Nucleon vector strangeness form factors: multi-pion continuum and the OZI rule

1 January 1998

Ž .Physics Letters B 416 1998 5–15

Nucleon vector strangeness form factors:multi-pion continuum and the OZI rule

H.-W. Hammer a,b, M.J. Ramsey-Musolf a,1

a Institute for Nuclear Theory, UniÕersity of Washington, Seattle, WA 98195, USAb UniÕersitat Mainz, Institut fur Kernphysik, J.-J.-Becher Weg 45, D-55099 Mainz, Germany¨ ¨

Received 25 March 1997; revised 21 July 1997Editor: J.-P. Blaizot

Abstract

We estimate the 3p continuum contribution to the nucleon strange quark vector current form factors, including the effectof 3plrp and 3plv resonances. We find the magnitude of this contribution to be comparable to that the lighteststrange intermediate states. We also study the isoscalar electromagnetic form factors, and find that the presence of a rp

resonance in the multi-pion continuum may generate an appreciable contribution. q 1998 Published by Elsevier ScienceB.V.

PACS: 14.20.Dh; 11.55.-m; 11.55.FvKeywords: Strange vector formfactors; OZI-violation; 3p continuum; Dispersion relations

1. Introduction

The role of the ss sea in the low-energy structureof the nucleon has received considerable attention

w xrecently 1 . In particular, several experiments – bothin progress as well as in preparation – will probe thestrange sea by measuring the nucleon’s strange quark

Ž s. Ž s. w xvector current form factors, G and G 2 . Moti-E M

vated by these prospective measurements, a numberof theoretical predictions for GŽ s. and GŽ s. haveE M

been made. While attempts to carry out a first princi-

1 On leave from the Department of Physics, University ofConnecticut, Storrs, CT 08629, USA. National Science Founda-tion Young Investigator

ples QCD calculation using the lattice are still inw xtheir infancy 3 , the use of effective hadronic mod-

els have provided a more tractable approach to treat-ing the non-perturbative physics responsible for the

w xform factors 4–12 . Unfortunately, the connectionbetween a given model and the underlying dynamicsof non-perturbative QCD is usually not transparent.Hence, one finds a rather broad range of predictionsfor the strange-quark form factors.

One may hope, nevertheless, to derive some qual-itative insights into the mechanisms governing GŽ s.

E

and GŽ s. by using models. Indeed, one issue whichM

models may address is the validity of using the OZIw xrule 12 as a guide to the expected magnitude of the

strange-quark form factors. With the OZI rule inmind, most model calculations have assumed that

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII S0370-2693 97 01322-1

( )H.-W. Hammer, M.J. Ramsey-MusolfrPhysics Letters B 416 1998 5–156

Ž . Ž .Fig. 1. Schematic illustration of a OZI-allowed and b OZI-violating mesonic contributions to the strange-quark vector current

Žmatrix element in the nucleon. Curly lines denote gluons 3 is the.minimum number required for a vector current insertion . Un-

marked lines and closed loops denote meson and baryon valencequarks.

GŽ s. and GŽ s. are dominated by OZI-allowedE M

hadronic proceses in which the nucleon fluctuatesinto intermediate states containing valence s- and

Ž .s-quarks Fig. 1a . In a recent quark model calcula-w xtion, Geiger and Isgur 6 have shown – at second

order in the strong hadronic coupling – that perform-ing a sum over a tower of such OZI-allowed statesleads to small values for the leading strangenessmoments. While conclusions based on a second-order

w xcalculation may be questioned 13 , the assumptionthat such OZI-allowed processes dominate the formfactors is largely un-tested 2.

In what follows, we analyze the validity of thisassumption by studying the contribution from the 3p

Ž .intermediate state Fig. 1b . The three pion state isthe lightest state carrying the same quantum numbersas the strange quark vector current. As this state alsocontains no valence s- or s-quarks, it represents thesimplest case by which to test the assumption thatthe OZI-allowed processes indicated above are themost important. On general grounds, one might ex-pect the 3p contribution to be suppressed for two

Ž .reasons: a its contribution depends on the OZI-² < < : Ž .violating matrix element 3p sg s 0 and b chiralm

2 Pole model treatments have incorporated the possibility ofOZI-violation. We discuss these analyses in more detail below.

power counting implies the presence of additionalŽfactors of pr4p F where p is a small momentump

.or mass as compared to contributions from twoparticle intermediate states. The latter expectation is

w xsupported by the recent calculation of Ref. 14 , inwhich the isoscalar EM form factor spectral func-

Ž 7.tions were analyzed to OO q in heavy baryon chiralŽ .perturbation theory CHPT . In that analysis, the 3p

contribution was found to lie well below the corre-sponding 2p continuum contribution to the isovectorEM spectral functions. Moreover, the isoscalar spec-

w xtral functions computed in Ref. 14 rise smoothlyŽ 2 .from zero at threshold ts9m and show no evi-p

dence of of a near-threshold singularity enhancementas occurs in the 2p isovector continuum.

