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Physics Letters B 559 (2003) 229–234

www.elsevier.com/locate/np

The quark–antiquark asymmetry of the strange sea of the nuc

Fu-Guang Cao, A.I. Signal

Institute of Fundamental Sciences, PN461, Massey University, Private Bag 11 222, Palmerston North, New Zealand

Received 19 February 2003; accepted 3 March 2003

Editor: T. Yanagida

Abstract

The strange sea of the proton is generally assumed to have quark–antiquark symmetry. However, it has been knowtime that non-perturbative processes involving the meson cloud of the proton may break this symmetry. Recently thisof interest as it affects the analysis of the so-called ‘NuTeV anomaly’, and could explain the large discrepancy betwNuTeV measurement of sin2 θW and the currently accepted value. In this Letter we re-examine strange–anti-strange asyusing the meson cloud model. We calculate contributions to the strange sea arising from fluctuations in the proton wavto states containing either Lambda or Sigma hyperons together with either Kaons or pseudovector K∗ mesons. We find thawe should not ignore fluctuations involving K∗ mesons in this picture. The strange sea asymmetry is found to be small,unlikely to affect the analysis of the Llewellyn–Smith cross section ratios or the Paschos–Wolfenstein relationship. 2003 Published by Elsevier Science B.V.

PACS: 14.20.Dh; 12.39.Ba; 12.15.Ff

Keywords: Meson cloud; Parton distributions; Strangeness

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1. Introduction

There has been interest for some time inquestion of whether non-perturbative processeslead to a difference between the strange and thestrange quark distribution functions of the protoThis possibility was first pointed out by Signal aThomas [1], and has been subsequently investigby other authors [2–4]. Recently there has been frinterest in this topic prompted by the measuremof sin2 θW by the NuTeV Collaboration [5]. The largdifference between the NuTeV result sin2 θW|NuTeV =

E-mail addresses: f.g.cao@massey.ac.nz (F.-G. Cao),a.i.signal@massey.ac.nz (A.I. Signal).

0370-2693/03/$ – see front matter 2003 Published by Elsevier Sciendoi:10.1016/S0370-2693(03)00342-3

0.2277±0.0013(stat)±0.0009(syst) and the acceptevalue sin2 θW = 0.2228± 0.0004 [6] of around threestandard deviations could arise, or be partly explaiby, a positive value of the second moment ofstrange–anti-strange distributions

⟨x(s − s)

⟩ =1∫

0

dx x[s(x)− s(x)

],

as has been pointed out by Davidson and co-wor[7].

As yet there is no direct experimental evidenceany asymmetry in the strange sea, however, BarPascaud and Zomer [8] found that, when performinglobal fit of unpolarized parton distributions, allowin

ce B.V.

230 F.-G. Cao, A.I. Signal / Physics Letters B 559 (2003) 229–234

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ry,nd

heasi.e.,

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onte

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ne

ron

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as aceFor

s(x) �= s(x) gave a small improvement to the fiTheir best fit result gave the second moment ofasymmetry as〈x(s − s)〉 = 0.002± 0.0028 atQ2 =20 GeV2. NuTeV have also looked for an asymmetand found a small negative value for the secomoment with a large uncertainty [9]. However, tfunctional form of the distributions used for this fit wnot constrained to give the first moment to be zero (zero net strangeness).

