The quark - antiquark asymmetry of the strange sea of the nucleon

The strange sea of the proton is generally assumed to have quark - antiquark symmetry. However it has been known for some time that non-perturbative processes involving the meson cloud of the proton may break this symmetry. Recently this has been of interest as it affects the analysis of the so-called `NuTeV anomaly', and could explain the large discrepancy between the NuTeV measurement of $\sin^{2} \theta_{W}$ and the currently accepted value. In this paper we re-examine strange - anti-strange asymmetry using the meson cloud model. We calculate contributions to the strange sea arising from fluctuations in the proton wavefunction to states containing either Lambda or Sigma hyperons together with either Kaons or pseudovector $K^{*}$ mesons. We find that we should not ignore fluctuations involving $K^{*}$ mesons in this picture. The strange sea asymmetry is found to be small, and is unlikely to affect the analysis of the Llewellyn-Smith cross section ratios or the Paschos-Wolfenstein relationship.


I. INTRODUCTION
There has been interest for some time in the question of whether non-perturbative processes can lead to a difference between the strange and the anti-strange quark distribution functions of the proton. This possibility was first pointed out by Signal and Thomas [1], and has been subsequently investigated by other authors [2][3][4]. Recently there has been fresh interest in this topic prompted by the measurement of sin 2 θ W by the NuTeV collaboration [5]. The large difference between the NuTeV result sin 2 θ W | NuTeV = 0.2277 ± 0.0013(stat) ± 0.0009(syst) and the accepted value sin 2 θ W = 0.2228 ± 0.0004 [6] of around three standard deviations could arise, or be partly explained by, a positive value of the second moment of the strange -anti-strange distributions x(s −s) = 1 0 dxx[s(x) −s(x)], as has been pointed out by Davidson and co-workers [7].
As yet there is no direct experimental evidence for any asymmetry in the strange sea, however Barone, Pascaud and Zomer [8] found that in performing a global fit of unpolarized parton distributions allowings(x) = s(x) gave a small improvement to the fit. Their best fit result gave the second moment of the asymmetry as x(s −s) = 0.002 ± 0.0028 at Q 2 = 20 GeV 2 . NuTeV have also looked for an asymmetry, and found a small negative value for the second moment with a large uncertainty [9]. However the functional form of the distributions used for this fit was not constrained to give the first moment to be zero (i.e. zero net strangeness).
The mechanism for breaking the quark -anti-quark symmetry of the strange sea comes from the kaon cloud that accompanies the proton. As shown by Sullivan [10] in the case of the pion cloud, there is a contribution to the parton distributions of the proton from amplitudes where the virtual photon is scattered from the meson. In this case the scaling contribution to the parton distribution of the proton can be written as a convolution of the parton distribution of the meson with a fluctuation function that describes the momentum probability distribution of the meson. In a similar vein there is a contribution to the parton distribution from amplitudes where the virtual photon scatters from the recoil baryon, and the meson is a spectator. Contributions to the strange sea can come from fluctuations such as p(uud) → Λ(uds) + K + (us). In this case we see that the contribution to the antistrange distribution, which we denote δs, comes from the anti-strange quark in the kaon, whereas the contribution to the strange distribution δs comes from the strange quark in the Lambda baryon. While the valence parton distributions of the kaon and Lambda have not been determined by experiment, they can be expected to differ from one another considerably, as thes in the kaon carries a larger fraction of the 4-momentum of its parent hadron than that which is carried by the s quark in the Λ. This is certainly the case in comparing the parton distributions of the pion with those of the proton, where the pion valence distributions are harder than the proton valence distributions [11]. While the convolution of the meson or baryon valence distribution with the appropriate fluctuation function can be expected to decrease the difference between them, this difference will lead to a difference between the quark and anti-quark distributions in the strange sea of the proton.
In this letter we re-examine the asymmetry between the strange and anti-strange distributions. We work within the context of the meson cloud model (MCM) [12], which describes proton → meson + baryon fluctuations using an effective Lagrangian, with amplitudes calculated using time-ordered perturbation theory. We consider fluctuations to ΛK * and ΣK * in addition to the ΛK and ΣK fluctuations of the proton. In the non-strange sector, fluctuations involving pseudovector mesons have been seen to have significant effects on sea distributions [13,14], so we include these fluctuations here to observe whether they make any contribution to the asymmetry of the strange sea.