Given the foregoing arguments, one would expectthe 3p continuum contributions to the strangenessform factors to be negligible. We find, however, thatthe presence of 3plrp and 3plv resonances in

² < : ² < < :the amplitudes NN 3p and 3p sg s 0 may en-m

hance the 3p continuum contribution to a levelcomparable to that of typical two-particle, OZI-al-lowed continuum effects. Our estimate of the 3plv

resonance contribution is similar in spirit to that ofw xprevious studies 8–10 . The impact of the 3plrp

resonance, however, has not been considered previ-ously. Our estimate of this contribution dependscrucially on the measured partial width for the decayf™rp , which signals the presence of non-negligi-

² < < :ble OZI-violation in the matrix element 3p sg s 0 .m

The inclusion of a 3plrp resonance itself iswell-founded on phenomenological grounds, as wediscuss below. Consequently, we also consider itsrole in the isoscalar EM channel. In this channel, therp structure in the 3p continuum appears to have aparticularly significant impact on the isoscalar mag-netic moment, and suggests that conventional poleanalyses of the isoscalar form factors may needmodification to account explicitly for continuum ef-fects 3.

In arriving at these conclusions, we have relied ona calculation requiring model-dependent assump-tions. In the absence of a more tractable first princi-

3 w xThe recent work of Ref. 12 reaches a similar conclusionregarding conventional pole analyses.

( )H.-W. Hammer, M.J. Ramsey-MusolfrPhysics Letters B 416 1998 5–15 7

Ž .ples QCD or effective theory approach, the use of amodel allows us to compare the 3p contribution tothe lowest lying OZI-allowed contribution computed

w xusing the same model as Refs. 5,13 . In making thiscomparison, we draw on the framework of dispersionrelations, which affords a systematic means for iden-tifying various hadronic contributions to the formfactors of interest. In focusing on the 3p contribu-tion, we discuss the phenomenological justificationfor including the v and rp resonances, and presentour model calculation for these contributions. Weconclude with a discussion of the results and theirimplications.

2. Dispersion relations

The framework of dispersion relations is wellsuited for the study of multi-meson contributions to

w xthe nucleon form factors. Following Refs. 15,13 ,we work in the NN production channel, where theseform factors are defined as

Ža.² < < :N p ;N p J 0Ž . Ž . m

Ža.iF tŽ .2Ža. nsU p F t g q s P V p ,Ž . Ž . Ž .1 m mn2mN

1Ž .

m m 2Ž . Ž .where P s pqp , tsP , U V is a nucleonŽ . Ž .anti-nucleon spinor, and ‘‘ a ’’ denotes the flavor

EMw Ž . xchannel J Is0 or sg s . The Dirac and Paulim m

strangeness form factors are related to GŽa. and GŽa.E M

as

GŽa.sF Ža.yt F Ža. , 2Ž .E 1 2

GŽa.sF Ža.qF Ža. , 3Ž .M 1 2

where tsytr4m2 . For elastic processes involvingNŽ X.2 Xthe nucleon, one has ts pyp , where p and p

are the initial and final nucleon momenta.Since the value of F Ža. at ts0 is rigorously1

known – it is just the nucleon’s net isoscalar EMŽ . Ž .charge s1r2 or net strangeness s0 – we use a

subtracted dispersion relation for F Ža.. Since we1

want to determine the value of F Ža. at ts0, we use2

an un-subtracted dispersion relation for the latter 4.Hence, we write

`Ža. Xt Im F tŽ .1XŽa. Ža.F t yF 0 s dt , 4Ž . Ž . Ž .H X X1 1

p t t y ty ieŽ .t0

`Ža. X1 Im F tŽ .2XŽa.F t s dt , 5Ž . Ž .H X2

p t y ty iet0

where t begins a cut along the real t-axis associated0

with the threshold for a given physical state. In whatfollows, we will be particularly interested in themean square radius and anomalous magnetic mo-ment associated with the currents J Ža.:m

Ža.dF1 2Ža. 2 2² :r s sy m r 6Ž .Ža.N3dt ts0

k Ža.sF Ža. ts0 . 7Ž . Ž .2

The corresponding dispersion integrals for the di-mensionless mean square radius and magnetic mo-ment are

2`

Ža.4m Im F tŽ .N 1Ža.r sy dt , 8Ž .H 2p tt0

`Ža.1 Im F tŽ .2Ža.k s dt . 9Ž .H

p tt0

The spectral decomposition of the spectral func-Ža.Ž .tions Im F t is readily obtained by applying thei

4 The use of additional subtractions in the dispersion relationfor F Ža. would require knowledge of the second and higher1

moments of the form factor. Since we wish to study the meansquare Dirac radius, we restrict ourselves to a single subtraction inthis case. For a discussion of theoretical considerations regarding

w xsubtractions and convergence, see Ref. 13 and references therein.


