Strangeness form factors of the proton in the chiral quark model

PHYSICAL REVIEW C, VOLUME 62, 045204

Strangeness form factors of the proton in the chiral quark model

L. Hannelius and D. O. RiskaDepartment of Physics, FIN-00014 University of Helsinki, Finland

~Received 1 February 2000; published 14 September 2000!

The chiral quark model describes the strangeness components of the light quarks as fluctuations into strangemesons and quarks. The single strange pseudoscalar and vector meson loop fluctuations of the constituentuandd quarks give rise to only very small strangeness form factors for the proton. This result is in line withrecent experimental results, given their wide uncertainty range.

PACS number~s!: 12.39.Ki, 14.20.Dh

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I. INTRODUCTION

The HAPPEX experiment@1# shows the combinationGE

s 10.39GMs of the strange charge and magnetic form fa

tors of the proton atQ250.48 (GeV/c)2 to be consistentwith 0~0.02360.03460.02260.026!. Similarly the SAMPLEexperiment@2# shows thatGM

s at Q250.1 (GeV/c)2 is con-sistent with 0, modulo uncertainties in the calculated valuethe weak axial form factor of the nucleon@3#. The experi-mental result that the strangeness form factors of the proare small may be used to constrain or test theoretical mofor nucleon structure, as the theoretical predictions for thobservables have covered a fairly wide range@5–11#.

A calculation of the strangeness form factors of the probased on the chiral quark model is reported here. Theproach considers the strangeness component of the protloop fluctuations, with intermediate strange mesons ansquarks, of the constituent quarks that form the proton. Tconstituent quark model approach represents an alternatithe hadronic approach, in which the strangeness componare considered as fluctuations of the nucleon into intermate strange mesons and hyperons. The chiral quark mapproach brings the advantages of much smaller coupconstants and consequently the possibility of a convergloop expansion@11#, which takes all baryonic intermediatstates into account. Moreover, as the amplitudes of the lomainly scale with the inverse squared mass of the interdiate meson, heavy meson contributions are suppressedamples of this are the desirably small meson loop contritions to the neutron charge radius@12# and the recentdemonstration that the strange meson loop contributionthe proton give rise to but a very small strangeness magnmoment@11,13#.

It is shown here that the kaon andK* loop fluctuations ofthe light quarks, which are illustrated in Fig. 1, and whilead to but a small value for the strangeness magneticment of the proton, also lead to strange form factors ofproton with very small magnitudes. The results are foundbe fairly insensitive to the value of the cutoff scale for tloop integrals, provided that this is taken to be about 1 Gor larger, i.e., values commensurate with the chiral symmrestoration scale;4p f p;1.2 GeV, at which scale thepseudoscalar mesons are expected to decouple from thestituent quarks. This insensitivity is due on the one handthe smallness of the net contribution from loops with int

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mediateK* mesons and strange quarks, and on the othand to the tendency for cancellation between the cutoffpendence of the contribution from loops with intermediateKmesons and strange quarks against that of the contribufrom loops withKK* transition couplings. The smallness othe the diagonalK* s loop contribution is due to a stroncancellation between the two loop amplitudes in whichelectromagnetic~e.m.! coupling is to the intermediateK*meson ands quark, respectively@13#.

The magnitude of the strange loop contributions tostrangeness form factors of the proton is of the or(g2/4p2)@mq

2/(mq21mM

2 )#, whereg is the meson-quark coupling, andmq andmM are the masses of the light constituequarks and strange mesons, respectively. Asg2/4p;0.7 forK and K* mesons, it follows withmq;300 MeV that theloop amplitude in the case of kaons is expected to onlyabout;0.06, and smaller still in the case ofK* mesons. Incomparison the expected magnitude of a typical loop amtude in the case of the hadronic approach, whereg2/4p.10 andmq is replaced by the proton mass, is more thanorder of magnitude greater. This is also revealed by a coparison of the calculated values for the strangeness magmoment of the proton in Refs.@9,13#. It suggests that a smanet loop contribution in the hadronic approach only cansult as a consequence of strong cancellations between selarge amplitudes unless strong cutoffs are invoked.

