Electric polarizabilities of neutral baryons in the QCD sum rule

12 March 1998

Ž .Physics Letters B 422 1998 26–32

Electric polarizabilities of neutral baryons in the QCD sum rule

Tetsuo Nishikawa a,1, Sakae Saito a,2, Yoshihiko Kondo b,3

a Department of Physics, Nagoya UniÕersity, Nagoya 464-01, Japanb Kokugakuin UniÕersity, Tokyo 150, Japan

Received 3 September 1997Editor: W. Haxton

Abstract

We investigate the electric polarizabilities of neutral baryons using the method of QCD sum rules. The diagrams in theoperator product expansion are taken into account up to dimension 6. We obtain different values of the polarizabilities

00 0among hyperons, the ordering of which is given a )a )a . The polarizability of S , a , has possibly a very smalln J L S

value. q 1998 Elsevier Science B.V.

PACS: 14.20.Dh; 13.40.-f; 12.38.LgKeywords: Electromagnetic polarizabilities of Baryons; QCD sum rules

Electric and magnetic polarizabilities, labeled a and b , are fundamental constants characterizing the hadronstructure. These constants measure the ease with which a hadron acquires electric and magnetic dipole moments

w xin external electromagnetic fields 1 . While for the proton and neutron they have long been studied throughw xtheoretically and experimentally and recent measurements have established certain values 2–5 , very little is

known about hyperon polarizabilities. However with the hyperon beams at CERN and Fermilab, the situation isw xexpected to change. In particular the S polarizabilities will be soon measured 6 .

This has induced a number of theoretical investigations. In fact predictions have been made in thew x w xnon-relativistic quark model 7 , the heavy baryon chiral perturbation theory 8 and the bound state soliton

y4 3w x q ymodel 9 . In the non-relativistic quark model the results are a s20.8, a s5.7, all in units of 10 fm . TheS S

large difference in them can be traced back to the different quark structure of the Sq and Sy. While in the Sq

the two quark flavors have opposite electric charges, all valence quarks in the Sy have the same charges,which inhibit internal excitation. In agreement with the non-relativistic quark model, the heavy baryon chiral

Ž . Ž . Ž .q y q yperturbation theory HBChPT at leading order finds a )a ; namely, a s9.4 8.8 and a s6.3 5.9 inS S S S

Ž . Ž .the same units for the axial-vector coupling constants Ds0.8 0.75 and Fs0.5 0.5 . Here the reason for thisresult is the kaon cloud contribution, which is strongly suppressed for the Sy. In addition hyperon polarizabili-

1 E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected].

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0370-2693 97 01567-0

( )T. Nishikawa et al.rPhysics Letters B 422 1998 26–32 27

Ž Ž . Ž ..ties are smaller than the nucleon polarizabilities a s13.0 12.1 , a s11.4 10.5 in the HBChPT. The largep n

value of a in the HBChPT is due to the large contribution of pion cloud. On the other hand the hyperonN

polarizabilities mainly come from the contribution of kaon cloud, whose mass is larger than that of pion andinhibit internal excitation under the external field.

w x w xWe use a completely alternative approach, the method of QCD sum rules 10,11 . In a previous work 12 weconstructed the QCD sum rule for the electromagnetic polarizabilities of the neutron and found that they arerelated with the vacuum condensates of quark-gluon fields in a weak external constant electromagnetic field. Wehave seen that the neutron electric polarizability is well described with the method. In this paper we extend themethod to hyperons and calculate the electric polarizabilities of neutral hyperons.