LSZ formalism to the absorptive part of the matrixŽ . w xelement in Eq. 1 15 :

Ža.² < < :Im N p ;N p J 0Ž . Ž . m

p 3r2™ 2p NNŽ .'Z

Ža.² < < :² < < := N p J 0 n n J 0 V pŽ . Ž . Ž .Ý N m

n

4=d pqpyp , 10Ž .Ž .n

where NN is a nucleon spinor normalization factor, Zis the nucleon’s wavefunction renormalization con-stant, and J is a nucleon source.N

< :The states n of momentum p are stable withnŽrespect to the strong interaction i.e., no vector me-

.son resonances and carry the same quantum num-EM G P C y yyŽ . Ž .bers as J and sg s: I J s0 1 . Them m

lightest such state is the three pion state, with a

Ž .Fig. 2. Schematic illustration of a the 3p contribution to theŽ . Ž .spectral function, as given in Eq. 10 , and b the model approxi-

Ž .mation given by Eq. 11 . Single solid lines denote pions andŽ .double line in b denotes a vector meson. Right hand parts of

GŽ P C .diagrams represent matrix element to produce a I J sy Ž yy . < : Ž < : < : < :.0 1 state n s 3p , v , or rp from the vacuum

through the isoscalar EM or strangeness current. Left hand sidedenotes n™ NN scattering amplitude.

physical threshold of t s9m2 . The contribution0 p

from this state to the spectral function via the de-Ž .composition of Eq. 10 is illustrated diagramatically

in Fig. 2a. In order of successive thresholds, the nextallowed purely mesonic states are the 5p , 7p , KK ,9p , KKp , . . . . In the baryonic sector, one has NN,LL, . . . . One may also consider states containingboth mesons and baryons.

Most existing hadronic calculations of theŽ .strangeness form factors have either a included

only the KK and YY states in the guise of loops –even though they are not the lightest such allowed

Ž .states – or b approximated the entire sum of Eq.Ž .10 by a series of poles. In the case of loops, it was

w xshown in Ref. 13 that YK loop calculations –which treat the KK and YY contributions together –are equivalent to the use of a dispersion relation in

Ž .w h ic h i th e s c a tte r in g a m p litu d e² Ž . < Ž . < : Ž .N p J 0 n V p is computed in the Born ap-N

Ž . ² < Ža. < :proximation and ii the matrix element n J 0 ism

taken to be point like, that is, the t-dependence ofthe associated form factor is neglected. As discussed

w xin Refs. 13,15 , both approximations are rather dras-tic. Indeed, considerations of unitarity imply thepresence of important meson rescattering correctionsŽ .higher-order loop effects in the scattering ampli-tude. Moreover, the amplitude may also containvector meson resonances, such as KKlf. In thecase of the KK contribution to the leadingstrangeness moments, it was also shown that theinclusion of a realistic kaon strangeness form factor

w xcan significantly affect the results 13 .Our aim at present is to compare the contributions

< : < :arising from the states KK and 3p as they enterŽ .the decomposition of Eq. 10 . With regard to the 3p

contribution, a careful study of NN™3p or Np™

Npp scattering data would be needed to give arealistic determination of both rescattering and reso-nance contributions. The resonances include 3plV

X yŽ yy.and 3plV p , where V is a 0 1 vector mesonX qŽ yy.such as the v or f, and where V is 1 1 vector

meson such as the r. Such a study would involvefitting the amplitudes in the physical region andperforming an analytic continuation to the un-physi-cal region appropriate to the dispersion relation. Thefeasibility of carrying out this analysis, given thecurrent state of multipion-nucleon scattering data, is

Ž .unclear. In addition, the use of Eq. 10 requires the


² < Ža. < :matrix element 3p J 0 , for which – in them

strangeness channel – no data currently exists. Inorder to make any statement about the scale of the3p contribution, then, it appears that one must intro-duce model assumptions.

3. Model calculation

In the present context, we seek to determinewhether any structure exists in the 3p continuumwhich might enhance its contribution over the scale

w xone expects based on chiral symmetry 14 and theŽ .OZI-rule. Pure resonance processes 3plV, etc.

have been considered previously in the pole modelw xanalyses of Refs. 4,8–11 . In these models, the form

factors are approximated by a sum over poles associ-yŽ yy.ated with the lightest 0 1 vector mesons; no

explicit mention is made of the intermediate statescattering amplitudes or form factors. When appliedto the isoscalar EM form factors, this approximationyields a value for the f-nucleon coupling signifi-cantly larger than suggested by the OZI rule:g rg fy1r2. The corresponding predictionsfNN v NN

for the strangeness form factors are surprisingly large,especially in the case of the strangeness radius 5.

