Nuclear PhysicsANorth-HnIland

2(1992)587-615

asielastic neutrino (antineutrino) reactions in nuclei

S.K. Singh' and E. Oset

Received 6 January 1992

1. Introduction

the axial-vector form factor of the nucle

Departarnento de Fisica Tedrica and IFIC, Centro Mixto Universidad de Valencia-CSIC,46100 Burjassot, Valencia, Spain

Abstract : Quasielastic neutrino and antineutrino reactions in nuclei are studied to investigate the effectof the nuclear medium on the determination of axial dipole mass M, in nuclei . The calculationsare done in the local density approximation and var;ous nuclear effects like Pauli blocking, Ftrmimotion of nucleons and strong renormalization effects in the spin-isospin channel are taken intoaccount. The nuclear effects are found to be quite large at low q', but the data in this region aretoo meagre to influence the determination of M, . The experimental data on differential and totalcross sections are found to be consistent with MA as determined from deuterium experimcats andno conclusive evidence for quenching of MA in nuclei is found. On the other hand, the latestneutrino experiment by SKAT collaboration is found to be inconsistent with our theoretical resultsas well as with other neutrino experiments.

The axial-vector form factor FA(q2) of the nucleon is parametrized in a dipoleform, ie .,

FA(q2) =FA(O)/(l - q2/M2)2,A

where MA is the dipole mass and FA(O) is the axial charge . FA(O) has been determinedfrom the A-decay experiments of the neutron) and the dipole mass MA has beendetermined either from the threshold electropion-production experiments z'3) orfrom the quasielastic neutrino and antineutrino reaciioiis . The neuirino and iindne-utrino reactions are done in hydrogen, deuterium or heavy nuclei . While the experi-ments with hydrogen and deuterium targets are preferred due to the simplicity intheir analysis 4-11) the experiments in heavy nuclei were the first ones to be performedand are still being done 12-24) . The analysis of neutrino and antinuetrino reactionsin nuclei, to determine the dipole mass MA involves some additional model depen-dence due to the nuclear structure . Nevertheless these experiments provide anopportunity to study the axial vector form factor of the nucleon inside the nuclearmedium . In recent years the study of nucleon properties and their modifications inthe nuclear medium has attracted much attention as it is intimately related with the

' On leave of academic pursuit from Aligarh Muslim University, Aligarh, India.

0375-9474/92/$05 .00 ©1992 - Elsevier Science Publishers B.V. All rights reserved

588

vacuum in the nuclear environment . This QCD vacuum in the nucleus mightrent from its bee-space value and thus change the nucleon properties in the

nuclear medium 25) . These changes are expected to be in addition to the changesinduced by the nuclear correlations, many-body elects etc., which have beencalculated in a fairly reliable way for various quantities . For example, the quenchingof Q and the magnetic moment jA of the nucleon inside nuclei due to nuclearany-body effects are well known 26-29) . The experimental data on moon capture

rates for various nuclei are explained very well when the quenching of weakinteraction coupling constants fA , Fm , etc ., are taken into account in a reasonablemodel Such an analysis has not been done for the neutrino nucleus reactionsand the effect of quenching has not been taken into account in the determinationf the dipole mass 114A -

revise value Of MA is very desirable as it plays an important role in the analysisof neutral current reaction of y and Pp scattering to determine the isoscalar formfactors

Such an analysis has given rise to a strangeness component As of theproton spin in agreement with the EMC experiment . However this value of As isvery sensitive to MAas a 5% change inMA would imply As ::=0 [ref. 32)] . Also, a10-151,16 reduction in the value of MA determined from the nuclear experimentsafter nuclear many-body effects have been taken into account in the analysis, wouldlend support to the recent suggestion or the scaling of effective chiral lagrangians,in a nuclear medium "') .

recently available values of A4A as determined from the nuclear experiments areconsistently smaller than the values quoted from the deuterium experiments whencentral values of MA are compared. However, the quoted error bars are such thatthe two values can be interpreted to be in agreement with each other. The existinganal~ , ses of nuclear experiments use a simple Fermi-gas model and free-nucleonvalues of the weak-interaction coupling constants without taking into account theeffect of the nuclear medium . The major uncertainty in the existing analyses of theseexperiments is attributed to the poor knowledge of the neutrino (antineutrino)spectrum and of the nuclear structure . In this paper we have studied neutrino andantineutrino reactions in a model where nuclear effects are taken into account in abetter way than the simple Fermi-gas model . The effect of renormalization due touclear medium is explicitly calculated . We apply the local ,density approximation(LDA) to calculate the inclusive neutrino (antinueutrino) cross sections in nuclei .The effect of Pauli blocking and Fermi momentum is taken into account by calculat-ing the imaginary part of the Lindhard function using relativistic kinematics whichis appropriate for large momentum transfers . The effect of quenching of the weak-interaction coupling constants in the nuclear medium is taken into account bycalculating the ph and Ah excitations using -7r- and p-exchange potentials andcorrelated nuclear wave functions and summing the iteration of these excitationsto all orders . Such a model applied to moon capture, radiative pion capture andhypernuclear decay in nuclei has given satisfactory results

We believe that

S. . Singh, E Oset / Quasielastic P( as) reactions

The

matrix

element

for

the

neutrino-nucleon

reaction

v~(k)+ n(p) ->g-(k')+p(p') is given by

where

S.IC. Singh, E. Oset / Quasielastic v(v) reactions 589

by this method nuclear erects are properly taken into account . These are found tobe quite large at small q2 where, unfortunately, there is very little data . At high q'a comparison ofthe experimental data, on differential cross sections with our presentcalculations, does not provide a compelling evidence for a reduced A . The totalcross-section data however can support a reduced MA but the error bars are toolarge to draw definite conclusions . We, however, find that a recent neutrino experi-ment 24) is inconsistent with our results as it does not see any nuclear effects evenat very low q`, where theoretically large effects are expected. However, one shouldnote that this experiment is also in disagreement with other experiments .The plan of this paper is as follows. In sect. 2, we review the existing theoretical

and experimental results for MA from neutrino and antinueutrino reactions . In sect .3 we describe our calculations and in sect . 4 compare our results with the experi-mental data . In sect . 5, we conclude by summarising our findings and their implica-tions for future experiments .