We consider a two-point correlation function in a weak external constant electromagnetic field F asmn

follows:

4 i p x² :P p s i d x e 0NT h x h 0 N0 , 1Ž . Ž . Ž . Ž .H FF B B

Ž .where h x is an interpolating field constructed out of quark fields such that it has the quantum numbers of aBw xbaryon under consideration. Following Ioffe and Smilga 13 , we expand the correlation function in powers of

F up to order F 2 terms:mn

P p sP Ž0. p qP Ž1. p F mn qP Ž2. p F mn F rs qOO F 3 . 2Ž . Ž . Ž . Ž . Ž . Ž .F mn mnrs

Ž0.Ž . Ž1.Ž .The leading term P p is the correlation function when the external field is absent. P p is known to bemnŽ2. Ž .related with the magnetic moments of baryons. It is P p that is related with the electromagneticmnrs

Ž2. Ž .polarizabilities of baryons. Requiring parity invariance, we can decompose P p into the following Lorentzmnrs

structure:

P Ž2. p sg g P p2 qp p g P p2 qg g puP p2Ž . Ž . Ž . Ž .mnrs m r ns S m r ns S m r ns V1 2 1

qp g g P p2 qp p g puP p2 qs plp g P p2Ž . Ž . Ž .m r ns V m r ns V ml r ns T2 3 1

l 2 l 2q ig e g P p qp p P pŽ . Ž .5 mnrl s P s P1 2

l 2 l 2 l 2 l 2q ig e g puP p qp g P p qg p P p qp p puP p . 3Ž .Ž . Ž . Ž . Ž .5 mnrl s A s A s A s A1 2 3 4

The invariant functions in the right hand side satisfy dispersion relations:

`1 Im P tŽ .2P p s dt qsubtraction terms, 4Ž .Ž . H 2p typ0

ˆwhere the subscripts S etc. are suppressed for brevity. One can apply the Borel transform B defined by1

nq1 n2yp dŽ .2 2B̂ P p ' lim y P p . 5Ž .Ž . Ž .22 n! d ypŽ .yp ™`

n™`2yp

ssn

Here s denotes the squared Borel mass. We make a Borel transform on both sides of the dispersion relations:

`12 ytr sB̂ P p s dt e Im P t . 6Ž . Ž .Ž . H

p 0

Ž .Evaluating the left hand side by an operator product expansion OPE in the deep Euclidean region we obtainthe Borel sum rules.

Ž .Let us next consider the physical content of the invariant functions in the right hand side of Eq. 6 . The

( )T. Nishikawa et al.rPhysics Letters B 422 1998 26–3228

Ž . Ž .lowest intermediate state of the correlation function P p is expressed by the baryon propagator: P p s2 w Ž .xyl r pu yM yS p,F , where M is the baryon mass, and l the coupling strength of the interpolatingB B B B

Ž .field to the baryon. S p,F is the self-energy part under the influence of the background electromagnetic field,and can be decomposed into possible types of Lorentz structure. The invariant functions of the self energy on

Ž 2 2 .the mass-shell p sM are related to physical quantities of a baryon, such as the magnetic moment, theBw xpolarizabilities, via an effective Lagrangian introduced by L’vov 1 . This Lagrangian, which was constructed by

requiring Lorentz and gauge invariance, and the discrete symmetries, describes the low-energy interaction ofspin-1r2 composite particles with a soft electromagnetic field. The amplitude of low energy Compton scattering

w xis reproduced from the Lagrangian. For the details see our preceding paper Ref. 12 . We find that the invariantfunctions P and P are related with the polarizabilities in the following way:S Vi i

1 1 12 3 2 w xŽ . w Ž .x w Ž . Ž .x Ž .P t syl 4M S t q 3M S t q2 M S t q S t , 7Ž .S B B 1 B 1 B S S3 2 21 ½ 52 2 tyMBtyM tyMŽ . Ž .B B

1 1 12 3w x w xŽ . Ž . w Ž . Ž . Ž .x Ž .P t syl y8 M S t q M S t qM S t q2 S t q M S t , 8Ž .S B B 1 B 2 B 3 V B 33 2 22 ½ 52 2 tyMBtyM tyMŽ . Ž .B B

1 12 2 w xŽ . w Ž .x Ž . Ž .P t syl 4M S t q S t q2 M S t , 9Ž .V B B 1 1 B S3 21 ½ 52 2tyM tyMŽ . Ž .B B

