Strange mesonic transition form factor

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PHYSICAL REVIEW C JANUARY 1996VOLUME 53, NUMBER 1

0556-2

Strange mesonic transition form factor

J. L. GoityDepartment of Physics, Hampton University, Hampton, Virginia 23668

and Theory Group, MS 12H2, Continuous Electron Beam Accelerator Facility, Newport News, Virginia 236

M. J. MusolfDepartment of Physics, Old Dominion University, Norfolk, Virginia 23529

and Theory Group, MS 12H2 Continuous Electron Beam Accelerator Facility, Newport News, Virginia 236~Received 1 May 1995!

The strange-quark vector currentr-to-p meson transition form factor is computed at one-loop order usingstrange meson intermediate states. A comparison is made with af-meson dominance model estimate. We findthat one-loop contributions are comparable in magnitude to those predicted byf-meson dominance. It ispossible that the one-loop contribution can make the matrix element as large as those of the electromagcurrent mediating vector meson radiative decays. However, due to the quadratic dependence of the oneresults on the hadronic form factor cutoff mass, a large uncertainty in the estimate of the loops is unavoidaThese results indicate that non-nucleonic strange quarks could contribute appreciable in moderate-uQ2u parity-violating electron-nucleus scattering measurements aimed at probing the strange-quark content of the nuc

PACS number~s!: 25.30.Bf, 12.40.Vv, 13.10.1q, 14.40.2n

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I. INTRODUCTION

There has been considerable interest recently in the ussemileptonic weak neutral current scattering to study tstrange-quark ‘‘content’’ of the nucleon@1–12#. In particular,several parity-violating~PV! electron scattering experimentare planned and/or underway at MIT-Bates, CEBAF, aMAINZ whose objective is to measure the nucleon’s strangquark vector current form factors@13–17#. In a similar vein,a low-uQ2u determination of the nucleon’s axial vectostrangeness form factor will be made using neutrino scating at LAMPF @18#, following on the higher-uQ2u measure-ment made at Brookhaven@19#. These experiments are ointerest in part because they provide a new window on trole played by nonvalence degrees of freedom~specifically,virtual ss pairs! in the nucleon’s response to a low- omedium-energy external probe. In contrast to the theoretianalysis of scattering in the deep inelastic regime, for whiperturbative methods are applicable, the interpretationlow-to-medium energy scattering can be carried outpresent only within the context of effective hadronic modeIn the case of the nucleon’s strangeness form factors, sevmodel calculations have been performed yielding a rathbroad spectrum of results@20–26#. It is desirable, then, thatexperimental determinations of these form factors be carrout at a level of precision allowing one to distinguish amonvarious models and the physical pictures on which they abased.

As discussed elsewhere in the literature, semileptomeasurements performed with proton targets alone wonot be sufficient for this purpose@1,2,6#. A program of mea-surements which includesA.1 targets appears to be warranted@1,2#. The interpretation of neutrino-nucleusand PVelectron-nucleusscattering observables naturally introducea new level of complication associated with many-bodnuclear dynamics not encountered in the case of protongets. These many-body effects are interesting in two

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spects. On the one hand, if one wishes to extract the sinnucleon strangeness form factors from nuclear form factoone requires knowledge of the many-body contributionsthe nuclear strangeness form factors. On the other, the rplayed by non-nucleonic strangeness~i.e., non-nucleonic,nonvalence quark degrees of freedom! in the nuclear re-sponse is interesting in its own right.

Recently, one class of many-body contributions to thnuclear strangeness form factors—meson exchange curr~MEC’s!—were analyzed for the case of4He @20,21#. A he-lium target will be used in a future CEBAF PV electronscattering experiment designed to study the nucleostrangeness electric form factor at moderateuQ2u @14#. In Ref.@21#, it was shown that MEC’s give a non-negligible contribution to the4He strangeness form factor at the kinematicsthe approved experiment. Of particular note is ther-to-p-meson strange-quark ‘‘transition current.’’ Using a simplf-meson vector dominance model and the knownf→rpbranching ratio, this MEC was estimated to give a 15% cotribution to the PV asymmetry. Given the magnitude of thresult and its potential importance in the interpretation of th4He experiment, one would like a more detailed analysisthis strangeness transition current. The goal of the prespaper is to improve upon the vector dominance estimateincluding loop effects. In so doing, our objective is not smuch to provide an airtight theoretical prediction as to arrivat an order of magnitude for, and quantify the theoreticuncertainty associated with, loop effects. Section II givesbrief discussion of the nuclear physics context for our calclation as well as a review of thef-dominance estimate of ther-p strangeness form factor. In Sec. III we present the formframework in which we carry out our analysis. The loocalculation is presented in Sec. IV. Section V gives our rsults and a discussion of their significance, and in Sec. VI wsummarize our work. Technical details may be found in thAppendix.

399 © 1996 The American Physical Society

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400 53J. L. GOITY AND M. J. MUSOLF

II. NUCLEAR STRANGENESS FORM FACTORS

The 4He PV asymmetry,ALR , can be written as@1,2,6#

ALR5GmuQ2u4A2pa

F4 sin2uw1FC

~s!~q!

FCT50~q!

G , ~1!

where Gm is the Fermi constant measured inm decay,Q25v22uqW u2 is the four-momentum transfer squared, anwhere electroweak radiative corrections have been ignorfor simplicity. The quantityF C

T50(q) is the4He electromag-netic elastic charge form factor~T50 for isoscalar targets!.The4He elastic strangeness form factor,F C

(s)(q), is given by

FC~s!~q!5E d3x eiqW •xW^g.s.us†~xW !s~xW !ug.s.&, ~2!

where ug.s.& is the nuclear ground state ands(xW ) is thestrange-quark field operator. Note that sinces†s is just thecharge component of the strangeness vector current,sgms,and since4He has not net strangeness, the form factorF C

(s)

must vanish at zero momentum transfer. For nonzero mmentum transfer,F C

(s)(q) receives a number of contributions,some of which are illustrated in Fig. 1.