This paper falls into five sections. In Sec. II the contribtion of the strange loop amplitudes to the proton form factthat contain intermediate kaons are derived. The correspoing results for the loop amplitudes that involve intermediaK* mesons are derived in Sec. III. The contribution froloop fluctuations withK* →Kg vertices is derived in SecIV. The numerical results for the strangeness form factorsthe proton are calculated in Sec. V.

II. KAON LOOP CONTRIBUTIONS

The strangeness form factors of the nucleon are definethe invariant coefficients of the matrix elements of the oeratorssgms in the proton. In standard notation the strangness current is

^p8u j ms ~0!up&5 i u~p8!FF1

s~Q2!gm2F2s~Q2!

smnqn

2mNGu~p!.

~2.1!

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L. HANNELIUS AND D. O. RISKA PHYSICAL REVIEW C62 045204

Hereq5p82p andQ25q25q22q02.0, andmN is the pro-

ton mass. The two form factors are calculated here inconstituent quark model from the strangeness loop fluctions illustrated by the Feynman diagrams in Fig. 1, whthe meson lines representsK andK* mesons.

Consider first the kaon loop diagrams. To calculate thewe consider the kaon-quark pseudovector coupling

L5 if Kqs

mKcg5gm (

a54

7

la]mKac. ~2.2!

The pseudovector kaon-quark coupling constant is obtaas @13#

f Kqs5gA

a

2

mK

f K, ~2.3!

wheregAq50.87 for quarks@14,15# and f K is the kaon decay

constant (f K5113 MeV). The numerical value forf Kqs isthen f Kqs51.9.

The kaon and strange quark current density operahave the form

j m5 ie$]mK†K1H.c.%, ~2.4a!

j m52 ie

3csgmcs . ~2.4b!

The standard convention on the strangeness form factorsigns thes quark a strangeness charge of11 and the kaon,which contains ans quark, a strangeness charge of21. In thecalculation of the loop amplitudes thes-quark current~2.4b!should therefore be multiplied by23 and the kaon curren~2.4a! by 21.

The pseudovector coupling term~2.2! requires the intro-duction of a contact coupling term for current conservatiThis contact current term gives rise to two seagull diagrawhich exactly cancel the corresponding seagull diagrawhich arise in the evaluation of the amplitudes of the lodiagrams in Fig. 1, upon application of the Dirac equatfor the external quarks. The remaining loop amplitudesequivalent to those, which arise if the loop amplitudescalculated using the pseudoscalar coupling

LKqs5 igKqscg5(a54

7

laKac, ~2.5!

FIG. 1. Kaon andK* loop fluctuations of constituent quarkswhich contribute to the strangeness form factors of the proton.

04520

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e,

d

rs

as-

.s,s,

ee

where the pseudoscalar coupling constantgKqs is defined as

gKqs5mq1ms

mKf Kqs . ~2.6!

In these expressionsmq represents the constituent massthe light flavor quarks (u,d) and ms represents the strangquark mass. For these masses we shall use the vamq5340 MeV and ms5460 MeV, respectively @13#.With these mass values we obtain the valuegKqs

2 /4p50.75for the ~squared! kaon-quark pseudoscalar couplinconstant.

The kaon loop contributions to the Dirac form factorsF1s

of the quarks are logarithmically divergent. We regularithese loop amplitudes by cutting off the loop momentuintegrals smoothly at the chiral restoration scaleLx54p f p

51.2 GeV. The cutoff is implemented by replacing thmeson propagator v(k2)51/(k21mK

2 ) in the loopdiagram, which contains thes-quark current coupling@Fig. 1~a!#, by the propagator multiplied by a dipole formfactor

v~k2!→ 1

mK2 1k2 FL22mK

2

L21k2 G 2

. ~2.7!

Current conservation then demands that the productthe two meson propagators in the loop amplitude tcorresponds to the kaon current loop@Fig. 1~b!# be modifiedas

1

mK2 1k1

2

1

mK2 1k2

2→ v~k2

2!2v~k12!

k122k2

2. ~2.8!

FIG. 2. The Dirac strangeness form factorF1s of the proton as a

function of momentum transfer.