2 . Axial-vector for

factor from neutrino reactions

I" = a(k')y'(1 - YO u(k)v

J, = ~(P I )

Fi(g2)y~. + F2(g2)1~N~

+ FÂ(q 2)yjLY5+ Fp(g2)qa'Ys u(P) -

(2.2)

In eq. (2.2), the isovector form factors Fi,2(g 2) are determined in terms of theelectromagnetic form factor F,, 2(g2) for the nucleons assuming conserved vectorcurrents (c.v.c .), which are given in terms of the experimentally determined Sachsform factors GE(g2 ) and GM(g2 ) by the following

Fi(g 2 ) =

1- q2

-1

GF(g 2 ) -

q2

Giw(g2)4M'` 4M`

where

2 -4F~(q2) - (

_

)'

4M2

CGNI(g 2 ) - GF(g2)l ~

_

1

q`AnGE(g2)

(1 - q2/M~)' 4M

_ 1+ "p- hc.n

GM(q~) -

21M2)2 (2 .3)

with gp- p n = 3 .70+6, q = p'- p and M~ = 0.71 GeV2.

The pseudoscalar form factor F~p(q2) is dominated by the pion pole and is givenin terms of the Goldberger-Treirran relation near q2-0 if p .c.a.c . is assumed . We

S90

Ss, Sitkgh, E. Osee / Quasielashe 0 ,) reactions

Or relation is valid Nr high q2 as well and witeqI MV ( 2 )A

ass-1111l8 that

If notatithe ass

factresults on dieutrino nucle

in a dipole form in aand the axial dipole

In the literature, anvector dominance-quarke axial vector form fact

structure form factor givet

v ere

r is t

(q 2 ) =

licity

s wit

will omit the superscript v in what follows.tions oF c.v.,c . and p.c.a.c ., the fairly undetermined form

e axial-vector form factor which is determined from the experimentalerential cross section dadq2 and the total cross section ca( B,) from

s reactions. The axial-vector form factor is generally parametrizedalogy with the Sachs form factor defined in equation (1 .1)ass MAis determined from the experimental data .ther parametrization for FA( q,2 ), which is motivated by the

et considerations is also used. In this para etrizationr is given in terms of the A, pole multiplied by a nucleon

y the quark mode1 36 ).

I

q-[ (20)ex 2R2(1- 1)]q.1 A,) p 1(I-q M

4M

si

du Gdq2

where AfAl is the mass ofdetermined from quark

ass MA is determined frdu

0 do-

1.= I

--

'P(qI

-q

FA(q2)[ F, (q2)+ il~,(q 2)],(q

tos't9c [A(q 22

) :t:8M,,

"C(q2)

F2(q2)+F2(q2)_

q

2 (2A)

I particle and R2 =6GeV-2 is taken as a parameterel phenomenology. The form factor FA(q2 ) or the dipoledo-/dq2 given in the laboratory system by -

I I I TrS(E,+(2 .6)

where p,,, is the muon momentum and a complete expression for Il i (MilEi ) Y, Y' I Tj2

is given in the appendix . We use this exact expression throughout in the presentwork However, at relatively high neutrino energies where the experiments areerformed, the muon mass can be neglected and then the pseudoscalar form factoroes not contribute . In this limit, the expression for the differential cross section

dor/dq 2 is considerably simplified and one gets

2)

_U

2) I~S~_U(q

s

+C(qm2 W (2.7)

F22(9z)- q2

2)F2(q2)~~ FI(qM

I I

(2A)

and the + (-) sign refer to the neutrino (antineutrino) scattering. The expressiongiven in eq . (2 .8) has been used to analyze the experimental data in deuterium andin various other nuclei. In the following we summarise the experimental results .

2 .1 . NEUTRINO EXPERIMENTS IN DEUTERIUM

The experiments done with deuterium targets utilize only neutrino beams .process studied, therefore, is P, + d --> 1A - + p + P. 11 n order to get the w uss sectionsfor the v+ n -> A - + p process from deuterium reactions one has to theoreticallyestimate the corrections due to the use of deuterium targets and the final-stateinteractions due to the two nucleons in the final state . Such calculations have beenmade by using closure over the final dinucleon states and using realistic potentkils

- -for the deuteron such as the Reid potential or the Paris pte"Jitial 47 412 ) .

'_O

The effectof using deuterium as target reduces the - free-nucleon cross section to _35-40% atsmall q2 but these effects become small with increasing q`. At about q

I== -0.2 Clef`'

the various deuterium effects due to Pauli principle, Fermi motion, etc., becomenegligible and the deuterium cross section approaches the free-nucleon cross section .Thus deuterium effects are not imperLant around q ' ::z- -0.3--0.4 GeV2/C2 wheredeviations from dipole parametrization for F

A(q2) have been noticed . The dipoleparametrization does not describe very well the experimental data at very love q2,but in this range of q', the experimental data are sparse and have large uncertainties .In the relevant range of q2 the simple calculations done in the closure approximationare good enough to describe the data . The various corrections to the closureapproximations have been calculated and found to be small in the range of q2 >

-12 GeV. These include corrections due to relativistic effects, final state interactionsand meson exchange current effect 39,40-42 ) and have been included in the analysis

of the experimental data ") . Thus the determination of MA from the neutrinoexperiments in deuterium are quite reliable . We present the results for MA fromvarious deuterium experiments in table 1 .

The value of M, in GeV as determined from neutrino and antineutrino experiments in deuteriumand nuclei

S.K. Singh, E Oset / Quasielastic v( ;) reactions

TABLE I

Av . 1 .05 _t 0.03

1.06:t 0.05± 0.14a

1 ) 24)

0.69±0.4418)0.94=0.17")

') See text for a discussion .b

) These experiments use only very high q 2 data .

59 1

e

deuterium freon freon-propane others )

1 .05 ± 0.14 0.7S±0.22") O.î±0.2'3) 0.99t0.11`')1 .00 ± 0.05 0.65±0.40!") 0.87t0.17'9) 1.00+0.04 22)

1.07t0.05') 0.88±0.20' 8) 0.99t0.12'9)1.07t0.04 0.96t0.16"') 0.91 t0.40 ' 6)I.00±0.35'2) 0.71 f0.1 t0.2 ,1 'l)

592

2.1 NEUTRIN

e axial dipole mass AlAhas also ken determined from neutrino and antine-utrino experiments in nuclei . The experiments have been done in freon (CF3Br),,freon-propane mixture (CF-,Br-CH,,), Fe, Ne and Al with neutrino beams and infreom, freon-propane mixture, and Fe nuclei with antineutrino beams. In analyses

these experiments, a simple Fermi-gas model has been used in most cases. Inis model the nuclear effects are taken into account by multiplying the free-nucleon

cress section given in ed. (2-8) by a nuclear factor RN (q2 ) given by refs . 43,44) .t

w

a

ere

table 1 .