The mechanism for breaking the quark–anti-qusymmetry of the strange sea comes from the kcloud that accompanies the proton. As shownSullivan [10] in the case of the pion cloud, therea contribution to the parton distributions of the protfrom amplitudes where the virtual photon is scattefrom the meson. In this case the scaling contributto the parton distribution of the proton can be writtas a convolution of the parton distribution of tmeson with a fluctuation function that describesmomentum probability distribution of the meson.a similar vein there is a contribution to the partdistribution from amplitudes where the virtual photscatters from the recoil baryon, and the mesona spectator. Contributions to the strange seacome from fluctuations such as p(uud) → �(uds) +K+(us). In this case we see that the contributito the anti-strange distribution, which we denoδs, comes from the anti-strange quark in the kawhereas the contribution to the strange distributδs comes from the strange quark in the Lambbaryon. While the valence parton distributionsthe kaon and Lambda have not been determinedexperiment, they can be expected to differ from oanother considerably, as thes in the kaon carries alarger fraction of the 4-momentum of its parent hadthan that carried by thes quark in the�. This iscertainly the case in comparing the parton distributiof the pion with those of the proton, where tpion valence distributions are harder than the provalence distributions [11]. While the convolutionthe meson or baryon valence distribution with tappropriate fluctuation function can be expecteddecrease the difference between them, this differewill lead to a difference between the quark and aquark distributions in the strange sea of the proton

In this Letter we re-examine the asymmetry btween the strange and anti-strange distributions.work within the context of the meson cloud mod

(MCM) [12], which describes proton→ meson+baryon fluctuations using an effective Lagrangiwith amplitudes calculated using time-ordered perbation theory. We consider fluctuations to�K∗ and K∗ in addition to the�K and K fluctuations of theproton. In the non-strange sector, fluctuations invoing pseudovector mesons have been seen to havenificant effects on sea distributions [13,14], so weclude these fluctuations here to observe whethermake any contribution to the asymmetry of the strasea.

2. Strangeness in the meson cloud model

In the meson cloud model (MCM) the nucleocan be viewed as a bare nucleon plus some mebaryon Fock states which result from the fluctuatN → BM. The wavefunction of the nucleon canwritten as [14],

|N〉physical= Z|N〉bare

+∑BM

∑λλ′

∫dy d2k⊥ φλλ′

BM

(y, k2⊥

)

(1)× ∣∣Bλ(y,k⊥);Mλ′(1− y,−k⊥)

⟩,

whereZ is the wave function renormalization costant,φλλ′

BM(y, k2⊥) is the wave function of the Focstate containing a baryon (B) with longitudinal mmentum fractiony, transverse momentumk⊥, and he-licity λ, and a meson (M) with momentum fractio1−y, transverse momentum−k⊥, and helicityλ′. Themodel assumes that the lifetime of a virtual baryomeson Fock state is much longer than the interactime in the deep inelastic process, thus the quarkanti-quark in the virtual meson–baryon Fock statescontribute to the parton distributions of the nucleon

For spin independent parton distributions thenon-perturbative contributions can be expressedconvolution of fluctuation functions with the valenparton distributions in the baryon and/or meson.the strange and anti-strange distributions we have

x δs(x)=1∫

x

dy fBM/N(y)x

ysB

(x

y

),

F.-G. Cao, A.I. Signal / Physics Letters B 559 (2003) 229–234 231

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thefrom

ve

rt

rs,ize,nen

lsolts.bend

sock

ea.

-

c-

hattum

c-

ex-agetu-tothatme-in-

essrmceofize.

e

(2)x δs(x)=1∫

x

dy fMB/N(y)x

ysM

(x

y

),

where

(3)fBM/N(y)=∑λλ′

∞∫0

dk2⊥ φλλ′BM

(y, k2⊥

)φ∗λλ′

BM

(y, k2⊥

),

(4)fMB/N(y)= fBM/N(1− y)

are the fluctuation functions. These are the probaties to find the baryon or meson respectively with frtion y of the longitudinal momentum. We note that trelation (4) ensures that while the shapes ofδs(x) andδs(x) are different, their integrals are equal, so thatstrangeness of the dressed nucleon is not changedthat of the bare nucleon.

The fluctuation functions are derived from effectimeson–nucleon Lagrangians [14]

LNHK = igNHKNγ5πH,

LNHV = gNHVNγµθµH

(5)+ fNHVNσµνH(∂µθν − ∂νθµ

)whereN andH are spin-1/2 fields,π a pseudoscalafield, and θ a vector field. The fluctuations thawe consider are p→ �K, K,�K∗ and K∗. Thecoupling constants that we use are [15],

gN�K = −13.98,

gN K = 2.69,

gN�K∗ = −5.63, fN�K∗ = −4.89 GeV−1,

(6)gN K∗ = −3.25, fN K∗ = 2.09 GeV−1.