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STRANGENESS FORM FACTORS OF THE PROTON IN . . . PHYSICAL REVIEW C62 045204

The calculation then proceeds by first calculatingstrangeness form factorsF1

s and F2s for the constituent

quarks. These form factors are the same for theu and dquarks. The relation of these form factors to the correspoing strangeness form factors of the proton is simonly under the assumption thatmq5mp/3, which implies anequipartition of the total proton momentum betweenquarks:

F1s~Q2!53F1q

s ~Q2!, F2s~Q2!5

mp

mqF2q

s ~Q2!. ~2.9!

From these relations we obtain the charge and magnetic ffactors of the proton as

GEs ~Q2!5F1

s~Q2!2Q2

4mp2 F2

s~Q2!, ~2.10a!

GMs ~Q2!5F1

s~Q2!1F2s~Q2!. ~2.10b!

The magnetic form factor atQ250 then yields the strangeness magnetic moment in units of nuclear magnetons.

Charge conservation requires thatF1s(0)50. This re-

quirement is satisfied by subtracting the value ofF1s(0) from

the expression below.The contribution to the strangeness form factorF1

s ofthe proton from the two loop diagrams in Fig. 1 are obtainas

FIG. 3. The Pauli strangeness form factorF2s of the proton as a


04520

e

d-e

e

m

d

F1qs ~Q2!$a,K%5

gKqs2

8p2E0

1

dx~12x!E0

1

dyH @~ms2mq!2

12mq~ms2mq!~12x!1mq2~12x!2

2Q2~12x!2~12y!y#K1~Q2!

1 lnH1~Lx

2!

H1~mK2 !

2xLx

22mK2

H1~Lx2!J , ~2.11a!

F1qs ~Q2!$b,K%5

gKqs2

8p2E0

1

dx xE0

1

dyH 2mq~ms2mqx!

3~12x!K2~Q2!2 lnH2~Lx

2!

H2~mk2!

1xL22mk

2

H2~Lx2!J . ~2.11b!

Here the functionsK1(Q2) and K2(Q2) have been definedas

K1~Q2!51

H1~mK2 !

21

H1~Lx2!

2xLx

22mK2

H12~Lx

2!, ~2.12a!

K2~Q2!51

H2~mK2 !

21

H2~Lx2!

2xLx

22mK2

H22~Lx

2!.

~2.12b!

The denominator functionsH1(m2) andH2(m2) are definedas

H1~m2!5ms2~12x!2mq

2x~12x!1m2x

1Q2~12x!2y~12y!, ~2.13a!

H2~m2!5ms2~12x!2mq

2x~12x!1m2x1Q2x2y~12y!.~2.13b!

After subtraction of the corresponding values atQ2

50 @F1s(0)# from the form factorsF1

s(Q2) the integrals re-main finite even in the limitLx

2→`.The corresponding contributions to the strangeness P

form factors of the quarks are

FIG. 4. Strangeness fluctuations of the constituent quarks, winvolve an intermediate radiativeK* →Kg transition.

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F2qs ~Q2!$a,K%52

g2

4p2E0

1

dx~12x!2

3E0

1

dy mq~ms2mqx!K1~Q2!, ~2.14a!

F2qs ~Q2!$b,K%52

g2

4p2E0

1

dx x~12x!

1E0

1

dy mq~ms2mqx!K2~Q2!. ~2.14b!

These expressions reduce to those given in Ref.@13# inthe limit Q2→0, if multiplied by the factormp /mq togive strangeness magnetic moments in units of nucmagnetons.

In Fig. 2 the kaon loop contributions to the protostrangeness Dirac form factorF1

s are shown as functions omomentum transfer, after subtraction of the irrelevant cstantF1

s(0). This loop contribution to the proton strangeneform factor is very small and negative, and forQ2

<1 (GeV/c)2 it decreases slowly from 0 to;20.01. Asshown below, the magnitude of this contribution is smalthan that of the strange vector meson loops.

The contributions from the kaon loop amplitudes to tstrangeness Pauli form factorF2

s are shown in Fig. 3. Theresult atQ250 agrees with the value found in Ref.@11#,once allowance is made for the slightly different choiceparameter values. These contributions, while small, aretably larger than the corresponding vector meson loop ctributions that are derived in Sec. III below. The momentudependence of the kaon loop contribution toF2

s(Q2) is fairlyweak forQ2 values below 1 (GeV/c)2.

