4k p pŽ .1 2VVfy VV H0 2 24p ip k kp kpŽ . Ž .1 2

a dx dy1 1yxfy VV 1Ž .H H02p x y0 0

We can see here the two logarithmic divergencesthat arise from the integration over the x, y Feyn-

w xmann parameters. As is well known 1 , one of themis of collinear origin and the other one is a proper IRdivergence. When we take some of the externalsquared momenta andror masses different from zero,they serve as cutoffs for the divergences. Let usconsider now some simple cases that will be usefulin the following, where the cutoff is given by asingle scale M. The behavior for C in the asymp-0

Ž .2 2totic region s' p qp 4M is as follows:1 2

12 2C m ,m , M , p , p ,s 'Ž .0 1 2 1 2 2ip

d4k=H 2 22 2 2 2w xkqp ym k yM kyp ymŽ . Ž .1 1 2 2

2Ž .

( )P. Ciafaloni, D. ComellirPhysics Letters B 446 1999 278–284280

Re C 0,0, M ,0,0,0,s� 4Ž .0

1 s2 2™ log for s4M 3Ž .2ž /2 s M

Re C M , M , M ,0,0,0,s� 4Ž .0

1 s2 2™ log for s4M 4Ž .2ž /2 s M

Ž .Then we can use 3 for the vertex of Fig. 1 in theasymptotic region finding:

a s2 2VVfy VV log for s4M 5Ž .0 2ž /4p M

where we can see the double logarithm behavior ofthe vertex correction for s4M 2.

The dependence on the IR logs simply factorizesfor the cross section:

s1 dt a s2 2< <sA MM 1y2 logH 0 2s s 4p M0

a s2ss 1y2 log0 24p M

Now let us consider the ‘‘susy-like’’ case inwhich a fermion is exchanged and a scalar couples to

Ž .the external gauge boson Fig. 1 . In supersymmetrythe internal fermion and scalar, for instance a neu-tralino and a selectron, have masses of the sameorder and can, for our purposes, be taken to have thesame mass M. In fact the distinction between thetwo masses is irrelevant as long as they are of the

Ž 2 . Ž 2 . 2same order, since log srm log srM s logŽ 2 . Ž 2 2 . Ž 2 . 2Ž 2 .srM q log M rm log srM f log srM

Žand we are interested only in double logs single logs. Ž .are neglected . Expression 5 is in this case substi-

tuted by:

a d4k M 2

VVfy VV H0 2 24p ip k kp kpŽ . Ž .1 2

s4M 2 2M a s2™ y VV log 6Ž .0 2ž /s 4p M

where we still have the double log behavior comingŽfrom the integration over the region kf0 remem-

2 .ber that always s4M . In this case however,2 p p ss is substituted by M 2 in the numerator, so1 2

2Ž 2 . Ž 2 . 2Žthat the we have log srM ™ M rs log sr2 .M . In the end the double logarithm behavior is

strongly suppressed by a factor M 2rs for the SUSYvertex with respect to the SM case. This is due to thedifferent couplings that appear in the vertex correc-tions: fermion-gauge boson coupling in the ‘‘SMlike’’ case and fermion-scalar in the ‘‘susy like’’case. In the first case the coupling is, at high energy,proportional to pi where i is the label of the exter-m

nal fermion the exchanged boson couples to. Thenwe have a factor p Pp where i and j are thei j

fermions connected by the exchanged boson. In the‘‘susy like’’ case where scalars and fermions areexchanged, no such factor is present and p Pp getsi j

substituted by M 2, generally subdominant at highenergies.

3. Sudakov logarithms in the boxes

Let us consider the exchange of a vector boson ofŽ .mass M in the s-channel see Fig. 2 . In the limit

Fig. 2. Box contribution for the SM and effective Feynman diagrams in the IR region.


ŽFig. 3. Box contribution for supersymmetry the crossed diagram.is also shown .

M™0 and in the IR region, the amplitude is givenby:

a d4kMMf MM H0 24p ip

=p p p pŽ . Ž .1 3 2 4

q 7Ž .2 2½ 5k p k p k k p k p kŽ . Ž . Ž . Ž .1 3 2 4

As is shown schematically in Fig. 2, the two terms inthis equation come from two different region of

Ž .2integration. When kf0, then kqp qp fs and1 2

we can think the ‘‘upper’’ boson line to be shrunk,like shown in the figure. The mirror situation iskqp qp f0,k 2 fs. This makes evident the fact1 2

that the IR structure of the box is the same of theŽ . Ž .vertex. Expression 7 is identical with 1 but with

the difference that 2 p p ss gets substituted by1 2

y2 p p sy2 p p s t. So for SM boxes we have1 3 2 4

an exchange of s and t variables with respect to SMvertices. In the end for the box contribution in the IRregion we can write:

a t2MMfy MM log 8Ž .0 2ž /2p M

It must be stressed however that this expression isvalid only in the asymptotic region t4M 2 wherethe double log behavior is generated, while we as-sumed s4M 2.

Let us now consider the ‘‘susy like’’ box where aŽscalar particle is exchanged in the t-channel see Fig.

.3 . In this case the amplitude is:

a d4kMMf MM H0 24p ip

=M 2 M 2

q 9Ž .2 2½ 5k p k p k k p k p kŽ . Ž . Ž . Ž .1 3 2 4

Ž . Ž .Comparing Eqs. 7 and 9 we note that the susyamplitude has a factor M 2rt with respect to the SM

2 Ž .one. In the IR region t4M we have, using 4 :

a M 2 t2MMf MM log 10Ž .0 24p t M

Care must be taken when we compute cross sectionsŽ . Ž .since, as noted above, Eqs. 8 and 10 are valid

only when t4M 2. Let us then consider a region ofthe phase space from a certain fixed value of t oforder s on, let’s say ys- t-ysr2. Then, if s4

M 2, we can use the expressions valid for t4M 2.Neglecting unessential factors, the leading box cor-rections to the tree level cross sections are given by:

a dt t sysr2 2 2SM Dsf log fs a logH 02 2s s M Mys

a dt M 2 tysr2 2SUSY Dsf logH 2s s t Mys

M 2 s2fs a log0 2s M

Again, SUSY boxes are depressed by a power factorwith respect to SM ones.

To conclude, we expect double logs of IR andcollinear origin to give at high energies large oneloop corrections to observables in the SM. This istrue both for box and vertex corrections. On theother hand, in a susy theory, due to the differentspins of the particles exchanged in the loops, thesedouble logs are expected to be power suppressed.For this reason, in the following we will consider indetail only SM electroweak corrections.

4. Sudakov logarithms in the Standard Model

We study the purely electroweak double logarith-mic corrections in the Standard Model coming fromthe exchange of the W and Z gauge bosons to the

q yprocess e e ™ f f in the massless case.For the moment we consider only the massless

external fermions m for leptons, u, c and d, s forquarks, and we neglect, for the moment, the bottomquark whose corrections contain a non trivial flavor

Ž .dependence on the top mass future analyses .This kind of contributions, as explained before,

come from only vertex corrections in which one


gauge boson it is exchanged and from the boxesŽ .direct and crossed with two Zs or two Ws.

Ž .The effective vertices g Z f f including tree levelŽ .and dominant double logs are given by Õ p ge 1 m

Ž g ŽZ . g ŽZ . . Ž .V P qV P u p withf L L f R R e 2

1 g 2 sg 2 2V s ig s Q 1y g logf L W f f L2 2 2ž 16p c mW Z

1 g 2 Q X sf 2y log 11Ž .2 2 /2 Q16p mf W

1 g 2 sg 2 2V s ig s Q 1y g log 12Ž .f R W f f R2 2 2ž /16p c mW Z

and

g 1 g 2 sZ 2 2V s i g 1y g logf L f L f L2 2 2žc 16p c mW W Z

1 g 2 g X sf L 2y log 13Ž .2 2 /2 g16p mf L W

g 1 g 2 sZ 2 2V s i g 1y g log 14Ž .f R f R f R2 2 2ž /c 16p c mW W Z

Here f is the external fermion and f X its isospinpartner. Moreover, g X syQ X s2 and g Xf Ž f .R f Ž f . W f Ž f .L

sT f Ž f X .yQ X s2 .3 f Ž f . W

Defining

Õ p g P u p u p g P Õ pŽ . Ž . Ž . Ž .e 1 m L , R e 2 f 3 m L , R f 4

'P mP 15Ž .L , R L , R

the corrections from box diagrams come from directand crossed diagrams as a sum of projected ampli-tudes on the left-right chiral basis:

B g P mg mP qB g P mg mPL L m L L L R m L R

qB g P mg mP qB g P mg mPR L m R L R R m R R

where

i g 4 g 2 g 2 sq t te L f L 2 2B s log y logL L 2 4 2 2ž /žs 8p c m mW Z Z

sq t t1 2 2q u log yu log2 f 1 f4 2 2ž / /m mW W

i g 4 g 2 g 2 sq t te L f R 2 2B s log y logL R 2 4 2 2ž /s 8p c m mW Z Z

i g 4 g 2 g 2 sq t te R f L 2 2B s log y logR L 2 4 2 2ž /s 8p c m mW Z Z

i g 4 g 2 g 2 sq t te R f R 2 2B s log y logR R 2 4 2 2ž /s 8p c m mW Z Z

with the above expressions obtained in the limits,t4M 2 andZ,W

u s1 for fsm, d and zero otherwise;1 f

u s1 for fsn , u and zero otherwise;2 f

The positive double log contributions come fromthe crossed box, while the negative ones from thedirect diagrams.

It is clear that the interference between the twoamplitudes, for the exchange of Z bosons, leads to adepression of the full contribution due to the fact that

sq t t s 1qcosu2 2log y log s2log log2 2 2 1ycosum m mZ Z Z

q finite 16Ž .

where finite means contributions not increasing aslog s. In such a way we lose the leading log2s factorand we remain with a single log that we neglect. Soin leading approximation, box diagram contributionscome only from W exchange.

To obtain the physical observables we must squarethe full amplitude:

MsM g P mg mP qM g P mg mPL L m L L L R m L R

qM g P mg mP qM g P mg mPR L m R L R R m R R

17Ž .


where

ig g Z ZM sy V V qV V qBŽ .L L e L f L e L f L L Ls

ig g Z ZM sy V V qV V qBŽ .R L e R f L e R f L R Ls

ig g Z ZM sy V V qV V qBŽ .L R e L f R e L f R L Rs

ig g Z ZM sy V V qV V qBŽ .R R e R f R e R f R R Rs

and compute the differential cross section

ds s 22 2f < < < <s N M q M 1qcosuŽ .Ž .c L L R R2dV 256p

22 2< < < <q M q M 1ycosu 18Ž . Ž .Ž .R L L R

f Ž . Ž .with N s1 3 for final state leptons quarks andc

y1q2m2rs-cosu-1y2m2rs to be consistentZ ZŽ 2 .with the above approximations t4ym . In anyZ

case we can extend the integration region to the full"1 range without modifying the leading results.

5. Sudakov logs in the cross section and in theH Iforward backward asymmetry for e e ™ f f

We define s and s respectively as the treeB TŽ .level Born cross section and as the total cross

section containing only the one loop double loga-rithms. The explicit expressions for differentfermionic final states are given by:

q ys rs e e ™m mŽ .T B

s1q y1.345 q0.282 a y0.330 aŽ .Box W Z

19Ž .q ys rs e e ™u uŽ .T B

s1q y2.139 q0.864 a y0.385 aŽ .Box W Z

20Ž .q ys rs e e ™d dŽ .T B

s1q y3.423 q1.807 a y0.557 aŽ .Box W Z

21Ž .where

g 2 s s2 y3 2a s log ,2.7 10 logW ,Z 2 2 216 p m mW ,Z W ,Z

With the index ‘‘Box’’ we give the contributionscoming from box diagrams, the rest is from vertexcorrections.

Ž q yFor the forward-backward asymmetry A e eFB.™ f f the analytic expressions are:

T B q yA rA e e ™m mŽ .FB FB

s1q y0.807 q0.770 a y0.002 aŽ .Box W Z

22Ž .T B q yA rA e e ™u uŽ .FB FB

s1q y0.521 q0.454 a y0.023 aŽ .Box W Z

23Ž .T B q yA rA e e ™d dŽ .FB FB

s1q y0.620 q0.508 a y0.029 aŽ .Box W Z

24Ž .' Ž .We see that already at s s1 0.5 TeV the

Ž . y2parameter a , 6 3 10 so that the above cor-Z,WŽ .rections can exceed the ten six percent for the cross

Žsections and a resummation technique which is un-.der study is needed.

In the limit a ,a we can summarize theZ W

above results in:sT

m m ,1y1.39 a ;Ž . Z ,WsB

ATFB

m m ,1y0.04 a ; 25Ž . Ž .Z ,WBAFB

sTu u ,1y1.66 a ;Ž . Z ,W

sB

ATFB

u u ,1y0.09 a ; 26Ž . Ž .Z ,WBAFB

sTd d ,1y2.17 a ;Ž . Z ,W

sB

ATFB

d d ,1y0.11 a ; 27Ž . Ž .Z ,WBAFB

We can make several comments to these results:Ž .Ø Z boson exchange is negative photon-like in the

vertex corrections: it decreases both left and righteffective vertices. In the boxes, Z exchange con-tribution does not gives a double log behavior due


to a cancellation between direct and crossed dia-grams.

Ø W boson exchange, due to his chiral structure,affects only the left gamma vertex proportionallyto yQ XrQ and the left Z vertex to yg X rg ,f f f L f L

giving always contributions that are positive withrespect to the tree level values. Also box dia-grams are peculiar because they affect only theleft-left structure of the amplitude and they al-ways give a negative contribution.

ŽØ In s rs box corrections are dominant moreT B.than three times the vertex ones . Since, as noted

above, box corrections are given only by W ex-change, the e.w. Sudakov corrections are a pecu-liar signature of the left-left structure of the fullamplitude.

Ø In AT rAB Z corrections almost cancel. W con-FB FB

tributions from vertex are accidently almost equaland opposite to the box’s ones leaving a negligi-ble contribution. As a result, the double logsrelative effect is more than one order of magni-tude smaller than for the full cross sections.

Ø The total effect from virtual double logs is nega-tive both for the cross sections and for the asym-metries.

6. Conclusions

We have investigated, in one loop electroweakcorrections, the IR origin of double logs that wedenote as e.w. Sudakov corrections. These Sudakoveffects can be important for next generation of col-liders running at TeV energies since they grow withenergy like the square of a logarithm. In supersym-metric models, loops containing the supersymmetricpartners of the usual particles do not have double log

Žasymptotical behavior i.e., the double logs are pre-.sent but power suppressed . In the SM the e.w.

Sudakov corrections are present with a peculiar chi-ral structure due to W boson exchange dominance; itshould be possible to test the different chiral contri-butions with colliders with polarized beams. In anycase, already for TeV machines, proper resummationof such large contributions seems to be needed; infact for the various cross sections we find thatcontributions of order 5–8% are present for the

q y w xplaned 500 GeV e e NLC 6 . The corrections tothe asymmetries considered in this paper, due to theaccidental cancellation between box and vertices

Žcontributions, are almost negligible one order ofmagnitude smaller with respect to the cross sections

.relative corrections . Sudakov effects in other kind ofŽ .asymmetries for instance polarized asymmetries and

in general in other observables, are currently understudy.

Acknowledgements

The authors are indebted to M. Ciafaloni and C.Verzegnassi for clarifying discussions. A special ac-knowledgement goes to F. Renard for discussionsand a check of some of the computations.

References

w x1 Landau-Lifshits: Relativistic Quantum Field theory IV tome,ed. MIR.

w x Ž .2 V.V. Sudakov, Sov. Phys. JETP 3 1956 65.w x3 M. Kuroda, G. Moultaka, D. Schildknecht, Nucl. Phys. B 350

Ž .1991 25.w x Ž .4 G. Degrassi, A. Sirlin, Phys. Rev. D 46 1992 3104.w x5 M. Beccaria, G. Montagna, F. Piccinini, F.M. Renard, C.

Verzegnassi, hep-phr9805250.w x q y6 Physics with e e Linear Colliders, ECFArDESY LC

Physics Working Group, hep-phr9705442.w x Ž .7 G. Passarino, M. Veltman, Nucl. Phys. B 160 1979 151.

28 January 1999


Critical basis dependence in bounding R-parity breakingcouplings from neutral meson mixing

Katri Huitu, Kai Puolamaki, Da-Xin Zhang¨Helsinki Institute of Physics, P.O. Box 9, FIN-00014 UniÕersity of Helsinki, Finland

Received 21 August 1998; revised 23 October 1998Editor: P.V. Landshoff

Abstract

Assuming one nonzero product of two lX-type couplings and working in two different bases for the left-handed quark

superfields, the neutral meson mixings are used to bound these products. We emphasize the strong basis dependence of thebounds: in one basis many products contribute to neutral meson mixings at tree level, while in the other these productsexcept one contribute at 1-loop level only and the GIM mechanism takes place. Correspondingly, these bounds differbetween bases by orders of magnitudes. q 1999 Elsevier Science B.V. All rights reserved.

1. In the supersymmetric extensions of the stan-dard model, the lepton and baryon numbers are notnecessarily conserved even at the tree-level. Theseviolations of the fermion numbers can be achievedby the breakdown of the so-called R-parity which is

Ž .Ž3 BqLq2 S .defined by Rs y1 , where S is spin ofw xthe field 1 . An important question is then to what

extent the R-parity violating effects can be consistentwith the present experimental data. One class ofthese R-parity and lepton number violating interac-tions is described by the superpotential

W slX L Q Dc i , j,ks1,2 or 3 , 1Ž . Ž .R r i jk i j k

Ž . cwhere under the gauge group SU 2 L , Q and DL i j k

are superfields for the lepton doublet, the left-handedquark doublet and the right-handed quark singlet,respectively. With these interactions special care is

needed since they contain quark mixing effects. An-other complication related to the basis is that in ageneral case the superpotential contains both bilinearand trilinear lepton number violating interactions,among which the R-parity violation can be moved

w xby suitable field redefinitions 2 . Here we will as-sume that the bilinear terms vanish. Effects of leptonmixing with the Higgs sector have been extensively

w x Ž .studied. E.g. in 3 lepton-Higgs -basis independentmeasures of R-parity violation have been discussed.

In this work we will concentrate on the couplingsX Ž . X

l in 1 and bound several products of two l ’si jk i jk

from neutral meson mixings. We will follow thew xpioneering work by Hall and Suzuki 2 who con-

sider the box diagrams which involve a scalar leptonŽ .and a weak or a Higgs boson which corresponds to

the decoupling limit of the chargino or the squarks.w xRecently it has been argued in 4 that these box


( )K. Huitu et al.rPhysics Letters B 446 1999 285–289286

diagrams give strict bounds on many of the productsof the l

X ’s.i jk

We will emphasize that these bounds are sensitiveto separate rotations of the upper and lower compo-nents of the quark doublet. We will demonstrate thehuge effect on the bounds depending on the basischosen. The rotations of the up- and down-type

w xfields are also considered in 5 , but there the strongdependence on the basis is somewhat concealed bythe assumption that only one of the couplings is

w xnonzero. In 5 single couplings are bounded byconsidering neutral meson mixing through boxes in-volving sneutrinos and right-handed squarks usingtwo different bases. In the first basis chosen, there iscontribution to the KK mixing, while in the secondone the contribution to the KK mixing is zeroassuming only one nonzero coupling. In the secondbasis one obtains instead bounds from DD mixing.The bounds from DD mixing in the second basisand from KK mixing in the first basis are of the

Ž y1 .same order of magnitude, OO 10 .The box contribution to the KK mixing induced

by the R-parity violating interactions considered hereis GIM suppressed, as will be shown.

2. Before calculating the diagrams directly, wespecify our conventions for the quark superfields.We will work in a basis in which the right-handedquarks in the superfield Dc’s are the mass eigen-k

states. The left-handed quarks in Q are weak eigen-j

states which are related to the mass eigenstates bythe general rotations

uI sV † u , d I sV † d , 2Ž .L u L L L d L L

and the quark mixing matrix is V'V V † . At thisu L d L

stage we must specify the basis in which we are

working, if we assume that only one product of twol

X’s is nonzero. Although there is no difference inphysics between these bases, assuming that only onecoupling or one product of the couplings is nonzerocorresponds to different assumptions made in differ-

Ž .ent bases in bounding the coupling s . Instead ofdiscussing the most general basis, we choose twodifferent bases: basis I with V s1 and basis IIu L

with V s1. In the basis I, all the products of thed LX Xform of l l contribute at tree-level to KKim1 i n2

mixing. In the basis II, only lX

lX contributes ati21 i12

tree-level, while all the other lX

lX ’s contribute atim1 i n2

1-loop level to the same process. Similar observationfollows in the cases of BB and B B mixings. Whens s

assuming that only one product is nonzero, the dif-ference between these two bases is critical.

3. First we take the basis I: V s1 and thusu L†VsV . We take KK mixing as an example. In thisd L

basis, the tree level FCNC process can be induced byany product of the form l

Xl

X ’s, and we willim1 i n2X Xdenote the coupling l in this basis by l . Theim n im n

Xtree level neutral couplings induced by l can beim nŽ . Ž .read from 1 and 2 , and thus KK mixing is

X Xinduced at tree level by any product of l l asim1 i n2

1X X ) )HH DSs2 sl l V V s d s d . 3Ž . Ž .tree im2 i n1 m1 n2 R L L R2mn

Consequently, if we assume that only one of theseproducts is nonzero, the bounds on these productsare very strong, and they are given in Table 1.

4. Next we take the basis II: V s1 and thusd L

VsV . We start with KK mixing. Assuming thatu L

Table 1X X X XBounds on all the products l l from DM , and on l l from DM . Numbers are given for m s100 GeVn j1 n k 2 K n j1 n k 3 B nd

y9 y8 y7Ž .Ž . Ž .Ž . Ž .Ž .n11 n12 4.5=10 n11 n22 2.0=10 n11 n32 8.1=10y1 0 y9 y7Ž .Ž . Ž .Ž . Ž .Ž .n21 n12 9.8=10 n21 n22 4.5=10 n21 n32 1.7=10y8 y7 y6Ž .Ž . Ž .Ž . Ž .Ž .n31 n12 2.3=10 n31 n22 1.1=10 n31 n32 4.3=10

y6 y5 y3Ž .Ž . Ž .Ž . Ž .Ž .n11 n13 6.0=10 n11 n23 2.7=10 n11 n33 1.1=10y7 y6 y4Ž .Ž . Ž .Ž . Ž .Ž .n21 n13 5.2=10 n21 n23 2.4=10 n21 n33 1.0=10y8 y7 y6Ž .Ž . Ž .Ž . Ž .Ž .n31 n13 2.1=10 n31 n23 1.0=10 n31 n33 3.8=10

( )K. Huitu et al.rPhysics Letters B 446 1999 285–289 287

only one lX product contributes, the box diagrams

give 1

g 22 X Xm n )HH DSs2 s l l s d s dŽ .box im2 i n1 R L L R2 232p mW

3) ) k hV V V V FÝ Ý k1 k m hn h2 X

XsW ,G , H k ,hs1

g 22 X X )s l l s d s dim2 i n1 R L L R2 232p mW

) ) 33 31= V V V V F yF� ŽÝ 3n 32 31 3m X XX s W ,G , H

yF13 qF11 qV V ) F 23 yF 21. ŽX X 21 2 m X X

13 11 ) )yF qF qV V V V.X X 2 n 22 31 3m

= F 32 yF 31 yF12 qF11Ž .X X X X

) 22 21 12 11qV V F yF yF qFŽ .21 2 m X X X X

) 31 11qd V V F yFŽ .n2 31 3m X X

) 21 11qV V F yFŽ .21 2 m X X

) 13 11qd V V F yFŽ .m1 3n 32 X X

) 12 11 11qV V F yF qd d F , 4Ž .4Ž .2 n 22 X X n2 m1 X

where the F i j’s are the contributions from the boxXŽ .diagram Xyeyu yu XsW,G, H . In the sec-˜ i j

Ž .ond equality of 4 , we have used the unitaritycondition VV † sV †Vs1. Note that the last term ofthe second equality has a tree-level correspondence

w xgiven in 6 , which differs from the contribution fromX X

)Ž .l l in basis I by only a factor of 1r V V ;i21 i12 22 11

1. The effect of the box diagram on lX

lX is thusi21 i12

1 The complete calculation is given elsewhere.

negligible 2. We will discard this term below. Ana-lytically,

lnc lne21 11F yF sc qž /e e ey1Ž .

ln t lne31 11F yF s t qž /tye ty1 tye 1yeŽ . Ž . Ž . Ž .

c22 21 12 11F yF y F yF sŽ . Ž .

e

F 32 yF 31 y F12 yF11Ž . Ž .lnc 1q t ln tŽ .

sc y qe tye ty1Ž . Ž .

t 1qe lneŽ .y

e tye ty1Ž . Ž .ln t

2ytcot b yž tye tyhŽ . Ž .lne

qtye eyhŽ . Ž .

lnhy /tyh eyhŽ . Ž .

F 33 yF 31 y F13 yF11Ž . Ž .21q t t 1q2 ey2 ty t ln tŽ .

s t q 2 2tye ty1Ž . Ž . tye ty1Ž . Ž .

t 1qe lne 1Ž .2q y tcot b y2 ž tye tyhŽ . Ž .tye ey1Ž . Ž .

t 2 yeh ln t elneŽ .q y2 2 2tye tyh tye eyhŽ . Ž . Ž . Ž .

hlnhq , 5Ž .2 /tyh eyhŽ . Ž .

2 X X Ž .Our bound for the product l l given in Table 1 fromi13 i31y8 y8w xBB mixing, differs from 3.3=10 given in 4 , and 8=10

w xgiven in 3 from the same process.

( )K. Huitu et al.rPhysics Letters B 446 1999 285–289288

where we have used the following dimensionlessquantities:

csm2rm2 , tsm2rm2 , esm2 rm2 ,c W t W e W˜n

2 2 ² 0: ² 0:"hsm rm , tanbs H r H . 6Ž .H W u d

We have summed over the W,G, H contributions inŽ .Eq. 5 .

Ž . Ž .Eqs. 4 and 5 show explicitly that the GIMcancellations take place also in the FCNC processesinduced by the R-parity violating interactions. TheŽ 33 31. Ž 13 11. 31 11F yF y F yF and F yF terms de-X X X X X X

pend only on the large masses m , m , etc., butt W

these terms are suppressed by the small entries of theCKM matrix V. The other contributions are sup-pressed by the small csm2rm2 . This is a generalc W

feature of the chosen basis. Consequently, thoseproducts of l

X’s which have large CKM factors forthe contributions and which are free from the masssuppression will get strong bounds from KyK mix-ing. In the case of B yB mixing, the mass sup-d d

pressed terms are usually related to the small CKMmatrix elements and are thus less relevant.

The amplitude for KyK mixing can be calcu-lated using vacuum insertion method and equation of

w xmotions for the quarks 7 . The relevant matrix ele-² <Ž .Ž . < :ment turns out to be K dP s dP s K sR L

1 12 2 2 2Ž Ž . .B f M M r M q M q , and the massK K K K s d2 12

Ž² < < :.splitting is Dm sRe K HH K rM . The corre-K K

sponding amplitude for B yB mixing isd d

² < < :B dP b dP b BŽ . Ž .d R L d

21 12 2 2sB f M M r M qM q ,Ž .ž /B B B B b d2 12d d d d

where f s0.15 GeV and f s0.2 GeV. We willK Bd

also use B s0.75, B s1, M qM s0.175 GeVK B s dd

and M qM s4.8 GeV.b d

Our constraints in the basis II are summarized inTable 2, where we use DM s3.49=10y12 MeVK

y1 w xand DM s0.474ps 8 . We also take the charmBd

quark mass as m s1.3 GeV, the top mass as m sc t

175 GeV, and three CKM angles to be sinu s12w x0.219, sinu s0.041 and sinu s0.0035 8 . In23 13

our numerical results we take the slepton masses tobe m sm s100 GeV. We use two representativen e˜ ˜n n

values of tanb : tanbs1 and tanbs50. We are left

Table 2New bounds on the products l

Xl

X from DM , and onn j1 n k 2 K

lX

lX from DM . These bounds are stronger than those givenn j1 n k 3 Bd

in the literature. Numbers are given for m s100 GeV, m qs100e H˜GeV and tanb s1 and 50, separated by a slash. Numbers in theparentheses are for m qs1000 GeVH

d s0 d sp r2 d sp

y3 y3 y4Ž .Ž .n11 n12 1.2=10 1.0=10 8.8=10y3 y3 y4Ž .Ž .n21 n22 1.3=10 1.1=10 9.5=10y4 y5 y5Ž .Ž .n21 n32 1.5=10 9.5=10 6.9=10y5 y5 y5Ž .Ž .n31 n12 2.1=10 2.1=10 2.3=10

Ž .Ž .n31 n32 0.0040r0.027 0.0026r0.018 0.0019r0.013Ž . Ž . Ž .0.022r0.027 0.014r0.018 0.010r0.013

Ž .Ž .n11 n13 0.0035 0.0021 0.0015y4 y4 y4Ž .Ž .n21 n13 4.7=10 4.8=10 4.9=10

Ž .Ž .n21 n33 0.092r0.62 0.058r0.39 0.043r0.29Ž . Ž . Ž .0.51r0.63 0.32r0.40 0.24r0.30

Ž .Ž .n31 n23 0.036r0.059 0.025r0.041 0.019r0.031Ž . Ž . Ž .0.058r0.059 0.040r0.041 0.031r0.031

Ž .Ž .n31 n33 0.0019r0.0031 0.0011r0.0019 0.00081r0.0013Ž . Ž . Ž .0.0030r0.0031 0.0019r0.0019 0.0013r0.0013

with two free parameters: the charged Higgs bosonmass m " and the complex phase d of the CKMH

matrix.Although the mixing in the case of basis II is in

the up-quark sector, note that the DD mixing cannotgive tree level bounds, since in the superpotential theinteractions with l

X-type couplings always containdown-type quarks.

5. We have investigated two different bases of thel

X-type R-parity breaking model. A comparison ofX X X Xthe bounds on ll in Table 1 and ll in Table 2 is

impressive showing the importance of specifying thebasis in which the bounds on couplings are found.Assuming only one nonzero product of the l

X’s butworking in different bases, the bounds can arise fromeither tree or 1-loop diagrams, and the numbers candiffer between the bases by orders of magnitudes.

Acknowledgements

This work is partially supported by the AcademyŽ .of Finland no. 37599 .

( )K. Huitu et al.rPhysics Letters B 446 1999 285–289 289

References

w x1 See, for example, H. Dreiner, hep-phr9707435, and refer-ences therein.

w x Ž .2 L. Hall, M. Suzuki, Nucl. Phys. B 231 1984 419.w x Ž .3 S. Davidson, J. Ellis, Phys. Lett. B 390 1997 210; Phys.

Ž .Rev. D 56 1997 4182.

w x Ž .4 G. Bhattacharyya, A. Raychaudhuri, Phys. Rev. D 57 19983837.

w x Ž .5 K. Agashe, M. Graesser, Phys. Rev. D 54 1996 4445.w x Ž .6 D. Choudhury, P. Roy, Phys. Lett. B 378 1996 153.w x Ž .7 B. Williams, O. Shanker, Phys. Rev. D 22 1980 2853.w x Ž .8 Particle Data Group, Phys. Rev. D 54 1996 1.

28 January 1999


Magnetic monopoles and topology of Yang-Mills theory inPolyakov gauge

M. Quandt 1, H. Reinhardt, A. Schafke 1¨Institut fur Theoretische Physik, UniÕersitat Tubingen, D-72076 Tubingen, Germany¨ ¨ ¨ ¨

Received 15 October 1998Editor: P.V. Landshoff

Abstract

We express the Pontryagin index in Polyakov gauge completely in terms of magnetically charged gauge fixing defects,namely magnetic monopoles, lines, and domain walls. Open lines and domain walls are topologically equivalent tomonopoles, which are the genuine defects. The emergence of non-genuine magnetically charged closed domain walls can beavoided by choosing the temporal gauge field smoothly. The Pontryagin index is then exclusively determined by themagnetic monopoles. q 1999 Published by Elsevier Science B.V. All rights reserved.

Keywords: Yang-Mills theory; Magnetic monopoles; Pontryagin index

1. Introduction

w xRecent lattice calculations 1 give evidence thatconfinement is realized as a dual Meissner effect, at

w xleast in the so-called Abelian gauges 2 . In thesegauges magnetic monopoles arise as obstructions tofixing the coset GrH of the gauge group G, whereH is the Cartan subgroup which is left invariant.Lattice calculations indicate that these monopoles are

w xin fact condensed 3 , a necessary condition for theQCD vacuum forming a dual superconductor.

Magnetic monopoles are long ranged fields andshould hence contribute to the topological propertiesof gauge fields. Furthermore topologically non-trivialfield configurations can explain spontaneous break-

w xing of chiral symmetry 4 . It is therefore interesting

1 Supported by ‘‘Graduiertenkolleg: Hadronen und Kerne’’.

to clarify to which extent magnetic monopoles con-tribute to the topology of gauge fields.

Previously it was shown by one of us that in thePolyakov gauge,

!V x sPexp y dx A s diag , 1Ž . Ž .H 0 0ž /which is a particular Abelian gauge, magneticmonopoles completely account for the non-trivialtopology of Yang-Mills fields. To be specific, con-sider a pure Yang-Mills theory with colour group

Ž .SU 2 . In Polyakov gauge the magnetic monopolesarise at those points in three-space where the

Ž .Polyakov loop V x becomes an irregular element ofŽ . Ž .nithe gauge group, V x s y| s"|, i.e. a cen-i

Ž .tre element for SU 2 . Here n is an integer. Further-i

more, in order for the Pontryagin index to be welldefined and the action finite, the gauge fields have tobecome asymptotically a pure gauge, which in turn


( )M. Quandt et al.rPhysics Letters B 446 1999 290–299 291

implies that the Polyakov loop approaches an angleindependent value at spatial infinity,

n0lim V r ,x s y| . 2Ž . Ž . Ž .ˆr™`

The following expression was derived for the Pon-w xtryagin index 5

nsy ll m , 3Ž .Ý iii

where m denotes the magnetic charge of thei

monopole and ll sn yn is an integer which cani 0i

be interpreted as the invariant length traced out bythe Dirac string in group space.

Ž .The boundary condition 2 allows us to compact-ify our spatial manifold

3 ˙ 3 3 3� 4R ™R sR j ` ,S . 4Ž .In this way, the surface at spatial infinity becomes a

˙ 3 3point of the compactified manifold, R ,S , whichŽ .hosts a magnetic monopole due to the b.c. 2 .

Furthermore on a compact manifold the net magneticcharge of all monopoles has to vanish, Ý m s0.i i

Given this setting, the expression for the Pontryaginw xindex found in Ref. 5 simplifies to

nsy m n , 5Ž .Ý i ii

where the summation is now over all magneticmonopoles including the one at the infinitely distantpoint.

Subsequently the same problem has been treatedw xin a somewhat different fashion in Refs. 6,7 result-

ing in the following expression for the Pontryaginindex,

nsy m , 6Ž .Ý ii

Vsy|

where the summation is performed over the chargesm of magnetic monopoles corresponding to thei

Ž .irregular element Vsy| while in Eq. 5 thesummation is over all monopoles. In addition theinvariant length of the Dirac string enters only in Eq.Ž . Ž . Ž .5 . Formally Eq. 6 results from Eq. 5 by restrict-ing the integers n to n s0,1.k k

In the present paper we will summarize the resultof a thorough investigation of the topological chargein the presence of gauge fixing defects. In particular

Ž .we will show that Eq. 5 is more general than Eq.Ž .6 . While the latter formula gives the correct wind-

ing number only in the absence of domain walls 2,Ž .Eq. 5 includes already the effect of non-genuine

domain walls, which arise when lnV is restricted tofirst Weyl alcove 3.

2. Abelian gauge fixing

Ž .The starting point of the canonical quantizationw xof Yang-Mills theory is the Weyl-gauge 8

A s0 . 7Ž .0

It is generally assumed that in this gauge the dynam-ical fields, i.e. the spatial field components

Ž .A x are smooth functions of space-time. Theis1,2,3

quantity of interest is the gauge invariant partitionfunction for which it is straightforward to derive the

w xfollowing functional integral representation 9,10 :

w x yi nw V xuZs DDm V PeHG

= w xDDA exp yS A s0,A . 8Ž .Ž .H YM 0Ž .b.c. V

Gauge invariance requires here the spatial gaugefields to satisfy the twisted boundary condition

VA ts0,x s A tsb ,x , 9Ž . Ž . Ž .Ž .where AV sV A V † qV E V † is the gauge trans-formed field, and further requires to integrate over

Ž . Ž .all gauge functions V x with the invariant HaarŽ .measure m. Like the dynamical fields A x ,is1,2,3

Ž .the gauge rotation V x can be assumed to besmooth.

Topologically the gauge fields are classified bythe Pontryagin index

1w xn A sy tr FnFŽ .H28p MM

14 a) as d x F F , 10Ž .H mn mn232p MM

where MM is the space-time manifold which wew xchoose to be MMs 0,b =M with the spatial three-

2 w xIn Ref. 6 domain walls were explicitly excluded by assump-tion.

3 A Weyl alcove is a fundamental domain in the Cartan algebrawith respect to the extended Weyl group, i.e. any of the discrete

Ž .symmetries displacements or Weyl symmetries leads out of theŽ . � i xs 34alcove. For GsSU 2 as an example, the Cartan group is e ,

the displacements are x ™ x q2p m and the Weyl symmetry isw xx ™y x whence the Weyl alcove is found to be x g 0,p .

( )M. Quandt et al.rPhysics Letters B 446 1999 290–299292

manifold M. We shall specify M as the one-point3 ˙ 3 3� 4compactification MsR j ` sR ,S .

In the Weyl gauge and with the twisted boundaryŽ .condition 9 the Pontryagin index is given by the

winding number

1w xn V sy tr LnLnL ;Ž .H224p M

LsVPdVy1 11Ž .Ž .of the gauge function V x , i.e.

w x w xn A s0,A sn V . 12Ž .0

Note that this number enters with the vacuum angleŽ .u in the partition function 8 .

For many purposes the twisted boundary condi-tions are inconvenient and it is useful to convertthem to periodic ones,

A tsb ,x sA ts0,x , 13Ž . Ž . Ž .by performing the following time-dependent gauge

w xtransformation 9,10t

y1b

U t ,x sV x , 14Ž . Ž . Ž .which introduces a time-independent temporal gaugefield component

UX y1 y1A s A s0 sUE U syb lnV x .Ž . Ž .0 0 0

15Ž .

From the gauge defined by the previous equation,which is equivalent to

E A s0 , 16Ž .0 0

one arrives at the PolyakoÕ gauge by a time-inde-Ž .pendent gauge transformation V x diagonalizing

Ž .V x and hence A ,0

Vy1V x sV x Pv x PV x ™ v x , 17Ž . Ž . Ž . Ž . Ž . Ž .where v and V live in the Cartan subgroup H;Gand in the coset GrH, respectively. The coset ele-ment VgGrH which diagonalizes VgG is obvi-ously defined only up to an element of the normal-

Ž .izer N H of H in G,

V™gPV , ggN H . 18Ž . Ž .Ž .The normalizer N H is related to the Cartan sub-

Ž .group H by N H sW=H where W denotes theŽ .Weyl group. For the gauge group GsSU N the

Weyl group W is isomorphic to the permutationgroup S . In fact, the Weyl transformations wgWN

permute the diagonal elements of v and are not partof the Cartan subgroup.

Topological obstructions to implementing thePolyakov gauge occur, and these are of three differ-ent types:

Ž .1. The gauge function V x may take values corre-sponding to irregular elements of the gauge groupin which two eigenvalues coincide and the diago-nalization, i.e. the coset element VgGrH, is notwell defined 4. In this case we have local gaugefixing defects, which manifest themselves asmagnetic charges in the induced gauge field AA si

VE V †.i

2. In the diagonalization VsVy1vV the elementsv and V may not be globally defined and smooth

Ž .on M even if V x is smooth and everywhereregular. The point is that the compactification

Ž .imposes certain boundary conditions on V x ,and v or V may fail to obey these conditions.

Ž .3. Functions of matrices like V x are generallydefined by the spectral theorem, i.e. even if diag-

Ž .onalization problems on M are absent, f V canonly be smooth if the function f is holomorphic

Ž .on the spectrum of V x . For the fractional powerŽ . Ž .in Eq. 14 or the logarithm in Eq. 15 this may

be impossible due to the branch cut of the loga-rithm in the complex plane.

The second type of obstructions has been discussedw xin Ref. 11 and we will not consider it here. Further-

more we will see below that the third type of ob-struction is automatically resolved by a proper treat-ment of the defects arising in item one, and hence wewill concentrate in the following on the investigationof the local gauge fixing defects. We will in particu-lar show that in the Polyakov gauge the Pontryaginindex arises entirely from these gauge defects.

Note that the gauge potentials in both the gaugeŽ . Ž .16 and the Polyakov gauge 17 fulfill the periodic

Ž .boundary conditions 13 and thus live on the closedcompact manifold MMsS1 =M. It is then easy to seethat the Pontryagin index arises exclusively from thedefects. As is well known, the integrand in the

4 Ž .For the gauge group GsSU 2 the irregular elements coin-cide with the centre elements "|.


Pontryagin index is a total derivative for non-singu-lar gauge fields,

w xtr FnF sd K A , 19Ž . Ž .where

1Ks tr FnAy AnAnA 20Ž .Ž .3

is the topological current. From E MMs0 and Stokes’theorem we find

1 1nsy d Ksy Ks0 . 21Ž .H H2 28p 8pMM E MM

The crucial observation is here that the gauge fixingdefects for which the coset element VgGrH isill-defined give rise to singular connections AV in

Ž Ž ..Polyakov gauge cf. Eq. 17 , so that at the gaugeŽ .defects Eq. 19 does not apply.

A similar conclusion may be drawn from thew xresults of Ref. 5 where it was shown that the

Pontryagin index, which is trivially invariant undersmall gauge rotations, also does not change under

Ž . Ž .both transformations 14 and 17 , i.e.Ž .12VU Uw x w x w xn A sn A sn A s n V . 22Ž . Ž .

In the Polyakov gauge, however, the gauge functionŽ . Ž .V x is diagonalized and from 17 we infer

w x w y1 x w x w x w xn V sn V qn v qn V sn v s0Ž .unless the coset transformation V x is singular

somewhere on the spatial manifold M. Again weconclude that the winding number of V and thus thePontryagin index in Polyakov gauge arises exclu-sively from the defects.

3. Gauge fixing defects

A group element is called irregular when two ofŽ .its eigenvalues are degenerate. For irregular V x we

can always consider the two degenerate eigenvaluesŽ . Ž .of V x to belong to an SU 2 subgroup of the full

Ž .gauge group GsSU N . Therefore it suffices toŽ .consider the gauge group SU 2 where the irregular

elements are given by the centre elements Vs"|.We define an individual defect D , as usual, as ai

connected set of points for which the smooth map-Ž .ping V x takes on an irregular element,

D s xgM , V x sconsts"| 9M ,� 4Ž .i

D connected. 23Ž .i

Ž .Since V x is time-independent, all defects are staticand it suffices to investigate the three-dimensionalspace M. According to the dimensionality we distin-guish the following defects:

Ž . œ ŽØ p M_D /0: Isolated point defects magnetic2 i.monopoles

Ž . œØ p M_D /0: Closed line defects1 iŽ . œØ p M_D /0: Closed domain walls0 i

Open line and wall defects are topologically equiva-lent to isolated point defects. Similarly, three-dimen-sional defects give merely rise to additional internal

Ž .boundaries of M where the gauge function V xtakes an irregular element. The volume of the three-dimensional defects does not contribute to the wind-

Ž .ing number, since V x is a constant "| there.Such volume defects can therefore be treated analo-gously to the point defects and will not be consid-ered here.

To proceed further we wrap the defects by closedsurfaces which are infinitesimally close to the de-fects, see Fig. 1. The defect D together with itsi

wrapping is denoted by D e. We then cut out thei

defects together with their wrappings from our spa-tial manifold M giving rise to the punctured space

eM sM _ D D , 24Ž .e i i

where each defect gives rise to an internal surfaceenclosing the defect, see Fig. 1. We will assume that

� 4the punctured space M admits a covering X bye a

closed contractible sets Xa

M sD X , 25Ž .e a a

with

X lX sE X lE X 26Ž .a b a b

Ž .such that on each topologically trivial patch X wea

Ž . Ž .have smooth diagonalization maps V x and v xa ay1 Ž .with VsV v V . The triangulation 25 impliesa a a

Ž .that, according to 26 , the closed oriented patchesX intersect precisely on their boundaries. Note thata

the patches X are oriented, and so is their intersec-a

tion. Contrary to ordinary sets, the intersection oper-ator does therefore not commute,

X lX syX lX . 27Ž .a b b a

In the presence of closed domain walls the manifoldM consists of disconnected pieces M , MsD Me a a a


Ž . Ž .Fig. 1. Wrapping of some generic diagonalisation defects. a Point defects and open or closed line defects, b wrapping of these defects byclosed surfaces with infinitesimal volume e .

separated by the domain walls. In each connectedcomponent M the induced gauge field a sa 0

1 Ž .y lnv x can be chosen smoothly, but some careb

Ž . Ž .has to be taken when extending v x and lnv xover different domains M .a

Ž .To see this, recall that V x is smooth over thecommon boundary of two patches X and X , anda b

we have

V x sVy1 x v x V xŽ . Ž . Ž . Ž .a a a

sVy1 x v x V x 28Ž . Ž . Ž . Ž .b b b

Ž .so that the diagonal maps v x are related bya

v x sh x v x hy1 x 29Ž . Ž . Ž . Ž . Ž .a a b b a b

with the transition functions

h x sV x PVy1 x . 30Ž . Ž . Ž . Ž .ab a b

They obviously satisfy the co-cycle condition

h Ph sh . 31Ž .ab bg ag

Ž .From Eq. 29 we infer that h takes values in theab

normalizer NsW=H of the Cartan subgroup Hand consequently, the diagonalizations v and va b

coincide up to a Weyl transformation. Since ourŽ Ž .. œcolor group is simply connected, p SU N s0,1

the picture P;H of a Weyl alcove under the expo-nential map represents a fundamental domain for theCartan group, i.e. any Weyl transformation leads outof P. Thus, by restricting our diagonalization v toP;H, it is possible to choose

v sv 32Ž .a b

smoothly on the overlap of two patches. Further-more, since the branch cut of the logarithm is situ-

ated at the defects, which are excluded from ourmanifold M , and since the subset P;H is simplye

connected, the same is true for the logarithm of ourdiagonalization,

lnv s lnv . 33Ž .a b

On the other hand, there are no overlapping patchesŽbetween different domains M disconnected bya

. Ž . Ž .closed domain walls , and Eqs. 32 and 33 do nolonger hold necessarily. In fact, there is an ambiguityin the choice of fundamental subsets P;H for thediagonalization in the various connected regions M .a

As a consequence, the maps v and v at infinitesi-a b

mally close points on opposite sides of a closeddomain wall are related by

v sv or v sv†a b a b

and

lnv s"lnv q2p iks , kgZ .a b 3

4. The winding number expressed by defects

w xThe integrand in the winding number n V canŽ .be locally i.e. within a patch X expressed as aa

total derivative

tr LnLnL sd G v ,V , 34Ž . Ž . Ž .a

where

w x y1G v ,V sy6 tr AA nv dvŽ .a a

q3 tr AA vy1 nAA v ,Ž .a a

AA sV dVy1 . 35Ž .a a a


Applying Stokes’ theorem, we find

1w x w xn V sy G v ,V . 36Ž .ÝH a224p E Xaa

A patch X can have a common border with anothera

patch X or with a defect D e whence we find forb i

the surface of a patch 5

E X s X lX q X lD e . 37Ž .Ý Ýa a b a ib/a i

Using this decomposition of the surface E X wea

obtain

1w xn V sy 248p

= w xG v ,V yG v ,VŽ .Ý H a bX lXa ba ,b

1w xy G v ,V , 38Ž .Ý H a2

e24p X lDa ia , i

Ž .where we exploited the different orientation 27 ofthe intersection X lX , as seen from X and X ,a b a b

Ž . Ž .respectively. From 35 and 30 we find

†w xG v ,V yG v ,V sy6 d tr h dh lnv .Ž .a b a b a b

39Ž .Ž .Using the cocycle condition 31 , the first integral in

Ž .38 can be rewritten by means of Stokes’ theorem,

w xG v ,V yG v ,VŽ .H a bX lXa b

sy6 tr h dh† lnv . 40Ž .Ž .ÝH ab a beX lX lDa b ii

Here we have decomposed the boundary of X lXa b

Ž Ž ..as cf. Eq. 37

E X lX sD X lX lXŽ .a b g a b g

qD X lX lD e . 41Ž .i a b i

Ž .Turning to the second integral in 38 we observeŽ .that the last term in Eq. 35 does not contribute,

5 Note that this equation defines an orientation for the intersec-Ž .tion operator of the oriented patches.

since it vanishes for e™0, i.e. v™"|. On theŽ .other hand the first term in 35 can be written as

y6 tr AA nv dvy1 sy6 d tr AA lnvŽ .Ž .a a

q6 tr d AA lnv . 42Ž . Ž .a

Hence by using Stokes’ theorem, we obtain for theŽ .second integral in 38

w xG v ,VÝH aeX lDa ia , i

sy6 tr AA lnvŽ .ÝH aeŽ .E X lDa ia , i

q6 tr d AA lnv . 43Ž . Ž .ÝH aeX lDa ia , i

Ž e .Expressing the surface E X lD analogously toa iŽ .Eq. 41 and taking care of the proper orientation of

Ž Ž ..the intersections cf. 27 , we have

E X lD e sy X lX lD e , 44Ž .Ž . Ýa i a b ib

Ž . 6and the first term on the r.h.s. of 43 becomes

y6 tr AA lnvŽ .ÝH aeŽ .E X lDa ia , i

sq3 tr AA yAA lnv . 45Ž .Ž .Ž .Ý H a beX lX lDa b ia ,b , i

Ž . Ž .Using Eq. 30 this term is seen to cancel Eq. 40 soŽ .that the winding number 38 receives a non-vanish-

Ž .ing contribution only from the second term in 43 ,

13 3w xn V sy tr lnvPT Pd AA ,Ž .Ý H a2

e4p X lDa ia , i

AA sV PdVy1 . 46Ž .a a a

Ž . Ž .On a defect D , V x and hence v x are irregulariŽ . Ž .ni Ž Ž ..i.e. v x takes values y| s"| for GsSU 2

and is constant. Hence we obtain

3etr lnvPT sp n . 47Ž . Ž .D ii

6 The relative sign in the last integral is again due to theopposite orientation of the common boundary, as seen from theadjacent patches.


Furthermore the quantity

13m s d AA 48Ž .ÝHi a

e4p X lDa ia

is the magnetic flux through the wrapping surface ofthe defect 7 and hence represents the magnetic chargeof the defect. Applying again Stokes’ theorem and

Ž .Eq. 44 the flux can be expressed as

13 3w xm sy AA V yAA VŽ .Ý Hi a b

e8p X lX lDa b ia ,b

1y1 3s tr h dh PT . 49Ž .Ž .Ý H ab a b

e4p X lX lDa b ia ,b

Ž . Ž . Ž .Inserting Eq. 47 and 48 into Eq. 46 we obtainfor the winding number

w xn V sy n m . 50Ž .Ý i ii

w xThis is precisely the result derived in Ref. 5 for acompact spatial manifold M. Let us stress that, asthe above derivation reveals, all defects with n /0i

carrying non-zero magnetic charge m /0 contributei

to the topological charge.Since a shift of lnv by 2p i leads to the sameŽ .v x , we have a freedom in the choice of xs

Ž .yilnv. Choosing x smooth at the domain walldefect 8 where vs"| will in general lead x

outside the first Weyl alcove. As we will illustratebelow in this case closed domain walls will carry nomagnetic charge and hence do not contribute to thewinding number.

Alternatively we can restrict x to the first Weylw xalcove xg 0,p . Then at the defect the integer

7 Note that the normal vector on the wrapping surface aroundethe defect D points out of the punctured space M , i.e. towardsi e

the defect. By contrast, the intersection X l D e is orienteda ie Ž Ž .. Ž .opposite to ED cf. Eq. 37 , so that our definition 48 yieldsi

the usual sign of the magnetic charge, i.e. the magnetic fluxemanating from the defect.

8 One might think that this is impossible e.g. for a point defectsituated on top of a domain wall. Since the mapping V is smooth,however, such a domain wall is necessarily open and this configu-ration is topologically equivalent to the case of two point defects.

Ž .defined in Eq. 47 is restricted to n s0,1 and Eq.iŽ .50 becomes

w xn V sy m , 51Ž .Ý ii

Vsy|

i.e. only the defects with Vsy| contribute. Re-w xstricting x to the first Weyl alcove xg 0,p im-

plies that x is continuous but not necessarily smoothat the defect, see Fig. 3. In this case closed domainwalls with vsy| now carry twice the magnetic

w xcharge of a monopole and hence contribute to n V

as will be illustrated below.Thus in the generic case where only magnetic

Ž .monopoles and domain walls are present, Eq. 51can be more explicitly written as

w xn V sy m y m .Ý Ýk kŽ . Ž .k magnetic monopoles k domain walls

Vsy|Vsy|

52Ž .We observe that for the smooth parameterization

the magnetic monopoles fully account for the topo-Ž .logical charge 50 , while there is an extra contribu-

Ž .tion in Eq. 52 due to magnetically charged domainw xwalls. In Refs. 6,7 this domain wall contribution to

w xn V was not included although xsyilnv wasrestricted to first Weyl alcove.

5. Hedgehog field as generic example

Let us finally illustrate our result for the twoŽ . Ž .different methods Eqs. 50 and 51 by means of a

specific example. The prototype of smooth maps˙ 3 Ž .V : R ™SU 2 with non-vanishing winding number

is provided by the well-known hedgehog configura-tion,

V x sexp ix r Px sŽ . Ž .Ž .ˆs|Pcos x r q i x sPsin x r ;Ž . Ž .ˆ

< <r' x . 53Ž .For this map to be smooth, we have to avoid thesingularity at the origin where x is ill-defined,ˆsin x 0 s0 m x 0 sn Pp ; n gZ .Ž . Ž . 0 0

54Ž .3 ˙ 3Furthermore, the compactification R ™R implies

Ž . < <that V x be angle-independent as rs x ™`, i.e.


˙ 3 Ž . Ž . Ž . Ž .Fig. 2. Smooth assignment of alcoves for a hedgehog type of mapping V : R ¨SU 2 . a Defect structure of V . b Profile x rŽ . < < Ž .lnvsyixs as a function of rs x . The Weyl alcove is changed between the defects, such that x r becomes globally smooth.3

the x-dependent part of V must vanish as r™`,ˆwhence

sin x ` s0 m x ` sn Pp ; n gZ .Ž . Ž . ` `

55Ž .

The winding number of such a hedgehog map isdetermined by the difference of the two integers

w xfrom the profile boundary conditions via n V sn`

yn . For definiteness we choose0

w xn s0 , i.e. n sn V . 56Ž .0 `

As expected, there is a continuous diagonalisation ofthis map,

v x sv r sexp ix r s , 57Ž . Ž . Ž . Ž .Ž .˜ 3

where the profile x reflects our choice of Weyl˜alcoves in the connected regions separated by the

domain walls. There are two basically differentchoices:1. Choose the alcoves in every connected region

Ž . Ž .such that x r is globally smooth, i.e. x r s˜ ˜Ž .x r , see Fig. 2.

2. Exploit the arbitrariness x™xq2p and theWeyl symmetry x™yx to fix the Weyl alcove

˙ 3w xxg 0,p for all connected regions of R , i.e.˜ e

Ž .x r is the profile obtained from the original˜Ž .x r by reflections at the boundary of the alcove,

Ž . Ž .see Fig. 3. While both v x and lnv x are stillcontinuous, they fail to be smooth across thedomain walls.

Note that we may enclose both the monopoles andthe domain walls by simple 2-spheres S2 so that it isconvenient to switch to spherical coordinates x™

˙ 3Ž .r,q ,w on R . We may cover every connected˙ 3region in the punctured space R by just two coordi-e

nate patches X , so that their intersections with the"

˙ 3 Ž . Ž . Ž . Ž .Fig. 3. Fixed assignment of alcoves for a hedgehog type of mapping V : R ¨SU 2 . a Defect structure of V . b Profile x r˜Ž . < < w x Ž .lnvsyixs as a function of rs x . The Weyl alcove is rigidly fixed to xg 0,p such that x r is reflected at the alcove boundary,˜ ˜ ˜3

i.e. the profile is continuous, but not smooth.


2 Ž 2 .wrapping sphere S yield the northern S andqŽ 2 .southern S hemisphere. In each of these con-y

tractible patches, we find a smooth diagonalisingŽ .coset lift V r,q ,w ."

Ž . ŽIn the case of a smooth choice of x r see Fig..2 , the hedgehog V can be diagonalized by the same

set of coset lifts V for all r. An explicit form is"

w xgiven by 5 ,q

V x sexp i e s ;Ž .q wž /2

V x sh x PV x , 58Ž . Ž . Ž . Ž .y " q

where the transition function

h sexp iw sŽ ." 3

has support in the overlap between the two hemi-p1spheres, i.e. on the equator S defined by qs .2

From these expressions, we can easily calculate themagnetic flux of the induced Abelian potential AA3 s"

V dVy1 3. The magnetic field is always directedŽ ." "

Ž .radially outwards pointing to infinity , and to everyŽ .wrapping surface, we assign a magnetic charge 48

1 13 3 3msy d AA sy AA yAAŽ .ÝH H" q y

2 2 24p 4pS S lS" q y"

1s dws"1 .H

12p equatorS

Recall that the sign of this charge is determined bythe orientation of the intersection X lD e, which isa i

opposite to the orientation of the wrapping surface.Thus, if the surface E D e encloses the defect from thei

Ž .outside i.e. it is closer to infinity than the defect ,the intersection X lD e is aligned with the mag-a i

Ž .netic field whence the defect has charge q1 . Onthe other hand, a wrapping surface closer to the

Ž . Žorigin than the defect is given a charge y1 see.also Fig. 4 .