In the present study, we seek to determine theexistence of important OZI-violating contributions to

Ž s.Ž .the F t without relying exclusively on the polei

approximation and the associated large fNN cou-pling. Instead, we adopt the following model approx-imation for the 3p contribution to the spectral func-tion:

rp3pŽa. Ža. Ža.Im F t sp a d ty t q Im F t ,Ž . Ž . Ž .i i v i

11Ž .

where t sm2 . The first term constitutes a modelv v

for the 3plv contribution in which a narrowresonance approximation is made for the 3p spectral

Ža.² < :² < < :function in the vicinity of t : NN 3p 3p J 0v m

5 w xWe note the study of Ref. 12 , in which the effect ofcorrelated rp exchange was included in the isoscalar EM spectralfunctions. The authors find a significant reduction in the value ofg from the pure pole analyses. In contrast to the present study,fNN

however, they assumed that the rp state does not couple to thestrangeness vector current.

Ža.² < :² < < :™ NN v v J 0 . The second term models them

3plrp contribution. The latter, which becomesŽ .2non-zero only for tG m qm , assumes that ther p

3p continuum can be approximated by a rp state inŽa.² < :² < < :this kinematic regime: NN 3p 3p J 0 ™m

Ža.² < :² < < :NN rp rp J 0 . The content of this modelmŽa.Ž .3pfor Im F t is illustrated diagramatically in Fig.i

2b.The rationale for our model rests on several ob-

servations. We first note that of the states appearingŽ .in the spectral decomposition of Eq. 10 , only the

< : < :3p and 5p can resonate entirely into an v, sincethe cuts for the other states occur for t )m2 and0 v

since G rm <1. Experimentally, it is well knownv v

that one finds a strong peak in eqey™3p near' w xs sm 16 . On the other hand, there does notv

exist – to our knowledge – convincing evidence fora strong coupling of the v to five pions. Moreover,pqpyp 0 invariant mass distributions for eqey™

w x5p events display a peak near m 17 , indicatingv

that the presence of the v in the 5p channel arisesŽprimarily through the 3plv resonance for t)1

2 .GeV . Hence, one has reason to assume that theŽa.Ž .presence of a pure v resonance in Im F t arisesi

dominantly through the 3p intermediate state. Giventhe relatively narrow width of the v, the narrow

Ž .resonance approximation appearing in Eq. 11 alsoappears to be justified.

Several experimental facts motivate the inclusionof the rp term in our model. First, there exists a

q y 'strong peak in e e ™3p near s sm and a largef' w xrp continuum for s )1.03 GeV 16 . Second, theŽ .f has a non-negligible partial width G f™rp

whereas the partial width for decay into an un-corre-lated 3p state is a more than a factor of four smallerw x18 . Together, these observations suggest that, for

2 ² < Ža. < :tR1 GeV , the matrix element 3p J 0 is domi-m

nated by processes in which the 3p resonate to a rp

and then – for tfm2 – to the f. Since the f isf

almost a pure ss state, one would expect the 3pl

rplf mechanism to generate a non-negligible² < < :contribution to 3p sg s 0 .m

Regarding the NN™3p scattering amplitude, oneobserves a pronounced rp resonance in the 3p

channel for pp annihilation at rest. The latter hasw xbeen seen in bubble chamber data 19 and at LEAR

w x Ž w x.20 for a review, see Ref. 21 . From analyses ofthese data, one finds the rp branching ratio to be


Ž .roughly three times that for the direct non-resonant3p final state. In addition, plots of 3p yield vs.pqpy invariant mass display peaks in the vicinity

Ž . Ž .of the f 1270 and f 1565 vector mesons as well2 20Ž .as the r 770 . While these data exist only at the ppŽ 2 .threshold ts4m , they suggest the presence ofN

significant V Xp resonance structure elsewhere along

the 3p cut. In our model, then, we consider only theŽ .lightest such resonance rp and approximate the

< : < : Ž .2state 3p as the state rp for tG m qm .r p

While this approximation entails omission ofhigher-lying V X

p structure, it is nevertheless suffi-cient for estimating the possible scale of the 3p

contribution to the strangeness moments. We alsonote in passing that our approximation here is similar

w xto the one employed in Ref. 6 , where OZI-allowedŽ .multi-pion states, e.g., KKp were approximated as

)Ž .two particle states viz., K K .To calculate the aŽa. and Im F Ža. we require thei i

strong meson-nucleon vertices. For the NNp cou-pling, we employ the linear s model, and for theNNV interaction, we use the conventional vector and

Ž w x.tensor couplings see, e.g. Ref. 22 . The corre-sponding Lagrangians are

LL syig NtPpg N 12Ž .NNp NNp 5

fNN V n mLL syN g g y s E V N , 13Ž .NN V NN V m mn2mN

where V m stPr m and V m sv m for the r and v,Ž .respectively, and where the derivative in Eq. 13

acts only on the V m field.The pole approximation for the v-resonance con-

tribution yields

aŽa.sg rf Ža. 14Ž .1 NNv v

aŽa.s f rf Ža. 15Ž .1 NNv v

where the f Ža. are defined viaV

M 2VŽa.² < < :0 J V q ,´ s ´ . 16Ž . Ž .m mŽa.fV

Ž s.A non-zero value for f arises from the small ssv

component of the v; it may be determined using thew xarguments of Ref. 8 as discussed below. For the

strong couplings, we employ values taken from fitsw xto NN scattering 22 : g s15.853 and f s0.NNv NNv