III. STRANGE VECTOR MESON LOOP FLUCTUATIONS

The coupling ofK* mesons to constituent quarks is dscribed by the Lagrangian

LK* qs5 igK* qscsS gm1ms2mq

mK*2 ]m1 i

kK* qs

ms1mqsmn]nD

3 (a54

7

laKma c1H.c. ~3.1!

This coupling is a generalization to fermions of unequmass of the conventional transverse Proca coupling for vtor mesons.

The coupling constantsgK* qs and kK* qs may be deter-mined from the corresponding couplings ofK* mesons tothe baryon octet by the quark model relations@13#

gK* qs5gK* BB ,

04520

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gK* qs~11kK* qs!53

5

ms1mq

MgK* BB~11kK* BB!. ~3.2!

Here M represents the average of the nucleon andS521hyperon (L,S) masses.

A recent comprehensive boson exchange potential mofit to nucleon-nucleon scattering data givesgK* BB52.97 andkK* BB54.22, with a liberal uncertainty margin@16#. Thesevalues yield gK* qs

2 /4p.0.7 and kK* qs.0.21. The smallvalue of the tensor couplingkK* qs and its large uncertaintyrange suggest that it is consistent with 0. At this stage itherefore justified to neglect the Pauli term in Eq.~3.1! alto-gether.

The current density operator for theK* mesons takes theform

j m56 ie$Kn*†]mKn* 2Kn*

†]nKm* %1H.c. ~3.3!

The contributions to the strangeness from factorsF1s of theu

and d quarks from theK* meson loops described by thFeynman diagrams in Figs. 1~a! and 1~b! ~when the mesonline represents aK* meson! are

F1qs ~Q2!$a,K* %

5gK* qs

2

4p2 E0

1

dx~12x!E0

1

dyH $ms224mqmsx1mq

2x2

2Q2@x1~12x!2y~12y!#%K1~Q2!

1 lnH1~Lx

2!

H1~mK*2

!2x

Lx22mK*

2

H1~Lx2!

J 1OS 1

mK*2 D . ~3.4a!

F1qs ~Q2!$b,K* %

5gK* qs

2

8p2 E0

1

dx xE0

1

dyH $6mq2x~12x!26mqms~12x!

1Q2x@122xy~12y!#%K2~Q2!

16F lnH2~Lx

2!

H2~mK*2

!2x

Lx22mK*

2

H2~Lx2!

G J 1OS 1

mK*2 D . ~3.4b!

Here the auxiliary functionsK1(Q2) andK2(Q2) have beendefined as the functionsK1(Q2) andK2(Q2) in Eqs.~2.12!,with the replacement ofmK

2 by mK*2 .

The terms of ordermK*22 and higher powers ofmK*

22 in Eq.~3.3! arise from the terms proportional tomK*

2 in the vectormeson propagator (dmn1kmkn /mK*

2 )/(mK*2

1k2) and thecoupling ~3.1!. These terms are small in comparison to tterms of Eqs.~3.3! at low values ofQ2. The explicit expres-sions for the contributions of ordermK*

22 to F1qs that are in-

dicated in Eqs.~3.4a! and ~3.4b! from the two loop diagramamplitudes illustrated in Figs. 1~a! and 1~b! are

4-4


F1qs ~Q2!$a,K* ,O~mK*

22!%5

gK* qs2

8p2

mq2

mK*2 E

0

1

dx~12x!E0

1

dyH Fms2~12x!22

ms2

mq2 Q2~12x!2y~12y!1msmq

3S 2x~12x!1Q2

mq2 ~12x!2@112xy~12y!# D 1mq

2x2~12x!21Q2~12x!2@2x2y~12y!

2y~12y2!1x~123y14y22y3!#1Q4

mq2 ~12x!2~12y!y@x1~12x!2~12y!y#G K1~Q2!

1Fms2

mq2 12

ms

mq~123x!1116x~12x!1

Q2

mq2 @223x26~12x!2y~12y!#G

3F lnH1~Lx

2!

H1~mK*2

!2x

Lx22mK*

2

H1~Lx2!

G16

mq2 FH1~mK*

2!ln

H1~mK*2

!

H1~Lx2!

2H1~mK*2

!1H1~Lx2!G J ,

~3.5a!