Ž . ŽThus, for a smooth choice of profile x r seeŽ ..Figs. 2 and 4 left , the magnetic charges of the two

surfaces S2 wrapping the domain wall cancel andŽ .there is no net intrinsic magnetic charge on the

domain walls. Hence the magnetic field goessmoothly through the domain wall without noticing

Ž .its existence. Thus for a smooth choice of x r thedomain walls do not contribute to the Pontryaginindex. The Pontryagin index is entirely determinedby the two monopoles at rs0 and rs`. In fact

Ž .since x 0 s0, only the monopole at infinity con-Ž . w xtributes. From x ` sn V p and the magnetic

Ž .charge m sy1 of this monopole, Eq. 50 yields`

the correct winding number.Consider now the alternative case of a fixed

assignment of alcoves as in Fig. 3. In order to reflectŽ . w xx r in the alcove 0,p , we need to combine the˜

coset lift V from the smooth case above with a"

Weyl flip in the transition functions h . This in turn"

leads to a change in the orientation of the magneticfield inside the shaded region of Fig. 3: In this

Ž . Ž .Fig. 4. The ns3 hedgehog mapping with continuous profile left and restricted profile right . Solid circles represent domain wall defects,dashed circles represent the wrapping surfaces and the dot in the origin symbolises the magnetic monopole. The numbers in the boxes

Ždenote the magnetic charge of the respective wrapping, which is determined from the relative orientation of the surface not indicated for. Ž .clarity and the magnetic field the radial arrows .


domain, the field is directed radially inwards, i.e.Žtowards the origin, while we have the usual out-

.ward orientation in the remaining space. The orien-tation of the wrapping surfaces is still determined bya normal vector pointing towards the defect. Taking

Žthe specific example ns3 for simplicity see Fig. 4Ž ..right , we encounter the following defects:

Ž .Ø The monopole in the origin is a q| -defect andw xdoes not contribute to n V .

Ø The first domain wall has two wrapping surfaces.ŽThe inner one is directed outwards towards the

.domain wall and this coincides with the orienta-tion of the magnetic field. From the rules ex-plained above, it is assigned a charge mysy1.The outer wrapping surface is directed inwardsŽ .towards the defect and this also coincides withthe flipped orientation of the magnetic field in theshaded region. It thus also carries a charge mqsy1. Altogether, the first domain wall hence car-ries a total magnetic charge msy2.

Ž .Ø The next domain wall is a q| -defect and againdoes not contribute. However, it carries magneticcharge q2 and flips the magnetic field back topointing outwards.

Ø The monopole at infinity is a Vsy| defectŽ .and has charge y1 since the magnetic field has

again its standard orientation pointing to infinity.Ž .With these observations, our formula 52 gives the

correct result

w xn V sy y2 q y1 sq3 .Ž . Ž .On the other hand, leaving out the domain walls as

w xwas done in a similar analysis in Refs. 6,7 wouldw xpredict the incorrect result n V s1 for the presently

considered example.

6. Summary and conclusions

We have investigated the topological charge ofYang-Mills fields in Polyakov gauge. Our main re-

Ž . Ž .sults are given by Eqs. 50 and 51 and can besummarized as follows: If the temporal gauge field

Ž . ŽA x is chosen smoothly across closed domain0. w xwalls as in Ref. 5 , the Pontryagin index of a

generic field configuration is entirely given by mag-netic monopoles. On the other hand, if xsb A is0

restricted to the first Weyl alcove, in addition mag-netic charges for the closed domain walls arise which

Ž .also contribute to the Pontryagin index, see Eq. 52 .Other, open, magnetically charged defects, like

open domain walls or lines, are topologically equiva-lent to magnetic monopoles and can be treated in thesame way.

Although these results have been obtained inPolyakov gauge, we believe that they are generic forall Abelian gauges. In fact, recent lattice calculations

w xperformed in the maximum Abelian gauge 12 showŽalso clear correlations between the topological

.charge of the instantons and magnetic monopoles.

Acknowledgements

We thank M. Engelhardt for a careful reading ofthe manuscript and critical remarks.

References

w x1 G.S. Bali, V. Bornyakov, M. Mueller-Preussker, K. Schilling,Ž .Phys. Rev. D 54 1996 2863.

w x w x Ž .2 G. ’t Hooft, Nucl. Phys. B FS3 190 1981 455.w x3 A. Di Giacomo, hep-latr9809014, talk presented at QCD’98,

Montpellier.w x Ž .4 T. Banks, A. Casher, Nucl. Phys. B 169 1980 103.w x Ž .5 H. Reinhardt, Nucl. Phys. B 503 1997 505.w x6 C. Ford, U.G. Mitreuter, T. Tok, A. Wipf, J.M. Pawlowski,

hep-thr9802191, Monopoles, Polyakov-Loops and GaugeFixing on the Torus.

w x7 O. Jahn, F. Lenz, hep-thr9803177, Structure and Dynamicsof Monopoles in Axial-Gauge QCD.

w x Ž .8 R. Jackiw, Rev. Mod. Phys. 52r4 1980 661.w x9 K. Johnson, L. Lellouch, J. Polonyi, Nucl. Phys. B 367

Ž .1991 675.w x Ž .10 H. Reinhardt, Mod. Phys. Lett. A 11 1996 2451.w x Ž .11 M. Blau, G. Thompson, Comm. Math. Phys. 171 1995 639.w x12 H. Markum, W. Sakuler, S. Thurner, Nucl. Phys. Proc.

Ž .Suppl. 47 1996 254; M. Feuerstein, H. Markum, S. Thurner,talk presented at Hirschegg’97, hep-latr9702006.

28 January 1999


Absence of fifth-order contributions to the nucleon mass inheavy-baryon chiral perturbation theory

Judith A. McGovern, Michael C. BirseTheoretical Physics Group, Department of Physics and Astronomy, UniÕersity of Manchester, Manchester, M13 9PL, UK

Received 25 August 1998; revised 16 November 1998Editor: P.V. Landshoff

Abstract

We have calculated the contribution of order M 5 in the chiral expansion of the nucleon mass in two-flavourp

heavy-baryon chiral perturbation theory. Only one irreducible two-loop integral enters, and this vanishes. All othercorrections in the heavy-baryon limit can be absorbed in the physical pion-nucleon coupling constant which enters in theM 3, term, and so there are no contributions at M 5. Including finite nucleon mass corrections, the only contribution agreesp p

with the expansion of the relativistic one-loop graph in powers of M rm , and is only 0.3% of the M 3 term. This is anp N p

encouraging result for the convergence of two-flavour heavy-baryon chiral perturbation theory. q 1999 Published byElsevier Science B.V. All rights reserved.

The fundamental degrees of freedom of the stronginteraction are quarks and gluons, but in spite of themany successes of QCD in describing high energyphenomenology, a full description of the particlesthat constitute ordinary matter still eludes us. Thereis however increasing interest in the interactions ofsuch particles at low energies as new, precise, dataon nucleon properties and interactions becomesavailable. For low enough energy it has long beenknown that the interaction of pions is governed only

Ž .by the symmetries of QCD, in particular SU 2 =Ž .SU 2 chiral symmetry, and a successful systematic

effective field theory of pions, Chiral Perturbationw xTheory 1 , has been developed. In this theory the

pionic lagrangian is expanded as a power series, witheach subsequent term having two more derivatives orpowers of the pion mass than the previous one, andwith the lowest order being two: LL sLL Ž2.qLL Ž4.

pp pp pp

q . . . . The expansion is in powers of qrL, where q

is a momentum or pion mass, and L is the scale ofthe physics which has not been included explicitly,for instance the r meson. Because chiral symmetryrequires the pion-pion scattering length to vanish inthe chiral limit as the pion momentum goes to zero,loop diagrams with vertices from LL Ž2. only con-pp

tribute at fourth order; divergences are cancelled bycounterterms in LL Ž4., and so on. At fourth orderpp

eight a priori undetermined low energy constantsŽ .LEC’s enter, which have to be fit to data. Thecurrent state of the art is calculation to sixth order,

w xinvolving two-loop integrals 2 .Attempts to expand the success of the pionic

theory by including nucleons initially ran into diffi-culties; since the nucleon mass is not small, it spoilsthe expansion of physical amplitudes in powers of

Ž .small quantities masses and momenta , and the the-w xory is not systematic 3 . However if the nucleon

mass is taken to be infinite, it effectively decouples,


( )J.A. McGoÕern, M.C. BirserPhysics Letters B 446 1999 300–305 301

and the power counting is restored. The Lagrangianexpansion includes both odd and even numbers ofderivatives, LL sLL Ž1.qLL Ž2.q . . . . Only evenp N p N p N

powers of the pion mass enter, since the underlyingparameter is the quark mass; M 2 ;M . Furthermorep q

corrections for a finite mass can be included system-atically, with a simultaneous expansion in powers of1rm ; since m ;L;M these terms can naturallyN N r

be fitted into the expansion above; thus LL Ž2. con-p N

tains terms of order 1rm , and LL Ž3., of orderN p N

1rm2 , etc.N

The heavy-baryon expansion starts by splitting thenucleon momentum into two parts, p smÕ q l ,m m m

with Õ2 s1 and ÕP l<m. The projection operators" Ž .P s 1"Õu r2 are used to split the nucleon spinorÕ

yi mÕP xŽ .into two parts, C s e H q h , and the‘‘small-component’’ degrees of freedom h are inte-grated out. For details, the reader should consult, for

w xinstance, Ref. 4 .Ž .Heavy-baryon chiral perturbation theory HBCPT

has been applied to a number of problems involvingthe electromagnetic properties of nucleons and theinteraction of nucleons and pions. The number ofterms to a given order however is much larger thanin the pionic theory; there are for instance sevenLEC’s in LL Ž2. and twenty three in LL Ž3.. So far thep N p N

full Lagrangian has only been worked out to thirdorder, though individual terms have been consideredat higher order. By no means all of the LEC’s tothird order have been determined from experiment.Although many of the calculations which have beendone show the method to be promising, the conver-gence of the expansion of amplitudes in powers ofthe physical pion mass is hard to judge so far. Inparticular, almost all calculations so far have been toone-loop order only 1.

One of the quantities which, through its relativesimplicity, lends itself to such an analysis, is the

Ž .nucleon mass shift due to the finite quark pionmass, or equivalently the pion-nucleon sigma com-mutator, s sM 2E m rE M 2, estimated from scat-p N p N p

w x 3tering data to be 45"8 MeV 6 . To order q there

1 w xBernard et al. 5 consider the contribution of two-loop dia-grams to the imaginary part of the nucleon isoscalar electromag-netic formfactor, but this does not require the evaluation oftwo-loop integrals.

are two contributions, with one LEC which has nowbeen estimated independently from pion-nucleon

w x 4scattering data 8,7 . To order q more LEC’s willenter, for which there we as yet have no experimen-tal handle. Borasoy and Meißner have attempted toestimate fourth order LEC’s in three flavour HBCPT

w xthrough the principle of resonance saturation 9 .There are however good reasons to distrust threeflavour calculations, at least without an explicit de-cuplet, since nucleonic excitations which are being

Ž .kept such as SK are much higher in energy thanŽothers which have been integrated out the delta and

. w x Žhigher resonances 10 . Convergence of the three-w x .flavour result also appears to be poor 11 . However

the mass shift is one quantity where it is possible tolook at the order q5 term without knowing the q4

piece, and this is what we have done in this paper.In order to calculate most quantities to order q5,

the expansion of the nucleonic Lagrangian up toLL Ž5. would be required. The only contribution top NŽ .S 0 from the fifth order term would be a simple

counterterm of order M 5. However the Lagrangian isp

analytic in the quark masses, that is in M 2, so such ap

Žterm cannot exist. For the same reason any irre-ducible two-loop diagrams must give finite contribu-

Ž .tions to S 0 , since there can be no counterterm to.cancel divergences. Similarly, in the absence of

mass insertions of order M , LL Ž4. cannot contributep p N

at this order. All the relevant Feynman amplitudesfor these calculations can be found in the work of

w xMeißner et al. 4,7 or, using an alternative form ofŽ3. w x ŽLL , in that of Mojzis 8 who also gives the rele-ˇ ˇp N

Ž4..vant amplitudes from LL .pp

The heavy-baryon propagator is given by

Sy1 svyS v ,k , 1Ž . Ž .

where the nucleon momentum is written as psmÕ

qk, m is the bare mass, and vsÕPk. The massshift d msm ym is the value of v for which theN

propagator has a pole at zero three-momentum:

d myS d m ,0 s0. 2Ž . Ž .

ŽOf course the mass shift could be found from thepole of the propagator for any three-momentum, butas HBCPT is constructed to respect Lorentz invari-

.ance, the result will not change.

( )J.A. McGoÕern, M.C. BirserPhysics Letters B 446 1999 300–305302

Fig. 1. The one-loop diagram of S Ž3.. Solid lines are nucleons anddashed lines, pions.

Ž .In order to solve Eq. 2 to a given order in M ,p

Ž .both d m and S v must be expanded in powers of3 Ž . Ž . w xM . To order M , S d m sS 0 , so 4p p

3g 2 M 3A pŽ2. Ž3. 2d m qd m sy4c M y , 3Ž .1 p 232p Fp

where the second term comes from the diagram inŽn. Ž n.Fig. 1. Writing as S the expression for the O q

Žn.Ž .part of S, which will have an expansion S v sa M n qa M Žny1.vq . . . , we obtain1 p 2 p

d mŽ5.sS Ž5. 0 qd mŽ2.S Ž4. X 0 qd mŽ3.S Ž3. X 0Ž . Ž . Ž .2 XX1 Ž2. Ž3.q d m S 0 , 4Ž . Ž . Ž .2

where primes indicate derivatives with respect to v.Any calculation in HBCPT yields an answer in termsof the bare Lagrangian parameters, M, g,F and m tolowest order, which are the first terms in an expan-sion in powers of M of M , g ,F and m . It isp A p N

customary to replace the bare parameters by thephysical ones so that the lowest order predictions donot change as higher orders are added. This howevergives an extra contribution to the higher order calcu-lations. In this case, therefore, to the calculation of

Ž5.Ž .S 0 from the diagrams of Fig. 2 must be added aŽ3.Ž .piece from S 0 . Here the relevant parameters are

M and the pseudo-vector p N coupling f 'p p NNŽ Ž 2 ..g rm sgrF 1qO M .p NN N p

Ž5.Ž .The diagrams which contribute to S 0 areshown in Fig. 2. They consist of two-loop diagrams,and one-loop diagrams with either one insertion fromLL Ž3. or LL Ž4., or two insertions from LL Ž2.. In allp N pp p N

cases only zero external momentum is required. Thetwo-loop diagrams shown in Fig. 2f–i all vanishtrivially. The diagram in Fig. 2a can after some

Ž 2 Ž . 4.algebra be written as 3M r4 2 dy3 F I, where

ddl ddk 1Is .H 2 d 22 2 22p ÕPk M y l M y ky lŽ . Ž . Ž .Ž .

5Ž .

This integral can be done by using Feynman parame-ters, first to combine the mesonic propagators, and

Fig. 2. Contributions to S Ž5.. Solid dots represent insertions fromŽ3. Ž4. Ž2. ŽLL and LL , and crosses from LL both include fixed termsp N p p p N

.from the expansion in 1rm .N


Fig. 3. Contributions to S Ž4..

then to include the heavy-baryon propagator; thisyields

52 dy5 1r2M p G ydž / 1 Ž .2 1yd r22Isy xyx dxŽ .Hd

04pŽ .5 3yd

2 dy5 dy2M 2 pG yd Gž / ž /2 2sy 6Ž .dd4p G 2yŽ . ž /2

which tends to zero as d™4.The same integral also appears in the evaluation

of Fig. 2b and 2d, and it is the only non-separabletwo-loop integral which does. Since it vanishes, allthe two-loop diagrams 2a–e are proportional to

Ž . w x Ž .D J 0 , in the notation of 4 . The integral J 0 sp 0 0

yMr8p is the one which enters the one-loop dia-2 Ž .gram, Fig. 1, and D s2 M L M is just the inte-p

Žgral of the meson propagator and diverges as 1r d.y4 . These divergences are cancelled by the graphs

of Fig. 2j–l with counterterm insertions from LL Ž3.p N

and LL Ž4., which however bring in the low energypp

constants 2 d yd from the baryonic, and l and16 18 3Žl from the mesonic Lagrangians. The notation is4

w xthat of 7 , with LEC’s defined to absorb the usualŽ . w xfactors of log Mrm ; in 8 d ™d and l ™n nq1 n

2 Ž2..l r16p . The graphs with two insertions from LL ,n p NŽFig. 2o–q, are all finite. Counterterm graphs 2m-o

.all give vanishing contributions. The final contribu-w xtion of Fig. 2, with the conventions of 7 , is

S Ž5. 0Ž .2y loopqCT

2 53g M 2 l y3l 4 2 d ydŽ .4 3 16 18s y2 2ž g32p F F

g 2 1 6c1 2q q q q24c . 7Ž .12 2 2 /m32p F 8m

The next contribution to d mŽ5. is from the lastŽ .three terms of Eq. 4 ; the relevant graphs are shown

in Fig. 3 and Fig. 1, and the integrals are finite. Theresult is

2X X XX1Ž3. Ž3. Ž2. Ž4. Ž2. Ž3.d m S 0 qd m S 0 q d m S 0Ž . Ž . Ž . Ž .2

3g 2M 5 3g 2 3s y4c q12c1 12 2 2 ž /ž 2m32p F 32p F

q24c2 . 8Ž .1 /It may be seen that all terms involving the LEC c1

Ž . Ž .vanish from the sum of Eqs. 7 and 8 .Ž5.Ž .Finally we need the contribution to S 0 from

Ž3.Ž .replacing the bare constants in S 0 with theirŽ Ž3.Ž .physical values. The expression for S 0 may be

Ž3. Ž .found from that for d m in Eq. 3 by reinstating.the bare coupling constants. The diagrams which

Žcontribute to the renormalised p N coupling f sp NN.g rm are given in Fig. 4.p NN N

The last diagram of Fig. 4 indicates the contribu-tion from the expansion of the pion and nucleonwavefunction renormalisation to order M 2. Somep

comment is required about Z , which is defined asN

the residue of the nucleon propagator at the pole.This has recently been the subject of a paper by

w xEcker and Mojzis 12 , who point out a correctionˇ ˇwhich is necessary in order to reproduce the resultsof a relativistic calculation in HBCPT. It arisesbecause a nucleon can also be created by the elimi-nated ‘‘small’’ component of the relativistic nucleonfield, and so the normalisations of the relativistic andheavy baryons do not match. The same correctionwas included through the spinor normalisation by

Fig. 4. Contributions to the pion-nucleon vertex of order M 2.p

( )J.A. McGoÕern, M.C. BirserPhysics Letters B 446 1999 300–305304

w xFearing et al. 13 . The net effect of including thesmall-component sources, to order q3, is to give

Z s 1qk 2r4m2 Z HB , 9Ž .Ž .N N N

where Z HB is calculated purely from the HBCPTNŽLagrangian. Whereas in the relativistic theory Z isN

a constant, in HBCPT it may depend on the on-shell. w xthree-momentum k. In the framework of Ref. 7 ,

which we have been using here,

Z HB s1yk 2r4m2 q . . . , 10Ž .N N

where other terms of order q2 have been suppressed.Thus in this framework, the dependence on k can-cels to this order 2. With this small-component-

Ž 2 .source correction, we find that there are no O 1rmŽ w xcorrections to f . In 12 such terms are alsop NN

shown to be absent from g , so the usual expressionAw xfor the Goldberger-Treiman discrepancy 4,7 , pro-

.portional only to d , holds. Thus we obtain fror the16

physical pion mass and p N coupling constant,2 2 2 2M sM 1q2 l M rF 11Ž .Ž .p 3

2g g l42f s 1yM qp NN 2 2 2žF 16p F F

4d y2 d16 18y /g

Ž3.Ž .and substituting in s 0 to obtain the final contri-bution to d mŽ5. gives

2 5 23g M g 2 l y3l4 3Ž5.S 0 s y yŽ .1- loop 2 2 2 2ž32p F 8p F F

4 2 d ydŽ .16 18q . 12Ž ./g

Ž . Ž . Ž .Collecting all contributions, Eqs. 7 , 8 and 12 ,we obtain our final result,

3g 2 M 3 M 2p NN p pŽ3. Ž5.d m qd m sy 1y . 13Ž .2 2ž /32p m 8mN N

Thus in the heavy-baryon limit the order M 5 contri-p

bution vanishes, with all corrections being absorbed

2 w xIn 12 , Z in this framework is given wrongly, since theNŽ 2 . Ž . w xO 1rm kinetic energy insertion, Eq. C1 of 7 , has been

missed. We have repeated our calculations in Ecker and Mojzis’sˇ ˇframework, and obtained the same final result.

in the physical pion mass and pion-nucleon couplingconstant in the M 3 contribution. For finite nucleonp

mass, the correction is just that obtained obtained ifthe relativistic one-loop contribution is expanded in

w x Žpowers of M rm 3 . Since the 1rm terms in thep N N

HBCPT Lagrangian are constructed to respectLorentz invariance, this agreement is reassuring but

.certainly not surprising.There remains the question of why there is no

other fifth order correction, apart from the renormali-sation of the coupling constants of the lowest-ordertheory and 1rm corrections. In fact the only thingN

which could give such corrections would be irre-ducible two-loop integrals from the diagrams of Fig.2a, 2b and 2d. As detailed above, only one suchintegral enters, and in four dimensions it vanishes.This vanishing seems to be essentially accidental. Itdoes not occur in odd dimensions, nor in all proba-

Žbility for unequal meson masses. We have been ableto calculate it for one massless and one massivemeson in four dimensions, with a finite but non-zero

.result.Interestingly, it has been conjectured that the main

contributions at order q5 would come from inser-tions of vertices from LL 2,3 in one-loop diagramsNp

w xand not from genuine two-loop diagrams 9 . Sincethe two-loop integral vanishes, this is trivially satis-fied; however the extent to which other terms areabsorbed in the physical coupling constants has nothitherto been realised.

Ž .As set out in Eq. 13 , the third and fifth orderpieces in the expansion of the nucleon mass inpowers of M are now known. For a full picture ofp

the convergence, we need to know the second andfourth order pieces. The latter have now been calcu-

w xlated, 14 and fall into two categories; pieces involv-ing new LEC’s from the fourth order Lagrangian,and fixed pieces. The latter are very small, but theLEC’s in the former are not known. As well asentering the nucleon mass, these affect the pion-nucleon sigma term and p N scattering. If c were1

well determined from scattering, the analysis ofwhich has currently been carried out only to orderq3, then comparison with the sigma term would tellus whether or not there was still a significant dis-crepancy to be made up from fourth or higher ordercontributions. Unfortunately the analyses are not sen-sitive to c , though the preferred value would leave a1


substantial M 4 contribution. None-the-less it wouldp

appear at least that the odd and even power series areconverging separately.

References

w x Ž . Ž .1 J. Gasser, H. Leutwyler, Ann. Phys. NY 158 1984 142;Ž .Nucl. Phys. B 250 1985 465.

w x2 For a review see J. Bijnens, hep-phr9710341.ˇw x Ž .3 J. Gasser, M.E. Sainio, A. Svarc, Nucl. Phys. B 307 1988

779.w x4 V. Bernard, N. Kaiser, U-G. Meißner, Int. J. Mod. Phys. E 4

Ž .1995 193.

w x5 V. Bernard, N. Kaiser, U-G. Meißner, Nucl. Phys. A 611Ž .1996 429.

w x6 J. Gasser, H. Leutwyler, M.E. Sainio, Phys. Lett. B 253Ž .1991 252, 260.

w x7 N. Fettes, U-G. Meißner, S. Steininger, Nucl. Phys. A 640Ž .1998 199.

w x Ž .8 M. Mojzis, Eur. Phys. J. C 2 1998 181.ˇ ˇw x Ž . Ž .9 B. Borasoy, U-G. Meißner, Ann. Phys. NY 254 1997 19.

w x Ž .10 M.K. Banerjee, J. Milana, Phys. Rev. D 52 1995 6451.w x11 J.F. Donoghue, B.R. Holstein, hep-phr9803312.w x Ž .12 G. Ecker, M. Mojzis, Phys. Lett. B 410 1997 266.ˇ ˇw x13 H.W. Fearing, R. Lewis, N. Mobed, S. Scherer, Phys. Rev. D

Ž .56 1997 1783.w x14 S. Steininger, U-G. Meißner, N. Fettes, J. High Energy Phys.

Ž .9809 1998 8.

28 January 1999


Finite-size effects and operator product expansions in a CFT ford)2

Anastasios C. Petkou 1, Nicholas D. Vlachos 2

Department of Physics, Aristotle UniÕersity of Thessaloniki, Thessaloniki 54006, Greece

Received 2 April 1998Editor: L. Alvarez-Gaume

Abstract

Ž . Ž .The large momentum expansion for the inverse propagator of the auxiliary field l x in the conformally invariant O Nvector model is calculated to leading order in 1rN, in a strip-like geometry with one finite dimension of length L for

Ž .2-d-4. Its leading terms are identified as contributions from l x itself and the energy momentum tensor, in agreementwith a previous calculation based on conformal operator product expansions. It is found that a non-trivial cancellation takesplace by virtue of the gap equation. The leading coefficient of the energy momentum tensor contribution is shown to berelated to the free energy density. q 1999 Elsevier Science B.V. All rights reserved.

The effects of finite geometry in systems nearsecond order phase transitions points are of greatimportance for statistical mechanics and quantumfield theory. These effects are largely explained by

w xthe theory of finite size scaling 1 . On the otherhand, from a purely field theoretical point of view,second order phase transitions are connected to con-

Ž .formal field theories CFTs . In ds2 spacetimedimensions, there is an almost complete understand-ing of CFTs in terms of operator product expansionsŽ . w xOPEs 2 . Recently, there has been some progresstowards understanding CFTs for d)2 in terms ofOPEs plus some additional algebraic conditions such

w xas the cancellation of ‘‘shadow singularities’’ 3–5 .


The CFT approach to finite size scaling at criticalityis based on the observation that the OPE, being ashort distance property of the critical theory, is in-sensitive to finite size effects. This means, for exam-ple, that the finite size corrections of two-pointfunctions are directly related to specific terms in theOPE. Such an approach has been very successful

w xwhen applied in ds2 6 .In the present work, the conformally invariantŽ .O N vector model is investigated in 2-d-4, in

order to demonstrate that OPE techniques can beuseful for studying finite size effects of critical sys-tems in d)2 as well. The geometry is taken to bestrip-like with one finite dimension of length L andperiodic boundary conditions. In this case, for 2-dF3 the critical gap equation has only one ‘‘massive’’solution, while, for 3-d-4 the gap equation hasboth a ‘‘massive’’ and a ‘‘massless’’ solution. We


( )A.C. Petkou, N.D. VlachosrPhysics Letters B 446 1999 306–313 307

study the large momentum expansion for the inverseŽ .propagator of the auxiliary field l x , which is

equivalent to an OPE. The leading scalar and tensorcontributions to this OPE for the ‘‘massive’’ and‘‘massless’’ solutions of the critical gap equationabove are identified. In both cases the results are inagreement with what would be expected from ab-stract CFT in d)2. Most importantly, for the‘‘massive’’ solution of the gap equation, a non-triv-ial cancellation is found to take place inside the

Ž .inverse propagator of l x . This is similar to thew x‘‘shadow singularities’’ cancellation found in 4,5 .

Finally, it is shown that the leading coefficient of theŽ .energy momentum tensor T x contribution is re-mn

lated to the free energy density of the model and alsoto C , the latter being the overall universal scale inT

Ž .the two-point function of T x .mn

Ž .The Euclidean partition function of the O Nvector model is given by

1a d aZs DDf DDs exp y d x f xŽ . Ž . Ž .H H

2

=1

2 a dyE qs x f x q d x s x ,Ž . Ž . Ž .Ž . Hx 2 f

1Ž .

a Ž . Ž .where f x , as1, . . . , N is the basic O N -vec-Ž . Ž . Ž .tor, O d -scalar field, s x is the O d -scalar auxil-

iary field and f is the coupling. Integrating outaŽ .f x , we obtain

NZs DDs exp y S s , g ,Ž . Ž .H eff2

12 dS s , g sTr ln yE qs y d x s x ,Ž . Ž . Ž .Heff g

2Ž .

Ž . 2for the rescaled coupling gsNf. Setting s x sm2'Ž . Ž .q ir N l x , where m is the stationary value of

Ž .the functional integral 2 given by the gap equation

ddp 1 1s , 3Ž .H d 2 2 gp qm2pŽ .

Ž . Ž .and expanding 2 in powers of l x , we get theusual large N expansion. The momentum space in-

Ž .verse propagator of l x is

Py1 p2Ž .

1 ddq 1s .H d 22 2 22 2p q qm qqp qmŽ . Ž .Ž .

4Ž .

The partition function is then given by

ZseyN Seff Žm2 , g .r2

=1

d d Ž .w x 'Ž . Ž . Ž .y d x d y l x D xyy l y qO 1r NHDDl e ,Ž . 2H5Ž .

Ž . Ž .where D x is the x-space Fourier transform of 4 .The effective theory above describes the ‘‘propa-gation’’ and ‘‘interactions’’ of the composite fieldŽ .l x . The critical theory, which is a non-trivial CFT

for 2 - d - 4, is obtained for 1rg ' 1rg s)

Ž .yd d 22p Hd prp and m'M s0.)

When the system is put in a strip-like geometryhaving one finite dimension of length L and periodicboundary conditions, the momentum along the finitedimension takes the discrete values v s2p nrL,n

ns0,"1,"2, . . . , and the relevant integrals be-come infinite sums. For example, the gap equation

Ž .and the inverse l x propagator become now

` dy11 1 d p 1s , 6Ž .Ý H dy1 2 2 2g L p qv qm2pŽ . n Lnsy`

Py1 p2 ,v 2Ž .L n

` dy 11 d qs HÝ d y12 L Ž .2pm s y`

=1

. 7Ž .2 22 2 2 2Ž . Ž .q q v q m qq p q v q v q mw xŽ .m L m n L

Since renormalisation of the bulk theory is sufficientfor the renormalisation of the theory in finite volumew x1 , the critical coupling in the latter case holds its

Ž .yd d 2bulk critical value 1rg s 2p Hd prp . The)

value of the critical mass parameter M however,)

( )A.C. Petkou, N.D. VlachosrPhysics Letters B 446 1999 306–313308

Žmay be different from zero. Indeed, the renormal-.ised gap equation for the finite system at criticality

Ž .can be found from 6 to be

1 1 1G dy G 1y dŽ . Ž .2 2 2dy20sM qII , 8Ž .

) 0'4 p

where

1Ž .dy3 qn

22` t y1Ž .

II s d t . 9Ž .Hn L M t)e y11

Ž .For 2-d-4, 8 has a solution with M /0. The)

dimensionless quantity M L is plotted for this case)

as a function of d in Fig. 1. For 2-dF3, theŽ .massless phase is saturated, since 8 diverges for

Ž .M zero, e.g. for 2-dF3 only the O N -symmet-)

ric and critical theories exist. However, for 3-d-4,Ž .8 is also satisfied for M s0 and the model retains

)

the two-phase structure which has in the bulk. Fords3, which is a special case as discussed below,

Ž . Ž . 2the solution of 8 takes the value M s 1rL lnt)'w x Ž .7,8 , where ts 5 q1 r2.

Before attempting the evaluation of the inverseŽ .l x propagator, we briefly discuss the CFT ap-

proach to the finite size scaling of the conformallyŽ .invariant O N vector model in 2-d-4. This

approach is based on the OPE structure of the model,

Fig. 1. The dimensionless quantity M L as a function of the)

spacetime dimensionality d.

w xstudied in a number of works 3–5 . The OPE of thea Ž .basic field f x with itself takes the form

C gf ff OOa b a bf x f 0 s d qŽ . Ž . 2h Cx OO

=1

abOO 0 d q PPP ,Ž .1hy ho2 2xŽ .

10Ž .where the dots stand for terms related to the energy

Ž . Ž .momentum tensor T x , the O N conserved cur-mnab Ž .rent J x and other fields with less singular coef-m

Ž .ficients as x™0. The field OO x has dimension'Ž . Ž .h s2qO 1rN , the coupling g is O 1r No ff OO

a Ž .and the dimension of f x is hsdr2y1qŽ .O 1rN . C and C are the normalisation constantsf OO

a Ž . Ž .of the two-point functions of f x and OO xŽ .respectively. Clearly, OO x may readily be identified

Ž . Ž .with l x in 5 , we prefer however for the sake ofgenerality to make this identification at a later stage.

Ž .The OPE of OO x with itself takes the form

C g 1OO OO

OO x OO 0 s q OO 0Ž . Ž . Ž .2h 1o Cx hOO o22xŽ .dh C x xo OO m n

q 1dy1 S CŽ . h y dq1d T o2 2xŽ .=T 0 q PPP , 11Ž . Ž .mn

w x Ž . 2 Ž .where 4 C sN dr dy1 S qO 1rN is the nor-T dŽ .malisation of the two-point function of T x , withmn

d r2 Ž .S s2p rG dr2 the area of the dydimensionaldŽ .unit sphere. The dots stand for derivatives of OO x

Ž .and T x , as well as for other fields not relevant tomn

the present work. The coefficient in front of theŽ .leading T x contribution is exactly determinedmn

from conformal invariance and the Ward identities² :satisfied by the three-point function T OOOO andmn

Ž .we also know that g C s2 dy3 g C to lead-OO f ff OO OO

w xing N 4 .Consider now the CFT having the OPEs above in

the strip-like geometry. Taking the expectation valueŽ .of 11 yields the leading finite size corrections to

the bulk-form 1rx 2ho of the two-point function ofŽ .OO x . These arise from the fact that the expectation

² : ² :values OO and T may in general be non-zero inmn


Ž .a finite geometry. Given the explicit form for OO xa Ž .in terms of the fundamental fields f x , one may

² :explicitly calculate OO in a 1rN expansion. Alter-² :natively, the leading N value for OO can be also

obtained by means of a consistency argument as wew xshow below. On the other hand, Cardy 10 showed

that for a general conformal field theory the diagonal3 ² :matrix elements of T are related to the finitemn

size correction of its free energy density f y f . His` L

result reads

² : ² :T sy dy1 TŽ .11 i i

2z dŽ .s dy1 cŽ . ˜dS Ld

s dy1 f y f no summation on i ,Ž . Ž . Ž .` L

12Ž .where, c is a universal number which has been˜considered to be a candidate for a possible generali-

w xsation of Zamolodchikov’s C-function for d)2 11 .Ž .Upon transforming 11 to momentum space for

Ž .h s2 and using 12 , the leading finite size correc-oŽ .tions for the OO x propagator can be written as

1d

2 1dy4C p 2 G dy2Ž .OO 22 2OO p ,v sŽ .L n 1dy2

22 2p qvŽ .n

=d g 1OO ² :1q4 y2 OO2 2 2ž /2 C p qvOO n�

2 dq1G dy1 S z d cŽ . Ž . ˜dq 1 dL Nd

12p G 2y dŽ .2

=

1dy1

2C yŽ .2q . . . , 13Ž .1

d 022 2p qvŽ .n

lŽ .where C y are the Gegenbauer polynomials andn

2 2(ysv r p qv . For 2-d-4, the terms shownn n

3 ² :The off-diagonal matrix elements of T vanish from re-mn

flection symmetry.

Ž .in 13 are the most singular ones in the largeŽ 2 2 .momentum expansion of OO p ,v .L n

Ž .Similarly, taking the expectation value of 10yields the leading finite size corrections to the two-

a Ž . Ž .point function of f x . Then, by transforming 10to momentum space we obtain for the propagatorŽ 2 2 . a Ž .P p ,v of f xL n

1d

24p 12 2P p ,v sCŽ .L n f 1 2 2G dy1 p qvŽ .2 n

=d g 1ff OO

1q4 y2 2 2ž /ž 2 C C p qvf OO n

=² :OO q PPP . 14Ž ./Ž .If OO x is to be identified now with the compos-

Ž . Ž . Ž . Ž .ite field l x in 5 , formulae 13 and 14 must beconsistent with what is expected by explicitly calcu-

Ž . a Ž .lating the propagators of l x and f x , in thecontext of the standard 1rN expansion at the critical

Ž .point of the O N vector model. Here, we will onlybe concerned with leading N calculations and wechoose to work in momentum space as it is custom-

w xary for finite size scaling studies 8,9 .Ž 2 2 .We start our consistency analysis from P p ,vL nŽ .which, to leading N, is easily found from 1 to be

12 2P p ,v sŽ .L n 2 2 2p qv qMn )

1 M 2)

s 1y q . . . .2 2 2 2ž /p qv p qvn n

15Ž .

Note that, to this order there is no contribution fromŽ . Ž . Ž .T x to the rhs of 15 . Consistency of 15 withmn

Ž . Ž . d r210 requires that C sG dr2y1 r4p and alsof

M 2 C C M 2 dy3 C 2Ž .) f OO ) OO² :OO sy sy .1 14 g dy2 2 g dy2Ž . Ž .ff OO OO2 2

16Ž .


y1Ž 2 2 . ŽNext, we turn to the propagator P p ,v 'L ny1Ž 2 2 .. Ž . Ž .OO p ,v of l x , in 7 . Using standard algebraL n

we obtain

1d 1

1 22yd d y22 G 2y d pŽ . 22y1 2 2 2 2OO p ,v s p q vŽ . Ž .L n n 2dŽ .2p

=p2 q v 2

n1 1 3F 2y d , ; ;1 2 2 2 2 2 2ž /p q v q4Mn )

ddy 1q 1qH d y1 2 22 2 L q q M'Ž .2p

)q q M e y1( w x)

=p2 q v 2 q2 qPpn

.22 2 2 2 2p q v q2 qPp q4v q q MŽ . Ž .n n )

17Ž .

Introducing the quantities

1 1dq

2 2 1 12yd2 p G 2y d G dy1Ž . Ž .2 2A sd d 1 12p G dyŽ . Ž .2 2

4M 2)2and Õ s , 18Ž .2 2p qvn

Ž .the first term on the rhs of 17 can be written in aform more suitable for large momentum expansionas,

1 1 3dy2 dy

2 2 22 2 2A p qv 1qÕŽ .Ž .d n

dy1

21 1 2G dy ÕŽ .Ž .2 2q 2' 1qÕp G d 2Ž .

=

2Õ1 1 1F dy ,1; d ; . 19Ž .1 2 2 2 2ž /1qÕ

Ž .The second term on the rhs of 17 can be written as

`dy2S q dqdy1 Hdy1 1

02pŽ . d2 222 2 L q qM'

)q qM e y1Ž .)

= 2 2Re I p ,v ,q , 20Ž .Ž .n

I p2 ,v 2 ,qŽ .n

k k` y2 i rŽ .s Ý kq12 2p qvks0 Ž .n

=p kdy3sin f cos ty isin t cosf df , 21Ž . Ž .H

0

where we have defined

2 2(v q qM sr cos t , q psr sin t ,n )

2 2 2 2 2rs p qv q qv M . 22( Ž .Ž .n n )

Ž . w xThe angular integral in 20 can be evaluated 12 interms of Gegenbauer polynomials and, taking thereal part, we obtain

2 2Re I p ,v ,qŽ .n

k 12 dy3` y4 G dy1 2 2k !Ž . Ž .Ž .2s Ý

G dy2q2kŽ .ks0

=

12 k dy1r2C cos t . 23Ž . Ž .2 k2 kq12 2p qvŽ .n

Ž .It is evident that 23 is a large momentum expan-y1Ž 2 2 .sion. The leading terms of OO p ,v can now beL n

Ž . Ž . Ž .evaluated after substituting 23 , 19 in 17 . Appar-Ž .ently, the hypergeometric function in 19 con-

tributes terms which do not seem to be present inŽ . Ž .13 . The second term of 19 has to be interpreted asa dimension dy2 scalar field contribution. This

4 Ž .field corresponds to the ‘‘shadow field’’ of OO x .Ž .Note that the dynamics of the O N vector model

4 Ž .The ‘‘shadow field’’ of a scalar field A x with dimension lis a scalar field with dimension dy l. For the ‘‘shadow symme-

w xtry’’ of the conformal group in d)2 see 13 and referencestherein.


requires that ‘‘shadow singularities’’ cancel out fromw xfour-point functions 4 . In order to clarify the role of

the ‘‘shadow singularities’’ inside the inverse two-Ž . Ž .point function 17 , we must reexpress 23 in terms

d r2y1Ž . lŽ .of C y . Given that C y for integer n are2 k n

orthogonal polynomials of order n , an expansion ofthe form

1 2 2dy1 y q qMŽ .k )22 2 2q qy M CŽ .) 2 k 2 2 2ž /(q qy M

)

12k dy1

22 2s B q , M C y 24Ž . Ž .Ž .Ý l ) lls0

Ž 2 2 . Ž 2 .exists. The scalar contribution B q , M 'B M0 ) 0 )

can be easily evaluated and reads

B M 2Ž .0 )

d 1G G dy2q2k G kqŽ .ž / ž /2 2 2 ks M .

)d'p G dy2 G 2kq1 G kqŽ . Ž . ž /2

25Ž .Ž .Substituting B in 17 we obtain, after some alge-0

bra,

1dy1

224 ÕŽ .y 1 1 2G d G 1y d 1qÕ 2Ž . Ž .2 2

=Õ2

1 1 1F 1, dy ; d ; II , 26Ž .1 02 2 2 2ž /1qÕ

Ž .which cancels exactly the second term in 19 byŽ .virtue of the gap Eq. 8 .Ž .Thus, the gap Eq. 8 , being a necessary and

sufficient condition for this cancellation to occur,suggests an interesting alternative approach forstudying this model. Namely, we could have started

Ž .with 7 as the definition of the inverse two-pointŽ .function of the field l x . Requiring then that this is

an inverse two-point function of a CFT obtainablefrom an OPE, and, postulating that no ‘‘shadowsingularities’’ appear we are led, by means of the

Ž .same calculations, to the gap Eq. 8 . Consequently,Ž .8 , which is the basic dynamical equation of theŽ .O N model, could be viewed as a precondition for

the ‘‘shadow singularities’’ cancellation. Such analgebraic approach to CFT in d)2 was initiated inw x5 and it is now demonstrated to work for finitegeometries as well.

Eventually, the first few terms in the large mo-Ž .mentum expansion of 17 are

OOy1 p2 ,v 2Ž .L n

12dy2 M)22 2sA p qv 1q2 dy3Ž .Ž .d n 2 2p qvn

1dy1

21 12 d2 G dy C yŽ .Ž . 22 2q 1 1'p G d G 2y dŽ . Ž . d2

22 2p qvŽ .n

1ydd=M II yII q PPP . 27Ž .) 0 1ž /d

Ž . Ž .Now, 27 must be consistent with 13 . It is thenŽ 2 . Ž .easy to see that the O M terms on the rhs of 13

)

Ž . Ž .and 27 coincide by virtue of 16 . This is a non-trivial consistency check, since it requires the non-trivial relation between the couplings g and g .OO ff OO

Next, consistency of the angular terms proportionald r2y1Ž . Ž . Ž .to C y in 13 and 27 requires that2

c dy1˜dG d z d sM II qII . 28Ž . Ž . Ž .) 0 1d ž /dL N

In order to prove that, we calculate the free energyŽ .density of the theory 1 to leading order in 1rN

2z d SŽ . dy1df y f ' csN M˜` L )d dy1S L 2pŽ .d

=

1 3dy`1 1 2 22II y x x y1Ž .H0d LMž 1

)

=ln 1yeyL M)

x d x . 29Ž . Ž ./


Now, for 2-d-4, a partial integration yields

1 3dy`dy1 2 22II sy x x y1Ž .H1 LM 1

)

=ln 1yeyL M)

x d x . 30Ž . Ž .

Ž .dy1 ŽThen, using the identity S S s2 2p rG dd dy1. Ž .y1 one can see that 28 is satisfied by means of

Ž . Ž .29 and 30 . To the best of our knowledge, this isw xthe first time that the validity of Cardy’s result 10

is explicitly demonstrated for CFTs in d)2. Forcompleteness, a plot of crN as a function of d for˜2-d-4 is depicted in Fig. 2. The evaluation of c

Ž .to O 1rN , and its relation to possible generalisa-tions of Zamolodchikov’s C-function for d)2, are

w xgiven in 14 .Ž . Ž .The case ds3 is special. From 13 or 17 , we

Ž .see that there in no self-contribution from OO x toits two-point function, which is related to the factthat the bulk theory respects the reflection symmetry

Ž . Ž .property OO x ™yOO x to leading order in 1rN.Ž . 2Also, for ds3, M s 1rL lnt and then it can be

)

w x Ž .shown 14 that the integrals involved in 29 arerelated to polylogarithms at the special point 2yt ,leading to the surprisingly simple result crNs4r5˜w x8 .

Fig. 2. Plot of crN as a function of the spacetime dimensionality˜d.

Ž .For 3-d-4, the gap Eq. 8 has also the solu-tion M s0. In this case, the corresponding expres-

)

y1Ž 2 2 .sion for OO p ,v is given byL n

OOy1 p2 ,v 2Ž .L n

11 12 dy3dy2 2 G dyŽ .2 222 2sA p qv 1qŽ .d n 1'p G 2y dŽ .2�

1k` dy1y4 2k ! z dy2q2kŽ . Ž . Ž . 2= C yŽ .Ý 2 k1

dy1qkks0 022 2 2L p qvŽ .n

1dy2

22 2sA p qvŽ .d n

=

1 12 dy32 G dyŽ .2 21q 1'p G 2y dŽ .2�

=z dy2Ž .

1dy1

22 2 2L p qvŽ .n

1 12 d2 G dyŽ .2 2y 1'p G 2y dŽ .2

1dy1z dŽ . 2

= C y q PPP .Ž .21d 022 2 2L p qvŽ .n

31Ž .

a Ž . Ž 2Since now the propagator of f x is simply 1r p2 . Ž . ² :qv , consistency with 12 requires OO s0 ton

leading order in 1rN which means that we do notŽ .expect to get any contributions from OO x in

y1Ž 2 2 2 . Ž 2P p ,v ;M . Indeed, a term A 1r p qL n )

2 .d r2y1 Ž .v is absent from the rhs of 31 . Instead, wenŽ 2 2 .find a term A1r p qv which has the correctn

dimensions to be interpreted as the leading contribu-Ž .tion of the ‘‘shadow field’’ of OO x . This term is not


cancelled out and this is a characteristic feature ofw xfree field theories as discussed in 4 .

Next, we require consistency of the coefficients ofd r2y1Ž . Ž . Ž .C y in 31 and 13 which yields, after some2

algebra, csN . This is exactly the result one obtains˜by calculating f y f to leading N at the critical` L

Ž .point with M s0 , i.e. the result for free massless)

Ž .scalar fields. Note that 31 looks like a free fieldtheory decomposition for the two-point function ofŽ . Ž . Ž .OO x , OO x which has in the place of OO x the

2Ž .dimension dy2 composite scalar field :f x :.Our results above provide evidence that two-point

functions of CFTs in d)2 and in finite geometryare determined by bulk conformally invariant OPEs.In fact, the dynamical equations for the finite sizetheory are just the conditions for the cancellation of

Ž . Ž .‘‘shadow’’ singularities as shown in 19 and 26 .Furthermore, we have explicitly shown that the lead-ing angular correction to the scalar two-point func-tion is proportional to crC . It would be interesting˜ T

to look for non trivial cancellations inside the twoa Ž .point function of f x where the calculations will

w x w xbe related to those in 8 and 15 . Extension of theOPE approach to fermionic, CP Ny1 and supersym-metric CFTs in 2-d-4 would further clarify theconnection between ‘‘shadow singularity’’ cancella-tions and dynamics. Another issue might be whetherthe conformal theory with M s0 is a free field

)

theory, which could be clarified by next-to-leadingorder in 1rN calculations.

Acknowledgements

This work was supported in part by PENEDr95K.A. 1795 research grant.

References

w x1 J. Zinn-Justin, Quantum Field Theory and Critical Phenom-ena, 2nd ed., Clarendon, Oxford, 1993.

w x Ž .2 P. Ginsparg, in: E. Brezin, J. Zinn-Justin Eds. , Champs,´Cordes et Phenomenes Critiques, North-Holland, Amster-´ `dam, 1989.

w x Ž .3 K. Lang, W. Ruhl, Nucl. Phys. B 402 1993 573; Z. Phys. C¨Ž .61 1994 495.

w x Ž .4 A.C. Petkou, Ann. Phys. 249 1996 180.w x Ž . Ž .5 A.C. Petkou, Phys. Lett. B 359 1995 101; B 389 1996 18.w x Ž .6 J.L. Cardy, in: E. Brezin, J. Zinn-Justin Eds. , Champs,´

Cordes et Phenomenes Critiques, North-Holland, Amster-´ `dam, 1989.

w x7 B. Rosenstein, B.J. Warr, S.H. Park, Nucl. Phys. B 336Ž .1990 435.

w x Ž .8 S. Sachdev, Phys. Lett. B 309 1993 285; A.V. Chubucov,Ž .S. Sachdev, T. Senthill, Phys. Rev. B 49 1994 1919.

w x Ž .9 M. Modugno, G. Pettini, R. Gatto, Phys. Rev. D 57 19984995; D.M. Danchev, N.S. Tonchev, cond-matr9806190.

w x Ž .10 J.L. Cardy, Nucl. Phys. B 290 1987 355.w x Ž .11 A.H. Castro Neto, E. Fradkin, Nucl. Phys. B 400 1993 525;

M. Zabzine, hep-thr9705015.w x12 I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and

Products, 5th ed. Academic Press, New York, 1980.w x Ž .13 S. Ferrara, G. Parisi, Nucl. Phys. B 42 1972 281.w x14 A.C. Petkou, N.D. Vlachos, work in progress.w x15 M. Dilaver, P. Rossi, Y. Gunduc, hep-latr9710041.¨ ¨

28 January 1999


The linear vector-tensor multiplet with gauged central charge

Norbert Dragon, Ulrich TheisInstitut fur Theoretische Physik, UniÕersitat HannoÕer, 30167 HannoÕer, Germany¨ ¨

Received 4 June 1998Editor: P.V. Landshoff

Abstract

We derive consistent superfield constraints for the linear vector-tensor multiplet with gauged central charge. The centralcharge transformations and the action turn out to be nonpolynomial in the gauge field. q 1999 Published by Elsevier ScienceB.V. All rights reserved.

PACS: 11.15.yq; 11.30.Pb

Keywords: Vector-tensor multiplet; Central charge

Ž . w xThe Ns2 vector-tensor VT multiplet 1 is a combination of an Ns1 vector- and an Ns1 linearmultiplet, and as such contains among its components a 1-form and a 2-form gauge field. Since its rediscovery

w x w xby string theorists 2,3 , much effort has been made to derive consistent interactions both with itself 4–6 andw x w xwith background Ns2 vector multiplets 4,7–9 . In 4 the VT multiplet was coupled to an abelian vector

multiplet that gauges the central charge of the Ns2 supersymmetry algebra, resulting in an interaction of threegauge fields.

Such interactions of higher degree gauge fields have recently been classified by cohomological meansŽ . w xoutside the framework of supersymmetry in 10 . As a particular example an interaction of two 1-forms and a

w x2-form was studied in 11 which turned out to be nonpolynomial and resembled the bosonic sector of the VTmultiplet with gauged central charge.

In the present letter, we show that this model is indeed contained in the linear VT multiplet with gaugedcentral charge, which is derived from a set of constraints on the basic superfield, thus providing the ground for a

w x w xsuperspace formulation of the component results of 4 . Here we follow an approach already applied in 5,6 ,Ž .where the nonlinear VT multiplet with global central charge was formulated in harmonic superspace, namely

to determine consistent deformations of the flat superfield constraints.Our considerations are based on the Ns2 supersymmetry algebra

i i a w xDD , DD syid s DD , DD , DD sF d ,½ 5a a j j a a a a b ab z˙ ˙

ii j i j i iDD , DD s´ ´ Zd , DD , DD s s l d ,� 4 1Ž . Ž .a b a b z a a a z2 a

iDD , DD s´ ´ Zd , DD , DD s l s d .Ž .˙ ˙½ 5a i b j a b i j z a i a i a z2 a˙ ˙ ˙ ˙


( )N. Dragon, U. TheisrPhysics Letters B 446 1999 314–320 315

The real generator d is central and commutes with all generators of the supersymmetry algebra,z

) i w xd sd , d , DD s0 , d , DD s0 , d , DD s0 . 2Ž .z z z a z a j z a˙

DD is the gauge covariant derivativea

DD sE qA d 3Ž .a a a z

and F the abelian field strength of the gauged central chargeab

F sE A yE A . 4Ž .ab a b b a

Ž .The algebra Eq. 1 might as well be read as an algebra with a gauged abelian transformation and no centralcharge. However, the scalar field Z has to have a nonvanishing constant background value as we will see.

iZ, Z, l and A are component fields of the vector multiplet which is completed by an auxiliary tripleta ai j ji Ž .)Y sY s Y ,i j

i il sDD Z , l sDD Z ,a a a i a i˙ ˙

1 1i j i j i jY s DD DD Zs DD DD Z ,2 2

1 i iF s DDs DD ZyDD s DD Z . 5Ž .ž /ab ab i i ab4

They are tensor fields and carry no central charge. So they are unchanged by gauged central chargeŽ .transformations s with a real gauge parameter C x with arbitrary spacetime dependence. Under thesez

transformations the vector field A is changed by the gradient of the gauge parameter,a

i i js Zs0 , s l s0 , s Y s0 , s A syE C . 6Ž .z z a z z a a

w xTo construct invariant actions, we make use of the linear multiplet with gauged central charge as in 12 . Onestarts from a field

)i j jiLL sLL s LL 7Ž .Ž .i j

satisfying the constraints

Ž i jk . Ž i jk .DD LL s0sDD LL . 8Ž .a a

ia a i jIt contains among its components a real vector V sy DD s DD LL , which is constrained byi j6

1a i jDD V s d Z DD DD q3Y q4l DD LL qh.c. 9Ž .a z i j i j i j12

From the identity

s A V a sy E C V a qA Cd V a syE CV a qC DD V aŽ . Ž .Ž .z a a a z a a

it follows immediately that

1 a i jLLs Z DD DD q3Y q4l DD q i A DD s DD LL qh.c. 10Ž .i j i j i j a i j12

is invariant under gauged central charge transformations upon integration over spacetime. This action is alsoNs2 supersymmetric, as can be checked by explicit calculation. So once we have constructed a multiplet on

Ž . i j Ž .which the algebra Eq. 1 is realized we try to construct a composite field LL which satisfies Eq. 8 and useŽ .Eq. 10 as the Lagrangian.

For the VT multiplet gauging the central charge is quite involved. This multiplet is obtained from a realscalar field L, which for ungauged central charge satisfies the constraints

) Ž i j. Ž i j.LsL , D D Ls0 , D D Ls0 . 11Ž .a a

( )N. Dragon, U. TheisrPhysics Letters B 446 1999 314–320316

i j i jSimply replacing the flat derivatives D and D with gauge covariant derivatives DD and DD leads toa a a a˙ ˙inconsistencies that show up at a rather late stage in the process of evaluating the algebra on the multiplet

Ž w x .components see 13 for details . This problem can be avoided, however, by a suitable change of the aboveconstraints. They must preserve the field content as compared to the case of ungauged central charge. Aconsistent set of constraints for the VT multiplet with gauged central charge is given by

21Ž i j. Ž i j. Ž i j. Ž i j. Ž i j.DD DD Ls0 , DD DD Ls DD Z DD LqDD Z DD Lq L DD DD Z . 12Ž .a a 2ž /˙ ZyZ

Obviously, this requires the imaginary part of Z to have a nonvanishing background value. For Zs i theŽ .constraints reduce to the flat case, eq. Eq. 11 .

Let us show how these constraints were obtained. We considered the following Ansatz, with the coefficientsbeing arbitrary functions of Z and Z,

Ž i j. Ž i j. Ž i j. Ž i j.DD DD Lsa DD Z DD Lqa DD L DD ZqbL DD Z DD Z , b real,a a a a a a a a˙ ˙ ˙ ˙

Ž i j. Ž i j. Ž i j. Ž i j. Ž i j. Ž i j.DD DD LsA DD Z DD LqB DD Z DD LqCL DD DD ZqDL DD Z DD ZqEL DD Z DD Z . 13Ž .It is linear in the components of the VT multiplet and hence invariant under a rescaling of L with a constantparameter. The constraints have to satisfy the necessary consistency conditions

Ž i j k . Ž i j k . Ž i j k .DD DD DD Ls0 , DD DD DD LsDD DD DD L , 14Ž .a a a˙ ˙

which yield a system of differential equations for the coefficient functions. The first condition requires1 1 10sCy A , 0sE CyDy AC , 0sEqaB , 0sE Ey AEqbB ,Z Z2 2 2

1 1 20sE BqB ay A , 0sE Ay A y2 D , 15Ž .Ž .Z Z2 2

while the second leads to

1 10sCy Bya , 0sE CyEy BCyaCyb ,Z2 2

10sE aqa ayA yD , 0sE DyE by BEyaDyb ayA ,Ž . Ž .Z Z Z 2

1 10sE AyE ay BByaayb , 0sE ByB aq A y2 E . 16Ž .Ž .Z Z Z2 2

ˆŽ . Ž .If we perform a more general rescaling Ls f Z,Z L in Eq. 13 , the coefficients transform as

ˆ ˆasayE frf , bs bfqaE fqaE fyE E f rf , AsAy2E frf ,ˆ Ž .Z Z Z Z Z Z

2ˆ ˆ ˆ ˆBsB , CsCyE frf , Ds DfqAE fyE f rf , Es EfqBE f rf . 17Ž .Ž .Ž .Z Z Z Z

Ž . Ž .We can choose f such that a vanishes, which simplifies the conditions Eq. 15 and Eq. 16 considerably, andˆŽ .obtain omitting the hats

2CsBsA , asbsDsEs0 , 18Ž .where A has to satisfy the differential equations

1 12E As A , E As AA . 19Ž .Z Z2 2

Ž .Their general nonvanishing solution is

2 eyi w

A Z,Z s , r ,wgR . 20Ž . Ž .i w yi we Zye Zq2i r


The parameters can be removed by a redefinition

Z™ei w Zq i r , DD i ™eyi w r2 DD i , 21Ž . Ž .a a

Ž . Ž .which eventually leads to the constraints given in eq. Eq. 12 . Of course, the conditions Eq. 14 do notguarantee the consistency of these constraints, we now have to carefully evaluate the algebra on each componentfield.

