Lattice Charge Overlap II. Aspects of Charged Pion Polarizability

File: 595J 564901 . By:DS . Date:04:00:00 . Time:21:25 LOP8M. V8.0. Page 01:01Codes: 4136 Signs: 2601 . Length: 51 pic 3 pts, 216 mm

Annals of Physics � PH5649

annals of physics 255, 60�74 (1997)

Lattice Charge OverlapII. Aspects of Charged Pion Polarizability

Walter Wilcox

Department of Physics, Baylor University, Waco, Texas 76798-7316

Received July 12, 1996

Formulas are developed for use in lattice studies of charged hadron polarizabilities. In par-ticular, the valence quark different-flavor component of the charged pion polarizability isexamined on a 163_24 lattice at ;=6.0 using Wilson fermions. Using the elastic limit resultsof Part I of this series, it is concluded that this represents a small negative component of thetotal charged polarizabilty. � 1997 Academic Press

I. INTRODUCTION

In the previous paper of this series, referred to here as Part I, the elastic limitsof pions and rho mesons were examined and the results were compared to latticesimulations using three-point function techniques. The elastic ``baseline'' set in PartI makes the measurement of nonelastic properties of these mesons possible.

Neutral hadron polarizabilities have been examined previously using the tech-niques of lattice quantum chromodynamics (QCD) [1]. The authors there used anexternal field method to study the effect of a uniform electric field on the correlationfunctions of mesons and baryons. However, as of yet there have been no inves-tigations of charged hadron polarizabilities using the lattice techniques. This isunfortunate since there is now a growing body of experimental information on thissubject. Of course, external field methods would be very difficult or impossible to usein the case of charged hadrons because charged particles accelerate in an electricfield. The definition of charged hadron polarizability takes the form of the responseof the particle to an applied oscillating electromagnetic field. For this purpose a singlephoton will do and the effect being considered is then just Compton scattering.

The subject of charged pion polarizability will be considered from the point ofview of an effective relativistic field theory. Comparing the general and phenomeno-logical forms of the Compton scattering amplitude to second order in the photonmomentum will give us the form of the polarizability coefficient. Next, the methodfor extracting polarizability on the lattice is explained. It is pointed out that the fullpion polarizabilty requires the separate evaluation of five types of connected anddisconnected fermion diagrams. In the fourth section the details of our numerical

article no. PH965649

600003-4916�97 �25.00Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.


simulation of the valence different-flavor component of the charged pion polariza-bilty will be given and the results from Part I will be extensively used. It will beestablished that the resulting contribution is a small negative component of the fullcharged polarizabilty.

II. THEORY

Charged hadron polarizability has been examined previously in Ref. [2] usingthe methods of charge overlap in the context of lattice QCD simulations. However,the methods of Ref. [2] are coordinate-space based; the present techniques [3]are momentum-space based. First, let us review the definition of charged pionpolarizability as measured in Compton scattering. The definition of chargedpolarizability in terms of relativistic matrix elements in the context of an effectivemeson theory will then be established.

A. Review

The general form of the charged pion Compton scattering amplitude,

T+&=i | d 4xeik 2 } x(?( p2)|T ( j+(x) j&(0))|?( p1)), (1)

where ``T '' denotes the time ordering and

j+(x)=qu j u+(x)+qd j d

+(x) (2)

(qu= 23 , qd=&1

3) is an electromagnetic current density for up and down quarks, tosecond order in the photon momentum is given by [4, 5]

- 2E12E2 T+&=&T+( p1+k1 , p1) T&( p2 , p2+k2)

( p1+k1)2&m2

&T+( p2 , p2&k1) T&( p1&k2 , p1)

( p1&k2)2&m2

+2g+&+A(k21 g+&&k1+k1++k2

2g+&&k2+k2+)

+B(k1 } k2 g+&&k2+ k1&)+Ct+& . (3)

In this context A, B, and C are simply numerical coefficients. A noncovariant con-tinuum state normalization,

:n|

d 3p(2?)3 |n( p))(n( p)|=1, (4)

61LATTICE CHARGE OVERLAP, II


is being used which is responsible for the square root factor involving pion energiesappearing on the left-hand side of Eq. (3). The pion electromagnetic vertex(q+#p$+&p+),