Ž .1 1 S tV2 2Ž . w Ž . Ž .x Ž .P t syl 4S t yM S t q yS t y , 10Ž .V B 1 B 2 22 22 ½ 52 MtyM BBtyMŽ .B

Ž .1 1 2 S t 1V2 2w x w xŽ . Ž . Ž . Ž . Ž .P t syl y8 S t q 2 S t qM S t q q S t . 11Ž .V B 1 2 B 3 33 2 23 ½ 52 2 M tyMB BtyM tyMŽ . Ž .B B

S and S on the mass-shell are related with the polarizabilities a and b :V S B B

2 2S M s a qb r2, S M syb r4, 12Ž .Ž . Ž .Ž .V B B B S B B

w xwhile S , S and S with other quantities 12 . Defining the following combination:1 2 3

Ža . 2 2 2 2 2 2 2 2BP p sy2 P p yp P p q2 M P p qM P p qp M P p , 13Ž .Ž . Ž . Ž . Ž . Ž . Ž .S S B V B V B V1 2 1 2 3

Ža .Bwe find the imaginary part of P to be related solely with a :B

1 aBŽa . 2 2BIm P t syl d tyM . 14Ž . Ž .Ž .B Bp 2

For the magnetic polarizabilities b , it seems difficult to derive relations free from the constants which are notB

known well. Numerical estimation of b is therefore left for a future work.B0 0 w xWe write the interpolating fields for the n, J , S , and L, respectively, as follows 14 :

aT b m ch x se d x Cg d x g g u x , 15Ž . Ž . Ž . Ž . Ž .n abc m 5

aT b m c0h x sye s x Cg s x g g u x , 16Ž . Ž . Ž . Ž . Ž .J abc m 5

aT b m c aT b m c'0h x s 2 e u x Cg s x g g d x q d x Cg s x g g u x , 17Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .½ 5S abc m 5 m 5

2aT b m c aT b m ch x s e u x Cg s x g g d x y d x Cg s x g g u x , 18Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .½ 5(L abc m 5 m 53

w xwhere C denotes the charge conjugation and a, b and c color indices. Following Ref. 12 the calculations ofthe Wilson coefficients are straightforward.

Let us list the vacuum expectation values of interest and then classify them according to their dimensions.


Ž .We are interested in the quadratic terms in F . Hence, the vacuum expectation value VEV of themn

lowest-dimensional operator qq is the second order term of the quark condensate in the external field, when we4 a aexpand it in powers of F . Four-dimensional operators are F F itself, iq SSg D q and G G , where themn mn rs m n mn rs

Ž a . acovariant derivative is given by D sE q ig l r2 A , and SS is a symbol which makes the operatorsm m m

symmetric and traceless with respect to the Lorentz indices. The contribution of the VEV of the last twow xoperators are so small that they can be safely neglected 12 . The following three operators: qs qPF ,mn rs

a aŽ .gqs l r2 G q, q SS D D q have dimension 5. The contribution of the second operator turns out to vanish,mn rs m n

as is for the sum rule for the masses of octet baryons. The Wilson coefficient of the last operator is constant anda aŽ .vanishes after a Borel transform. There are three six-dimensional operators; that is, gqg D l r2 G , qqqqm n rs

and qs qqs q. The VEV of the first operator cannot be induced by the second-order interaction of themn rsa aŽ .vacuum with an electromagnetic field, since qg D l r2 G q has odd C-parity. To estimate the VEV’s of them n rs

last two operators we have applied the factorization hypothesis, which is firmly based in the case of the four² : ² : ² : ² : ² : ² : ² :quark operators: qqqq s2 qq P qq , qs qqs q s qs q P qs q , where PPP de-F F 0 F F F Fmn rs mn rs

² :notes the VEV in the electromagnetic field and PPP the VEV when the external field is absent.0

For the calculation up to six-dimensional terms, we are then left with the two external-field-induced VEV’s:² : ² : w xqs q , qq . It is assumed in Ref. 13 that the first of them can be written, in general,F Fmn