Ž .The linearity of the constraints Eq. 12 in L and its supersymmetry transformations implies that we will notencounter any self-interactions if LL i j is quadratic in L. To find constraints that underlie the nonlinear VT

w xmultiplet as described in 4 , one has to start from a more general Ansatz.Ž .We now investigate the consequences of the constraints Eq. 12 . The independent components of the

multiplet can be chosen as

i iL , c s i DD L , c syi DD L ,a a a i a i˙ ˙

1 1i iUsd L , G s DD , DD L , G s DD , DD L ,˙ ˙z a b a b i a b i a b2 2˙ ˙

1 iW sy DD , DD L . 22Ž .aa a a i2˙ ˙

In addition some abbreviations will prove useful in the following,

Is Im Z , RsRe Z ,1 i iL sLE Rq ls c qc s l ,Ž .a a a i a i2

i iS sLF q i l s c yc s l . 23Ž .ž /ab ab i ab i ab

R, L and S vanish in the case of a global central charge, i.e. when Z is reduced to its background valuea ab

Zs i.Ž .From the algebra Eq. 1 we obtain the supersymmetry transformations

iii j i j ab Ž i j. Ž i j. i jDD c s ´ ´ ZUqs G q ´ l c yl c q iY L ,Ž .ž /a b a b a b ab a b2 2 I1i j i j aDD c s ´ s DD Ly iW ,Ž .a a a a a a2˙ ˙

1i i i iDD W s iZ s d c q Us l yDD c ,Ž .a a a z a a2 a

i i i c d iDD G s 2 I s d c qUs l q i´ s DD c , 24Ž .Ž .a ab ab z ab abcd a

and the central charge transformations. These contain covariant derivatives which in turn again contain d , eq.zŽ .Eq. 3 , so the transformations are given only implicitly. One could solve for d of the fields, however at thisz

Žstage it is more advantageous to use the manifestly covariant expressions here a tilde denotes the dual of a1ab abcd˜ .2-form, i.e. G s ´ Gcd2

ii a i id c s s DD c yl UŽ .z aZ

ii 1 ii i j a i a i a i ab i ab iq Zl UyY c q is c E ZqLs E l q DD Ly iW s l yF s c y G s lŽ .j a a a a ab ab2 2 22ZI

iic j ic j i jŽ . Ž .y l l yl q iY L ,Ž .j2 I

d IW yL sDDbG ,Ž .z a a ab

˜ c dd IG qRG qS sy´ DD W . 25Ž .Ž .z ab ab ab abcd


As in the case of the ordinary VT multiplet, the vector W and the antisymmetric tensor G are subject toa ab

some constraints. Their solvability is the final consistency check. Closure of the algebra requires1a abDD IW yL s F G ,Ž .a a ab2

a ˜ ˜ cDD IG qRG qS sF W . 26Ž .Ž .ab ab ab bc

With the help of the central charge transformations we obtain from these covariant expressions the equations

E a IW yG Ab yL s0 ,Ž .a ab a

a ˜ c dE IG qRG q´ A W qS s0 , 27Ž .Ž .ab ab abcd ab

which identify the brackets as duals of the field strengths of a 2-form B and a 1-form V , respectively,ab a

1a abcd a˜IW s ´ E B yA G qL , 28Ž .Ž .b cd b cd2

˜ c d c dIG qRG s´ E V yA W yS . 29Ž . Ž .ab ab abcd ab

Ž .We cannot read off W and G , however, since the equations are coupled. Solving eq. Eq. 29 for G anda ab abŽ .inserting this into eq. Eq. 28 , we find

a a b < < 2 a abI Ed qA A W s Z H qV A , 30Ž .Ž . Ž .b b b

where we used the abbreviations

< < 2 aEs Z yA A ,a

1a abcd aH s ´ E B qL ,b cd2

I Rc d˜V s E V yE V qS q ´ E V yS . 31Ž .Ž .Ž .ab a b b a ab abcd ab2 2< < < <Z Z

Ž a a .The inverse of the matrix Ed qA A isb b

1 y2b b< <d y Z A A , 32Ž .Ž .a aE

which gives

12 2a a a b ab< < < <W s Z H yA A H q Z V A , 33Ž .Ž .b bIE

Ž .and finally, from eq. Eq. 29 ,

2 Rc c d deG sV y A H qV A y ´ A H qV A . 34Ž . Ž .Ž .ab ab w a b b xc abcd exE IE

These expressions are nonpolynomial in the gauge field A due to the appearance of E in the denominator.a

Although it appears as if they would transform inhomogeneously under the central charge transformation s ,zŽ .explicit calculation shows that the gradient of the gauge parameter, eq. Eq. 6 , cancels, so W and G area ab

indeed tensors as assumed from the beginning.Ž . Ž .It remains to give the transformations of the potentials V and B . As the brackets in Eq. 28 and Eq. 29a ab

already indicate, the central charge transformations read

˜d V syW , d B syG , 35Ž .z a a z ab ab

while the supersymmetry generators act as

1i i i iDD V s A c y Ls l y iZs c ,ž /a a a a a2 a

1i i i iDD B sy2i Is c q Ls l qA s c . 36Ž .ž /a ab ab ab w a b x2 a


Then the algebra closes up to gauge transformations of the form

V ™V qE j , B ™B qE j yE j 37Ž .a a a ab ab a b b a

when evaluated on the potentials.i j Ž .Next, we construct a linear multiplet LL . Given the constraints Eq. 12 on L it is easy to find a field with

Ž Ž i j. .the required properties note that DD DD L is imaginary ,

ii j Ž i j. Ž i j. Ž i j. i j i j Ž i j. Ž i j. i jLL s i DD L DD LyDD L DD LyL DD DD L s i c c yc c y L l c yl c q iY L .Ž . Ž . Ž .

I38Ž .

Ž .Applying the rule Eq. 10 we eventually obtain the Lagrangian

1 2 1 1a a a 2 2 a ab ˜< <LLs I E L E LyW W q Z yA A U y L E E Iy G IG yRG q4 A WŽ .ž / ž /a a a a ab ab w a b x2 2 4

1i j 2y Y Y L q fermion terms , 39Ž .i j4 I

Ž .where we used the identities Eq. 27 to combine terms into a total derivative which was dropped thereafter. Tow xcompare with the action found in 11 , we insert the expressions for W and G ,a ab

< < 21 Z 21 1 1 1a 2 a 2 i j 2 ab a ab˜LLs IE L E Ly LE E Iq IE U y Y Y L y V IV yRV y H qV AŽ .Ž .a a i j ab ab b2 2 2 44 I 2 IE

1 2aq A H q fermion terms . 40Ž .Ž .a2 IE

With L and U set to zero and Zs i, which breaks supersymmetry but keeps central charge invariance, thew xbosonic Lagrangian indeed coincides with the one given in Ref. 11 :

2 2a ac a1 1 H qV A 1 A H 1Ž . Ž .c aab abLLsy V V y q y F F , 41Ž .ab abb b 24 2 21yA A 1yA A 4 gb b

1a abcdwhere now H s ´ E B and V sE V yE V , and a kinetic term for A has been added. The secondb cd ab a b b a a2

term contains a vertex involving all three gauge fields A , V and B , which has been identified as aa a abw xFreedman-Townsend coupling in 10 .

Acknowledgements

We thank F. Brandt, P. Fayet and S. Kuzenko for stimulating and enjoyable discussions.

References

w x Ž .1 M.F. Sohnius, K.S. Stelle, P.C. West, Phys. Lett. B 92 1980 123.w x Ž .2 B. de Wit, V. Kaplunovsky, J. Louis, D. Lust, Nucl. Phys. B 451 1995 53. hep-thr9504006.¨w x Ž .3 R. Siebelink, Nucl. Phys. B 524 1998 86. hep-thr9709129.w x Ž .4 P. Claus, B. de Wit, M. Faux, B. Kleijn, R. Siebelink, P. Termonia, Phys. Lett. B 373 1996 81, hep-thr9512143; P. Claus, B. de Wit,

Ž .M. Faux, P. Termonia, Nucl. Phys. B 491 1997 201, hep-thr9612203.w x Ž .5 N. Dragon, S. Kuzenko, Phys. Lett. B 420 1998 64. hep-thr9709088.w x Ž .6 E. Ivanov, E. Sokatchev, Phys. Lett. B 429 1998 35. hep-thr9711038.w x Ž .7 R. Grimm, M. Hasler, C. Herrmann, Int. J. Mod. Phys. A 13 1998 1805. hep-thr9706108.


w x Ž .8 I. Buchbinder, A. Hindawi, B.A. Ovrut, Phys. Lett. B 413 1997 79. hep-thr9706216.w x Ž .9 N. Dragon, S. Kuzenko, U. Theis, Eur. Phys. J. C 4 1998 717. hep-thr9706169.

w x Ž .10 M. Henneaux, B. Knaepen, Phys. Rev. D 56 1997 6076. hep-thr9706119.w x Ž .11 F. Brandt, N. Dragon, in: H. Dorn, D. Lust, G. Weigt Eds. , Theory of Elementary Particles, Proceedings of the 31st International¨

Symposium Ahrenshoop, Wiley-VCH, Weinheim, 1998, p. 149, hep-thr9709021.w x Ž .12 B. de Wit, J.W. van Holten, A. van Proeyen, Phys. Lett. B 95 1980 51.w x13 N. Dragon, E. Ivanov, S. Kuzenko, E. Sokatchev, U. Theis, Ns2 rigid supersymmetry with gauged central charge. hep-thr9805152,

to appear in Nucl. Phys. B.

28 January 1999


Baryon stopping at HERA: evidence for gluonic mechanism

Boris Kopeliovich a,b, Bogdan Povh a

a Max-Planck Institut fur Kernphysik, Postfach 103980, 69029 Heidelberg, Germany¨b Joint Institute for Nuclear Research, Dubna 141980, Moscow Region, Russia

Received 2 November 1998; revised 7 December 1998Editor: P.V. Landshoff

Abstract

wRecent results from the H1 experiment The H1 Collaboration, C. Adloff et al., Measurement of the Baryon-AntibaryonAsymmetry in Photoproduction at HERA, Submitted to the 29th Int. Conf. on High-Energy Phys. ICHEP98, Vancouver,

xCanada, July 1998. Abstract 556. http:rrichep98.triumf.carprivaterconvenorsrbody.asp?abstractIDs556 have con-wfirmed the existence of a substantial baryon asymmetry of the proton sea at small x with magnitude predicted in B.Z.

Ž . x wKopeliovich, B. Povh, Z. Phys. C 75 1997 693 . This is strong support for the idea B.Z. Kopeliovich, B.G. Zakharov, Z.Ž . xPhys. C 43 1989 241 that baryon number can be transferred by gluons through a large rapidity interval without

attenuation. In this paper we calculate the dependence of baryon asymmetry on associated multiplicity of produced hadronswwhich turns out to be very sensitive to the underlying dynamics. Comparison with data The H1 Collaboration, C. Adloff et

al., Measurement of the Baryon-Antibaryon Asymmetry in Photoproduction at HERA, Submitted to the 29th Int. Conf. onHigh-Energy Phys. ICHEP98, Vancouver, Canada, July 1998. Abstract 556. http:rrichep98.triumf.carprivater

xconvenorsrbody.asp?abstractIDs556 confirms the dominance of the gluonic mechanism of baryon number transfer andexcludes any substantial contribution of valence quark exchange. q 1999 Published by Elsevier Science B.V. All rightsreserved.

A sizable baryon-antibaryon asymmetry in photon-proton interaction was recently observed by the H1Collaboration for prp with small momentum in the laboratory frame produced in g p collisions at HERA. The

w xpreliminary data presented at the Vancouver Conference 1 show that

N yNp pAs2 s 8.0"1.0"2.5 % . 1Ž . Ž .

N qNp p

Here N and N are the numbers of detected protons and antiprotons respectively.p p

Obviously, the observed excess of protons is a consequence of the presence of the proton baryon numberŽ .BN in the initial state of the reaction. Nontrivial is, however, the very large rapidity interval of about 8 unitsbetween the initial and final protons. One could expect an exponential attenuation of the BN flow over such along rapidity interval and a vanishing baryon asymmetry. In contrast, a baryon asymmetry A of about 7% was

w x w xpredicted in 2 . The calculations are based on the gluonic mechanism of BN transfer first suggested in 3 . Thismechanism provides a rapidity independent probability of BN stopping, which is natural for gluonic exchanges.


( )B. KopelioÕich, B. PoÕhrPhysics Letters B 446 1999 321–325322

w xIn topological classification 4 BN is associated with the string junction for a star-shaped string configurationw xin the baryon. Therefore, baryon stopping is stopping of the string junction. It is argued in 4 that at asymptotic

energies the string junction alone is stopped without valence quarks, i.e. only gluonic fields are involved. Thisestablishes a correspondence to our gluonic mechanism of BN stopping.

w xAnother mechanism of baryon stopping also suggested and calculated within pQCD in 3 is associated with aprobability to find in the proton a low-x valence quark accompanied by the string junction. The dependence of

Ž .BN transfer on the rapidity interval D y is proportional to exp yD yr2 since it is related to the well knownw x w xx-distribution of valence quarks dictated by Regge phenomenology 5 . Evaluated in pQCD 3 this mechanism

well explains both the magnitude and energy dependence of baryon asymmetry observed at central rapidity inw x w xpp collisions at the ISR 12–14 . The contribution of the asymptotic gluonic mechanism calculated in 2 is too

small to show up at such a small rapidity interval D yF4.w xThe source of baryon asymmetry can be understood in the parton model 2 . In the infinite momentum frame

of the proton one can attribute a partonic interpretation to the carrier of BN, the string junction, since it carries aw xfraction of the proton momentum 4 . In the rest frame of the proton all the partons in the initial state of the g p

interaction belong to the photon. Obviously, the parton distribution of the photon is BN symmetric. However,the interaction with the proton target breaks up this symmetry due to the possibility of annihilation of anti-BN inthe projectile parton cloud of the photon with BN of the proton. This leads to a nonzero BN asymmetry in thefinal state. The rapidity distribution of the produced net BN is related to the energy behaviour of the annihilationcross section.

The annihilation cross section at very high energies was predicted to be nearly energy independent, inw x w xnonperturbative 6 and perturbative 7–9 approaches. In both cases the magnitude was predicted to be

Ž . w xs pp f1–2 mb. Such a contribution was indeed found in the analysis 9,11 of data on multiplicityannŽ .distribution in pp and pp collision. The corresponding cross section s pp f1.5"0.1 mb agrees well withann

the theoretical expectation. This asymptotic mechanism of annihilation results in a rapidity independent BNw xtransfer according to the above partonic picture. Its contribution to baryon asymmetry was estimated in 2 .

Experimental data for pp annihilation are available only at low energy up to 12 GeV. In this energy range theannihilation cross section is much larger that the asymptotic value 1.5 mb and decreases with energyapproximately as sy1r2. This is to be explained by the preasymptotic mechanism corresponding to the exchangeof a valence quark accompanied with the string junction. This assignment explains in a natural way the energy

w xdependence of annihilation, moreover, a parameter-free evaluation of the cross section 10 is also in a goodagreement with the data.

w xThe same preasymptotic mechanism nicely explains 3 the energy dependence and the absolute value for thecross section of net BN production in the central rapidity region in pp collisions at ISR. Baryon stopping in

w xheavy ion collisions measured at the SPS is also well explained by this mechanism 15,16 without freeparameters.

w xThis valence quark-exchange contribution to BN transfer in g p collisions was also calculated in 2 . It turnsw xout that the rapidity interval between the initial and final protons in the experiment 1 is not large enough to

w xexclude possibility of this contribution. Both, the gluonic and quark exchange mechanisms are estimated in 2Ž w x.to give about the same asymmetry at rapidity hs0 see Fig. 5 in 2 and are able to explain the data within

uncertainty of calculations. One needs more detailed information to discriminate between the two mechanisms.It worth noting that a nice topological classification for these mechanisms was suggested by Rossi and

w xVeneziano 4 . However, their prediction for the energy dependence has no reasonable justification and is in aŽsevere contradiction with the standard high-energy Regge phenomenology for total cross sections see

w x.discussion in review 11 , therefore we discard it. Particularly, if the difference between pp and pp total crosssections is due to annihilation, one has to eliminate the v exchange from the elastic pp amplitude, but keep itas the major Reggeon term in the Kp elastic scattering. Another problem arises with relation between v and r

exchanges, since the latter is predicted by quark models to be much smaller than the former in agreement withw xresults of the standard Regge phenomenology. According to Eylon and Harari 17 annihilation is related via

( )B. KopelioÕich, B. PoÕhrPhysics Letters B 446 1999 321–325 323

Fig. 1. Multiparticle production corresponding to the gluonic mechanism of BN transfer in gyp interaction. The BN is produced with theŽ .rapidity of the sea quark left . The same as on the left, but for the valence quark mechanism. BN has the rapidity of the valence quark

Ž .right .

Ž w x.unitarity to the Pomeron at least a substantial part of it, see 10 and does not contribute to the pp and pp totalcross section difference.

An important signature of the asymptotic gluonic mechanism is a higher mean multiplicity of producedw xparticles. This is due to three sheet topology of the final state according to classification in 4 , which is

Ž .illustrated in Fig. 1 left . The gluon is replaced by a sea qq pair. The valence quark exchange mechanism alsoŽ .shown in Fig. 1 exhibits a two-sheet two-string topology, i.e. the same multiplicity as in the Pomeron. It is

easy to see in Fig. 1 that the mean multiplicity of produced particles in the rapidity interval where the baryonŽ .asymmetry is measured is 5r4 times larger for the gluonic mechanism left compared to the quark exchange

Ž .mechanism right . This fact makes baryon asymmetry dependent on the multiplicity of the produced hadrons.w xFirst of all, we should describe the multiplicity distribution measured in 1 . We use the standard AGK

w xcutting rules 18 which relate via unitarity inelastic processes with multi-pomeron exchanges in the elasticamplitude. Keeping only the double-Pomeron corrections the multiplicity distribution normalized to unity reads,

nn² : ² :n 2 nŽ .y² n: y2²n:N s 1y4d e q4d e . 2Ž . Ž .n n ! n !

² :Here n is the mean number of produced particles corresponding to one cut pomeron; the parameter d is theweight of the double-pomeron contribution to the total cross section. We use them as free parameters to adjust

² : Ž .to the data by eye. The result with n s5.2 and ds0.02 is shown in Fig. 2 left by full circles. It isremarkable that the value of d is about an order of magnitude smaller than what follows from eikonal model,

w xFig. 2. Multiplicity distribution of charged hadrons produced in photon-proton interaction as measured in 1 . The histogram represents theŽ .data, the black points are the result of our calculation left Baryon asymmetry as function of multiplicity of charged hadrons. The crosses

w xare the results of measurements in 1 . The solid and dashed curves show our predictions for the gluonic and quark exchange mechanismsŽ .respectively right . See the text for details.

( )B. KopelioÕich, B. PoÕhrPhysics Letters B 446 1999 321–325324

which is reasonably good for hadronic collisions. We interpret it as suppression of gluons and sea quarks, whichare responsible for the multi-pomeron terms, in a photon compared to a hadron. This would also naturallyexplain a substantially steeper growth with energy of the photon-photon total cross section measured by the L3

w xcollaboration 19 . A further development of this issue goes too far beyond the scope of present paper. Whateverthe reason of smallness of d is, it has no effect on our results for multiplicity dependence of baryon asymmetry.

Ž .Now we are in position to predict the variation of the baryon asymmetry 1 with associated multiplicity.Ž .Most of the p and p contributing to the denominator of 1 are due to production of sea baryon-antibaryon

pairs, number of which is proportional to the multiplicity of hadrons. Therefore, the shape of n-dependence ofŽ . Ž . Ž .N qN in 1 is given by 2 . The same is true for the numerator in 1 , except the mean multiplicity associatedp p

Ž .with net BN production according to Fig. 1 is larger for the gluonic mechanism. Using 2 we can represent theŽ .asymmetry 1 as function of n,

nn yŽKy1.²n: yK²n:1y4d K e q4d Kq1 eŽ . Ž .A sA . 3Ž .n n y²n:1q4d 2 e y1Ž .

According to the previous discussion we assume that the mean associated multiplicity for net BN productionŽ .see Fig. 1 is K times larger than the mean multiplicity in BN symmetric events, where Ks5r4 or Ks1 for

Ž .gluonic and quarks exchange mechanisms respectively. The results for A are depicted in Fig. 2 right by solidnŽ . Ž . w x w xKs5r4 and dashed Ks1 lines. We use As0.07 as it is predicted in 2 . The data 1 shown by crossesagree well with the assumption that the baryon asymmetry is dominated by the contribution of the gluonicmechanism, but reject any sizable contribution of the preasymptotic quark exchange mechanism which leads toa constant A .n

Note, that the systematical uncertainty not included in the error bars in Fig. 2 can affect only the overallw xnormalization 1 , but does not change the shape of n-dependence. The kinematical restrictions imposed by the

² :experimental set up do not affect our calculations, which depend only on parameters n and d measured in thesame experiment.

Summarizing, an exciting possibility that baryon number can be transferred by gluons through a largew x w xrapidity interval without attenuation 3 is strongly supported by measurements 1 of baryon asymmetry in g p

w xcollisions. Although the magnitude of the effect agrees with the prediction made in 2 , this fact alone cannotexclude a large contribution of the preasymptotic valence quark exchange mechanism at this rapidity intervalD yf7–8. We have found that the dependence of baryon asymmetry on associated particle multiplicity is

w xextremely sensitive to the underlying mechanism. Comparison with corresponding data from 1 stronglysupports dominance of the gluonic mechanism and excludes a large contribution of BN transfer by valencequarks.

Acknowledgements

We are grateful to Andrei Rostovtsev for useful discussions.

References

w x1 The H1 Collaboration, C. Adloff et al., Measurement of the Baryon-Antibaryon Asymmetry in Photoproduction at HERA, Submitted tothe 29th Int. Conf. on High-Energy Phys. ICHEP98, Vancouver, Canada, July 1998. Abstract 556. http:rrichep98.triumf.carprivaterconvenorsrbody.asp?abstractIDs556.

w x Ž .2 B.Z. Kopeliovich, B. Povh, Z. Phys. C 75 1997 693.w x Ž .3 B.Z. Kopeliovich, B.G. Zakharov, Z. Phys. C 43 1989 241.w x Ž . Ž .4 G.C. Rossi, G. Veneziano, Nucl. Phys. B 123 1977 507; Phys. Rep. 63 1980 149.

( )B. KopelioÕich, B. PoÕhrPhysics Letters B 446 1999 321–325 325

w x5 R.G. Roberts, The structure of the proton, Cambridge University Press, 1990, p. 35.w x Ž .6 E. Gotsman, S. Nussinov, Phys. Rev. D 22 1980 624.w x Ž .7 B.Z. Kopeliovich, Sov. J. Nucl. Phys. 45 1987 1078.w x Ž .8 B.Z. Kopeliovich, B.G. Zakharov, Sov. J. Nucl. Phys. 48 1988 136.w x Ž .9 B.Z. Kopeliovich, B.G. Zakharov, Phys. Lett. B 211 1988 221.

w x Ž .10 B.Z. Kopeliovich, B.G. Zakharov, Sov. J. Nucl. Phys. 49 1989 674.w x Ž .11 B.Z. Kopeliovich, B.G. Zakharov, Sov. Phys. Particles and Nuclei 22 1991 140.w x Ž .12 B. Alper, Nucl. Phys. B 100 1975 237.w x Ž .13 T. Akesson, Nucl. Phys. B 228 1983 409.w x Ž .14 L. Camilleri, Phys. Rep. 53 1987 144.w x Ž .15 A. Capella, B.Z. Kopeliovich, Phys. Lett. B 381 1996 325.w x16 S.E. Vance, M. Gyulassy, X.-N. Wang, nucl-thr9806008.w x Ž .17 Y. Eylon, H. Harari, Nucl. Phys. B 80 1974 349.w x Ž .18 V. Abramovsky, V.N. Gribov, O.V. Kancheli, Sov. J. Nucl. Phys. 18 1974 308.w x19 The L3 Collaboration, Cross section of hadron production in gg collisions at LEP, paper presented at XXIX Int. Conf. on High Energy

Phys., Vancouver, 1998. L3 note 2280, http:rrl3www.cern.chranalysisrJoachimMnichrabstracts_ichep.html.

28 January 1999


Inverse slope systematics in high-energy pqp and AuqAureactions 1

A. Dumitru a, C. Spieles b

a Physics Department, Yale UniÕersity, P.O. Box 208124, New HaÕen, CT 06520, USAb Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Received 17 October 1998Editor: P.V. Landshoff

Abstract

We employ the Monte Carlo PYTHIA to calculate the transverse mass spectra of various hadrons and their inverse slopes) )' Ž .T at m yms1.5–2 GeV in pqp reactions at s s200 GeV. Due to multiple minijet production T in generalT

)Ž .increases as a function of the hadron mass. Moreover, the T m systematics has a ‘‘discontinuity’’ at the charm threshold,i.e. the inverse slope of D-mesons is much higher than that of non-charmed hadrons and even of the heavier L baryon. TheC

experimental observation of this characteristic behaviour in AuqAu collisions would indicate the absence of c-quarkrescattering. In contrast, the assumption of thermalized partons and hydrodynamical evolution would lead to a smoothly

)Ž .increasing T m , without discontinuity at the charm threshold. The degree of collective transverse flow, indicated by the)Ž .slope of the T m systematics, depends strongly on whether kinetic equilibrium is maintained for some time after

hadronization or not. q 1999 Elsevier Science B.V. All rights reserved.

Experimental data on single-inclusive hadron pro-'duction in pqp reactions at s s23–63 GeV show

nearly exponential transverse mass spectra at loww xtransverse momentum 1 . Moreover, the inverse

Ž .slopes ‘‘apparent temperatures’’ are practically thesame for pions, kaons, protons and their antiparti-cles, i.e. they do not depend on the hadron mass. Theobserved deviation from this behaviour in nucleus–nucleus collisions has been interpreted as a signature

w xfor collective transverse flow 2 .In this letter, we discuss how the inverse slope

systematics extends to higher energies, i.e. pqp'reactions at s s200 GeV, and transverse momenta

on the order of a few GeV. In this kinematic domain,

1 Supported by AvH Foundation and DAAD.

minijet production and fragmentation gives an im-portant contribution.

We shall also discuss AuqAu collisions, whereminijets might rescatter substantially. This, in turn,should reflect in a characteristic mass dependence ofthe inverse slopes. We will confront the predictionsof two extreme scenarios: superposition of minijet

Ž .production and fragmentation without final-stateŽ .interactions as in pqp reactions versus local ther-

malization of parton matter undergoing hydrodynam-ical expansion.

'In pqp reactions at s s200 GeV, we expectthat the hadrons with transverse masses on the orderof a few GeV are dominantly produced via fragmen-tation of minijets. As a first step, we estimate thetransverse momentum distribution of c-quarks atmidrapidity employing the well-known expression


( )A. Dumitru, C. SpielesrPhysics Letters B 446 1999 326–331 327

for inclusive single-jet production within perturba-Ž .tive QCD pQCD in leading-logarithm approxima-

Ž . w xtion LLA 3 ,

d3sE pp™ccqXŽ .3d p

s dx dx G x ,m2 G x ,m2Ž . Ž .H a b a b

=

g g ™ ccs dsˆ2ˆd sq tquy2m . 1Ž .ˆ ˆŽ .c

p dt

ˆs, t, u are the usual Mandelstam variables of theˆ ˆparton – parton scattering subprocess, andds ab™ cdrdt denotes its differential cross section inlowest order of perturbative QCD. For simplicity, wetake into account only the contribution from thegg™qq process. This is sufficient to illustrate ourpoint.

The expression for dsrdt that accounts for thefinite quark-mass is rather lengthy and can be found

w xin the literature, cf. e.g. 4 . We therefore omit ithere. We assume m s1.5 GeV, L s300 MeV,c QCD

and evaluate the strong coupling constant at themomentum transfer scale given by the transverse

2 2 2 2 w xmass of the produced quark, Q sm sm qp 5 .T c TŽ 2 .G x,m denotes the LO gluon distribution func-

w xtion in the proton, which we take from Ref. 6 .Since we work only in LLA, we employ the samemomentum scale in the parton distribution functionsas in the strong coupling constant, i.e. m2 'Q2.

In principle, the hadron spectra could be calcu-lated by convoluting the expression for jet produc-

w xtion with fragmentation functions 3 . However, fortransverse masses of a few GeV such an analysiswould at best be qualitative since, e.g., multi jetproduction and initial state radiation are not in-cluded. Also, in this domain the fragmentation func-tions suffer from logarithmic infrared divergenceswhich have to be regulated by a model for softparticle production.

Therefore, to calculate the hadron transverse massspectra we rather employ the PYTHIA event genera-

w x Žtor 7 , using the default parameter settings version.6.115 . PYTHIA simulates high energy hadronic and

leptonic interactions by implementing a large num-Ž .ber of hard and soft sub- processes, and in particu-

lar, a scheme for the nonperturbative hadronization

mechanism. The model goes significantly beyondpQCD in LLA. It describes not only single-inclusiveminijet production but also includes multi-jet produc-tion and initial state radiation. PYTHIA is designed

Žto model the complete event structure like jet pro-files, multiplicity fluctuations, various types of corre-

.lations etc. and it has been shown to agree reason-ably well with experimental observations at collider

w xenergies 8 .w xThe Lund string scheme 9 is an integral part of

the model used to describe the fragmentation ofŽ .minijets. They are modeled as one dimensionalcolor flux tubes which decay into hadrons: quark–antiquark pairs tunnel in the color field and the fieldenergy is transformed into the sum of the transversemasses m 2.T

The tunnel probability is proportional toŽ 2 .exp yp m rk , where k is the string tension. Thus,T

the creation of quarks with high transverse momen-tum is heavily suppressed. As a consequence, alsothe produced hadrons cannot acquire large values ofm . The above formula also leads to a strong sup-T

pression of heavy quarks. The probability for produc-ing a light quark as compared to a charm quark is

y11 w xabout 1:10 9 . The energy E and the longitudinalmomentum p of the produced hadrons are deter-z

mined by an iterative scheme: for each hadron theŽ .fragmentation function f z determines the probabil-

ity that the hadron picks a fraction z out of theavailable Eqp . The default fragmentation func-z

Ž . y1Žtion used in PYTHIA reads f z ; z 1 y.0.3 Ž y2 2 .z exp y0.58 GeV m rz .T

The Lund string model has been shown to suc-cessfully describe the nonperturbative hadronization

q y w xin e e annihilation events 9 . Moreover, the con-cept of a color flux tube, fragmenting according to auniversal fragmentation scheme, has been carriedover to hadron–hadron interactions. The microscopic

w xmodels FRITIOF, RQMD, and UrQMD 10 utilizestring fragmentation routines for the simulation ofsoft particle production in pqp, pqA and AqAreactions. The only difference to strings from eqey

Ž .annihilation are the leading valence di- quarks – the

2 Note that ‘transverse’ is defined with respect to the stringaxis. Nothing is said here about the orientation of the string withrespect to the beam axis.

( )A. Dumitru, C. SpielesrPhysics Letters B 446 1999 326–331328

remnants of the incident hadrons – as string end-points. The excitation of the strings is due to singleor double diffractive interactions which can be un-derstood and parametrized in the framework of Regge

Ž w x.theory see e.g. 11 . These nonperturbative pro-cesses account for the major part of the total cross

'sections at CERN-SPS energies or higher, s )20GeV. Strings which are excited in these processesare preferentially oriented along the beam axis of theincident hadrons, leading to much higher typicallongitudinal than transverse momenta of producedhadrons.

In the kinematic region where perturbative minijetproduction becomes important, the color flux tubesmay no longer be oriented longitudinally, but accord-ing to the pQCD subprocess that produces the mini-jets acquire significant transverse momentum. In theframe were the z-axis is parallel to the flux tube, thehadrons are still produced according to the abovefragmentation function. However, the string axis andthe particles’ momenta are now rotated with respectto the lab frame.

Fig. 1 depicts the resulting transverse mass spec-tra. One observes that the PYTHIA-spectra follownonexponential distributions, remnant of the pertur-bative QCD processes that describe the minijet pro-duction. Also, the ‘‘stiffness’’ of the spectra in-creases with the mass of the hadron. This will bediscussed in more detail below. In particular, theslope of the D-meson spectrum equals that of thec-quarks, which are produced purely by perturbativeparton–parton scattering 3. The reason is that D-me-sons can only be produced as the leading hadronfrom a crc quark jet since the tunneling probabilityof a c–c pair in a color flux tube is practically zeroŽ .as discussed above . The m -distribution of L -T C

baryons, on the other hand, is slightly ‘‘softer’’ sinceit involves tunneling of a diquark–antidiquark pair

Žout of the vacuum besides the perturbative produc-.tion of a c-quark .

In Fig. 2, we show the inverse slopes as a func-tion of hadron mass. We compute the inverse slope

3 To obtain the number distribution of c-quarks we have simplyŽ .divided the differential cross section, Eq. 1 , by 40 mb. If we

Ž .multiplied by two to include also c-quarks the quark and D-me-son spectra would coincide.

Ž .Fig. 1. Transverse mass spectra at midrapidity, ys0 of various'hadrons in pqp reactions at s s200 GeV, as calculated with

PYTHIA 6.115. The spectra of the individual hadron speciesinclude all isospin projections and charge conjugated states. The

Ž .c-quark spectrum without c-quarks is calculated within pQCD inLLA.

by a fit of the transverse mass spectrum to a Boltz-mann distribution,

1 d2N)Aexp ym rT , 2Ž .Ž .T2 dm dym TT

w xWe restrict the fit to the range m ymg 1.5,2T

GeV.As already mentioned in the introduction, in pqp

reactions at lower energies, the apparent tempera-tures at small p were found to be about the sameT

w xfor pions, kaons, protons and their antiparticles 1,2 .'This changes at higher p and s due to the contri-T

bution from minijets.The single inclusive cross section at midrapidity

and m ymG1 GeV is dominated by color fluxT

tubes that are no longer oriented longitudinally, butŽhave significant transverse momentum due to the

.pQCD subprocess . This leads to much higher mT

than in case of longitudinally oriented strings. More-over, the inverse slopes of the m ym distributionsT

in the lab frame are strongly m-dependent. Such anincrease of the inverse slope with particle mass canalso be extracted from pqp collider experiments


Ž .Fig. 2. Inverse slopes at midrapidity, ys0 as a function of'hadron mass; PYTHIA 6.115 predictions for pqp at s s200

GeV, and results from hydrodynamics of pqp and AuqAuŽcalculated on the boundary between mixed and hadronic phase

.and on the T s130 MeV isotherm, respectively .

'w x12 at s s540 GeV, and from the parametrizationw xof the p -distributions given in Ref. 13 . AlthoughT

that parametrization was restricted to the energy'region s -63 GeV, it yields slopes for the pi-,

phi-, and D-mesons that agree with those obtainedfrom PYTHIA to within 20%, if one simply extrapo-

'lates it to s s200 GeV.The qualitative reason for this behaviour can be

traced back to eqey annihilation processes whichcan be modeled as the fragmentation of a string withfixed energy. The observed kinetic energy distribu-tion of produced hadrons show a considerably harder

w xspectrum for kaons and protons than for pions 14 .Thinking of the initial state in AqA collisions as

a superposition of pqp reactions one could attributeany change of the particle slopes to final state inter-actions. The multiple soft p -kicks that a projectileT

nucleon can experience while propagating throughthe target, i.e. the Cronin effect, has been shown to

'be small for pions produced in AuqAu at s sw x200 A GeV 15 . However, this does not automati-

cally hold true for heavier hadrons. Since we havenot made any attempt to include such multiple scat-tering effects, nor modifications of the parton distri-bution functions in nuclei, we restrict the application

of the minijetrstring fragmentation model to pqpreactions.

In ultrarelativistic head-on collisions of heavyŽ .nuclei A;200 , the situation might, however, be

very different as compared to the pqp case. Trans-w xport calculations 16 suggest that the initially di-

rected momenta of the partons could be quicklyredistributed through rescattering. This would then

Ž .lead to the formation of a very hot Tf300 GeVŽ 3.vacuum of ‘‘macroscopic’’ size volume ;100 fm ,

offering the opportunity to study hot QCD.Indeed, if minijets with transverse masses of a

few GeV thermalize quickly, energy densities in3 w x Žexcess of 10 GeVrfm can be reached 16 for

' .AuqAu at s s200 A GeV . This energy density ismuch higher than in pqp since the contribution due

2r3 w xto minijets increases as A 17 .For definitness we consider AuqAu collisions at

's s200 A GeV that will be studied at the Relativis-Ž .tic Heavy Ion Collider RHIC in Brookhaven in the

near future. As the extreme case we assume that theproduced minijets rescatter so frequently that theythermalize locally. The subsequent evolution is thendescribed within hydrodynamics.

The pQCD processes that produce the minijets inthe central region occur on a time scale of 1rm ,T

0.1 fm. We assume that one to two rescatterings perparticle are necessary for local thermalization, andtake t s0.6 fm. This seems also reasonable in view0

of the fact that with t s1 fm one is able to0

reproduce the measured single particle spectra for'central PbqPb reactions at s s18 A GeV, cf. e.g.

w x18 . At the higher center of mass energy of RHIC,the parton density in the central region increases, andtherefore a smaller thermalization time is expected,

w xcf. also 16,19 .As initial conditions we assume a net baryon

rapidity density of dN rdys25, and an energyB

density of e s17 GeVrfm3. Employing the for-0w x 2 Ž .mula of Bjorken 20 , e t p R sdE t rdy, with0 0 T T 0

a nuclear radius of R s6 fm, we obtain an initialT

transverse energy at midrapidity of dE rdys1.2T

TeV. We have extracted this value for dE rdy forT'Ž .central b-2 fm AuqAu collisions at s s200 AGeV from the minijetrstring fragmentation model

w xFRITIOF 7.02 21 . It is also compatible with thew xprediction of HIJING 22 . Note that for an isen-

tropic hydrodynamical expansion dE rdy decreasesT

( )A. Dumitru, C. SpielesrPhysics Letters B 446 1999 326–331330

with time. On the hadronization hypersurface, wew xobtain dE rdys640 GeV 18 .T

Ž .The initial net baryon density at midrapidity, r ,0

is given by a similar expression as e above, except0

that dE rdy is replaced by dN rdy. These densitiesT BŽ .are initially assumed to be distributed in the trans-

verse plane according to a so-called ‘‘wounded nu-Ž . Ž . Ž .cleon’’ distribution, e t s e f r , r t si 0 T i

3 2 2Ž . Ž . (r f r , with f r s 1yr rR .0 T T T T2

In order to respect boost-invariance, we requirethe longitudinal flow to have a ‘‘scaling flow’’ pro-

w xfile, Õ szrt 20,23 . Cylindrically symmetric trans-zw xverse expansion 18,24,25 is superimposed. For T)

T s160 MeV we employ the well-known MITC

bagmodel equation of state. For simplicity we as-Žsume an ideal gas of quarks, antiquarks with masses

.m sm s0, m s150 MeV , and gluons.u d s

For T-T we assume an ideal hadron gas thatC

includes the complete hadronic spectrum up to amass of 2 GeV. At TsT we require that bothC

pressures are equal, which fixes the bag constant toBs380 MeVrfm3. The normalization is such thatfor T™0 the pressure of the nonperturbative vac-

Ž .uum i.e. that of the hadronic phase vanishes. Byconstruction the EoS exhibits a first-order phasetransition. This ‘‘softening’’ of the EoS in the transi-tion region strongly reduces the tendency of matter

w xto expand on account of its pressure 25,26 . For amore detailed discussion of the initial conditions andexpansion dynamics in AuqAu at RHIC energy

w xplease refer to Ref. 18 .For comparison, we have also extracted the in-

'verse slopes from hydrodynamics in pqp at s s200 GeV. In this case, we employ R s1.18 fm,TŽ . Ž .f r s Q R y r , dN rdy s 0, dE rdy s 2.8T T T B T

Ž . ŽGeV as obtained from PYTHIA , and for simplic-.ity the same t as for AuqAu.0

In Fig. 2 we compare the PYTHIA predictions forpqp with those of hydrodynamics of pqp andAuqAu. Within the hydrodynamical solution forAuqAu, strong collective flow of quark–gluonmatter 4 Doppler-shifts T ) far above the real emis-sion temperature T . If kinetic equilibrium in the hotC

4 The average flow velocity on the phase boundary to purelyhadronic matter is approximately one third of the velocity of lightw x18 .

hadron gas is maintained for some time afterŽhadronization say, until the temperature drops to

.130 MeV , the collective transverse flow can in-crease even further, and T ) ;400 MeV can be

Žreached for the charmed hadrons cf. open squares in.Fig. 2 . Hadrons produced in the expanding QGP can

thus reach comparably ‘‘stiff’’ m -spectra as thoseT

produced in pqp via minijets.One also observes that in hydrodynamics T ) is

nearly proportional to m. In particular, there is nojump in T ) at the charm threshold and the inverseslope of the L is larger than that of the D, unlikeC

in the minijetrstring fragmentation model. In a ther-Ž .mal environment without interactions , the mass is

the only hadron-specific quantity that enters its mo-mentum distribution. Thus, if the perturbatively pro-duced c–c pairs equilibrate with the QGP, the in-verse slope of the D-mesons is significantly smallerthan in pqp reactions at the same energy per nu-

Ž ² :cleon cf. also the discussion of p in Refs.DTw x w x18,27 , and 28 for the effect of c-quark energy losson lepton radiation in ultrarelativistic heavy-ion col-

.lisions .In pqp reactions, on the other hand, the initial

energy density e s1.1 GeVrfm3 is much smaller0

than in AuqAu. In fact, for our choice of initialconditions the initial state is not in the pure QGPphase but in the phase coexistence region. Conse-quently, on the hadronization hypersurface there ispractically no collective transverse flow, and theinverse slopes of the various hadrons are similar, andequal to the real emission temperature TsT . IfC

freeze-out occurs deeper in the hadronic phase, e.g.on the Ts130 MeV isotherm, a small flow iscreated due to rescattering in the purely hadronic

'phase. The pressure in pqp reactions at s of a fewhundred GeV can not exceed that of the nonperturba-tive vacuum by far, and we thus find no significant

w xtransverse expansion 29 . Therefore, in contrast tothe minijetrstring fragmentation model hydrodynam-ics can not reproduce the experimentally observedw x12 bending of the m distributions and the increaseT

of T ) with m.In summary, we have shown that in pqp reac-

tions at high energy the inverse slopes T ) of theŽtransverse mass spectra at midrapidity and m ymT

.of a few GeV of various hadrons are correlated totheir mass. This is due to the underlying pQCD


Ž .subprocess i.e. minijet production that producesfragmenting color strings that are not parallel to the

) Ž .beam axis. T m shows a very strong increase atthe charm threshold, i.e. the inverse slope of theD-mesons is much higher than that of the V-baryons.

If a dense QGP is created in central AuqAucollisions, in which u-,d-,s-, and c-quarks, and the

Ž .gluons up to a few GeV of p equilibrate kineti-T

cally, the inverse slope of the D-mesons decreasessubstantially as compared to the pqp case, and thatof the V-baryons increases. Collective transverseflow of such a hypothetical quark–gluon fluid wouldestablish a nearly linear relationship between theinverse slopes and the hadron masses. Thus, the

) Ž .T m systematics at m ym,1–3 GeV providesT

an opportunity to experimentally determine the de-gree of heavy quark rescattering and equilibration inrelativistic heavy ion collisions.

Acknowledgements

We thank M. Gyulassy, B. Muller, and T. Ullrich¨for helpful discussions. A.D. gratefully acknowl-edges a postdoctoral fellowship by the German Aca-

Ž .demic Exchange Service DAAD . C.S. thanks theNuclear Theory Group at the LBNL for support andkind hospitality. C.S. is supported by the Alexandervon Humboldt Foundation through a Feodor LynenFellowship.

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28 January 1999


ž /ss dominance of the f 980 meson0

R. Delbourgo, Dongsheng Liu, M.D. Scadron 1

School of Physics and Mathematics, UniÕersity of Tasmania, GPO Box 252-21, Hobart, Tasmania 7001, Australia

Received 22 October 1998Editor: M. Cvetic

Abstract

Ž .We prove that recent data demonstrates unequivocally that the scalar meson f 980 is mostly composed of ss quarks0

and that the coupling of f to photons and mesons is in agreement with expectations from the linear sigma model. q 19990

Published by Elsevier Science B.V.All rights reserved.

PACS: 13.20.Gd; 13.25.qm; 14.40.Cs

w x Ž .The observation 1 of the decay mode f 1020™p 0p 0g was reported very recently for the firsttime. The experiment clearly shows that the process

Ž .is dominated by the f 980 g channel. This may be0

contrasted with the decay process Jrc™vpp ,w xwhich was measured a decade ago 2 and where it

Ž .was found that the v f 980 channel was highly0Ž . Ž .suppressed but that the vs 500 and v f 12702

channels predominated. In this letter we wish to givean interpretation of these results and show that theŽ .f 980 scalar meson is mostly composed of strange0

quarks.First we recall that, according to quark models,

Ž . Žthe vector meson v 782 is 97% nonstrange viz.'Ž . . Ž .NSs uuqdd r 2 and 3% strange Ssss . Con-

Ž .versely, the other vector meson f 1020 has theopposite composition: 3% NS and 97% S. We will

w xshow that the above two experiments 1,2 stronglyŽ .suggest that the f 980 is largely S. Independently0

1 Permanent address: Physics Department, University of Ari-zona, Tucson, AZ 85721, USA.

and for different reasons the phenomenological studyw xof low-energy pp scattering 3 and the phase shift

w xanalysis 4 of p K scattering have led to the sameŽ .conclusion about the composition of f 980 .0

The effective coupling constants, extracted fromw xRef. 1 , are based upon an inferred f total decay0

width of 188q48 MeV:y33

g 2 effq yr4ps0.51q0 .13 GeV 2 orf p p y0.090

< eff <0 q yg ,2.5"0.3 GeV, 1Ž .f p p

g 2 effq yr4ps2.10q0 .88 GeV 2 orf K K y0.560

< eff <0 q yg ,5.1"0.9 GeV. 2Ž .f K K

These values roughly follow from a dynamicallyŽ .generated theory of the SU 3 linear sigma model

Ž . w xLs M Lagrangian 5 where a scalar meson nonetŽ . Ž . Ž .pattern is demanded: s 670 ,k 810 ,s 940 ,NS S

Ž . w xa 984 .A related description 6 is that the scalar0Ž .mesons are composed of 4 quarks qqqq , which is

w xthe description adopted by Ref. 1 .

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01538-X

( )R. Delbourgo et al.rPhysics Letters B 446 1999 332–335 333

w xIn order to bring the observed 7 isoscalar mesonsŽ . Ž .s 600 and f 980 into the Ls M picture, we must0

first consider the NS–S mixing basis,< : < : < :s scosf s ysinf s ,s NS s S

< : < : < :f ssinf s qcosf s 3Ž .0 s NS s SX w xin a manner similar to h–h mixing 8 . For the

² < : w xorthogonal mixed states s f s0, we find 50Ž .from 3 ,

m2 sm2 cos2f qm2 sin2f ,s s s f sNS 0

m2 sm2 sin2f qm2 cos2f . 4Ž .s s s f sS 0

Inserting the dynamically generated NJL-type massesw x5 ,

m s2m,670 MeV, m s2m ,940 MeV,ˆs s sNS S

5Ž .along with m ,980 MeV, one getsf 0

1r22 2 2m s m qm ym ,610 MeV,s s s fNS S 0

1r22 2m ymf s0 NSf sarccos ,208. 6Ž .s 2 2m ymf s0

Such a value for the scalar mixing angle was pro-w xposed several years ago 8 and it is worth noting

Ž .that the proximity in mass between s 940 andSŽ .f 980 has its counterpart in the vector mesons with0Ž . Ž .f 985 and f 1020 . Thus one should not be sur-S

Ž . Ž .prised that the f 980 meson with cos208,0.94 is0

principally an ss state.In order to link up to the recently extracted inter-

Ž . Ž .actions 1 and 2 , we first state the predicted Ls Mw x ŽLagrangian couplings 5 for brevity we refer to

either the pqpy or KqKy final states as pp or.KK ,

g s m2 ym2 r2 f ,2.3 GeV,Ž .ps p s p pNS NS

g s m2 ym2 r4 f ,0.45 GeV,Ž .K s K s K KNS NS

2 2 'g s m ym r2 2 f ,2.1 GeV, 7Ž .Ž .K s K s K KS S

where we have substituted the experimental valuesf ,93 MeV, f rf ,1.22. From these and thep K p

mixing angle f ,208 we may compute the effectivesw x9 Ls M couplings,

g eff s2sinf g ,1.6 GeV, 8Ž .f pp s ps p0 NS

g eff s2sinf g q2cosf g ,4.3 GeV.f K K s K s K s K s K0 NS S

9Ž .

Ž . Ž .Although these predictions 8 and 9 seem shy ofŽ . Ž .1 and 2 by 64% and 84% respectively, we havenot yet considered other data.

Specifically, the detailed Particle Data GroupŽ . w xPDG Tables 7 quote the average decay rate f ™0

gg to be 0.56"0.11 keV. In combination with thew x Ž . y5f ™gg branching ratio 7 of 1.19"0.33 =10 ,0

one may derive a total f width of 47"16 MeV.0

Given the f ™pp branching ratio, one may deduce0w xthe PDG effective coupling 9 ,

< eff <g s1.1"0.4 GeV. 10Ž .PDGf pp0

Ž < <g is unknown, because of the negligiblePDGf K K0

.phase space.Ž .This said, we note that our theoretical values 8

Ž .and 9 lie between the Novosibirsk and PDG values.See Table 1. Regardless of the final phenomenologi-cal resolution of the discrepancy, we must emphasizethat the important result is the contrast between the

w xmeasured pp invariant mass spectra of Refs. 1,2 .For the reader’s convenience, we display the ob-

served invariant p 0p 0 mass spectrum in f™p 0p 0g

in Fig. 1; our quark model interpretation is givenalongside, in Fig. 2. The observed pp mass spectrafor Jrc™vpp are displayed in Fig. 3 to show thestark difference from Fig. 1; again we provide thequark model interpretation alongside, in Fig. 4. The

Ž . 0 0 Ž .resonance bumps f 980 in f™p p g and s 5000

in Jrc™vpp along with the near absence of anŽ . Ž .f 980 bump in Jrc™vpp and of s 500 in0

0 0 Ž .f™p p g is telling us that s 500 is mostly NSwhile f is mainly S. This is in consonance with the0

linear sigma model and a scalar mixing angle off ;208.s

However, to stress that this small mixing angle off is not the central issue, we close this letter bys

considering f ™2g decays which proceed via a0

Table 1Ž .Comparison of effective couplings in GeV of f to the pseu-0

doscalar mesons

New data Ls M theory Particle Data Groupeffg 2.5"0.3 1.6 1.1"0.4f p p0

effg 5.1"0.9 4.3 –f K K0

( )R. Delbourgo et al.rPhysics Letters B 446 1999 332–335334

Fig. 1. The measured p 0p 0 invariant spectrum in f ™p 0p 0g .Reprinted from DM2 group by kind permission from Dr. ACalcaterra.

quark loop. We may compare this radiative channelwith p 0 ™gg , which is accurately estimated by anonstrange quark triangle; the latter provides theeffective amplitude aN r3p f ,0.025 GeVy1 andc p

agrees with the data to within 2% for N s3, f ,93c p

MeV.Ž .If f were purely nonstrange too, the isoscalar0

f gg effective amplitude would be given by0

< < y1M f ™2g s5aN r9p f ,0.042 GeV ,Ž .0NS c p

11Ž .

predicting a decay rate

3 < < 2G f ™2g sm M r64p;8 keV. 12Ž . Ž .0NS f NS0

On the other hand, if f were a pure ss scalar, its0

effective gg amplitude would be

< <M f ™2gŽ .0S

saN g r9p m ;0.0081 GeVy1 , 13Ž .c f ss s0

for a constituent mass m ,490 MeV and a stranges

Fig. 2. Theoretical interpretation of f ™g f ™gp 0p 0, as due to0

three gluon exchange.

Fig. 3. Fit of the pp distribution in Jrc ™ vpp . Reprintedfrom hep-exr9807016 by kind permission from Budker Institute.

' ' 'w x 0f ss coupling 5 of 2 g s 2 2pr 3 . The0 p qq

decay rate would then be3 < < 2G f ™2g sm M r64p;0.3 keV. 14Ž . Ž .0S f S0

Ž .From 12 we see that the pure f ™gg decay0NSw xrate is about 15 times larger than the PDG 7

average rate of 0.56 keV, while the pure f ™gg0SŽ .rate 14 is within striking range of the experimental

Fig. 4. Theoretical interpretation of Jrc ™ vs ™ vpp .

( )R. Delbourgo et al.rPhysics Letters B 446 1999 332–335 335

value. The comparison can be improved by includingthe small mixing angle f . But in any case the quarks

triangle description of radiative meson decays rein-Ž .forces the fact that f 980 is mostly ss rather than a0

Ž .uuqdd scalar meson.

Acknowledgements

This research was partially supported by the Aus-tralian Research Council. M.D.S. appreciates thehospitality of the University of Tasmania where thiswork was carried out.

References

w x1 V.M. Aulchenko et al., Budker Institute, Novosibirsk, hep-exr9807016, 17r07r1998.

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Ž .Theor. Phys. 98 1997 621; M. Ishida, S. Ishida, hep-phr9712231.

w x Ž .5 R. Delbourgo, M.D. Scadron, Int. J. Mod. Phys. A 13 1998Ž .657; also see M. Levy, Nuovo Cim. A 52 1967 23; S.

Ž .Gasiorowicz, D. Geffen, Rev. Mod. Phys. 41 1969 531.w x Ž .6 R.L. Jaffe, Phys. Rev. D 15 1977 267, 281, 4084; N.N.

Ž .Achasov, V.N. Ivanchenko, Nucl. Phys. B 315 1989 465.w x Ž .7 Particle Data Group, C. Caso et al., Eur. Phys. J. C 3 1998 1.w x Ž .8 H.F. Jones, M.D. Scadron, Nucl. Phys. B 155 1979 409;

Ž .M.D. Scadron, Phys. Rev. D 29 1984 2079; A. Bramon,Ž .M.D. Scadron, Phys. Lett. B 234 1990 346; see also N.

Ž .Isgur, Phys. Rev. D 12 1975 3770.w x < eff < 29 Effective couplings are defined via G s3 p g rf p p f p p0 0

16p m2 and they yield g eff ,1.12 GeV for G s0.781f f p p f p p0 0 0

=47 MeV. We need to double the Lagrangian couplings by afactor of two to compare them with g eff.

28 January 1999


ž /Is the resonance D 2637 really a radial excitation?

D. Melikhov 1,2, O. Pene 3´Laboratoire de Physique Theorique et Hautes Energies, UniÕersite de Paris XI, Batiment 211, 91405 Orsay Cedex, France 4´ ´ ˆ

Received 15 September 1998; revised 16 November 1998Editor: R. Gatto

Abstract

Ž .We consider various possible identifications of the quantum numbers of the resonance D 2637 recently observed byDELPHI in the D)pp channel. We argue that in spite of a good agreement of the measured mass with the quark-modelprediction for the radial excitation, a total width as small as F15 MeV makes problematic its identification as a radialcharm excitation. The J Ps2y,3y orbitally excited mesons with such a mass could have widths of the observed order ofmagnitude. However in this case one would expect two neighbouring states with the mass difference of about 30–50 MeVcorresponding to the nearly degenerate components of the heavy-meson multiplet with light-quanta angular momentumjs5r2, and moreover, according to the quark-model predictions the mass of the orbital excitation should be more than 50MeV larger than 2637 MeV. Thus we conclude that, at present, we find no fully convincing understanding of the quantumnumbers of the observed resonance. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

Recently, DELPHI has observed a narrow reso-Ž . )nance D 2637 in the D pp channel with a total

width of less than the detector resolution: 15 MeVw x1 . The mass of the observed resonance turns out tobe in perfect agreement with predictions of the quark

w x )XŽ Pmodels 2,3 for the charm radial excitation D J

y.s1 . This coincidence has led to a quick identifi-cation of the discovered resonance state with theradially excited vector charm meson.

1 On leave of absence from Nuclear Physics Institute, MoscowState University, Moscow, 119899, Russia.

2 E-mail: [email protected] E-mail: [email protected] Laboratoire associe au Centre National de la Recherche Scien-´

tifique - URA D00063.

In this letter we reconsider the identification ofthe quantum numbers of the observed resonance bysubmitting it to the following criteria:Ø the resonance mass should be ,2637"6 MeV;Ø the resonance width should be F15 MeV;Ø the resonance should have a sizable branching

Ž . )ratio of the channel D 2637 ™D pp in whichit has been observed.We estimate the decay rate of a radial excitation

of the reported mass and find that, although thepartial widths are strongly model-dependent, the totalrate is conservatively estimated to be not less thanabout 50 MeV and presumably it should be muchbroader. We thus conclude that the observed widthof only F15 MeV is hardly compatible with itsidentification as a radial charm excitation.

An identification of the observed resonance withan orbital charm excitation seems to be morefavourable: two mesons with the quantum numbers


( )D. MelikhoÕ, O. PenerPhysics Letters B 446 1999 336–341´ 337

P y y Ž .J s2 ,3 quark orbital momentum Ls2 couldhave a width of the observed order of magnitude.However, theoretical estimates yield a mass for thelatter orbitally-excited states approximately G50MeV above the reported value. In addition, in thiscase, one would expect two neighbouring states with

Ža mass difference of about 30–50 MeV similar toŽ .qthe mass difference between D 2460 and2

Ž . .qD 2420 in the js3r2 positive-parity sector ,1

while only one resonance has been reported. Alto-gether we conclude that, at the moment, there is nofully convincing understanding of the quantum num-bers of the observed resonance.

Our analysis is based on combining the heavy-quark symmetry relations for the transition ampli-tudes between heavy mesons through the emission ofthe light hadrons with the quark-model estimates.Namely, we estimate the decay rates of radially andorbitally excited charm resonances into Dp , D)p ,Dpp and D)pp assuming the resonance mass of2637 MeV as measured by DELPHI.

