Consistent determination of the strange and antistrange quark distributions of the nucleon and their asymmetry

The strange quark and antiquark distributions of the proton, $s(x)$ and $\bar{s}(x)$, remain the least well determined light parton distributions, and the determinations obtained from neutrino dimuon production, semi-inclusive kaon production, and LHC $W/Z$ and $W+c$ data remain in tension. The problem, whose concrete unresolved symptoms were raised by Cao, Chen and Signal (2010), is to determine $s(x)$ and $\bar{s}(x)$ — equivalently their sum $S^{+}(x)=s+\bar{s}$ and difference $\delta s(x)=s-\bar{s}$ — with controlled uncertainties in the face of mutually inconsistent direct extractions and global PDF fits, such that the result is physically admissible (non-negative $S^{+}$) and it is settled whether the strange–antistrange asymmetry $\delta s(x)$ is nonzero and what its shape is. A concrete unresolved symptom is the unphysical negative $xS^{+}(x)$ for $x>0.25$ produced when the meson-cloud-model $\Delta(x)$ is combined with the CTEQ6.6 $\bar{d}+\bar{u}$ distribution: whether the model overestimates $\Delta(x)$, the global fit underestimates $\bar{d}+\bar{u}$, or both has never been determined. Despite substantial progress — NNLO theory for dimuon production, ATLAS and CMS strangeness measurements at 7 and 13 TeV, simultaneous PDF–fragmentation analyses, and first lattice-QCD constraints on $s(x)-\bar{s}(x)$ — no consistent determination exists; the ATLAS-enhanced-strangeness preference conflicts with NuTeV/CMS/SIDIS-based determinations in current global fits.

Background

Parton distribution functions (PDFs) give the probability density for finding a quark, antiquark, or gluon carrying momentum fraction $x$ (Bjorken $x$) of a fast-moving proton, and are the essential input for predicting any hard scattering process at hadron colliders. While the light ($u$, $d$) and gluon distributions are tightly constrained by deep-inelastic scattering (DIS) and Drell–Yan data, the strange sector is constrained only indirectly, because no fully inclusive process isolates it: strange quarks appear in the sea and strange antiquarks must be inferred from processes in which an $s$ or $\bar{s}$ fragments into strange hadrons or produces a charm quark.

Three classes of observables carry direct sensitivity. First, charged-current neutrino DIS with opposite-sign dimuon final states, $\nu(\bar{\nu})N\to\mu^{\mp}\mu^{\pm}X$, tags the subprocess $\nu s\to\mu c$ (or $\bar{\nu}\bar{s}\to\mu\bar{c}$) and so separately constrains $s$ and $\bar{s}$; the CCFR and NuTeV dimuon data were long the dominant constraint, and the strange–antistrange difference $\delta s(x)=s(x)-\bar{s}(x)$ matters because it shifts the neutrino and antineutrino cross sections differently and thereby affects the extraction of the weak mixing parameter $\sin^{2}\theta_{W}$ — a contribution to the long-standing NuTeV anomaly (Zeller et al. 2002). Second, semi-inclusive DIS (SIDIS) with identified charged kaons measures $s$ and $\bar{s}$ through strange-to-kaon fragmentation functions, whose correlations with the strange PDF limit the separation (HERMES 2008, 2013). Third, at the LHC, $W$ and $Z$ boson production rates and especially associated $W+c$ production probe the strange sea at small and intermediate $x$ (ATLAS 2012, 2017; CMS 2014, 2024).