In conventional~nonrelativistic! nuclear models, the pro-cesses of Figs. 1~a! and 1~b! are sensitive to the nucleon’sstrangeness vector current form factors. Those of type 1~b!which involve an exchange of a mesonM must be includedin any nuclear calculation which respects vector current coservation and in which the two-nucleon potential arises frothe exchange of mesonM. In the present paper, we are concerned with contributions of type 1~c!, which involve matrixelements of the strangeness vector current between mestatesM andM8. It is straightforward to show usingG-parityinvariance thatM8usgmsuM & vanishes whenM5M 8. Henceonly transition matrix elements contribute. The corresponing MEC’s are referred to as ‘‘transition currents.’’ For moderate values of momentum transfer, one expects the lightmesons to give the largest effect, since they have the longrange and experience the least suppression from theN-Nshort range repulsion. For this reason, we restrict our attetion to the transition current involving the lightest possiblallowed pair of states:M5p andM 85r.

FIG. 1. Contributions to the nuclear strangeness charge fofactor,F C

(s)(q). One-body~a! and ‘‘pair current’’ ~b! contributionsdepend on the nucleon’s strange-quark vector current form facto‘‘Transition current’’~c! contributions arise from strange-quark vector current matrix elements between meson statesuM& and uM8&.Here, the cross indicates the insertion of the strangeness chaoperator.

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For on-shell mesons, ther-to-p transition matrix ele-ment has the structure

^ra~k1 ,«!usgmsupb~k2!&5grp

~s!~Q2!

M remnabk1

nk2a«b* dab,

~3!

where«b is the r-meson polarization vector,k1 and k2 arethe meson momenta, anda andb are isospin indices. In thecase of nuclear processes, where the typical momenta of hrons inside the nucleus have magnitudes less than the Femomentum~;200 MeV!, the virtualr meson will be ratherfar off its mass shell. Consequently, the dimensionless forfactorg rp

(s) ought also to depend onk 12 andk 2

2 as well asQ2.It is conventional in nuclear calculations, however, to neglethe off-shell dependence of transition form factors, so fopurposes of making contact with this framework, we wilquote results for the on-shell case.

In previous work@20,21#, an estimate ofg rp(s)(Q2) was

made based on the assumption off-meson dominance, asillustrated in Fig. 2. Under this assumption, one has

^ra~k1 ,«!usgmsupb~k2!&

5^0usgmsuf~Q,«f!&1

Q22Mf2 ^f~Q,«f!

3ur~2k1 ,«!p~k2!&, ~4!

where «f is the virtual f-meson polarization vector andwhere a sum over all independent polarizations is implieNoting that the f is nearly a puress state, so that^0uugmuuf&'0'^0udgmduf&, one has~see the Appendixfor details!

^0usgmsuf~Q,«!&'23^0uJmEMuf~Q,«!&523

Mf2

f f«m ,

~5!

where one obtainsf f'13 from an analysis ofG~f→e1e2!@22#. From the experimental value for thef→rp branchingratio @22#, one may obtain a value for the magnitude of thdecay amplitude,rpuf&. From these inputs one obtains forthe transition form factor~see the Appendix!

grp~s!~Q2!uf2dom5

grps

Q22Mf2 , ~6!

with ugrpsu'0.20 GeV2 @23#.

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FIG. 2. f-meson dominance picture of the strangeness curretransition matrix element.V refers to a vector meson,P to a pseu-doscalar, and the cross represents the insertion of the current.

53 401STRANGE MESONIC TRANSITION FORM FACTOR

FIG. 3. One-loop diagramswhich contribute to the strange-ness current transition matrix ele-ment. The cross represents thestrangeness vector current.

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In what follows, we consider additional contributions tg rp(s)(Q2) arising from loops, as in Fig. 3.The rationale for considering these contributions is sim

lar to that for including loops in studies of other form factorFrom a dispersion integral standpoint,g rp

(s)(Q2) receivescontributions not only from poles~identified with vector me-sons!, but also from the multi-meson, intermediate-state cotinuum. For values ofQ2 sufficiently far from the poles~e.g.,M f

2 !, the continuum contributions need not be negligible.the present instance, the lightest allowed intermediate stacontain one pseudoscalar and one vector meson~parity re-quires the vanishing of the strong interactionp→two K am-plitude!. Hence we will consider contributions involving onK and oneK* in the intermediate state.

III. EFFECTIVE LAGRANGIANS

In this section we derive the effective interactions relevato the calculation of one-loop contributions to the matrelement^ra~k1 ,«!usgmsup

b~k2!& and determine the effectivecoupling constants associated with them. In the derivatwe use an effective chiral Lagrangian, which takes into acount the global symmetries of QCD. This Lagrangian ivolves the octet of light pseudoscalar mesons@the quasi-Goldstone bosons of spontaneously broken chiSUL~3!3SUR~3!#, the nonet of vector mesons, and externgauge sources needed to obtain the relevant currents~elec-tromagnetic and strangeness! which play a role in our analy-sis.

The framework is that of the nonlinears model. Thequasi-Goldstone fields are coordinates of the coset spSUL~3!3SUR~3!/SUV~3!;SU~3!, and appear in the follow-ing form:

U~x!5expS 2 iP

F0D , P5 (

a51

8

pala, ~7!

wherela are the Gell-Mann matrices normalized accordinto Tr~lalb!52dab, pa are the members of the octet of pseudoscalar quasi-Goldstone fields, andF0;93 MeV is the piondecay constant in the chiral limit. The transformation proerties of the quasi-Goldstones are determined by the w

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U~x! responds to a chiral transformation: U~x!→RU~x)L†,whereL(R) belongs to SUL~3!@SUR~3!#.

The chiral transformation law for the octet of vector mesons is given by

Vm→hVmh†, ~8!

whereVgm5V mala, u(x)[AU(x), and h5h[L,R,u(x)] is

determined by the equations

Lu~x!5u8~x!h, Ru†~x!5u8†~x!h, ~9!

whereu8 results from the action of the chiral transformationon u.