To obtain estimates of the branching ratios of thethree-body decays with two pions in the final state

Ž .we treat them as cascade two-body decays D 2637Ž ) . Ž ) .™ D, D R™ D, D pp through the intermedi-

ate Breit-Wigner resonance with relevant quantumŽ .numbers. The status of the pp channel isls0, Is0

not well-defined, and we varied the corresponding s

resonance mass in the range 400–800 MeV and theŽ .width in the range G s ,700–900 MeV. The

Ž .low-energy pp partial-wave is dominated byls1

r. Higher partial waves of the pp system givenegligible contributions.

Table 1 lists the candidate charm states and theirallowed decay modes as given by the spin-parityconservation.

For heavy-meson decays, additional constraintsŽ . w xare given by the heavy-quark HQ symmetry 4 .

Namely, in the heavy quark limit the heavy quarkspin decouples from other degrees of freedom andremains conserved in hadron transitions. Thus, instrong decays of heavy hadrons, the total angularmomentum of the heavy and light degrees of free-dom are conserved separately, in addition to theconservation of the parity and total angular momen-tum. Hence, with respect to strong decays, heavyhadrons can be assigned an additional conservedquantum number, j, which is the total angular mo-

Table 1Decay modes of possible candidate states allowed by spin-parityconservation. Modes allowed also by the HQ symmetry, i.e.corresponding also to a separate conservation of the total angularmomentum of the light degrees of freedom, are listed in bold andHQ symmetry relations for the corresponding amplitude squared

w x Ž .from 5 are given without phase-space factors includedX

y y yD D D1 , js1r2 2 , js5r2 3 , js5r2

1 32 2w x w xDp L s1 b – L s3 a3 72 4) 2 2 2w x w x w xD p L s1 b L s1,3 a L s3 a3 7

) )D p :qD p – L s2 –0 , js1r2

qD p L s0,2 L s2 L s2,41 , js1r2

qD p L s0,2 L s2 L s2,41 , js3r2

qD p L s2 L s0,2,4 L s2,42 , js3r23 2Ž . w xD pp – L s2 d –ls0 52) 2 2Ž . w x w xD pp L s0,2 L s2 d L s2 d ,4ls0 53 2Ž . w xD pp L s1 L s1 j ,3 L s3ls1 52) 2 2Ž . w x w xD pp L s1,3 L s1 j ,3 L s1 j ,3,5ls1 5

mentum of the light degrees of freedom. The conse-quences of the HQ symmetry for hadron transitions

w xhave been worked out by Isgur and Wise 5 . Namely,the HQ symmetry allows one to relate to each otherdifferent amplitudes of strong transitions between thestates with fixed j and jX, the latter being the angularmomenta of the light degrees of freedom in theinitial and final hadronic states, respectively. Table 1also presents the HQ symmetry allowed transitionsin terms of the few independent amplitudes. TheŽ .O 1rm corrections in the effective HamiltonianQ

yield corrections to these HQ symmetry relations.However, for our order-of-magnitude analysis thesecorrections are generally unimportant and will beneglected unless explicitly specified.

For the calculation of the independent amplitudesŽ .a , b , d , and j in Table 1 one needs a non-per-turbative approach. We apply here a naive quark-

Ž3 . w xpair-creation P model 6 which, in spite of its0

simplicity, has proven to provide a reasonable quan-titative description of the two-body hadronic decays.The model is based on the assumption that the the

Ž .spectator quarks do not change their SU 3 quantumnumbers, nor their momenta and spins. The createdquark-antiquark pair should be therefore in a 3P0Ž P C qq. Ž .J s0 SU 3 singlet state of zero total 3-momentum. More details concerning the model can

w xbe found e.g. in Ref. 6 . To compute the transitionamplitudes one needs an overall transition-strength

( )D. MelikhoÕ, O. PenerPhysics Letters B 446 1999 336–341´338

constant which determines the amplitude of the pro-duction of the light qq pair from the vacuum and thewave functions of the initial and final mesons. Forthe overall strength constant we use the value gs2.2as found from the analysis of the hadronic decays in

w xthe light sector 6 . For the nonperturbative mesonwave functions we assume an harmonic-oscillatorapproximation. The remaining parameters to be fixedare the size of the light-light and heavy-light wavefunctions. They are given by two radii the light-lightŽ . Ž .R and heavy-light R . For the ground state theseD

22 2²Ž . :radii are simply R s r yr .q q3

The heavy meson has to be smaller than the lightone. From estimates with different potentials, weassume the radii to satisfy the relation R2 rR2 ,D

0.5–0.7 and allowed R2 s6–9 GeV 2: such valuesof R are compatible with previous descriptions of

w xthe spectrum and decay rates 6 and in addition, wecheck, Table 3, that the 3P model with these param-0

eters describes correctly the experimentally observedŽ ) . )

qD ™ D, D p decay rates. For the D ™Dp2

transition, Table 2, we also express the decay widthin terms of a dimensionless coupling constant de-fined as follows² 0 q < )q : m

)D p p q D p sg q e , 1Ž . Ž . Ž . Ž .2 1 D Dp m 1

mŽ .where e p is the vector-meson polarization vec-1

tor, the states being normalized covariantly and wherewe have omitted the momentum conservation deltafunction. The g ) of the 3P model agrees withD Dp 0

the experimental bound although seems to be a bitsmall compared with other theoretical estimates. No-tice however that the 3P model is non-relativistic0

and as such does not describe properly the soft pionlimit. In the D) ™Dp decay the pion is producedalmost at rest and the model is indeed expected tounderestimate the coupling constant as observed. Butthe model has proven to work rather well for hadronicresonance decays in the usual domain of the emittedpion energies, i.e. E , 300–600 MeV. Indeed, asp

we will see later, the model estimates for the decay

Table 2The D) ™ Dp transition

3w xExp. 7 P model Other estimates0Ž w x.see refs in 8

)g -21 7"1 7–21D Dp)Ž .G D ™ Dp -89 keV 7–10 keV 7–90 keV

Table 3Ž ) .qDecay rate of the D ™ D, D p transition2

3w xExp. 7 P model0

)Ž . Ž .G Dp rG D p 2.3"0.9 2.6G 23"5 MeV 11–22 MeVtot

of a heavier radially-excited DX are in better agree-ment with covariant methods.

Having thus fixed the ranges of the basic parame-Ž ) .qters from the light sector and D ™ D, D p de-2

cays, we apply the model to the analysis of the decayof radially and orbitally excited negative-parity states.

2. Radial excitation D I1

The decay of the vector radially-excited D y into1

Dp and D)p is governed by the following ampli-tudes

² 0 q < )Xq : m

X)D p p q D p sg q e ,Ž . Ž . Ž .2 1 D Dp m 1

² ) 0 q < )Xq :D p p q D pŽ . Ž . Ž .2 1

s ig )X

) e p mpn e ae b 2Ž .D D p mna b 1 2 1 2

with the coupling constants g )X and g )

X) toD Dp D D p

be determined on the basis of a dynamical approach.In the heavy-quark limit the constants are related toeach other as follows

g )X sM )

X g )X

) . 3Ž .D Dp D D D p

To estimate these coupling constants one canapply various theoretical approaches. For instance,one may use the PCAC definition of the pion fieldŽ .although the pion is not soft at all in this decay inwhich case a complicated problem of calculating thecoupling constants of interest is reduced to a rela-

)X Ž ) .tively simpler one of calculating the D ™ D, D

transition form factors through the axial-vector cur-rent. Namely, one finds

1g s M qM A 0Ž . Ž .V Pp V P 1fp

q M yM A 0 . 4Ž . Ž . Ž .V P 2

For estimating the meson transition form factors fand a we used the relativistic dispersion approachq

w xof Ref. 9 adopting wave functions of the ground-state and radially-excited D which provide the val-


Table 4X Ž ) . 3

yDecay rates of the D ™ D, D p transition in the P model1 0X yŽ .D 1 ™ Dp 150–220 MeVX y )Ž .D 1 ™ D p 200–300 MeVX y )Ž .D 1 ™ D pp 120–160 MeV

ues f ,200 MeV, f ) ,240 MeV and f )X F400D D D

MeV.Actually, one observes a suppression of the form

factor A in the D)X™D transition, as compared1

with the D) ™D one, because of the orthogonalityof the wave functions of the orbitally-excited and the

5 D ) ™ DŽ .ground states : namely, A 0 , 0.5 and1D )X ™ DŽ .A 0 ,0.2–0.3. In the absence of the second1

Ž . )X Ž ) .term of 4 the D ™ D, D transition would thus

be suppressed by a factor 2 to 4 in rate. Howeversuch a suppression due to orthogonality in the softpion limit is expected to be reduced when the mo-

Ž .mentum recoil which is large in this case is taken)

X Ž ) .into account. The transition D ™ D, D p wouldonly produce a soft pion if the mass of the D)

X

would be close to that of the DŽ) ., thus cancelingŽ .the second term in 4 . But we are not in such a

situation, the second term is not negligible sinceD )X ™ DŽ .A 0 ,0.9–2.5 and one ends up with the rela-2

Ž .X) ) )tion g s 0.5–1.5 g where g ,15.D Dp D Dp D Dp

Notice that the large uncertainties in g )X areD Dp

connected with a strong sensitivity of the latter to thesubtle details of the sign-changing wave function ofthe radially-excited state. Altogether, and using the

Ž .HQ symmetry relation 3 we find a rather largeŽ )

X .range for our estimate: G D ™Dp s20–200Ž )

X) .MeV and G D ™D p s25–250 MeV.

To obtain more precise estimates we also used the3P model and found values in the region g )

X s0 D Dp

13–15 which are compatible with PCAC based esti-mates. Table 4 presents the 3P model estimates of0

X Ž .ythe D 2637 decay rates.1

So for the sum of the decay rates of the channelsŽ )

X . Ž )X

) .G D ™Dp qG D ™D p one finds ratheruncertain estimates ranging from 40–50 MeV, whichis already a factor of 3 above the reported width of

5 We are indebted to Damir Becirevic and Alain Le yaouanc forattracting our attention on this suppression which they find even

w xstronger in a related approach 10 based on the Dirac equation.

the DELPHI resonance, to several hundreds MeV,which is far above.

Notice that anyway an important contribution tothe total width of the radially-excited D)

X is given)

X) Ž .by the decay channel D ™D pp allowedls0

Ž .by the HQ symmetry in the S-wave Table 1 . Thishas the double effect of predicting a large branchingratio for the observed D)pp channel, but unhappilyalso a large total width. The 3P model estimate of0

its rate is 120–160 MeV. So, conservatively, onecannot expect the total width of the radially excited

)X Ž .ystate D 2637 to be less than 50 MeV and presum-1

ably it should be much broader. Thus we conclude( )that identification of the resonance D 2637 as a

radial excited J P s1y charmed meson is problem-atic taking into account the resonance total widthF15 MeV.

3. Orbital excitations D I I2 ,3

To proceed with these states, we assume the HQsymmetry at the level of the transition amplitudesbut use the physical masses to compute the relevantphase-space factors. Namely, we calculate the transi-tion amplitudes of the modes D y ™DŽ) .p3 , js5r2

Ž) .Ž .yand D ™D pp and determine all3 , js5r2 ls0,1

other related amplitudes through the HQ symmetryrelations listed in Table 1.

Notice that the transition of D y into the2 , js5r2

four positive parity states D y ™D) )

qp2 , js5r2 js1r2,3r2

™DŽ) .pp , is allowed, in the heavy-quark limit,only in the D-wave, since the latter positive paritystates with js1r2 or js3r2 cannot be producedwith an S-wave pion from a js5r2. A roughestimate convinced us that, due to the small finalmomenta, these decays will be even more suppressed

Table 5Branching ratios of the D y decays in the 3P model.2 , js5r2 0

Ž .Br Dp –)Ž .Br D p 0.65–0.7

Ž Ž . .Br D pp -0.01ls0Ž Ž . .Br D pp 0.2–0.3ls1

)Ž Ž . .Br D pp -0.003ls0)Ž Ž . .Br D pp 0.02–0.03ls1

G 6–14 MeVtot

( )D. MelikhoÕ, O. PenerPhysics Letters B 446 1999 336–341´340

Table 6Branching ratios of the D y in the 3P model3 , js5r2 0

Ž .Br Dp 0.65–0.74)Ž .Br D p 0.23–0.27

Ž Ž . .Br D pp –ls0Ž Ž . .Br D pp -0.001ls1

)Ž Ž . .Br D pp -0.003ls0)Ž Ž . .Br D pp 0.03–0.06ls1

G 8–22 MeVtot

than the other DŽ) .pp channels to be discussedlater.

Taking into account the phase-space factors yieldsthe results listed in Tables 5 and 6. Some commentson the presented numbers are in order. It can be seen

Ž) .Ž .from Table 1 that the decay into D pp canls1

go through a P-wave between pp and the charmedmeson. Compared to the F-wave for DŽ) .p and the

Ž) .Ž .D-wave for D pp , this decay channel has tols0

be considered even though only a small part of ther-meson Breit-Wigner tail is included in the phase

Ž) .Ž .space. The D pp is suppressed by one orderls0Ž) .Ž .of magnitude as compared to the D pp duels1

to the centrifugal barrier suppression.The physical D y state is dominantly the3

Ž .y yD one, and one can safely estimate G D3 , js5r2 3Ž .y,G D , since a possible small admixture3 , js5r2

of the narrow D y state does not practically3 , js7r2

change its width. For the D y state however this is2

not the case: one might expect a sizable increase ofthe width of the physical D y with respect to the2

width of the D y . For example, even a small2 , js5r2

admixture of the D y to the dominant2 , js3r2

D y may increase the total width of the physi-2 , js5r2Ž .y ycal D meson, since one expects G D 42 2 , js3r2

Ž .yG D due to the P-wave decay mode2 , js5r2

D y ™D)p allowed by the HQ symmetry.2 , js3r2

A similar situation has been observed in the posi-tive-parity D multiplet: namely, the experimen-js3r2

Ž .qtal ratio of the decay rate of D 2420 which is1

dominantly D q and the decay rate of1 , js3r2Ž .qD 2460 which is practically the rate of a pure2

D q is:2 , js3r2

G D q 2420 rG D q 2460 s0.71 5Ž . Ž . Ž .Ž . Ž .1 2

that is about twice larger than the leading-order HQsymmetry estimate

G D q rG D q s0.3. 6Ž .Ž . Ž .1 , js3r2 2 , js3r2

The latter value is obtained by assuming theleading-order HQ symmetry relations between theamplitudes and taking the physical masses of thecorresponding states for the calculation of the rele-vant phase-space factors.

This discrepancy can be solved by invoking the1rm corrections e.g. by assuming a small admix-c

ture of a broad D q to a narrow D q in1 , js1r2 1 , js3r2Ž . w xqthe physical D 2420 as proposed in Ref. 11 .1Ž w x .Another possibility see 12 and refs therein is to

have a rather strong increase in the decay rate of apure D q due to the 1rm corrections to the1 , js3r2 c

Žeffective Hamiltonian A combination of these two.variants is of course also possible . Anyway the

D positive-parity sector prompts that the netjs3r2

effect of the 1rm corrections is a doubling of thecŽ . Ž .ratio G D rG D .Js jy1r2, j Jsjq1r2, j

Hence the obtained value of the D y width2 , js5r2

should be considered as a lower bound of the physi-cal D y width. Still, from the comparison with the2

js3r2 sector, we expect the physical D y width2

not to exceed the HQ estimate of the D y state2 , js5r2

by much more than a factor 2.Similarly, the subleading 1rm effects can influ-c

ence also the rate of the transition D y ™2 , js5r2

D) )

qp . A rough estimate based on the Eqs.js1r2,3r2Ž . Ž . Ž .5 and 6 , where an S-wave O 1rm decay leadsc

to an increase of about 10 MeV of the width, andconsidering that the phase space is smaller in the

Ž .decay of a D 2634 into positive parity resonancesthan in the decay of the latter into the ground state,we are not too worried. Still, a closer scrutiny of

Ž .these O 1rm effects would be welcome.c

Finally, we conclude that the orbitally excitedD y and D y charm resonances with the mass in the2 3

region of 2640 MeV can haÕe the width of the orderreported by DELPHI. However, an identification of

Ž .the resonance D 2637 with an orbital excitation inthe charm system is not straightforward since thereported mass seems to be significantly smaller thanthe theoretical expectations.

Ž . w xFor example, the Godfrey-Isgur GI model 2which describes with a good accuracy nearly allknown mesons predicts the mass of the D y to be3

2830 MeV. Taking into account that for the D q2

state the GI model gives 2500 MeV which is 40MeV heavier than the observed value of 2460 MeV,we can expect for the D y state the mass F 28003


MeV. From the typical mass-splitting between thestates with neighbouring js2 and js3 in the GImodel, one could expect the D y mass near 27502

MeV.w xThe quark-gluon string model 13 has predicted

Ž .ythe masses of the charm resonances M D s26602Ž .y"70 MeV and M D s2760"70 MeV. The3

Ž ypartial rates of the latter were estimated to be G D3. Ž ) .y™ Dp s 1.3–2.0 MeV and G D ™ D p s3

3.5–7.0 MeV yielding the total width of D y in the3

necessary range. On the other hand a large masssplitting between the 2y and 3y charm states signalsthat the D y in the quark-gluon string model con-2

tains a big admixture of the lighter state D y .2 , js3r2

So, for the mass of the pure D y state one2 , js5r2

would expect a higher value.Altogether, from the above theoretical analyses

we could expect the mass of the D y in the2 , js5r2

region of 2650–2750 MeV which is only marginallycompatible with the reported resonance mass of 2637MeV.

The branching ratios of the D)pp channels inTables 5 and 6 are in the range of a few percent,which may seem a little small for these channels tohave been observed. Still some non-resonant pp

contributions, which are difficult to estimate, willadd up.

Ž .Summing up, if the resonance D 2637 with thewidth of F15 MeV is confirmed by further analysesŽ w x.at the moment CLEO and OPAL do not see it 14then the theoretical understanding of its quantumnumbers is not clear: in spite of the coincidence ofthe observed mass with the predicted mass of theradial charm excitation DX

y, its interpretation as a1

radial excitation is hardly compatible with a smallobserved width.

On the other hand, although an identification ofthe state with the D y orbital excitation seems to be2

appropriate from the viewpoint of the total width, itsmass seems to be too low compared with the quark-model theoretical estimates. Its D)pp branchingratio of a few percent is a little small. More impor-tantly in the case of the orbital excitations in thismass region, one would expect two neighbouringresonances with the width of order several MeV eachand with the mass difference of about 30–50 MeV,corresponding to the D y and D y states. The pub-2 3

lished plots do not show any sign of a neighbouring3y resonance.

Thus we conclude that the experimental confirma-tion of this resonance would put forward a challengeof its proper theoretical understanding, unless aneighbouring slightly heavier resonance was found.

Acknowledgements

Wa are grateful to Claire Bourdarios and PatrickRoudeau for attracting our attention to this problemand for inspiring discussions, to Damir Becirevic,Dominique Pallin and Alain Le Yaouanc for fruitfuldiscussions, and to Damir Becirevic and Alain LeYaouanc for presenting their results prior to publica-tion. D.M. acknowledges financial support from theNATO Research Fellowships Program.

References

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w x Ž .2 S. Godfrey, N. Isgur, Phys. Rev. D 32 1985 189.w x Ž .3 D. Ebert, R. Faustov, V. Galkin, Phys. Rev. D 57 1998

5663.w x Ž .4 N. Isgur, M.B. Wise, Phys. Lett. B 232 1989 113; B 237

Ž .1990 527.w x Ž .5 N. Isgur, M. Wise, Phys. Rev. Lett. 66 1991 1130.w x6 A. Le Yaouanc, L. Oliver, O. Pene, J.-C. Raynal, Phys. Rev.´

Ž . Ž .D 8 1973 2223; D 11 1975 1272; Hadron transitions inthe quark model, Gordon and Breach Science PublishersS.A., New-York, 1988.

w x Ž .7 Particle Data Group, Phys. Rev. D 54 1996 1.w x8 V.M. Belyaev, V.M. Braun, A. Khodjamirian, R. Ruckl,¨

Ž .Phys. Rev. D 51 1995 6177.w x Ž . Ž .9 D. Melikhov, Phys. Rev. D 53 1996 2460; 56 1997 7089.

w x10 D. Becirevic, A. Le Yaouanc, paper in preparation.w x Ž .11 M.-L. Lu, M.B. Wise, N. Isgur, Phys. Rev. D 45 1992

1553.w x12 A. Falk, in: M. Jamin, O. Nachtmann, G. Domokos, S.

Ž .Kovesi-Domokos Eds. , Non-Perturbative Particle Theoryand Experimental Tests, Proceedings of 20th Johns HopkinsWorkshop on Current Problems in Particle Theory, Heidel-

Žberg, Germany, 1996, World Scientific, 1997, p. 79 e-Print.Archive: hep-phr9609380 .

w x Ž .13 A. Kaidalov, A. Nogteva, Sov. J. Nucl. Phys. 47 1988 321.w x14 P. Krieger, Talk given at the ICHEP XXIX, July 1998,

Ž .Vancouver, Canada; I. Shipsey for the CLEO Collaboration ,Talk given at the ICHEP XXIX, July 1998, Vancouver,Canada.

28 January 1999


A study of the centrally produced baryon-antibaryon systems inpp interactions at 450 GeVrc

WA102 Collaboration

D. Barberis d, W. Beusch d, F.G. Binon f, A.M. Blick e, F.E. Close c,d,K.M. Danielsen j, A.V. Dolgopolov e, S.V. Donskov e, B.C. Earl c, D. Evans c,

B.R. French d, T. Hino k, S. Inaba h, A.V. Inyakin e, T. Ishida h, A. Jacholkowski d,T. Jacobsen j, G.T. Jones c, G.V. Khaustov e, T. Kinashi l, J.B. Kinson c, A. Kirk c,

W. Klempt d, V. Kolosov e, A.A. Kondashov e, A.A. Lednev e, V. Lenti d,S. Maljukov g, P. Martinengo d, I. Minashvili g, T. Nakagawa k, K.L. Norman c,

J.P. Peigneux a, S.A. Polovnikov e, V.A. Polyakov e, V. Romanovsky g,H. Rotscheidt d, V. Rumyantsev g, N. Russakovich g, V.D. Samoylenko e,

A. Semenov g, M. Sene d, R. Sene d, P.M. Shagin e, H. Shimizu l,´ ´A.V. Singovsky a,e, A. Sobol e, A. Solovjev g, M. Stassinaki b, J.P. Stroot f,

V.P. Sugonyaev e, K. Takamatsu i, G. Tchlatchidze g, T. Tsuru h, M. Venables c,O. Villalobos Baillie c, M.F. Votruba c, Y. Yasu h

a LAPP-IN2P3, Annecy, Franceb Athens UniÕersity, Physics Department, Athens, Greece

c School of Physics and Astronomy, UniÕersity of Birmingham, Birmingham, UKd CERN – European Organization for Nuclear Research, GeneÕa, Switzerland

e IHEP, ProtÕino, Russiaf IISN, Belgium

g JINR, Dubna, Russiah ( )High Energy Accelerator Research Organization KEK , Tsukuba, Ibaraki 305, Japan

i Faculty of Engineering, Miyazaki UniÕersity, Miyazaki, Japanj Oslo UniÕersity, Oslo, Norway

k Faculty of Science, Tohoku UniÕersity, Aoba-ku, Sendai 980, Japanl Faculty of Science, Yamagata UniÕersity, Yamagata 990, Japan

Received 4 December 1998Editor: L. Montanet

Abstract

0 q yA study of the centrally produced pp, ppp , ppp p and LL channels has been performed in pp collisions using anincident beam momentum of 450 GeVrc. No significant new structures are observed in the mass spectra, however,


( )D. Barberis et al.rPhysics Letters B 446 1999 342–348 343

important new information on the production dynamics is obtained. A systematic study of the production properties of thesesystems has been performed and it is found that these systems are not produced dominantly by double Pomeron exchange.q 1999 Published by Elsevier Science B.V. All rights reserved.

Experiment WA102 is designed to study exclu-sive final states formed in the reaction

pp™p X 0 p 1Ž . Ž .f s

at 450 GeVrc. The subscripts f and s indicate thefastest and slowest particles in the laboratory respec-tively and X 0 represents the central system that ispresumed to be produced by double exchange pro-cesses. The experiment has been performed using theCERN Omega Spectrometer, the layout of which is

w xdescribed in Ref. 1 . In previous analyses of otherchannels it has been observed that when the centrallyproduced system has been analysed as a function ofthe parameter dP , which is the difference in theT

transverse momentum vectors of the two exchangew xparticles 1,2 , all the undisputed qq states are sup-

pressed at small dP , in contrast to glueball candi-T

dates.0This paper presents a study of the pp, ppp ,

q yppp p and LL final states at a centre of massq y'energy of s s29.1 GeV. The pp, ppp p and

'LL channels have previously been studied at s sw x12.7 GeV 3 . In recent years there have been claims

of the observation of two different resonant signalsin the pp channel. The first claim is for the observa-

Ž .tion of the j 2220 , with a width of 20 MeV, inradiative Jrc decays made by the BES collabora-

w x Ž .tion 4 . The j 2220 is claimed to be a goodcandidate for the tensor glueball. To date, everyestablished J P C s0qq and 2qq glueball candidatehas been observed in central pp collisions. There-fore, it is important to look for these new states incentral production in order to learn more about the

Ž .nature andror existence of the j 2220 .The second claimed observation is for a state with

a mass of 2.02 GeV and a width of less than 10 MeVobserved in central baryon exchange by the WA56

w xexperiment 5 . It is claimed that this state could beinterpreted as a baryonium candidate. Although thecurrent experiment does not study baryon exchangeit does study central production and hence a searchfor this state may be of interest.

In addition to these searches, an analysis of theproduction kinematics of baryon-antibaryon produc-tion is presented which can give information on themechanism of the formation of these final states incentral production.

The reaction

pp™p pp p 2Ž . Ž .f s

has been isolated from the sample of events havingfour outgoing charged tracks, by first imposing thefollowing cuts on the components of the missing

< < <momentum: missing P -14.0 GeVrc, missingx< < <P -0.16 GeVrc and missing P -0.08 GeVrc,y z

where the x axis is along the beam direction. Acorrelation between pulse-height and momentum ob-tained from a system of scintillation counters wasused to ensure that the slow particle was a proton.

In order to select the pp system, information fromˇthe Cerenkov counter was used. One centrally pro-

duced charged particle was required to be identifiedˇas a p or an ambiguous Krp by the Cerenkov

counter and the other particle was required to beconsistent with being a proton. The method of Ehrlich

w xet al. 6 , has been used to compute the mass squaredof the two centrally produced particles assumingthem to have equal mass. The resulting distributionis shown in Fig. 1a where a clear peak can be seen atthe proton mass squared. This distribution has beenfitted with Gaussians to represent the contributions

q y q yfrom the p p , K K and pp channels. From thisfit the number of pp events is found to be 6256 "

220.A cut on the Ehrlich mass squared of 0.65FM 2

X2F1.15 GeV has been used to select a sample of pp

events. The resulting pp effective mass distributionis shown in Fig. 1b. There are no significant struc-tures in the mass spectrum, in particular there is no

Ž .evidence for the j 2220 that has been claimed inw xthe pp channel by the BES collaboration 4 nor is

there evidence for the narrow structure at 2.02 GeVclaimed to have been observed in central baryon

w xexchange reactions 5 . The mass resolution of theWA102 experiment in each region is better than 10

( )D. Barberis et al.rPhysics Letters B 446 1999 342–348344

Ž . Ž . Ž .Fig. 1. a The Ehrlich mass squared distribution and b the pp mass spectrum for the pp channel. c The Ehrlich mass squared0 0Ž .distribution and d the ppp mass spectrum for the ppp channel.

MeV. We can calculate an upper limit for the cross-sections for the production of these claimed reso-

Ž Ž ..nances in central pp interactions to be s j 2220Ž .-1.6 nb and s 2.02 GeV -1.4 nb at 95 % confi-

dence level.The reaction

0pp™p ppp p 3Ž .Ž .f s

where the p 0 has been observed decaying to gg ,has been isolated from the sample of events havingfour outgoing charged tracks plus two g s each withenergy greater than 0.5 GeV reconstructed in theelectromagnetic calorimeter 1, by first imposing the

1 The showers associated with the impact of the charged trackson the calorimeter have been removed from the event before therequirement of only two g s was made.

following cuts on the components of the missing< < <momentum: missing P -17.0 GeVrc, missingx

< < <P -0.16 GeVrc and missing P -0.12 GeVrc.y z

One centrally produced charged particle was re-quired to be identified as a p or an ambiguous Krp

ˇby the Cerenkov counter and the other particle wasrequired to be consistent with being a proton. TheEhrlich mass distribution is shown in Fig. 1c where aclear peak can be seen at the proton mass squared.This distribution has been fitted with Gaussians torepresent the contributions from the pqpyp 0,

q y 0 0K K p and ppp channels. From this fit the0number of ppp events is found to be 877 " 85. A

cut on the Ehrlich mass squared of 0.65FM 2 F1.15X2 0GeV has been used to select a sample of ppp

0events. The resulting ppp effective mass distribu-tion is shown in Fig. 1d where no significant struc-


0ture can be observed. Similarly, the pp, pp and0pp mass spectra do not show any significant struc-Ž .tures not shown .

The reaction

q ypp™p ppp p p 4Ž . Ž .f s

has been isolated from the sample of events havingsix outgoing charged tracks, by first imposing thefollowing cuts on the components of the missing

< < <momentum: missing P -14.0 GeVrc, missingx< < <P -0.12 GeVrc and missing P -0.08 GeVrc.y z

q yIn order to select the ppp p system an eventwas accepted if a positive or negative particle was

ˇidentified as a p or a Krp by the Cerenkov systemand the other particle with the same charge wasconsistent with being a p . A modified method of

w xEhrlich et al. 6 , has been used to compute the masssquared of the two highest momentum central parti-cles assuming the other two particles to be pions.The resulting distribution is shown in Fig. 2a wherea clear peak can be seen at the proton mass squared.This distribution has been fitted with Gaussians torepresent the contributions from the pqpypqpy ,

q y q y q yK K p p and ppp p channels. From this fitq ythe number of ppp p events is found to be 2076

" 160. A cut on the Ehrlich mass squared of0.65FM 2 F1.15 GeV 2 has been used to select aX

q y q ysample of ppp p events. The resulting ppp p

effective mass spectrum is shown in Fig. 2b andshows a broad distribution with a maximum nearthreshold.

A study has been performed of the various twoand three body subsystems but no structure has been

q y q yŽ . Ž . Ž .Fig. 2. For the ppp p channel a the Ehrlich mass squared distribution, b the ppp p mass spectrum and c the scatter plot ofy qŽ . Ž .M pp versus M pp .


q yŽ . Ž . Ž . Ž . Ž . Ž .Fig. 3. For the LL channel a the scatter plot of M p p versus M pp , b the M pp mass distribution and c the LL massspectrum.

qq yy qobserved except D and D in the pp andypp mass spectra respectively. Fig. 2c shows a

y qŽ . Ž .scatter plot of M pp versus M pp where aclear accumulation of events can be observed in the

qq yyD D region. However, it is not possible toqq yyextract a reliable measure of the D D contribu-

tion due to the difficulties in establishing the level ofbackground.

The reaction

pp™p LL p 5Ž . Ž .f s

has been isolated from the sample of events having

Table 1Ž .Production of the channels as a function of dP expressed as a percentage of its total contribution and the ratio R of events produced atT

dP F 0.2 GeV to the events produced at dP G 0.5 GeVT T

dP F0.2 GeVTdP F0.2 GeV 0.2FdP F0.5 GeV dP G0.5 GeV RsT T TdP G0.5 GeVT

pp 14.4"0.4 44.6"0.7 41.0"0.7 0.35"0.010ppp 14.5"1.0 43.3"1.6 42.2"1.6 0.34"0.03

q yppp p 13.9"0.5 43.3"0.9 42.8"0.9 0.32"0.01

LL 15.0"3.5 39.4"5.6 45.6"6.1 0.33"0.09


Table 2The cross-sections and slope parameter of the four momentum

0 q ytransfer squared for the pp, ppp , ppp p and LL channels

Channel Cross-section_nb Slope by2GeV' 's s12.7 GeV s s29.1 GeV

pp 400"60 186"19 5.5"0.30ppp – 43" 5 5.9"0.5

q yppp p 226"42 82" 7 5.4"0.3

LL 29" 8 12" 2 4.5"2.0

two outgoing charged tracks plus two V 0s, by firstimposing the following cuts on the components of

< <the missing momentum: missing P -14.0 GeVrc,x< < < <missing P -0.16 GeVrc and missing P -0.02y z

0 Pqy PyL LGeVrc. For each V the value of as , wasq yP q PL L

q Ž y.calculated, where P P is the longitudinal mo-L LŽ .mentum of the positive negative particle from the

decay of the V 0 with respect to the V 0 momentumŽ . Ž .vector. For a L L a is positive negative and

hence events which were compatible with being LL

were selected by requiring that the product a a1 2

was negative.2Ž .The quantity D, defined as DsMM p p yf s

2 2Ž . Ž .M LL , where MM p p is the missing massf s

squared of the two outgoing protons, was then calcu-< < Ž .2lated for each event and a cut of D F 2.0 GeV

was used to select the LL channel. In order to studyany possible residual K 0 contamination a scatterS

Ž . Ž .plot of M pp versus M pp is shown in Fig. 3afor the case when the other V 0 is compatible with

Ž Ž . .being a L 1.09FM pp F1.14 GeV . The result-

0 q yŽ . Ž . Ž . Ž . Ž .Fig. 4. a , c , e and g the azimuthal angle f between the two outgoing protons for the pp, ppp , ppp p and LL channel0 q yŽ . Ž . Ž . Ž . Ž < <.respectively. b , d , f and h the four momentum transfer squared t from one of the proton vertices for the pp, ppp , ppp p and

LL channel respectively.


y qŽ .ing pp pp mass distribution is shown in Fig.Ž .3b where a clear L L signal can be seen over little

background, with negligible contribution from the0K . The resulting LL effective mass spectrum isS

shown in Fig. 3c and consists of 123 events.0 q yA study of the pp, ppp , ppp p and LL

systems has been performed as a function of theparameter dP , which is the difference in the trans-T

verse momentum vectors of the two exchanged parti-w xcles 1,2 . After acceptance corrections the results are

shown in Table 1 together with the value of the ratioŽ .R of events at small dP to large dP . In previousT T

w xstudies 7 of the ratio R we have observed that allsystems fall into three distinct classes. Firstly, thereare all the undisputed qq states which can be pro-

Ž .duced in Double Pomeron Exchange DPE , namelythose with positive G parity and Is0, which have a

Ž .small value for this ratio -0.1 . Secondly, there arethose states with Is1 or G parity negative, whichcannot be produced by DPE, which have a slightly

Ž .higher value f0.25 . Finally, there are the stateswhich could have a gluonic component, which have

Ž .a large value for this ratio )0.6 . It is interesting tonote that the baryon-antibaryon systems have a valueof R consistent with the second class, i.e. that theyare not produced by DPE. This fact can be investi-gated by studying the cross-section dependence as afunction of centre of mass energy.

After correcting for geometrical acceptances, de-tector efficiencies and losses due to selection cuts,

'the cross-sections for the channels at s s29.1 GeV< <in the x interval x F0.2 have been calculatedF F

and are shown in Table 2. These can be compared,'where possible, to the cross-sections found at s s

12.7 GeV which are also shown in Table 2. As canbe seen the cross-sections are decreasing with in-creasing centre of mass energy. This is not consistentwith them being produced dominantly by DPE andsuggest that these systems are produced by double

w xRegge or Regge-Pomeron exchanges 8 .Ž .The acceptance corrected azimuthal angle f

Žbetween the p vectors of the two protons p andT f.p is shown in Fig. 4a, c, e and g. The distributionss

in all cases are consistent with being flat. Althoughnaively a flat distribution would be expected, this isthe first time that a system or resonance has been

w xobserved to have a flat f distribution 9 .

Fig. 4b, d, f and h shows the four momentumtransfer squared at one of the proton vertices. Thedistributions have been fitted with a single exponen-

Ž < <.tial of the form exp yb t and the results are pre-sented in Table 2. The first bin in the distributionshas been excluded from the fit due to the fact thatthe uncertainties in the acceptance correction aregreatest in this bin.

In conclusion, a study of the centrally produced0 q ypp, ppp , ppp p and LL channels has been

performed. There is no evidence for resonance pro-qq yyduction with the exception of D and D in the

q yppp p channel. In the pp channel there is noŽ .evidence for the j 2220 and an upper limit on the

cross-section for its production in central pp colli-sions has been calculated to be 1.6 nb. A study of thecentre of mass energy dependence for the productionof central baryon-antibaryon systems shows that theyare not produced dominantly by double Pomeronexchange.

Acknowledgements

This work is supported, in part, by grants fromthe British Particle Physics and Astronomy ResearchCouncil, the British Royal Society, the Ministry ofEducation, Science, Sports and Culture of JapanŽ .grants no. 04044159 and 07044098 , the Pro-gramme International de Cooperation ScientifiqueŽ .grant no. 576 and the Russian Foundation for Basic

Ž .Research grants 96-15-96633 and 98-02-22032 .

References

w x Ž .1 D. Barberis et al., Phys. Lett. B 397 1997 339.w x Ž .2 F.E. Close, A. Kirk, Phys. Lett. B 397 1997 333.w x Ž .3 T.A. Armstrong et al., Zeit. Phys. C 35 1987 167.w x Ž .4 J.Z. Bai et al., Phys. Rev. Lett. 76 1996 3502; K.T. Chao,

Ž .Comm. Theor. Phys. 24 1995 373; F.E. Close, G. Farrar,Ž .Z.P. Li, Phys. Rev. D 55 1997 5749.

w x5 A. Ferrer et al., CERNrEPr98-66.w x Ž .6 R. Ehrlich et al., Phys. Rev. Lett. 20 1968 686.w x7 A. Kirk, hep-exr9803024.w x Ž .8 S.N. Ganguli, D.P. Roy, Phys. Rep. 67 1980 203.w x Ž . Ž .9 D. Barberis et al., Phys. Lett. B 427 1998 398; B 422 1998

Ž . Ž . Ž .399; B 432 1998 436; B 436 1998 204; B 440 1998 225.

28 January 1999


0 0Evidence for a ph-P-wave in pp-annihilations at rest into p p h

Crystal Barrel Collaboration

A. Abele h, J. Adomeit g, C. Amsler n, C.A. Baker e, B.M. Barnett c,1, C.J. Batty e,M. Benayoun k, A. Berdoz l, K. Beuchert b, S. Bischoff h, P. Blum h, K. Braune j,¨T. Case a, O. Cramer j, V. Crede c, K.M. Crowe a, T. Degener b, N. Djaoshvili h,´

S.v. Dombrowski n,2, M. Doser f, W. Dunnweber j, A. Ehmanns c, D. Engelhardt h,¨M.A. Faessler j, P. Giarritta n, R.P. Haddock i, F.H. Heinsius a,3, M. Heinzelmann n,

A. Herbstrith h, M. Herz c,4, N.P. Hessey j, P. Hidas d, C. Holtzhaußen h,K. Huttmann j, D. Jamnik j,5, H. Kalinowsky c, B. Kammle g, P. Kammel a,¨ ¨

J. Kisiel f,6, E. Klempt c, H. Koch b, C. Kolo j, M. Kunze b, U. Kurilla b,M. Lakata a, R. Landua f, H. Matthay b, R. McCrady l, J. Meier g, C.A. Meyer l,¨

R. Ouared f, F. Ould-Saada n, K. Peters b, B. Pick c, C. Pietra n, C.N. Pinder e,M. Ratajczak b, C. Regenfus j,7, S. Resag c, W. Roethel j, P. Schmidt g,R. Seibert g, S. Spanier n, H. Stock b, C. Straßburger c, U. Strohbusch g,¨

M. Suffert m, J.S. Suh c, U. Thoma c, M. Tischhauser h, I. Uman j, C. Volcker j,¨ ¨S. Wallis-Plachner j, D. Walther j,8, U. Wiedner j, K. Wittmack c

a UniÕersity of California, LBNL, Berkeley, CA 94720, USAb UniÕersitat Bochum, D-44780 Bochum, FRG¨

c UniÕersitat Bonn, D-53115 Bonn, FRG¨d Academy of Science, H-1525 Budapest, Hungary

e Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UKf CERN, CH-1211 GeneÕa 4, Switzerland

g UniÕersitat Hamburg, D-22761 Hamburg, FRG¨h UniÕersitat Karlsruhe, D-76021 Karlsruhe, FRG¨

i UniÕersity of California, Los Angeles, CA 90024, USAj UniÕersitat Munchen, D-80333 Munchen, FRG¨ ¨ ¨k LPNHE Paris VI, VII, F-75252 Paris, France

l Carnegie Mellon UniÕersity, Pittsburgh, PA 15213, USAm Centre de Recherches Nucleaires, F-67037 Strasbourg, France´

n UniÕersitat Zurich, CH-8057 Zurich, Switzerland¨ ¨ ¨

Received 4 September 1998Editor: K. Winter


( )A. Abele et al.rPhysics Letters B 446 1999 349–355350

Abstract

A partial wave analysis is presented of two high-statistics data samples of protonium annihilation into p 0p 0h in liquidand 12 atm gaseous hydrogen. The contributions from the 1S ,3P and 3P initial atomic fine structure states to the two data0 1 2

sets are different. The change of their fractional contributions when going from liquid to gaseous H as calculated in a2

cascade model is imposed in fitting the data. Thus the uncertainty in the fraction of S-state and P-state capture is minimized.Both data sets allow a description with a common set of resonances and resonance parameters. The inclusion of a ph

Ž .P-wave in the fit gives supportive evidence for the r 1405 , with parameters compatible with previous findings. q 1999ˆElsevier Science B.V. All rights reserved.

Meson resonances with exotic quantum numbersidentifying their non-qq nature are of special interestin meson spectroscopy. Particular attention has beengiven to the ph system which appears to be resonantin the partial wave with orbital angular momentumlls1 carrying exotic quantum numbers. First evi-dence for an exotic ph resonance was claimed by

w xthe GAMS collaboration 1 in the charge exchangereaction pyp ™ hp 0 n; the findings were, how-

w xever, ambiguous in later analyses 2 . Contributionsfrom an exotic ph P-wave were also reported from

w x w xVES 3 . At KEK 4 , observation of a ph resonancewas claimed but the mass and width coincided with

Ž .the a 1320 parameters, and a feedthrough from the2

dominant D-wave into the P-wave cannot be ex-cluded. Evidence for a resonant ph P-wave wasreported at BNL with parameters which differ signif-

Ž . w xicantly from those of the a 1320 5 . In all these2

studies the ph P-wave is seen in a forward-back-ward asymmetry of the hp-system produced in pyp™hpyp or pyp ™hp 0 n which evidences interfer-ence between even and odd hp partial waves. Con-tributions from odd partial waves were already re-

w xported in 6 but with no resonant phase motion inthe ph P-wave.

The Crystal Barrel Collaboration found evidenceGŽ PC . yŽ yq. w xfor an I J s 1 1 exotic state 7 with

1 Now at University of Mainz, Mainz, Germany.2 Now at Cornell University, Ithaca, USA.3 Now at University of Freiburg, Freiburg,Germany.4 This work comprises part of the PhD thesis of M. Herz.5 University of Ljubljana, Ljubljana, Slovenia.6 University of Silesia, Katowice, Poland.7 Now at University of Zurich, Zurich, Switzerland.¨ ¨8 Now at University of Bonn, Bonn, Germany.

mass and width of 1400"20 "20 MeV andstat systŽq50.310"50 MeV, respectively. The reso-stat y30 syst

y 0nance was produced in the reaction pn™p p h

obtained by stopping antiprotons in liquid deuterium.On the other hand, data on pp annihilation at rest in

Ž . 0 0liquid hydrogen LH into p p h had been used by2w x w xus 8 and by Bugg and coworkers 9 to search for

this state. These analyses found strong evidence for aŽ .new scalar isovector resonance, the a 1450 . A0

weak ph P-wave contribution was found but therewas no evidence for a resonant phase motion. Nei-ther analyses included contributions of annihilationsfrom atomic P-states. In this letter we report on ananalysis of data on pp annihilation in liquid and ingaseous H at 12 atm. In H gas, the probability of2 2

w xannihilation from atomic P-states is much larger 10and can certainly no longer be neglected. The addi-tion of data for annihilation in H gas and the2

inclusion of P-state capture in the analysis will nowŽgive supportive evidence in favor and no longer

.against a resonant P-wave in the ph system.The data we discuss here were recorded with the

Crystal Barrel detector at LEAR. It has been de-w xscribed in detail elsewhere 11 , therefore only a

short summary is given. A beam of slow antiprotonswas extracted from LEAR; the p’s stopped in atarget at the center of the detector. Liquid and gaseous

Ž Ž12 atm . .hydrogen LH and G H have been used as2 2

targets. The target was surrounded by a pair ofŽ .cylindrical multiwire proportional chambers PWC’s

Ž .and a cylindrical jet drift chamber JDC with 23layers. The JDC was surrounded by a barrel consist-

Ž .ing of 1380 CsI Tl crystals in pointing geometry.The CsI calorimeter covers the polar angles between128 and 1688 with full coverage in azimuth. Theoverall acceptance for shower detection is 0.95=4p

( )A. Abele et al.rPhysics Letters B 446 1999 349–355 351

sr. Typical photon energy resolutions are s rEsE

2.5% at 1 GeV, and s s1.28 in both the polarF ,Q

and azimuthal angles.From previous run periods a high-statistics data

sample on pp annihilation in a LH target is avail-2

able. For the present analysis we recorded data on ppannihilation in gaseous hydrogen at 12 atm, againwith an all neutral trigger requiring no hits either inthe PWC’s or in the JDC. After rejection of eventswith residual charged particles we were left with13 239 623 all neutral events. In addition we recordedminimum-bias data, only requiring an antiprotonstopping in the target. These data are used for nor-malisation of the p 0p 0h branching ratio.

As a first step we select events with exactly sixelectromagnetic showers in the calorimeter with anenergy deposit in the central crystal exceeding 10MeV. This cut reduces spurious photons due toshower fluctuations. No accepted photon should haveits maximum energy deposit in a crystal adjoiningthe beam pipe since part of its energy may haveescaped detection. Subsequently the data are sub-jected to a series of kinematic fits. In a first step weimpose energy and momentum conservation by ap-

Ž .plying a four-constraint 4C fit; events with a proba-bility for the pp™6g hypothesis exceeding 1% arekept. This sample is then submitted to a 6C kine-

0 0matic fit to the hypothesis pp™p p 2g and finally0 0to a 7C kinematic fit to the hypothesis pp™p p h.

Events having a probability to combine the photonsto p 0p 0h of less than 10% are rejected so as tominimize background contaminations from p 0p 0p 0,

0 0 0 Žp hh or p p v where one soft photon is missing0 .in the decay v™p g . In addition we applied an

anticut at 1% in the confidence level on the mostfrequent 3p 0 final state. These cuts lead to 269 087

0 0events of the type pp™p p h. The backgroundcontribution determined from Monte Carlo simula-tions is less than 1%. Quality and statistics areidentical to the data set we have for this reactionfrom annihilation in liquid H .2

The p 0p 0h decay branching ratio is calculatedusing

N 0 0 N 1p p h M B0 0BR pp™p p h s = = 1Ž .Ž .N N eA Nyt r i g A N

where N represents the number of triggered allAN - trig

neutral data and N 0 0 is the number of recon-p p h

structed events corrected by the decay probabilitiesŽ 0 . Ž .for P p ™gg s 0.988 and P h™gg s

0.3925. The fraction of all neutral events in ppannihilations is determined by selecting p 0p 0-eventsin all neutral triggered data and by comparing to aselection of the same channel in minimum-bias-tri-

Ž .ggered data. We find N rN s 3.32"0.28 %.AN MB

The detection efficiency of the electromagneticcalorimeter and its performance were determined byMonte Carlo simulations. The simulated eventspassed the same selection chain as real data. TheMonte Carlo Dalitz plot was used to define anacceptance correction function. It proved to be uni-form to better than "4.5% except for edge bins. A

Ž .mean detection efficiency of e s 28.7"1.7 %gas

was derived. Thus the branching ratio for the reac-0 0tion pp™p p h for annihilations in gaseous hydro-

gen was determined to be

0 0 y3BR pp™p p h s 6.14"0.67 =10 . 2Ž . Ž .Ž .This value is similar to the values found in LH ,2Ž . y3 w x Ž . y3 w x6.7"1.2 =10 8 and 6.5"0.72 =10 12 ,respectively.

The p 0p 0h Dalitz plot for antiprotons annihilat-ing in gaseous H is shown in Fig. 1, and the p 0p 0-2

and p 0h-mass projections including fit results in Fig.2 and Fig. 3, respectively. The data with antiprotons

0 0Fig. 1. Dalitz plot of pp annihilations at rest into p p h forŽantiprotons stopping in gaseous hydrogen at 12 atm 2 entries per

.event .


0 0Ž .Fig. 2. The hp mass distribution for pp™p p h for data fromgaseous hydrogen. The shaded area represents the fit.

w x Žstopping in liquid H were presented in 8 280 0002.events .

We first comment on general features of the data.The most prominent signals are a sharp-edged band

Ž .structure due to the a 980 and a w-shaped structure0Ž . 0 0at the a 1320 mass. In the p p mass projection2

Ž .see Fig. 3 the intensity rises slowly and peaks atŽ .the f 980 ; then, at the KK threshold, the intensity0

falls off rapidly. The high-mass peak is not due toŽ . Ž .the f 1270 but is a reflection from the a 980 and2 0

Ž . 0a 1320 decay angular distributions. The hp -mass2Ž .distribution along the a 980 band exhibits a slight0

depletion at a squared mass of 2.2 GeV 2. This deple-tion requires the introduction of the second isovector

Ž . w x Ž .state, the a 1450 8 . The a 1450 in the vertical0 0Ž .band interferes destructively with the a 980 in a0

horizontal band and vice versa.We now turn to a discussion of the partial wave

analysis. The atomic cascade of antiprotons capturedby protons to form protonium atoms plays an impor-tant role in this analysis. Therefore a short outline ofthe cascade processes is given.

Antiprotons stopping in H are captured by pro-2

tons and form pp atoms in high-n Rydberg states. Incollisions with neighbouring H molecules, Stark2

mixing between levels of different orbital angularmomenta occurs leading to annihilation from high-nS- and P-states. The fractional contributions of S-stateand P-state capture depends on the target density andon the specific channel under consideration. Glob-

Ž .ally, 13"2 % of all annihilations proceed viaatomic P-states when antiprotons are stopped in liq-

w xuid H 10 . The neglect of the small fraction P-state2

capture is well justified when strong signals areobserved in the final state. The inclusion of P-statecapture would then increase the number of freeparameters in the partial wave analysis and oftenresults in unphysical solutions for the P-state anni-hilation dynamics. In this paper we present data on

0 0pp annihilation at rest into p p h in liquid and inpressurised hydrogen gas at 12 atm. In H gas we2

expect a significant contribution from P-state cap-ture. The two sets of data thus give better informa-tion to include annihilation from S- and P-states inthe analysis. Of course, the inclusion of P-statecapture not only gives access to P-state annihilationdynamics but also alters the fit results for the data settaken in LH which would be obtained when P-state2

capture is neglected.While the exact fraction of P-state capture for a

specific final state is unknown, at least the ratioswith which capture rates from S-states and P-stateschange can be estimated reliably when the targetdensity is varied. These ratios can be imposed in thepartial wave analyses thus minimizing the uncertain-ties due to the atomic cascade which precedes anni-hilation. From now on we will refer to these ratios ascascade ratios. A cascade ratio of 0.564 for the 1S0

level of the pp system indicates that the contributionof the 1S state to the annihilation process decreases0

by the factor 0.564 when the target is changed fromliquid H to 12 atm H gas. The cascade ratios are2 2

assumed not to depend on the specific final state.

0 0Fig. 3. The pp-mass distribution for pp™p p h for data fromgaseous hydrogen. The shaded area represents the fit.


w xBatty 10 determined these cascade ratios for thecontributions of individual protonium states to theannihilation process in a model of the protoniumcascade. The model allows one to calculate the yieldof X-rays emitted during the atomic cascade and thefraction with which atomic levels contribute to anni-hilation. The hadronic widths of the atomic states arepartially known from experiment, partially frommodel calculations. But since the agreement betweendata and model calculation is good, the calculationcan be trusted also for those hadronic widths whichare not measured directly. The free parameters of thecascade model are determined by a fit to experimen-tal branching ratios for two-body annihilations deter-mined over a wide range of target densities.

The same kind of ratios were determined also byw xGastaldi and Placentino 13 for pp annihilation in

liquid H and gaseous H at various densities. They2 2

do not use a cascade model but only known ratios ofbranching ratios. From their curves we estimated theratios we should expect for 12 atm H gas, and also2

estimated the errors. Of course, this procedure iscrude but may help to convince us that the cascaderatios are reliably estimated. We shall use only thecascade ratios as calculated in the cascade model ofBatty.

In Table 1 we give the cascade ratios for the 3initial states from which annihilation into p 0p 0h isallowed. These have the quantum numbers 2Sq1L J

s1S , 3P and 3P .0 1 2

The two p 0p 0h Dalitz plots were analysed usingan isobar model. The scalar p 0p 0 and p 0h interac-tions are described in the K-matrix formalism using

w x 0 0the P-vector approach of Aitchison 14 . The p p

scattering amplitude is constrained to be consistentwith results on phase shift analyses of pp interac-

w xtions 15,16 . Tensor waves and the ph P-wave areparametrized by relativistic Breit-Wigner amplitudes.

The same model has been used to fit the Dalitz plotŽ . w xalso used here for p’s stopping in liquid H 8 .2

w xFurther details of the method can be found in 17respectively.

w xIn 8 P-state capture was neglected. In spite ofthis approximation we found a good description ofthe data with the following amplitudes:

Ž . Ž .Ø a scalar pp wave with f 980 and f 13700 0Ž .Ø a weak tensor pp wave with f 12752

Ž . Ž .Ø a scalar ph wave with a 980 and a 14500 0Ž .Ø a tensor ph wave with a 1320 and some2

Ž .a 16502ŽØ a weak non-resonant isovector ph wave or with

.a width of ;400 MeV or moreFirst we demonstrate the need for P-state capture.

If we fit both data sets with S-state capture alone thex 2 is 14 200 for 5219 data points or 2.75 per degreeof freedom. Obviously, S-state capture alone cannotdescribe the data with sufficient accuracy, it is neces-sary to introduce P-state capture. Now we impose thecascade ratios but allow them to vary freely withinthe limits given in Table 1. The best fit with freecascade ratios gives a x 2rN of 1.33 which we findF

Žacceptable in view of the very large statistics ;0 0 .550 000 p p h events . The fit requires a resonant

ph P-wave with a finite width; the optimum isreached for Ms1382 MeV and Gs245 MeV. Thex 2 gets worse by 149 when the ph P-wave is

Ž .chosen to be nonresonant the width set to 1 GeV ,and by 367 when the ph P-wave is suppressedcompletely. However, the P-state capture rate forannihilation in liquid H of ;35% derived from2

this fit seems high. However, these ratios are stronglycorrelated with the cascade ratios.

Ž . 2A reasonable fit Figs. 2 and 3 with x rN sF

1.37 is also obtained when the cascade ratios areŽfixed to the central values given by Batty see Table

.1 . The ph P-wave is then seen with a mass of 1350

Table 1Ratio of annihilation rates for annihilation in LH and 12 atm H gas from specific initial states of the protonium atom. The first line shows2 2

two errors. The first one is the error due to uncertainties in the cascade parameters and in the branching ratios used to determine theseconstants; the second error gives the variation of the constants when the pressure is changed by "3 atm. The second line shows estimatedvalues and errors using pp annihilation branching ratios2Sq1 1 3 3L S P PJ 0 1 2

w xBatty 10 0.564"0.062"0.080 4.961"1.605"0.692 4.046"1.309"0.582q3 .0w xGastaldi 13 0.60" 0.10 2.0 5.0" 3.0y1 .4


MeV and a width of 270 MeV. The 1S state con-0

tributes 92% of all annihilations in liquid and 60% ofall annihilations in gaseous H at 12 atm. The x 2

2

difference for excluding the ph P-wave is now 600.Obviously, the cascade ratios play an important

role in the interpretation of the data. Therefore wemade a series of fits in which the cascade ratios were

Ž 1varied stochastically over a wide range for S from0

0.5 to 1; for 3P from 2.3 to 10; for 3P from 0.8 to1 2.10 . Since the cascade ratios are correlated their

effect on the results of the partial wave analysis isexplored over a wider range than would be requiredminimally. All fits consistently required a resonantph wave. The extreme values we found were 1335MeV to 1385 MeV for the mass and 130 MeV to310 MeV for the width.

We conclude that there is evidence for the exis-tence of a resonant ph P-wave in the data which is

Ž . w x w xcalled r 1405 in 18 in accordance with 7 . Theˆstatistical significance is weak if we allow compara-

Ž .tively large ;30% P-state capture rates in liquidH . For fits with ;10% P-state capture probabili-2

ties the evidence is much stronger. Independent ofthis uncertainty we find in all fits a resonant ph

P-wave and masses and widths lying in the range

M s 1360" 25 MeV ,Ž .rŽ1405.ˆ

G s 220" 90 MeVŽ .rŽ1405.ˆ

A relativistic Breit-Wigner amplitude is the simplestdescription of the ph P-wave but may not be unique.With the data presented here we cannot exclude thepossibility that the phase variation required in the fitcould be introduced through threshold effects due to

Ž . Ž .the opening of the f 1285 p or b 1235 p channel.1 1Ž .The r 1405 is produced at a very small rateˆ

Ž 0 0 . 1;1% of p p h in liquid hydrogen via the S0

Ž 0 0state but significantly ;4% of p p h in gaseous. 3hydrogen from the P state. This is a rather large1

rate considering the fact that the total contribution of3 Ž . w xthe P state is 20 to 25 %. In 7 we reported1

y 0Ž .evidence for the r 1405 in pn™p p h annihila-ˆŽ . 3tion where the r 1405 production is larger from Sˆ 1

than from the 1P states. We conjecture that the1Ž .r 1405 is produced more abundantly from spinˆ

triplet states and not from spin singlet states.A resonance with quantum numbers JPC s1yq

cannot possibly be a qq state; it is exotic. Since itsw xisospin is Is1 it cannot be a glueball 19 . The two

Ž .alternative interpretations are that the r 1405 is aˆw xhybrid 20 , a state in which a color-octet qq system

is neutralised in color by a constituent gluon; or itw xcould be a four-quark state 21,22 . The latter possi-

bility seems to be more likely: as a hybrid it wouldŽ .be a SU 3 octet state which is forbidden to decay

Ž .into two SU 3 octet states since the orbital angularŽ .momentum requires antisymmetry, the SU 3

isoscalar factor symmetry with respect to the ex-w xchange of the p and the h meson 23 . A hybrid

Ž .state could couple to the SU 3 -singlet component ofŽ .the h. In this case the r 1405 should decay promi-ˆ

X X0 0nantly into ph . A study of the reaction pp™p p h

did, however, not show any evidence for the r

Ž . w x1405 24 . As a four-quark resonance it can be aŽ .SU 3 decuplet state with allowed coupling to two

octet states. These arguments would suggest that theŽ .r 1405 is likely to be a four-quark state.ˆ

Acknowledgements

We would like to thank the technical staff of theLEAR machine group and of all the participatinginstitutions for their invaluable contributions to thesuccess of the experiment. We acknowledge finan-cial support from the German Bundesministerium fur¨Bildung, Wissenschaft, Forschung und Technologie,the Schweizerischer Nationalfonds, the British Parti-cle Physics and Astronomy Research Council, theUS Department of Energy and the National Science

ŽResearch Fund Committee of Hungary contract No.DE-FG03-87ER40323, DE-AC03-76SF00098, DE-

.FG02-87ER40315 and OTKA T023635 . K.M.C. andF.H.H. acknowledge support from the A. von Hum-boldt Foundation, and N. Djaoshvili from the DAAD.