The meson–baryon vertices require form factowhich reflect the fact that the hadrons have finite sand act to suppress large momenta. We use expotial form factors

(7)GBM(y, k2⊥

) = exp

[m2

N −m2BM(y, k2⊥)

2Λ2c

],

though monopole or dipole form factors could abe used with no significant difference to our resuHereΛc is a cut-off parameter, which appears tothe same for fluctuations involving octet baryons ahas a value of 1.08 GeV, consistent with data on�

production in semi-inclusive p–p scattering [14]. Alm2

BM is the invariant mass squared of the BM Fo

-

Fig. 1. The fluctuation functions contributing to the strange sThe thick and thin solid curves aref�K/N(y) and f K/N(y),respectively. The thick and thin dashed curves aref�K∗/N(y) andf K∗/N(y).

state,

(8)m2BM

(y, k2⊥

) = m2B + k2⊥y

+ m2M + k2⊥1− y

.

In Fig. 1 we show the four fluctuation functions of interestf�K/N(y), f K/N(y), f�K∗/N(y) andf K∗/N(y), wherey is the longitudinal momentumfraction of the baryon. We note that the kaon flutuation functions peak aroundy = 0.6, whereas theK∗ fluctuation functions peak aroundy = 0.5 andare fairly symmetrical about the peak, indicating tthe meson and baryon share the proton momenequally. We also note that the K∗ fluctuation func-tions are of similar size to the kaon fluctuation funtions. Thus the higher mass of the K∗ does not leadto a suppression of these fluctuations, as might bepected on kinematic grounds, but to a smaller averbaryon momentum. In fact, the least probable flucation is p→ K, which is suppressed mainly duethe smaller coupling constant. We can concludeany analysis of strange quark or anti-quarks in theson cloud model needs to include the fluctuationsvolving the K∗ meson.

The fluctuation functions depend upon the hardnof the form factor that is used. Using a softer fofactor (e.g.,Λc = 0.8 GeV as suggested by referen[16]) does not change the position of the peaksthe fluctuation functions, but does decreases their sThe higher mass fluctuations involving K∗ mesonsare more sensitive to the value ofΛc, and decreas

232 F.-G. Cao, A.I. Signal / Physics Letters B 559 (2003) 229–234

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2].s ofheisby

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in size faster asΛc decreases. The first momenta fluctuation function gives the probability of thparticular fluctuation occurring. ForΛc = 1.08 GeVwe find the total probabilities of finding a kaon or K∗to beP(K) = 1.6%,P(K∗) = 2.7%, whereas forΛc =0.80 GeV these probabilities reduce toP(K) = 0.2%,P(K∗) = 0.09%.

In order to calculate the MCM contributionsδs(x)and δs(x) in Eq. (2) we need to know the partodistributions of the� and baryons and the K anK∗ mesons. Previous studies [1,3,4] in the MCM hausedSU(3) flavour symmetry to relate these partdistributions to those of the proton and pion. HowevSU(3) flavour symmetry is known to be broken in thsea distributions of the proton, so it may not be a goapproximation to assume that it holds for the valedistributions of octet baryons or the pseudoscaor pseudovector nonet mesons. A phenomenologparameterization of the kaon distributions, basedthose of the pion, is available [17], but therenothing similar for the baryons. An alternative isuse some model of the required distributions. Opossibility would be to use a Gaussian light-cowavefunction to calculate the parton distributionwhich is an approach used by Brodsky and Ma [Another approach is to generalise the calculationthe parton distribution functions of the nucleon in tMIT bag model by the Adelaide group [18,19]. Thhas been done in the case of baryon distributionsBoros and Thomas [20], and also in the case of thρmeson by ourselves [21].