S. . Singh, E. Oset / Quasielastic vj v) reactions

qANTINEUTRINO) EXPERIMENTS IN NUCLEI

Z, A are the neutron, proton, nucleon number and q is the momentum transferthe nucleon corresponding to the free-nucleon kinematics, and kF is the Fermi

momentum of the struck nucleon. Improvements in this simple Fermi-gas modelcalculations exist in literature, but only for some simple nucleus like 12C and "0

have not been extended to the nuclei, relevant for neutrino reactions 45,46) .

Neutrino-nucleus reactions in this energy range have also been calculated in theclosure approximation with harmonic-oscillator potentials for some closed-shellnuclei but have not been used in the analysis of neutrino experiments 43,44) Theaxial dipole mass JW, as determined from these experiments, is summarised in

We see from table I that M, as determined from the nuclear experiments, isconsistently lower than the , as determined from the deuterium experiments.oes it signal P change of MA in nuclei? Any conclusion based in this observation

.ao_'.d tie taken with some caution due to the large errors on the values of A , asdetermined prom the nuclei, which are mainly due to the poor knowledge of neutrinospectra*. This observation however becomes quae important in view of the recenttheoretical interest in the change of nucleon properties inside the nuclear mediumand search for experimental support for it in electroweak interactions in nuclei .There have been some attempts to look for this change through modification of

for Y<U-V

(2N

1/3

12Z\ 1/3u-

A )

k A )

for u-v<X<u+v

(2.9)

for X>u+v,

(2-10)

* The uncertainly due to poor knowledge of nuclear structure effect as compared to deuterium targetis expected to be large only at small q2 , where there are very few experimental results . See sect . 4 for details .

S.IC. Singh, E. Oset / Quasielastic v(v) reactions 593

electromagnetic form factors in nuclei"-"), and one could expect from there alowering of MA . A lower value of MA would result in the lowering of the neutrinonuclear cross sections per nucleon as compared to the free nucleon .

3 . Neutrino nucleus scattering i

the local density a

roi

do

We assume that the neutrino, while moving through the nucleus, scatters fromthe neutron moving in the finite nucleus of density pn(r) . The neutrino cross sectionin the nucleus is then given by

o,(A)= I pn(r) d3rQ(v+n --~- ac - +p) ,

(3.1)

where o,(v + n -> /u, - + p) is given by eq. (2.6) . However, we have to consider thatthere is a momentum distribution for the neutrons and protons in the nucleus . Inthe local density approximation it is given by a Fermi distribution with local Fermimomenta

N

1/3

2 Z

1/3PFn=

37r` N p(r)

PFp=

3~

p(r)

-

(3.2)

The cross section in the nucleus is then given by3

-

pn(r) d3r_

d P

MEL 1 TÎ 2S(Ev+En__E,, - Ep) .

(3.3)(2?r) i Ei

The energy conserving 8-function has to be treated with some caution as the initialand final nucleons are no longer free but they are moving in the Fermi sea ofneutrons and protons . Accordingly the neutron and proton momenta given by pnand pn+ q have to satisfy the constraint that

Pn CPFn 9

I pn + q I> PFp ,)

where PFn and PFp are the Fermi momentum of neutrons and protons. In addition,an integration over the initial momentum has to be performed. These constraintsare incorporated by replacing -7rpn6 (E v + En - E,,, - Ep) (Mp1 Ep)(Mn1 En) ineq. (3 .3) by the imaginary part of the Lindhard function corresponding to the phexcitation shown in fig . 1 .The Lindhard function U(p, - p,,) for fig. 1, is given by 5° )

2 d3Pn

nl(Pn)(1 °n2(Pn+q»

Mp np -P

-

(2 ,U)3

E � -Eu.+En(Pn)-Ep(q+P)+iE Ep En,

(3.4)

where n, and n 2 are ocupation numbers for neutron and proton and q = pv - p". .Taking the imaginary part of the Lindhard function in eq. (3.4) corresponds toputting the intermediate particles in fig . 1 on sheil, thereby describing the processv+n --3-, ,u-+p. In the static limit for the neutron (En -->M) and neglecting any Pauli

$94

S!

ex

0

serti

Fig. 1 . Feynmann diagram for the neutrino self energy related to vn - ju -p processes in nuclei .

blockingprescri

erstun

ti

lifying eq. (3.5) and performing the phase-space integration, we get the followingressio

for the proton (n2 --> 0), one recovers the result for the free nucleons . Thisn therefore adequately describes the effect of Pauli blocking and Fermifor the process v + n- --> .u - + p inside the nucleus. The neutrino cross

er nuc!i!on in the nucleus is then given by

kinemstir constraints

S. X Si

2

where t=( q2_M2M +2E,E,,,)12Eîp, = 'Al

gh, E

[3 7TIpjr)] 2/3 _

3 .1 . STRONG RENORMALIZATION EFFECTS

set / Quasielastic P( ,) reactions

c 11 j ~L i Fi,

3E 11 T12 lm Ù(p,3N 4,ff Ei

[3,rr2pp(r )I-PI3,2M

P

(3-5)

for the differential cross section:

da 1

P".111

i -

Jq2=-

7r

r2 dr

r~ indp,

A.

,~H

Y, Y_ I T12 1M Cg p"' _

PIA

(3.6)p,, i = 1.