On the theory side, the sea can be generated perturbatively by gluon splitting $g\to q\bar{q}$ — flavour symmetric apart from masses — or non-perturbatively by the meson cloud: the physical nucleon fluctuates into baryon–meson Fock states (a Fock state is a component of the light-front wave function with a definite parton content, here e.g. $|\Lambda K\rangle$, $|\Sigma K\rangle$, $|N\pi\rangle$), which naturally break flavour symmetries, as observed in the $\bar{d}\neq\bar{u}$ asymmetry. Cao, Chen and Signal (Cao–Signal 2003; Chen–Cao–Signal 2010; Cao–Chen–Signal 2010) computed in the meson cloud model (MCM) both $\delta s(x)$ and the combination $\Delta(x)=\bar{d}(x)+\bar{u}(x)-s(x)-\bar{s}(x)$, which isolates the non-perturbative sea generated by these meson–baryon fluctuations and is independent of any global PDF set. Combining the calculated $\Delta(x)$ with the light antiquark distribution $\bar{d}+\bar{u}$ from a global fit then yields the total strange sea $S^{+}(x)=s(x)+\bar{s}(x)$ through $xS^{+}(x)=x[\bar{d}+\bar{u}]_{\mathrm{fit}}-x\Delta(x)$. With the CTEQ6.6 fit this construction gives an unphysical negative $xS^{+}(x)$ for $x>0.25$, and the 2010 papers could not decide whether the fault lies in the model (overestimated $\Delta$) or in the fit (underestimated $\bar{d}+\bar{u}$), or both. At the same time the LO HERMES kaon extraction and the NLO NuTeV dimuon extraction of $x\Delta(x)$ and $xS^{+}(x)$ disagreed with each other and with the spread of MSTW2008/CTEQ6.5/6.6/6L, and the uncertainty of the MCM calculation was only partially estimated (by comparison with a truncated Fock-state space omitting $K^{*}$ mesons), while the cutoff and parameter dependence and the time-ordered-perturbation-theory systematics were left unquantified. Whether $\delta s(x)$ is nonzero at all — with direct consequences for the NuTeV $\sin^{2}\theta_{W}$ anomaly and for determinations of the CKM (Cabibbo–Kobayashi–Maskawa) quark-mixing matrix element $|V_{cs}|$, which governs the strange-to-charm charged-current transition tagged in the dimuon process — remains unsettled.

Problem Statement

Deliver a consistent, physically admissible determination of the strange and antistrange quark distributions of the nucleon — the pair $s(x)$, $\bar{s}(x)$, equivalently the sum $S^{+}(x)=s(x)+\bar{s}(x)$ and the difference $\delta s(x)=s(x)-\bar{s}(x)$ — with quantified uncertainties, such that all of the following are simultaneously satisfied and mutually consistent within those uncertainties:

1. Cross-experiment consistency: the determination agrees within its quantified uncertainties with the direct experimental constraints — opposite-sign dimuon production in neutrino DIS (CCFR/NuTeV/NOMAD/CHORUS), semi-inclusive kaon production in charged-lepton DIS (HERMES/COMPASS, with a controlled treatment of strange-to-kaon fragmentation functions), and LHC $W/Z$ and $W+c$ production (ATLAS, CMS) — and its relation to the global QCD analyses (CT, MSHT, NNPDF, ATLAS-epWZ) is made explicit: either the long-standing disagreement between extractions that prefer a suppressed strange sea and those (notably ATLAS $W/Z$) that prefer $r_{s}\approx1$ is resolved, or, where direct datasets or fits cannot be made mutually consistent within the determined uncertainties, the origin of the incompatibility (correlated systematics, nuclear corrections, fragmentation-function uncertainty, or irreducible dataset tension) is identified. Here $r_{s}=(s+\bar{s})/(\bar{d}+\bar{u})$ is the ratio of the strange sea to the light-antiquark sea (ATLAS quotes $r_{s}=1.00^{+0.25}_{-0.28}$ at $Q^{2}=1.9\,\mathrm{GeV}^{2}$ and $x=0.023$).

2. Physical admissibility and resolved origin of the MCM pathology: the obtained $S^{+}(x)$ is non-negative over the full $x$ range, and in particular the origin of the unphysical negative $xS^{+}(x)$ for $x>0.25$ that arises when the meson-cloud-model $\Delta(x)=\bar{d}+\bar{u}-s-\bar{s}$ is combined with the CTEQ6.6 (or a modern) $\bar{d}+\bar{u}$ distribution is identified — i.e. it is determined whether the non-perturbative model overestimates $\Delta(x)$, the global fit underestimates $\bar{d}+\bar{u}$ at moderate $x$, or both — with a quantified uncertainty budget for the model calculation (Fock-state truncation, cutoff and parameter dependence, perturbative-theory systematics).

3. The asymmetry: it is settled whether $\delta s(x)=s(x)-\bar{s}(x)$ is nonzero, its sign and shape in $x$ (including whether it changes sign), and its moment $\int x\,\delta s(x)\,dx$, whose magnitude bears on the analysis of the NuTeV $\sin^{2}\theta_{W}$ anomaly.

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