Adapting the argument of the Adelaide group,have the expressions for the strange quark distribuof a baryon and the anti-strange quark distributiona meson:

sB(x) = mB

(2π)3

∫dpn

|φ2(pn)|2|φ3(0)|2 δ

(mB(1− x)− p+

n

)

(9)× ∣∣Ψ+,s(pn)∣∣2,

sM(x) = mM

(2π)3

∫dpn

|φ1(pn)|2|φ2(0)|2 δ

(mM(1− x)− p+

n

)

(10)× ∣∣ ˜Ψ+,s(pn)∣∣2.

Here we have defined+ components of momentby p+ = p0 + p3, pn is the 3-momentum of th

2(1)-quark intermediate state,Ψ ( ˜Ψ ) is the Fouriertransform of the MIT bag wavefunction for thes

quark (anti-quark) in the ground state, andφm(p) isthe Fourier transform of the Hill–Wheeler overlafunction betweenm-quark bag states

(11)

∣∣φm(p)∣∣2 =

∫dR e−ip·R

[∫drΨ †(r − R)Ψ (r)

]m

.

The input parameters for the bag model calcutions of the parton distributions are the bag radiusR,the mass of the intermediate statemn, the mass ofthe strange quark (anti-quark)ms and the bag scalµ2—at this scale the model is taken as a good appimation to their valence structure of the hadron. Fthe baryons (�, ) we use the same parameter setBoros and Thomas [20] (R = 0.8 fm, mn = 800 MeVbefore hyperfine splitting of scalar (for�) and vec-tor (for ) states,ms = 150 MeV,µ2 = 0.23 GeV2).For the mesons (K, K∗) we use a parameter set bason the baryon set and our earlierρ meson calculation[21] (R = 0.7 fm, mn = 425 MeV, ms = 150 MeV,µ2 = 0.23 GeV2).

The valence distributions calculated using Eq. (do not satisfy the straightforward normalisation codition as they ignore intermediate states with mquarks and anti-quarks than the initial state, i.equarks plus one anti-quark for the baryon distributior 2 quarks plus one anti-quark for the meson disbution. We can parameterise the effects of such stby adding to the distributions a piece proportiona(1−x)7 for the baryon distributions or(1−x)5 for themeson distributions, consistent with the Drell–YaWest relation, such that the normalisation conditis satisfied. While this ansatz for the shape is sowhat arbitrary, it has little effect at medium and larx, especially after the distributions are evolved upexperimental scales. From the calculated fluctuafunctions, we can see that the convolution (Eq. (is most sensitive to parton distributions in the mediand largex regions, which are little affected by ouansatz for the contributions from intermediate stawith larger mass than the parent hadron. In Figwe show the calculated parton distributions after Nevolution toQ2 = 16 GeV2, which is the region of theNuTeV data. We can see that the valance distributof the K and K∗ mesons are harder than that of the�

and baryons, as expected.Having calculated both the MCM fluctuation fun

tions and the strange valence parton distributions

F.-G. Cao, A.I. Signal / Physics Letters B 559 (2003) 229–234 233

to

sults

ate

eand

utlv-ommfore

ibu--n

--

s to-the

nti-

f thensthe

ononin

onter

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etrythe

seahat

Fig. 2. The valences(x) distributions of the K and K∗ mesons (solidcurve), and the valences(x) distributions of the� (dashed curve)and the (dotted curve) hyperons. All distributions are evolvedQ2 = 16 GeV2.