In order to build a chirally invariant Lagrangian for theoctet of vector mesons we further need to introduce a covriant derivative

¹m5]m2 iGm , Gm5i

2~u†]mu1u]mu

†!, ~10!

which operates as follows onVm : ¹mVn5]mVn2 i [Gm ,Vn].Similarly, we require the axial vector connection

vm5i

2~u†]mu2u]mu

†!. ~11!

Both ¹m andvm have the same chiral transformation law aVm .

In addition, we will consider two external source gaugfields, the electromagnetic potentialAm and the potentialS mwhich couples to the strangeness vector current. In this wone can obtain the most general form of the respective curents in the effective theory. At the appropriate stage we wshow how they enter in the effective Lagrangian.

The piece of the effective Lagrangian containing the knetic and mass terms of the vector meson octet is given b

LV521

8Tr~VmnV

mn!1MV

2

4Tr~VmV

m!, ~12!

whereVmn5¹mVn2¹nVm . The LagrangianLV only con-tains terms with two vector mesons~V! and even numbers ofquasi-Goldstone bosons~P!. There is no SU~3! breaking at

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this level in the interactions. For simplicity, we will onlyinclude the dominant SU~3! breaking effects which are generated by the mass splittings in the octets of pseudoscand vector mesons.

The vertex of typeVVPneeded in our analysis is specifically rK*K. In order to determine the corresponding copling constant we need to consider in addition theVV0Pvertices, whereV0 is the singlet component of the nonet ovector mesons. The corresponding effective Lagrangiansleading order in chiral power counting are ofO ~p!, wherepdenotes the generic small momentum carried by the quaGoldstone bosons, and read

LVVP5R8emnrsTr~$¹mVn,Vr%vs!,

LVV0P5R0emnrsTr~¹mVnvr!V0s. ~13!

Here, we have used the identityemnrs¹rvs50 to simplify theexpressions. For the vertices of interest we need only kethe first term in the expansion ofvm:vm51/2F0]

mP1O ~P3!.The only observable strong interaction process of this typef→rp. In order to be able to determineR8 , we must simul-taneously pin downR0 , and this requires further informationwhich we obtain by considering the radiative decaysV→Pgsupplemented with the hypothesis of vector meson domnance. From this analysis, presented in the Appendix,obtainR850.2760.08.

The vertices of typeVPPwe need are of the typeK*Kp.They are determined in terms of a single effective couplinand the corresponding effective Lagrangian is

LVPP5j8Tr~¹mVn@vm,vn#!, ~14!

wherej8 can be determined from either of the following twdecay widths:

G~r0→p1p2!5j82

192pF04 M r

2~M r224Mp

2 !3/2,

G~K*1→K1p0!5j82

96pF04 EKEpkfFMK*

22S 11

Ep

EKDMK

2

2S 11EK

EpDMp

2 G , ~15!

kf51

2MK*Al@MK*

2 ,MK2 ,Mp

2 #,

where l@x,y,z]5x21y21z222xy22xz22yz. The firstwidth givesj850.175 and the second givesj850.140. No-tice that we have used the same decay constant forp andKmesons. Under the assumption of SU~3! symmetry in theinteractions, the use of either value ofj8 is justified.

Now we turn to those terms in the effective Lagrangiawhich determine the electromagnetic and strangenessrents. The electric charge operator contains only octet pieand is explicitly given by Q5(e/2)[l31~1/A3!l8#5~e/3)diag~2,21,21!, while the strangeness charge operatcontains a singlet and an octet piece and readsS5 1

3 ~1

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2A3l8!5diag~0,0,1!. In the following, q denotes either ofthese charge operators, andvm represents either of the twosource fieldsAm andS m.

Let us first consider theV→vm amplitudes relevant forthe VMD analysis. The effective Lagrangian contains an otet and a singlet piece

LVv5C8vmTr~ qVm!1C0v

mTr~ q!Vm0 . ~16!

The leptonic widths ofr0, v, andf determineC8 accordingto

G~V→e1e2!54p

3aMV

f V2 , ~17!

where

f r5A2M r

2

C8,

f v5A6Mv

2

cosuC8, ~18!

ff5A6Mf

2

sinuC8.

In practice, thev2f mixing angle is taken to be that of ideamixing, tanu5A2, which leads toC8.0.16 GeV2. In orderto determine the transition mediated by the strangeness crent, we also need to know the singlet couplingC0 . As thereis no direct experimental determination of this coupling, onmust rely on some hypothesis. By assuming exact OZI supression in thev-to-vacuum current matrix elements~i.e.,^0usgmsuv&50), we obtainC052C8/A3520.092 GeV2.

The transitions within the octet of pseudoscalar andvector mesons mediated by the currents are obtainedminimal substitution into the chiral covariant derivative¹min Eq. ~12!, plus a gauge invariant term involving the fieldstrength tensor of the gauge field in the case of the vecmesons. Both electromagnetic and strangeness current tsitions are determined by the following effectiveLagrangians:

LPPv52i

2Tr~]mP@ q,P#1••• !vm,

LVVv5i

4Tr~Vmn@ q,Vn#!vm1 i

z

2Tr~@Vm ,Vn#q!vmn.

~19!

Only one new unknown effective coupling~z! enters, whichis related to the magnetic moments of the vector mesons.we mention in Sec. V, the lack of experimental access tozdoes not affect our results, as it appears only in a loop dgram @diagram 3~c!# which turns out to be numerically sub-stantially smaller than the other diagrams.

Vertices of typeVPPv are obtained from Eq.~14! byminimal substitution in¹m andvm , and by a term propor-tional to the field strength tensor of the gauge field. Theffective Lagrangian reads~for the sake of simplicity wekeep only terms relevant to our calculation!:

l


LVPPv52 ij84F0

2 Tr„]mVn~†]mP,@ q,P#‡vn2m↔n!1vm@ q,Vn#@]mP,]nP#…1 i

z84F0

2 Tr~]mVn†@ q,P#,P‡!vmn1••• .

~20!