The resultant v contributions to the strangeness

radius and magnetic moment do not differ apprecia-bly if we use, instead, the strong couplings deter-mined from pole analyses of isoscalar EM form

w xfactors 23 . We note that this method for includingthe v contribution is identical to that followed in

w xRefs. 8–10,4 .Ža.Ž . rpWhen evaluating the function Im F t , wei

w xfollow the philosophy of Ref. 6 and evaluate theŽNN™rp amplitude in the Born approximation Fig.

.3 , neglecting possible meson-meson correlationsw x24 . In addition, we require knowledge of the matrix

² < Ža. < :element rp J 0 , which one may parameterizem

as

² b c < Ža. < :r ´ ,k p q J 0Ž . Ž . m

g Ža.rp bc a b l)s d e q k ´ . 17Ž .ma blmr

In the case of the isoscalar EM current, one mayIs0Ž .obtain the value of g ts0 from the radiativerp

decay of the r. For purposes of computing thedispersion integrals, one also requires the behavior of

Is0Ž .g t away from the photon point. Since thererp

exist no data for t/0, one must rely on a model forthis kinematic region. A similar statement applies inthe strangeness channel, since there exist no data forŽ s.Ž .g t for any value of t. We therefore follow Ref.rp

w x Ž .25 and employ a vector meson dominance VMDmodel for the transition form factors. The VMDmodel yields

g Ža. t G rf Ža.Ž . Ž .rp Vrp Vs , 18Ž .Ý 2 3m 1y trM y iG trMr V V VVsv ,f

where the G are strong Vrp couplings, the 1rf Ža.Vrp V

² < Ža. < :give the strength of the matrix element V J 0 ,m

Fig. 3. Born diagrams for the rp ™ NN amplitude.


and G trM 2 gives an energy-dependent width forV Vw xvector meson V as used in Ref. 26 . The G haveVrp

w xbeen determined in Ref. 25 using the measuredŽ .partial width G f™rp and a fit to radiative

decays of vector mesons in conjunction with theVMD hypothesis. The isoscalar EM constants f Is0

v ,fw xare well known. Using the arguments of Ref. 8 , we

may determine the corresponding values in thestrangeness sector: 1rf Ž s.fy0.2rf Is0 and 1rf Ž s.

v v f

fy3rf Is0. The value for 1rf Ž s. is essentiallyf f

what one would expect were the f to be a pure ssw xstate 25 , while the small, but non-zero, value of

Ž s.1rf accounts for the small ss component of thevŽa.Ž .v. The resulting values for the g 0 are 0.53 andrp

0.076 in the EM and strangeness channels, respec-tively.

Ž . Ž .Using Eqs. 12 – 18 as inputs, we obtain theŽa.Ž . rp 2spectral functions Im F t for tG4m , follow-i N

ing the procedures outlined in the Appendix of Ref.w x13 :

rpŽa.Im F tŽ .1

)Ža. 2g t g QŽ .rp NNpsRe 'm 16p t Pr

2m t f m tN r NN N= g Q z q Q zŽ . Ž .NNr 2 22 2½2m2 P PN

tq 2Q z qQ zŽ . Ž .0 23

2 2m t PN 2y 2 K y 1q Q z , 19Ž . Ž .12ž / 5ž /PQ 2 mN

rpŽa.Im F tŽ .2

)Ža. 2g t g QŽ .rp NNpsRe 'm 16p t Pr

2= g m Q z q2Q zŽ . Ž .NNr N 2 03½t f m2 trNN N

y Q z y Q zŽ . Ž .2 22 25 ½2m2 P PN

2m tN 2y 2 K y Q z , 20Ž . Ž .1 5ž /PQ 2

where

2 2 2 2 2 2 2m qm y2Q y ty2 Q qm Q qm(Ž . Ž .p r p r

zs8 PQ

21Ž .

2(Ps tr4ym , 22Ž .N

1 22 2 2 2 2(Qs t q m ym y2 t m qm ,Ž . Ž .r p p r'2 t23Ž .