T+( p$, p)=( p$++p+) F?(q2)+q+p$2&p2

q2 (1&F?(q2)), (5)

has been written to obey

q+ T+( p$, p)=p$2&p2 (6)

for off-shell pions, which is needed in order to satisfy the scalar electromagneticWard identity [5]. To the fourth order in momentum (needed later),

F?(q2)=1&(r2)

6q2+

(r4)120

q4. (7)

In the above m is the pion mass and the tensor t+& is given by

t+&=(k1 } k2) Q+Q&+(Q } k1)(Q } k2) g+&&(Q } k2) Q+k1&&(Q } k1) Q&k2+ . (8)

(Q+=p1++p2+ .) The Compton amplitude is conserved (k+1 T+&=k&

2 T+&=0), whichcan be shown to result in A=(r2)�3 to this order. The first two terms on the right-hand side of Eq. (3) represent the Born contribution to scattering from the pionwhile the others are contact terms. (A metric with g00=1, gii=&1, (i=1, 2, 3) isbeing used.) In order to define electric and magnetic polarizabilities, let us examinethe Compton scattering amplitude :=+

1 T+&=&*2 , where : is the fine structure constant

and =&1 and =&*

2 are the initial and final photon polarization vectors, respectively. Inthe lab frame where the inital pion is at rest:

k1 } =1=k2 } =2*=0, (9)

p1 } =1=p1 } =2*=0. (10)

Choose =1 and =2* to have only spatial components. Defining k0(1, 2)=|(1, 2) , in the

nonrelativistic limit ( |k9 (1, 2) |, |(1, 2)<<m), one can show for off-shell photons that

:=+1 T+&=&*

2 ==̂1 } =̂2* _&:m \1+

(r2)6

(k21+k2

2)++:E|1|2&+;M( =̂1_k9 ) } ( =̂2*_k9 2), (11)

where

:E#&: \ B2m

+2mC+ , (12)

;M#:B

2m(13)

62 WALTER WILCOX


are by definition the electric and magnetic polarizabilities, respectively. An explicitform for :E will now be exhibited and an explanation of how this coefficient can bemeasured in a lattice simulation will be given.

B. Derivation

First of all, let us use the continuum to lattice correspondences,

|n( p)) � V1�2 |n( p)) , (14)

j+(x) �jL

+(x)a3 , (15)

| d 4x � Va |�

&�dt :

x�

, (16)

where V=Nsa3, Ns being the number of spatial sites in the lattice and a being thespacing, to rewrite Eq. (1) as

T+&=iN 2s a | dt :

x�

eik2 } x (?( p2)| T ( jL+(x) jL

& (0)) |?( p1)) . (17)

Note that time remains continuous (and is dimensionless) in this formalism andthat a possible finite renormalization factor has been neglected in Eq. (15) since theexactly conserved lattice charge density will be used here.

Next, a point about the amplitude being calculated needs to be made. On the lat-tice, the appropriate object to consider is actually a normal ordered product ofcurrents:

T eff+&#iN 2

s a | dt :x�

eik2 } x (?( p2)| : jL+(x) jL

& (0): |?( p1)) . (18)

This is because the vacuum amplitude lattice Green function used (see Eq. (36)below) contains a disconnected part when p2=p1 which must be removed by hand.The form of the disconnected amplitude being subtracted in Eq. (18) will later beexhibited explicitly.

The methods of charge overlap will be used to calculate the necessary amplitudes.Thus, we consider the four-point function corresponding to T00 with the kinematicconditions:

|1=|2=0; p10=p20=m; |k9 1 |=|k9 2 |<<m. (19)

The relation between normal-ordered and time-ordered currents is

: j L+(x) j L

& (0):=T ( j L+(x) j L

& (0))&(0| T ( jL+(x) j L

& (0)) |0). (20)



Using lattice completeness,

:n, p� n

|n( p� n))(n( p� n)|=1, (21)

as well as the space-time translations,

j L+(x)=eip } xj L

+(0) e&ip } x, (22)

the space sum and time integral in Eq. (18) now results in

T eff00 | ``kinematics''=2N 2

s :n

|(?(09 )| \L(0) |n(k9 )) | 2

(En&m)

&2Ns

a3 :n

|(0| \L(0) |n(k9 )) | 2

En. (23)

Contact between Eq. (23) and Eq. (3) above now needs to be made. Separate offthe elastic part in Eq. (23),