The new coupling constantz8 is fixed by considering the radiative decayr0→p1p2g. Using the expression for the partiawidth

G~r0→p1p2g!5aM r

24p2F04E

Mp

MrdE1E

Mp

MrE1dE2 Q@4~E1

22Mp2 !~E2

22Mp2 !2„M r

212Mp222~E11E2!M r12E1E2…

2#

3„2z8M r2~2z82j8!~E11E2!…2 ~21!

rmi-thethefore,sse-ge-e-,

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e

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if-

and using the previously determined valuej850.175, we ob-tain two solutions:z850.156 andz8520.082. Unfortunately,there are no further measured observables to discriminbetween these two solutions. In our results we will includboth possibilities.

The final vertices we need are those of the typeVVPv. Asin the previous case, we only need those involving membof the meson octets. They are entirely determined by mimal substitution into Eq.~13!, and the relevant pieces read

LVVPv52 iR8

2F0emnrsTr~v

m$@ q,Vn#,Vr%]sP

1vs$]mVn,Vr%@ q,P#!. ~22!

We notice here that the effective Lagrangians~19!, ~20!, and~22! only couple the octet components of the vector sourcto the mesons. The SU~3! singlet piece of the vector currenis the baryon number current which cannot couple to msons.

From the effective Lagrangians given in this section itstraightforward to derive the Feynman rules and calculateone-loop diagrams in the following section.

IV. ONE-LOOP CALCULATION

In this section we discuss some salient features ofcalculation of the one-loop contributions to^r0~k1 ,«!usgmsup0~k2!&, which on grounds of isospin symmetry are the same as the contributions t^r1~k1 ,«!usgmsup1~k2!&.

There are only four diagrams, depicted in Fig. 3, whiccontribute at one-loop order. As expected, all diagramsultraviolet divergent. Diagrams 3~a!, 3~b!, and 3~d! are qua-dratically divergent, while diagram 3~c! is only logarithmi-cally divergent. At this point we must warn that, quite igeneral, the present calculation shows lack of an approprchiral expansion as an expansion in the number of loops; tis due to the presence of a heavy meson~K* ! in the loop.Hence, in contrast to calculations involving light mesonsstable heavy baryons carried out using chiral perturbattheory, in the present case multiloop contributions contapieces which are leading in the low energy expansion.

We regulate the loops by using a convenient form factOne could instead have used dimensional regularization ausually done in chiral perturbation theory. This procedurhowever, will require the introduction of counterterms whic

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are unknown in the present case. Instead, the use of fofactors is more likely to give a good estimate of loop contrbutions even when counterterms are disregarded. Inpresent case, we are concerned with a matrix element ofstrangeness vector current operator, and as mentioned beonly matrix elements of the octet piece of the strangenecurrent are affected by the one loop corrections. Consquently, one could have related the octet piece of the stranness vector current matrix elements to similar matrix elments involving the electromagnetic current. This ishowever, not good enough due to important SU~3! breakingeffects in the loops produced by the mass spittings in toctets of pseudoscalar and vector mesons. For this reasonchoose in this work to perform the calculation of the loocorrections to both strangeness and electromagnetic curmatrix elements using the aforementioned form factors. Asis made clear later, vector current Ward identities will bproperly restored while implementing the form factor.

For purposes of simplicity, we choose the form factor tdepend only onk2, the momentum squared of the virtualKmeson, and use the same cutoff mass parameter for all tyof vertices. We expect this choice to be of little significancconcerning the generality of our results. The hadronic forfactor is taken to be

F~k2!5MK

22L2

k22L21 i e, ~23!

where the scaleL will be chosen within a reasonable rangas discussed below. Fork25M K

2 , one hasF(M K2 )51.

Hence this choice for the form factor is consistent with thvalues of the mesonic coupling constants (R0 ,R8 ,j8! ex-tracted from on-shell amplitudes.

The implementation of this form factor in the diagramwhere the current is inserted in the meson line@3~c!,3~d!# isperformed by the following straightforward procedure:G(MK

2 ,MK*2 ) denotes the diagram for pointlike hadronic ver

tices@F(k2![1#, then the corresponding diagram withF(k2!as given in Eq.~23! is

F ~L!@G~MK2 ,MK*

2!2G~L2,MK*

2!#, ~24!

where the operatorF is given by

F ~L!52~L22MK2 !2

]

]L2

1

MK22L2 . ~25!

e

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d,

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ds,

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We note that, on general grounds, the introduction ofadditional momentum dependence at the vertices alsoquires the inclusion of new ‘‘seagull’’ vertices in order tomaintain gauge invariance@24–26#. In the present case,these seagull terms generate additional terms in theVPPvandVVPv interactions given in Eqs.~20! and~22!. Althoughthere exists no unique prescription for maintaining gauinvariance in the presence of hadronic form factors, we flow the minimal prescription of Refs.@24–26# to derive ourseagull terms. To this end, one may consider the form factappearing at the hadronic vertices as arising from a coornate space interaction of the form

C~f1 ,...,fk!lF~2]2!]lP, ~26!

whereP is the octet of pseudoscalar meson fields definedEq. ~7!, Cl is a Lorentz vector constructed from the othepseudoscalar and vector meson fields, and]2 is theD’Alembertian. Transforming to momentum space yields tsame hadronic vertices as generated by the effectLagrangians in Eqs.~13! and ~14! multiplied by the formfactorF(k2!, wherek is the momentum of the pseudoscalameson associated with the fieldP. The gauge invariance ofthis interaction can be restored by making the minimal sustitution ]m→Dm5]m2i qvm , where q is either the EM orstrangeness charge operator andvm is the correspondingsource field. Expanding the resultant interaction to first ordin vm yields the seagull interaction

C~f1 ,...,fk!lH v•~Q12i ]!FF~2D2!2F~2]2!

~2D2!2~2]2! G3]l@ q,P#2 ivlF~2D2!@ q,P#J . ~27!

Here,D25]222iQ•]2Q2, whereQ is the momentum car-ried by the source. Transforming to momentum space,placing the source by the associated vector boson polartion vector«m~Q!, and taking the specific form for the formfactor given in Eq.~23! leads to the vertex structure

2 i ClH 7F~k2!F ~Q62k!m

~Q6k!22L2Gkl«m~q!