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28 January 1999


ž q y / 1Partial wave analysis of Jrc™g hp p

BES Collaboration

J.Z. Bai a, Y. Ban f, J.G. Bian a, G.P. Chen a, H.F. Chen b, J.C. Chen a, Y. Chen a,Y.B. Chen a, Y.Q. Chen a, B.S. Cheng a, X.Z. Cui a, H.L. Ding a, L.Y. Dong a,Z.Z. Du a, C.S. Gao a, M.L. Gao a, S.Q. Gao a, J.H. Gu a, S.D. Gu a, W.X. Gu a,

Y.F. Gu a, Y.N. Guo a, S.W. Han a, Y. Han a, J. He a, J.T. He a, K.L. He a, M. He c,G.Y. Hu a, H.M. Hu a, J.L. Hu a,g, Q.H. Hu a, T. Hu a, X.Q. Hu a, Y.Z. Huang a,

C.H. Jiang a, Y. Jin a, Z.J. Ke a, Y.F. Lai a, P.F. Lang a, C.G. Li a, D. Li a, H.B. Li a,J. Li a, P.Q. Li a, R.B. Li a, W. Li a, W.G. Li a, X.H. Li a, X.N. Li a, H.M. Liu a,J. Liu a, R.G. Liu a, Y. Liu a, F. Lu a, J.G. Lu a, X.L. Luo a, E.C. Ma a, J.M. Ma a,H.S. Mao a, Z.P. Mao a, X.C. Meng a, J. Nie a, N.D. Qi a, X.R. Qi a, C.D. Qian e,

J.F. Qiu a, Y.H. Qu a, Y.K. Que a, G. Rong a, Y.Y. Shao a, B.W. Shen a, D.L. Shen a,H. Shen a, X.Y. Shen a, H.Y. Sheng a, H.Z. Shi a, X.F. Song a, F. Sun a, H.S. Sun a,

Y. Sun a, Y.Z. Sun a, S.Q. Tang a, G.L. Tong a, F. Wang a, L.S. Wang a,L.Z. Wang a, Meng Wang a, P. Wang a, P.L. Wang a, S.M. Wang a, T.J. Wang a,Y.Y. Wang a, C.L. Wei a, Y.G. Wu a, D.M. Xi a, X.M. Xia a, P.P. Xie a, Y. Xie a,

Y.H. Xie a, G.F. Xu a, S.T. Xue a, J. Yan a, W.G. Yan a, C.M. Yang a, C.Y. Yang a,J. Yang a, X.F. Yang a, M.H. Ye a, S.W. Ye b, Y.X. Ye b, C.S. Yu a, C.X. Yu a,

G.W. Yu a, Y.H. Yu d, Z.Q. Yu a, C.Z. Yuan a, Y. Yuan a, B.Y. Zhang a,C.C. Zhang a, D.H. Zhang a, Dehong Zhang a, H.L. Zhang a, J. Zhang a,J.W. Zhang a, L.S. Zhang a, Q.J. Zhang a, S.Q. Zhang a, X.Y. Zhang c,

Y.Y. Zhang a, D.X. Zhao a, H.W. Zhao a, Jiawei Zhao b, J.W. Zhao a, M. Zhao a,W.R. Zhao a, Z.G. Zhao a, J.P. Zheng a, L.S. Zheng a, Z.P. Zheng a, B.Q. Zhou a,

G.P. Zhou a, H.S. Zhou a, L. Zhou a, K.J. Zhu a, Q.M. Zhu a, Y.C. Zhu a, Y.S. Zhu a,B.A. Zhuang a

a Institute of High Energy Physics, Beijing 100039, People’s Republic of Chinab UniÕersity of Science and Technology of China, Hefei 230026, People’s Republic of China

c Shandong UniÕersity, Jinan 250100, People’s Republic of Chinad Hangzhou UniÕersity, Hanzhou 310028, People’s Republic of China

1 Data analyzed were taken prior to the participation of US members of the BES Collaboration.


( )J.Z. Bai et al.rPhysics Letters B 446 1999 356–362 357

e Shanghai Jiaotong UniÕersity, Shanghai 200030, People’s Republic of Chinaf Peking UniÕersity, Beijing 100871, People’s Republic of China

g ( )China Center for AdÕanced Science and Technology CCAST , World Laboratory, Beijing 100080, People’s Republic of China

D.V. Bugg h, A.V. Sarantsev h, B.S. Zou a,h

h Queen Mary and Westfield College, London E1 4NS, UK


Abstract

Ž q y.A partial wave analysis of BES Jrc™g hp p data has been performed in the mass region 1.1 to 2.0 GeV. A peakŽ . q y P y qis observed due to h 1440 in the hp p invariant mass distribution; J s0 is preferred over 1 . Its mass is

Ž . w Ž . x Ž .Ms1385"7 MeV and it decays into both hs and a 980 p , with BR a 980 p , a ™hprhs s0.70"0.12 stat "0 0 0Ž . q y0.20 syst . Destructive interference between the two decay modes is observed. In addition, in the higher hp p mass

region, there is a definite 2yq signal, with Ms1840"15 MeV and Gs170"40 MeV and an additional 0yq signal atqq Ž .1760"35 MeV. We also find a possible contribution from a 1 signal decaying dominantly through a 980 p ; its mass0

and width are fitted to be Ms1505"20 MeV and Gs130q50 MeV. q 1999 Elsevier Science B.V. All rights reserved.y20

PACS: 14.40.Cs; 12.39.Mk; 13.25.Jx; 13.40.Hq

Radiative Jrc decays provide a good laboratoryfor the study of glueballs and hybrids, especially inthe mass range 1 to 2 GeV. Candidates have been

q yobserved in hp p , KKp and 4p channels. TheŽ q y.Jrc™g hp p channel is an interesting one to

study, especially in the 1400 MeV region, whereŽ .h 1440 has been found. Mark III and DM2 Collab-

orations made PWA analyses on this channel in 1992w x1,2 . They focused only on the low part of thehpqpy invariant mass spectrum, and both observeda narrow signal near 1400 MeV decaying mainly

Ž .through a 980 p . Mark III claimed it to be a0

pseudoscalar state, while DM2 preferred but notfirmly a 1qq spin-parity assignment. The discrep-ancy might be explained to some extent by thedifferent treatment of the pp S-wave interactioninvolved in the PWA. Mark III used the amplitude of

w xRef. 3 for the pp S-wave interaction, whereasDM2 simply ignored it. In the last few years, the

w xunderstanding of the pp S-wave has improved 4and that parametrization is used here. The existence

Ž .of a broad f 400;1200 is now well established0w x5 .

In order to clarify the spin-parity of the signalnear 1.4 GeV, BES has restudied the decay Jrc™Ž q y.g hp p . In our analysis we include the hs am-

Ž .plitude as well as the a 980 p , where s is short-0

hand for the entire pp S-wave amplitude. Our anal-ysis is conducted up to hpp masses of 2.0 GeV.

The analysis in this paper uses 8.6=106 Jrc

Ž .triggers collected by the Beijing Spectrometer BES .The BES detector has been described in detail in

w xRef. 6 . Here we only briefly describe detectorelements crucial to this measurement. Tracking isprovided by a 10 superlayer main drift chamberŽ .MDC . Each superlayer contains four layers of sensewires measuring both the position and the ionization

Ž .energy loss d Erd x of charged particles. The mo-2'mentum resolution is s rPs1.7% 1qP , whereP

P is the momentum of charged tracks in GeVrc.The resolution of the d Erd x measurement is about9% for hadron tracks. This provides good prKseparation and proton identification in the low mo-mentum region. An array of 48 scintillation counterssurrounding the MDC measures the time-of-flightŽ .TOF of charged tracks with a resolution of 330 psfor hadrons. Outside the TOF system is an electro-magnetic calorimeter composed of streamer tubesand lead blocks with a z positional resolution of 4cm. The energy resolution scales as s rE sE'22%r E , where E is the energy in GeV. Beyondthe shower counter is a solenoidal magnet producing

( )J.Z. Bai et al.rPhysics Letters B 446 1999 356–362358

Ž . q y Ž . "Fig. 1. Mass spectra for a p p , b hp .

a 0.4 tesla magnetic field in the central trackingregion of the detector.

The h is detected here in its gg decay mode.Therefore, much effort has been devoted to theselection of the events in the 3gpqpy final state.Each candidate event is required to have two oppo-sitely charged tracks with a good helix fit in thepolar angle range y0.8-cosu-0.8 and at leastthree reconstructed g ’s in the barrel shower counter.A minimum energy cut of 80 MeV is imposed on thephotons. Showers associated with charged tracks arealso removed. Events are fitted kinematically to the4C hypotheses Jrc™3gpqpy. If the number ofthe selected photons is larger than three, the fit isrepeated using all permutations of the photons. Forevents with a good fit, the three-photon combinationwith the largest probability is selected. Meanwhile,the events are also fitted to Jrc™2gpqpy and4gpqpy. We require

x 2 2gpqpy )20, x 2 4gpqpy )20Ž . Ž .to reject the vp , vpp and rp backgrounds. Inorder to suppress further the backgrounds with a p 0,a 5C fit is performed on the selected events. Theextra constraint is that of the h mass. x 2 -15 is5C

required. This 5C fit helps improve the mass resolu-tion for combinations of charged particles. Thepqpy and hp " mass distributions from the decay

Ž q y. Ž . Ž .Jrc™g hp p are shown in Fig. 1 a and b ,together with the fit. Crosses denote the real data,and the full line the maximum likelihood fit.

The amplitudes in the PWA analysis are con-structed from Lorentz-invariant combinations of the

momenta and the photon polarization four-vectorsfor Jrc initial states with helicity "1. Cross sec-tions are summed over photon polarizations. Therelative magnitudes and phases of the amplitudes aredetermined by a maximum likelihood fit to the data.Based on the study of pqpy and hp " invariantmass distributions in our data, the decay chain Jc™Ž .g hpp is analyzed taking into account the hs ,Ž . Ž . Ž .a 980 p , h f 1270 and a 1320 p intermediate0 2 2

Ž . Ž .processes. The h f 1270 and a 1320 p amplitudes2 2

are only used to fit the data in the high mass region.The background under the h signal is ;30% in the4C fit. We have included a phase space backgroundin the PWA fit to allow for this; it agrees with whatis observed experimentally in sidebands outside theh peak. It peaks strongly at high masses, and doesnot influence much the region below 2 GeV. All

Fig. 2. The hpqpy mass spectrum.


possible spin-parity assignments up to Js2 havebeen tried. We shall use L to denote the orbital

angular momentum between the photon and hpp

states in the production reaction. Because this is an

Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Fig. 3. Scans for a the mass of h 1440 , b the mass of f 1505 , c the width of f 1505 , d the mass of h 1840 and e the width of1 1 2Ž .h 1840 .2


electromagnetic transition, the same phase is used foramplitudes with different L but otherwise the same

Ž .final state, e.g. f 1285 ™a p ; different phases are1 0

allowed for decays of one resonance to differentŽ .channels, e.g. f 1285 ™a p and hs .1 0

We now discuss the features of the data and theoutcome of fits. The hpqpy mass projection fittedto the real data is shown in Fig. 2, where the dashedline represents the fitted background. There are nar-row peaks in the hpp mass spectrum at 1285 and1400 MeV and a further possible peak at ;1505

Ž .MeV. Parameters of the Particle Data Group PDGŽ . P C qqare used for f 1285 . The data favor J s11

over 0yq. For the 1qq assignment, log likelihood,Ž .log L, improves by 11.8 when f 1285 is included1

Ž .using two L s 0 amplitudes with f 1285 ™1Ž . Ža 980 p or hs . Our definition of log L is such that0

.a change of 0.5 corresponds to 1s . The dominantŽ . Ž .process is decay of f 1285 to a 980 p . With our1 0

four fitted parameters, the statistical significance ofthe peak is 3.9s . The inclusion of Ls2 amplitudes

Ž Ž . .affects log likelihood very little D log L -1.0 . Afit with J P C s0yq instead gives ln L worse by 6.6than for 1qq.

The peak at ;1400 MeV optimizes at Ms1385Ž ."7 MeV as shown in Fig. 3 a . This is consistent

with recent determinations cited by the PDG. We fixGs45 MeV from recent results of the Crystal Barrel

w xcollaboration 7 . Our data demand a similar narrowwidth, but are less accurate. The fit prefers 0yq over1qq. Log L improves by 16.6 for 0yq using fourfitted parameters, a 4.7s effect; 1qq instead givesŽ .D log L s5.8 for four fitted parameters. The reso-

Ž .nance decays through both hs and a 980 p modes,0

and the hs decay dominates. The ratio of the decaybranching fractions is

BR h 1440 ™a p ,a ™hpŽ . 0 0s0.70"0.12 .

BR h 1440 ™hsŽ .

ŽThe systematic error dependent on which ampli-.tudes are included in the fit is ;"0.20. The two

decay modes interfere destructively.w xIn Ref. 8 , a Breit-Wigner amplitude with s-de-

Ž .pendent width for channels a 980 p , KK and0 0) Ž .KK was used to fit the h 1440 signal observed in

KqKyp 0 mass spectrum. In our analysis of Jrc™

ghpqpy, we have also tried this Breit-Wigner am-

Ž .plitude for h 1440 , and found that the fit, comparedwith the results got by using a simple Breit-Wigneramplitude, became ;2.5 worse in log L, which is

Ž . y Ž . q Ž . yFig. 4. Component contributions from a 0 , b 1 and c 2 .


Table 1Product branching ratios for the resonances in Jrc™ghpqpy decays. Errors are assessed from systematic variations over a variety offits. There is a "15% uncertainty in overall normalization arising from uncertainties in the net number of Jrc data and the event selection

Process Branching ratiosq y y4Ž Ž .. Ž Ž . . Ž .BR Jrc™g f 1285 =BR f 1285 ™hp p 1.5"0.3 =101 1

q y y4Ž Ž .. Ž Ž . . Ž .BR Jrc™gh 1440 =BR h 1440 ™hp p 2.6"0.7 =10q y y4Ž Ž .. Ž Ž . . Ž .BR Jrc™g f 1505 =BR f 1505 ™hp p 4.5"1.0 =101 1

q y y4Ž Ž .. Ž Ž . . Ž .BR Jrc™gh 1800 =BR h 1800 ™hp p 7.2"0.3 =10q y y4Ž Ž .. Ž Ž . . Ž .BR Jrc™gh 1840 =BR h 1840 ™hp p 6.2"2.2 =102 2

q y y3Ž Ž .. Ž Ž . . Ž .BR Jrc™gh 1760 =BR h 1760 ™hp p 1.2"0.5 =10

negligible. This could be well explained by the factŽ .that the h 1440 resonance is very narrow and the

Ž .a 980 p , hs phase spaces hardly change around0

1.5 GeV. Therefore, as a good approximation, weemploy an ordinary Breit-Wigner amplitude with aconstant width instead in this paper.

At ;1505 MeV, there is a further wider peak.This is just the mass at which a narrow 1qq reso-nance decaying to KK ) is reported in the ParticleData Tables. We have included a 1qq resonance andlog L improves by 19.8 for four fitted parameters, a5.4s effect. The mass is fitted to be Ms1505"20MeV. However, our data require a broader width,

q50 Ž .Gs130 MeV as shown in Fig. 3 c . The decayy20

to hs is roughly half of that to a p in strength. We0

have tried alternative fits with J P s0y, 2q and 2y

but the improvement in log L is -8.0.Now we comment on the broad components. Un-

derneath the narrow peaks in Fig. 2 is a broad 0y

Ž .contribution shown in Fig. 4 a . It gives an improve-ment in log L of 31.3 for four fitted parameters. It isparametrized according to the formula developed by

w x Ž .Bugg and Zou 9 to fit data on Jrc™g 4p andŽ .we shall refer to it as h 1800 . This broad compo-

nent falls away above ;1500 MeV because ofsuppression caused by strong decays to rr, vv and

) )K K .A large improvement to the fit is given by includ-

ing a J P C s2yq resonance, which optimizes atMs1840"15 MeV, Gs170"40 MeV, as shown

Ž . Ž .by scans in Fig. 3 d and e . The production processis with Ls1. We find branching fractions toŽ . Ž .f 1270 h, a 1320 p and hs in the ratios 4:1:2.2 2

The improvement in log L is 50.6 for a fit to threecomplex coupling constants, mass and width; this is

w xan 8.1s effect. Earlier, Crystal Ball 10 , CELLO

w x w x11 and Crystal Barrel 12 have reported evidencefor a 2yq resonance in hpp at ;1875 MeV. It ispossible that this accounts for the resonance weobserve. We have tried J P s2q instead of 2y, butthe fit is much worse.

Ž .The Crystal Barrel group also reports an h 1645 .2

We find an insignificant improvement of 2.4 in log Lwith five fitted parameters when this is included. Wedo, however, find an improvement of 36.5 in log LŽ . y7.2s using a 0 resonance at 1760"35 MeV,produced with Ls0; in this case, only four fittedparameters are needed. Its decays to a p and hs0

are in the ratio 1:0.57. Its width is large, ;250MeV, but not well determined.

Ž q y.The branching ratio for Jrc™g hp p forŽ . y3hpp masses up to 2 GeV is 3.4"0.43 =10 .

Branching ratios for individual components are sum-marized in Table 1. The errors cover not only statis-tics, but also systematic errors dependent on whichcomponents are included in the fit.

In summary, we find evidence for narrow peaksŽ . Ž .due to the presence of f 1285 and h 1440 . There1

is a definite requirement for a 2y component atMs1840"15 MeV with Gs170"40 MeV andstrong evidence for a 0yq component around 1760MeV, though its width is not well determined. Inaddition, a large contribution to our fit is also ob-

Ž .served from a broad f 1505 .1

Acknowledgements

We wish to acknowledge financial support fromthe Royal Society for collaboration between Chineseand UK groups. We also would like to thank Prof. L.Montanet for his helpful discussions. The BES group


are grateful to the staff of IHEP for technical supportin running the experiment. This work is supported inpart by Chinese National Science Foundation undercontract No. 19605007.

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28 January 1999


Search for exotic muon decays 1

R. Bilger a,2, K. Fohl b, H. Clement a, M. Croni a, A. Erhardt a, R. Meier a,¨ ¨J. Patzold a, G.J. Wagner a¨

a Physikalisches Institut der UniÕersitat Tubingen, Auf der Morgenstelle 14, D-72076 Tubingen, Germany¨ ¨ ¨b Department of Physics and Astronomy, UniÕersity of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road,

Edinburgh EH9 3JZ, UK


Abstract

Recently, it has been proposed that the observed anomaly in the time distribution of neutrino induced reactions, reportedby the KARMEN Collaboration, can be interpreted as a signal from an exotic muon decay branch m

q™eqX. It has beenshown that this hypothesis gives an acceptable fit to the KARMEN data if the boson X has a mass of m s103.9 MeVrc2,X

close to the kinematical limit. We have performed a search for the X particle by studying for the first time the very lowenergy part of the Michel spectrum in m

q decays. Using a HPGe detector setup at the mE4 beamline at PSI we findŽ q q . y4 Ž . 2 2branching ratios BR m ™e X -5.7P10 90% C.L. for most of the region 103 MeVrc -m -105 MeVrc .X

q 1999 Elsevier Science B.V. All rights reserved.

PACS: 13.35.Bv

Keywords: Rare decay of muon; Non-standard-model boson

1. Introduction

Ž .At the Rutherford Appleton Laboratory RAL theKARMEN Collaboration is studying neutrino-nuclearreactions, induced from the decay products of posi-tive pions, which are produced and stopped in theproton beam dump. In 1995 KARMEN for the first

w xtime reported 1 an anomaly in the time distributionof single prong events concerning the time interval

1 Ž . ŽSupported by the BMBF 06 TU 886 , DFG Mu 705r3,.Graduiertenkolleg and the UK Engineering and Physical Sciences

Research Council.2 E-mail: [email protected]

corresponding to muon decay. Even with a muchimproved active detector shielding the anomaly has

w xpersisted in new KARMEN data 2 .This anomaly has been suggested to originate

from the observation of a hitherto unknown weaklyinteracting neutral and massive fermion, called x,from a rare pion decay process p

q™mqx. After a

mean flight path of 17.5 m x is registered in theŽ .KARMEN calorimeter after t s 3.60"0.25 msTOF

beam on target by its decay resulting in visibleenergies of typically T s11–35 MeV. The ob-vis

served velocity and the two-body kinematics of theassumed pion decay branch lead to a mass m sx

33.9 MeVrc2, extremely close to the kinematicallimit.

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01507-X

( )R. Bilger et al.rPhysics Letters B 446 1999 363–367364

The hypothetical decay pq™m

qx has beensearched for at PSI in a series of experiments usingmagnetic spectrometers by studying muons from pion

w xdecay in flight 3–5 , the latest measurement result-ing in an upper limit for the branching ratio of

Ž q q . y8 Ž . w xBR p ™m x -1.2P10 95% C.L. 5 . Com-bined with theoretical constraints which assume no

w xnew weak interaction 6 this result rules out theexistence of this rare pion decay branch if x is anisodoublet neutrino. However, if x is mainly isosin-

Ž .glet sterile , the branching ratio can be considerablyw xlower 7 . From the number of observed x events in

comparison with the total number of pq decays the

KARMEN Collaboration gives a lower limit for thebranching ratio of 10y16.

Very recently Gninenko and Krasnikov have pro-w xposed 8 that the observed time anomaly can also be

explained by an exotic m decay branch mq™eqX

resulting in the production of a new, weakly interact-ing neutral boson with mass m s103.9 MeVrc2.X

They show that a second exponential in the KAR-MEN time distribution with time constant equal tothe muon lifetime and shifted by the flight time ofthe X-particle t s3.60 ms gives an acceptable fitTOF

to the KARMEN data. Considering three possibleX-boson phenomenologies, they predict branchingratios for m

q™eqX in the order of 10y2 , if X is ascalar particle; 10y5, if X decays via a hypotheticalvirtual charged lepton; and 10y13, if X decays viatwo additional hypothetical neutral scalar bosons.

In this paper we present a direct experimentalsearch for the X particle by studying the low energypart of the Michel spectrum looking for a peak from

Ž 2mono-energetic positrons with energy T s m qe m2 2 . Ž .m ym r 2m ym s1.23 MeV resulting frome X m e

the two-body decay mq™eqX.

In the past, searches for exotic two-body m decayw xmodes have already been performed 9 motivated by

predictions about the existence of light, weakly inter-acting bosons like axions, majorons, Higgs particles,familons and Goldstone bosons resulting in upperlimits for the branching ratio of approximately 3P

y4 Ž .10 90% C.L. . However, these searches are notsensitive to the suggested X boson with m sX

103.9 MeVrc2 since the lowest positron energyregion studied was between 1.6 and 6.8 MeV,corresponding to the X mass region 103.5 to98.3 MeVrc2.

2. The experiment

The basic idea is to stop a mq beam inside agermanium detector. The low energy decay positronsof interest also deposit their entire kinetic energy inthe detector volume. For a sizeable fraction of eventsthe subsequent annihilation radiation does not inter-act with the detector thus preserving the positronenergy information.

This experiment has been performed at the mE4Ž .channel at PSI see Fig. 1 . The beam line is opti-

mized for intense polarized muon beams in the mo-mentum range between 30 and 100 MeVrc withvery low pion and positron contamination. Pionsfrom the production target are collected at an angleof 908 relative to the primary proton beam and areinjected into a long 5 T superconducting solenoid inwhich they can decay. The last part of the beam lineis the muon extraction section which allows theselection of a central muon momentum differentfrom that of the injected pions.

ŽThe detector setup consists of a large 120=2 .200 mm 2 mm thick plastic scintillator counter S1

followed by a 35 mm diameter hole in a 10 cm thickŽ 2 .lead shielding wall and a small 20=20 mm 1 mm

thick plastic scintillator counter S2 directly in frontof a 9 mm thick planar high purity germaniumŽ . 2HPGe detector with an area of 1900 mm . In addi-

Ž .tion, we have placed a 127 mm 5 inch diameter,127 mm thick NaI detector shielded against the m-fluxadjacent to the HPGe for detecting 511 keV g raysfrom positron annihilation.

The coincidence S1=S2=HPGe was used as atrigger which generated – in addition to a promptgate – a delayed gate 2.2–7.2 ms after the promptmuon signal for the expected decay events. Duringthe time period for the delayed gate, S1 was used asa veto detector to discriminate against further beamparticles. Timing and energy information from thedetectors utilizing several different methods for sig-nal discrimination, amplification, shaping and digiti-zation were recorded for both prompt and delayedsignals using the MIDAS data acquisition systemw x10 .

For the energy calibration of signals occurringduring the prompt gate, g rays from 22 Na and 60Cosources were used. In order to derive the energyinformation from the HPGe detector signal, both

( )R. Bilger et al.rPhysics Letters B 446 1999 363–367 365

Fig. 1. Schematical layout of the experimental setup. The mE4 low energy m channel at PSI is shown in the left part of the figure togetherwith a sketch of the detector setup, which is shown in more detail in the right part of the figure.

spectroscopy amplifiers and peak-sensitive ADCs asŽ .well as a timing filter amplifier TFA connected to a

charge sensitive QDC were employed. In addition,sample signals from the HPGe detector, both before

Fig. 2. Left: Energy spectrum of prompt signals resulting from muons stopping in the HPGe detector with an additional constraint requiringthe presence of an afterpulse arriving during the delayed gate. Right: Spectrum for the time difference between delayed and prompt signals.The time constant ts2.21"0.02ms of the exponential shape is in very good agreement with the muon life time.

( )R. Bilger et al.rPhysics Letters B 446 1999 363–367366

and after amplification, were recorded and storedwith a digital oscilloscope. It turned out that everyspectroscopy amplifier available during the course ofthe experiment showed a significantly varying base-line shift for a few microseconds following a promptsignal. The variations of the baseline level just afterthe prompt signal were due to fluctuations in timefor the onset of the baseline restoration circuitry.Thus, for spectroscopy amplifiers, a sufficiently ac-curate energy calibration for the delayed signal wasnot possible.

The TFA branch did not have such baseline prob-lems, however the energy resolution for the delayedsignal in this branch is 100 keV FWHM only. Ashort shaping time of 0.25 ms and low amplificationto avoid saturation from the high-amplitude promptsignal had to be used to be ready in time for thedelayed pulse.

During 12 hours of data taking 1.3P107 eventswere recorded on tape. Saturation of the HPGe pre-

Ž . 3 y1amplifier at a singles rate of 5–6 P10 s waslimiting the event rate.

3. Results

The energy deposition of the stopped muons inŽ .the HPGe detector is 11.3"0.7 MeV see Fig. 2 .

The cut on the energy of the prompt signal is 9.9–12.7 MeV. The delayed signal has to occur withinthe time interval of 3.4–7.2 ms after the prompt

Ž .signal. The time distribution see Fig. 2 nicelyshows the expected exponential shape with ts2.21"0.02 ms. For shorter times the tail of the promptsignal still causes a varying effective discriminatorthreshold thus the TDC spectrum deviates from anexponential shape. The information from the NaIdetector is used to check the consistency of theanalysis, but is not used for the determination of thebranching ratio.

After energy and time cuts 1.32P106 events re-main. Accounting for high energy positrons frommuon decay causing a signal in the veto counter S1,a 3% correction results in 1.36P106 good muondecays for normalization.

w xGEANT 11 based Monte Carlo studies haveprovided an understanding of the shape of the de-

Ž .layed signal energy spectrum see inset in Fig. 3 .

Fig. 3. Plots showing the energy deposition during the delayedŽ .gate in the HPGe detector top and fit results leading to upper

limits for the branching ratio for the decay mq™eq X. For the

abscissa two corresponding scales, which are the same for allŽ qgraphs except for the inset at the top, which shows the full e

.energy range recorded , are drawn, one is the positron kineticenergy T , the other the X boson mass m . In the graph at the tope X

the Gaussian centered at 1.23 MeV gives the expected detectorresponse if m

q™eq X would contribute with a branching ratio of5P10y3. The second graph shows the reduced x 2, dashed line fora polynomial-only fit, solid line for a combined polynomial andGaussian fit. The third graph, with ordinate units already con-

Ž .verted into branching ratio, shows the contents solid line and theŽ .error dashed line of the Gaussian from this fit. The graph at the

bottom gives the upper limit for a mq™eq X decay branch at

90% confidence level by applying the Bayesian method to the fitresults.

The two peaks are due to an asymmetric m stopdistribution with respect to the symmetry plane per-pendicular to the beam axis of the cylindrical HPGedetector resulting in different energy distributions forMichel positrons emitted in the backward and for-ward hemispheres of the detector, respectively.

The interaction of the annihilation g rays with thedetector has also been studied. For positrons in theconsidered energy range the double escape probabil-

( )R. Bilger et al.rPhysics Letters B 446 1999 363–367 367

Žity is 40–44% no 511 keV g rays interacting in the.HPGe , the single escape probability being a factor 4

lower. The search for mq™eqX events as described

below concentrates on double escape events.Assuming a smooth and gently varying back-

ground as confirmed by the Monte Carlo studies, thesearch for a peak structure in the delayed signal

Ž .energy spectrum see Fig. 3 has been done forenergies from 0.3 to 2.2 MeV. The lower energylimit is given by the effective discriminator thresh-old, the upper energy limit from the positron zerotransmission range in germanium. Since the beam

Ž .muons are stopped after 2–3 mm 2s in the HPGedetector, and since the 2.2 MeV electrons have a zerotransmission range of 2 mm, this is the highest en-ergy for which all positrons remain within the detec-tor volume thus completely depositing their kineticenergy.

For all positron energies between 0.3 and 2.2 MeVa typically 1.2 MeV wide energy interval is chosenand a polynomial fitted to this part of the spectrum.For a polynomial of low order the fit has an unrealis-tically high x 2. Increasing the order of the polyno-mial, the resulting x

2rD.F. first decreases and thenremains roughly constant with values around one.

A polynomial of order seven was chosen as the2 Žlowest order to have a suitable reduced x second

.graph in Fig. 3 . Then a simultaneous fit of a Gauss-Ž .ian position and width fixed and a polynomial

provides the area and error for a possible peak. In thethird graph of Fig. 3 these results have already been

Ž .converted in branching ratio BR units. With aw xBayesian approach 12 one can derive from these

results an upper limit with a given confidence level.Shown on the bottom of Fig. 3 is the 90% C.L.upper limit.

For the positron energy T s1.23 MeV correspon-e

ding to an X particle with mass m s103.9 MeVrc2X

w xas suggested by Gninenko and Krasnikov 8 the90% C.L. upper limit for the branching ratio in thedecay m

q™eqX is BRs4.9P10y4.

4. Summary and outlook

Following the proposition that a new, weaklyinteracting boson X with mass m s103.9 MeVrc2

X

produced in mq™eqX might be the reason for the

observed anomaly in the KARMEN data, we havesearched for this two-body m decay branch by in-spection of the low energy end of the Michel spec-trum. Utilizing a clean m beam from the mE4 chan-nel at PSI and stopping the muons in a planar HPGedetector this work is the first direct search for suchan exotic m decay process for X boson masses103 MeVrc2 -m -105 MeVrc2 corresponding toX

positron energies 0.3 MeV-T -2.2 MeV. Our firsteŽ q q .results give a branching ratio BR m ™e X -5.7P

y4 Ž .10 90% C.L. over most of the accessible region,Žexcluding therefore the simplest scenario X being a

.scalar for the X boson phenomenology suggested inw xRef. 8 . By refining the experimental method used

in this experiment it will be feasible to improve onthis result.

Acknowledgements

We gratefully acknowledge valuable support fromand discussions with D. Branford, M. Daum, T.Davinson, F. Foroughi, C. Petitjean, D. Renker, U.Rohrer, and A.C. Shotter. We also would like tothank the Paul Scherrer Institut for assistance insetting up this experiment in a very short time.

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a I. Physikalisches Institut, RWTH, D-52056 Aachen, Germany, and III. Physikalisches Institut, RWTH, D-52056 Aachen, Germany 5

b National Institute for High Energy Physics, NIKHEF, and UniÕersity of Amsterdam, NL-1009 DB Amsterdam, The Netherlandsc UniÕersity of Michigan, Ann Arbor, MI 48109, USA

d Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux CEDEX, Francee Johns Hopkins UniÕersity, Baltimore, MD 21218, USA

f Institute of Physics, UniÕersity of Basel, CH-4056 Basel, Switzerlandg Louisiana State UniÕersity, Baton Rouge, LA 70803, USA

h Institute of High Energy Physics, IHEP, 100039 Beijing, China 6

i Humboldt UniÕersity, D-10099 Berlin, Germany 5

j UniÕersity of Bologna and INFN-Sezione di Bologna, I-40126 Bologna, Italyk Tata Institute of Fundamental Research, Bombay 400 005, India

l Boston UniÕersity, Boston, MA 02215, USAm Northeastern UniÕersity, Boston, MA 02115, USA

n Institute of Atomic Physics and UniÕersity of Bucharest, R-76900 Bucharest, Romaniao Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary 7

p Massachusetts Institute of Technology, Cambridge, MA 02139, USAq INFN Sezione di Firenze and UniÕersity of Florence, I-50125 Florence, Italy

r European Laboratory for Particle Physics, CERN, CH-1211 GeneÕa 23, Switzerlands World Laboratory, FBLJA Project, CH-1211 GeneÕa 23, Switzerland

t UniÕersity of GeneÕa, CH-1211 GeneÕa 4, Switzerlandu Chinese UniÕersity of Science and Technology, USTC, Hefei, Anhui 230 029, China 6

v SEFT, Research Institute for High Energy Physics, P.O. Box 9, SF-00014 Helsinki, Finlandw UniÕersity of Lausanne, CH-1015 Lausanne, Switzerland

5 Supported by the German Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie.¨6 Supported by the National Natural Science Foundation of China.7 Supported by the Hungarian OTKA fund under contract numbers T019181, F023259 and T024011.


x INFN-Sezione di Lecce and UniÕersita Degli Studi di Lecce, I-73100 Lecce, Italyý Los Alamos National Laboratory, Los Alamos, NM 87544, USA

z Institut de Physique Nucleaire de Lyon, IN2P3-CNRS, UniÕersite Claude Bernard, F-69622 Villeurbanne, France´ áa Centro de InÕestigaciones Energeticas, Medioambientales y Tecnologicas, CIEMAT, E-28040 Madrid, Spain 8

ab INFN-Sezione di Milano, I-20133 Milan, Italyac Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russiaad INFN-Sezione di Napoli and UniÕersity of Naples, I-80125 Naples, Italyae Department of Natural Sciences, UniÕersity of Cyprus, Nicosia, Cyprus

af UniÕersity of Nijmegen and NIKHEF, NL-6525 ED Nijmegen, The Netherlandsag California Institute of Technology, Pasadena, CA 91125, USA

ah INFN-Sezione di Perugia and UniÕersita Degli Studi di Perugia, I-06100 Perugia, Italyái Carnegie Mellon UniÕersity, Pittsburgh, PA 15213, USA

aj Princeton UniÕersity, Princeton, NJ 08544, USAak INFN-Sezione di Roma and UniÕersity of Rome, ‘‘La Sapienza’’, I-00185 Rome, Italy

al Nuclear Physics Institute, St. Petersburg, Russiaam UniÕersity and INFN, Salerno, I-84100 Salerno, Italyan UniÕersity of California, San Diego, CA 92093, USA

ao Dept. de Fisica de Particulas Elementales, UniÕ. de Santiago, E-15706 Santiago de Compostela, Spainap Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgaria

aq Center for High Energy Physics, AdÕ. Inst. of Sciences and Technology, 305-701 Taejon, South Koreaar UniÕersity of Alabama, Tuscaloosa, AL 35486, USA

as Utrecht UniÕersity and NIKHEF, NL-3584 CB Utrecht, The Netherlandsat Purdue UniÕersity, West Lafayette, IN 47907, USA

au Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerlandav DESY-Institut fur Hochenergiephysik, D-15738 Zeuthen, Germany¨

aw Eidgenossische Technische Hochschule, ETH Zurich, CH-8093 Zurich, Switzerland¨ ¨ äx UniÕersity of Hamburg, D-22761 Hamburg, Germanyay National Central UniÕersity, Chung-Li, Taiwan, ROC

az Department of Physics, National Tsing Hua UniÕersity, Taiwan, ROC

Received 21 September 1998Editor: K. Winter

Abstract

A search for pair-produced charged Higgs bosons is performed with the L3 detector at LEP using data collected atcentre-of-mass energies from 130 to 183 GeV, corresponding to an integrated luminosity of 88.3 pby1. The Higgs decaysinto a charm and a strange quark or into a tau lepton and its associated neutrino are considered. The observed candidates areconsistent with the expectations from Standard Model background processes. A lower limit of 57.5 GeV on the charged

Ž " .Higgs mass is derived at 95% CL, independent of the decay branching ratio Br H ™tn . q 1999 Published by ElsevierScience B.V. All rights reserved.

1. Introduction

w xIn the Standard Model 1 , the Higgs mechanismw x2 is used to generate the masses of W and Z bosonsvia spontaneous breaking of the local gauge symme-try. The Higgs sector requires one doublet of com-

8 Supported also by the Comision Interministerial de Ciencia y´´Technologia.

plex scalar fields which leads to the prediction of asingle neutral scalar Higgs boson.

There are more general models, e.g. those derivedfrom supersymmetry, that contain more than one

w xHiggs doublet 3 . A minimal extension to the Stan-dard Model has a two-doublet Higgs sector, whichleads to five physical Higgs bosons: three neutralŽ 0 0 0. Ž ".A , h , H and two charged H . The discoveryof a charged Higgs particle would be clear evidencefor physics beyond the Standard Model.


Charged Higgs bosons can be produced in eqey

q y Ž . q yinteractions via the process e e ™ Zrg ™H H .The Born cross section in the framework of twodoublet models contains the mass of the charged

w xHiggs boson as the only free parameter 4 . In thisletter, we describe the analysis of the data taken atLEP from 1995 to 1997 at centre-of-mass energiesbetween 130–183 GeV. The sensitivity of this datacovers the Higgs mass region below the mass of thecharged heavy gauge boson, m . Charged HiggsW

bosons are expected to decay mainly into the heavi-est lepton that is kinematically allowed and its asso-ciated neutrino, or into the heaviest kinematicallyallowed quark pair whose decay is not Cabibbo-sup-pressed. Thus there are three possible decay modes:

q y q y yH H ™t n t n , cst n and cscs. The relativet t t

branching ratio is model dependent. Therefore threedifferent analyses are optimised for each of thepossible final states. The results include and super-sede previous lower limits to the mass of chargedHiggs bosons established by L3 using the data col-

w xlected at the Z peak 5 . Results from other LEPw xexperiments are published in Ref. 6 .

2. Data analysis

w xThe data were collected with the L3 detector 7 atLEP, corresponding to an integrated luminosity of88.3 pby1 ; where 12.0 pby1 were collected at acentre-of-mass energy of 130–136 GeV, 10.8 pby1

at 161 GeV, 10.2 pby1 at 172 GeV and 55.3 pby1 at183 GeV.

The signal cross section is calculated using thew xPYTHIA Monte Carlo program 8 . For the effi-q y Ž .ciency estimates, samples of e e ™ Zrg ™

HqHy events are generated for Higgs masses be-tween 40 and 80 GeV in mass steps of 5 GeV. About1000 events for each final state are generated at eachHiggs mass. For the background studies the follow-ing Monte Carlo generators are used: PYTHIA for

q y q yŽ . w xe e ™qq g and e e ™ZZ, KORALW 9 forq y q y q y q yw xe e ™W W , PHOJET 10 for e e ™e e qq,

w x q y q y q yŽ .DIAG36 11 for e e ™e e ll ll llse,m,t ,w x q y q y q y q yKORALZ 12 for e e ™m m , e e ™t t

w x q y q yand BHAGENE3 13 for e e ™e e . The L3detector response is simulated using the GEANT

w xprogram 14 which takes into account the effects of

energy loss, multiple scattering and showering in thedetector.

q y q y2.1. Search in the H H ™t n t n channelt t

The signature for the leptonic decay channel is apair of tau leptons with large missing energy andmomentum, giving rise to low multiplicity eventswith low visible energy. Such events are selected by

'requiring a visible energy of less than 0.5 s , be-tween 2 and 20 calorimetric clusters and a chargedtrack multiplicity of between 2 and 8. Dilepton final

q y q y Ž .states from e e ™ ll ll llse,m,t are rejectedby requiring the maximum angle between any pair oftracks to be less than 1658 and the event thrust to beless than 0.98. Radiative dilepton production is re-duced by rejecting events with one or more recon-structed photons with energy greater than 20 GeV.Background from two-photon interactions is reducedby rejecting events where the sum of the energydeposited in the luminosity monitor and the activelead rings exceeds 1 GeV. Remaining two-photoninteraction events are rejected by requiring the ratioof the missing transverse momentum and the visibleenergy to be greater than 0.2 and the ratio of themissing longitudinal momentum and the visible en-ergy to be less than 0.7, and by rejecting eventswhere there is no reconstructed jet with a momentumtransverse to the beam axis exceeding 15 GeV.Cosmic muons are rejected by requiring tracks to

Ž .Fig. 1. Energy spectra, after pre-selection, for events with a'Ž .electrons and b muons in the final state for s s183 GeV. The

dotted line indicates the signal of a 60 GeV charged Higgs bosonŽ " .at Br H ™tn s1 multiplied by a factor 5. The background is

dominated by W decays into leptons.


originate from the eqey interaction region and atleast one scintillator hit in time with the beam cross-ing.

Tau leptons are identified via their decay intoisolated electrons or muons, or as a low multiplicityjet comprising 1, 2 or 3 tracks within a 108 half-opening angle around the jet direction. This allowsacceptance of 1-prong t decays with a spurious trackor a photon conversion and 3-prong decays in whichone track is not reconstructed. Muons must have amomentum of at least 5% of the beam energy inorder to reduce the number of fake signatures fromhadrons that escape the hadron calorimeter. Forhadronically decaying t candidates, the ratioE rE must be less than 1.3, where E and E30 10 30 10

are the energy depositions in a 308 and 108 half anglecone around the direction of the decay particles ofthe t respectively.

Events that are consistent with the signal areselected by requiring the presence of at least two t

decay candidates. At centre-of-mass energies above136 GeV, additional criteria are applied to the t

candidates to reduce contamination from WW™Ž .qq lln llse,m . More energetic leptons are more

likely to come directly from a W decay than from acharged Higgs because of the greater number ofunobserved neutrinos in the latter case. For t decaysto electrons or muons, the observed lepton energy

Ž .must be less than 0.45 of the beam energy Fig. 1 .Further reduction of W background is achieved byrequiring the event to have at least one hadronicallydecaying t candidate.

q y q yThe efficiency of the H H ™t n t n selec-t t

tion for the different Higgs masses is shown in Table1. Table 2 shows the number of events selected in

Table 1q yŽ .Selection efficiencies in% for the t ntt nt final state for

different masses m " at different centre-of-mass energiesH

' Ž .s GeV

Ž ."m GeV 130–136 161 172 183H

45 29 20 21 2350 33 22 23 2555 35 24 26 2660 38 27 29 2765 - 28 29 2870 - 30 30 30

Table 2Expected background and number of events selected in data in the

q yt ntt nt final state at each centre-of-mass energy

' Ž .s GeV 130–136 161 172 183

Expected background 0.3 0.5 1.3 9.2Data 0 0 1 6

the data and the expected background for the differ-ent centre-of-mass energies. The total number ofevents selected in data is 7, where 11.3 backgroundevents are expected from Standard Model processes.Almost all of the remaining background comes fromW pair production.

Systematic uncertainties in the signal efficienciesand the expected number of background events wereinvestigated by comparing the distributions of sev-eral signal-sensitive variables in the data and theMonte Carlo. We assign a systematic error of 0.8events in the total predicted background and 1.5% inthe expected signal efficiencies.

q y y2.2. Search in the H H ™cst n channelt

q y y 9The semileptonic final state H H ™cst n ist

characterised by two hadronic jets, a t lepton andmissing momentum. High multiplicity events areselected by requiring more than 5 charged tracks andmore than 10 calorimetric clusters. Tau leptons areidentified in the same way as for the HqHy™

q yt n t n selection with the additional constraint thatt t

hadronically decaying t candidates must have one orthree tracks and unit charge. The latter requirementon the t leptons reduces the contamination fromŽ .qq g events. Selected events are forced into two

w xjets using the DURHAM algorithm 15 , after sub-tracting the t candidate.

The kinematic cuts differ slightly for the differentcentre-of-mass energies. As an example, we describe

'here the cuts for the s s183 GeV data where wehave the largest search sensitivity due to the highcentre-of-mass energy and the large integrated lumi-nosity.

9 The charge conjugated decay is also considered.


q y yFig. 2. Distributions for the H H ™cst nt channel after theŽ .pre-selection and the t identification: a ratio of the missing

'transverse momentum and the visible energy for s s183 GeV,Ž .b sum of the electron energy and the absolute value of themissing momentum in the rest frame of the leptonically decayingparent particle for events with an identified electron in the final

q y'state for s s183 GeV, the background process W W ™qqen

Ž .is clearly separated from the signal, c polar angle distribution of' Ž .the negative parent particle for s s183 GeV, d reconstructed'mass spectrum after all cuts for s s130–183 GeV. The dotted

lines indicate the signal for a 60 GeV charged Higgs boson atŽ " . Ž . Ž .Br H ™tn s0.5 multiplied by a factor 100 a – c and by a

Ž .factor one d .

The missing transverse momentum must be atleast 10% of the visible energy in order to reject

q y Ž .background from the reactions e e ™qqqq g andŽ . Ž .qq g Fig. 2a . The background contribution from

q y Ž .e e ™qq g is further reduced by requiring themissing momentum parallel to the beam axis to besmaller than 50% of the visible energy. The polarangle of the missing momentum vector must satisfy< <cosQ -0.9. Furthermore, the visible mass, aftermiss

subtraction of the t candidate, must be less than 90GeV and the opening angle of the two jets must beless than 1608 in the plane perpendicular to the beamaxis. The energy deposition in a cone of 258 aroundthe missing momentum vector projected in the sameplane must be smaller than 40 GeV and the sum ofthe opening angles of the t candidate and the miss-ing momentum vector to the closest jet is required tobe larger than 808.

A kinematic fit is performed imposing energy andmomentum conservation for an assumed productionof a pair of equal mass particles with one decayinginto two jets and the other into a t and a neutrino.The directions of the jets, of the t and of the missingmomentum vector are kept at their measured values.Using this method, a resolution of about 4 GeV isobtained in the distribution of the effective mass ofthe two jets and of the t and the neutrino.

ŽSemileptonically decaying W-pairs WW ™.qq lln ; llse,m are suppressed in the following way:

the four momenta are transformed into the rest frameof the leptonically decaying parent particle. In thisframe, the lepton energy E) is greater if the leptonll

comes from a prompt W decay than from a t decay.< ) <The missing momentum P is also larger in themiss

first case because the neutrinos from the t decay arealmost oppositely directed to the t neutrino comingdirectly from the W. For the selection, the sum

) < ) <E q P is used, which should be smaller than 60ll miss

GeV for an electron and smaller than 50 GeV for amuon in the final state. The discriminating power ofthis variable is shown in Fig. 2b.

Ž .To further reject qq g , two-photon interactionsand W pair events, the flight direction of the parentparticle is considered. The production of the chargedHiggs follows a sin2Q dependence whereas the ma-jor fraction of the background is collected in the

Ž .forward-backward region of the detector Fig. 2c .Events with NcosQNF0.9 are accepted.

The selection efficiencies for the different centre-of-mass energies are shown in Table 3. The back-ground expectation together with the selected dataevents are given in Table 4. The total number ofevents selected in data is 39, where 39.8 background

Table 3yŽ .Selection efficiencies in% for the cst nt final state for differ-

ent masses m " at different centre-of-mass energiesH

' Ž .s GeV

Ž ."m GeV 130–136 161 172 183H

45 41 36 34 3850 41 40 35 4155 39 42 46 4260 33 47 46 4165 – 44 44 4270 – 41 42 42


Table 4Background expectation and observed data at the investigated

ycentre-of-mass energies for the cst nt channel

' Ž .s GeV 130–136 161 172 183


events are expected from Standard Model processes.The background is dominated by the process WW™

Ž . Ž .qqtn f70% and other WW decays f22% ; theŽ .remaining contributions are qq g and neutral cur-

rent four-fermion events. For the final mass distribu-tion, we use the average of the masses of the jet-jetand the t n pairs respectively, calculated after thekinematic fit. Fig. 2d shows the mass distribution fordata and background events for all investigated cen-tre-of-mass energies combined.

The main contribution to the systematic errorcomes from the t identification. Systematic uncer-tainties in the t identification were studied usinghigh statistic eqey™eqey, eqey™mqmy, eqey

q y q y Ž .™t t and e e ™qq g data and MC samplesat 91 GeV centre-of-mass energy. A systematic errorof 2% for the signal efficiency and 2.5% for thebackground expectation is derived.

q y2.3. Search in the H H ™cscs channel

q yEvents of the channel H H ™cscs have highmultiplicity and are balanced in transverse and longi-tudinal momenta. Their total centre-of-mass energyis deposited in the detector and they are character-ized by four hadronic jets. The cut values differslightly at the different centre-of-mass energies. The

'cuts described here are for s s 183 GeV.Candidate events are selected by requiring more

than 15 charged tracks and more than 45 calorimetric'clusters. The visible energy must be between 0.6 s

'and 1.4 s and the transverse and longitudinal nor-malised missing energy less than 0.3.

Ž .Radiative qq g events are suppressed by reject-ing events that contain an isolated photon with an

'energy greater 0.1 s . Furthermore, the event sphe-rocity must be within 0.14 and 0.74.

The events are subject to the DURHAM algo-rithm with Y s0.008. Events with less than 4 jetscut

Table 5Ž .Selection efficiencies in% for the cscs final state for different

masses m " at different centre-of-mass energiesH

' Ž .s GeV

Ž ."m GeV 130–136 161 172 183H

45 36 37 35 2950 41 45 45 3655 44 51 45 3960 46 44 43 4065 y 45 41 3870 y 46 39 34

are rejected and the remaining ones are forced intofour jets. The jet energies are rescaled with a com-

'mon factor so that their sum is equal to s .The four jets are grouped into three possible

pairings and the differences between the invariantmasses of all pairings are calculated. Choosing thepair with the minimum invariant mass difference, the

< <polar angle of the parent particle must satisfy cosQ

-0.8. The opening angle between the two jets origi-nating from the same parent particle must be be-tween 538 and 1308. Considering the jet pairing withthe medium invariant mass difference, events arerejected if the average of the two masses is within 2GeV equal to m and their difference is less than 20W

GeV. With these cuts the number of WW events isfurther reduced.

A five-constraint kinematic fit is then appliedassuming the production of a pair of equal massparticles each decaying into two jets. The x 2 perdegree-of-freedom of the fit must be smaller than5.5. This further suppresses the qq background.

The selection efficiencies are shown in Table 5.The expected background and the selected data areshown in Table 6. The total number of events se-lected in data is 145, where 159.5 background eventsare expected from Standard Model processes. The

Table 6Expected background and number of events selected in data in thecscs final state at each centre-of-mass energy

' Ž .s GeV

130–136 161 172 183



main contribution to the background comes from Wpair decays into four jets. In Fig. 3, the average dijetinvariant mass distribution is shown for the data and

'the expected background at s s130–183 GeV. Thelow mass tail for the WW background is due toincorrectly assigned jet pairs.

Systematic errors are assigned to the signal effi-ciencies and the expected number of backgroundevents by comparing the distributions of signal-sensi-tive variables in the data and the Monte Carlo simu-lation. The main contribution to the systematic errorcomes from the fact that the number of reconstructedjets per event is not perfectly simulated in the MonteCarlo. We assign a systematic error of 4.5 events inthe total predicted background and 0.6% in the ex-pected signal efficiencies.

3. Results

The number of selected events in data is consis-tent, in each decay channel, with the number ofevents expected from Standard Model processes. Noindication of pair-produced charged Higgs bosons isobserved. Mass limits as a function of the branching

Ž " .fraction Br H ™tn are derived at 95% confidencelevel, where the confidence level is calculated using

w x q ythe same technique described in 16 . For the H Hq y y™cscs and the H H ™cst n channels we uset

the reconstructed mass distribution in the limit calcu-

Fig. 3. Distribution of the mass resulting from a kinematic fit,with assumed production of a pair of equal mass particles, for data

'and background events in the cscs channel at s s130–183 GeV.The dotted line indicates the signal of a 60 GeV charged Higgs

Ž " .boson at Br H ™cs s1.

Fig. 4. Excluded regions for the charged Higgs boson at 95% CLŽ " .in the plane of the branching fraction Br H ™tn versus mass.

q y q ylation, whereas for the H H ™t n t n channelt t

the total number of data, expected background andexpected signal events are used.

Systematic uncertainties are taken into accountusing the same procedure as in the Standard Model

w xHiggs search 17 . In addition to the systematicerrors resulting from the selection, an error of 0.3%on the luminosity measurement, an error of 5% onthe background normalisation and an error of 2% onthe signal cross section are taken into account.

Fig. 4 shows the excluded mass regions of chargedHiggs bosons at 95% CL for the analyses of eachfinal state and their combination as function of the

Ž " .branching fraction Br H ™tn . A lower limit onthe mass of the charged Higgs boson of

m ")57.5 GeV 1Ž .H

independent of the branching fraction is obtained.

Acknowledgements

We wish to express our gratitude to the CERNaccelerator divisions for the excellent performance ofthe LEP machine. We acknowledge the efforts of allengineers and technicians who have participated inthe construction and maintenance of the experiment.