T el00#2N 2

s

|(?(09 )| \L(0) |?(k9 )) | 2

(E?&m), (24)

(?(09 )| \L(0) |?(k9 )) =1

Ns

(E?+m)

- 2E?2mF?(q2), (25)

where F?(q2) is the usual continuum pion form factor. Using Eq. (7), one can showafter some algebra that

T el00=

4mk9 2

+{ 1m

&43

m(r2)=+k9 2 {&(r2)3m

+19

m(r2) 2+1

15m(r4)= , (26)

up to second order in the photon momentum. For the nonelastic part one mayshow at low momentum ( |k9 |<<m) for vector or axial vector states that

(?(09 )| \L(0) |n(k9 )) ==0(k) d?nmn

Ns

, (27)

(0| \L(0) |n(k9 )) ==0(k) d0nmna1�2

- Ns

, (28)

which serves to define the constants d?n and d0n , and =0(k) is the 0th componentof a polarization vector for the state |n(k9 )) with mass mn . The d?n , d0n factors aredefined in these expressions so that they are independent of the lattice size andspacing. Interestingly, vectors and axial vectors are the only types of intermediatestates which may contribute to the polarizability. See the Appendix for furtherdetails. If we choose a helicity basis where the =+(k, *) with *=+, & are purely

64 WALTER WILCOX


spatial and the positive z-axis is along k9 , then only the longitudinal polarizationstate, *=0, contributes in Eqs. (27) and (28),

=0(k, 0)=|k9 |mn

, (29)

and the expansion to second order in photon momentum for these kinematics thenyields

T eff00 | ``kinematics''=T el

00+2k9 2 \ :n{?

|d?n | 2

(En&m)&V :

n

|d0n | 2

En + , (30)

where T el00 is given by Eq. (26).

The phenomenological expression, Eq. (30), is to be compared with the +=&=0component of the general Compton amplitude, Eq. (3), also to order k9 2 with thesame kinematics. One may show that

2mT00 | ``kinematics''=&T0( p1+k1 , p1) T0( p2 , p2+k2)

( p1+k1)2&m2 &T0( p2 , p2&k1) T0( p1&k2 , p1)

( p1&k2)2&m2

+2&2 (r2)

3k9 2&Bk9 2&4m2k9 2C. (31)

For these kinematics, one has

1( p1+k1)2&m2=&

1( p1&k2)2&m2=&

1

k9 2, (32)

and also, for example,

T0( p1+k1 , p1)=2m \1&(r2)

6q2+

(r4)120

q4+ . (33)

When the pieces are put together, one obtains

T00 | ``kinematics''=4mk9 2

+{ 1m

&43

m (r2)=+k9 2 {&(r2)3m

+19

m (r2) 2

+1

15m (r4) &\ B

2m+2mC+= . (34)

As the last step in the derivation, compare Eq. (30) and Eq. (34) and identify

:E=&: \ B2m

+2mC+=2: { :n{?

|d?n | 2

(En&m)&V :

n

|d0n | 2

En = . (35)



This formula relates :E to properties of relativistic matrix elements, which wasthe goal. Of course, the d?n and d0n factors are nothing but appropriate derivativesof the matrix elements in Eq. (28) with respect to |k9 |, evaluated at k9 =0. There areseveral other aspects of this formula worthy of mention. First, the explicitappearance of a volume factor makes it clear that neither term above is finite, butthe whole expression must be regulated (as on the lattice) to extract a finite answer.Second, it does not have the classical charge radius term, : (r2)�3m. This isbecause it comes from comparing it to an effective theory of point mesons. (Eventhe pion is viewed as a point object in this case; its ``size'' simply comes from addi-tional local interactions with photons.) Third, it explicitly includes the vacuumpolarization contribution first explained by Terent'ev in Ref. [6]. The negative signin front of this term makes it abundantly clear why in relativistic field theories thecharged particle's polarizability can be positive or negative. The polarizability ofthe pion, no longer considered the groundstate as in nonrelativistic quantummechanics, is actually defined relative to the vacuum, a result remininiscent of theCasimir effect. Finally, even though we are only keeping terms to order k9 2, it isclear that the above is an exact formula for the polarizability, which is simply thenumerical coefficient of the second order energy term in Eq. (11).