1F@~Q6k!2#«l~q!J @ q,la#, ~28!

where the SU~3! index ‘‘a’’ is associated with the pseudo-scalar meson carrying the momentumk, Cl is themomentum-space form for theCl ~without the field opera-tors!, and the upper~lower! sign corresponds to an incoming~outgoing! pseudoscalar meson.

Inclusion of this ‘‘minimal’’ seagull vertex@via diagrams3~a!, 3~b!# is sufficient to preserve the gauge invariancethe loop calculation in the presence of hadronic form facto~for a demonstration of this feature for the case of mesobaryon loops; see Refs.@26–28#. One may, of course, in-clude additional transverse seagull terms, which are serately conserved and which, therefore, do not modify tgauge invariance of the calculation. At leading order in tlow energy expansion we have already included the opossible seagull term, namely the term proportional toz8 in

anre-

geol-

orsdi-

inr

heive

r

b-

er

re-iza-

ofrsn-

pa-hehenly

LVPPv . Thus, other transverse seagull terms will be ofhigher order, and for this reason they can be disregarded.

In general, the inclusion of hadronic form factors alsodestroys the chiral invariance of the calculation, since thinteraction in Eq.~26! is not invariant under a local chiralrotation. Restoration of chiral invariance would necessitatreplacement of the derivatives]m acting onP by the chiralcovariant derivative¹m introduced in Eq.~10!. Since thevector connectionGm appearing in¹m can be expanded in apower series in the pseudoscalar meson fieldP, this prescrip-tion for maintaining chiral symmetry would generate newchiral seagull vertices containing additional pseudoscalamesons. The chiral seagull vertex of lowest order in 1/F0would contain two additional pseudoscalar meson fields anconsequently, would first contribute to the transition formfactor at two-loop order. Since we restrict our attention toone-loop results, we do not consider contributions from theschiral seagull interactions.

In the notation introduced in the previous section, theVMD piece of this form factor is given by

1

M rgrp

~s!~Q250!Uf2dom

523

F0S 4

A3R8sinu2R0cosu D 1

f f.

~29!

It is convenient to use the measured rate forf→rp, whichgives

1

M rgrp

~s!~Q250!Uf2dom

523

f fGrfpphen, ~30!

with Grfpphen51.08 GeV21. As shown in the Appendix we

can use Eq.~30! to determine the combination ofR0 andR8which appears in Eq.~29!.

The one-loop contributions turn out to affect only the oc-tet piece of the strangeness current. This is easily inferrefrom the effective Lagrangian containing the strangenessourceS m : the source always appears in a commutatorwhich obviously eliminates its coupling to the SU~3! singletcomponent of the strangeness current. We denote the onloop contributions by grp

(s)(Q2,k12,k1•Q)u12 loop

( j ) , wherej5a,b,c,d refers to the different diagrams, and where thedependence onk 1

2 and k1Q allow for off-shell initial andfinal state mesons. In order to give the expressions of thecontributions in a convenient form, we introduce an operatoto project out the coefficients multiplying terms proportionalto k 1

a or Qa resulting from the loop integrals:

Pa~p,q!5p2qa2pqpa

p2q22~pq!2. ~31!

We then have thatPa~p,q)pa50, Pa~p,q)qa51.The expressions for the different diagrams then read a

follows:

gneticic


1

M rgrp

~s!~Q2,k12,k1Q!U

12 loop

~a!

54R8

F03 E d4k

~2p!41

~k22MK2 !@~k12k!22MK*

2#H 12 ~j82z8!F~k!2k22j8F~k!F~k1Q!

3Fk24 S 2112k~k12k!

k22L2 D1 Pa~k1 ,Q!ka~k11Q!~k12k!G J ,1

M rgrp

~s!~k12,k1Q!U

12 loop

~b!

52R8

F03 j8E d4k

~2p!4F~k!F~k2Q!

~k22MK2 !@~k11Q2k!22MK*

2#H k~k11Q2k!S 12

1

2

k2

k22L2D2 Pa~k1 ,Q!ka

„~k11Q!21k@k22~k11Q!#…2 Pa~Q,k1!ka„2~k11Q!21k@k23~k11Q!#…J ,

1

M rgrp

~s!~Q2,k12,k1Q!U

12 loop

~c!

524R8j8F03 E d4k

~2p!4F2~k!

1

~k22MK2 !@~k12k!22MK*

2#@~k11Q2k!22MK*

2#

3H ~114z!Pa~Q,k1!ka@Qk~p11Q!2Q~k11Q!k#~k12k!1

k2

4@2~114z!~k11Q!2

1~314z!k~k11Q!#J ,1

M rgrp

~s!~Q2,k12,k1Q!U

12 loop

~d!

54R8j8F03 E d4k

~2p!4F2~k!k2~k12k!2

~k22MK2 !@~k1Q!22MK#@~k12k!22MK*

2#, ~32!

where the vector meson propagator used is as usual:2 i [gmn2~kmkn/M V2)#/~k22M V

21 i e).For purposes of comparison we also consider the one-loop contributions to the matrix elements of the electroma

current ^r(k1,«)uJ mEMup(k2)& and ^K* (k1,«)uJ m

EMuK(k2)&. The one-loop contributions in the case of the electromagnetcurrent are related to those already obtained above in a simple manner. With obvious notation@f andd are structure constantsof SU~3!#, one has

gVP~g!u12 loop52

eMV

M r(

a,b,c51

8

dVabf PacS f 3bc1 1

A3f 8bcD grp

~s!U12 loop

~MK*→MVa, MK→MPb!. ~33!

nte

n-a

e

.

It is interesting to notice that the one-loop contributionsg rp(g) are due purely to strange particles in the loop. Only t

isosinglet piece of the electromagnetic current contribuhere, allowing us to writeg pp

(g)u12loop52~e/2)g rp(s) u12loop. This

one-loop correction clearly cannot be the source of explation for the difference betweengr0p0

(g) andgr1p1(g) experimen-

tally observed. On the other hand, the contributionsgK* K(g) u12 loop are more complicated, with contributions fromboth pions and kaons in the loop. Moreover, both, isosingand isovector pieces of the electromagnetic current contute, leading to different one loop corrections for charged aneutral cases.