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Cumulative author index to volumes 441–446

Abbaneo, D., 445, 239Abbiendi, G., 443, 394; 444, 539Abe, T., 443, 394Abel, S., 444, 427Abel, S.A., 444, 119Abele, A., 446, 349Abraham, K.J., 446, 163Abramowicz, H., 443, 394Abreu, M.C., 444, 516Abreu, P., 441, 479; 444, 491; 446, 62, 75Acciarri, M., 444, 503, 569; 445, 428; 446, 368Acebal, J.L., 445, 94Ackerstaff, K., 444, 531, 539Acosta, D., 443, 394Adam, W., 441, 479; 444, 491; 446, 62, 75Adamczyk, L., 443, 394Adamus, M., 443, 394Adler, J.-O., 442, 43Adomeit, J., 446, 349Adriani, O., 444, 503, 569; 445, 428; 446, 368Adye, T., 441, 479; 444, 491; 446, 62, 75Adzic, P., 441, 479; 444, 491; 446, 62, 75Afanasiev, S., 445, 14Affholderbach, K., 445, 239Agashe, K., 444, 61Aglietti, U., 441, 371Agodi, C., 442, 48Aguilar-Benitez, M., 444, 503, 569; 445, 428;

446, 368Ahlen, S., 444, 503, 569; 445, 428; 446, 368Ahn, C., 442, 109Ahn, J.K., 444, 267Ahn, S.H., 443, 394Airapetian, A., 442, 484; 444, 531Aitala, E.M., 445, 449Ajaltouni, Z., 445, 239Ajinenko, I., 446, 75Akama, K., 445, 106Akeroyd, A.G., 441, 224; 442, 335Akhmedov, E.T., 442, 152Akopov, N., 442, 484; 444, 531Akushevich, I., 442, 484; 444, 531Alamanos, N., 442, 48Alba, R., 442, 48Alcaraz, J., 444, 503, 569; 445, 428; 446, 368

Aldeweireld, T., 444, 491; 446, 62, 75Alekseev, G.D., 441, 479; 444, 491; 446, 62,

75Alemanni, G., 444, 503, 569; 445, 428; 446,

368Alemany, R., 441, 479; 444, 491; 445, 239;

446, 62, 75ALEPH Collaboration, 445, 239Aleppo, M., 445, 239Alexander, G., 444, 539Alexandre, J., 445, 351Aliev, T.M., 441, 410Allaby, J., 444, 503, 569; 445, 428; 446, 368Allison, J., 444, 539Allmendinger, T., 444, 491; 446, 62, 75Allport, P.P., 441, 479; 444, 491; 446, 62, 75Almehed, S., 441, 479; 444, 491; 446, 62, 75Aloisio, A., 444, 503, 569; 445, 428; 446, 368Altegoer, J., 445, 439Altekamp, N., 444, 539Alviggi, M.G., 444, 503, 569; 445, 428; 446,

368Amaldi, U., 441, 479; 444, 491; 446, 62, 75Amarian, M., 442, 484; 444, 531Amato, S., 441, 479; 444, 491; 445, 449; 446,

62, 75Ambjørn, J., 445, 307Ambrosi, G., 444, 503, 569; 445, 428; 446, 368Amelung, C., 443, 394Amsler, C., 446, 349Amzal, N., 443, 69, 82An, S.H., 443, 394Anassontzis, E.G., 441, 479; 444, 491; 446, 62,

75Anderhub, H., 444, 503, 569; 445, 428; 446,

368Andersen, J., 446, 117Anderson, K.J., 444, 539Anderson, S., 444, 539Andersson, P., 441, 479; 444, 491; 446, 62, 75Andreazza, A., 441, 479; 444, 491; 446, 62, 75Andreev, V.P., 444, 503, 569; 445, 428; 446,

368Andringa, S., 441, 479; 444, 491; 446, 62, 75Angelantonj, C., 444, 309

CumulatiÕe author index to Õolumes 441–446380

Angelescu, T., 444, 503, 569; 445, 428; 446,368

Angelini, C., 445, 439Angelopoulos, A., 444, 38, 43, 52Anjos, J.C., 445, 449Annand, J.R.M., 442, 43Anselmino, M., 442, 470Anselmo, F., 443, 394; 444, 503, 569; 445,

428; 446, 368Antilogus, P., 441, 479; 444, 491; 446, 62, 75Antonelli, A., 445, 239Antonelli, M., 445, 239Antoniadis, I., 444, 284Antonioli, P., 443, 394Antoniou, N.G., 444, 583Antonov, D., 444, 208Antonuccio, F., 442, 173Anzivino, G., 446, 117Aoki, S., 444, 267Apel, W.-D., 441, 479; 444, 491; 446, 62, 75Apostolakis, A., 444, 38, 43, 52Appel, J.A., 445, 449Appelshauser, H., 444, 523¨Aranda, A., 443, 352Arcelli, S., 444, 539Arcidiacono, R., 446, 117Ardouin, D., 446, 191Arefiev, A., 444, 503, 569; 445, 428; 446, 368Argyres, P.C., 441, 96; 442, 180Arima, A., 445, 1Arkhipov, V., 445, 14Armbruster, P., 444, 32Armesto, N., 442, 459Armstrong, S.R., 445, 239Arneodo, M., 443, 394Arnoud, Y., 441, 479; 444, 491; 446, 62, 75Arutyunov, G.E., 441, 173Asai, M., 443, 409Asai, S., 444, 539Aschenauer, E.C., 442, 484; 444, 531Ashby, S.F., 444, 539Ashery, D., 445, 449Aslanides, E., 444, 38, 43, 52Asman, B., 441, 479; 444, 491; 446, 62, 75Astier, P., 445, 439Aubert, J.J., 445, 239Auger, F., 442, 48Augustin, I., 446, 117Augustin, J.-E., 441, 479; 444, 491; 446, 62, 75Augustinus, A., 441, 479; 444, 491; 446, 62, 75Autiero, D., 445, 439Avakian, H., 442, 484; 444, 531Avakian, R., 442, 484; 444, 531Avetissian, A., 442, 484; 444, 531Avila, C., 445, 419Axen, D., 444, 539Axenides, M., 444, 190Azcoiti, V., 444, 421

Azemoon, T., 444, 503, 569; 445, 428; 446,368

Aziz, T., 444, 503, 569; 445, 428; 446, 368Azuelos, G., 444, 539Azzurri, P., 445, 239

Bachler, J., 444, 523¨Back, T., 443, 69, 82¨Backenstoss, G., 444, 38, 43, 52Bacon, T.C., 443, 394Badaud, F., 445, 239Badgett, W.F., 443, 394Bagliesi, G., 445, 239Baglin, C., 444, 516Bagnaia, P., 444, 503, 569; 445, 428; 446, 368Bai, J.Z., 446, 356Bailey, D.C., 443, 394Bailey, D.S., 443, 394Bailey, S.J., 444, 523Bailin, D., 443, 111Baillon, P., 441, 479; 444, 491; 446, 62, 75Bains, B., 442, 484; 444, 531Bakas, I., 445, 69Baker, C.A., 446, 349Baker, W.F., 445, 419Baksay, L., 444, 503, 569; 445, 428; 446, 368Baldini, R., 444, 111Baldisseri, A., 445, 439Baldit, A., 444, 516Baldo-Ceolin, M., 445, 439Ball, A.H., 444, 539Ballesteros, H.G., 441, 330Ballocchi, G., 445, 439Bambade, P., 441, 479; 444, 491; 446, 62, 75Bamberger, A., 443, 394Ban, Y., 446, 356Bando, M., 444, 373Banerjee, S., 444, 503, 503, 569; 445, 428, 449;

446, 368Banerjee, Sw., 444, 569; 445, 428; 446, 368Banicz, K., 444, 503, 569; 445, 428; 446, 368Banner, M., 445, 439Barao, F., 441, 479; 444, 491; 446, 62, 75Barate, R., 445, 239Barbagli, G., 443, 394Barberio, E., 444, 539Barberis, D., 446, 342Barbero, C., 445, 249Barbiellini, G., 441, 479; 444, 491; 446, 62, 75Barbier, R., 441, 479; 444, 491; 446, 62, 75Barbieri, R., 445, 407Barczyk, A., 444, 503, 569; 445, 428; 446, 368Bardin, D.Y., 441, 479; 444, 491; 446, 62, 75Barenboim, G., 443, 317Bargassa, P., 444, 38, 43, 52Barger, V., 442, 255Bari, G., 443, 394Barillere, R., 444, 503, 569; 445, 428; 446, 368`

CumulatiÕe author index to Õolumes 441–446 381

Bark, R., 443, 69Barker, G., 441, 479; 444, 491; 446, 62, 75Barlow, R.J., 444, 539Barna, D., 444, 523Barnby, L.S., 444, 523Barnea, N., 446, 185Barnett, B.M., 446, 349Baroncelli, A., 441, 479; 444, 491; 446, 62, 75Barone, L., 444, 503, 569; 445, 428; 446, 368Barr, G., 446, 117Barreiro, F., 443, 394Barreiro, T., 445, 82Barret, O., 443, 394Barrow, J.D., 443, 104Barrow, S., 444, 531Bartalini, P., 444, 503, 569; 445, 428; 446, 368Bartels, J., 442, 459Barth, J., 444, 555; 445, 20Bartke, J., 444, 523Bartoldus, R., 444, 539Barton, R.A., 444, 523Basa, S., 445, 439Baschirotto, A., 444, 503, 569; 445, 428; 446,

368Bashindzhagyan, G.L., 443, 394Bashkirov, V., 443, 394Basile, M., 443, 394; 444, 503, 569; 445, 428;

446, 368Bass, S.A., 442, 443; 446, 191Bassetto, A., 443, 325Bassompierre, G., 445, 439Batalin, I., 441, 243Batalin, I.A., 446, 175Batley, J.R., 444, 539Battaglia, M., 441, 479; 444, 491; 446, 62, 75Battiston, R., 444, 503, 569; 445, 428; 446, 368Batty, C.J., 446, 349Baubillier, M., 441, 479; 444, 491; 446, 62, 75Bauer, W., 444, 231Bauerdick, L.A.T., 443, 394Baulieu, L., 441, 250Baumann, S., 444, 539Baumann, T., 444, 32Baumgarten, C., 442, 484; 444, 531Bay, A., 444, 503, 569; 445, 428; 446, 368Beane, S.R., 444, 147Becattini, F., 444, 503, 569; 445, 428; 446, 368Bechtluft, J., 444, 539Becirevic, D., 444, 401Becker, H.G., 446, 117Becker, U., 444, 503, 569; 445, 239, 428; 446,

368Beckmann, M., 442, 484; 444, 531Becks, K.-H., 441, 479; 444, 491; 446, 62, 75Bediaga, I., 445, 449Bedjidian, M., 444, 516Bednarek, B., 443, 394Bednyakov, V., 442, 203

Begalli, M., 441, 479; 444, 491; 446, 62, 75Behner, F., 444, 503, 569; 445, 428; 446, 368Behnke, O., 444, 38, 43, 52Behnke, T., 444, 539Behrend, R.E., 444, 163Behrens, U., 443, 394Behrndt, K., 442, 97Beier, H., 443, 394Beilliere, P., 441, 479; 444, 491; 446, 62, 75Belitsky, A.V., 442, 307Bell, K.W., 444, 539Bella, G., 444, 539Bellagamba, L., 443, 394Bellerive, A., 444, 539Bellia, G., 442, 48Belokopytov, Yu., 441, 479; 444, 491; 446, 62,

75Belostotski, S., 444, 531Belostotski, St., 442, 484Belous, K., 441, 479; 444, 491Belz, E., 442, 484Belz, J.E., 444, 531Benayoun, M., 446, 349Benchouk, C., 445, 239Bencivenni, G., 445, 239Benczer-Koller, N., 446, 22Bender, M., 446, 117Benedetti, R., 441, 60Beneke, M., 443, 308Benelli, A., 444, 38, 43, 52Benisch, T., 444, 531Benisch, Th., 442, 484Benlliure, J., 444, 32Bennhold, C., 445, 20Benslama, K., 445, 439Bentvelsen, S., 444, 539Benvenuti, A.C., 441, 479; 444, 491; 446, 62,

75Berat, C., 441, 479; 444, 491; 446, 62, 75Berdoz, A., 446, 349Berdugo, J., 444, 503, 569; 445, 428; 446, 368Berezinsky, V., 444, 387Berg, B.A., 444, 487Berges, P., 444, 503, 569; 445, 428; 446, 368Berggren, M., 441, 479; 444, 491; 446, 62, 75Bergman, O., 441, 133Bering, K., 446, 175Berlich, R., 445, 239Bern, Z., 444, 273; 445, 168Bernas, M., 444, 32Bernreuther, S., 442, 484; 444, 531Bernstein, A.M., 442, 20Bertani, M., 444, 111Bertin, V., 444, 38, 43, 52Bertini, D., 441, 479; 444, 491; 446, 62, 75Bertini, M., 442, 398Bertolin, A., 443, 394Bertram, I., 443, 347


Bertrand, D., 441, 479; 444, 491; 446, 62, 75Bertucci, B., 444, 503, 569; 445, 428; 446, 368Bes, D.R., 446, 93Besancon, M., 441, 479; 444, 491; 446, 62, 75BES Collaboration, 446, 356Besson, N., 445, 439Betev, B.L., 444, 503, 569; 445, 428; 446, 368Bethke, S., 444, 539Betsch, A., 446, 179Bettarini, S., 445, 239Betteridge, A.P., 445, 239Betts, S., 444, 539Beuchert, K., 446, 349Beusch, W., 446, 342Beuselinck, R., 445, 239Bhadra, S., 443, 394Bhattacharya, S., 444, 503, 569; 445, 428; 446,

368Białkowska, H., 444, 523Bian, J.G., 446, 356Bianchi, F., 441, 479; 444, 491; 446, 62, 75Bianchi, N., 442, 484; 444, 531Biasini, M., 444, 503, 569; 445, 428; 446, 368Bicudo, P., 442, 349Biebel, O., 444, 539Bienlein, J.K., 443, 394Bigi, M., 441, 479; 444, 491; 446, 62, 75Biguzzi, A., 444, 539Biino, C., 446, 117Bijnens, J., 441, 437Biland, A., 444, 503, 569; 445, 428; 446, 368Bilei, G.M., 444, 503, 569; 445, 428; 446, 368Bilenky, M.S., 441, 479; 444, 491; 446, 62, 75Bilenky, S.M., 444, 379Bilger, R., 443, 77; 446, 179, 363Billmeier, A., 444, 523Bimonte, G., 441, 69Binetruy, P., 441, 52, 163´Bing, O., 445, 423Bini, C., 444, 111Binnie, D.M., 445, 239Binon, F.G., 446, 342Bird, I., 445, 439Bird, S.D., 444, 539Birse, M.C., 446, 300Bischoff, S., 446, 349Bizouard, M.-A., 441, 479; 444, 491; 446, 62,

75Bizzeti, A., 446, 117Black, S.N., 445, 239Blaikley, H.E., 443, 394Blair, G.A., 445, 239Blaising, J.J., 444, 503, 569; 445, 428; 446, 368Blanc, F., 444, 38, 43, 52Blanchard, S., 444, 531Blaylock, G., 445, 449Bleicher, M., 442, 443Blick, A.M., 446, 342

Blobel, V., 444, 539Bloch, D., 441, 479; 444, 491; 446, 62, 75Bloch-Devaux, B., 445, 239Bloch, P., 444, 38, 43, 52Blom, H.M., 441, 479; 444, 491; 446, 62, 75Blondel, A., 445, 239Bloodworth, I.J., 444, 539Blouw, J., 442, 484; 444, 531Blum, P., 446, 349¨Blumenfeld, B., 445, 439Blumenfeld, Y., 442, 48Blumer, H., 446, 117¨Blyth, C.O., 444, 523Blyth, S.C., 444, 503, 569; 445, 428; 446, 368Bobbink, G.J., 444, 503, 569; 445, 428; 446,

368Bobinski, M., 444, 539Bobisut, F., 445, 439Boccali, T., 445, 239Bock, P., 444, 539Bock, R., 444, 503, 523, 569; 445, 428; 446,

368Bockhorst, M., 445, 20Bocquet, G., 446, 117Bohm, A., 444, 503, 569; 445, 428; 446, 368¨Bohme, J., 444, 539¨Bohnet, I., 443, 394Bohrer, A., 445, 239¨Boivin, M., 445, 423Boix, G., 445, 239Bokel, C., 443, 394Boldizsar, L., 444, 503, 569; 445, 428; 446,

368Bologna, G., 445, 239Bolz, M., 443, 209Bonacorsi, D., 444, 539Bondarev, V., 445, 14Bonesini, M., 441, 479; 444, 491; 446, 62, 75Bonissent, A., 445, 239Bonivento, W., 441, 479; 444, 491; 446, 62, 75Boonekamp, M., 441, 479; 444, 491; 446, 62,

75Booth, C.N., 445, 239Booth, P.S.L., 441, 479; 444, 491; 446, 62, 75Bordalo, P., 444, 516Borgia, B., 444, 503, 569; 445, 428; 446, 368Borgland, A.W., 441, 479; 444, 491; 446, 62,

75Borisov, G., 441, 479; 444, 491; 446, 62, 75Borissov, A., 442, 484; 444, 531Bormann, C., 444, 523Bornheim, A., 443, 394Bortignon, P.F., 444, 1Borzemski, P., 443, 394Boscherini, D., 443, 394Bosio, C., 441, 479; 444, 491; 446, 62, 75Bossi, F., 445, 239Botje, M., 443, 394


Botner, O., 441, 479; 444, 491; 446, 62, 75Bottcher, H., 442, 484; 444, 531¨Botterill, D.R., 445, 239Boucaud, Ph., 444, 401Bouchez, J., 445, 439Boucrot, J., 445, 239Boudinov, E., 444, 491; 446, 62, 75Bouquet, B., 441, 479; 444, 491; 446, 62, 75Bourdarios, C., 441, 479; 444, 491; 446, 62, 75Bourilkov, D., 444, 503, 569; 445, 428; 446,

368Bourquin, M., 444, 503, 569; 445, 428; 446,

368Bourrely, C., 442, 479Boutemeur, M., 444, 539Bowcock, T.J.V., 441, 479; 444, 491; 446, 62,

75Bowdery, C.K., 445, 239Bowser-Chao, D., 441, 468Boyd, S., 445, 439Boyko, I., 441, 479; 444, 491; 446, 62, 75Boyle, O., 446, 117Bozovic, I., 441, 479; 444, 491; 446, 62, 75Bozzi, C., 445, 239Bozzo, M., 441, 479; 444, 491; 446, 62, 75Braccini, S., 444, 503, 569; 445, 428; 446, 368Brack, J., 442, 484; 444, 531Bracker, S.B., 445, 449Bradley, D.A., 442, 38Brady, F.P., 444, 523Braghin, F.L., 446, 1Braibant, S., 444, 539Brambilla, N., 442, 349Branchina, V., 445, 351Branchini, P., 441, 479; 444, 491; 446, 62, 75Branco, G.C., 442, 229Brandenberger, R., 445, 323Brandt, F., 443, 147Brandt, S., 445, 239Branson, J.G., 444, 503, 569; 445, 428; 446,

368Bratkovskaya, E.L., 445, 265Brauksiepe, S., 442, 484; 444, 531Braun, B., 442, 484; 444, 531Braun, M.A., 442, 459; 444, 435Braun, V.M., 443, 308Braun, W., 444, 555; 445, 20Braune, K., 446, 349Bravina, L., 442, 443Bray, B., 444, 531Brecher, D., 442, 117Breitweg, J., 443, 394Bremer, J., 446, 117Bremner, C.A., 444, 260Brenke, T., 441, 479; 444, 491; 446, 62, 75Brenner, R.A., 441, 479; 444, 491; 446, 62, 75Brient, J.-C., 445, 239Bright-Thomas, P., 444, 539

Brigliadori, L., 444, 539Brigljevic, V., 444, 503, 569; 445, 428; 446,

368Brihaye, Y., 441, 77Briskin, G., 443, 394Broadhurst, D.J., 441, 345Brock, I., 443, 394Brock, I.C., 444, 503, 569; 445, 428; 446, 368Brockmann, R., 444, 523Brodie, J.H., 445, 296Brodowski, W., 446, 179Brons, S., 442, 484; 444, 531Brook, N.H., 443, 394Broude, C., 446, 22Brown, R.M., 444, 539Browning, F., 444, 142Bruckman, P., 441, 479; 444, 491; 446, 62, 75Bruckner, W., 442, 484; 444, 531¨Brugnera, R., 443, 394Bruins, E.E.W., 444, 531Brull, A., 442, 484; 444, 531¨Brummer, N., 443, 394¨Brun, R., 444, 523Brunet, J.-M., 441, 479; 444, 491; 446, 62, 75Bruni, A., 443, 394Bruni, G., 443, 394Brustein, R., 442, 74Buchbinder, E.I., 446, 216Buchbinder, I.L., 446, 216Buchel, A., 442, 180Buchholz, P., 446, 117Buchmuller, O., 445, 239¨Buchmuller, W., 443, 209; 445, 399¨Buck, P.G., 445, 239Budzanowski, A., 445, 20Bueno, A., 445, 439Buffini, A., 444, 503, 569; 445, 428; 446, 368Bugge, L., 441, 479; 444, 491; 446, 62, 75Buijs, A., 444, 503, 569; 445, 428; 446, 368Bulten, H.J., 442, 484; 444, 531Buncic, P., 444, 523ˇ ´Bunyatov, S., 445, 439Buran, T., 441, 479; 444, 491; 446, 62, 75Burbach, G., 445, 20Burchat, P.R., 445, 449Burckhart, H.J., 444, 539Burgard, C., 443, 394; 444, 539Burger, J.D., 444, 503, 569; 445, 428; 446, 368Burger, W.J., 444, 503, 569; 445, 428; 446, 368Burgin, R., 444, 539¨Burgsmueller, T., 441, 479; 444, 491; 446, 62,

75Burgwinkel, R., 444, 555; 445, 20Burnstein, R.A., 445, 449Burow, B.D., 443, 394Buscher, V., 445, 239¨Buschmann, P., 441, 479; 444, 491; 446, 62, 75Busenitz, J., 444, 503, 569; 445, 428; 446, 368


Buskulic, D., 445, 239Bussey, P.J., 443, 394Bussiere, A., 444, 516`Busson, P., 444, 516Butterworth, J.M., 443, 394Button, A., 444, 503, 569; 445, 428; 446, 368Bylsma, B., 443, 394Bytsenko, A.A., 443, 121

Cabrera, S., 441, 479; 444, 491; 446, 62, 75Caccia, M., 441, 479; 444, 491; 446, 62, 75Cacciatori, S., 444, 332Cadman, R., 444, 531Cai, X.D., 444, 503, 569; 445, 428; 446, 368Caines, H.L., 444, 523Calabrese, R., 444, 111Calafiura, P., 446, 117Calderini, G., 445, 239Caldwell, A., 443, 394Calen, H., 446, 179´Callot, O., 445, 239Calvetti, M., 446, 117Calvi, M., 441, 479; 444, 491; 446, 62, 75Camacho Rozas, A.J., 441, 479; 444, 491; 446,

62, 75Cameron, W., 445, 239Camilleri, L., 445, 439Campana, P., 445, 239Campanelli, M., 444, 503, 569; 445, 428; 446,

368Camporesi, T., 441, 479; 444, 491; 446, 62, 75Campos, A., 443, 338Canale, V., 441, 479; 444, 491; 446, 62, 75Capell, M., 444, 503, 569; 445, 428; 446, 368Capiluppi, P., 444, 539Capitani, G.P., 442, 484; 444, 531Capon, G., 445, 239Capua, M., 443, 394Cara Romeo, G., 443, 394; 444, 503, 569; 445,

428; 446, 368Cardini, A., 445, 439Carena, F., 441, 479; 444, 491; 446, 62, 75Carena, M., 441, 205Carlin, R., 443, 394Carlino, G., 444, 503, 569; 445, 428; 446, 368Carlson, C.E., 441, 363Carlson, P., 444, 38, 43, 52Carman, T.S., 444, 252Carnegie, R.K., 444, 539Carone, C.D., 441, 363, 443, 352Carosi, R., 446, 117Carr, J., 445, 239Carr, L.D., 444, 523Carroll, L., 441, 479; 444, 491; 446, 62, 75Carroll, M., 444, 38, 43, 52Cartacci, A.M., 444, 503, 569; 445, 428; 446,

368Carter, A.A., 444, 539

Carter, J.R., 444, 539Carter, P., 442, 484; 444, 531Carter, T., 445, 449Cartiglia, N., 443, 394Cartwright, S., 445, 239Carvalho, H.S., 445, 449Casado, M.P., 445, 239Casas, J.A., 445, 82Casaus, J., 444, 503, 569; 445, 428; 446, 368Case, T., 446, 349Cashmore, R.J., 443, 394Caso, C., 441, 479; 444, 491; 446, 62, 75Castellini, G., 443, 394; 444, 503, 569; 445,

428; 446, 368Castillo Gimenez, M.V., 441, 479; 444, 491;

446, 62, 75Castor, J., 444, 516Catani, S., 446, 143Catford, W.N., 444, 32Cattai, A., 441, 479; 444, 491; 446, 62, 75Cattaneo, M., 445, 239Cattaneo, P.W., 445, 439Catterall, C.D., 443, 394Catterall, S., 442, 266Cavallari, F., 444, 503, 569; 445, 428; 446, 368Cavallo, F.R., 441, 479; 444, 491; 446, 62, 75Cavallo, N., 444, 503, 569; 445, 428; 446, 368Cavanaugh, R., 445, 239Cavasinni, V., 445, 439Cawley, E., 444, 38, 43, 52Cebra, D., 444, 523Cecchi, C., 444, 503, 569; 445, 428; 446, 368Ceccucci, A., 446, 117Cederwall, B., 443, 69, 82Celikel, A., 443, 359Cenci, P., 446, 117Cerrada, M., 444, 503, 569; 445, 428; 446, 368Cerri, C., 446, 117Cerruti, Ch., 441, 479Cerutti, F., 445, 239Cervera-Villanueva, A., 445, 439Cesaroni, F., 444, 503, 569; 445, 428; 446, 368Chabaud, V., 441, 479; 444, 491; 446, 62, 75Chacko, Z., 442, 199Chaichian, M., 442, 192Chalmers, M., 445, 239Chambers, J.T., 445, 239Chambon, T., 444, 516Chamizo, M., 444, 503, 569; 445, 428; 446,

368Chamseddine, A.H., 442, 97Chang, C.Y., 444, 539Chang, D., 444, 142Chang, L.N., 441, 419Chang, Y.H., 444, 503, 569; 445, 428; 446, 368Chankowski, P.H., 441, 205Chapin, D., 443, 394Chapkin, M., 444, 491; 446, 62


Chappell, S.P.G., 444, 260Charalambous, S., 444, 43Charity, R.J., 446, 197Charles, E., 445, 239Charlot, C., 444, 516Charlton, D.G., 444, 539Charpentier, Ph., 441, 479; 444, 491; 446, 62,

75Chaturvedi, U.K., 444, 503, 569; 445, 428; 446,

368Chaurand, B., 444, 516Chaussard, L., 441, 479; 444, 491; 446, 62, 75Chaves, A.S., 445, 94Chazelle, G., 445, 239Checchia, P., 441, 479; 444, 491; 446, 62, 75Chekanov, S., 443, 394Chelkov, G.A., 441, 479; 444, 491; 446, 62, 75Chemarin, M., 444, 503, 569; 445, 428; 446,

368Chen, A., 444, 503, 569; 445, 428; 446, 368Chen, F.-z., 442, 223Chen, G., 444, 503, 569; 445, 428; 446, 368Chen, G.M., 444, 503, 569; 445, 428; 446, 368Chen, G.P., 446, 356Chen, H.F., 444, 503, 569; 445, 428; 446, 356,

368Chen, H.S., 444, 503, 569; 445, 428; 446, 368Chen, J.C., 446, 356Chen, Q., 444, 252Chen, S., 445, 239Chen, Y., 446, 356Chen, Y.B., 446, 356Chen, Y.Q., 446, 356Cheng, B.S., 446, 356Chereau, X., 444, 503, 569; 445, 428; 446, 368Chernodub, M.N., 443, 244Chertok, M.B., 444, 38, 43, 52Cheze, J.B., 446, 117Chiarella, V., 445, 239Chiba, T., 442, 59Chiefari, G., 444, 503, 569; 445, 428; 446, 368Chien, C.Y., 444, 503, 569; 445, 428; 446, 368Chierici, R., 441, 479; 444, 491; 446, 62, 75Chishtie, F., 446, 267Chiu, T.-W., 445, 371Chliapnikov, P., 444, 491; 446, 62, 75Chmeissani, M., 445, 239Chochula, P., 441, 479; 444, 491; 446, 62, 75Chollet, J.C., 446, 117Chomaz, Ph., 442, 48Chorowicz, V., 441, 479; 444, 491; 446, 62, 75Chrisman, D., 444, 539Christiansen, H.R., 441, 185; 445, 8Chudoba, J., 441, 479; 444, 491; 446, 62, 75Chumney, P., 442, 484; 444, 531Chung, K.S., 444, 267Chung, M.S., 444, 267Chwastowski, J., 443, 394

Ciafaloni, P., 446, 278Ciborowski, J., 443, 394Cifarelli, L., 443, 394; 444, 503, 569; 445, 428;

446, 368Cindolo, F., 443, 394; 444, 503, 569; 445, 428;

446, 368Ciocca, C., 444, 539Cirilli, M., 446, 117Cirio, R., 443, 394Cisbani, E., 442, 484; 444, 531Ciuchini, M., 441, 371Ciulli, V., 445, 239Civinini, C., 444, 503, 569; 445, 428; 446, 368Civitarese, O., 446, 93Cizewski, J.A., 446, 22Clare, I., 444, 503, 569; 445, 428; 446, 368Clare, R., 444, 503, 569; 445, 428; 446, 368Clarke, P.E.L., 444, 539Clavelli, L., 446, 153Clay, E., 444, 539Clement, H., 443, 77; 446, 179, 363Clifft, R.W., 445, 239Close, F.E., 446, 342Cloth, P., 443, 394Coarasa, J.A., 442, 326Coboken, K., 443, 394¨Cocks, J.F.C., 443, 69, 82Coffin, J.-P., 446, 191Cogan, J., 446, 117Cohen, I., 444, 539Coignet, G., 444, 503, 569; 445, 428; 446, 368Colaleo, A., 445, 239Colangelo, G., 441, 437Colas, P., 445, 239Coldewey, C., 443, 394Cole, J.E., 443, 394Coles, J., 445, 239Colijn, A.P., 444, 503, 569; 445, 428; 446, 368Colino, N., 444, 503, 569; 445, 428; 446, 368Collaboration, Z.E.U.S., 443, 394Collazuol, G., 445, 439Collins, P., 441, 479; 444, 491; 446, 62, 75Colo, G., 444, 1`Colomer, M., 441, 479; 444, 491Colonna, M., 442, 48Colrain, P., 445, 239Combley, F., 445, 239Comelli, D., 446, 278Conboy, J.E., 444, 539Conforto, G., 445, 439Coniglione, R., 442, 48Conta, C., 445, 439Contalbrigo, M., 445, 439Contardo, D., 444, 516Contin, A., 443, 394Contogouris, A.P., 442, 374Contreras, J.G., 446, 158Contri, R., 441, 479; 444, 491; 446, 62, 75


Cooke, O.C., 444, 539Cooper, G.E., 444, 523Cooper-Sarkar, A.M., 443, 394Copeland, E.J., 443, 97Coppola, N., 443, 394Copty, N.K., 445, 449Coquereaux, R., 443, 221Corbo, G., 441, 371`Corcella, G., 442, 417Corden, M., 445, 239Cormack, C., 443, 394Corradi, M., 443, 394Corriveau, F., 443, 394Cortes, J.L., 444, 451Ćortina, E., 441, 479; 444, 491; 446, 62, 75Cortina-Gil, D., 444, 32Cosme, G., 441, 479; 444, 491; 446, 62, 75Cossutti, F., 441, 479; 444, 491; 446, 62, 75Costa, M., 443, 394Costantini, F., 446, 117Costantini, S., 444, 503, 569; 445, 428; 446,

368Cotorobai, F., 444, 503, 569; 445, 428; 446,

368Cottingham, W.N., 443, 394Cougo-Pinto, M.V., 446, 170Court, G.R., 442, 484; 444, 531Courtat, P., 445, 423Cousins, R., 445, 439Couyoumtzelis, C., 444, 539Cowan, G., 445, 239Coward, D., 446, 117Cowell, J.-H., 441, 479; 444, 491; 446, 62, 75Cowin, R.L., 444, 260Coxe, R.L., 444, 539Coyle, P., 445, 239CPLEAR Collaboration, 444, 38, 43, 52Cramer, J.G., 444, 523Cramer, O., 446, 349Craps, B., 445, 150Crawford, G., 445, 239Crawford, G.I., 442, 43Crawley, H.B., 441, 479; 444, 491; 446, 62, 75Creanza, D., 445, 239Crede, V., 446, 349Ćremaldi, L.M., 445, 449Crennell, D., 441, 479; 444, 491; 446, 62, 75Crepe, S., 446, 117´ Ćrespo, J.M., 445, 239Cristinziani, M., 444, 523Crittenden, J., 443, 394Croni, M., 446, 363¨Crosetti, G., 441, 479; 444, 491; 446, 62, 75Cross, R., 443, 394Crowe, K.M., 446, 349Crystal Barrel Collaboration, 446, 349Csato, P., 444, 523Csikor, F., 441, 354

Csilling, A., 444, 503, 569; 445, 428; 446, 368Csizmadia, P., 443, 21Csorgo, T., 443, 21¨ ˝Cuevas Maestro, J., 441, 479; 444, 491; 446,

62, 75Cuffiani, M., 444, 539Cui, X.Z., 446, 356Cullen, D.M., 443, 69, 82Cundy, D., 446, 117Curtis, L., 445, 239Czellar, S., 441, 479; 444, 491; 446, 62, 75

Dado, S., 444, 539Dagan, S., 443, 394D’Agostini, G., 443, 394Dai, T.S., 444, 503, 569; 445, 428; 446, 368Dal Corso, F., 443, 394D’Alessandro, R., 444, 503, 569; 445, 428;

446, 368Dallavalle, G.M., 444, 539Dalpiaz, P., 446, 117Dalpiaz, P.F., 442, 484; 444, 531Damgaard, G., 441, 479; 444, 491; 446, 62, 75Damgaard, P.H., 445, 366; 446, 175Daniels, D., 445, 439Danielsen, K.M., 446, 342Danielsson, M., 444, 38, 43, 52Dann, J.H., 445, 239Dardo, M., 443, 394Darling, C., 445, 449Das, S.R., 445, 142Dasgupta, A., 445, 279Da Silva, W., 441, 479; 444, 491; 446, 62, 75Daskalakis, G., 445, 239Daugas, J.M., 444, 32Davenport, M., 441, 479; 444, 491; 446, 62, 75Davidson, S., 445, 191Davier, M., 445, 239Davis, A.-C., 446, 238Davis, R., 444, 539De Angelis, A., 441, 479; 444, 491; 446, 62, 75De Asmundis, R., 444, 503, 569; 445, 428;

446, 368De Beer, M., 446, 117De Boer, W., 441, 479; 444, 491; 446, 62, 75De Brabandere, S., 441, 479; 444, 491; 446, 62,

75Debruyne, D., 441, 1Debu, P., 446, 117Decamp, D., 445, 239De Carlos, B., 445, 82De Clercq, C., 441, 479; 444, 491; 446, 62, 75De Conti, C., 444, 14De Falco, A., 444, 111Deffayet, C., 441, 52, 163Deffner, R., 443, 394Degaudenzi, H., 445, 439Degener, T., 446, 349


Deghorain, A., 441, 479; 444, 491; 446, 62, 75Degl’Innocenti, S., 441, 291; 444, 387Degre, A., 444, 503, 569; 445, 428; 446, 368´Deiters, K., 444, 503, 569; 445, 428; 446, 368Dejardin, M., 444, 38, 43, 52De Jong, P., 444, 503, 569; 445, 428; 446, 368De Jong, S., 444, 539De la Cruz, B., 444, 503, 569; 445, 428; 446,

368Delbourgo, R., 446, 332Del Duca, V., 445, 168De Leo, R., 442, 484Delepine, D., 442, 229´Delfino, M., 445, 239Delheij, P.P.J., 444, 531Delius, G.W., 444, 217Della Ricca, G., 441, 479; 444, 491; 446, 62,

75Della Volpe, D., 444, 503, 569; 445, 428; 446,

368Dell’Orso, R., 445, 239De Lotto, B., 441, 479; 444, 491; 446, 62, 75Del Peso, J., 443, 394DELPHI Collaboration, 441, 479; 444, 491;

446, 62, 75Delpierre, P., 441, 479; 444, 491; 446, 62, 75Del Pozo, L.A., 444, 539Del Prete, T., 445, 439Del Zoppo, A., 442, 48Demaria, N., 441, 479; 444, 491; 446, 62, 75De Mello Neto, J.R.T., 445, 449De Min, A., 441, 479; 444, 491; 446, 62, 75De Miranda, J.M., 445, 449Dempsey, J.F., 446, 197Denes, P., 444, 503, 569; 445, 428; 446, 368Deng, C., 441, 285Denig, A., 443, 77Denisenko, K., 445, 449DeNotaristefani, F., 444, 503, 569; 445, 428;

446, 368De Oliviera, F., 444, 32De Palma, M., 445, 239De Pasquale, S., 443, 394De Paula, L., 441, 479; 444, 491; 446, 62, 75Deppe, O., 443, 394De Rafael, E., 443, 255De Roeck, A., 444, 539Derre, J., 444, 38, 43, 52Derrick, M., 443, 394DeSalvo, R., 445, 419De Sanctis, E., 442, 484; 444, 531De Santo, A., 445, 439Desch, K., 444, 539Deschamps, O., 445, 239De Schepper, D., 442, 484; 444, 531Descroix, E., 444, 516Deshpande, A., 443, 394Desler, K., 443, 394

Dessagne, S., 445, 239DESY-Munster Collaboration, 446, 209¨Devaux, A., 444, 516Devenish, R.C.E., 443, 394Devitsin, E., 442, 484; 444, 531De Vivie de Regie, J.-B., 445, 239´De Wit, B., 443, 153De Witt Huberts, P.K.A., 442, 484; 444, 531De Wolf, E., 443, 394Dey, J., 443, 293Dey, M., 443, 293Dhamotharan, S., 445, 239Dhawan, S., 443, 394Diakonos, F.K., 444, 583Dias de Deus, J., 442, 395Dıaz, M.A., 441, 224´Dibon, H., 446, 117Di Cecio, G., 441, 319Di Ciaccio, L., 441, 479; 444, 491; 446, 62, 75Di Diodato, A., 441, 479Diemoz, M., 444, 503, 569; 445, 428; 446, 368Dienes, B., 444, 539Dieperink, A.E.L., 441, 17; 446, 15Dietl, H., 445, 239Dignan, T., 445, 439Dijkstra, H., 441, 479; 444, 491; 446, 62, 75Di Lella, L., 445, 439Dillon, G.K., 444, 260Di Lodovico, F., 444, 503, 569; 445, 428; 446,

368Dimopoulos, K., 446, 238Dimopoulos, S., 441, 96Dine, M., 444, 103Di Nezza, P., 442, 484; 444, 531Ding, H.L., 446, 356Dinh Dang, N., 445, 1Dinius, J., 446, 197Dionisi, C., 444, 503, 569; 445, 428; 446, 368Di Salvo, E., 441, 447Dissertori, G., 445, 239Dittmaier, S., 441, 383Dittmar, M., 444, 503, 569; 445, 428; 446, 368Dixit, M.S., 444, 539Dixon, L., 444, 273Djaoshvili, N., 446, 349Doble, N., 446, 117Do Couto e Silva, E., 445, 439Dolbeau, J., 441, 479; 444, 491; 446, 62, 75Dolgopolov, A.V., 446, 342Dolgoshein, B.A., 443, 394Dolgov, A.D., 442, 82D’Oliveira, A.B., 445, 449Dominguez, A., 444, 503, 569; 445, 428; 446,

368Dong, L.Y., 446, 356Donnelly, I.J., 445, 439Donskov, S.V., 446, 342Dorey, N., 442, 145


Doria, A., 444, 503, 569; 445, 428; 446, 368Dornan, P.J., 445, 239Doroba, K., 441, 479; 444, 491; 446, 62, 75Doser, M., 446, 349Dos Reis, A.C., 445, 449Dosselli, U., 443, 394Dova, M.T., 444, 503, 569; 445, 428; 446, 368Doyle, A.T., 443, 394Draayer, J.P., 442, 7Dracos, M., 441, 479; 444, 491; 446, 62, 75Dragon, N., 446, 314Drapier, O., 444, 516Drees, J., 441, 479; 444, 491; 446, 62, 75Drevermann, H., 445, 239Drews, G., 443, 394Dris, M., 441, 479; 444, 491; 446, 62, 75Drummond, I.T., 442, 279Du, Z.Z., 446, 356Dubbert, J., 444, 539Duchesneau, D., 444, 503, 569; 445, 428; 446,

368Duchovni, E., 444, 539Duckeck, G., 444, 539Duclos, J., 446, 117Dudas, E., 441, 163Duerdoth, I.P., 444, 539Duflot, L., 445, 239Duinker, P., 444, 503, 569; 445, 428; 446, 368Dulinski, Z., 443, 394´Dumarchez, J., 445, 439Dumitru, A., 446, 326Dunn, J., 444, 523Dunnweber, W., 446, 197, 349¨Duperrin, A., 441, 479; 444, 491; 446, 62, 75Duran, I., 444, 503, 569; 445, 428; 446, 368Durand, J.-D., 441, 479; 444, 491; 446, 62, 75Duren, M., 442, 484; 444, 531¨Durkin, L.S., 443, 394Dvoredsky, A., 442, 484; 444, 531Dyring, J., 446, 179

E-811 Collaboration, 445, 419Ealet, A., 444, 38, 43, 52; 445, 239Earl, B.C., 446, 342Eartly, D.P., 445, 419Easo, S., 444, 503, 569; 445, 428; 446, 368Eatough, D., 444, 539Ebersberger, C., 446, 117Ebert, D., 444, 208Echols, R., 444, 103Eckardt, V., 444, 523Ecker, G., 441, 437Eckert, M., 443, 394Eckhardt, F., 444, 523Edgecock, T.R., 445, 239Edmonds, J.K., 443, 394Ehara, M., 445, 14

Ehmanns, A., 446, 349Ehret, R., 441, 479Eigen, G., 441, 479; 444, 491; 446, 62, 75Eisenberg, Y., 443, 394Eisenhardt, S., 443, 394Ekelof, T., 441, 479; 444, 491; 446, 62, 75Ekspong, G., 441, 479; 444, 491; 446, 62, 75Ekstrom, C., 446, 179¨Elbakian, G., 442, 484; 444, 531Eleftheriadis, C., 444, 38, 43, 52Elias, V., 446, 267Ellert, M., 441, 479; 444, 491; 446, 62, 75Ellis, G., 445, 239Ellis, J., 444, 367Ellis, M., 445, 439Ellis, P.J., 443, 63El Mamouni, H., 444, 503, 569; 445, 428; 446,

368Elmer, P., 445, 239Elsing, M., 441, 479; 444, 491; 446, 62, 75Ely, J., 442, 484; 444, 531Emerson, J., 444, 531Emmanuel-Costa, D., 442, 229Engel, J.-P., 441, 479; 444, 491; 446, 62, 75Engelen, J., 443, 394Engelhardt, D., 446, 349Engler, A., 444, 503, 569; 445, 428; 446, 368En’yo, H., 444, 267Epperson, D., 443, 394Eppling, F.J., 444, 503, 569; 445, 428; 446, 368Erhan, S., 445, 455Erhardt, A., 446, 363Ermolov, P.F., 443, 394Erne, F.C., 444, 503, 569; 445, 428; 446, 368´Ernst, C., 442, 443Ernst, J., 444, 555; 445, 20Erzen, B., 441, 479; 444, 491; 446, 62, 75Escribano, R., 444, 397Eskreys, A., 443, 394Espagnon, B., 444, 516Espirito Santo, M., 441, 479; 444, 491; 446, 62,

75Estabrooks, P.G., 444, 539Etzion, E., 444, 539Evans, D., 446, 342Evans, H.G., 444, 539Extermann, P., 444, 503, 569; 446, 368Eyal, G., 441, 191

Fabbri, F., 444, 539Fabbro, B., 445, 239Fabre, M., 444, 503, 569; 446, 368Fabre, P.E.M., 445, 428Faccini, R., 444, 503, 569; 445, 428; 446, 368Faessler, A., 442, 203, 443, 7Faessler, M.A., 446, 349Fagerstroem, C.-P., 443, 394


Faıf, G., 445, 239¨Falagan, M.A., 444, 503, 569; 445, 428; 446,

368Falciano, S., 444, 503, 569; 445, 428; 446, 368Faldt, G., 445, 423¨Falk, E., 441, 479; 444, 491; 446, 62, 75Falk, T., 444, 367, 427Falvard, A., 445, 239Fanourakis, G., 441, 479; 444, 491; 446, 62, 75Fantechi, R., 446, 117Fanti, M., 444, 539Fanti, V., 446, 117Fantoni, A., 442, 484; 444, 531Fantoni, S., 446, 99Faraggi, A.E., 445, 77, 357Faravel, L., 444, 38, 43, 52Farchioni, F., 443, 214Fargeix, J., 444, 516Farina, C., 446, 170Fassouliotis, D., 441, 479; 444, 491; 446, 62,

75Faust, A.A., 444, 539Favara, A., 444, 503, 569; 445, 428; 446, 368Fay, J., 444, 503, 569; 445, 428; 446, 368Fayard, L., 446, 117Fayot, J., 441, 479; 444, 491; 446, 62, 75Fazio, T., 445, 439Fechtchenko, A., 442, 484; 444, 531Fedin, O., 444, 503, 569; 445, 428; 446, 368Feindt, M., 441, 479; 444, 491; 446, 62, 75Feinstein, A., 441, 40Felcini, M., 444, 503, 569; 445, 428; 446, 368Feldman, G.J., 445, 439Fenyuk, A., 441, 479; 444, 491; 446, 62Ferdi, C., 445, 239Ferguson, D.P.S., 445, 239Ferguson, M.I., 444, 523Ferguson, T., 444, 503, 569; 445, 428; 446, 368Fermilab E791 Collaboration, 445, 449Fernandez, A., 445, 449Fernandez-Bosman, M., 445, 239Fernandez, E., 445, 239Fernandez, J.P., 443, 394´Fernandez, L.A., 441, 330´Fernando, S., 445, 52Ferrante, I., 445, 239Ferrari, F., 444, 167Ferrari, P., 441, 479; 444, 491; 446, 62, 75Ferrari, R., 445, 439Ferreira, R., 444, 516Ferrer, A., 441, 479; 444, 491; 446, 62, 75Ferrer, F., 446, 111Ferrer, M.L., 444, 111Ferrer-Ribas, E., 441, 479; 444, 491; 446, 62,

75Ferrere, D., 445, 439`Ferrero, M.I., 443, 394Ferroni, F., 444, 503, 569; 445, 428; 446, 368

Ferstl, M., 442, 484; 444, 531Fesefeldt, H., 444, 503, 569; 445, 428; 446, 368Fetscher, W., 444, 38, 43, 52Feverati, G., 444, 442Fiandrini, E., 444, 503, 569; 445, 428; 446, 368Fichet, S., 441, 479; 444, 491; 446, 62, 75Fick, D., 442, 484; 444, 531Fidecaro, M., 444, 38, 43, 52Fiedler, F., 444, 539Fiedler, K., 442, 484; 444, 531Field, J.H., 444, 503, 569; 445, 428; 446, 368Fierro, M., 444, 539Figiel, J., 443, 394Figueiredo, J.M., 445, 94Fil’chenkov, M.L., 441, 34Filges, D., 443, 394Filipcic, A., 444, 38, 43, 52ˇ ˇFilipov, G., 445, 14Filippone, B.W., 442, 484; 444, 531Filthaut, F., 444, 503, 569; 445, 428; 446, 368Finch, A.J., 445, 239Finocchiaro, P., 442, 48Fiorentini, G., 441, 291; 444, 387Firestone, A., 441, 479; 444, 491; 446, 62, 75Fischer, G., 446, 117Fischer, H., 442, 484; 444, 531Fischer, H.G., 444, 523Fischer, P.-A., 441, 479; 444, 491; 446, 62, 75Fisher, P.H., 444, 503, 569; 445, 428; 446, 368Fisk, I., 444, 503, 569; 445, 428; 446, 368Fissum, K.G., 442, 43Flagmeyer, U., 441, 479; 444, 491; 446, 62, 75Flaminio, V., 445, 439Fleck, I., 444, 539Fleuret, F., 444, 516Flierl, D., 444, 523Flohr, M.A.I., 444, 179Floratos, E., 444, 190Floratos, E.G., 445, 69Focardi, E., 445, 239Fodor, Z., 441, 354; 444, 523Foeth, H., 441, 479; 444, 491; 446, 62, 75Fohl, K., 443, 77; 446, 363¨Foka, P., 444, 523Fokitis, E., 441, 479; 444, 491; 446, 62, 75Folman, R., 444, 539Fontanelli, F., 441, 479; 444, 491; 446, 62, 75Foot, R., 443, 185Force, P., 444, 516Forconi, G., 444, 503, 569; 445, 428; 446, 368Forgacs, P., 441, 275´Formica, A., 446, 117Fort, H., 444, 174Fortune, H.T., 444, 531Forty, R.W., 445, 239Foss, J., 445, 239Foster, B., 443, 394Foster, F., 445, 239


Fouchez, D., 445, 239Foudas, C., 443, 394Fox, B., 442, 484; 444, 531Fox, G.F., 445, 449Fox, H., 446, 117Frabetti, P.L., 446, 117Frabetti, S., 442, 484; 444, 531Francis, D., 444, 38, 43, 52Franco, E., 441, 371Franek, B., 441, 479; 444, 491; 446, 62, 75Frank, M., 445, 239Frankel, S., 441, 425Fransson, K., 446, 179Franz, J., 442, 484; 444, 531Frascaria, N., 442, 48Fraternali, M., 445, 439Frati, W., 441, 425Fredj, L., 444, 503, 516, 569; 445, 428; 446,

368Freer, M., 444, 260French, B.R., 446, 342Frere, J.-M., 444, 397`Freudenreich, K., 444, 503, 569; 445, 428; 446,

368Freund, P., 444, 523Fricke, U., 443, 394Fries, R.J., 443, 40Friese, V., 444, 523Frisken, W.R., 443, 394Frodesen, A.G., 441, 479; 444, 491; 446, 62, 75Froggatt, C.D., 446, 256Frolov, S.A., 441, 173Fruhwirth, R., 441, 479; 444, 491; 446, 62, 75Frullani, S., 442, 484; 444, 531Fry, J., 444, 38, 43, 52Fuchs, J., 441, 141Fuchs, M., 444, 523Fukuda, T., 444, 267Fukui, S., 445, 14Fukushima, Y., 443, 409Fulda-Quenzer, F., 441, 479; 444, 491; 446, 62,

75Funahashi, H., 444, 267Funk, M.-A., 442, 484; 444, 531Funk, W., 446, 117Furetta, C., 444, 503, 569; 445, 428; 446, 368Furtjes, A., 444, 539¨Furuno, K., 442, 53Fusayasu, T., 443, 394Fuster, J., 441, 479; 444, 491; 446, 62, 75Futyan, D.I., 444, 539

Gabathuler, E., 444, 38, 43, 52Gaberdiel, M.R., 441, 133Gabler, F., 444, 523Gacougnolle, R., 445, 423Gadaj, T., 443, 394Gaff, S., 446, 197

Gagnon, P., 444, 539; 445, 449Gago, J., 444, 516Gagunashvili, N.D., 442, 484; 444, 531Gaillard, J-M., 445, 439Gal, J., 444, 523Galaktionov, Yu., 444, 503, 569; 445, 428; 446,

368Galante, A., 444, 421Galea, R., 443, 394Galeao, A.P., 444, 14˜Galindo-Uribarri, A., 443, 89Gallo, E., 443, 394Galloni, A., 441, 479; 444, 491; 446, 62, 75Galumian, P., 444, 531; 445, 439Gamba, D., 441, 479; 444, 491; 446, 62, 75Gamblin, S., 441, 479; 444, 491; 446, 62, 75Gamboa, J., 444, 451Gamet, R., 444, 38, 43, 52Gandelman, M., 441, 479; 444, 491; 446, 62,

75Gangler, E., 445, 439Ganguli, S.N., 444, 503, 569; 445, 428; 446,

368Ganis, G., 445, 239Ganz, R., 444, 523Gao, C.S., 446, 356Gao, H., 442, 484; 444, 531Gao, M.L., 446, 356Gao, S.Q., 446, 356Gao, Y., 445, 239Garattini, R., 446, 135Garber, Y., 442, 484; 444, 531¨Garcia-Abia, P., 444, 503, 569; 445, 428; 446,

368Garcıa, A.O., 443, 221´Garcia, C., 441, 479; 444, 491; 446, 62, 75Garcıa, G., 443, 394´Garcia, J., 441, 479; 444, 491; 446, 62, 75Garcia, J.A., 441, 198Gardestig, A., 445, 423˚Garfagnini, A., 443, 394Garibaldi, F., 442, 484; 444, 531Garrido, L., 445, 239Gary, J.W., 444, 539Gascon, J., 444, 539Gascon-Shotkin, S.M., 444, 539Gaspar, C., 441, 479; 444, 491; 446, 62, 75Gaspar, M., 441, 479; 444, 491; 446, 62, 75Gasparini, U., 441, 479; 444, 491; 446, 62, 75Gataullin, M., 444, 503, 569; 445, 428; 446,

368Gatignon, L., 446, 117Gau, S.S., 444, 503, 569; 445, 428; 446, 368Gauzzi, P., 444, 111Gavillet, Ph., 441, 479; 444, 491; 446, 62, 75Gavrilov, G., 442, 484; 444, 531Gay, P., 445, 239Gaycken, G., 444, 539


Gazdzicki, M., 444, 523´Gazis, E.N., 441, 479; 444, 491; 446, 62, 75Geich-Gimbel, C., 444, 539Geiger, P., 442, 484; 444, 531Geiser, A., 444, 358; 445, 439Geissel, H., 444, 32Geist, W., 444, 523Gelao, G., 445, 239Gelbke, C.K., 446, 197Gele, D., 441, 479; 444, 491; 446, 62, 75Gendner, N., 443, 394Genovese, M., 442, 398Gentile, S., 444, 503, 569; 445, 428; 446, 368Georgiopoulos, C., 445, 239Geppert, D., 445, 439Gerber, H.-J., 444, 38, 43, 52Gerber, J.-P., 441, 479Gerdyukov, L., 446, 62, 75Gerschel, C., 444, 516Gevaert, A., 444, 397Geweniger, C., 445, 239Gharib, T., 444, 231Gharibyan, V., 442, 484; 444, 531Gheordanescu, N., 444, 503, 569; 445, 428;

446, 368Gherghetta, T., 446, 28Ghete, V.M., 445, 239Ghez, P., 445, 239Ghilencea, D., 442, 165Ghodbane, N., 441, 479; 444, 491; 446, 62, 75Giacconi, P., 441, 257Giacomelli, G., 444, 539Giacomelli, P., 444, 539Giagu, S., 444, 503, 569; 445, 428; 446, 368Gialas, I., 443, 394Giannini, G., 445, 239Gianoli, A., 446, 117Gianotti, F., 445, 239Giarritta, P., 446, 349Giassi, A., 445, 239Gibbons, G.W., 443, 138Gibbs, W.R., 444, 252Gibin, D., 445, 439Gibson, B.F., 444, 252Gibson, M., 446, 256Gibson, V., 444, 539Gibson, W.R., 444, 539Giehl, I., 445, 239Gil, I., 441, 479; 444, 491; 446, 62, 75Gillibert, A., 442, 48Gilmore, J., 443, 394Gingrich, D.M., 444, 539Ginsburg, C.M., 443, 394Giordjian, V., 442, 484Girone, M., 445, 239Girtler, P., 445, 239Giudice, G.F., 446, 28Giudici, S., 446, 117

Giunti, C., 444, 379Giusti, P., 443, 394Gladilin, L.K., 443, 394Gładysz, E., 444, 523Glander, K.H., 444, 555; 445, 20Glashow, S.L., 445, 412Glasman, C., 443, 394Glege, F., 441, 479; 444, 491; 446, 62, 75Glenzinski, D., 444, 539Gluck, M., 443, 298¨Gninenko, S., 445, 439Go, A., 444, 38, 43, 52Gobbo, B., 445, 239Gobel, C., 445, 449Godley, A., 445, 439Goebel, F., 443, 394Goers, S., 444, 555; 445, 20Gokieli, R., 441, 479; 444, 491; 446, 62, 75Goldberg, H., 444, 68Goldberg, J., 444, 539Goldfarb, S., 444, 503, 569; 445, 428; 446, 368Goldstein, J., 444, 503, 569; 445, 428; 446, 368Golendoukhin, A., 444, 531Golendukhin, A., 442, 484Golob, B., 441, 479; 444, 491; 446, 62, 75Golubkov, Yu.A., 443, 394Gomez-Cadenas, J-J., 445, 439Gomez-Ceballos, G., 444, 491; 446, 62, 75Gomis, J., 443, 147Goncalves, P., 441, 479; 444, 491; 446, 62, 75Gong, Z.F., 444, 503, 569; 445, 428; 446, 368Gonidec, A., 446, 117Gonzalez-Arroyo, A., 442, 273´Gonzalez, S., 445, 239´Gonzalez Caballero, I., 441, 479; 444, 491; 446,

62, 75Gonzalez Felipe, R., 442, 229´Goodsir, S., 445, 239Gopal, G., 441, 479; 444, 491; 446, 62, 75Gorini, B., 446, 117Gorn, L., 441, 479; 444, 491; 446, 62, 75Gorn, W., 444, 539Gorodetzky, P., 444, 516Gorska, M., 444, 32´Gorski, M., 441, 479; 444, 491; 446, 62, 75Gosdzinsky, P., 441, 265Goßling, C., 445, 439¨Gosset, J., 445, 439Goto, Y., 444, 267Gottlicher, P., 443, 394¨Gouanere, M., 445, 439`Gougas, A., 444, 503, 569; 445, 428; 446, 368Gounder, K., 445, 449Gourio, D., 446, 191Gouz, Yu., 441, 479; 444, 491; 446, 62, 75Govi, G., 446, 117Goy, C., 445, 239Grabosch, H.J., 443, 394


Gracco, V., 441, 479; 444, 491; 446, 62, 75Graciani, R., 443, 394Grafstrom, P., 446, 117¨Grahl, J., 441, 479; 444, 491; 446, 62, 75Grandi, C., 444, 539Granier-De-Cassagnac, R., 446, 117Grant, A., 445, 439Grater, J., 443, 77¨Gratta, G., 444, 503, 569; 445, 428; 446, 368Grauges, E., 445, 239`Graw, G., 442, 484; 444, 531Grawe, H., 444, 32Gray, J.P., 444, 103Graziani, E., 441, 479; 444, 491; 446, 62, 75Graziani, G., 445, 439; 446, 117Grazzini, M., 446, 143Grebeniouk, O., 442, 484; 444, 531Grebeniuk, M.A., 442, 125Grebieszkow, J., 444, 523Green, C., 441, 479; 444, 491; 446, 62, 75Green, M.G., 445, 239Green, P.W., 442, 484; 444, 531Greenhalgh, B., 444, 260Greeniaus, L.G., 442, 484; 444, 531Greening, T.C., 445, 239Greenlees, P.T., 443, 69, 82Gregorio, A., 445, 239Greiner, C., 446, 191Greiner, W., 442, 443Grifols, J.A., 446, 111Griguolo, L., 443, 325Grimm, H.-J., 444, 491; 446, 62, 75Gris, P., 441, 479; 444, 491; 446, 62, 75Grispos, G., 442, 374Grivaz, J.-F., 445, 239Groot Nibbelink, S., 442, 185Große-Knetter, J., 443, 394Gross, E., 444, 539Grosshauser, C., 442, 484; 444, 531Grossiord, J.Y., 444, 516Grothe, M., 443, 394Grozin, A.G., 445, 165Gruenewald, M.W., 444, 503, 569; 445, 428;

446, 368Grunhaus, J., 444, 539Grupen, C., 445, 239Gruwe, M., 444, 539´Grzelak, G., 443, 394Grzelak, K., 441, 479; 444, 491; 446, 62, 75Grzywacz, R., 444, 32Gu, J.H., 446, 356Gu, S.D., 446, 356Gu, W.X., 446, 356Gu, Y.F., 446, 356Guadagnini, E., 441, 60Guasch, J., 442, 326Gubarev, F.V., 443, 244Guglielmi, A., 445, 439

Guichard, A., 444, 516Guicheney, C., 445, 239Guidal, M., 442, 484Guidal, M.G., 444, 531Guillaud, J.P., 444, 516Gunion, J.F., 444, 136Gunther, J., 444, 523¨Gunther, M., 441, 479; 444, 491; 446, 62, 75Guo, Y.N., 446, 356Gupta, V.K., 444, 503, 569; 445, 428; 446, 368Gurtu, A., 444, 503, 569; 445, 428; 446, 368Guse, B., 445, 20Guss, C., 445, 419Gustafsson, L., 446, 179Gutay, L.J., 444, 503, 569; 445, 428; 446, 368Gute, A., 442, 484; 444, 531Gutperle, M., 445, 296Guy, J., 441, 479; 444, 491; 446, 62, 75Gyulassy, M., 442, 1, 443, 45Gyurjyan, V., 444, 531, 531Gyurusi, J., 441, 275¨ ¨