Ej

where gin and p"' are the minimum and maximum muon moments, fixed by theA

AL

abs(t) ---- I ,(3 .7)

In nuclei, the strength of electroweak couplings may change from their free-nucleon values due to the presence of strongly interacting nucleons . Though theexact conservation of vector current forbids any change in the charge coupling,other couplings like magnetic moment, axial charge and current and pseudoscalar

coupling of the nucleon may undergo changes . Due to the partially conserveaxial-current hypothesis (p.c.a.c .), the axial current is strongly coupled to the piofield in the nuclear medium and therefore axial couplings are more likely to changedue to pionic effects . To get an idea of these effects, if we look at the nonrelativisticlimit of eq. (2.2) we see the occurrence of FA r, F2 x ,r and Fp " qr terms inthe weak current which are linked to spin--isospin excitation . While F2 and Fp arecoupled to transverse and longitudinal channels, FA is coupled to both.

e couplingof these external probes to the mesonic effects can be depicted through the diagramshown in fig . 2, where the wavy line stands for the spin-isospin ph and Ah interactionand can be described by 7r, p exchanges modulated by the effect of nuclear shortrange correlations .

Generally this interaction V(q) can be represented by2

V(q) =

2 [ V1 i j + VL(gij -

i j)]UiojT ° -r

(3.8)

for the ph case and a similar one for the case of ph-®h interaction by substituting--> S, ,r --> T andf-*f* = 2.15f, V, is the strength of the potential in the longitudinal

channel and Vt is the strength of the potential in the transverse channel .

erepresentation into longitudinal and transverse channel is useful when one tries tosum the geometric series in fig . 2 where the longitudinal and transverse channelsdecouple and can be summed independently 5 ' ) . The renormalization of the variousterms in eq . (A.1) in the appendix due to the Feynman diagrams shown in fig . 2can be calculated in a straightforward way. We demonstrate this for the term FA-

This term contains contribution from the longitudinal and transverse part, which

S.K. Singh, E. Oset / Quasielastic v(v) reactions 595

Fig . 2 . Many-body Feynmann diagrams accounting for the medium polarization in the spin-isospinchannel driven by the vn -+ 1,-p transition .

ee

aking t

fIenoincoa

arate

sic diagram ofe lo it

(30 - Qqj )/ (I) and we

2, t

eq. A

d

re give

e imaginary pan, we get

Sin

, EI.S. Oset / Quasidastic P(;) reactionsgh

y

dcr

I [[

lr,~d°2 dr

7r j

j P-,.L

Im F2 C ---> F2 Im 0

1-A A

[3 11 -

-6Tr crp,U[qqj + 0, - qjqj ) II for ph excitation . After summing over the geometric

"al

~l(I - UVI ) and the transverse termQ%-Mqmter) where fJ is now 2

_to account for the isospin

. 30)

1 2 1U I-UY+ 3 1 - UY

~7r-2 1

_+-Ir

3 11 -- UV,1 21 -

Similarly, the term FA(q)2F,(q2) and Aq2) get only transverse contribution while2the terms F2(q2 ) and Fp(q2)FN,'q 2) get only longitudinal contributions and arePTenormalized as follows :

)) --> (F- F~FF A, A 2 UVJ2

(39)

he effects of _1h excitation are taken into account by including the Lindhardnotion for the _1h excitation in U and replacing U by U "-- UN + U.1 in the

inator of eqs . (3.10) and (3 .11) . The different couplings for N and A areorated in UNand U.., and then the same interaction I and Y is used for ph

excitations 50"). However, we only have Ian U for ph excitation in theerwor of eqs. (3A 0) and (3 .1 Y since w are looking at the quasielastic reaction

,ad not at -excitation or pion production . The Ah excitation i% only operative asa source of nuclear polarization . With these modifications, the renormalization

acts in the quasielastic neutrino-nucleus reaction can be calculated by using thecation

dp,,,L'-'-

f]

-'-"Y_Y_IT' 12 IM Cj( PV _P11) ,(3 .12)

P, i=g,v Ejwhere

I'1`

is

calculated

from

eq .

(A.1)

by replacing

FA2 - Ft2 , FA F2, 1A2~ I t2

AQ, F2

F:! , etc ., according to eqT (3AO) and (111) .In actual calculation from eqv (3h) and (3 .12), expressions of U for the asym-etric nuclear matter have been used retaining the relativistic expressions for the

energies in eq. (3.4) . However, while calculating the differential cross section fromeq. (3 .12), the expression of U for symmetric matter has been used but only in the

denominator to simplify the calculation . T " s approxi ation has we+r e well i~~other case ~°~ ;~) and is expected to be quite good in the present case . The followingforms have been used for the potentials V, and Va

_ ,-q _ + ~ ..

~ - q_

S. ~C. Singh, E. ®set / Quasielassie v(v) reacti®ns

~ =1 .3 CBeV, C,, = 2, ~,,, = 2.5 Ge`~!, ~ and rnF, are the pion and P-masses and ~° isthe Landau-

igdal parameter, taken to be 0.7 in the present caQ~a!~tions .

exults a

isc ssi®

:~Jc !gave used `âjc fu~~~narisYi`~ cle~re;opeû in s~ca . 3 to study the neutrino reactionsin various nuclei .

e have chosen to anaflyze those neutrino and antineutrinoreactions which have good statistics and are sensitive to nuclear effects .

e detailwdcalculations in deuterium and a simple Fermi-gas model of nuclei show that nucleareffects are important below q2 = -0.3 ~e~I2/ c` . VVe have therefore selected thoseexperiments which include the data below q2 = - 0.3 C~e~2/c2 and have been quotedroost often in'diterature. These include the CERN neutrino and antineutrino experi-ments of ~onnetti et cad. '~), l'ohl ~t cal. '9~ and Armanise et cal. 2 ") . Also analyzed arethe most recf;nt data of the S1{~;.T Collaboration 23,24) .

e recent data of

elikove: ~~ 22) or Asratyan ~t cal. 2' ) use only high-q2 d;~ta and the older CERN experi-rraents' 2- ' S ) have poor statistics and have not beeii analyzed .