V. RESULTS

In this section we present and discuss the results forone-loop contributions to the transition matrix elementsboth electromagnetic and strangeness currents. Throughwe useR850.27, j850.18,z50, andz850.16 ~case 1! andz8520.08~case 2!. Since diagram 3~c! turns out to give onlya modest contribution, the choicez50 has little impact onthe results @we have explicitly checked that by settingz5O ~1! the overall one loop corrections considered in th

tohetes

na-

to

letrib-nd

theofout

e

following are affected by less than 15%#. On the other hand,if we instead usej850.14, as obtained by consideringK*decay, the results obtained forj8 and the final results of theloop calculation are only slightly affected.

We discuss first the case of the electromagnetic currematrix elements. To this end, it is convenient to define thratio of the one-loop contribution to the form factor to themeasured form factor~‘‘phen’’ !

RVP~g![

gVP~g!~ loop!

gVP~g!~phen!

. ~34!

The overall sign of this ratio is undetermined, since the sigof the low energy coupling constants involved cannot be established from available observables. Hence we considerparticular choice of the overall sign, keeping in mind that inreality the sign might be the opposite. The ratioRVP

(g) isshown in Figs. 4 and 5 for the cases ofr-to-p andK* -to-Kradiative transitions, respectively. At this point it is conve-nient to stress that for almost the whole allowed rangof the cutoff, g rp

(s)~loop!, g rp(g)~loop!, gK*1K1

(g) ~loop!, and

gK* 0K0(g) ~loop! have relative signs as follows:21,11,11,11.For L,0.5 GeV the two latter signs change in case 2 only

-on-ato

n a

nshateoon

of

ithf-

.o

or

ly

e

re

e

lf--

ffwurs.

le.or,

tio


As shown in the Appendix, the corresponding VMD piecehave the relative signs:21,11,11,21.

In each case, the choice of a ‘‘reasonable range’’ for tmass parameterL was dictated by two criteria. First, in ordeto maintain consistency with our use of VMD in extractinsome of the coupling constants from radiative transitions,requireL to fall within a range such thatRrp

(g)<1/2 andRK* K(g) <1/2 ~RVP

(g)51/2 corresponds to loop and vector mesopoles giving equal contributions if the signs of the contribtions are the same!. This condition gives an upper bound oroughly one GeV. Second, to obtain a lower bound onL, we

FIG. 4. RatioRrp(g) for the matrix element of the electromagneti

current betweenr andp mesons. The solid curve correspondscase 1~z50.16! and the dashed one to case 2~z520.08!. In eachcase we have usedk 1

25M r2 , Q250, andk 2

25M p2 .

FIG. 5. RatioRK* K(g) for the matrix element of the electromag

netic current betweenK* andK mesons. The solid lines corresponto charged kaons and the dashed ones to neutral kaons. The losolid and upper dashed curves correspond to case 1, and the reming two to case 2. The kinematic invariants used arek1

25MK*2 ,

Q250, andk 225M K

2 .

s

hergwe

nu-f

refer to a ‘‘cloudy bag’’ picture of hadrons in which the pseudoscalar Goldstone bosons live outside a hadronic bag ctaining quarks. In this picture, the virtual meson must havewavelength longer than the bag radius, so as to be unablepenetrate the bag interior. From this requirement, we obtailower bound ofL;1/Rbag'0.2 GeV for a bag radius of oneFermi. For this choice ofL, the form factor in Eq.~23! willsuppress contributions from virtual pseudoscalar mesowith wavelengths shorter than one Fermi. We emphasize talthough we do not perform this calculation within thcloudy bag framework, we simply turn to that picture tobtain a physical argument for a reasonable lower boundthe cutoff mass.

ForL falling within our ‘‘reasonable range,’’ the diagramscontributing to therpg form factor display a zero atL5MK .This zero results from the numerator in Eq.~23!, which waschosen to give the normalizationF(M K

2 )51. Hence this zeroshould be taken as an unphysical artifact of the choiceform factor. We believe that the values ofRVP

(g) at L;0.2GeV give a realistic lower bound, since this value onL issufficiently far from the artificial zero atMK . In order tocheck this assumption, we also computed the diagrams wa slightly different form factor, replacing the numerator oEq. ~23! with 2L2. Such a form factor also displays pointlike behavior [F(k2!51# asL→`. The resulting values ofRVP

(g) for L in the vicinity of MK do not differ significantlyfrom those obtained with the form in Eq.~23! andL;0.2GeV. In the case of theK*→Kg form factor, no zero appearsatL5MK since loops involving virtual pions also contributeThe latter enter with hadronic form factors normalized tunity at k25mp

2 and, thus, do not vanish atL5MK .The one-loop contribution in therpg case is about 2–5 %

whenL;0.7 GeV, and grows to 20–35 % forL;1 GeV~seeFig. 4!. Thus we take roughly 1 GeV as an upper bound four reasonable range forL. For theK*Kg form factor, theloops containing ap and aK* in the intermediate state arethe dominant ones. The loop contributions are relativemore important than in the case of therpg form factor. Forcharged kaons we find the one-loop contribution to b0–15 % forL;0.7 GeV and 20–60 % forL;1 GeV, whilefor neutral kaons the corresponding contributions a0–10 % and 20–50 %~see Fig. 5!. Notice that our analysisrelies on VMD to determine some low energy constants, likR8 andR0 ~see the Appendix!. Large one-loop contributionto the radiative decays would demand performing a seconsistent fitting procedure, in which both VMD and oneloop amplitudes are included in the determination ofR8 andR0 . Such an analysis would become imperative if the cutowould be taken larger than about 1 GeV. Notice that if a nefit would be required, this should be done twofold due to oignorance about the overall sign of the one-loop correctionAlthough a shift in the value used forR8 will result from theloop corrections, we expect that for our choice of cutoff scasuch a shift will not qualitatively alter our main conclusions

For the strangeness vector current transition form factwe must define a somewhat different ratio sinceg rp

(s)(Q2)has not been measured. We compute instead raRrp

(s)of one-loop to vector meson dominance contributions

Rrp~s![

grp~s!~ loop!

grp~s!~VMD !

