The strangeness content of the nucleon: determining the asymmetry $s_-(x)$ and the normalization of the strange sea

The CTEQ6.5S global QCD analysis (Lai et al., 2007) gave the first dedicated determination of the strange parton distributions of the proton, finding that the symmetric combination $s_+(x)=s(x)+\bar{s}(x)$ carries a proton momentum fraction of 0.018–0.040 at 90% C.L. while the strangeness asymmetry $s_-(x)=s(x)-\bar{s}(x)$ is only loosely constrained ($\langle x\rangle_{s_-}\in(-0.001,0.005)$), with its magnitude and shape hostage to the relative normalization of the neutrino and antineutrino dimuon data and to parametrization restrictions (a single zero-crossing). In the subsequent literature, full NNLO predictions for $W+c$ production now exist, LHC $W+c$ and $W/Z$ data have sharpened — but also exposed persistent tensions in — the determination of $s_+(x)$, and first lattice-QCD strangeness distributions have appeared. Nevertheless, no analysis to date establishes a nonzero $s_-(x)$ or tightly bounds its shape, multi-crossing structure, or small-$x$ behavior, and intrinsic charm has still not been constrained jointly with strangeness in a single global fit. The problem — obtaining a complete and precise picture of the nucleon's strangeness content, centered on the strangeness asymmetry — remains open.

Background

Parton distribution functions (PDFs), $q(x,Q)$, describe the probability density for finding a quark of flavor $q$ carrying a fraction $x$ of the proton's longitudinal momentum when the proton is probed at resolution scale $Q$; they are nonperturbative inputs extracted from global fits to deep-inelastic scattering (DIS) and collider data, and they feed every prediction for processes at hadron colliders. While the non-strange light-quark sea is well measured, the strange and antistrange distributions, $s(x)$ and $\bar{s}(x)$, are poorly known: for decades global analyses imposed the ansatz $s(x)=\bar{s}(x)=r\,(\bar{u}(x)+\bar{d}(x))/2$, assuming the strange sea has the same shape as the isoscalar non-strange sea with only the ratio $r$ loosely constrained.

The strange sector is most directly probed by charged-current neutrino DIS: opposite-sign dimuon events, $\nu_\mu N\to\mu^-\mu^- c X$ and $\bar{\nu}_\mu N\to\mu^+\mu^+ \bar{c} X$ (observed by the CCFR and NuTeV fixed-target experiments at Fermilab), tag charm production through the weak $Wsc$ vertex, so the neutrino and antineutrino cross sections separately constrain $s(x)$ and $\bar{s}(x)$. It is conventional to fit the symmetric and antisymmetric combinations $s_+(x)=s(x)+\bar{s}(x)$ and $s_-(x)=s(x)-\bar{s}(x)$; the strangeness number sum rule $\int_0^1 s_-(x)\,dx=0$ holds exactly, since the proton carries zero net strangeness. At colliders, associated $W+c$ production, dominated by the gluon–strange subprocess $g\,s\to W\,c$ (computed in perturbative QCD at leading order, LO, next-to-leading order, NLO, and next-to-next-to-leading order, NNLO), constrains mainly $s_+$.

The CTEQ6.5S analysis (Lai, Nadolsky, Pumplin, Stump, Tung, Yuan, JHEP 04 (2007) 089) performed the first systematic global analysis of the strange sector without the shape assumption, using a general-mass perturbative QCD formalism, HERA (the electron–proton collider at DESY) and fixed-target DIS data, and the NuTeV and CCFR dimuon data. It found that $s_+(x)$ has a shape distinct from $\bar{u}+\bar{d}$, with the proton momentum fraction $\langle x\rangle_{s_+}$ in $0.018$–$0.040$ at 90% C.L., but that the strangeness asymmetry $s_-(x)$ is only loosely constrained: $\langle x\rangle_{s_-}\in(-0.001,+0.005)$, consistent with zero, and strongly dependent on the relative normalization of the $\nu$ and $\bar{\nu}$ dimuon data sets, which was held fixed in the fit. The data could not distinguish multiple oscillatory (multi-zero-crossing) shapes of $s_-(x)$, forcing a single-crossing parametrization. The analysis flagged further open issues: unresolved inconsistencies among fixed-target inclusive neutrino data sets (CDHSW, CCFR, NuTeV, CHORUS) at large $x$; normalization systematics of the dimuon data (experimental normalization of 2–5%, and charm fragmentation functions — phenomenological functions describing how a produced charm quark hadronizes into the charmed hadron that is actually observed — together with semileptonic branching ratios of about 10%) limiting $s_+$; the need for full NLO $W+c$ predictions for collider strangeness studies; and the degeneracy between strangeness and a possible nonperturbative intrinsic charm (IC) component of the proton — relevant, for example, to charged-Higgs production $c\,\bar{s}\to H^+$ — calling for a combined strange–charm analysis using processes such as associated $Zc$ production.