'² : ² :qs q s 4pa F x e qq , 19Ž .F 0mn mn q

where a is the fine structure constant and x the so-called magnetic susceptibility of quark condensate. Weextend this hypothesis to the case of the second order in F and define a ‘‘susceptibility’’ f as follows:mn

mn 2² : ² :qq s 1q4paF F f e qq . 20Ž .Ž .F 0mn q

Ž . Ž .In order to obtain the desired sum rule, we substitute Eq. 14 into Eq. 6 . The contributions to the imaginaryparts from higher mass states are approximated using the logarithmic terms in the OPE, starting at a threshold

Ž .S . We calculate the left hand side of Eq. 6 by an OPE up to dimension 6. For hyperons, we further introduce0

0the effect of the finite mass of the strange quark, m . In the sum rule for a the leading terms of the masss J

0correction appear in the dimension 6 terms. In the sum rules for a and a we consider only the leadingL S

² :correction, which appears in the dimension 4 terms, namely m qq . One finally obtain the following results:s

a 42 2 2 2 2² : ² :8p f e uu s E S ,s q p x 6e e qe uu sE S ,sŽ . Ž .� 4Ž .0 0u 1 0 u d u 0 03 ½ 34p

32 12 24 2 2 2 2 yM r sn² :yM 2 e e sE S ,s y p 4f e yx e dd sy l a e , 21Ž . Ž .� 4 Ž .0N u d 0 0 d d n n53 2

a 42 2 2 2 2² : ² :8p f e uu s E S ,s q p x 6e e qe uu sE S ,sŽ . Ž .� 4Ž .0 0u 1 0 u s u 0 03 ½ 34p

32 1 224 2 2 2 2 yM r sJ² : 0yM 2 e e sE S ,s y p 4f e yx e ss sy l a e , 22Ž . Ž .� 4 Ž .0J u s 0 0 s s J J53 2

4 w x ² : ² :In a recent work 15 , it is argued that qq cannot be expanded in powers of F for massless quarks. We find, however, that qqF Fmn

w xcan be expanded in the NJL model for massless quarks 16 . In this paper we assume that the expansion is possible.


a 42 2 2 2 2² : ² :8p f e ss s E S ,s q p x 3e e q3e e qe ss sE S ,sŽ . Ž .Ž .0 0s 1 0 u s d s s 0 03 ½ 34p

322 2 4 2 2 2² : ² : ² :yM e qe e y8p fm e ss sE S ,s y p 2f e qe yx e e uu ddŽ . Ž . � 4� 4 Ž .0 0 0S u d s s s 0 0 u d u d 53

1 22 yM r sS0sy l a e , 23Ž .S S2

a 42 2 2 2 2 2² : ² : ² : ² :8p f 2 e uu q2 e dd ye ss s E S ,s q p x 2 e q3e q3e e uuŽ . Ž .�0 0 0 0½ 5u d s 1 0 u d s u3 ½ 312p

² : ² :q2 e q3e q3e e dd y e q3e q3e e ss sE S ,sŽ . Ž . Ž .40 0d u s d s u d s 0 0

2 2 2 2² : ² : ² :yM e e qe e q4e e q8p fm 2 e uu q2 e dd y3e ss sE S ,sŽ .0 0 0½ 5ž /L s u s d u d s u d s 0 0

644 2 2 2 2 2 2² : ² : ² : ² : ² : ² :y p f 2 e qe uu ss q2 e qe dd ss y e qe dd uuŽ . Ž . Ž .½ 50 0 0 0 0 0u s d s d u3

32 1 24 2 2 yM r sL² : ² : ² : ² : ² : ² :q p x 2 e e uu ss q2 e e dd ss ye e uu dd sy l a e ,0 0 0 0 0 0½ 5u s d s u d L L53 2