In calculaatiog the differential çross sectior:, cqs. (3 .6) and (3.11) nave been used ."Two-parameter harmonic-oscillator density for ' zC, '~® and two-parameter Fe idensity far '9F and 8 'ßr which are determined from the electron scattering experi-ments have; been used 52) in eqs . (3.2) and (3 .7) . No .approximations in the kinematicshave been. made . The muon mass rn~, and therefore FP( q 2 ) has been retained in(A.1) . ïn a separate calculation we have studied the effect of neglecting rn~, andFp{q2 ) . The sensitivity of d~/dg2 to CIA is not aÉ~;cted by this approximation andeq. {2.7) can be used to find 1VIA from experiments . However in the results presentedbelow we have used the full expression given in eq. (A.1) .

before we present our results for these experiments, we will like to show the effectof nuclear effects in a simple way. For this we have plotted the ratio IZ(g2) definedas

(q2~~l 1

d~

da~

( vN),

2 --

q2

dq vA d9

~ 7

(3.13)

where vA denotes the neudrino-nucleus interaction, as a function of q2 and haveshown it in fig . 3a for '~(J where a comparison with other calculations has beenmade.

e see that the effect of 'auli blocking and Fermi motion of the nucleon in

9"

Sirgh, E Oset / Quasielastic vt~,) reactions

.

I

1. 2

v-T--T--T

I

I

I

I

I

I

I

I

:

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

i

2 Q eV2).2

.25

-4

.3

Fig. 3. (a) R,,( 22) for 160 : W closure approximation of Bell and Llwelynsmith 44 ) (dotted), (ii) Fermi-gasModel (dot-dashed), (iii) present calculations without renormalization (dashed) and (iv) with renormaliz-

ation (solid) ; (b) Same as (a) (ii), (iii) and (iv), for "Br.

4.1 . DIFFERENTIAL CROSS SECTION

S.K. Singh, E. Oset / Quasielastic P( er) reactions

599

the local density approximation ib similar to the results of calculations done in theclosure approximation with harmonic oscillator potential 43,44 ) except for very lowg2. When the effects of strong renormalizations are included, the cross sections areconsiderably reduced specially at low q

2 and become similar to the results ofcalculation done in a simple Fermi-gas model. The q2 dependence of this suppressionis however quite different in the two models. Similar results are obtained forantineutrino scattering with the difference that the q2 dependence for (9'z ) isdifferent for P and P scattering in our calculations .

In the following we will present results for the differential cross section as afunction of q2 and the total cross section a(Ej as a function of neutrino energy inthe local density approximation with and without the strong renormalization effects .Along with these two results, we also present the results in the simple Fermi-gasmodel and compare all the theoretical results with the experimental data for theexperiments considered . In a separate set of curves, we also study the sensitivity ofthe da/dq2 and QE.) for various values ofI consistent with the MA determfinedfrom deuterium experiments for the dipole form factor. Finally we also presentresults for the VDM-QM parametrization for the axial-vector form factor withMAI= 1 .14 GeV, as determined from the deuterium experiments .

The differential cross section as calculated from eqs . (3.6) and (3.11) has beenfolded with the neutrino and antineutrino spectrum given by the authors of therespective experiments . We have taken the published neutrino (antineutrino) spec-trum as such and tried to see whether the differential cross sections as obtainedfrom the neutrino reactions in nuclei are consistent with the value of MA derivedfrom the deuterium experiments . For this purpose a world average of MA =

1 .05 + 0.03 GeV for the dipole parametrization and A, = 1 . 14 GeV for the VIAM-QM parametrization has been used .

In fig . 4a-c we show the results for do-/dg2 versus q 2 for the neutrino experimentsof Bonnetti et al. "'), Pohl et a/. ' 9 ) and SKAT collaboration 23,24 ) . The experimentsof Bonnetti et al. and SKAT collaboration are in freon which are dominated by ' 9 Fand "'Br while the experiments of Pohl et al. are in freon-propane which aredominated by '`C . As a result the nuclear effects are expected to be larger in thefreon case than in the freon-propane case . Moreovcr, the freon experiments ofSKAT collaboration have high energy neutrinos (antineutrinos) with 3.0<E,,<30.0 GeV as compared with to the lower neutrino (antineutrino) beams of Bonnettiet al. which have neutrino (antineutrino) energies in the range of 1 .0 < E, < 15 .0 GeV.The result for differential cross sections are therefore expected to be somewhatdifferent for individual experiments even though it is independent of energy athigher energies, i .e . ., E, >M. However, in all cases we see a large reduction due tonuclear effects such as Pauh blocking, Fermi motion, etc., at smaller q

2 which

6010

Fig . 4 . dcr/dq :! versus q :! compared with neutrino experiments of (a) Pohl et al ' y ), (b) Bonnetti et al.' S )and (c) SKATcollaboration 2

.) . Fermi-gas model (long dash) ; present calculation withou! renormalization

q short dash) and with renormalization (solid) . Also shown ii-j (a) and (c) are the results for free-nucleoncase for Lomparision (dotted) . Differential cross section in units of [10- "s cm2/GeV2 ] .

ecomes neghgib!e for q2> 0.3 GeV/c2 (compare with free-nucleon case) . The effectof strong Tenormalization due to ph and Ah correlations is quite large in this regionwhere nuclear effects are important. When comparing the present results with thesimple Fermi-gas model, we see that the dimerential cross sections in the Fermi-gas

et are surprisingly close to our present results with strong renormalization takeninto account . The strong suppression in the differential cross section due to nucleareffects seems to be quite we!l simulated by the simple Fermi-gas calculations, Animproved permi-gas calculation with local density approximation will overestimatethe differential cross section especially at lower q2 . In comparing these results withthe experimental results of Bonnetti et al and Pohl et al. we find a good agreementexcept at very low q2 , where the experiments show still stronger suppression . Theeffect of strong renormalization helps in imMoving the agreement, but the quality

0

b0

. Singh, E. Oset / Quasielastic P(T,) reactions

.5

- 1

1 .5

2Q'-(GeV**2)

tua

'Ilb10

Z5

L-

S.K . Singh, E Osel / Quasielasfic P(~) reactions

601

1 " 1 -r- [--F-"

FREON(SKAT)

01 1 1 1 1 1 1 1 1 1 1 1 1

.4

.6 .8

1 .2W(GeV2)

Fig. 4-co&nued

t

exIl

IergCleo

M

13"IbV

&K. Singh, E. Oser / Quasielastic s ,6,-) reactions

to in the low-q -' region of nuclear sensitivity is not good enough to make aled comparison . The data at low q2 can support a smaller MA without disturbing

the agreement at higher q-, but as we show in fig . 5 the range allowed for MA bye deuteriurn experiments does give a fair description of the data when dipole form

factors am use& The description with the VDM-QM parametrization with the worldaverage of I'VIA , = 1 . 14 GeV gives slightly higher value of the da/dq2

, specially atsmaller q'. In fact the results of VDM-QM parametrization with MAI =1 .14 GeVare quite similar to the results of dipole parametrization with MA = 1 .08 GeV. Based

this data these is no compelling evidence to prefer one over the other.c, we see the complete absence of any nuclear effects in the most recent

ant of SKAT collaboration . These results are even larger than the free-cross section in the low-q 2 region, which is quite surprising . This is similar

Fig . 5 . daldq2 for MA = 1 .05 j-- .03 GeV (shaded region) compared with neutrino experiments of (a) Pohlei a!.' 9 ) (b) Bonnetti et al . "') . Also shown are the results with VDM-QM parametrization (solid line) .