Fig. 3. The strange sea asymmetryx[s(x) − s(x)] calculated in themeson cloud model. The solid and dashed curves are the rewithout and with K∗ contributions, respectively.

the constituents of the cloud we can now calculthe MCM contribution to the protons and s distrib-utions using the convolution in Eq. (2). In Fig. 3 wshow our calculated difference between strangeanti-strange parton distributions usingΛc = 1.08 GeV.We show the difference calculated with and withoincluding the contributions from Fock states invoing K∗ mesons. We can see that the contributions fr�K∗ and K∗ are of similar magnitude to those frothe lower mass Fock states. As has been noted be[22], considering only K states gives a small (s − s)difference because the harder of the parton distrtions (sK(x)) is convoluted with the softer of the fluctuation functions (fKB/N), whereas the softer parto

distribution (sB(x)) is convoluted with the harder fluctuation function (fBK/N). When K∗ states are considered, we see from Fig. 1 thatfK∗B/N(y)≈ fBK∗/N(y),so each fluctuation function gives equal emphasithe medium—largex region of the hadron parton distributions when they are convoluted together. Thusrelative hardness ofsK∗

(x) to sB(x) is manifested inthe largex region ass(x) > s(x) in the calculation.We calculate the second moment of the strange–astrange asymmetry both with and without K∗ contri-butions. We find that⟨x(s − s)

⟩(12)=

{1.43× 10−4 without K∗ states,−1.35× 10−4 including K∗ states.

We observe that omitting fluctuations involving K∗mesons changes the sign of the second moment oasymmetry, so it is important that these fluctuatioare included in any discussion of the asymmetry ofstrange sea.

Because the calculated asymmetry dependsthe difference between MCM contributions to partdistributions, it is relatively insensitive to changesthe form factor used to calculate the MCM fluctuatifunctions. If we had used the form factor parameΛc = 0.80 GeV instead ofΛc = 1.08 GeV, we wouldhave obtained values of 4.1 × 10−5 (3.4 × 10−5) forthe second moments of the asymmetry without (wK∗ fluctuations.

The non-zero value of the strange sea asymmaffects the experimentally determined value ofPaschos–Wolfenstein ratio

(13)RPW = σνNC − σ ν

NC

σνCC − σ ν

CC

= g2L − g2

R = 1

2− sin2 θW.

The effect of the strange sea asymmetry is to shiftRPWby an amount [7,22]

(14)-RPW = − 3b1 + b2

〈x(uV + dV )〉/2

⟨x(s − s)

⟩,

where

b1 = ∆2u = g2

Lu− g2

Ru,

(15)b2 = ∆2d = g2

Ld− g2

Rd.

At the NuTeV scale (Q2 = 16 GeV2) the coefficientin front of the second moment of the strangeasymmetry in Eq. (14) is about 1.3, which means t

234 F.-G. Cao, A.I. Signal / Physics Letters B 559 (2003) 229–234

on

nti-on-en-cru-re-heomton.in-oudhatpo-

thesions ofnd

irlytheoym-red.

-p-

oralnce

yal

.88

) 2.64

ys.

)

96)

00

59

99)

44

e,the2.

-RPW is of the order 1∼ 2×10−4. This is an order ofmagnitude too small to have any significant effectthe NuTeV result for the weak mixing angle.

3. Summary

Any asymmetry between strange quarks and aquarks in the nucleon sea must arise from nperturbative effects. This would make any experimtal observation of a strange sea asymmetry acial test for models of nucleon structure. We haveexamined this asymmetry within the context of tmeson cloud model, which gives an asymmetry frstrange hadrons in the meson cloud of the proA novel aspect of our calculation is that we havecluded the effects of components of the meson clinvolving the K∗ vector meson, and we have seen tthe contributions to the strange sea from these comnents are of similar magnitude to those involvingpseudoscalar kaon. Hence, any quantitative discusof the strange sea in the MCM requires that both setcontributions are considered. Overall, we have fouthat the strange sea asymmetry in the MCM is fasmall, and does not have any significant effect onNuTeV extraction of sin2 θW. However, we have alsseen that the sign of the second moment of the asmetry depends on which contributions are conside

Acknowledgements

We would like to thank Tony Thomas for a number of useful discussions. This work was partially suported by the Science and Technology PostdoctFellowship of the Foundation for Research Scie

and Technology, and the Marsden Fund of the RoSociety of New Zealand.

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