Haas, D., 444, 503, 569; 445, 428; 446, 368Haas, J.P., 444, 531Haas, K.-M., 445, 20Haas, T., 443, 394Hadad, M., 442, 74Haddock, R.P., 446, 349Haeberli, W., 442, 484; 444, 531Haggstrom, S., 446, 179¨ ¨Hagner, C., 445, 439Hahn, F., 441, 479; 444, 491; 446, 62, 75Hahn, S., 441, 479; 444, 491; 446, 62, 75Haidenbauer, J., 444, 25Haider, S., 441, 479; 444, 491; 446, 62, 75Hain, W., 443, 394Hall, L.J., 445, 407Hall-Wilton, R., 443, 394Halley, A.W., 445, 239Hallgren, A., 441, 479; 444, 491; 446, 62, 75Halling, A.M., 445, 449Ham, S.W., 441, 215Hamacher, K., 441, 479; 444, 491; 446, 62, 75Hamatsu, R., 443, 394Han, S.W., 446, 356Han, Y., 446, 356Hanhart, C., 444, 25Hanke, P., 445, 239Hanna, D.S., 443, 394Hannappel, J., 444, 555; 445, 20Hansen, J.-O., 442, 484; 444, 531Hansen, J.B., 445, 239Hansen, J.D., 445, 239Hansen, J.R., 445, 239Hansen, K., 442, 43Hansen, P.H., 445, 239Hanson, G.G., 444, 539Hansper, G., 445, 239


Hansroul, M., 444, 539Hapke, M., 444, 539Harder, K., 444, 539Harder, M.K., 443, 82Hargrove, C.K., 444, 539Harnew, N., 443, 394Haroutunian, R., 444, 516Harris, F.J., 441, 479; 444, 491; 446, 62, 75Harris, J.W., 444, 523Hart, A., 441, 319Hart, J.C., 443, 394Hartmann, B., 441, 77; 444, 503, 569; 445,

428; 446, 368Hartmann, C., 444, 539Hartmann, H., 443, 394Hartmann, J., 443, 394Hartner, G.F., 443, 394Harvey, J., 445, 239Hasan, A., 444, 503, 569; 445, 428; 446, 368Hasch, D., 442, 484; 444, 531Hasegawa, S., 445, 14Hasegawa, T., 445, 14Haselden, A., 444, 38, 43, 52Hasell, D., 443, 394Hashimoto, M., 441, 389Hass, M., 446, 22Hata, K., 442, 53Hattori, T., 445, 106Hatzifotiadou, D., 444, 503, 569; 445, 428;

446, 368Hauschild, M., 444, 539Hausser, O., 444, 531¨Hawkes, C.M., 444, 539Hawkings, R., 444, 539Hay, B., 446, 117Hayakawa, T., 442, 53Hayes, M.E., 443, 394Hayes, O.J., 445, 239Haymaker, R.W., 441, 319Hayman, P.J., 444, 38, 43, 52He, J., 446, 356He, J.T., 446, 356He, K.L., 446, 356He, M., 446, 356He, X.-G., 444, 75; 445, 344Heaphy, E.A., 443, 394Heath, G.P., 443, 394Heath, H.F., 443, 394Hebbeker, T., 444, 503, 569; 445, 428; 446,

368Hebbel, K., 443, 394Hedberg, V., 441, 479; 444, 491; 446, 62, 75Hegyi, S., 444, 523Heinloth, K., 443, 394; 445, 20Heinsius, F.H., 442, 484; 444, 531; 446, 349Heinz, L., 443, 394Heinzelmann, M., 446, 349Heising, S., 441, 479; 444, 491; 446, 62, 75

Heitger, J., 441, 354Helariutta, K., 443, 69, 82Heller, U.M., 445, 366Hellstrom, M., 444, 32¨Hemingway, R.J., 444, 539Hemmi, Y., 443, 409Henderson, R., 444, 531Henkel, T., 444, 523Henkes, T., 444, 531Henoch, M., 442, 484; 444, 531Henrard, P., 445, 239Henry-Couannier, F., 444, 38, 43, 52Hepp, V., 445, 239Herbstrith, A., 446, 349HERMES Collaboration, 442, 484; 444, 531Hernandez, J.J., 441, 479; 444, 491; 446, 62, 75Hernandez, J.M., 443, 394´Hernando, J., 445, 439Herndon, M., 444, 539Herquet, P., 441, 479; 444, 491; 446, 62, 75Herr, H., 441, 479; 444, 491; 446, 62, 75Herrera, G., 445, 449Herrera-Siklody, P., 442, 359´Herten, G., 444, 539Hertenberger, R., 442, 484; 444, 531Herve, A., 444, 503, 569; 445, 428; 446, 368´Herz, M., 446, 349Hessey, N.P., 446, 349Hessing, T.L., 441, 479; 444, 491; 446, 62, 75Heuer, R.D., 444, 539Heusch, C., 443, 394Heuser, J.-M., 441, 479; 444, 491; 446, 62, 75Heusse, Ph., 445, 239Hibou, F., 445, 423Hidas, P., 444, 503, 569; 445, 428; 446, 349,

368Higashi, A., 444, 267Higon, E., 441, 479; 444, 491; 446, 62, 75Hikami, K., 443, 233Hildreth, M.D., 444, 539Hilger, E., 443, 394Hill, J.C., 444, 539Hill, L.A., 444, 523Hillier, S.J., 444, 539Hino, T., 446, 342Hip, I., 443, 214Hirose, T., 443, 394Hirschfelder, J., 444, 503, 569; 445, 428; 446,

368Hisano, J., 445, 316Ho, J., 445, 27Hobson, P.R., 444, 539Hochman, D., 443, 394Hocker, A., 444, 539Hofer, H., 444, 503, 569; 445, 428; 446, 368Hoffmann, C., 445, 239Hoistad, B., 446, 179¨Holden, J., 446, 22


Holder, M., 446, 117Holland, K., 443, 338Hollander, R.W., 444, 38, 43, 52Holler, Y., 442, 484; 444, 531Hollik, W., 442, 326Holm, U., 443, 394Holmgren, S.-O., 441, 479; 444, 491; 446, 62,

75Holt, P.J., 441, 479; 444, 491; 446, 62, 75Holt, R.J., 442, 484; 444, 531Holthuizen, D., 441, 479; 444, 491; 446, 62, 75Holtzhaußen, C., 446, 349Homer, R.J., 444, 539Homma, K., 443, 394Hong, D.K., 445, 36Hong, S.J., 443, 394Hong Tuan, R., 445, 100Honma, A.K., 444, 539Honscheid, K., 445, 20Hoorani, H., 444, 503, 569; 445, 428; 446, 368Hoorelbeke, S., 441, 479; 444, 491; 446, 62, 75Hoprich, W., 442, 484; 444, 531Horikawa, N., 445, 14Horowitz, C.J., 443, 58Horvath, D., 444, 539´Hossain, K.R., 444, 539Hou, S.R., 444, 503, 569; 445, 428; 446, 368Hou, W.-S., 445, 344Houlden, M., 441, 479; 444, 491; 446, 62, 75Howard, R., 444, 539Howe, P.S., 444, 341Howell, C.R., 444, 252Howell, G., 443, 394Hristov, P., 446, 117Hrubec, J., 441, 479; 444, 491; 446, 62, 75Hu, G., 444, 503, 569; 445, 428; 446, 368Hu, G.Y., 446, 356Hu, H., 445, 239Hu, H.M., 446, 356Hu, J.L., 446, 356Hu, Q.H., 446, 356Hu, T., 446, 356Hu, X.Q., 446, 356Huang, C.-G., 441, 285Huang, C.-S., 442, 209Huang, H.-W., 441, 396Huang, M.J., 446, 197Huang, X., 445, 239Huang, Y.Z., 446, 356Hubbard, D., 445, 439Huehn, T., 445, 239Huet, K., 441, 479; 444, 491; 446, 62, 75Hufner, J., 445, 223¨Hughes, G., 445, 239Hughes, G.J., 446, 62, 75Hughes, V.W., 443, 394Huitu, K., 445, 394; 446, 285Hultqvist, K., 441, 479; 444, 491; 446, 62, 75

Hummler, H., 444, 523¨Huntemeyer, P., 444, 539¨Hurst, P., 445, 439Hurvits, G., 445, 449Hussein, A., 444, 252Huttmann, K., 445, 239; 446, 349¨Hyett, N., 445, 439Hyun, S., 441, 116

Iacobucci, G., 443, 394Iacopini, E., 445, 439; 446, 117Iannotti, L., 443, 394Iaselli, G., 445, 239Iashvili, I., 444, 503, 569; 445, 428; 446, 368Iconomidou-Fayard, L., 446, 117Ieiri, M., 444, 267Iga, Y., 443, 394Igo, G., 444, 523Igo-Kemenes, P., 444, 539Ihssen, H., 442, 484; 444, 531Iijima, T., 444, 267Iinuma, M., 444, 267Ikegami, Y., 443, 409Ilgenfritz, E.-M., 443, 244Imai, K., 444, 267Imbo, T.D., 441, 468Imrie, D.C., 444, 539Inaba, S., 446, 342Inuzuka, M., 443, 394Inyakin, A.V., 446, 342Iodice, M., 442, 484; 444, 531Ireland, D.G., 442, 43Irmscher, D., 444, 523Isaksson, L., 442, 43Ishida, T., 446, 342Ishii, K., 444, 539Ishii, T., 443, 394Ishiyama, H., 442, 53Isupov, A., 445, 14Itakura, K., 442, 217Ito, K., 441, 155Itow, Y., 444, 267Ivanov, D.Yu., 442, 453Ivanov, E., 445, 60Ivanov, M.A., 442, 435Ivashchuk, V.D., 442, 125Iwasa, N., 444, 32Iwata, T., 445, 14Izotov, A., 442, 484; 444, 531

Jacholkowska, A., 445, 239Jacholkowski, A., 446, 342Jack, I., 443, 177Jackson, H.E., 442, 484; 444, 531Jackson, J.N., 441, 479; 444, 491; 446, 62, 75Jacob, F.R., 444, 539Jacobs, P., 444, 523Jacobsen, T., 446, 342


Jacobsson, R., 441, 479; 444, 491; 446, 62, 75Jaffe, D.E., 445, 239Jahnen, T., 445, 20Jaimungal, S., 441, 147Jakob, H.-P., 443, 394Jakobs, K., 445, 239Jakovac, A., 446, 203´Jalocha, P., 441, 479; 444, 491; 446, 62, 75James, C., 445, 449Jaminon, M., 443, 33Jamnik, D., 446, 349Janas, Z., 444, 32Janik, R., 441, 479; 444, 491; 446, 62, 75Janik, R.A., 442, 300; 446, 9Janot, P., 445, 239Janssens, R.V.F., 446, 22Jarlskog, Ch., 441, 479; 444, 491; 446, 62, 75Jarlskog, G., 441, 479; 444, 491; 446, 62, 75Jarry, P., 441, 479; 444, 491; 446, 62, 75Jawahery, A., 444, 539Jean-Marie, B., 441, 479; 444, 491; 446, 62, 75Jeitler, M., 446, 117Jelen, K., 443, 394´Jeoung, H.Y., 443, 394Jeremie, H., 444, 539Jezequel, S., 445, 239Jgoun, A., 442, 484; 444, 531Jiang, C.H., 446, 356Jimack, M., 444, 539Jin, B.N., 444, 503, 569; 445, 428; 446, 368Jin, H.Q., 443, 89Jin, S., 445, 239Jin, Y., 446, 356Jing, Z., 443, 394Jinghua, F., 444, 563Joepen, N., 444, 555Johanson, J., 446, 179Johansson, A., 446, 179Johansson, E.K., 441, 479; 444, 491; 446, 62,

75Johansson, T., 446, 179Johnson, K.F., 443, 394Jon-And, K., 444, 38, 43, 52Jones, C., 444, 531Jones, C.R., 444, 539Jones, D.R.T., 443, 177Jones, G.T., 446, 342Jones, L.W., 444, 503, 569; 445, 428; 446, 368Jones, P., 443, 82Jones, P.G., 444, 523Jones, P.M., 443, 69Jones, R.W.L., 445, 239Jones, T.W., 443, 394Jonsson, P., 441, 479; 444, 491; 446, 62, 75Jopen, N., 445, 20¨Joram, C., 441, 479; 444, 491; 446, 62, 75Josa-Mutuberria, I., 444, 503, 569; 445, 428;

446, 368

Joseph, C., 445, 439Jost, B., 445, 239Jostlein, H., 445, 419Jouan, D., 444, 516Jousset, J., 445, 239Jovanovic, P., 444, 539Juget, F., 445, 439Juillot, P., 441, 479; 444, 491; 446, 62, 75Julin, R., 443, 69, 82Jumatsu, T., 442, 53Junghans, A.R., 444, 32Jungst, H., 445, 20¨Jungst, H.G., 444, 555¨Junk, T.R., 444, 539Juste, A., 445, 239Juutinen, S., 443, 69, 82

Kadija, K., 444, 523Kado, M., 445, 239Kageya, T., 445, 14Kaiser, R., 442, 484; 444, 531Kalinowsky, H., 444, 555; 445, 20; 446, 349Kallosh, R., 443, 143Kalter, A., 446, 117Kammel, P., 446, 349Kammle, B., 446, 349¨Kamrad, D., 444, 503, 569; 445, 428; 446, 368Kananov, S., 443, 394Kang, G., 445, 27Kang, K., 442, 249Kang, S.K., 442, 249Kankaanpaa, H., 443, 69, 82¨¨Kantar, M., 443, 359Kappes, A., 443, 394Kapusta, F., 441, 479; 444, 491; 446, 62, 75Kapusta, J.I., 443, 63Kapustinsky, J.S., 444, 503, 569; 445, 428;

446, 368Karafasoulis, K., 441, 479; 444, 491; 446, 62,

75Karat, E., 445, 337Karch, A., 441, 235Karlen, D., 444, 539Karny, M., 444, 32Karsch, F., 442, 291Karshon, U., 443, 394Kartvelishvili, V., 444, 539Kasemann, M., 443, 394Kashirin, V., 445, 14Kasper, P.A., 445, 449Kato, M., 442, 53Katsanevas, S., 441, 479; 444, 491; 446, 62, 75Katsoufis, E.C., 441, 479; 444, 491; 446, 62, 75Katz, U.F., 443, 394Kaur, M., 444, 503, 569; 445, 428; 446, 368Kaw, P.K., 446, 104Kawagoe, K., 444, 539Kawamoto, T., 444, 539


Kawamura, Y., 446, 228Kawano, M., 445, 14Kayal, P.I., 444, 539Kcira, D., 443, 394Ke, Z.J., 446, 356Keeler, R.K., 444, 539Keenan, A., 443, 69, 82Kehagias, A., 444, 190; 445, 69KEK-PS E224 Collaboration, 444, 267Kekelidze, V., 446, 117Kellogg, R.G., 444, 539Kelly, M.S., 445, 239Kennedy, B.W., 444, 539Keranen, R., 441, 479; 444, 491; 446, 62, 75Kerger, R., 443, 394Kernan, P.J., 445, 412Kersevan, B.P., 444, 491; 446, 62, 75Kesseler, G., 446, 117Kettle, P.-R., 444, 38, 43, 52Kettunen, H., 443, 69, 82Keung, W.-Y., 444, 142Khakzad, M., 443, 394Khan, R.A., 444, 503, 569; 445, 428; 446, 368Khaustov, G.V., 446, 342Khein, L.A., 443, 394Khokhlov, Yu., 441, 479Khomenko, B.A., 441, 479; 444, 491; 446, 62,

75Khovanski, N.N., 441, 479; 444, 491; 446, 62,

75Khoze, V.V., 442, 145Khrenov, A., 445, 14Khriplovich, I.B., 444, 98Kienzle-Focacci, M.N., 444, 503, 569; 445,

428; 446, 368Kiiskinen, A., 441, 479; 444, 491; 446, 62, 75Kilian, K., 446, 179Kim, B.R., 441, 215Kim, C.L., 443, 394Kim, C.S., 441, 410Kim, C.W., 444, 204Kim, D., 444, 503, 569; 445, 428; 446, 368Kim, D.H., 444, 503, 569; 445, 428; 446, 368Kim, H.Y., 445, 239Kim, J.E., 442, 249Kim, J.K., 444, 503, 569; 445, 428; 446, 368Kim, J.Y., 443, 394Kim, N.J., 441, 83Kim, S.C., 444, 503, 569; 445, 428; 446, 368Kinashi, T., 446, 342King, B., 441, 479; 444, 491; 446, 62, 75King, B.A., 441, 468King, S.F., 442, 68; 445, 191King, S.L., 443, 69, 82Kinney, E., 442, 484; 444, 531Kinnison, W.W., 444, 503, 569; 445, 428; 446,

368Kinson, J.B., 446, 342

Kinvig, A., 446, 62, 75Kirch, U., 444, 555; 445, 20Kirchner, R., 446, 209Kirk, A., 446, 342Kirkby, A., 444, 503, 569; 445, 428; 446, 368Kirkby, D., 444, 503, 569; 445, 428; 446, 368Kirkby, J., 444, 503, 569; 445, 428; 446, 368Kirsanov, M., 445, 439Kirsch, M., 444, 531Kisiel, J., 446, 349Kisielewska, D., 443, 394Kiss, D., 444, 503, 569; 445, 428; 446, 368Kisselev, A., 442, 484; 444, 531Kisslinger, L.S., 445, 271Kitamura, S., 443, 394Kitching, P., 442, 484; 444, 531Kittel, W., 444, 503, 569; 445, 428; 446, 368Kivel, N., 443, 308Kjaer, N.J., 441, 479; 444, 491; 446, 62, 75Klanner, R., 443, 394Klapp, O., 441, 479; 444, 491; 446, 62, 75Klein, F., 444, 555; 445, 20Klein, F.-J., 444, 555Klein, F.J., 445, 20Klein, H., 441, 479; 444, 491; 446, 62, 75Kleinknecht, K., 445, 239; 446, 117Klempt, E., 444, 555; 445, 20; 446, 349Klempt, W., 446, 342Klier, A., 444, 539Klimek, K., 443, 394Klimentov, A., 444, 503, 569; 445, 428; 446,

368Klimov, O., 445, 439Kluberg, L., 444, 516Kluge, E.E., 445, 239Kluge, W., 443, 77Kluit, P., 441, 479; 444, 491; 446, 62, 75Kluth, S., 444, 539Knaepen, B., 441, 198Knecht, M., 443, 255Kneringer, E., 445, 239Knoblauch, D., 441, 479Knowles, I.G., 446, 117Ko, C.M., 444, 237; 445, 265Ko, P., 442, 249Ko, U., 443, 394Kobayashi, H., 442, 484; 444, 531Kobayashi, T., 441, 235; 442, 192; 444, 539Kobel, M., 444, 539Koch, H., 446, 349Koch, N., 442, 484; 444, 531Koch, U., 446, 117Koch, W., 443, 394Koetke, D.S., 444, 539Koffeman, E., 443, 394Kogan, I.I., 442, 136Kokkas, P., 444, 38, 43, 52Kokkinias, P., 441, 479; 444, 491; 446, 62, 75


Kokkonen, J., 445, 439Kokott, T.P., 444, 539Kolesnikov, V., 445, 14Kolesnikov, V.I., 444, 523Kolo, C., 446, 349Kolosov, V., 446, 342Kolrep, M., 444, 539Kolstein, M., 442, 484; 444, 531Kolster, H., 442, 484; 444, 531Komamiya, S., 444, 539Komatsubara, T., 442, 53Konashenok, A., 444, 523Kondashov, A.A., 446, 342Konig, A.C., 444, 503, 569; 445, 428; 446, 368¨Konigsmann, K., 442, 484; 444, 531¨Konopelchenko, B.G., 444, 299Konopliannikov, A., 441, 479Konstantinidis, N., 445, 239Kooijman, P., 443, 394Koop, T., 443, 394Kopeliovich, B., 446, 321Kopeliovich, B.Z., 445, 223Kopke, L., 446, 117¨Kopp, A., 444, 503, 569; 445, 428; 446, 368Koratzinos, M., 441, 479; 444, 491; 446, 62, 75Korchin, A.Yu., 441, 17Korner, J.G., 442, 435¨Korolko, I., 444, 503, 569; 445, 428; 446, 368Korotkov, V., 442, 484; 444, 531Korotkova, N.A., 443, 394Korpa, C.L., 446, 15Korsch, W., 442, 484; 444, 531Korzhavina, I.A., 443, 394Kosmas, T.S., 443, 7Kossakowski, R., 444, 516Kostioukhine, V., 441, 479; 444, 491; 446, 62,

75Kostov, I.K., 444, 196Kostrewa, D., 445, 20Kotanski, A., 443, 394´Kotikov, A.V., 441, 345Kourkoumelis, C., 441, 479; 444, 491; 446, 62,

75Koutsenko, V., 444, 503, 569; 445, 428; 446,

368Kouznetsov, O., 441, 479; 444, 491; 446, 62,

75Kovacs, T.G., 443, 239´Kovalenko, S., 442, 203Kovzelev, A., 445, 439Kowal, A.M., 443, 394Kowalewski, R.V., 444, 539Kowalski, H., 443, 394Kowalski, M., 444, 523Kowalski, T., 443, 394Kozela, A., 444, 555; 445, 20Kozlov, V., 442, 484; 444, 531

Kraemer, R.W., 444, 503, 569; 445, 428; 446,368

Krakauer, D., 443, 394Kramer, L.H., 442, 484; 444, 531Kramer, M., 441, 383¨Krammer, M., 441, 479; 444, 491; 446, 62, 75Kraniotis, G.V., 443, 111Krasnitz, A., 445, 366Krasnoperov, A., 445, 439Krause, B., 444, 531Krauss, L.M., 445, 412Krcmar, M., 442, 38ˇKrecak, Z., 442, 38ˇKrehl, O., 444, 25Krenz, W., 444, 503, 569; 445, 428; 446, 368Kress, T., 444, 539Kreuger, R., 444, 38, 43, 52Kreuter, C., 441, 479; 444, 491; 446, 62, 75Kreyerhoff, G., 441, 215Krieger, P., 444, 539Krishnaswami, G.S., 441, 429Krivokhijine, V.G., 442, 484; 444, 531Kriznic, E., 441, 479; 444, 491; 446, 62, 75Krmpotic, F., 444, 14; 445, 249´Krocker, M., 445, 239¨Krstic, J., 441, 479; 444, 491; 446, 62, 75Krumstein, Z., 441, 479; 444, 491; 446, 62, 75Ktorides, C.N., 444, 583Kubinec, P., 441, 479; 444, 491; 446, 62, 75Kubischta, W., 446, 117Kubo, J., 441, 235Kucewicz, W., 441, 479; 444, 491; 446, 62, 75Kuckes, M., 444, 531¨Kuhl, T., 444, 539Kuhn, C., 446, 191Kuhn, D., 445, 239Kuiroukidis, A., 443, 131Kullander, S., 446, 179Kumbartzki, G., 446, 22Kummell, F., 442, 484; 444, 531¨Kunde, G.J., 446, 197Kunin, A., 444, 503, 569; 445, 428; 446, 368Kunz, J., 441, 77Kunze, M., 446, 349Kupsc, A., 446, 179´´Kuraev, E.A., 442, 453Kurashige, H., 443, 409Kurilla, U., 446, 349Kurisuno, M., 442, 484Kurosawa, K., 445, 316Kurowska, J., 446, 62, 75Kurvinen, K., 441, 479; 444, 491; 446, 62, 75Kurz, G., 446, 179Kuusiniemi, P., 443, 69, 82Kuze, M., 443, 394Kuzenko, S.M., 446, 216Kuzmin, V.A., 443, 394


Kuznetsov, A.V., 446, 378Kuznetsov, V.E., 445, 439Kwan, S., 445, 449Kyberd, P., 444, 539Kyle, G., 442, 484; 444, 531Kyriakis, A., 445, 239

Labarga, L., 443, 394Lacaprara, S., 445, 439Lacentre, P., 444, 503, 569; 445, 428; 446, 368Lachnit, W., 442, 484; 444, 531Ladron de Guevara, P., 444, 503, 569; 445,

428; 446, 368Ladygin, V., 445, 14Lafferty, G.D., 444, 539LaFosse, D., 443, 89Lahanas, M., 444, 583Lai, A., 446, 117Lai, Y.F., 446, 356Lakata, M., 446, 349Laktineh, I., 444, 503, 569; 445, 428; 446, 368Lamberti, L., 443, 394Lampe, B., 446, 163Lamsa, J.W., 441, 479; 444, 491; 446, 62, 75Lancon, E., 445, 239Landaud, G., 444, 516Landi, G., 444, 503, 569; 445, 428; 446, 368Landolfi, G., 444, 299Landua, R., 446, 349Lane, D.W., 441, 479; 444, 491; 446, 62, 75Lane, J.B., 443, 394Lang, C.B., 443, 214Lang, P.F., 446, 356Langefeld, P., 441, 479; 444, 491; 446, 62, 75Langs, D.C., 445, 449Lanske, D., 444, 539Lanza, A., 445, 439Lapin, V., 441, 479; 444, 491; 446, 62, 75Lapoint, C., 444, 503, 569; 445, 428; 446, 368La Rotonda, L., 445, 439Lasiuk, B., 444, 523Lassila-Perini, K., 444, 503, 569; 445, 428;

446, 368Lauber, J., 444, 539Laugier, J.-P., 441, 479; 444, 491; 446, 62, 75Lauhakangas, R., 441, 479; 444, 491; 446, 62,

75Laurelli, P., 445, 239Laurenti, G., 443, 394Laurikainen, P., 444, 503, 569; 445, 428; 446,

368Lauritsen, T., 446, 22Lautenschlager, S.R., 444, 539Laveder, M., 445, 439Lavorato, A., 444, 503, 569; 445, 428; 446, 368Lavoura, L., 442, 390Lawall, R., 444, 555; 445, 20Lawson, I., 444, 539

Layter, J.G., 444, 539Lazarides, G., 441, 46Lazic, D., 444, 539Lazzeroni, C., 445, 439Lazzizzera, I., 444, 167L3 Collaboration, 444, 503, 569; 445, 428; 446,

368Leader, E., 445, 232Lebeau, M., 444, 503, 569; 445, 428; 446, 368Lebedev, A., 444, 503, 569; 445, 428; 446, 368Lebedev, O., 441, 419Le Bornec, Y., 445, 423Lebrun, P., 444, 503, 569; 445, 428; 446, 368Lecomte, P., 444, 503, 569; 445, 428; 446, 368Lecoq, P., 444, 503, 569; 445, 428; 446, 368Le Coultre, P., 444, 503, 569; 445, 428; 446,

368Leder, G., 441, 479; 444, 491; 446, 62, 75Lednev, A.A., 446, 342Lednicky, R., 446, 191Ledroit, F., 441, 479; 444, 491; 446, 62, 75Lee, A.M., 444, 539Lee, D., 444, 474Lee, H.J., 444, 503, 569; 445, 428; 446, 368Lee, H.W., 441, 83Lee, I.Y., 446, 22Lee, J.H., 443, 394Lee, J.M., 444, 267Lee, K., 445, 387Lee, S.B., 443, 394Lee, S.W., 443, 394Lee, U.W., 444, 204Leenhardt, S., 444, 32Lees, J.-P., 445, 239Le Faou, J.H., 442, 48Lefebure, V., 441, 479; 444, 491; 446, 62, 75Lefrancois, J., 445, 239Le Gac, R., 444, 38, 43, 52Le Goff, J.M., 444, 503, 569; 445, 428; 446,

368Lehraus, I., 445, 239Lehto, M., 445, 239Leimgruber, F., 444, 38, 43, 52Leino, M., 443, 69, 82Leinonen, L., 441, 479; 444, 491; 446, 62, 75Leisos, A., 441, 479; 444, 491; 446, 62, 75Leiste, R., 444, 503, 569; 445, 428; 446, 368Leitner, R., 441, 479; 444, 491; 446, 62, 75Lellouch, D., 444, 539Lemaire, M.-C., 445, 239Lemmon, R., 443, 69, 82Lemmon, R.C., 446, 197Lemonne, J., 441, 479; 444, 491; 446, 62, 75Lenti, M., 446, 117Lenti, V., 446, 342Lenzen, G., 441, 479; 444, 491; 446, 62, 75Leonardi, E., 444, 503, 569; 445, 428; 446, 368Lepeltier, V., 441, 479; 444, 491; 446, 62, 75


Leroy, J.P., 444, 401Leroy, O., 445, 239Lesiak, T., 441, 479; 444, 491; 446, 62, 75Leslie, J., 445, 449Letessier-Selvon, A., 445, 439Lethuillier, M., 441, 479; 444, 491; 446, 62, 75Letts, J., 444, 539Leung, C.N., 443, 185Levai, P., 442, 1; 444, 523Levi, G., 443, 394Levinson, L., 444, 539Levman, G.M., 443, 394Levtchenko, P., 444, 503, 569; 445, 428; 446,

368Levy, A., 443, 394Levy, J-M., 445, 439Lewitowicz, M., 444, 32Li, C., 444, 503, 569; 445, 428; 446, 368Li, C.G., 446, 356Li, C.S., 444, 224Li, D., 446, 356Li, G., 443, 58Li, H.B., 446, 356Li, J., 446, 356Li, P.Q., 446, 356Li, R.B., 446, 356Li, W., 446, 356Li, W.G., 446, 356Li, X.H., 446, 356Li, X.N., 446, 356Li, Z., 445, 271Lianshou, L., 444, 563Liao, Y., 441, 383Libanov, M.V., 442, 63Libby, J., 441, 479; 444, 491; 446, 62, 75Lidsey, J.E., 443, 97Liebisch, R., 444, 539Ligabue, F., 445, 239Liko, D., 441, 479; 444, 491; 446, 62, 75Lim, H., 443, 394Lim, I.T., 443, 394Limentani, S., 443, 394Lin, C.H., 444, 503, 569; 445, 428; 446, 368Lin, J., 445, 239Lin, W.T., 444, 503, 569; 445, 428; 446, 368Lin, Z., 444, 245Linde, F.L., 444, 503, 569; 445, 428; 446, 368Lindemann, L., 443, 394; 445, 20Ling, T.Y., 443, 394Link, J., 444, 555; 445, 20Linssen, L., 445, 439Lipkin, H.J., 445, 403Lipniacka, A., 441, 479; 444, 491; 446, 62, 75Lippi, I., 441, 479; 444, 491; 446, 62, 75Lissia, M., 441, 291List, B., 444, 539Lista, L., 444, 503, 569; 445, 428; 446, 368Litke, A.M., 445, 239

Littlewood, C., 444, 539Litvinenko, A., 445, 14Liu, D., 444, 539; 446, 332Liu, F., 444, 523Liu, H.M., 446, 356Liu, J., 446, 356Liu, R., 441, 473Liu, R.G., 446, 356Liu, W., 443, 394Liu, Y., 446, 356Liu, Z.A., 444, 503, 569; 445, 428; 446, 368Ljubicic, A., 442, 38; 445, 439ˇ ´Lloyd, A.W., 444, 539Lloyd, S.L., 444, 539Locci, E., 445, 239Loebinger, F.K., 444, 539Loerstad, B., 441, 479; 444, 491; 446, 62, 75Lohmann, W., 444, 503, 569; 445, 428; 446,

368Lohr, B., 443, 394¨Lohrmann, E., 443, 394Loinaz, W., 445, 178Lokajicek, M., 441, 479; 446, 62Loken, J.G., 441, 479; 444, 491; 446, 62, 75Long, G.D., 444, 539Long, J., 445, 439Long, K.R., 443, 394Longo, E., 444, 503, 569; 445, 428; 446, 368Loomis, C., 445, 239Lopes, J.H., 441, 479; 444, 491; 446, 62, 75Lopez-Duran Viani, A., 443, 394Lopez-Fernandez, R., 441, 479; 444, 491; 446,

62, 75Lopez, J.M., 441, 479; 444, 491; 446, 62, 75Lorenzon, W., 442, 484; 444, 531Losty, M.J., 444, 539Loukachine, K., 442, 48Loukas, D., 441, 479; 444, 491; 446, 62, 75Lourenco, C., 444, 516Love, A., 443, 111Lu, C.-D., 445, 394¨Lu, D.H., 441, 27, 443, 26Lu, F., 446, 356Lu, J.G., 446, 356Lu, J.X., 443, 167Lu, W., 444, 503, 569; 445, 428; 446, 368Lu, Y.S., 444, 503, 569; 445, 428; 446, 368Lubelsmeyer, K., 444, 503, 569; 445, 428; 446,¨

368Lubicz, V., 444, 401Lubrano, P., 446, 117Luci, C., 444, 503, 569; 445, 428; 446, 368Luckey, D., 444, 503, 569; 445, 428; 446, 368Luckmann, S., 446, 209Lucotte, A., 445, 239Ludwig, J., 444, 539Luitz, S., 446, 117Lukacs, B., 443, 21´


Lukina, O.Yu., 443, 394Luminari, L., 444, 503, 569; 445, 428; 446, 368Lundberg, B., 445, 449Lundin, M., 442, 43Lung, A., 444, 531Lunin, O., 442, 173Luo, X.L., 446, 356Lupi, A., 445, 439Luppi, E., 444, 111Lusiani, A., 445, 239Lustermann, W., 444, 503, 569; 445, 428; 446,

368Lutjens, G., 445, 239¨Lutz, P., 441, 479; 444, 491; 446, 62, 75

l’Yi, W.S., 445, 134

Lynch, J.G., 445, 239Lynch, W.G., 446, 197Lyons, L., 441, 479; 444, 491; 446, 62, 75Lyuboshitz, V.L., 446, 191Lyubovitskij, V.E., 442, 435

Ma, B.-Q., 441, 461Ma, E., 442, 238; 444, 391Ma, E.C., 446, 356Ma, J.M., 446, 356Ma, K.J., 443, 394Ma, W.G., 444, 503, 569; 445, 428; 446, 368Maccarrone, G., 443, 394Macchiavelli, A.O., 446, 22Macchiolo, A., 444, 539Macdonald, N., 443, 394Machefert, F., 445, 239Machleidt, R., 445, 259MacNaughton, J., 444, 491; 446, 62Macpherson, A., 444, 539Mader, W., 444, 539Maedan, S., 442, 217Maggi, G., 445, 239Maggi, M., 445, 239Magierski, P., 443, 69Magill, S., 443, 394Magnin, J., 445, 8Magueijo, J., 443, 104Mahanthappa, K.T., 441, 178Mahon, J.R., 441, 479; 444, 491; 446, 62, 75Maio, A., 441, 479; 444, 491; 446, 62, 75Maiolino, C., 442, 48Maity, M., 444, 503, 569; 445, 428; 446, 368Majumdar, P., 445, 129Majumder, G., 444, 503, 569; 445, 428; 446,

368Makeenko, Y.M., 445, 307Makino, S., 444, 267Makins, N.C.R., 442, 484; 444, 531Malakhov, A., 445, 14Malakhov, A.I., 444, 523

Malek, A., 441, 479; 444, 491; 446, 62, 75Maley, P., 445, 239Malgeri, L., 444, 503, 569; 445, 428; 446, 368Malinin, A., 444, 503, 569; 445, 428; 446, 368Maljukov, S., 446, 342Mallik, U., 443, 394Malmgren, T.G.M., 441, 479; 444, 491; 446,

62, 75Maltoni, F., 441, 257Malychev, V., 441, 479; 444, 491; 446, 62, 75Mana, C., 444, 503, 569; 445, 428; 446, 368˜Manaenkov, S.I., 442, 484; 444, 531Mandic, I., 444, 38, 43, 52´Mandl, F., 441, 479; 444, 491; 446, 62, 75Mandry, R., 444, 516Manduci, L., 446, 197Mangeol, D., 444, 503, 569; 445, 428; 446, 368Mannelli, I., 446, 117Mannelli, M., 444, 539Manner, W., 445, 239¨Mannert, C., 445, 239Mannocchi, G., 445, 239Manns, J., 445, 20Manola-Poggioli, E., 445, 439Mansouri, F., 445, 52Manthos, N., 444, 38, 43, 52Manvelyan, R., 444, 86Mao, H.S., 446, 356Mao, Z.P., 446, 356Marcellini, S., 444, 539March-Russell, J., 441, 96Marchesini, P., 444, 503, 569; 445, 428; 446,

368Marchetto, F., 446, 117Marchionni, A., 445, 439Marciniewski, P., 446, 179Marco, J., 441, 479; 444, 491; 446, 62, 75Marco, R., 441, 479; 444, 491; 446, 62, 75Marechal, B., 441, 479; 444, 491; 446, 62, 75Marel, G., 444, 38, 43, 52Margetis, S., 444, 523Margoni, M., 441, 479; 444, 491; 446, 62, 75Margotti, A., 443, 394Marian, G., 444, 503, 569; 445, 428; 446, 368Mariano, A., 445, 249Marin, A., 444, 503, 569; 445, 428; 446, 368Marin, J.-C., 441, 479; 444, 491; 446, 62, 75Marinelli, N., 445, 239Marini, G., 443, 394Mariotti, C., 441, 479; 444, 491; 446, 62, 75Markert, C., 444, 523Markopoulos, C., 444, 539Markou, A., 441, 479; 444, 491; 446, 62, 75Markou, C., 445, 239Markun, P., 443, 394Markytan, M., 446, 117Marnelius, R., 441, 243Marras, D., 446, 117


Marrocchesi, P.S., 445, 239Mart, T., 445, 20Martell, E.C., 441, 468Martelli, F., 445, 439Martens, F.K., 444, 531Martens, K., 442, 484Marti i Garcia, S., 441, 479; 444, 491; 446, 62,

75Martin, A.D., 443, 301Martin, A.J., 444, 539Martin, E.B., 445, 239Martin, F., 445, 239Martin, J.F., 443, 394Martin, J.M., 445, 423Martin, J.P., 444, 503, 539, 569; 445, 428; 446,

368Martin, J.W., 442, 484; 444, 531Martın-Mayor, V., 441, 330´Martin, V.J., 446, 117Martinelli, G., 441, 371; 444, 401Martinengo, P., 446, 342Martinez, G., 444, 539Martınez, M., 443, 394; 445, 239´Martinez-Rivero, C., 441, 479; 444, 491; 446,

62, 75Martinez-Vidal, F., 441, 479; 444, 491; 446,

62, 75Martini, M., 446, 117Martins, C.J.A.P., 445, 43Marukyan, H., 442, 484Marzano, F., 444, 503, 569; 445, 428; 446, 368Marzulli, V., 446, 117Masaike, A., 444, 267Maselli, S., 443, 394Mashimo, T., 444, 539Masip, M., 444, 352Masoli, F., 442, 484; 444, 531Masoni, A., 444, 111Massam, T., 443, 394Massaro, G.G.G., 444, 503, 569; 445, 428; 446,

368Mastroberardino, A., 443, 394Mastroyiannopoulos, N., 441, 479; 444, 491;

446, 62, 75Matchev, K.T., 445, 331Mateos, A., 442, 484; 444, 531Mateos, D., 443, 147Matheys, J.P., 446, 117Mathur, N., 444, 7Mato, P., 445, 239Matone, M., 445, 77, 357Matono, Y., 443, 409Matorras, F., 441, 479; 444, 491; 446, 62, 75Matsuda, T., 445, 14Matsuda, Y., 444, 267Matsushita, T., 443, 394Matsuura, K., 442, 53Matsuyama, Y., 444, 267

Matsuzaki, M., 445, 254Matteuzzi, C., 441, 479; 444, 491; 446, 62, 75Matthay, H., 446, 349¨Matthiae, G., 441, 479; 444, 491; 446, 62, 75Mattig, P., 444, 539¨Mattingly, M.C.K., 443, 394Mattingly, S., 443, 394Mattis, M.P., 442, 145Mayer, R.H., 446, 22MayTal-Beck, S., 445, 449Mazik, J., 441, 479; 444, 491; 446, 62, 75Mazumdar, K., 444, 503, 569; 445, 428; 446,

368Mazur, P.O., 444, 284Mazzucato, E., 446, 117Mazzucato, F., 441, 479; 444, 491; 446, 62, 75Mazzucato, M., 441, 479; 444, 491; 446, 62, 75McAllister, S., 442, 43McAndrew, M., 442, 484; 444, 531McCance, G.J., 443, 394McCrady, R., 446, 349Mc Cubbin, M., 441, 479; 444, 491; 446, 62,

75McCubbin, N.A., 443, 394McDonald, W.J., 444, 539McFall, J.D., 443, 394McGeorge, J.C., 442, 43McGovern, J.A., 446, 300McIlhany, K., 442, 484; 444, 531Mc Kay, R., 441, 479; 444, 491; 446, 62, 75McKellar, B.H.J., 444, 75McKenna, J., 444, 539McKeown, R.D., 442, 484; 444, 531Mckigney, E.A., 444, 539McMahon, T.J., 444, 539McNabb, D.P., 446, 22McNamara III, P.A., 445, 239McNeil, M.A., 445, 239McNeil, R.R., 444, 503, 569; 445, 428; 446,

368Mc Nulty, R., 441, 479; 444, 491; 446, 62, 75Mc Pherson, G., 441, 479; 444, 491; 446, 62,

75McPherson, R.A., 444, 539Meadows, B., 445, 449Mechain, X., 445, 439´Medcalf, T., 445, 239Mehen, T., 445, 378Meier, J., 446, 349Meier, R., 443, 77; 446, 363Meijers, F., 444, 539Meißner, F., 442, 484Meissner, F., 444, 531Mele, S., 444, 569; 445, 428; 446, 368Melikhov, D., 442, 381; 446, 336Melikyan, A., 444, 86Melkumov, G.L., 444, 523Mellado, B., 443, 394


Menary, S., 443, 394Menden, F., 442, 484; 444, 531Mendiburu, J-P., 445, 439Meng, X.C., 446, 356Menichetti, E., 446, 117Menke, S., 444, 539Menze, D., 444, 555; 445, 20Mercer, D., 444, 531Merebashvili, Z., 442, 374Merino, G., 445, 239Merkel, H., 445, 20Merkel, R., 445, 20Merle, E., 445, 239Merola, L., 444, 569; 445, 428; 446, 368Merola, S.M.L., 444, 503Meroni, C., 441, 479; 444, 491; 446, 62, 75Merritt, F.S., 444, 539Mertens, G., 444, 252Mes, H., 444, 539Meschini, M., 444, 503, 569; 445, 428; 446,

368Mescia, F., 444, 401Messi, R., 444, 111Messineo, A., 445, 239Mestvirishvili, A., 446, 117Metz, A., 442, 484; 444, 531Metzger, W.J., 444, 503, 569; 445, 428; 446,

368Meyer, A., 443, 394Meyer, C.A., 446, 349Meyer, J., 444, 539Meyer, J.-P., 445, 439Meyer-Larsen, A., 443, 394Meyer, W.T., 441, 479; 444, 491; 446, 62, 75Meyners, N., 442, 484; 444, 531Mezzetto, M., 445, 439Miagkov, A., 441, 479; 446, 75Michel, B., 445, 239Michelini, A., 444, 539Michetti, A., 446, 117Migani, D., 444, 569; 445, 428; 446, 368Migliore, E., 441, 479; 444, 491; 446, 62, 75Migneco, E., 442, 48Mihalcea, D., 445, 449Mihara, S., 444, 267, 539Mihul, A., 444, 569; 445, 428; 446, 368Mihul, D.M.A., 444, 503Mikenberg, G., 444, 539Mikheev, N.V., 446, 378Mikloukho, O., 442, 484; 444, 531Mikulec, I., 446, 117Mikuz, M., 444, 38, 43, 52ˇMilburn, R.H., 445, 449Milcent, H., 444, 503, 569; 445, 428; 446, 368Milewski, J., 443, 394Milite, M., 443, 394Miller, C.A., 442, 484; 444, 531Miller, D.B., 443, 394

Miller, D.J., 444, 539Miller, J., 444, 38, 43, 52Miller, M.A., 442, 484; 444, 531Milner, R., 442, 484; 444, 531Minard, M.-N., 445, 239Minashvili, I., 446, 342Minic, D., 442, 102Minten, A., 445, 239Miquel, R., 445, 239Mir, L.M., 445, 239Mir, R., 444, 539Mirabelli, G., 444, 503, 569; 445, 428; 446,

368Mirabito, L., 441, 479; 444, 491; 446, 62, 75Mishra, S.R., 445, 439Mitaroff, W.A., 441, 479; 444, 491; 446, 62, 75Mitra, P., 441, 89Mitsyn, V., 442, 484; 444, 531Miyamura, O., 443, 331Mizoguchi, S., 441, 123Mjoernmark, U., 441, 479; 444, 491; 446, 62,

75Mkrtchyan, R., 444, 86Mnich, J., 444, 503, 569; 445, 428; 446, 368Moa, T., 441, 479; 444, 491; 446, 62, 75Mock, A., 444, 523Moeller, R., 441, 479; 444, 491; 446, 62, 75Moenig, K., 441, 479; 444, 491; 446, 62, 75Mohanty, S., 445, 185Mohapatra, R.N., 441, 299; 442, 199Mohr, W., 444, 539Molnar, J., 444, 523´Molnar, P., 444, 569; 445, 428; 446, 368Molodtsov, S.V., 443, 387Monaco, V., 443, 394Mondardini, M.R., 445, 419Monderen, D., 444, 397Moneta, L., 445, 239Monge, M.R., 441, 479; 444, 491; 446, 62, 75Monig, K., 443, 394¨Montanari, A., 444, 539Montanet, F., 444, 38, 43, 52Monteil, S., 445, 239Monteiro, T., 443, 394Monteleoni, B., 444, 569; 445, 428; 446, 368Monteleoni, P.M.B., 444, 503Montero, A., 442, 273Montret, J-C., 445, 239Montvay, I., 446, 209Moore, C.F., 444, 252Moore, K.N., 446, 117Moore, R., 444, 503, 569; 445, 428; 446, 368Moore, R.W., 446, 117Moorhead, G.F., 445, 439Morandin, M., 443, 394Morawitz, P., 445, 239Moreau, X., 441, 479; 444, 491; 446, 62, 75Moreno, J.M., 445, 82


Morettini, P., 441, 479; 444, 491; 446, 62, 75Mori, T., 444, 539Morosov, B., 446, 179Morris, C., 444, 252Morton, G., 441, 479; 444, 491; 446, 62, 75Mortsell, A., 446, 179¨Moser, H.-G., 445, 239Mossuz, L., 445, 439Most, A., 442, 484; 444, 531Mota, A.L., 445, 94Motsch, F., 445, 239Mottola, E., 444, 284Moulik, T., 444, 503, 569; 445, 428; 446, 368Mount, R., 444, 503, 569; 445, 428; 446, 368Moutoussi, A., 445, 239Mozzetti, R., 444, 531Muanza, G.S., 444, 503, 569; 445, 428; 446,

368Muccifora, V., 442, 484; 444, 531Mueller, A.C., 444, 32Mueller, U., 441, 479; 444, 491; 446, 62, 75Muenich, K., 441, 479; 444, 491; 446, 62, 75Muheim, F., 444, 503, 569; 445, 428; 446, 368Muijs, A.J.M., 444, 503, 569; 445, 428; 446,

368Muikku, M., 443, 69, 82Mukhopadhyay, N.C., 444, 7Mukhopadhyaya, B., 443, 191Mulders, M., 441, 479; 444, 491; 446, 62, 75Mulet-Marquis, C., 441, 479; 444, 491; 446,

62, 75Muller, A., 444, 38, 43, 52Muller-Kirsten, H.J.W., 444, 86; 445, 287¨Mullins, S.M., 443, 89Multamaki, T., 445, 199¨Munday, D.J., 446, 117Munoz Sudupe, A., 441, 330˜Murakami, K., 443, 409Muresan, R., 441, 479; 444, 491; 446, 62, 75Murgia, F., 442, 470Murray, W.J., 441, 479; 444, 491; 446, 62, 75Murray, W.N., 443, 394Murtas, F., 445, 239Murtas, G.P., 445, 239Muryn, B., 441, 479; 444, 491; 446, 62, 75Musa, L., 446, 117Musgrave, B., 443, 394Musto, R., 441, 69Muther, H., 445, 259¨Myatt, G., 441, 479; 444, 491; 446, 62, 75Myklebust, T., 441, 479; 444, 491; 446, 62, 75Myung, Y.S., 441, 83

Nachtman, J.M., 445, 239NA38 Collaboration, 444, 516, 523; 446, 117Nagai, K., 444, 539Nagaitsev, A., 442, 484; 444, 531Nagano, K., 443, 394

Nagoshi, C., 444, 267Nahn, S., 444, 503, 569; 445, 428; 446, 368Nakada, T., 444, 38, 43, 52Nakagawa, T., 446, 342Nakamura, I., 444, 539Nakamura, T.T., 443, 409Nakayama, H., 445, 14Nam, S.W., 443, 394Nania, R., 443, 394Napier, A., 445, 449Napolitano, M., 444, 503, 569; 445, 428; 446,

368Nappi, A., 446, 117Nappi, E., 442, 484Naraghi, F., 441, 479; 444, 491; 446, 62, 75Nardi, M., 442, 14Naryshkin, Y., 442, 484; 444, 531Nash, J., 445, 239Nassalski, J., 446, 117Natale, A.A., 442, 369Nathan, A.M., 442, 484; 444, 531Nauta, B.J., 444, 463Navarra, F.S., 443, 285Navarria, F.L., 441, 479; 444, 491; 446, 62, 75Navas, S., 441, 479; 444, 491; 446, 62, 75Nawrocki, K., 441, 479; 444, 491; 446, 62, 75Nayak, G.C., 442, 427Neal, H.A., 444, 539Nedelec, P., 445, 439´ ´Needham, M.D., 446, 117Nefedov, Yu., 445, 439Negri, P., 441, 479; 444, 491; 446, 62, 75Negus, P., 445, 239Nellen, B., 444, 539Nelson, J.M., 444, 523Nemecek, S., 446, 75Nemes, M.C., 445, 94Nersessian, A., 445, 123Nessi-Tedaldi, F., 444, 503, 569; 445, 428; 446,

368Neubert, M., 441, 403Neuerburg, W., 444, 555; 445, 20Neufeld, N., 441, 479; 444, 491; 446, 62, 75Neuhofer, G., 446, 117Neumeister, N., 441, 479; 444, 491; 446, 62, 75Neunreither, F., 442, 484; 444, 531Newman, H., 444, 569; 445, 428; 446, 368Ng, J.N., 441, 419Nguyen, A., 445, 449Nguyen-Mau, C., 445, 439Nicolaidou, R., 441, 479; 444, 491; 446, 62, 75Niczyporuk, M., 444, 531Nie, J., 446, 356Nief, J.-Y., 445, 239Nielsen, B.S., 441, 479; 444, 491; 446, 62, 75Nielsen, H.B., 446, 256Nielsen, J., 445, 239Nielsen, M., 443, 285


Niessen, H.N.T., 444, 503Niessen, T., 444, 569; 445, 428; 446, 368Nigro, A., 443, 394Nikiforov, A., 445, 14Nikitin, N., 442, 381Nikolaenko, V., 441, 479Nikolaev, N.N., 442, 398Nikolenko, M., 441, 479; 444, 491; 446, 62, 75Nilsson, B., 442, 43Nilsson, B.S., 445, 239Nippe, A., 444, 503, 569; 445, 428; 446, 368Nisati, A., 444, 503, 569; 445, 428; 446, 368Nishimura, M., 446, 37Nishimura, T., 443, 394Nisius, R., 444, 539Nogueira, F.S., 441, 339Nojiri, S., 443, 121; 444, 92NOMAD Collaboration, 445, 439Nomokonov, V., 441, 479; 444, 491; 446, 62,

75Nomura, I., 444, 267Nomura, T., 443, 409Nomura, Y., 445, 316Nordmann, D., 445, 439Norman, K.L., 446, 342Normand, A., 441, 479; 444, 491; 446, 62, 75Norrbin, E., 442, 407Norton, A., 446, 117Norton, P.R., 445, 239Notz, D., 443, 394Novikov, I.D., 442, 82Nowak, H., 444, 503, 569; 445, 428; 446, 368Nowak, M.A., 442, 300; 446, 9Nowak, R.J., 443, 394Nowak, W.-D., 442, 484, 443, 379; 444, 531Nowakowski, M., 446, 111Noyes, V.A., 443, 394Nunez, C., 441, 185´˜Nupieri, M., 444, 531Nurnberger, H.-A., 445, 239¨Nussinov, S., 441, 299Nuzzo, S., 445, 239Nygren, A., 441, 479; 444, 491; 446, 62, 75Nylander, P., 443, 394

Oakham, F.G., 444, 539Oblakowska-Mucha, A., 441, 479Obraztsov, V., 441, 479; 444, 491; 446, 62, 75Obst, A., 444, 252Ocariz, J., 446, 117Ochs, A., 443, 394Oda, I., 444, 127Ødegard, S., 443, 69˚Odintsov, S.D., 443, 121; 444, 92Odorici, F., 444, 539Odyniec, G., 444, 523Oelert, W., 446, 179Oelwein, P., 444, 531

Oevers, M., 442, 291Ogami, H., 444, 531Ogren, H.O., 444, 539Oh, B.Y., 443, 394Oh, K., 442, 109Oh, P., 444, 469Oh, S., 441, 178Oh, S.K., 441, 215Oh, Y.D., 444, 503, 569; 445, 428; 446, 368Ohlsson-Malek, F., 444, 516Ohta, N., 441, 123; 445, 287Okrasinski, J.R., 443, 394´Oldenburg, M., 444, 523Olive, K.A., 444, 367Olshevski, A.G., 441, 479; 444, 491; 446, 62,

75O’Neale, S.W., 444, 539O’Neill, T.G., 442, 484; 444, 531Onofre, A., 441, 479; 444, 491; 446, 62, 75OPAL Collaboration, 444, 539Openshaw, R., 444, 531Orava, R., 441, 479; 444, 491; 446, 62, 75Orazi, G., 441, 479; 444, 491; 446, 62, 75Orear, J., 445, 419Oreglia, M.J., 444, 539Orejudos, W., 445, 239Orestano, D., 445, 439Organtini, G., 444, 569; 445, 428; 446, 368Orito, S., 444, 539Orr, R.S., 443, 394Osada, E., 445, 14O’Shaughnessy, K., 445, 449O’Shea, V., 445, 239Osterberg, K., 441, 479; 444, 491; 446, 62, 75Ostonen, G.O.R., 444, 503Ostonen, R., 444, 569; 445, 428; 446, 368Ostrick, M., 444, 555Ouared, R., 446, 349Ould-Saada, F., 446, 349Ouraou, A., 441, 479; 444, 491; 446, 62, 75Ouyang, J., 442, 484; 444, 531Ouyang, Q., 445, 239Owen, B., 444, 531Owen, B.R., 442, 484Oz, Y., 444, 318

Pac, M.Y., 443, 394Pacheco, A., 445, 239Pachos, J., 444, 469Padhi, S., 443, 394Paganetti, M., 444, 555; 445, 20Paganoni, M., 441, 479; 444, 491; 446, 62, 75Page, R.D., 443, 69, 82Pagels, B., 444, 38, 43, 52Paiano, S., 441, 479; 444, 491; 446, 62, 75Pain, R., 441, 479; 444, 491; 446, 62, 75Paiva, R., 441, 479; 444, 491; 446, 62, 75Pajares, C., 442, 395; 444, 435


Palacios, J., 441, 479; 444, 491; 446, 62, 75Palestini, S., 446, 117Palinkas, J., 444, 539´ ´Palka, H., 441, 479; 444, 491; 446, 62, 75Palla, F., 445, 239Palla, G., 444, 523Pallavicini, M., 441, 447Pallin, D., 445, 239Palmonari, F., 443, 394Palomares, C., 444, 503, 569; 445, 428; 446,

368Pan, F., 442, 7Pan, Y.B., 445, 239Panagiotakopoulos, C., 446, 224Panagiotou, A.D., 444, 523Pandoulas, D., 444, 503, 569; 445, 428; 446,

368Panzer-Steindel, B., 446, 117Paoletti, S., 444, 503, 569; 445, 428; 446, 368Paolucci, P., 444, 569; 445, 428; 446, 368Paoluzi, L., 444, 111Papadopoulos, D.B., 443, 131Papadopoulos, G., 443, 159Papadopoulos, I., 444, 38, 43, 52Papadopoulou, T.D., 441, 479Papadopoulou, Th.D., 444, 491; 446, 62, 75Papageorgiou, K., 441, 479; 444, 491; 446, 62,

75Papavassiliou, V., 442, 484; 444, 531Pape, L., 441, 479; 444, 491; 446, 62, 75Papoyan, V.V., 444, 293Papp, G., 442, 300; 446, 9Parikh, J.C., 446, 104Park, C., 444, 156Park, H.K., 444, 569; 445, 428; 446, 368Park, I.C., 445, 239Park, I.H., 443, 394; 444, 503, 569; 445, 428;

446, 368Park, I.S., 444, 267Park, P.P.H.K., 444, 503Park, S.K., 443, 394Parker, M.A., 446, 117Parkes, C., 441, 479; 444, 491; 446, 62, 75Parodi, F., 441, 479; 444, 491; 446, 62, 75Parrini, G., 445, 239Parsons, H.L.C., 446, 117Parsons, J.A., 443, 394Parzefall, U., 441, 479; 444, 491; 446, 62, 75Pascale, G., 444, 503, 569; 445, 428; 446, 368Paschos, E.A., 443, 201Pascual, A., 445, 239Pasqualucci, E., 444, 111Pasquinucci, A., 444, 318Passalacqua, L., 445, 239Passaleva, G., 444, 503, 569; 445, 428; 446,