II. STRANGENESS IN THE MESON CLOUD MODEL
In the meson cloud model (MCM) the nucleon can be viewed as a bare nucleon plus some meson-baryon Fock states which result from the fluctuation N → BM. The wavefunction of the nucleon can be written as [14], are the fluctuation functions. These are the probabilities to find the baryon or meson respectively with fraction y of the longitudinal momentum. We note that the relation (4) ensures that while the shapes of δs(x) and δs(x) are different, their integrals are equal, ensuring that the strangeness of the dressed nucleon is not changed from that of the bare nucleon.
The fluctuation functions are derived from effective meson-nucleon Lagrangians [14] L N HK = ig N HKN γ 5 πH where N and H are spin-1/2 fields, π a pseudoscalar field, and θ a vector field. The fluctuations that we consider are p → ΛK, ΣK, ΛK * and ΣK * . The coupling constants that we use are [15], The meson-baryon vertices require form factors, which reflect the fact that the hadrons have finite size, and act to suppress large momenta. We use exponential form factors though monopole or dipole form factors could also be used with no significant difference to our results. Here Λ c is a cut-off parameter, which appears to be the same for fluctuations involving octet baryons and has a value of 1.08 GeV, consistent with data on Λ production in semi-inclusive p-p scattering [14]. Also m 2 BM is the invariant mass squared of the BM Fock state, In figure 1 we show the four fluctuation functions of interest f ΛK/N (y), f ΣK/N (y), where y is the longitudinal momentum fraction of the baryon.
We note that the kaon fluctuation functions peak around y = 0.6, whereas the K * fluctuation functions peak around y = 0.5 and are fairly symmetrical about the peak, indicating that the meson and baryon share the proton momentum equally. We also note that the One possibility would be to use a gaussian light-cone wavefunction to calculate the parton distributions, which is an approach used by Brodsky and Ma [2]. Another approach is to generalise the calculations of the parton distribution functions of the nucleon in the MIT bag model by the Adelaide group [18,19]. This has been done in the case of baryon distributions by Boros and Thomas [20], and also in the case of the ρ meson by ourselves [21].
Adapting the argument of the Adelaide group, we have the expressions for the strange quark distribution of a baryon and the anti-strange quark distribution of a meson: Here we have defined + components of momenta by p + = p 0 + p 3 , p n is the 3-momentum which is the region of the NuTeV data. We can see that the valance distributions of the K and K * mesons are harder than that of the Λ and Σ baryons, as expected.
Having calculated both the MCM fluctuation functions and the strange valence parton distributions of the constituents of the cloud we can now calculate the MCM contribution to the proton s ands distributions using the convolution in equation (2). In figure 3 we show our calculated difference between strange and anti-strange parton distributions using Λ c = 1.08 GeV. We show the difference calculated with and without including the contributions from Fock states involving K * mesons. We can see that the contributions from ΛK * and ΣK * are of similar magnitude to those from the lower mass Fock states.
As has been noted before [22], The non-zero value of the strange sea asymmetry affects the experimentally determined value of the Paschos -Wolfenstein ratio The effect of the strange sea asymmetry is to shift R P W by an amount [7,22] where At the NuTeV scale (Q 2 = 16 GeV 2 ) the coefficient in front of the second moment of the stange sea asymmetry in equation (14) is about 1.3, which means that ∆R P W is of the order 1 2 × 10 −4 . This is an order of magnitude too small to have any significant effect on the NuTeV result for the weak mixing angle.

III. SUMMARY
Any asymmetry between strange quarks and anti-quarks in the nucleon sea must arise from non-perturbative effects. This would make any experimental observation of a strange sea asymmetry a crucial test for models of nucleon structure. We have re-examined this asymmetry within the context of the meson cloud model, which gives an asymmetry from strange hadrons in the meson cloud of the proton. A novel aspect of our calculation is that we have included the effects of components of the meson cloud involving the K * vector meson, and we have seen that the contributions to the strange sea from these components are of similar magnitude to those involving the pseudoscalar kaon. Hence any qualitative discussion of the strange sea in the MCM requires that both sets of contributions are considered. Overall we have found that the strange sea asymmetry in the MCM is fairly small, and does not have significant effect on the NuTeV extraction of sin 2 θ W . However we have also seen that the sign of the second moment of the asymmetry depends on which contributions are considered.