24Ž .Ž . Ž . n Ž .k Ž0.Ž . Ž .where E S ,s s1yexp yS rs Ý S rs rk!. It is well known that from P p in Eq. 2 one can getn 0 0 ks0 0

the Borel sum rules for l2 listed below.B

a 64 2 2s 43 2 2 4 2 yM r sN² : ² :s E S ,s qp G sE S ,s q p dd s2 2p l e , 25Ž . Ž . Ž . Ž .Ž .0 02 0 0 0 Np 3

a 64 2 2s 43 2 2 4 2 yM r sJ² : ² :s E S ,s qp G sE S ,s q p dd s2 2p l e , 26Ž . Ž . Ž . Ž .Ž .0 02 0 0 0 Jp 3

a 64 22s 43 2 2 2 4 2 yM r sS² : ² : ² :s E S ,s q8p m ss sE S ,s qp G sE S ,s q p uu s2 2p l e ,Ž . Ž . Ž . Ž .Ž .0 0 02 0 s 0 0 0 0 Sp 3

27Ž .8 as3 2 2 2² : ² : ² : ² :s E S ,s y p m 2 uu q dd y3 ss sE S ,s qp G sE S ,sŽ . Ž . Ž .Ž .0 0 0 02 0 s 0 0 0 03 p

64 244 2 yM r sL² : ² : ² : ² : ² :q p 2 uu q dd ss y uu dd s2 2p l e , 28Ž . Ž .Ž .0 0 0 0 0 L9

²Ž . 2:where a rp G denotes the gluon condensate.0s0Ž . Ž .Here, let us compare the sum rules for a , Eqs. 21 ; 24 . In the n and J the left-hand sides are similarB

Ž . Ž .to one another because of the same isospin-1r2 structure of the interpolating fields in Eqs. 15 and 16 , whileŽ . Ž .the interpolating fields of Eqs. 17 and 18 have isospin-1 and 0, respectively and the above similarity is notŽ . 0seen. In the limit of flavor SU 3 symmetry, the n and J exactly coincide with each other, because of

e se . Furthermore, up to dimension 5 we find the generic results:s d

40 0a sa s a s4a . 29Ž .n J L S3


It is interesting to know that not only the leading terms but also higher dimensional terms satisfy the aboverelation.

2 Ž . Ž . Ž . Ž .Let us now estimate a numerically. We eliminate l in Eqs. 21 ; 24 by using Eqs. 25 ; 28 . For theB Bw xsusceptibilities we use the values evaluated in Ref. 12 :

xsy2.58 GeVy2 , fs1.66 GeVy4 . 30Ž .We note, however, that there is an uncertainty due to the variation of the susceptibilities with the change of s: Inthe reliable interval of s, x varies from y2.76 GeVy2 to y2.44 GeVy2 , f from 1.49 GeVy4 to 1.91 GeVy4.

w xThe vacuum condensates of the operators and the mass of the strange quark are taken from Ref. 14 , and areas3 42² : ² : ² : ² : ² :uu s dd s y225 MeV , ss s0.8 uu , G s 360 MeV , m s150 MeV.Ž . Ž .0 0 0 0 0 sp

0In Fig. 1 we show the squared Borel mass s dependence of a . We see that, for a , a and a , theB n J L

plateau develops for the suitable values of the continuum threshold parameter S . This implies that the sum rules0

0for a , a and a work very well and are reliable. Indeed, we see that dominant contributions in the OPEn J L

come from the leading terms and the higher dimensional terms are suppressed. The convergence of the OPEhence appears to be good. Taking the maximum value of the curves, we predict that

y4 3a s14.0=10 fm , 31Ž .n

y4 30a s10.8=10 fm , 32Ž .J

y4 3a s6.2=10 fm . 33Ž .L

These results have uncertainties arising from the errors in the susceptibilities. With the variation of the0susceptibilities, a varies from 12.1 to 16.8, a 9.1 to 13.5, and a 5.0 to 8.0.n J L

0On the other hand, the Borel curve of a are not stabilized for any choice of S due to apparently slowS 0

0convergence of the OPE. Indeed, the dimension 6 term is significantly large and contributes to a negatively.S

0 0In view of this, no definite conclusion for a can be drawn. However, we can say that a might take a smallS S

value.It is interesting to compare our results with those obtained in the heavy baryon chiral perturbation theory