Units as in hg . 4 .

S K. Singh, E. ®set / Quasielastic v(v) reactions

603

Fig. 5-continued

to the old experimental results of Kustom et ral. '6) done in Fe who also failed tosee such a suppression as present in other experiments and predicted by calculations .We believe that present SKAT collaboration 23,24) results, like the old Argonneresults are inconsistent with present understanding of these processes.

Similarly in figs . 6 and 7 we show the do,/dq2 versus q2 for antineutrino scatteringexperiments of onnetti et al. 'g), Armenise et rat '9) and S T collaboration 23,24) .

Again we see a large suppression due to the nuclear effects at lower q2, whichbecomes negligible for q2 > -0.3 GeV/ c2 . In this case also the strong renormalizationeffects due to ph and ®h correlations are quite large in the region where othernuclear effects are important . However, the q2 dependence of the reduction in thecase of antineutrino scattering is different as compared to the q2 dependence of thereduction in the neutrino scattering. This is expected because the interference termFA(g2)(F,(g2)+ F2(q2 )) enters with the opposite sign in the case of antineutrinoscattering as compared with the case of neutrino scattering. This offers an interesting

possibility of ftudying strong renormalization effects on the axial vector form factor,if the sum and/or difference of neutrino and antineutrino cross sections can beanalyzed, specially at lower energies . With the modification of Fm(g2) in the medium

determined from quasielastic electron scattering through the assumption of c.v.c .,

the quenching of FA(g 2 ) as a function of q2 can be determined . Experimentallysuch an analysis seems to be difficult and has not been done.

6N S.

Fig . 6. Same as fig . 4, for antineutrino experiments of Bonnetti et A "'), Armenise et al. -") and SKATcollaboration 24) .

comparision with the results in the simple Fermi-gas model shows that ourresults with the Tenormalization effect included are slightly higher than the Fermias model at very low q , , due to the reduced strong renormalization . A comparision

with the experimental result of Bonnetti et el., Armenise et al and SKAT collabor-ation shows a fair agreement with our results except at lower q`', where there is,again, a stronger suppression in the experimental results. The situation is quite

tsimilar to the neutrino case, where the low-q 2 data can support a smaller MA(MA <1 .05 CueV) without disturbing the good agreement at higher q 2 , but more data isneeded in the low-q' region to make a definite statement .

4.2 . TCTAL CROSS SECTIONS

. Singh, set / Quasielastic ip( ;) reactions

the total cross sections o- (E,.) i-, are calculated as a function of neutrino (anti-eutrino) energy E,,, i-,, by integrating over the allowed i-ange of q 2 including the

1-104

S

bIV

S.K. Singh, E. Oset / Quasielastic P(;) reactions

Fig."continued

Q2(GeV

2

605

effects of Fermi motion inside the nucleus. The inclusion of Fermi motion increasesthe range of q2 considerably beyond the range allowed by the free-nucleon kinematicsused in the simple Fermi gas or the closure approximation. The contribution to thetotal cross section from the increased range of q2 is however small and does notexceed 3-4% . In the following, we have integrated over the whole range of q 2 andpresented our results in figs . 8-11 .

In figs . 8 and 9 we show the total cross section for the neutrino-nucleus crosssection for the experiments of Bonnetti et A and SKAT collaboration for freon andthe experiments of Pohl et al. for freon-propane mixture. In fig. 8, we show thesensivity to !he nuclear effects, while in hg. 9, we show the sensitivity to the valuesof MA in the range 112 < M,,_ < 1 .08. The large cross sections of the SKAT collabor-ation due to increased do-ldq2 at low q2 can be seen to lie higher than the predictedcross section and the expected behavior of the energy dependence. When comparedto the free-nucleon cross section, the nuclear cross sections per nucleon are reduced,

606

IexT

is case,Value of j

to that inuclear struct

uence theoor

in

We have calcin nuclei at inter

. Shkqla, E. Oset / Qu sielasde 0;7) reactions

FREON(SKAT)

.2

.4

.6

.8

1

1 .2 1 .4

Q7 (Ge 'J 2 )

Fig. 6--continued

but the dafarminationOf MA from these data will have large uncertainties due tothe olliality of data. The total cross section results are in fair agreement with the MMAvalues an determined from dueterium experiments and any change due to change

e 111A or due to the nuclear cReas lies well within the expLrimentai errors .figs . 10 and I I we show the similar results for the antineutrino scattering

eriments of Bonnetti et al ' x), Armenie et al. 0) and SKAT Collaboration 23,24) .

sions are similar to those reached from neutrino-nucleus reactions. Inowever, the total cross-section data is better and can support a lowerA but a definite conclusion can not be drawn at present. It is important

neutrino and antineutrino reactions, the uncertainties due to there are mainly at low q', where there is not enough data to effectively,etermination Or MA . The major uncertainty therefore is due to the

etermination a the neutrino flux . It is interesting to point out that the totalcross sections in the intermediate-energy region 0.5-1 .5 GeV are more sensitive tovariations in M.-N than those at high energy.