368Passeri, A., 441, 479; 446, 62, 75Passon, O., 441, 479; 444, 491; 446, 62, 75

Pastore, F., 445, 439Pasyuk, E., 444, 252Pasztor, G., 444, 539´Pate, S.F., 442, 484; 444, 531Pater, J.R., 444, 539Patkos, A., 446, 272´Patrascioiu, A., 445, 160Patricelli, S., 444, 503, 569; 445, 428; 446, 368Patrick, G.N., 444, 539Patt, J., 444, 539Patzold, J., 443, 77; 446, 179, 363¨Paul, E., 443, 394; 444, 555; 445, 20Paul, T., 444, 569; 445, 428; 446, 368Pauluzzi, M., 444, 569; 445, 428; 446, 368Pauluzzi, T.P.M., 444, 503Paus, C., 444, 503, 569; 445, 428; 446, 368Pauss, F., 444, 503, 569; 445, 428; 446, 368Pavel, N., 443, 394Pavlopoulos, P., 444, 38, 43, 52Pawlak, J.M., 443, 394Pawlak, R., 443, 394Pawlowski, M., 444, 293Payre, P., 445, 239Peach, D., 444, 503, 569; 445, 428; 446, 368Peak, L.S., 445, 439Pearce, P.A., 444, 163Pedace, M., 444, 503, 569; 445, 428; 446, 368Peeters, K., 443, 153Pegoraro, M., 441, 479; 444, 491; 446, 62, 75Pei, Y.J., 444, 503, 569; 445, 428; 446, 368Peigneux, J.P., 446, 342Pelfer, P., 443, 394Pellegrino, A., 443, 394Pelucchi, F., 443, 394Pene, O., 446, 336´Peng, K.C., 445, 449Penin, A.A., 443, 264Pennacchio, E., 445, 439Pensotti, S., 444, 503, 569; 445, 428; 446, 368Pepe-Altarelli, M., 445, 239Pepe, M., 446, 117Peralta, L., 441, 479; 444, 491; 446, 62, 75Perelstein, M., 444, 273Perera, L.P., 445, 449Perez-Ochoa, R., 444, 539Perez, P., 445, 239Perez-Victoria, M., 442, 315´Peris, S., 443, 255Pernicka, M., 441, 479; 444, 491; 446, 62, 75,

117Peroni, C., 443, 394Perret-Gallix, D., 444, 503, 569; 445, 428; 446,

368Perret, P., 445, 239Perrodo, P., 445, 239Perrotta, A., 441, 479; 444, 491; 446, 62, 75Perroud, J-P., 445, 439Pervushin, V.N., 444, 293


Pesci, A., 443, 394Pessard, H., 445, 439Petcov, S.T., 444, 584Peter, P., 441, 52Peters, K., 446, 349Petersen, B., 444, 503, 569; 445, 428; 446, 368Petkou, A.C., 446, 306Petkova, V.B., 444, 163Petrak, S., 444, 503, 569; 445, 428; 446, 368Petreczky, P., 442, 291Petridis, A., 444, 523Petridou, C., 441, 479; 444, 491; 446, 62, 75Petrolini, A., 441, 479; 444, 491; 446, 62, 75Petrucci, F., 446, 117Petrucci, M.C., 443, 394Petti, R., 445, 439Petzold, S., 444, 539Pevsner, A., 444, 503, 569; 445, 428; 446, 368Peyaud, B., 446, 117Pfeifenschneider, P., 444, 539Pfeiffer, M., 443, 394Pfutzner, M., 444, 32¨Phillips, H.T., 441, 479; 444, 491; 446, 62, 75Piana, G., 441, 479Piattelli, P., 442, 48Piccioni, D., 443, 394Piccolo, D., 444, 503, 569; 445, 428; 446, 368Pick, B., 446, 349Pierazzini, G., 446, 117Pierce, D.M., 445, 331Pieri, M., 444, 503, 569; 445, 428; 446, 368Pierre, F., 441, 479; 444, 491; 446, 62, 75Pietra, C., 446, 349Pietrzyk, B., 445, 239Pilcher, J.E., 444, 539Pilipenko, Yu., 445, 14Pimenta, M., 444, 491; 446, 62, 75Pinciuc, C., 443, 394Pinder, C.N., 446, 349Pinfold, J., 444, 539Pinsky, S., 442, 173Piotrzkowski, K., 443, 394Piotto, E., 441, 479; 444, 491; 446, 62, 75Piper, A., 444, 523Piroue, P.A., 444, 503, 569; 445, 428; 446, 368´Pistolesi, E., 444, 503, 569; 445, 428; 446, 368Pitt, M., 444, 531Pivovarov, A.A., 443, 264Pizzi, J.R., 444, 516Placci, A., 445, 439Plane, D.E., 444, 539Plefka, J., 443, 153Pliszka, J., 444, 136Plotzke, R., 444, 555; 445, 20¨Plouin, F., 445, 423Plumacher, M., 443, 209¨Pluquet, A., 445, 439Plyaskin, V., 444, 503, 569; 445, 428; 446, 368

Podlyski, F., 445, 239Podobnik, T., 441, 479; 444, 491; 446, 62, 75Poelz, G., 443, 394Poffenberger, P., 444, 539Pohl, M., 444, 503, 569; 445, 428; 446, 368Pojidaev, V., 444, 503, 569; 445, 428; 446, 368Pokorski, S., 441, 205Pol, M.E., 441, 479; 444, 491; 446, 62, 75Polarski, D., 446, 53Polenz, S., 443, 394Polesello, G., 445, 439Policarpo, A., 444, 43Polini, A., 443, 394Polivka, G., 444, 38, 43, 52Poljsak, M., 444, 411ˇPollmann, D., 445, 439Polls, A., 445, 259Polok, G., 441, 479; 444, 491; 446, 62, 75Polok, J., 444, 539Polonyi, J., 445, 351Polovnikov, S.A., 446, 342Polyakov, V.A., 446, 342Poolman, H.R., 444, 531Popescu, R., 446, 197Popov, B., 445, 439Porcu, M., 446, 117Poropat, P., 441, 479; 444, 491; 446, 62, 75Porter, R.J., 444, 523Poskanzer, A.M., 444, 523Posocco, M., 443, 394; 444, 111Postema, H., 444, 503, 569; 445, 428; 446, 368Potashov, S., 442, 484; 444, 531Pothier, J., 444, 503, 569; 445, 428; 446, 368Potrebenikov, I., 446, 117Potterveld, D.H., 442, 484; 444, 531Poulsen, C., 445, 439Poves, A., 443, 1Povh, B., 446, 321Pozdniakov, V., 441, 479; 444, 491; 446, 62,

75Prange, G., 445, 239Pratt, S., 444, 231Prindle, D.J., 444, 523Prinias, A., 443, 394Privitera, P., 441, 479; 444, 491; 446, 62, 75Prodanov, E.M., 445, 112Produit, N., 444, 503, 569; 445, 428; 446, 368Prokofiev, D., 444, 503, 569; 445, 428; 446,

368Proskuryakov, A.S., 443, 394Pruss, S.M., 445, 419Przybycien, M., 443, 394; 444, 539´Przybycien, M.B., 443, 394´Przysiezniak, H., 445, 239Puga, J., 443, 394Puhlhofer, F., 444, 523¨Pukhaeva, N., 441, 479; 444, 491; 446, 62, 75Pullia, A., 441, 479; 444, 491; 446, 62, 75


Puolamaki, K., 446, 285¨Purohit, M.V., 445, 449Putz, J., 445, 239Putzer, A., 445, 239

Qi, N., 445, 239Qi, N.D., 446, 356Qi, X.R., 446, 356Qian, C.D., 446, 356Qiu, J.F., 446, 356Qu, Y.H., 446, 356Quadt, A., 443, 394Quandt, M., 446, 290Quarati, P., 441, 291Quartieri, J., 444, 503, 569; 445, 428; 446, 368Quast, G., 445, 239Que, Y.K., 446, 356Quinn, B., 445, 449

Raach, H., 443, 394Racca, C., 444, 516Radeztsky, S., 445, 449Radojicic, D., 441, 479; 444, 491; 446, 62, 75Rae, W.D.M., 444, 260Raeven, B., 445, 239Rafatian, A., 445, 449Ragazzi, S., 441, 479; 444, 491; 446, 62, 75Ragusa, F., 445, 239Rahal-Callot, G., 444, 503, 569; 445, 428; 446,

368Rahkila, P., 443, 69Rahmani, H., 441, 479; 444, 491; 446, 62, 75Rahmfeld, J., 443, 143Raine, C., 445, 239Raja, N., 444, 503, 569; 445, 428; 446, 368Rajeev, S.G., 441, 429Rakness, G., 442, 484; 444, 531Rakoczy, D., 441, 479; 444, 491; 446, 75Rames, J., 444, 491Ramond, P., 441, 163Ramos, E., 445, 123Ramos, S., 444, 516Rancoita, P.G., 444, 503, 569; 445, 428; 446,

368Rander, J., 445, 239Ranieri, A., 445, 239Ranjard, F., 445, 239Raso, G., 445, 239Raso, M., 443, 394Ratajczak, M., 446, 349Rathouit, P., 445, 439Ratoff, P.N., 441, 479; 444, 491; 446, 62, 75Rattaggi, M., 444, 503, 569; 445, 428; 446, 368Rauch, W., 444, 523Ravanini, F., 444, 442Raven, G., 444, 503, 569; 445, 428; 446, 368Ravindran, V., 445, 206, 214Razis, P., 444, 503, 569; 445, 428; 446, 368

Read, A.L., 441, 479; 444, 491; 446, 62, 75Reali, A., 442, 484; 444, 531Reay, N.W., 445, 449Rebecchi, P., 441, 479; 444, 491; 446, 62, 75Redaelli, N.G., 441, 479; 444, 491; 446, 62, 75Redondo, I., 443, 394Redwine, R., 442, 484; 444, 531Reeder, D.D., 443, 394Regan, P.H., 444, 32Regenfus, C., 446, 349Regler, M., 441, 479; 444, 491; 446, 62, 75Reid, D., 441, 479; 444, 491; 446, 62, 75Reid, J.G., 444, 523Reidy, J.J., 445, 449Reinhardt, H., 446, 290Reinhardt, R., 441, 479; 444, 491; 446, 62, 75Rejmund, M., 444, 32Rembser, C., 444, 539Ren, D., 444, 503, 569; 445, 428; 446, 368Renardy, J.-F., 445, 239Renfordt, R., 444, 523Renk, B., 445, 239; 446, 117Renken, R., 442, 266Rensch, B., 445, 239Renton, P.B., 441, 479; 444, 491; 446, 62, 75Renzoni, G., 445, 439Reolon, A.R., 442, 484; 444, 531Repond, J., 443, 394Resag, S., 446, 349Rescigno, M., 444, 503, 569; 445, 428; 446,

368Resvanis, L.K., 441, 479; 444, 491; 446, 62, 75Retamosa, J., 443, 1Retyk, W., 444, 523Reucroft, S., 444, 503, 569; 445, 428; 446, 368Reviol, W., 443, 89Reya, E., 443, 298Reznikov, S., 445, 14Ribeiro, E., 442, 349Ricci, B., 441, 291; 444, 387Richard, F., 441, 479; 444, 491; 446, 62, 75Richter, A., 443, 1Rick, H., 444, 539Rickenbach, R., 444, 38, 43, 52Ridky, J., 441, 479; 444, 491; 446, 62, 75Riedinger, L.L., 443, 89Riemann, S., 444, 503, 569; 445, 428; 446, 368Riles, K., 444, 503, 569; 445, 428; 446, 368Rinaudo, G., 441, 479; 444, 491; 446, 62, 75Riotto, A., 442, 68; 445, 323; 446, 28Riska, D.O., 444, 21Ristinen, R., 442, 484; 444, 531Rith, K., 442, 484; 444, 531Ritter, H.G., 444, 523Ritz, S., 443, 394Riu, I., 445, 239Riveline, M., 443, 394Rizzo, G., 445, 239


Roberts, B.L., 444, 38, 43, 52Roberts, R.G., 443, 301Robertson, A.N., 445, 239Robertson, S., 444, 539Robins, S.A., 444, 539Robohm, A., 444, 503, 569; 445, 428; 446, 368Robutti, E., 441, 447Roda, C., 445, 439Rodin, J., 444, 503, 569; 445, 428; 446, 368Rodning, N., 444, 539Rodrigues da Silva, P.S., 442, 369Roe, B.P., 444, 503, 569; 445, 428; 446, 368Roethel, W., 446, 349Rohde, M., 443, 394Rohne, E., 445, 239Rohne, O., 441, 479; 444, 491; 446, 62, 75Rohrich, D., 444, 523¨Roland, C.R.G., 444, 523Rolandi, L., 445, 239Roldan, J., 443, 394´Roloff, H., 442, 484; 444, 531Romana, A., 444, 516Romanovsky, V., 446, 342Romeo, A., 441, 265Romero, A., 441, 479; 444, 491; 446, 62, 75Romero, L., 444, 503, 569; 445, 428; 446, 368Ronchese, P., 441, 479; 444, 491; 446, 62, 75Rondio, E., 446, 117Roney, J.M., 444, 539Rong, G., 446, 356Ronningen, R., 446, 197Roose, F., 445, 150Roper, C.D., 444, 252Roper, G., 444, 531¨Roscoe, K., 444, 539Rosenberg, E.I., 441, 479; 444, 491; 446, 62,

75Rosier-Lees, S., 444, 503, 569; 445, 428; 446,

368Rosinsky, P., 441, 479; 444, 491; 446, 62, 75Rosner, J.L., 441, 403Rosowsky, A., 445, 239Ross, G.G., 442, 165Rossi, A.M., 444, 539Rossi, P., 442, 484; 444, 531Roth, S., 444, 503, 569; 445, 428; 446, 368Rothberg, J., 445, 239Rotscheidt, H., 446, 342Roudeau, P., 441, 479; 444, 491; 446, 62, 75Rouge, A., 445, 239´Rousseau, D., 445, 239Rovelli, T., 441, 479; 444, 491; 446, 62, 75Roy, D.P., 444, 391; 445, 185Roy, M.S., 443, 293Roy, S., 443, 167, 191Roynette, J.C., 442, 48Royon, Ch., 446, 62, 75Rozen, Y., 444, 539

Rozowsky, J.S., 444, 273Rubakov, V.A., 442, 63Rubbia, A., 445, 439Ruber, R.J.M.Y., 446, 179Rubin, H.A., 445, 449Rubinstein, R., 445, 419Rubio, J.A., 444, 503, 569; 445, 428; 446, 368Rudnitsky, S., 442, 484; 444, 531Rudolph, G., 445, 239Rudolph, H., 444, 523Ruf, T., 444, 38, 43, 52Ruggieri, F., 445, 239Ruh, M., 442, 484; 444, 531Ruhlmann-Kleider, V., 441, 479; 444, 491; 446,

62, 75Ruijter, H., 442, 43Ruiz, A., 441, 479; 444, 491; 446, 62, 75Ruiz Arriola, E., 443, 33Rukoyatkin, P., 445, 14Rulikowska-Zarebska, E., 443, 394Rumpf, M., 445, 239Rumyantsev, O.A., 443, 51Rumyantsev, V., 446, 342Runge, K., 444, 539Runolfsson, O., 444, 539Ruschmeier, D., 444, 503, 569; 445, 428; 446,

368Rusetsky, A.G., 442, 435Ruske, O., 443, 394Ruspa, M., 443, 394Russakovich, N., 446, 342Rust, D.R., 444, 539Ryan, J.J., 443, 394Rybicki, A., 444, 523Rychenkova, P., 443, 138Ryckbosch, D., 442, 484; 444, 531Ryckebusch, J., 441, 1Rykaczewski, H., 444, 503, 569; 445, 428; 446,

368Rykaczewski, K., 444, 32Ryskin, M.G., 446, 48

Saadi, Y., 445, 239Saarikko, H., 441, 479; 444, 491; 446, 62, 75Sabetfakhri, A., 443, 394Sabra, W.A., 442, 97Sacchi, R., 443, 394Sachs, K., 444, 539Sacquin, Y., 441, 479; 444, 491; 446, 62, 75Sadovsky, A., 441, 479; 444, 491; 446, 62, 75Sadrozinski, H.F.-W., 443, 394Saeki, T., 444, 539Sagawa, H., 444, 1Sahr, O., 444, 539Saito, K., 441, 27, 443, 26Saito, N., 444, 267Sajot, G., 441, 479; 444, 491; 446, 62, 75Sakamoto, H., 443, 409


Sakar, S., 444, 503, 569; 445, 428; 446, 368Sakemi, Y., 442, 484; 444, 531Salehi, H., 443, 394Salicio, J., 444, 503, 569; 445, 428; 446, 368Salinas, F., 444, 252Salt, J., 441, 479; 444, 491; 446, 62, 75Salvatore, F., 445, 439Samoylenko, V.D., 446, 342Sampson, S., 443, 394Sampsonidis, D., 441, 479; 444, 491; 446, 62,

75Sanchez, E., 444, 503, 569; 445, 428; 446, 368Sanchez, F., 445, 239Sandell, A., 442, 43Sander, H.-G., 445, 239Sanders, D.A., 445, 449Sanders, M.P., 444, 503, 569; 445, 428; 446,

368Sandhas, W., 442, 43Sandoval, A., 444, 523Sang, W.M., 444, 539Sanguinetti, G., 445, 239Sann, H., 444, 523Sannino, M., 441, 479; 444, 491; 446, 62, 75Santha, A.K.S., 445, 449Santoni, C., 444, 43Santonocito, D., 442, 48Santoro, A.F.S., 445, 449Santos, F.C., 446, 170SAPHIR Collaboration, 444, 555; 445, 20Sapienza, P., 442, 48Saradzhev, F.M., 442, 259Sarakinos, M.E., 444, 503, 569; 445, 428; 446,

368Sarantites, D.G., 443, 89Sarkar, U., 442, 243; 444, 391; 445, 185Sarkisyan, E.K.G., 444, 539Sartorelli, G., 443, 394Sasaki, S., 443, 331Sasaki, Y., 442, 53Sasao, N., 443, 409Sato, T., 441, 105Satteson, M., 446, 22Satuła, W., 443, 89Saturnini, P., 444, 516Satz, H., 442, 14Saull, P.R.B., 443, 394Savcı, M., 441, 410Savelius, A., 443, 69, 82Savin, A.A., 443, 394Savin, I., 442, 484; 444, 531Savoy, C.A., 444, 119Savrie, M., 446, 117´Saxon, D.H., 443, 394Sbarra, C., 444, 539Scadron, M.D., 446, 332Scarlett, C., 442, 484; 444, 531Scarpaci, J.A., 442, 48

Schael, S., 445, 239Schafer, A., 443, 40¨Schafer, C., 444, 503, 569; 445, 428; 446, 368¨Schafer, E., 444, 523¨Schafer, M., 444, 38, 43, 52¨Schafke, A., 446, 290¨Schahmaneche, K., 445, 439Schaile, A.D., 444, 539Schaile, O., 444, 539Schaller, L.A., 444, 38, 43, 52Schalm, K., 446, 247Schanne, S., 446, 117Schapler, D., 443, 77Schaposnik, F.A., 441, 185Scharf, F., 444, 539Scharff-Hansen, P., 444, 539Schegelsky, V., 444, 503, 569; 445, 428; 446,

368Scheidt, J., 446, 117Schenk, U., 445, 20Schepkin, M., 443, 77Schepkin, M.G., 446, 179Schieck, J., 444, 539Schietinger, T., 444, 38, 43, 52Schiller, A., 442, 453, 443, 244Schinzel, D., 446, 117Schioppa, M., 443, 394Schlatter, D., 445, 239Schlein, P.E., 445, 455Schlenstedt, S., 443, 394Schmeling, S., 445, 239Schmidke, W.B., 443, 394Schmidt, B., 445, 439Schmidt, C.R., 445, 168Schmidt, F., 442, 484; 444, 531Schmidt, I., 441, 461; 444, 451Schmidt, J., 446, 117Schmidt-Kaerst, S., 444, 503, 569; 445, 428;

446, 368Schmidt, K.E., 446, 99Schmidt, P., 446, 349Schmischke, D., 444, 523Schmitt, B., 444, 539Schmitt, H., 442, 484; 444, 531Schmitt, M., 445, 239Schmitt, S., 444, 539Schmitz, D., 444, 503, 569; 445, 428; 446, 368Schmitz, N., 444, 523Schneekloth, U., 443, 394Schneider, H., 441, 479; 444, 491; 446, 62, 75Schneider, O., 445, 239Schnell, G., 442, 484; 444, 531Schnurbusch, H., 443, 394Scholmann, J., 445, 20Scholten, O., 441, 17Scholz, N., 444, 503, 569; 445, 428; 446, 368Schonfelder, S., 444, 523¨Schonharting, V., 446, 117¨


Schoning, A., 444, 539¨Schopper, A., 444, 38, 43, 52Schopper, H., 444, 503, 569; 445, 428; 446,

368Schotanus, D.J., 444, 503, 569; 445, 428; 446,

368Schramm, S., 446, 191Schroder, B., 442, 43¨Schroder, M., 444, 539¨Schue, I., 446, 117´Schulday, I., 444, 555; 445, 20Schuler, K.P., 442, 484; 444, 531¨Schumacher, M., 444, 539; 445, 20Schutz, P., 445, 20¨Schwartz, A.J., 445, 449Schwarzer, O., 443, 394Schweigert, C., 441, 141Schwemling, Ph., 441, 479; 444, 491; 446, 62,

75Schwenke, J., 444, 503, 569; 446, 368Schwering, G., 444, 503, 569; 446, 368Schwering, J.S.G., 445, 428Schwick, C., 444, 539Schwickerath, U., 441, 479; 444, 491; 446, 62,

75Schwieger, J., 443, 7Schwille, W.J., 444, 555; 445, 20Schwind, A., 442, 484; 444, 531Schyns, M.A.E., 441, 479; 444, 491; 446, 62,

75Sciaba, A., 445, 239`Sciacca, C., 444, 503, 569; 445, 428; 446, 368Sciarrino, D., 444, 503, 569; 445, 428; 446, 368Sciulli, F., 443, 394Scoccola, N.N., 444, 21; 446, 93Scopetta, S., 442, 28Scott, I.J., 445, 239Scott, J., 443, 394Scott, W.G., 444, 539Scuri, F., 441, 479; 444, 491; 446, 62, 75Seager, P., 441, 479; 444, 491; 446, 62, 75Sedgbeer, J.K., 443, 394; 445, 239Sedykh, Y., 441, 479; 444, 491; 446, 62, 75Segar, A.M., 441, 479; 444, 491; 446, 62, 75Seibert, J., 442, 484; 444, 531Seibert, R., 446, 349Seiden, A., 443, 394Seiler, E., 445, 160Sekimoto, M., 444, 267Sekulin, R., 441, 479; 444, 491; 446, 62, 75Selonke, F., 443, 394Selvaggi, G., 445, 239Semenoff, G.W., 445, 307Semenov, A., 445, 14; 446, 342Semenov, A.Yu., 444, 523Semenova, I., 445, 14Sen, S., 445, 112Sene, M., 446, 342´

Sene, R., 446, 342´SenGupta, S., 445, 129; 446, 104Serbo, V.G., 442, 453Serin, L., 445, 239Serrano, M., 445, 439Servoli, L., 444, 503, 569; 445, 428; 446, 368Seth, K.K., 441, 479Settles, R., 445, 239Seuster, R., 444, 539Sevior, M.E., 445, 439Sevrin, A., 443, 153Seyboth, P., 444, 523Seyerlein, J., 444, 523Seymour, M.H., 442, 417Seywerd, H., 445, 239Sguazzoni, G., 445, 239Shabelski, Yu.M., 446, 48Shagin, P.M., 446, 342Shah, T.P., 443, 394Shao, Y.Y., 446, 356Shcheglova, L.M., 443, 394Sheaff, M., 445, 449Shears, T.G., 444, 539Sheikh, J.A., 443, 16Shellard, E.P.S., 445, 43Shellard, R.C., 441, 479; 444, 491; 446, 62, 75Shen, B.C., 444, 539Shen, B.W., 446, 356Shen, D.L., 446, 356Shen, H., 446, 356Shen, X.Y., 446, 356Sheng, H.Y., 446, 356Shepherd-Themistocleous, C.H., 444, 539Sheridan, A., 441, 479; 444, 491; 446, 62, 75Sherwood, P., 444, 539Shevchenko, S., 444, 503, 569; 445, 428; 446,

368Shi, H.Z., 446, 356Shibata, T.-A., 442, 484; 444, 531Shibatani, K., 442, 484Shimizu, H., 446, 342Shin, T., 442, 484; 444, 531Shin, Y.M., 444, 267Shirkov, D.V., 442, 344Shiromizu, T., 443, 127Shivarov, N., 444, 503, 569; 445, 428; 446, 368Shizuma, T., 442, 53Shmatikov, M., 446, 43Shoutko, V., 444, 503, 569; 445, 428; 446, 368Shovkovy, I.A., 441, 313Shukla, J., 444, 503, 569; 445, 428; 446, 368Shukla, S., 445, 419Shumilov, E., 444, 503, 569; 445, 428; 446,

368Shutov, V., 442, 484; 444, 531Shuvaev, A.G., 446, 48Shvedov, O.Yu., 443, 373Shvorob, A., 444, 503, 569; 445, 428; 446, 368


Sideris, D., 443, 394Sidorov, A.V., 445, 232Sidwell, R.A., 445, 449Siebel, M., 441, 479; 444, 491; 446, 62, 75Siedenburg, T., 444, 503, 569; 445, 428; 446,

368Sievers, M., 443, 394Sikler, F., 444, 523Sillou, D., 445, 439Silva-Marcos, J.I., 443, 276Silva, S., 444, 516Silva Neto, M.B., 441, 339Silvestre, R., 441, 479; 444, 491Silvestrini, L., 441, 371Silvestris, L., 445, 239Sim, K.S., 444, 267Simani, C., 442, 484Simani, M.C., 444, 531Simard, L., 441, 479; 444, 491; 446, 62, 75Simmons, D., 443, 394Simmons, E.H., 443, 347Simon, A., 442, 484; 444, 531Simon, J., 443, 147´Simonetto, F., 441, 479; 444, 491; 446, 62, 75Simopoulou, E., 445, 239Simpson, J., 443, 69, 82Sims, D.A., 442, 43Simula, S., 442, 381Sin, S.-J., 444, 156Sinclair, L.E., 443, 394Singer, P., 445, 394Singer, S.M., 444, 260Singh, H., 444, 327Singovsky, A.V., 446, 342Sinram, K., 442, 484; 444, 531Siroli, G.P., 444, 539Sisakian, A.N., 441, 479; 444, 491; 446, 62, 75Sittler, A., 444, 539Sjostrand, T., 442, 407¨Skaali, T.B., 441, 479; 444, 491; 446, 62, 75Skillicorn, I.O., 443, 394Skrzypczak, E., 444, 523Skuja, A., 444, 539Slaughter, A.J., 445, 449Slaus, I., 444, 252Slavich, P., 442, 484; 444, 531Smadja, G., 441, 479; 444, 491; 446, 62, 75Smalska, B., 443, 394Smend, F., 445, 20Smirichinski, V.I., 444, 293Smirnov, N., 444, 491; 446, 62Smirnova, O., 441, 479; 444, 491; 446, 62, 75Smith, A.M., 444, 539Smith, B., 444, 503, 569; 445, 428; 446, 368Smith, B.H., 443, 89Smith, D., 445, 239Smith, G.R., 441, 479; 444, 491; 446, 62, 75Smith, W.H., 443, 394

Smolik, L., 445, 239Smyrski, J., 444, 555; 445, 20Smythe, W.R., 444, 531Snellings, R., 444, 523Snigirev, A.M., 443, 387Snow, G.A., 444, 539Sobie, R., 444, 539Sobol, A., 446, 342Sobotka, L.G., 446, 197Soff, S., 446, 191Soffer, J., 441, 461; 442, 479Sofianos, S.A., 442, 43Sokatchev, E., 444, 341Sokoloff, M.D., 445, 449Sokolov, A., 446, 75Sola, J., 442, 326`Solano, A., 443, 394Solano, J., 445, 449Soldati, R., 441, 257Soldner-Rembold, S., 444, 539¨Soler, F.J.P., 445, 439Solomin, A.N., 443, 394Solovianov, O., 446, 75Soloviev, O.A., 442, 136Solovjev, A., 446, 342Solovtsov, I.L., 442, 344Sommer, J., 445, 239Son, D., 443, 394; 444, 503, 569; 445, 428;

446, 368Sonderegger, P., 444, 516Song, X.F., 446, 356Sonoda, H., 446, 58Sopczak, A., 441, 479; 444, 491; 446, 62, 75Sosnowski, R., 441, 479; 444, 491; 446, 62, 75Sowinski, J., 444, 531Sozzi, G., 445, 439Sozzi, M., 446, 117Spagnolo, P., 445, 239Spanderen, K., 446, 209Spanier, S., 446, 349Spassov, T., 441, 479; 444, 491; 446, 62, 75Spector, D., 442, 159Spengos, M., 442, 484; 444, 531Speth, J., 444, 25Speth, W., 445, 20Spieles, C., 442, 443; 446, 191, 326Spillantini, P., 444, 503, 569; 445, 428; 446,

368Spinetti, M., 444, 111Spira, M., 441, 383Spiriti, E., 441, 479; 444, 491; 446, 62, 75Sponholz, P., 441, 479; 444, 491; 446, 62, 75Sproston, M., 444, 539Squarcia, S., 441, 479; 444, 491; 446, 62, 75Squier, G.T.A., 444, 523St-Laurent, M., 443, 394Stahl, A., 444, 539Staiano, A., 443, 394


Stairs, D.G., 443, 394Stamenov, D.B., 445, 232Stampfer, D., 441, 479; 444, 491; 446, 62, 75Stanco, L., 443, 394Stanek, R., 443, 394Stanescu, C., 441, 479; 444, 491; 446, 62, 75Stanic, S., 441, 479; 444, 491; 446, 62, 75Stanton, N.R., 445, 449Stapnes, S., 441, 479; 444, 491; 446, 62, 75Stassinaki, M., 446, 342Steele, D., 445, 439Steele, T.G., 446, 267Stefanski, R.J., 445, 449Steffens, E., 442, 484; 444, 531Steininger, M., 445, 439Stenger, J., 442, 484; 444, 531Stenson, K., 445, 449Stenzel, H., 445, 239Stepaniak, J., 446, 179Stephan, F., 445, 239Stephens, K., 444, 539Stephenson Jr., G.J., 444, 75Sterbenz, S., 444, 252Stern, A., 441, 69Steuer, M., 444, 503, 569; 445, 428; 446, 368Steuerer, J., 444, 539Stevenson, K., 441, 479; 444, 491; 446, 62, 75Stewart, I.W., 445, 378Stewart, J., 442, 484; 444, 531Stickland, D.P., 444, 503, 569; 445, 428; 446,

368Stiegler, U., 445, 439Stifutkin, A., 443, 394Stipcevic, M., 442, 38; 445, 439ˇ ´Stirling, W.J., 443, 301Stocchi, A., 441, 479; 444, 491; 446, 62, 75Stock, F., 444, 531Stock, H., 446, 349¨Stock, R., 444, 523Stocker, H., 442, 443; 446, 191¨Stoesslein, U., 444, 531Stolarczyk, T., 445, 439Stoll, K., 444, 539Stone, A., 444, 503, 569; 445, 428; 446, 368Stone, H., 444, 503, 569; 445, 428; 446, 368Stonjek, S., 443, 394Stoßlein, U., 442, 484¨Stoyanov, B., 444, 503, 569; 445, 428; 446,

368Straessner, A., 444, 503, 569; 445, 428; 446,

368Straßburger, C., 446, 349Straub, P.B., 443, 394Strauss, J., 441, 479; 446, 62, 75Strickland, E., 443, 394Strobele, H., 444, 523¨Strohbusch, U., 446, 349Strohmer, R., 444, 539¨

Stroili, R., 443, 394Strom, D., 444, 539Strong, J.A., 445, 239Stroot, J.P., 446, 342Strub, R., 441, 479; 444, 491; 446, 62, 75Struck, Chr., 444, 523Strumia, A., 445, 407Stugu, B., 441, 479; 444, 491; 446, 62, 75Sudhakar, K., 444, 503, 569; 445, 428; 446,

368Suehiro, M., 443, 409Suffert, M., 446, 349Sugonyaev, V.P., 446, 342Suh, J.S., 446, 349Sukhanov, A., 446, 179Sultanov, G., 444, 503, 569; 445, 428; 446, 368Sultansoy, S., 443, 359Summerer, K., 444, 32¨Summers, D.J., 445, 449Sun, F., 446, 356Sun, H.S., 446, 356Sun, L.Z., 444, 503, 569; 445, 428; 446, 368Sun, Y., 446, 356Sun, Y.Z., 446, 356Suomijarvi, T., 442, 48¨Surguladze, L.R., 446, 153Surrow, B., 443, 394; 444, 539Susa, T., 444, 523Susinno, G., 443, 394Susinno, G.F., 444, 503, 569; 445, 428; 446,

368Susukita, R., 444, 267Suszycki, L., 443, 394Suter, H., 444, 503, 569; 445, 428; 446, 368Sutter, M., 442, 484; 444, 531Sutton, M.R., 443, 394Suzuki, I., 443, 394Svaiter, N.F., 441, 339Swain, J.D., 444, 503, 569; 445, 428; 446, 368Szczekowski, M., 441, 479; 444, 491; 446, 62,

75Szentpetery, I., 444, 523Szep, Zs., 446, 272´Szeptycka, M., 441, 479; 444, 491; 446, 62, 75Sziklai, J., 444, 523Szillasi, Z., 444, 503, 569; 446, 368Szleper, M., 446, 117

Tabarelli, T., 441, 479; 444, 491; 446, 62, 75Tachibana, M., 442, 217Tadic, D., 445, 249´Takabayashi, N., 445, 14Takach, S., 445, 449Takacs, G., 444, 442´Takamatsu, K., 446, 342Takashima, R., 444, 267Takeuchi, Y., 443, 409Takeutchi, F., 444, 267


Talbot, S.D., 444, 539Talby, M., 445, 239Tallini, H., 442, 484; 444, 531Tamvakis, K., 446, 224Tanabe, K., 445, 1Tanaka, R., 445, 239Tanaka, S., 444, 539Tang, H.-B., 443, 63Tang, S.Q., 446, 356Tang, X.W., 444, 503, 569; 445, 428; 446, 368Tanigawa, T., 445, 254Taniguchi, T., 443, 409Tanii, Y., 446, 37Tapper, A.D., 443, 394Tapper, R.J., 443, 394Taras, P., 444, 539Tarasov, Yu.A., 441, 453Tareb-Reyes, M., 445, 439Tarem, S., 444, 539Taroian, S., 442, 484; 444, 531Tarrago, X., 444, 516Tassi, E., 443, 394Tatar, R., 442, 109Tatichvili, G., 446, 117Tatischeff, B., 445, 423Tatsumi, T., 441, 9Taureg, H., 446, 117Taurok, A., 446, 117Tauscher, L., 444, 38, 43, 52, 503, 569; 445,

428; 446, 368Taxil, P., 441, 376Taylor, G., 445, 239Taylor, G.N., 445, 439Taylor, L., 444, 503, 569; 445, 428; 446, 368Tchlatchidze, G., 446, 342Tegenfeldt, F., 441, 479; 444, 491; 446, 62, 75Teixeira-Dias, P., 445, 239Tejessy, W., 445, 239Tempesta, P., 445, 239Tenchini, R., 445, 239Tereshchenko, S., 445, 439Terkulov, A., 442, 484; 444, 531Terranova, F., 441, 479; 444, 491; 446, 62, 75Terron, J., 443, 394´Teschendorff, A., 443, 159Teubert, F., 445, 239Teuscher, R., 444, 539Theis, U., 446, 314Thibault, C., 444, 38, 43, 52Thiergen, M., 444, 539Thiessen, D.M., 444, 531Thoma, U., 444, 555; 446, 349Thomas, A.W., 441, 27, 443, 26Thomas, E., 444, 531Thomas, J., 441, 479; 444, 491; 446, 62, 75Thompson, A.S., 445, 239Thompson, D.J., 443, 201Thompson, J.C., 445, 239

Thompson, L.F., 445, 239Thomson, E., 445, 239Thomson, M.A., 444, 539Thorne, K., 445, 449Thorne, R.S., 443, 301Thulasidas, M., 445, 239Tiator, L., 444, 555Tiecke, H., 443, 394Tilquin, A., 441, 479; 444, 491; 445, 239; 446,

62, 75Timmermans, C., 444, 503, 569; 445, 428; 446,

368Timmermans, J., 441, 479; 444, 491; 446, 62,

75Ting, S.C.C., 444, 503, 569; 445, 428; 446, 368Ting, S.M., 444, 503, 569; 445, 428; 446, 368Tinti, N., 446, 62, 75Tinyakov, P.G., 442, 63Tipton, B., 442, 484; 444, 531Tischhauser, M., 446, 349¨Tittel, K., 445, 239Tkabladze, A., 443, 379Tkatchev, A., 446, 117Tkatchev, L.G., 441, 479; 444, 491; 446, 62, 75Tlusty, P., 444, 267´Todorov, T., 441, 479Todorova, S., 441, 479; 444, 491; 446, 62, 75Toet, D.Z., 441, 479; 444, 491; 446, 62, 75Tokushuku, K., 443, 394Tomalin, I.R., 445, 239Tomaradze, A., 441, 479; 444, 491; 446, 62Tomboulis, E.T., 443, 239Tome, B., 441, 479; 444, 491; 446, 62, 75Tonazzo, A., 441, 479; 444, 491; 446, 62, 75Tong, G.L., 446, 356Tonwar, S.C., 444, 503, 569; 445, 428; 446,

368Toothacker, W.S., 443, 394Tormanen, S., 443, 69¨ ¨Tornow, W., 444, 252Toropin, A., 445, 439Torrence, E., 444, 539Torrente-Lujan, E., 441, 305Tort, A., 446, 170Tortora, L., 441, 479; 444, 491; 446, 62, 75Toth, J., 444, 503, 569; 445, 428; 446, 368´Touchard, A-M., 445, 439Touchard, F., 444, 38, 43, 52Touramanis, C., 444, 38, 43, 52Tournefier, E., 445, 239Tovey, S.N., 445, 439Towers, S., 444, 539Toy, M., 444, 523Trabelsi, A., 445, 239Traini, M., 442, 28Trainor, T.A., 444, 523Tran, H.N., 445, 20Tran, M-T., 445, 439


Tran, M.Q., 444, 555; 445, 20Transtromer, G., 441, 479; 444, 491; 446, 62,

75Trautmann, W., 446, 197Treille, D., 441, 479; 444, 491; 446, 62, 75Trentalange, S., 444, 523Tricomi, A., 445, 239Trigger, I., 444, 539Trinchero, R., 443, 221Tripathi, A.K., 445, 449Tristram, G., 441, 479; 444, 491; 446, 62, 75Trivedi, S.P., 445, 142Trochimczuk, M., 446, 62, 75Trocsanyi, Z., 444, 539´ ´Troncon, C., 441, 479; 444, 491; 446, 62, 75Trucks, M., 445, 117Trudel, A., 444, 531Tsang, M.B., 446, 197Tseng, B., 446, 125Tsesmelis, E., 445, 439Tsirou, A., 441, 479; 444, 491; 446, 62, 75Tsur, E., 444, 539Tsuru, T., 446, 342Tsurugai, T., 443, 394Tsushima, K., 441, 27, 443, 26Tuchming, B., 445, 239Tully, C., 444, 503, 569; 445, 428; 446, 368Tung, K.L., 444, 503, 569; 445, 428; 446, 368Tuning, N., 443, 394Turcot, A.S., 444, 539Turkot, F., 445, 419Turlay, R., 446, 117Turluer, M.-L., 441, 479; 444, 491; 446, 62, 75Turner-Watson, M.F., 444, 539Turowiecki, A., 446, 179Tyapkin, I.A., 441, 479; 444, 491; 446, 62, 75Tymieniecka, T., 443, 394Tytgat, M., 442, 484; 444, 531Tzamarias, S., 441, 479; 444, 491; 446, 62, 75

Uchida, Y., 444, 503, 569; 445, 428; 446, 368Uchiyama, K., 442, 53Ueberschaer, B., 441, 479; 444, 491; 446, 62,

75Ulbricht, J., 444, 503, 569; 445, 428; 446, 368Ullaland, O., 441, 479; 444, 491; 446, 62, 75Ullrich, T., 444, 523Ulrichs, J., 445, 439Uman, I., 446, 349Umemori, K., 443, 394Unal, G., 446, 117Urciuoli, G.M., 442, 484; 444, 531Urin, M.H., 443, 51Uros, V., 445, 439Uusitalo, J., 443, 69, 82Uvarov, V., 441, 479; 444, 491; 446, 62, 75

Vacavant, V., 445, 439Vaiciulis, A., 443, 394

Vaidya, R., 442, 243Vairo, A., 442, 349Valassi, A., 445, 239Valdata-Nappi, M., 445, 439Valent, G., 445, 60Valente, E., 444, 503, 569; 445, 428; 446, 368Valenti, G., 441, 479; 444, 491; 446, 62, 75Vallage, B., 445, 239; 446, 117Vallazza, E., 441, 479; 444, 491; 446, 62, 75Valle, J.W.F., 441, 224Valuev, V., 445, 439Van Apeldoorn, G.W., 441, 479; 444, 491; 446,

62, 75Vance, S.E., 443, 45Van Dam, P., 441, 479; 444, 491; 446, 62, 75Van den Brand, J.F.J., 442, 484; 444, 531Van der Steenhoven, G., 442, 484; 444, 531Vander Velde, C., 444, 491; 446, 62, 75Van de Vyver, R., 442, 484; 444, 531Van Dierendonck, D., 444, 503, 569; 445, 428;

446, 368Van Doninck, W.K., 444, 491; 446, 62, 75Vandoren, S., 442, 145Van Eijk, C.W.E., 444, 38, 43, 52Van Eldik, J., 441, 479; 444, 491; 446, 62, 75Van Gemmeren, P., 445, 239Van Gulik, R., 444, 503, 569; 445, 428; 446,

368Van Hoek, W.C., 444, 503, 569; 445, 428; 446,

368Van Holten, J.W., 442, 185Vanhove, P., 444, 196Van Hunen, J.J., 442, 484; 444, 531Van Isacker, P., 443, 16, 82Van Kooten, R., 444, 539Van Lysebetten, A., 441, 479; 444, 491; 446,

62, 75Van Mil, A.J.W., 444, 503, 569; 445, 428; 446,

368Van Neck, D., 441, 17Van Neerven, W.L., 445, 206, 214Vannerem, P., 444, 539Van Nespen, W., 441, 1Van Nieuwenhuizen, P., 446, 247Vannini, C., 445, 239Vannucci, F., 445, 439Van Pee, H., 444, 555; 445, 20Van Rhee, T., 444, 503, 569; 445, 428; 446,

368Van Sighem, A., 443, 394Van Vulpen, I., 441, 479; 444, 491; 446, 62, 75Van Weert, Ch.G., 444, 463Varela, J., 444, 516Varvell, K.E., 445, 439Vassilevskaya, L.A., 446, 378Vassiliou, M., 444, 523Vassilopoulos, N., 441, 479; 444, 491; 446, 62,

75


Vattolo, D., 446, 117Vayaki, A., 445, 239Vaz, C., 442, 90Vazeille, F., 444, 516Vazquez-Mozo, M.A., 441, 40´Von Dombrowski, S., 446, 349Vegni, G., 441, 479; 444, 491; 446, 62, 75Veillet, J.-J., 445, 239Velasco, M., 446, 117Veltri, M., 445, 439Venables, M., 446, 342Vento, V., 442, 28Ventura, L., 441, 479; 444, 491; 446, 62, 75Venturi, A., 445, 239Venus, W., 441, 479; 444, 491; 446, 62, 75Verbeure, F., 441, 479; 444, 491; 446, 62, 75Vercesi, V., 445, 439Verdini, P.G., 445, 239Veres, G., 444, 523Verkerke, W., 443, 394Verkindt, D., 445, 439Verlato, M., 441, 479; 444, 491; 446, 62, 75Veropoulos, G., 442, 374Vertogradov, L.S., 441, 479; 444, 491; 446, 62,

75Verzi, V., 441, 479; 444, 491; 446, 62, 75Verzocchi, M., 444, 539Vesztergombi, G., 444, 503, 523, 569; 445,

428; 446, 368Vetlitsky, I., 444, 503, 569; 445, 428; 446, 368Vetterli, M.C., 442, 484; 444, 531Videau, H., 445, 239Videau, I., 445, 239Vieira, J-M., 445, 439Viertel, G., 444, 503, 569; 445, 428; 446, 368Vilanova, D., 441, 479; 444, 491; 446, 62, 75Vilja, I., 445, 199Villa, S., 444, 503, 569; 445, 428; 446, 368Villalobos Baillie, O., 446, 342Vincter, M., 442, 484; 444, 531Vinogradova, T., 445, 439Virey, J.-M., 441, 376Vissani, F., 443, 191Visser, J., 442, 484; 444, 531Vitale, L., 441, 479; 444, 491; 446, 62, 75Vitale, P., 441, 69Vivargent, M., 444, 503, 569; 445, 428; 446,

368Vlachos, N.D., 441, 46; 446, 306Vlachos, S., 444, 38, 43, 52, 503, 569; 445,

428; 446, 368Vlasov, E., 441, 479; 444, 491; 446, 62, 75Vo, M-K., 445, 439Voci, C., 443, 394; 444, 111Vodopyanov, A.S., 441, 479; 444, 491; 446,

62, 75Vogel, H., 444, 503, 569; 445, 428; 446, 368Vogl, W., 445, 20

Vogt, H., 444, 503, 569; 445, 428; 446, 368Volcker, C., 446, 349¨Volk, E., 442, 484; 444, 531Volkov, S., 445, 439Vollmer, C., 444, 491; 446, 62, 75Von der Mey, M., 444, 503, 569; 445, 428;

446, 368Von Krogh, J., 444, 539Von Neumann-Cosel, P., 443, 1Von Torne, E., 444, 539¨Von Wimmersperg-Toeller, J.H., 445, 239Vorobiev, I., 444, 503, 569; 445, 428; 446, 368Vorobyov, A.A., 444, 503, 569; 445, 428; 446,

368Vorvolakos, A., 444, 503, 569; 445, 428; 446,

368Voss, H., 444, 539Vossebeld, J., 443, 394Vossnack, O., 446, 117Votano, L., 443, 394Votruba, M.F., 446, 342Voulgaris, G., 441, 479; 444, 491; 446, 62, 75Vranic, D., 444, 523´Vrba, V., 441, 479; 444, 491; 446, 62, 75Vreeswijk, M., 445, 239

Waananen, A., 445, 239¨¨ ¨Wachsmuth, H., 445, 239Wackerle, F., 444, 539¨WA102 Collaboration, 446, 342Wadhwa, M., 444, 503, 569; 445, 428; 446,

368Wagner, A., 444, 539Wagner, C.E.M., 441, 205Wagner, G.J., 443, 77; 446, 179, 363Wahl, H., 446, 117Wahlen, H., 441, 479; 444, 491; 446, 62, 75Wakai, A., 445, 14Walck, C., 441, 479; 444, 491; 446, 62, 75Walczak, R., 443, 394Walker, R., 443, 394Wallis-Plachner, S., 446, 349Wallraff, W., 444, 503, 569; 445, 428; 446, 368Walsh, J., 445, 239Walter, R.L., 444, 252Walther, D., 446, 349Wander, W., 442, 484; 444, 531Wands, D., 443, 97Wang, F., 444, 523; 446, 356Wang, J.C., 444, 503, 569; 445, 428; 446, 368Wang, L.S., 446, 356Wang, L.Z., 446, 356Wang, M., 446, 356Wang, P., 446, 356Wang, P.L., 446, 356Wang, S., 441, 473Wang, S.M., 443, 394; 446, 356Wang, T., 445, 239


Wang, T.J., 446, 356Wang, X.-N., 443, 45; 444, 245Wang, X.L., 444, 503, 569; 445, 428; 446, 368Wang, X.N., 444, 237Wang, Y.Y., 446, 356Wang, Z., 441, 473Wang, Z.M., 444, 503, 569; 445, 428; 446, 368Ward, C.P., 444, 539Ward, D.R., 444, 539Ward, J.J., 445, 239Warner, D.D., 443, 16Warner, S., 442, 266Waroquier, M., 441, 17Wasserbaech, S., 445, 239Watanabe, S., 445, 449Waters, D.S., 443, 394Watkins, P.M., 444, 539Watson, A.T., 444, 539Watson, D.L., 444, 260Watson, N.K., 444, 539Waugh, R., 443, 394Weathers, L., 446, 197Webber, B.R., 444, 81Weber, A., 443, 394; 444, 503, 569; 445, 428;

446, 368Weber, F., 445, 439Weber, H., 442, 443Weber, P., 444, 38, 43, 52Wedemeyer, R., 445, 20Weerasundara, D.D., 444, 523Wehnes, F., 444, 555; 445, 20Wei, C.L., 446, 356Weibe, S., 444, 267Weiler, T.J., 442, 255Weiser, C., 441, 479; 444, 491; 446, 62, 75Weiss-Babai, R., 445, 449Weisse, T., 445, 439Weissman, L., 446, 22Welch, T.P., 444, 531Wells, J.D., 443, 196; 445, 178Wells, P.S., 444, 539Wenig, S., 444, 523Werlen, M., 445, 439Wermes, N., 444, 539Werner, S., 445, 239West, P.C., 444, 341Westphal, D., 443, 394Westphalen, J., 446, 209Whisnant, K., 442, 255White, D., 446, 197White, J.S., 444, 539White, T.O., 446, 117Whiteley, C.R., 444, 252Whitmore, J.J., 443, 394Whitten, C., 444, 523Whitton, M., 444, 252Wichmann, R., 443, 394Wick, K., 443, 394

Wicke, D., 441, 479; 444, 491; 446, 62, 75Wickens, J.H., 441, 479; 444, 491; 446, 62, 75Wieber, H., 443, 394Wiedenmann, W., 445, 239Wiedner, U., 446, 349Wiegers, B., 444, 555; 445, 20Wieland, F.W., 444, 555; 445, 20Wiener, J., 445, 449Wienold, T., 444, 523Wiese, U.-J., 443, 338Wigger, O., 444, 38, 43, 52Wiggers, L., 443, 394Wildschek, T., 443, 394Wilhelm, R., 446, 117Wilhelmi, Z., 446, 179Wilkin, C., 445, 423Wilkinson, G.R., 441, 479; 444, 491; 446, 62,

75Williams, A.G., 441, 27Williams, D.C., 443, 394Williams, M.D., 445, 239Williams, M.I., 445, 239Williams, R.W., 445, 239Williamson, S.E., 442, 484; 444, 531Willis, N., 445, 423Wilson, F., 445, 439Wilson, G.W., 444, 539Wilson, J.A., 444, 539Wilson, J.N., 443, 89Wing, M., 443, 394Wingerter, I., 446, 117Winharting, A., 446, 117Winter, M., 441, 479; 444, 491; 446, 62, 75Winton, L.J., 445, 439Wirrer, G., 446, 117Wise, T., 442, 484; 444, 531Wißkirchen, J., 444, 555; 445, 20Wislicki, W., 446, 117Witchey, N., 445, 449Witek, M., 441, 479; 444, 491; 446, 62, 75Witten, L., 442, 90Wittgen, M., 446, 117Wittmack, K., 446, 349Wodarczyk, M., 443, 394Wolf, A., 445, 20Wolf, G., 441, 479, 443, 394; 444, 491; 445,

239; 446, 62, 75Wolfle, S., 443, 394¨Wolin, E., 445, 449Woller, K., 442, 484; 444, 531Wollmer, U., 443, 394Wolter, M., 444, 38, 43, 52Wood, L., 444, 523Wotton, S.A., 446, 117Wright, A.E., 445, 239Wroblewski, A.K., 443, 394´Wronka, S., 446, 117Wu, S.L., 445, 239


Wu, S.X., 444, 503, 569; 445, 428; 446, 368Wu, X., 445, 239Wu, Y.G., 446, 356Wu, Z.C., 445, 274Wunsch, M., 445, 239Wurzinger, R., 445, 423Wyatt, T.R., 444, 539Wynhoff, S., 444, 503, 569; 445, 428; 446, 368Wyss, R., 443, 69

Xi, D.M., 446, 356Xia, X.M., 446, 356Xie, P.P., 446, 356Xie, Y., 445, 239; 446, 356Xie, Y.H., 446, 356Xing, Z., 443, 365Xiong, C.-S., 441, 155Xu, G.F., 446, 356Xu, J., 444, 503, 569; 445, 428; 446, 368Xu, N., 444, 523Xu, R., 445, 239Xu, Z.Z., 444, 503, 569; 445, 428; 446, 368Xue, S., 445, 239Xue, S.T., 446, 356

Yabsley, B.D., 445, 439Yamada, S., 443, 394Yamashita, S., 444, 267, 539Yamashita, T., 443, 394Yamauchi, K., 443, 394Yamazaki, Y., 443, 394Yan, J., 446, 356Yan, Q.-S., 442, 209Yan, W.G., 446, 356Yanagida, T., 445, 399Yang, B.Z., 444, 503, 569; 445, 428; 446, 368Yang, C.G., 444, 503, 569; 445, 428; 446, 368Yang, C.M., 446, 356Yang, C.Y., 446, 356Yang, H.J., 444, 503, 569; 445, 428; 446, 368Yang, J., 446, 356Yang, M., 444, 503, 569; 445, 428; 446, 368Yang, S.-K., 441, 155Yang, S.M., 445, 449Yang, X.F., 446, 356Yasu, Y., 446, 342Yasuda, O., 443, 185Yasuhira, M., 441, 9Yates, T.A., 444, 523Ye, J.B., 444, 503, 569; 445, 428; 446, 368Ye, M.H., 446, 356Ye, S.W., 446, 356Ye, Y.X., 446, 356Yeh, S.C., 444, 503, 569; 445, 428; 446, 368Yekutieli, G., 444, 539Yi, D., 445, 449Yi, J., 441, 479; 444, 491; 446, 62, 75Yokkaichi, S., 444, 267

Yoneyama, S., 442, 484; 444, 531Yonnet, J., 445, 423Yoshida, K., 444, 267Yoshida, M., 444, 267Yoshida, R., 443, 394Yoshida, S., 445, 449Yoshida, T., 444, 267Yoshioka, K., 444, 373You, J.M., 444, 503, 569; 445, 428; 446, 368Youngman, C., 443, 394Yu, C.S., 446, 356Yu, C.X., 446, 356Yu, G.W., 446, 356Yu, Y.H., 446, 356Yu, Z.Q., 446, 356Yuan, C.Z., 446, 356Yuan, Y., 446, 356Yuanfang, W., 444, 563Yushchenko, O., 441, 479; 444, 491; 446, 62,

75

Zabierowski, J., 446, 179Zaccone, H., 445, 439Zacek, V., 444, 539Zachariadou, K., 445, 239Zahed, I., 442, 300; 446, 9Zaitsev, A., 441, 479; 444, 491; 446, 62Zajac, J., 443, 394Zakharov, B.G., 442, 398Zakrzewski, J.A., 443, 394Zalewska, A., 441, 479; 444, 491; 446, 62, 75Zalewski, P., 441, 479; 444, 491; 446, 62, 75Zalite, A., 444, 503; 446, 368Zalite, An., 444, 569; 445, 428Zalite, Yu., 444, 503, 569; 445, 428; 446, 368Zaliznyak, R., 445, 449Zamora Garcia, Y., 443, 394Zanelli, J., 444, 451Zanon, D., 444, 332Zapfe-Duren, K., 444, 531¨Zarnecki, A.F., 443, 394Zavrtanik, D., 441, 479; 444, 38, 43, 52, 491;

446, 62, 75Zawiejski, L., 443, 394Zeitnitz, C., 445, 239Zemp, P., 444, 503, 569; 445, 428; 446, 368Zeng, Y., 444, 503, 569; 445, 428; 446, 368Zer-Zion, D., 444, 539Zernov, A., 446, 179Zerwas, D., 445, 239Zerwas, P.M., 441, 383Zetsche, F., 443, 394Zeuner, W., 443, 394Zevgolatakos, E., 441, 479; 444, 491; 446, 62,

75Zghiche, A., 445, 423Zhang, B., 444, 237Zhang, B.Y., 446, 356


Zhang, C., 445, 449Zhang, C.C., 446, 356Zhang, D., 446, 356Zhang, D.-X., 445, 394; 446, 285Zhang, D.H., 446, 356Zhang, H.L., 446, 356Zhang, J., 445, 239; 446, 356Zhang, J.W., 446, 356Zhang, L., 445, 239Zhang, L.S., 446, 356Zhang, Q.J., 446, 356Zhang, S.Q., 446, 356Zhang, X.F., 444, 237Zhang, X.Y., 446, 356Zhang, Y.Y., 446, 356Zhang, Z.P., 444, 503, 569; 445, 428; 446, 368Zhao, D.X., 446, 356Zhao, H.W., 446, 356Zhao, J., 446, 356Zhao, J.W., 446, 356Zhao, M., 446, 356Zhao, W., 445, 239Zhao, W.R., 446, 356Zhao, Z.G., 446, 356Zheng, J.P., 446, 356Zheng, L.S., 446, 356Zheng, Z.P., 446, 356Zhou, B., 444, 503, 569; 445, 428; 446, 368Zhou, B.-R., 444, 455Zhou, B.Q., 446, 356Zhou, G.P., 446, 356Zhou, H.S., 446, 356Zhou, J.-G., 445, 287Zhou, L., 446, 356Zhu, G.Y., 444, 503, 569; 445, 428; 446, 368Zhu, K.J., 446, 356

Zhu, Q., 443, 394Zhu, Q.M., 446, 356Zhu, R.Y., 444, 503, 569; 445, 428; 446, 368Zhu, S.H., 444, 224Zhu, X.-Z., 444, 523Zhu, Y.C., 446, 356Zhu, Y.S., 446, 356Zhuang, B.A., 446, 356Zichichi, A., 443, 394; 444, 503, 569; 445, 428;

446, 368Ziegler, F., 444, 503, 569; 445, 428; 446, 368Ziegler, T., 445, 239Zilizi, G., 444, 503, 569; 445, 428; 446, 368Zimanyi, J., 444, 523Zimin, N.I., 441, 479; 444, 491; 446, 62, 75Zimmerman, D., 444, 38, 43, 52Zinchenko, A., 446, 117Zinovjev, G.M., 443, 387Ziolkowski, M., 446, 117Zioutas, K., 443, 201Zito, G., 445, 239Złomanczuk, J., 446, 179´Zobernig, G., 445, 239Zohrabian, H., 442, 484; 444, 531Zolin, L., 445, 14Zotkin, S.A., 443, 394Zoupanos, G., 441, 235Zuber, J.-B., 444, 163Zuber, K., 445, 439Zucchelli, G.C., 441, 479; 444, 491; 446, 62,

75Zuccon, P., 445, 439Zumerle, G., 441, 479; 444, 491; 446, 62, 75Zurmuhle, R., 444, 531¨Zybert, R., 444, 523

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