212 2 2 2 2 2(K s Q qm y Q qm yQ , 24Ž .( r p4 ž /

Ž .and the Q z , is0,1,2 are the Legendre functionsi

of the second kind. Since we have made the two-par-ticle rp approximation for the 3p continuum, thespectral functions become non-zero only for t) t s0Ž .2 2m qm , rather than t s9m . Moreover, be-r p 0 p

cause t lies below the physical NN production0Ž .threshold, the imaginary parts from Eqs. 19 and

Ž .20 have to be analytically continued into the un-Ž .2 2physical region m qm F tF4m . In order tor p N

evaluate the dispersion integrals using these spectralfunctions, we must make an additional assumptionregarding the relative phases of the quantities enter-

Ž . Ž .ing Eqs. 19 , 20 . In the absence of sufficientexperimental data for the NN™rp amplitude and

Ža.Ž .for the g t , we have no unambiguous way ofrp

determining the relative phases of these two quanti-w xties. We therefore follow Ref. 13 and replace the

Ž .scattering amplitude and form factors in Eqs. 19 ,Ž .20 by their magnitudes – a procedure which yieldsan upper bound on the magnitude of the spectralfunctions. We also take the overall sign for thespectral functions from our model, although thischoice has no rigorous justification. Explicit numeri-cal results obtained under these assumptions, to-

Ž .gether with the VDM hypothesis of Eq. 18 , aregiven in the following section.

4. Results and discussion

In Fig. 4 we plot the spectral functions of Eqs.Ž . Ž . Ža.Ž .19 , 20 , scaled to the value of g ts0 andrp


Ž . Ž .Fig. 4. Weighted spectral functions entering the dispersion integrals for the mean square radius a and anomalous magnetic moment b ,Ža.Ž . Ž . Ž .scaled to the value of g ts0 . In the case of scenario 1 solid line , the curves for the isoscalar EM and strangeness channels arerp

Ž . Ž . Ž .identical. For scenario 2 , the isoscalar EM dotted line and strangeness dashed line spectral functions differ. The weighted, un-scaledIs0Ž . Ž s.Ž .spectral functions are obtained from the curves by multiplying by g 0 s0.53 and g 0 s0.076 as appropriate.rp rp

weighted by powers of 1rt as they enter the radiusw Ž . Ž .xand magnetic moment Eqs. 8 , 9 . We do not

display the region containing the contribution fromŽ .the first term in Eq. 11 as it simply yields a

d-function at tf25m2 . We consider two scenariosp

Ž .for illustrative purposes: 1 a point-like form factorŽa.Ž . Ža.Ž .at the rp current vertex, i.e., g t sg 0 andrp rp

Ž . Ža.Ž . Ž . Ž .2 g t given by Eq. 18 . For scenario 1 , therp

plots for both the isoscalar EM and strangenesschannels are identical. In this case, the spectral func-tions rise smoothly from zero at the rp thresholdw Ž .2 xt s m qm and exhibit no structure suggest-0 r p

ing any enhancement over the un-correlated 3p con-

tinuum. The structure of the spectral functions inŽ .scenario 2 , where we include more realistic rp

form factors, is markedly different. In both the EMand strangeness channels, the spectral functions con-tain a strong peak in the vicinity of the f resonance,followed by a subsequent suppression for larger t

Ž .compared to scenario 1 . Although the rp formfactors contain both v- and f-pole contributions, theeffect of the v is rather mild since m lies belowv

the rp threshold. The f peak itself is stronger inthe strangeness as compared to the EM channel,since 1rf Ž s.rfy3rf Is0.f f

Using these spectral functions, we compute the


Table 1Isoscalar EM and strangeness dimensionless mean square Diracradius and anomalous magnetic moment. First row gives experi-mental values for isoscalar EM moments. Second and third rowsgive contributions to the EM and strangeness moments from therp intermediate state, computed using the Born approximationfor the NN™ rp amplitude and a vector meson dominancemodel rp form factor. Fourth and fifth rows list v-resonancecontribution to strangeness moments. Numbers in parenthesescorrespond to strong couplings obtained from fits to isoscalar EMform factors. Final row gives lightest OZI-allowed intermediatestate contribution to the strangeness moments, computed using the

² 2:same approximations as for the rp case. To convert r to r ,multiply r by -0.066 fmy2 .

Flavor channel Source r k

Is0, EM Exp’t y4.56 y0.06Is0, EM 3p l rp 0.60 y0.38

strange 3p l rp 0.86 y0.44strange 3p l v 1.08 0

Ž . Ž .1.41 0.04

strange KK 0.53 y0.16

contributions to the EM isoscalar and strangenessradii and magnetic moments. The results are given inTable 1. For the illustrative purposes, we have listedthe 3plv and 3plrp contributions to thestrangeness form factors separately. In the case of

Ž .scenario 1 , the dispersion integrals diverge. Conse-Ž .quently, we only quote results for scenario 2 , in

which the VMD rp form factors render the integralsfinite. We also give the contribution from the lightestintermediate state containing valence s- and s-quarksŽ .KK , computed using similar assumptions as in the

Ž .case of the rp contribution: a the Born approxima-Ž .tion for the NN™KK amplitude; b the linear

Ž . Ž .SU 3 s-model for the hadronic couplings; and c aVMD form factor at the KK-current vertex. Wereiterate that the phases of the rp contributions inTable 1 are not certain.