Ž . w xHBChPT 8 . The QCD sum rules give smaller values for the electric polarizabilities of hyperons as comparedwith that of neutron. This is in agreement with the HBChPT prediction. In the HBChPT, as already mentioned,the splitting is due to the contribution of kaon cloud. It can be seen that the ordering of magnitudes for strange

0 0baryons is reversal in the two calculations; namely, in the HBChPT we find a )a )a , whileS L J

0 0a )a )a in the QCD sum rules. It is known that the effect of the decouplet states is large, so that it isJ L S

w xhighly desired to obtain the result of the HBChPT with including the decouplet states 17 .

2Fig. 1. The squared Borel mass s dependence of a . The values of the continuum threshold parameter S are taken to be 3.6 GeV for theB 0

n, 4.2 GeV2 for the J 0, 6.7 GeV 2 for the L and ` GeV2 for the S 0.


For charged baryons, there newly appear in the phenomenological side a triple pole term which isproportional to squared charge of the baryon and involves large factor. Within the calculation up to dimension 6in the OPE, this term makes the Borel stability worse. We therefore need to evaluate the higher dimensionalterms. This task is presently under way.

In summary, we have presented a calculation of the electric polarizabilities of neutral baryons in theframework of the QCD sum rule. The operators up to dimension 6 were retained in the operator product

0 0expansion. The sum rule predicts that the ordering of the polarizabilities is a )a )a )a , and impliesn J L S

Ž .0that a has a possibility to be very small. In particular, we find that in the flavor SU 3 symmetric limit,S

0 0a sa s4a r3s4a , up to dimension 5 terms in the OPE. The calculations performed in the HBChPTn J L S

have led still different predictions. However, the future experimental data from CERN and Fermilab could be ofhelp to discriminate the different approaches.

Ž .One of the authors Y.K. would like to thank O. Morimatsu, M. Oka and D. Jido for helpful discussions.Y.K. thanks also the members of the theory group of Institute of Particle and Nuclear Studies for theirhospitality during his stay there.

References

w x Ž .1 A.I. L’vov, Int. J. Mod. Phys. A 8 1993 5267.w x Ž .2 F.J. Federspiel et al., Phys. Rev. Lett. 67 1991 1511.w x Ž .3 A. Zieger et al., Phys. Lett. B 278 1992 34.w x Ž .4 E. Hallin et al., Phys. Rev. C 48 1993 1497.w x Ž .5 J. Schmiedmeyer et al., Phys. Rev. Lett. 66 1991 1015.w x Ž .6 M.A. Moinester, in: A. Bernstein, B. Holstein Eds. , Proc. Workshop on Chiral Dynamics, July 1994, Los Alamos archive

hep-phr9409463.w x Ž .7 H.J. Lipkin, M.A. Moinester, Phys. Lett. B 287 1992 179.w x Ž . Ž .8 V. Bernard, N. Kaiser, J. Kambor, U.-G. Meissner, Phys. Rev. D 46 1992 R2756; Phys. Lett. B 319 1993 269.w x Ž .9 C. Gobbi, C.L. Schat, N.N. Scoccola, Nucl. Phys. A 598 1996 318

w x Ž .10 M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 1979 385.w x Ž .11 M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 1979 448.w x Ž .12 T. Nishikawa, S. Saito, Y. Kondo, Nucl. Phys. A 615 1997 417.w x Ž .13 B.L. Ioffe, A.V. Smilga, Nucl. Phys. B 232 1984 109.w x Ž .14 L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Reports 127 1985 1.w x Ž .15 I.A. Shushpanov, A.V. Smilga, Phys. Lett. B 402 1997 351.w x Ž .16 S.P. Klevansky, Rev. Mod. Phys. 64 1992 649; references therein.w x Ž .17 T.R. Hemmert, B.R. Holstein, J. Kambor, Phys. Rev. D 55 1997 5598.

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Electric polarizabilities of neutral baryons in the QCD sum rule