. ~35!

cto

-dwerain-

r-

en-

ec-n-aye

a-

on-e-

torne-r iner-

f-

to-e

re-esuron

i-sst-

ofr-

0.d-


The results for the ratioRrp(s) , shown in Fig. 6, turn out to

be significant almost for any cutoff mass value within thchosen range. The reason is thatug rp

(s)(Q250!uf2domu is verysmall ~a factor 3 smaller than 1/egrp

~g!phen!, while the absolutevalue of the one-loop correction is a factor 2 larger tha1/egrp

~g!u12loop; thus, the corresponding ratio is almost a fact6 larger than inrpg case. Consequently, forL;1 GeV, thestrangenessr2p transition matrix element can be as largemagnitude as the corresponding EM transition matrix ement, assuming the loop andf-pole contributions enter withthe same sign~uncertain at present!. For small values of theform factor mass~L;0.2 GeV!, however, the loop correctionto the f-pole contribution is 50% at most, in which casg rp(s)(0) is no more than half as large asg rp

(g)(0).In addition to giving theL dependence ofRrp

(s) , thecurves in Fig. 6 also illustrate the sensitivity of our resultsother parameters which enter. As in the case ofg rp

(g)(0), thedependence of the strangeness form factor on the choiclow-energy constantz8 is modest: the lower set of curves~case 1! and upper set~case 2! differ by less than a factor of2 over our reasonable range forL. Similarly, the dependenceon ther andp virtuality is negligible, as a comparison of thesolid curves (k 1

25mr2 , k 2

25mp2 ) and dashed curves~k 1

250,k 2250) indicates. For this reason, we conclude that a nucl

MEC calculation carried out usingk 125mr

2 andk 225mp

(2) ing rp(s) rather than allowing these momenta to vary introduc

negligible error.Finally, we refer to the dash-dotted curve, which gives t

ratio Rrp(s) ~for case 1! at the kinematics of the approved

CEBAF experiment@14#, 2Q250.6 GeV2. In this case thef-pole contribution tog rp

(s)(Q2) is down from its value at thephoton point by (12Q2/mf

2 )21'0.6. The ratioRrp(s) on the

other hand, is essentially unchanged from its value atphoton point. Thus we would expect the loop contributioto modify the strangeness MEC results of Refs.@20,21# by afactor of between 0.1 and 3.5~for L varying over our rea-

FIG. 6. Ratio Rrp(s) for the matrix element of the strangenes

current betweenr andp mesons. The solid curves correspondk 125M r

2 , Q250, and k 225M p

2 , and the dashed ones tok 1250,

Q250, andk 2250. The lower two curves correspond to case 1 a

the upper two to case 2. The dash-dotted curve corresponds to1 with k 1

25M r2 , Q2520.6 GeV2 andk 2

25M p2 .

e

nor

inle-

e

to

e of

ear

es

he

thens

sonable range! over the complete range ofQ2 considered inthose calculations.

These results have significant implications for the intepretation of CEBAF experiment@14#. Assuming, for ex-ample, that the nucleon’s strangeness form factors were idtically zero, PV4He asymmetry would still differ from its‘‘zero-strangeness’’ value [F C

(s)(0)50 in Eq.~1!# by roughly15–40 % due to ther-p strangeness MEC. Thus, for pur-poses of extracting limits on the nucleon’s strangeness eltric form factor, one encounters about a 25% theoretical ucertainty associated with non-nucleonic strangeness. By wof comparison, we note that models which give a largnucleon strangeness electric form factor,uGE

(s)/GEn u;1 gen-

erate a 20% correction toALR via the one- and two-bodymechanisms of Figs. 1~a! and 1~b!. On the other hand, giventhe projected 40% experimental error for the CEBAF mesurement, a statistically significant nonzero result forF C

(s)

would signal the presence of both a large strange-quark ctent of the nucleon as well as large, non-nucleonic strangquark component of the nucleus.

VI. SUMMARY AND CONCLUSIONS

We have calculated vector-to-pseudoscalar meson veccurrent form factors using a combination of vector mesopole and one one-loop contributions. As expected, the onloop results are strongly dependent on the mass parametethe hadronic form factor needed to regulate otherwise divgent loop integrals. For values ofL lying within a ‘‘reason-able range’’ whose upper limit is determined by selconsistency with VMD and lower limit by a cloudy bagpicture of hadrons, we find that the one-loop contributionstherpg andK*Kg amplitudes may introduce important corrections to the predictions of VMD model. In the case of thstrangeness vector currentr-to-p transition form factor, theloop contributions may enhance the total amplitude by mothan a factor ofthreeover the estimate based on VMD. Assuming loops involving heavier mesonic intermediate statdo not cancel the contribution from the lightest mesons, oresults could have serious implications for the interpretatiof the moderate-uQ2u, CEBAF PV electron scattering experi-ment with a4He target@14#. Indeed, this strangeness transtion form factor, which contributes to the nuclear strangenecharge form factorF C

(s)~q! via a meson exchange curren@20,21#, would induce a 15–40 % correction to the zerostrangeness4He PV asymmetry@Eq. ~1! with F C

(s), onewould have evidence that nonvalence quark degreesfreedom—both nucleonic and nonnucleonic—play an impotant role in the medium energy nuclear response.

ACKNOWLEDGMENTS

J.L.G. was supported by NSF Grant No. HRD-915408and by DOE Contract No. DE-AC05-84ER40150. M.J.Mwas supported by NSF NYI Award No. PHY-9357484 anDOE Contracts No. DE-AC05-84ER40150 and No. DEFG05-94ER0832.