The results for the strangeness radius and mag-netic moment indicate that the magnitude of the 3p

contribution, arising via the v and rp resonances,is similar to that of the KK contribution. At leastwithin the present framework, then, we find noreason to neglect the lightest OZI-violating interme-diate state contribution, as is done in nearly allexisting hadronic model calculations. Moreover, themagnitudes of the v and rp contributions are com-parable for the radius, while the rp term gives a

significantly larger contribution to k s. This resultappears to depend rather crucially on the presence of

Ž s.Ž .the f-meson pole in g t which enhances therp

spectral functions in the region where they areweighted most heavily in the dispersion integral.

We also find that while the rp contribution to theisoscalar EM charge radius is small compared toexperimental value, the isoscalar EM magnetic mo-ment is not. This results suggests that a re-analysis of

Is0Ž .the F t ought to be performed, including notiyŽ yy.only the effects of sharp 0 1 resonances but

also those from the rp .It is instructive to compare our results with those

obtained for the isoscalar EM spectral functions us-ing CHPT and those obtained in the pure pole ap-proximation. A direct comparison with the former isdifficult, since the chiral expansion is valid only for't <4p F , whereas the rp contribution to thep 'spectral function becomes non-zero only for t )mr

qm ;4p F . Nevertheless, our results point to ap p

different conclusion than the one reached in Ref.w x X14 . Specifically, we find that because of V p struc-ture in the continuum, the 3p state could play anon-trivial role in the isoscalar EM channel – evenapart from the effect of pure 3plV resonances.Regarding the strangeness channel, we also note that

w xthe 3p-current vertex employed in Ref. 14 , derivedfrom the Wess-Zumino-Witten term, gives no hint ofany OZI-violating hadronic structure effects which

² < < :would generate a non-zero 3p sg s 0 matrix ele-m

Ž 7.ment. To OO q in CHPT, then, the un-correlated 3pŽ s.Ž .continuum does not contribute to the F t . Ini

order to find such a contribution, we have relied onhadron phenomenology, including the observation ofthe OZI-violating f™rp decay and the presence ofa rp resonance in the pp™3p reaction.

w xWith respect to the pole analyses of Refs. 4,8–10 ,we have found evidence of non-negligible f reso-

Ž s.Ž .nance contribution to the F t without relyingi

exclusively on the validity of the pole approximationŽa.Ž .for the F t or on the presence of a strong fNNi

coupling obtained from pole model fits to theIs0Ž .F t . At this time, however, we are unable toi

determine whether the pure pole approximation ef-fectively and accurately includes the effect of the f

as it arises via a rp resonance in the 3p continuumas well as in the KK continuum as analyzed in Refs.w x13,27 . Within the framework employed here, nei-


ther the 3plrplf effect, nor the KKlf con-tribution, approaches in magnitude the predictions of

w xRefs. 4,8–10 for the strangeness radius. The corre-sponding predictions for the strange magnetic mo-ment, however, are commensurate. An independenttheoretical confirmation of the pole model predic-tions would require a more sophisticated treatment ofthe NN™3p , NN™KK , etc. amplitudes than un-dertaken here, including the effects of resonant andnon-resonant multi-meson rescattering. Indeed, stud-

w xies of KN scattering amplitudes 27 , as well asw xtheoretical models for NN™rp 24 , suggest that

rescattering corrections could significantly modifythe spectral functions employed in the present calcu-lation.

We make no pretense of having performed adefinitive analysis. Rather, within the level of ap-proximation employed in many model calculations,we have simply demonstrated the prospective impor-tance of contributions to the strangeness form factorsarising from mesonic intermediate states containingno valence s or s-quarks. Although a naive interpre-tation of the OZI-rule suggests that such contribu-tions ought to be negligible compared to those fromstrange intermediate states, we conclude that realistic

Ž s.Ž .treatments of the F t , using effective hadronici

approaches, ought to take the former contributionsinto consideration. Our results also agree with the

w xconclusions of Ref. 12 , suggesting that a morecareful treatment of the 3p continuum in the isoscalarEM form factors is warranted.

Acknowledgements

We wish to thank R.L. Jaffe, N. Isgur, U.-G.Meißner and D. Drechsel for useful discussions.HWH has been supported by the German Academic

Ž .Exchange Service Doktorandenstipendien HSP III .MJR-M has been supported in part under US Depart-ment of Energy contract a DE-FG06-90ER40561and under a National Science Foundation YoungInvestigator Award.

References

w x Ž .1 M.J. Musolf et al., Phys. Rep. 239 1994 1w x Ž .2 MIT-Bates proposal a 89-06 1989 , R.D. McKeown and

D.H. Beck spokespersons;Ž .MIT-Bates proposal a94-11 1994 , M. Pitt and E.J. Beise,

spokespersons;Ž .CEBAF proposal a PR-91-017 1991 , D.H. Beck,

spokesperson;Ž .CEBAF proposal a PR-91-004 1991 , E.J. Beise,

spokesperson;Ž .CEBAF proposal a PR-91-010 1991 , M. Finn and P.A.