III. LATTICE MEASUREMENT

Let us concentrate now on the lattice amplitudes necessary to directly extract thecharged pion polarizability. Some comments about indirect lattice measurements ofthis quantity will be made later.

For these purposes, one forms the usual charge overlap function, but with nor-mal ordering of the charge densities instead of time ordering:

P?(r� ; t3 , t2 , t1)#�x� 3 , z� (0| ,-(x3) :\L(x2) \L(x1): ,(z) |0)

�x� 3 , z� (0| ,-(x3) ,(z) |0), (36)

where t3>t1, 2>0, z=(0, z� ), and r#x2&x1 . Dimensionless times and distances arebeing used in this section. The formulas are now in discrete Euclidean space but oneimagines that the time axis is still continuous. The time limits t3>>t1, 2>>1 yield

P?(r� ; t3 , t2 , t1) � (?(09 )| :\L(x2) \L(x1): |?(09 )), (37)

where

(?(09 )| :\L(x2) \L(x1): |?(09 ))=(?(09 )| T (\L(r) \L0)) |?(09 ))

&(0| T (\L(r) \L(0)) |0). (38)

Consider now the finite Fourier transform of P?(r� ; t3 , t2 , t1),

Q(q� ; t3 , t2 , t1)#Ns :r�

e&iq� } r� P?(r� ; t3 , t2 , t1). (39)

66 WALTER WILCOX


In the same time limit,

Q(q� ; t3 , t2 , t1) � Ns :r�

e&iq� } r� [(?(09 )| T (\L(r) \L0)) |?(09 )))

&(0| T (\L(r) \L(0)) |0)]. (40)

Assuming t2&t1>0 and inserting a complete set of states now gives

Q(q� ; t3 , t2 , t1) � N 2s {:

n

|(?(09 )| \L(0) |n(q� )) | 2 e&(E n a&m? a)(t 2&t1)

&:n

|(0| \L(0) |n(q� )) | 2 e&E na(t 2&t 1)= . (41)

Separate off the elastic term (n=?),

Qel(q� ; t2&t1)#N 2s |(?(09 )| \L(0) |?(q� )) | 2 e&(E? a&m?a)(t 2&t1), (42)

above. Then, using Eqs. (27), (28), and (29) at low momentum transfer results inthe construction,

:a |�

&�d(t2&t1)[Q(q� ; t3 , t2 , t1)&Qel (q� ; t2&t1)]

=2:k9 2 \ :n{?

|d?n | 2

(En&m)&V :

n

|d0n | 2

En +=k9 2:E , (43)

(k9 =q� �a) when the relative time integral between the two charge densities isexplicitly determined. (One gets a doubling of the result when the other time rela-tionship, t2&t1<0, is assumed.) Of course, on a finite lattice the time integrals donot really extend to \� because of the fixed time locations of the particle sources,but instead extend to time limits such that the elastic limit is effectively established.Equation (43) now gives us the means of measuring the Compton scattering pionpolarizability coefficient. On the lattice one may take a numerical ``derivative'' ofthe left-hand side of Eq. (43) with respect to q� 2 by simply dividing it by q� 2 at thelowest lattice momentum. (The relative error in this procedure is of order q� 2.)

It is crucial to work in momentum space to extract the polarizability. In positionspace one would look at (image corrections are assumed)

R2(t3 , t2 , t1)#:r�

r� 2P?(r� ; t3 , t2 , t1), (44)

which does not project onto good momentum when a complete set of states isinserted [2]. Thus, the limitation to small momenta, necessary to use Eqs. (27) and(28) above, cannot be enforced.

Although Eq. (43) is a simple formula, its direct evaluation on the lattice viafour-point functions which arise from the field contractions in Eq. (36) is not easy.


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Fig. 1. Graphical representation of the four-point functions contributing to charged pion polariza-tion. The zero-momentum pion interpolating fields are representated by horizontal bars, the time axisruns horizontally, and the vertical direction represents one space dimension. (a) and (b) represent dif-ferent-flavor and same-flavor valence contributions, respectively. (c) is the contribution from a singleisolated quark loop, while (d) is the two quark loop contribution. (e) arises from a same-flavorcurrent�current contraction. It is assumed that gluon lines connect the various fermion lines together ineach diagram. Diagrams (d) and (e) have a disconnected part which must be subtracted.

