Direct determination of the strange-antistrange asymmetry $s(x)-\bar{s}(x)$ and of the separate strange helicity distributions $\Delta s(x)$, $\Delta\bar{s}(x)$ in the nucleon

The statistical parton model of Bourrely, Soffer and Buccella predicts that the strange-antistrange difference $s(x)-\bar{s}(x)$ in the nucleon is positive for $0.05<x<0.25$, changes sign near $x\approx0.25$, and yields a small negative momentum moment $[S^-]=-0.00194$, while both strange helicity distributions $\Delta s(x)$ and $\Delta\bar{s}(x)$ are negative for all $x$. Almost two decades later these predictions remain undecided: the only direct experimental separation of $s$ and $\bar{s}$, the NuTeV dimuon analysis, found a positive $[S^-]$ at $1.6\sigma$ on an iron target — opposite in sign to the prediction, but not conclusive — and the only existing separation of $\Delta s$ from $\Delta\bar{s}$, a leading-order COMPASS analysis of identified kaons, found both compatible with zero. The problem is to determine $s(x)$ and $\bar{s}(x)$, and $\Delta s(x)$ and $\Delta\bar{s}(x)$, separately over a broad $x$ range with precision sufficient to fix the sign structure of $s-\bar{s}$ and the signs of $\Delta s$, $\Delta\bar{s}$, and thereby to discriminate the competing mechanisms proposed for the strangeness asymmetry.

Background

Deep-inelastic scattering (DIS) probes parton distribution functions (PDFs) $q(x)$, the probability densities for quarks of flavor $q$ to carry a fraction $x$ (the Bjorken variable) of the nucleon's light-cone momentum. Most global QCD fits treat the strange sea as symmetric, $s(x)=\bar{s}(x)$, not because symmetry is required but because almost no data distinguish strange quarks from strange antiquarks. The exception is neutrino and antineutrino charged-current DIS producing opposite-sign dimuon events (a neutrino interacts, charm is produced from $s$ or $\bar{s}$, and its semileptonic decay yields a second muon of opposite sign): the $\nu$ and $\bar{\nu}$ cross sections weigh $s$ and $\bar{s}$ differently. Interest in a possible strangeness asymmetry sharpened with the NuTeV anomaly, a roughly $3\sigma$ deviation in the weak mixing angle measured in neutrino scattering, part of which could be absorbed by a nonzero momentum moment $[S^-]=\int_0^1 x[s(x)-\bar{s}(x)]dx$.

The statistical parton model of Bourrely, Soffer and Buccella builds unpolarized and polarized PDFs from Fermi-Dirac and Bose-Einstein distributions in $x$ and transverse momentum plus a small diffractive component, fitted to DIS and dimuon data. In Phys. Lett. B 648 (2007) it predicted: $s(x)-\bar{s}(x)$ positive for $0.05<x<0.25$ and negative for $x>0.25$ (a sign change near $x\approx0.25$, stable under QCD evolution up to $Q^2=100$ GeV$^2$), with $[S^-]=-0.00194$; and strange helicity distributions $\Delta s(x)$, $\Delta\bar{s}(x)$ (the differences of PDFs for quark spins aligned and antialigned with the nucleon spin) both negative for all $x$. These predictions sit among conflicting mechanisms. In the light-cone meson-baryon fluctuation model of Brodsky and Ma, the proton fluctuates into a $K^+\Lambda$ pair, the antistrange quark residing in the light kaon and the strange quark in the heavy $\Lambda$, giving $s(x)<\bar{s}(x)$ at small $x$ and $s(x)>\bar{s}(x)$ at large $x$, hence a positive $[S^-]$; the effective chiral quark model of Ding, Xu and Ma gives a positive $[S^-]$ large enough to absorb 60-100% of the NuTeV anomaly, and light-front holographic (AdS/QCD) models, which map the nucleon's light-front wave functions onto a higher-dimensional gravity dual, give $\langle x(s-\bar{s})\rangle\simeq+0.001$. Meson-cloud calculations instead give the opposite pattern, with $s(x)>\bar{s}(x)$ at small $x$ and a small negative $[S^-]$. Catani, de Florian, Rodrigo and Vogelsang showed that three-loop perturbative QCD evolution generates a sizable $s-\bar{s}$ from the valence $u,d$ densities alone, positive at small $x$ and negative at large $x$ with $\langle x(s-\bar{s})\rangle\approx-5\times10^{-4}$ — qualitatively the statistical model's pattern, but with the sign change at much smaller $x$.

On the polarized side, $\Delta s$ and $\Delta\bar{s}$ are accessed through double-spin asymmetries in semi-inclusive DIS (SIDIS), where identified final-state hadrons — kaons for strangeness — tag the struck quark; this requires fragmentation functions, the probabilities for a quark to produce the observed hadron, which are themselves uncertain. The only existing separation, a leading-order COMPASS analysis of identified kaons, found both $\Delta s$ and $\Delta\bar{s}$ compatible with zero, with no significant difference between them; global helicity fits either assume $\Delta s=\Delta\bar{s}$ (the inclusive DIS and $W$-production data they use do not separate them) or find the strange polarization consistent with zero, while fits imposing SU(3) flavor symmetry (which fixes the strange axial charge from baryon decay constants) prefer negative values. Whether $\Delta s$ and $\Delta\bar{s}$ are negative for all $x$, and whether $s-\bar{s}$ really changes sign near $x\approx0.25$, remain experimentally undecided.

Problem Statement

Determine, directly and with quantified uncertainty, the separate strange-quark and strange-antiquark distributions of the nucleon — unpolarized $s(x)$, $\bar{s}(x)$ and polarized $\Delta s(x)$, $\Delta\bar{s}(x)$ — over a broad range of Bjorken $x$, at a precision sufficient to: (i) establish or exclude a sign change of $s(x)-\bar{s}(x)$ near $x\approx0.25$ and fix the sign and magnitude of the momentum moment $[S^-]=\int_0^1 x[s(x)-\bar{s}(x)]dx$; (ii) establish the signs of $\Delta s(x)$ and $\Delta\bar{s}(x)$ and whether they differ from each other; and (iii) discriminate among the proposed generating mechanisms (chiral/meson-cloud, perturbative three-loop evolution, statistical, light-front holographic). These three results jointly test the strangeness sector of the statistical parton model and supply the missing flavor-separated input to global PDF and helicity fits; the problem is complete when the sign structure of $s-\bar{s}$ and the sign and magnitude of the momentum moment $[S^-]$, together with the signs of $\Delta s$ and $\Delta\bar{s}$, are settled with quantified uncertainties at a stated confidence level over the range $0.01<x<0.7$ of Bjorken $x$, so that the competing mechanisms can be accepted or excluded.

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