Souder, spokespersons;Ž .Mainz proposal A4r1-93 1993 , D. von Harrach, spokesper-

sonw x Ž .3 K.-F. Liu, U. of Kentucky preprint UKr95-11 1995 and

references therein;Ž .D.B. Leinweber, Phys. Rev. D 53 1996 5115

w x Ž .4 M.J Ramsey-Musolf and H. Ito, Phys. Rev. C 55 19973066

w x Ž .5 M.J. Musolf and M. Burkardt, Z. Phys. C 61 1994 433w x Ž .6 P. Geiger and N. Isgur, Phys. Rev. D 55 1997 299w x7 W. Koepf, S.J. Pollock, and E.M. Henley, Phys. Lett. B 288

Ž .1992 11;Ž .W. Koepf and E.M. Henley, Phys. Rev. C 49 1994 2219;

Ž .H. Forkel et al., Phys. Rev. C 50 1994 3108;T. Cohen, H. Forkel, and M. Nielsen, Phys. Lett. B 316Ž .1993 1;N.W. Park, J. Schechter, and H. Weigel, Phys. Rev. D 43Ž .1991 869;

Ž .S.-T. Hong and B.-Y. Park, Nucl. Phys. A 561 1993 525;Ž .S.C. Phatak and S. Sahu, Phys. Lett. B 321 1994 11;

Ž .H. Ito, Phys. Rev. C 52 1995 R1750;Ž .X. Ji and J. Tang, Phys. Lett. B 362 1995 182;

Ž .W. Melnitchouk and M. Malheiro, Phys. Rev. C 55 1997431;H.-C. Kim, T. Watabe, and K. Goeke, Nucl. Phys A 616Ž .1997 606

w x Ž .8 R.L. Jaffe, Phys. Lett. B 229 1989 275w x9 H.-W. Hammer, U.-G. Meißner, and D. Drechsel, Phys. Lett.

Ž .B 367 1996 323w x Ž .10 H. Forkel, Prog. Part. Nucl. Phys. 36 1996 229;

) w xH. Forkel, preprint ECT rSeptr95-04 hep-phr9607452w x11 M.J. Musolf, lectures given at the 11th Students Workshop

Žon Electromagnetic Interactions, Bosen, Germany Septem-.ber 4–9, 1994 ; Invited talk presented at the Conference on

ŽPolarization in Electron Scattering, Santorini, Greece Sep-.tember 12–17, 1995

w x Ž . w12 U.-G. Meißner et al., preprint KFA-IKP TH -1997-01 hep-xphr9701296

w x13 M.J. Musolf, H.-W. Hammer, and D. Drechsel, Phys. Rev. DŽ .55 1997 2741

w x14 V. Bernard, N. Kaiser, and U.-G. Meißner, Nucl. Phys. AŽ .611 1996 429

w x15 P. Federbush, M.L. Goldberger, and S.B. Treiman, Phys.Ž .Rev. 112 1958 642;

See also S.D. Drell and F. Zachariasen, ElectromagneticStructure of Nucleons, Oxford University Press, 1961

w x Ž .16 S. I. Dolinsky et al., Phys. Rep. 202 1991 99w x Ž .17 A. Cordier et al., Phys. Lett. B 106 1981 155;

B. Delcourt et al., Proceedings of the International Sympo-


sium on Lepton-Photon Interactions at High Energies, E.Ž .Pfeil, ed., Univ. of Bonn, Germany 1981 p. 205

w x18 Particle Data Group, Review of Particle Properties, Phys.Ž .Rev. D 54 1996 1

w x Ž .19 M. Foster et al., Nucl. Phys. B 6 1968 107w x Ž .20 B. May et al., Phys. Lett. B 225 1989 450;

Ž .B. May et al., Z. Phys. C 46 1990 191, 203w x21 C. Amsler and F. Myhrer, Ann. Rev. Nuc. Part. Sci. 41

Ž .1991 219

w x22 B. Holzenkamp, K. Holinde, and J. Speth, Nucl. Phys. A 500Ž .1989 485

w x23 P. Mergell, U.-G. Meißner, and D. Drechsel, Nucl. Phys. AŽ .596 1996 367

w x Ž .24 V. Mull et al., Phys. Lett. B 347 1995 193w x Ž .25 J.L. Goity and M.J. Musolf, Phys. Rev. C 53 1996 399w x Ž .26 F. Felicetti and Y. Srivastava, Phys. Lett. B 107 1981 227w x27 M.J. Ramsey-Musolf and H.-W. Hammer, INT Preprint a

w xDOErERr40561-323-INT97-00-170 hep-phr9705409

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Nucleon vector strangeness form factors: multi-pion continuum and the OZI rule