68 WALTER WILCOX


There are five types of diagrams, which may be classified by fermion line topology,as shown in Fig. 1. In the case of Figs. 1a and 1b, the photons interact directly withthe valence quarks. Figure 1a will be termed the same-flavor valence part andFig. 1b will be called the different-flavor valence part. Figures 1c, 1d and 1e representcontributions from disconnected fermion lines in which arbitrary numbers of gluonlines connect the various pieces together. These gluon connections are enforced inthe correlations between the pion propagator and the various fermion loops in thelattice Monte Carlo. (When isolated groups of fermion lines arise, ``connected part''means ``statistically correlated part'' on the lattice.) Figure 1c represents the con-tribution from correlations between a valence quark and a self-contracted quarkloop. (Flavors other than u and d can of course contribute to the quark loops.)Figure 1d has both currents, either same-flavor or different-flavor, self-contracted.In the case of Fig. 1e, one has a complete quark loop formed. It is the amplitudewhere the pion propagator is completely disconnected from the quark loops in (d)and (e) (when p2= p1) which needs explicit subtraction and is responsible for thenegative contribution on the right-hand side of Eq. (35). Figure 1c also has a dis-connected part, however, there is no need to do a subtraction since it can be shownthat the vacuum expectation value of the exactly conserved lattice charge densityvanishes configuration by configuration for color SU(2), and in the configurationaverage for SU(3).

The lattice formula for :E has now been derived and the agenda for evaluatingthe charged pion polarizability directly on the lattice has been set. In the next sec-tion we will examine one particular type of diagram and show that it actuallymakes only a small contribution to :E .

IV. LATTICE SIMULATION

As pointed out immediately above, the direct calculation of the charged pionpolarizability involves many diagrams which must be separately evaluated. In thissection the different-flavor valence piece, Fig. 1a, will be considered.

The full different-flavor piece involves Figs. 1a, 1c, and 1d and can be evaluatedon the lattice at low lattice momentum, q� , by

:u, dE #

:a3

q� 2 |�

&�d(t2&t1)[Qu, d (q� ; t3 , t2 , t1)&Qel; u, d (q� ; t2&t1)], (45)

where Qu, d (q� ; t3 , t2 , t1) and Qel; u, d (q� ; t2&t1) are identified from the Fourier trans-form of

Pu, d? (r� ; t3 , t2 , t1)#2quqd

�x� 3 , z� (0| ,-(x3) :\L; u(x2) \L; d(x1): ,(z) |0)�x� 3 , z� (0| ,-(x3) ,(z) |0)

, (46)

as above.



TABLE IValence Contribution to the Different-

Flavor Charged Pion Polarizabilityas a Function of }

} :E (fm3)

0.148 &0.29(6)_10&4

0.152 &0.23(6)_10&4

0.154 &0.15(10)_10&4

We now specialize to evaluating the valence u, d contribution from Fig. 1a. Werefer the reader to Part I for a discussion of the details of the numerical simulation,which was carried out on 20 quenched 163_24 lattices with Wilson fermions at;=6.0. The appropriate correlation functions for three quark masses are displayedin Fig. 5 of Part I. According to Eq. (45), we now need to evalute the area underthese functions and subtract the elastic contribution. The results of this procedureare presented in Table I, where the errors specified are purely statistical. In arrivingat these results, the numerical value :a3=1.06_10&44 cm3 has been used, where :is the fine structure constant. This value is inferred from the inverse lattice spacing,a&1=1740 MeV, which is taken from comparing the chirally extrapolated latticenucleon mass with the experimental value [8].

When the results in Table I are extrapolated linearly in quark mass, as defined inPart I, to the chiral limit at }cr=0.1564, we obtain (:u, d

E )valence=&0.12(11)_10&4 fm3.There are various systematic uncertainties in our final numerical result. One

systematic uncertainty is associated with the numerical evaluation of the discretenonelastic time correlation integral. This discrete time integral was determinedusing both the trapezoidal rule and Simpson's method [7]. The differences for allthree } values were less than 20. There is an additional systematic uncertaintyassociated with the numerical momentum derivative. As pointed out above, this isexpected to be of order q� 2, or about 100, the same as other finite lattice spacingerrors. This was examined numerically by repeating the simulation (with the samefits) on 10 configurations, also at ;=6.0, on a larger 203_30 lattice at }=0.152.The result on the larger lattices is &0.33(14)_10&4 fm3, which unfortunately haslarge statistical errors but is consistent with the value in Table I. In addition tothese uncertainties, there is also the systematic uncertainty associated with thesetting of the lattice scale. Since the pion polarizability scales as a3, it is rather sen-sitive to a change in the scale. In order to remain consistent with the low energylattice results for nucleon mass, pion decay constant, and proton electric form fac-tor in Ref. [8], a roughly 100 change in the present lattice scale could betolerated. This means a systematic scale uncertainty of approximately 330for :E .