F1qs ~Q2!$b,K* ,O~mK*

22!%5

gK* qs2

8p2

mq2

mK*2 E dx xE dyH F2msmqS ~12x!31

Q2

mq2 ~12x! D

1Q2x2~12x!@122y~12y!~11x!#1Q4

mq2 x2y~12y!@12x12x2y~12y!#G K2~Q2!

1F6ms

mq~12x!16~12x!22~12x!~325x!1

Q2

mq2 @2213x211x2y~12y!#G

3F lnH2~Lx

2!

H2~mK*2

!2x

Lx22mK*

2

H2~Lx2!

G19

mq2 FH2~mK*

2!ln

H2~mK*2

!

H2~Lx2!

2H2~mK*2

!1H2~Lx2!G J .

~3.5b!

The contributions from theK* loop diagrams in Fig. 1 to the strangeness factorsF2s of the u andd quarks are obtained as

F2q2 ~Q2!$a,K* %5

gK* qs2

4p2 E0

1

dx~12x!E0

1

dy 2mqx@2~ms2mq!1mq~12x!#K1~Q2!1OS 1

mK*2 D , ~3.6a!

F2qs ~Q2!$b,K* %5

gK* qs2

4p2 E0

1

dx xE0

1

dy mq@ms~3x22!2mqx~2x21!#K2~Q2!1OS 1

mK*2 D . ~3.6b!

Finally the corresponding terms that are proportional to 1/mK*2 have the expressions

F2q2 ~Q2!$a,K* ,O~mK*

22!%52

gK* qs2

4p2

mq2

mK*2 E

0

1

dx~12x!E0

1

dyH F ~12x!2~ms2mq!~ms1mqx!1Q2

mq~ms2mq!~12x2!~12x!

3~12y!yG K1~Q2!22S 12ms

mqD S 12

3

2xD S ln

H1~Lx2!

H1~mK*2

!2x

Lx22mK*

2

H1~Lx2!

D J , ~3.7a!

F2qs ~Q2!$b,K* ,O~mK*

22!%5

gK* qs2

4p2

mq2

mK*2 E

0

1

dx xE0

1

dyH Fmq~12x!2~ms2mqx2!1Q2x2y~12y!S 222x1x22ms

mqD G K2~Q2!

22F ms

mq22~12x!22xGF ln

H2~Lx2!

H2~mK*2

!2x

Lx22mK*

2

H2~Lx2!

G J . ~3.7b!

045204-5

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These expressions reduce to those derived in Ref.@13# in thelimit Q2→0, once multiplied bymp /mq in order to obtainthe results in units of nuclear magnetons.

The strange vector meson loop contributions tostrangeness Dirac form factorF1

s(Q2) of the proton are ob-tained after subtraction of the irrelevant valuesF1q

s (0) fromthe expressions~3.4! and~3.5! and multiplication of the sumof the remainders by a factor of 3, Eq.~2.9!. This contribu-tion is shown in Fig. 2 along with the corresponding kaloop contribution. In this case the vector meson loop conbution is larger in magnitude than the kaon loop contribtion. Even so the sum of theK and K* loop contributionsremains very small in magnitude, reaching only the va20.086 aroundQ251 (GeV/c)2.

The contribution of the strange vector meson loops tostrangeness Pauli form factor is very small because of acancellation between the two involved loop diagrams@13#.This contribution is shown in Fig. 3, with the much largcontribution from the kaon loop diagrams.

IV. K* -K LOOP CONTRIBUTION

We finally consider the strangeness loop fluctuation,which the e.m. coupling is to theK* K transition vertex~Fig.4!. The amplitude of this loop fluctuation may be calculatfrom the empirically known radiative widths of theK* me-sons. TheK* K transition current vertex has the form

^Ka~k8!uJmuKs*b~k!&52 i

gK* Kg

mK*emlnsklkn8d

ab. ~4.1!

The coupling constantgK* Kg depends on the charge statethe strange mesons. It may be determined from the radiadecay widths using the expression@13#

FIG. 5. The Sachs strangeness form factorGEs of the proton as a


04520

e

i--

e

ear

r

ve

G~K* →Kg!5agK* Kg

2

24pmK* F12S mK

mK*D 2G 3

. ~4.2!