5. Conclusions

fated the quasielastic neutrino- and antineutrino-nucleus reactionsediate and high energies in order to study the effect of 1 A . The

VIN,bIV

S.K. Singh, E. Oset / Quasielastic P(~,' rearm,° ns 607

Fig . 7 . Same as fig . 5 for antineutrino experiments of Bonnetti et al.' x), Armenise et al.`") and SKATcollaboration 24) (shaded region) .

calculations have been done in the local density approximation for finite nuclei,and medium effects due to particle-hole and delta-hole excitations are taken intoaccount through the renormalization of various terms which depend upon thelongitudinal and transverse modes of excitation . The logitudinal and transverseexcitations are described by the pion and rho exchanges and the effect of shortrange correlations are taken into account through the Landau-Migdal parameter.Inclusion of these effects is expected to improve the results of earlier calculationsdone in simple Fermi-gas model with no medium polarization effects. The resultsare then applied to analyze the most recent experiments done in medium and heavynuclei using a world average value of MA =1 .05-+-0 .03GeV determined fromdeuterium experiments. In particular the analysis of the neutrino and antineutrinoscattering data in freon, and freon-propane mixture by Bonnetti et at ' K .), Arrnenispet al. '9) and SKAT collaboration 23a4) has been made. From this analysis weconclude that :

~i ) The data on

i erential crass sections and integrated cross sections are in fairagree

ent wit

the theory wit

,~ = 1 .05t0.03 C~eV and there is no compellingreaso~~ to ex ect a lower

_a . T e

ata on total cross sections with antineutrinoscattering could su port a lower

~ but the data is not good enough to draw aehnite conclusio

( ài ~ Tl~~c various nuclear effects ie

auli blocking,

ermi motion, strong renormaliz-ati®n effect due to p and

h excitation are large only at low g', where there is~~erv little data . The experi ental determination of

,~ from the neutrino-nucleusexperiment is not much affected

y the unce~ainties in the nuclear structure. Themayor uncertainty i~~ the deter° ination is therefore due ~o the poor knowledge ofthe neutrino spectrum_

i iig ï The latest nP~~trino na~clear experiment d~~: :e in freon by S

AT collab®rationdoes not sec any suppression of the differential cross section due t® the nucleareffects. This is in strong disagreement wïth our calculations and other experiments .

.5

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2

W

3P "Fn

S. K. Singh, E. Oset / Ouasielastic P(;) reactions

609

T

FA_" f -IF -1- f F-F-F-I

FREON

(FREE.NRYERM1,R)

r-T-r-I

FREON-PROPANE

(FREE,NR .FERMI .R)

I

I

I

1 -1---F_T__I

-'- - ---__L

-- --- - - - - - - -

8

(b) -7

0

3 4 5 6 7E(GeV)

Fig . 8 . Total cross section o- versus E compared with neutrino experiments in (a) freon and (b)Aeon-propane, for M,, == 1 .05 GeV . Free-nucleon case (dotted),

- : model (long dash), presentcalculation without renormalization (short dash) and with renorr,°alizatio i (solid), data from Bonnettiet al. and SKAT Collaboration for freon and Pohl et al. for freon-propane . Cross section in units of

[10-3K CM21 .

6 it) . Singh, E Osey /

r1*11'knlwamll'

IIIIIIIIIIIIIIIII

zoasielastic v( ,) reactions

rich-

FREON

r ' - i rif

PFNQPM

FREON-PROPANE

(WITH

RENORM )

~'7~/ -

M, ,M/0K11-Z%71 W17

1 2 3 4 5 6 7

E(GeV)

Fig . 9 . Total cross section a versus E for MA = 1-05 :f-- .03 GeV (shaded region), for experiments in (a)Aeon and (b) Aeon-propane . Units as in fig. 8 .

t

b

S. K. Singh, E. Oset / Quasielastic v( v) reactions

611

E(GeV)

Fig. 10 . Same as fig . 8, for antineutrino experiments in (a) freon and (b) freon-propane . Data in freonfrom Bonnetti et al. and SKAT collaboration, in freon-propane from Armenise et al

!C~~ ~

!!üSlE"~tlS~9C !®(i~~ !'eS1C! ®®!!S

®

~ _ -

1~JI/ll/IIILI!/~lll7lj/!~

~f,~~!rlr~/~fe~Jlll!!!lU11lÏ/1~l/~!!!/!/!I!~_ , .$-

.S

2 3

_

_

_ _-/il~li~~~~iiilr_liil7U`iUlllllliTil/liii~lT~lil!_ïl~

/!Il/!>2~!Ilr/dilhllll/Ul~

® _ Lff~iul~1-I ~

t 1 I 1 I 1 i B 1 I I I I 1

F1-~E~ ~'~: - F'k~~P-~tiE

l fH kEt(}k',I i

fb) -1 I I 1 ( 1 1 I L.I

4E:

( 7 ~-5 6

Also sho~~n are the results for M,., = 0.9 GeV for comparison .

7 8

1=ig. 11 . Same as fig . 9 for antineutrino experiments in la) freon (b) and freon-propane (shaded region) .

The enhancement seen in this experiment at lowest q 2 disagrees with other dataand theoretical calculations .

(iv) Simple Fermi-gas model calculations seem to stimulate the strong reno

aliz-ation effects by giving too much suppression as compared to an improved Fermi-Gasmodel calculation . This simulation seems to be even better for 0-(v) than d~/dq2

as the q2 dependence of the difference of our results with the renormalization effectand the simple Fermi-gas model result gives rise to a cancellation when integratedover the entire range of q

- .Strong renormalization effects due to particle-hole and delta-hole excitations are

different for neutrino and antineutrino reactions due to different combinations ofthe weak interaction coupling occurring in the neutrino and antineutrino crosssections . This fact can be used to analyze the q 2 dependence of the quenching ofFA(g2), if data on v and v scattering in nuclei at low energies are available .

This work has been partly supported by CICYT, grant no. AEN 90-0049 . One ofus, S.K. Singh, wishes to acknowledge the support of the Ministerio de Educaci®ny Ciencia in his sabbatical stay at the University of Valencia.

EXPRESSION FOR IX IT12

The sum over final polarizations and average over initial ones of I T12 for theT-matrix is given by:

MpMnMvmg 211 T12IG 2

SK. Single, E. Oset / Quasielastic v(v) reactions

613

Appendix

= Fï[(P1 ' P2)(P3 ' P4)+(P1 ' P4)(P2 ' P3) - mpmn(P1 ' P3)]

+FA[(P1 ' P2)(P3 ' P4)+(Pl ' P4)(P2 ' P3)+Mpmn(P1 ' P3)]

+ 2F, FA[(pl ' P4)(P2 ' P3) - (Pl ' P2)(P3 ' P4)]

+( ;F2/M)`[( pl - P2)(p3 ' q)(P4 ' q) - (P1 ' P2)(P3 ' P4)g2

+(Pl ' P4)(P2 ' q)(P3 ' q) - (P1 ' P4)(P2 ' P3)q?