70 WALTER WILCOX


V. SUMMARY AND CONCLUSIONS

The establishment of the charge overlap elastic limit in Part I allows thepossibility of obtaining nonelastic hadron properties directly from lattice QCD. Inparticular, using the lattice formulas developed, we can see that the charged pionpolarizability can be measured on the lattice from low momentum charge overlapcorrelation functions.

We have obtained (:u, dE )valence=&0.12(11)_10&4 fm3 in the chiral limit from

the quark diagram in Fig. 1a. This is to be compared with the experimentalresult [9]

:expE =2.2(16)_10&4 fm3,

and the result from second order chiral pertubation theory [10],

:chiralE =2.4(5)_10&4 fm3.

Thus, the present results establish that the valence quark different-flavor diagram,Fig. 1a, contributes only a small, negative component to the total charged pionpolarizability. We must therefore look to the other diagrams in Fig. 1 for the bulkof this effect.

It is important to note that, in addition to the present, direct measurement of thecharged pion polarizability, there is also the Das, Mathur, and Okubo (DMO)current algebra sum rule [11] for :E . Although this formula is not exact, themeasurements required for its evaluation on the lattice are much simpler than thedirect method since it involves only mesonic two-point functions. A lattice evalua-tion of the so-called intrinsic part of the charged pion polarizability in the DMOsum rule should be quite interesting and is currently underway.

Finally, it was noted above that the lattice measurement of :E scales like a3 andis therefore quite sensitive to the lattice scale. This opens up the possibility of usingcharged pion polarizability as a place to accurately test QCD, or alternately, to setthe lattice scale, once the lattice and experimental results are both improved.

APPENDIX

In this appendix it is established that the only intermediate states which may con-tribute in Eq. (35) to the charged pion polarizability are vectors and axial vectors.

First, let us dispose of the spin-0 case. We have (n{?)

(?( p) | j:(0)| n( pn))=p: G1(q2)+pn: G2(q2)

(2E?2En)1�2 , (47)



where as usual q2=( p&pn)2. Charge conservation requires the right-hand side ofEq. (47) dotted into p+

n&p+ to be 0, so that

G2(q2)=G1(q2) \m2&p } pn

m2n&p } pn+ . (48)

Therefore (considering the pion at 0 spatial momentum)

(?(09 ) | j4(0)| n( pn))=G1(q2)

(2m2En)1�2 \m+Enm2&mEn

m2n&mEn+ . (49)

At low momentum, | p� n |<<m, we then find

(?(09 )| j0(0)| n( pn)) B G1((m&mn)2) p� 2n . (50)

Therefore the absolute square of this matrix element goes at least like p� 4n at low

momentum, assuming G1((m&mn)2) is nonsingular, and cannot contribute to thepion polarizability term in T eff

00 , which is proportional to p� 2n .

We now consider the general nonzero spin, s. Some fundamental properties ofpolarization tensors are (*s=&s, , , 0, , , s):

=+ 1 } } } + s ( p, *s) =*+ 1 } } } + s ( p, *$s)=$*s, *$s, (51)

g+ :+; =+ 1 } } } + : } } } + ; } } } + s( p, *s)=0, (52)

p+ : =+ 1 } } } + : } } } + s( p, *s)=0. (53)

Adopting a helicity basis, the Lorentz structure for spin s must be of the form

(?( p) | j:(0)| n(s, *s , pn))==+ 1 } } } +s( pn , *s) A+ 1 } } } + s:( p, pn)

(2E?2En)1�2 , (54)

where it is assumed that the tensor A+ 1 } } } + s :( p, pn) is nonsingular.First, we will need the following result:

=0 } } } 0( p, *s) B | p� | s,=0 } } } 0( p, *s)=0,

*s=0*s{0= . (55)

This can be established from the higher spin considerations in Ref. [12]. Thisreference shows that the polarization vectors for higher spin may be constructedfrom those of spin 1 via