Theoretically, a nonzero $s_-(x)$ is expected: nonperturbative chiral and recombination dynamics that generate the observed $\bar{d}-\bar{u}$ light-sea asymmetry also differentiate $s$ from $\bar{s}$ (the statistical parton model — a phenomenological framework that derives all parton distributions from statistical-mechanics distributions in quark momentum and an energy potential — for instance, predicts a definite nonzero strangeness asymmetry), and $s_-$ was discussed as a possible contributor to the NuTeV anomalous measurement of the weak mixing angle. Determining $s_-(x)$ is thus a benchmark problem for the nonperturbative structure of the QCD sea.

Problem Statement

Obtain a complete and precise determination of the nucleon's strangeness content — the strange and antistrange parton distributions $s(x)$ and $\bar{s}(x)$ — with quantified uncertainties, in a way that resolves the degeneracies the CTEQ6.5S analysis identified. Concretely, with a common scientific objective of pinning down the strange sector to the accuracy already achieved for the non-strange sea, a solution must address the following linked components:

1. The strangeness asymmetry $s_-(x)=s(x)-\bar{s}(x)$: establish whether it is nonzero, and determine or tightly bound its magnitude and shape in $x$ — including how many zero crossings it has (single- vs multi-crossing/oscillatory behavior) — in a way that ensures the answer is not an artifact of the fixed relative normalization of the $\nu$ and $\bar{\nu}$ dimuon data sets or of the parametrization family. The strangeness number sum rule $\int_0^1 s_-(x)\,dx=0$ must be respected.

2. The symmetric strange distribution $s_+(x)$: determine its magnitude and shape at a precision no longer dominated by the dimuon normalization systematics (experimental normalization, charm fragmentation functions, semileptonic branching ratios), consistently with both neutrino dimuon data and collider $W/Z$ and $W+c$ data — that is, either resolving or quantifying the documented tension between them — while being aware of the residual inconsistencies among fixed-target inclusive neutrino data sets (CDHSW, CCFR, NuTeV, CHORUS) at large $x$, which the source considers not to directly affect the strange-sector determination.

A recommended follow-up direction, not a mandatory component of a solution, concerns the strange–charm degeneracy: the source paper noted that a possible nonperturbative intrinsic charm (IC) component is degenerate with strangeness — relevant, for example, to charged-Higgs production $c\,\bar{s}\to H^+$ — and recommended constraining intrinsic charm simultaneously with the strange PDFs in a single global analysis using processes such as associated $W+c$ and $Zc$ production, work it explicitly deferred to a separate paper. Reducing this degeneracy would remove the combined, uncorrelated uncertainty of the two sectors from strangeness-dependent predictions, but a solution to this problem is not required to include a joint strange–intrinsic-charm global fit.

The problem is complete when the strange sector, and in particular the strangeness asymmetry, is determined with quantified uncertainty from consistent experimental and theoretical inputs, such that whether $s_-(x)$ is nonzero, together with its magnitude and shape in $x$ and the number of its zero crossings, is established or tightly bounded at each $x$, robustly with respect to the relative normalization of the $\nu$ and $\bar{\nu}$ dimuon data sets and to the choice of parametrization family.

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