APPENDIX

In this appendix we describe the determination ofR0 andR8 .

sto

ndcase

tedst

es

er-

s

g


Determination ofR0 and R8 : these two effective cou-plings can be determined by using the decay width off→rpand the radiative decays of vector mesons supplementedthe hypothesis of VMD.

The partial widthG~f→rp! is given by

G~f→rp!5uGfrpu2

12pukf u3.

From Eq.~13! we obtain

Gfrp51

F0S 2R0cosu1

4

A3R8sinu D ,

while the experimentally observed partial width gives

uGfrpphenu51.08 GeV21.

In practice we will takeu to correspond to idealf-v mixing.The radiative transition amplitudeV→Pg has the genera

form

A~V→Pg!52 igVP~g!emnrsPV

meVnPg

r«gs .

The radiative partial width is given by

G~V→Pg!5a

3UgVP~g!

MVU2ukf u3.

Using VMD we have that

gVP~g!~Q2!52MV (

V85r0,v,f

GVV8PCV8

Q22MV82 ,

whereCV5M V2/ f V , f r55.1, fv517, andff513. From Eq.

~13! we obtain

Grrp50,

Grvp51

F0S 4

A3R8cosu1R0sinu D ,

Grfp51

F0S 4

A3R8sinu2R0cosu D ,

GK*1rK152

F0R8 ,

GK*1vK151

F0S 2

2

A3R8cosu1R0sinu D ,

with

l

GK*1fK1521

F0S 2

A3R8sinu1R0cosu D ,

GK* 0rK052GK*1rK1,

GK* 0vK05GK*1vK1,

GK* 0fK05GK*1fK1.

The effective couplingsg VP(g) (Q250! determined from

the various radiative decays turn out to be

gr0p0~g!

50.753, gr1p1~g!

50.568, gK* 0K0~g!

521.13,

gK*1K1~g!

50.75, gvp0~g!

51.81, gfp0~g!

520.14,

where the relative signs are chosen to match those predicby VMD. Using these phenomenological results, our beVMD fit leads to

R0;1.0, R850.27.

These fitted values give the VMD results forg VP(g)

0.55, 0.55, 21.27, 0.76, 1.91,20.15

to be compared with the respective phenomenological valugiven above.

The VMD result for the transition matrix elements of thestrangeness current have the same structure, the only diffence is that nowCV has to be replaced bySV . In the follow-ing we assume thatSV is only nonvanishing forV5f. Thisholds wheneverC0 corresponds to exact OZI suppression amentioned in Sec. III. From Eq.~16! we obtain

Sf523Mf

2

f f.

This leads to the VMD result used in the text:

1

M rgrp

~s!~Q250!Uf2dom

51

Mf2 Grfp

phen Sf .

From the results quoted above, VMD predicts the followinpattern of signs:

grp~g!.0, gK* 0K0

~g!,0, gK*1K1

~g!.0,

gvp0~g!

.0, gfp0~g!

,0, grp~s!,0.

.
@1# M. J. Musolfet al., Phys. Rep.239, 1 ~1994!.@2# M. J. Musolf and T. W. Donnelly, Nucl. Phys.A546, 509
~1992!; A550, 564~E! ~1992!.@3# W. M. Alberico et al., Nucl. Phys.A569, 701 ~1994!.@4# T. W. Donnellyet al., Nucl. Phys.A541, 525 ~1992!.@5# D. H. Beck, Phys. Rev. D39, 3248~1989!.@6# M J. Musolf and T. W. Donnelly, Z. Phys. C57, 559 ~1993!.

@7# E. J. Beise and R. D. McKeown, Comments Nucl. Part. Phys20, 105 ~1991!.

@8# D. B. Kaplan and A. Manohar, Nucl. Phys.B310, 527 ~1988!;J. Napolitano, Phys. Rev. C43, 1473~1991!.

@9# G. T. Garveyet al., Phys. Lett. B289, 249 ~1992!.@10# G. T. Garvey, W. C. Louis, and D. H. White, Phys. Rev. C48,

761 ~1993!.

v.

o,


@11# G. T. Garveyet al., Phys. Rev. C48, 1919~1993!.@12# C. J. Horowitzet al., Phys. Rev. C48, 3078~1993!.@13# MIT-Bates Proposal No. 89-06, R. D. McKeown and D. H

Beck, contact persons~unpublished!.@14# CEBAF Proposal No. PR-91-004, E. J. Beise, spokespers

~unpublished!.@15# CEBAF Proposal No. PR-91-010, J. M. Finn and P. A. Soud

spokespersons~unpublished!.@16# CEBAF Proposal No. PR-91-017, D. H. Beck, spokespers

~unpublished!.@17# MIT-Bates Proposal No. 94-11, B. Mueller, M. Pitt, and E.

Beise, spokespersons~unpublished!.@18# LAMPF Proposal No. 1173, LSND Collaboration, Los Alamo

National Laboratory Report No. LA-UR-89-3764~1989!, W.C. Louis, spokesperson.

@19# L. A. Ahrenset al., Phys. Rev. D35, 785 ~1987!.

.

on

er,

on

J.

s

@20# M. J. Musolf and T. W Donnelly, Phys. Lett. B318, 263~1993!.

@21# M. J. Musolf, R. Schiavilla, and T. W. Donnelly, Phys. Rev. C50, 2173~1994!.

@22# Particle Data Group, Review of Particle Properties, Phys. ReD 50, 1173~1994!.

@23# The value ofgrps quoted in Refs.@20,21# is adimensional andit is related to the value given here, which has dimension twby a factormrmf .

@24# F. Gross and D. O. Riska, Phys. Rev. C36, 1928~1987!.@25# K. Ohta, Phys. Rev. C40, 1335~1989!.@26# M. J. Musolf and M. Burkardt, Z. Phys. C61, 433 ~1994!.@27# H. Forkelet al., Phys. Rev. C50, 3108~1994!.@28# W. Koepf, E. M. Henley, and J. J. Pollock, Phys. Lett. B288,

11 ~1992!; W. Koepf and E. M. Henley, Phys. Rev. C49, 2219~1994!.

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Strange mesonic transition form factor