:s

*s=&s

�s*s (!) =+ 1 } } } + s ( p, *s) x+ 1} } } x+s=_ :

1

*1=&1

�1* 1(!) =+( p, *1) x+&

s

, (56)

where

�s*s(!)#_ (2s)!(s+*s)! (s&*s)!&

1�2

!s+* s+ !s&*s

& . (57)

72 WALTER WILCOX


Using a purely timelike x+ and recalling that =0( p, *1){0 only when *1=0, it iseasy to see by matching coefficients of !+ and !& on both sides of Eq. (56) that

=0 } } } 0( p, *s) B (=0( p, 0))s,=0 } } } 0( p, *s)=0,

*s=0*s{0= . (58)

Now using =0( p, 0) B | p� | from Eq. (29) establishes Eq. (55) above. There are manyother ways of proving this result.

Working in the rest frame of the pion and considering only j0 , the matrix elementin Eq. (54) is now seen to be proportional to =0 } } } 0( p, *s):

(?(09 ) | j0(0)| n(s, *s , pn)) B =0 } } } 0( pn , *s). (59)

This comes about through the repeated use of Eqs. (52) and (53). For example,consider the s=2 case. Explicitly, one has

(?(09 ) | j0(0)| n(2, *2 , pn))=1

(2m2En)1�2 (=i 1 i2 Ai 1 i2 0+=i10Ai 100

+=0i2 A0i 20+=00A000). (60)

The remaining spatial O(3) symmetry of the matrix element requires thatAi 100 B ( pn) i1

, A0i2 0 B ( pn) i2, and Ai1 i 20 B ( pn) i1

( pn) i2or Ai 1 i2 0 B gi 1 i2

. Throughthe use of Eqs. (52) and (53) each term in Eq. (60) is seen to be proportional to=00( pn , *2). This in turn means that the matrix element couples only to *2=0 andbehaves like p� 2

n (or possibly higher power) at low momentum from Eq. (55). Thisargument can be extended to higher spin. Thus, s=1, *1=0 (axial) is the onlypossible contributor to the charged pion polarizability through this matrix element.

Although only the matrix element (?(09 ) | j0(0)| n(s, *s , pn)) has been discussed,similar conclusions hold for (0 | j0(0)| n(s, *s , pn)); namely, that the state n(s, *s , pn)contributes to charged pion polarizability only when s=1, *1=0 (vector).

ACKNOWLEDGMENTS

This work was supported in part by the NSF under Grant PHY-9401068. It was also supported bythe National Center for Supercomputing Applications (NCSA) at the University of Illinois atUrbana�Champaign and used the CM5 and SGI Power Challenge computers.

REFERENCES

1. H. R. Fiebig, W. Wilcox, and R. M. Woloshyn, Nucl. Phys. B 324 (1989), 47.2. M. Burkardt, J. M. Grandy, and J. W. Negele, Ann. Phys. 238 (1995), 441.3. W. Wilcox, Phys. Lett. B 289 (1992), 411.4. J. F. Donoghue and B. F. Holstein, Phys. Rev. D 40 (1989), 2378.5. M. V. Terent'ev, Sov. J. Nucl. Phys. 16 (1973), 87.



6. M. V. Terent'ev, Sov. J. Nucl. Phys. 16 (1973), 576.7. P. J. Davis and I. Polonsky, in ``Handbook of Mathematical Functions'' (M. Abramowitz and

I. Stegun, Eds.), Chap. 25, U.S. Dept. of Commerce, Washington, D.C., 1972.8. K. F. Liu, S. J. Dong, T. Draper, and W. Wilcox, Phys. Rev. Lett. 74 (1995), 2172.9. D. Babusci et al., Phys. Lett. B 277 (1992), 158.

10. U. Bu� rgi, ``Charged Pion-Pair Production and Pion Polarizabilities to Two Loops,'' Preprinthep-ph�9602429.

11. T. Das, V. S. Mathur, and S. Okubo, Phys. Rev. Lett. 19, (1967), 859.12. J. Schwinger, ``Particle, Sources, and Fields,'' Vol. 1, Section 2�5, Addison�Wesley, Reading, MA,

1970.

74 WALTER WILCOX

Lattice Charge Overlap II. Aspects of Charged Pion Polarizability

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