+ ;(P1 ' P3)(P2 ' P4)g2- '(P1 ' P3)

pmng2

- (P1 ' q)(P3 ' q)(P2 ' P4) - ( Pl ' q)(P3 ' q)mnmp

+(P1 ' q)(P2 ' q)(P3 ' P4)+(Pl ' q)(P4' q)(P2 ' P3)]

+ ;Fl (F2/1Ul)[-mn(pl

p3)(P4' q) - mn(P1 ' q)(P3 ' P4) - mn(P3 ' q)(P1 "P4)

+mp(pl'P3)(P2' q)+mp(pl' q)(P3'P2) +mp(P3' q)(P1'P2)]

+FA(F2/

)[ - mp(P3' q)(P1 ' P2)+mp (p2 ' P3)(P, ' q)

t

ere

e

(

)C

~ -ttt~(

r)(

~. ~â . Sirr~lr,

,

set /

rt~siel~stic ~f~1 r°eaeticaras

=( na n)

~)a

a=(

pa

p) a

e ere ces

A.G

e sa

e e~ cession can be used îor antineutrino scattering by changing sign toto

s

t

m an

.~,

a.

1 ! ~~1 . ~~lorita, Proc . 5th V Int. Symp . on mesons and nuclei, Prague 1991 (Springer Verlag, Berlin) inpress

2) 1\° . gourdin, Phvs . Reports C11 (1974) 203' 1~ . Amaldi, S. Pubini and ~. ~urlan, Pion electroproduction at low energies and hadron form factors,

Springer °Fracts in ~4odern Physics, vol . 83 (Springer Verlag, Berlin, 1979)"' .A. ~l~iann et al., Phys. Rev. Lett. 31 (1973 ) 844S.J . Barish et al., Phvs Re - .

16 (1977 ) 3103C~ . 1= :~noukaris et a.., Phys . Rev .

21 11980 562N .J . Baker et a!, Phy°s . Re`' .

~3 (19811 2499R.1... . Ailler et al., Phys . Rev . >_~26 119821 537"f' . I{itaaaki et al., Phys . Rev .

2g t 1983 4 436634 11986 ) 2554

-145464~)849)

1U 4 7~ . lvitamaki et al., Phys . ReV.114 î' . Mita®aki er al., Phys . Rev .

42 ;19901 13311 _' ~

~i.ßti . Block et al., Phys . Lett . 12 01964) 28134 A. ®rkin Lecourtous et al., Nuovo C'im . A50 (1967) 27144 R1 .

older et al., Nuovo C'im . A57 (1968) 338154 T. Budago`~ ~t al., L.ett . Nuovo C~im . 2 (1969) 689164 R.L. . ICustom et al., Ph`s . Re`~ . i.ett . 9 11969) 1014174 C' . Baltav, in 1~1uon physics, vol . ll, ed . V.W . 1-Iughes and C.S . Wu (Academic Press, New Yerk

19751 P. _'64iSo S. Bonnetei ~t cal., Nuo`~o C~im . A3

(1977) 2601~4 ~1 . Pohl et al., i..ett . Nuo~o Cim . 26 11979) 3322f44 N. Armenis et al., Nucl . Phys .

152 (19791 36521 4 A.~ . Asratyan et al., Sov. J . Nucl . Phys . 39 (19841 392"4 S.\!. Beliko~- ~t al., Z.Phys. A32® ! 1985) 625?3 4

1-i .J . CHrabosh et al., Sov. J . Nucl . Phvs . 47 (1988 ) 10322-14 J. Brunner et al., Z. Phvs . C45 ( 19904 55125 B Ci .~ . Bro~.vn, Nucl . Phvs . A552 o i991 1 397c,6 a

i~t . Rho, Phvs . Re~~ . Lett . 54 11985 4 76~

S. K. Singh, E. Oset / Quasielastic v(;) reacaions

27) M. Ericson, Nucl. Phys. A335 (1980) 309293) I .S. Towner and E.C . Khanna, Phys . Rev . Lett . 42 (1979) 5129) E. Oset and M . Rho, Phys . Rev . Lett. 42 (1979) 4730) H.C . Chiang et al., Nucl Phys. A510 (1990) 59131) L.A. Ahrens et al., Phyç . Rev. D35 (1987) 78532) B. Holstein, in Parity violation in electron scattering, ed . E . Beise and R. Mckeon (World Scientific,

Singapore, 1991)33) G.E. Brown and M. Rho, Phys . Rev. Lett . 66 (1991) 272034) H.C . Chiang et al., Nucl. Phys . A510 (1990) 57335) E. Cset et al., Phys . Reports 188 (1990) 7936) L.M . Sehgal, Proc. European Phys . Soc. Int . Conf. i n high energy physics (1979) ed . A . Zichichi, p. 9837) S.K . Singh, Nucl. Phys . B36 (1972) 41938) S.K . Singh, Phys . Rev. D10 (1974) 198839) J . Bernabéu and P . Pascual, Nuovo Cim. A10 (1972) 6140) R . Tarrach and P. Pascual, Nuovo Cim . A18 (1973) 76041) J . Be.nabéu, Phys . Lett . B39 (1972) 31342) S.K. Singh tnd H . Arenh6vel, Z . Phys. A324 (1986) 34743) C . H . Llwelynsmith, Phys . Reports C3 (1972) 26144) J.S . Bell and C.H . Llwelynsmith, Nucl . Phys . B28 (1971) 31745) R.A. Smith and E.J . Moniz, Nucl . Phys . B43 (1972) 60546) T.K. Gaisser and J.S. O'Connell, Phys . Rev . D34 (1986) 82247) A. Magnon et al., Phys . Lett . B222 (1989) 35248) S.A . Dytman et al., Phys . Rev . C38 (1988) 80049) M . Soyeur et al., Saclay preprint 199150) C. Garcia-Recio, E . Oset and L. Salcedo, Phys . Rev . C37 (1988) 19451) E . Oset, in SERC School of nuclear physics, ed . B.K . Jain (World Scientific, Singapore, 1987)52) C.W . de Jager et aL, At . Data Nucl . Data Tables 14 (197e) 479

615