Herea is the fine structure constant.Given the empirical radiative widthsG(K* 1→K1g)

550 keV and G(K* 0→K0g)5116 keV @17#, the corre-sponding coupling constant values are obtained asgK* 6K6g50.75 andgK* 0K0g51.14~with the sign convention of@13#!.

The K* K loop diagrams~Fig. 4! only contribute to thestrange Pauli form factors of the quarks. With the conventof assigning theK and K* mesons a ‘‘strangeness chargeof 21, the contribution to the strangeness Pauli form facF2q

s (Q2) from these loop diagrams is found to be

F2qs ~Q2!52

gKqsgK* qsgK* Kg

2p2

mq

mK*E

0

1

dx xE0

1

dy

3H mq~12x!~ms2mqx!

3S 1

G12

1

G22

1

G31

1

G4D2 lnS G2G3

G1G4D J .

~4.3!

The quantitiesGi here have been defined as

G15G~mK ,mK* !, G25G~Lx ,mK* !,~4.4!

G35G~mK ,Lx!, G45G~Lx ,Lx!.

The auxiliary functionG(m,m8) is defined as

FIG. 6. The Sachs strangeness magnetic form factorGMs of the

proton as a function of momentum transfer. The SAMPLE expement gives the experimental value atQ250.1 (GeV/c)2.

4-6

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th

e

-

o

rityha

as

blesel.sug-

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G~m,m8!5ms2~12x!2mq

2x~12x!1m2x~12y!

1m82xy1Q2x2y~12y!. ~4.5!

These expressions represent direct generalizations ofcorresponding expressions defined in Ref.@13# for the caseQ250.

To account for the different coupling constantsgK* Kgin the case ofu andd quarks, Eq.~2.9! is generalized to

F2s~Q2!5

mp

msF4

3F2u

s ~Q2!21

3F2d

s ~Q2!G . ~4.6!

TheK* K loop contribution toF2s(Q2) is shown in Fig. 3.

This loop contribution has the opposite sign to that ofdiagonal strangeness loop fluctuations.

V. SAMPLE AND HAPPEX

The SAMPLE @2,3# experiment measures the strangness magnetic form factor of the protonGM

s at Q250.1(GeV/c)2. The HAPPEX@1# experiment measures the combinationGE

s 10.39GMs at Q250.48 (GeV/c)2. The contribu-

tions to these observables from the kaon andK* loops inFigs. 1 and 4 are shown in Figs. 5 and 6 as a functionmomentum transfer.

The cutoff dependence is small@13#, and for Lx

51.2 GeV we obtain GMs (0.1)520.06 and GE

s (0.48)10.39GM

s (0.48)520.08. The latter value is slightly belowthe uncertainty range of the result of the HAPPEX expement. Similarly the former value is within the uncertainrange of the result of the SAMPLE experiment, provided t

. B

04520

he

e

-

f

-

t

GAZ is small and positive rather than large and negative

originally suggested@4,18#.The smallness of the calculated strangeness observa

of the proton is an inherent feature of the chiral quark modThe smallness of the measured strangeness observablesgests that this model may provide a useful frameworkdescribing those observables. The calculated momentumpendence ofGE

s agrees fairly well with that obtained bheavy baryon chiral perturbation theory@19#. The strange-ness radius is;0.02 fm2, which value is also obtained in thbaryon loop calculations in Ref.@10# if the cutoff scale is setto the chiral symmetry restoration scale. Because of the lanegative vector meson contribution to the Dirac form facF1

s , the calculatedGMs form factor grows more negative with

increasingQ2, whereas the third order chiral perturbatiotheory result, which does not consider vector mesons,GM

s is that it increases slowly with momentum transfer. Hiterto QCD lattice calculations have been made only forstrangeness magnetic momentGM

s (0), but not for theformfactors. The calculated values are negative, with substauncertainty limits ~20.3660.20 @20#, 20.1660.18 @21#!.The chiral quark model value forGM

s (0);20.06 @13# fallswithin the uncertainty range of the latter value.

ACKNOWLEDGMENTS

We are grateful for the hospitality of Professor R.McKeown at the W. K. Kellogg Radiation Laboratory of thCalifornia Institute of Technology where this work was completed. L.H. thanks the Waldemar von Frenckell foundatfor a stipend. This work was supported in part by the Acaemy of Finland under Contract No. 43982.

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Strangeness form factors of the proton in the chiral quark model