Quantitative determination of the strange–antistrange asymmetry $s(x)-\bar{s}(x)$ of the proton

Chiral effective theory predicts that the strange and antistrange parton distribution functions (PDFs) of the proton need not be equal: kaon–hyperon fluctuations generate a nonperturbative asymmetry $s(x)-\bar{s}(x)$, including a $\delta(x)$-type contribution to $\bar{s}$ at $x=0$ that makes the number integral of $s-\bar{s}$ nonzero whenever $x=0$ is experimentally inaccessible. Wang et al. (Phys. Rev. D 94, 094035, 2016) computed this asymmetry at the lowest nontrivial chiral order with octet baryon intermediate states only, finding a momentum moment $\langle x(s-\bar{s})\rangle$ of order $10^{-3}$, and identified as open issues the regulator and SU(3)-flavor-breaking systematics, the missing decuplet, higher-order, and multi-meson contributions, the absence of the $x=0$ singular component in global PDF fits, and the lack of decisive experimental constraints. Later work has extended the theory (decuplet baryons, gauge-invariant nonlocal regulators, polarized strange PDFs) and lattice QCD now provides first $x$-dependent constraints on $s-\bar{s}$; global fits that allow a nonvanishing strangeness asymmetry exist but give conflicting indications, with current collider and semi-inclusive data consistent with zero. The problem — to establish $s(x)-\bar{s}(x)$ quantitatively, its sign, shape, moments, its $x=0$ component, and a defensible uncertainty budget combining theory and global analysis of data — remains open.

Background

Parton distribution functions (PDFs) $q(x)$ and $\bar{q}(x)$ describe the number density of quarks and antiquarks of flavor $q$ carrying a fraction $x$ of the proton's longitudinal momentum; they are the universal nonperturbative input to all hard-scattering cross sections. Although the proton has zero net strangeness, gluon splitting and nonperturbative fluctuations populate a strange sea, and there is no symmetry requiring $s(x)=\bar{s}(x)$. The precedent is the established $\bar{d}-\bar{u}$ asymmetry revealed by violation of the Gottfried sum rule (a sum rule for the antiquark sea): in the meson-cloud picture the proton fluctuates into a meson plus a baryon, and because the two hadrons carry different quark content, the resulting antiquark distributions are flavor-asymmetric. For strangeness the analogous fluctuation is $p \to K + \Lambda$ (or $\Sigma$), where the kaon carries the $\bar{s}$ and the hyperon the $s$; the different masses, spectra, and spins of the two hadrons generically make $s(x)$ and $\bar{s}(x)$ differ. Perturbative QCD by itself generates only a tiny negative asymmetry at three loops, with momentum moment $S^- = \int dx\, x[s(x)-\bar{s}(x)] \approx -0.5\times10^{-3}$. The natural nonperturbative framework is chiral effective theory: an effective Lagrangian of the octet baryons (the spin-1/2 multiplet of the approximate SU(3) flavor symmetry of the light quarks: $N$, $\Lambda$, $\Sigma$, $\Xi$) interacting with the pseudoscalar meson octet ($\pi$, $K$, $\eta$), with kaon loop diagrams regularized (for example by Pauli–Villars subtractions — a subtraction procedure that renders divergent loop integrals finite while preserving chiral and gauge symmetries) and cutoff parameters constrained phenomenologically. Beyond leading order, gauge invariance requires off-shell terms (loop contributions that depend on the virtuality of the intermediate baryon) and Kroll–Ruderman contact interactions (local meson–baryon coupling terms demanded by gauge invariance) in addition to the usual on-shell rainbow diagrams (in which the baryon emits and later reabsorbs a meson). Moments of PDFs expanded in the kaon mass obey a leading nonanalytic (LNA) behavior fixed by chiral symmetry, which serves as an exact benchmark any consistent calculation must reproduce. Wang et al. (2016) performed the complete lowest-order kaon-loop calculation of $s$ and $\bar{s}$, constraining their Pauli–Villars regulator masses by inclusive hyperon production data. They found that kaon loops supply roughly 30%–50% of the empirical strange sea at $Q^2=1$ GeV$^2$, with the asymmetry about ten times smaller than the sum: $S^- = (0.42\text{–}1.12)\times10^{-3}$. A structurally novel result is that kaon bubble (tadpole) diagrams produce $\delta$-function contributions to $\bar{s}(x)$ at $x=0$ — independent of the ultraviolet regulator — so that in any observable where $x=0$ is kinematically inaccessible, $\int dx\,[s(x)-\bar{s}(x)]$ need not vanish; the corresponding splitting function (the probability density for the proton to fluctuate into a kaon plus a hyperon carrying momentum fraction $y$ of the proton momentum) is a $\delta$-function at hyperon momentum fraction $\bar{y}=1$, generating a valencelike strange component at larger $x$ that perturbative gluon radiation alone cannot produce. The phenomenological stakes include the long-standing NuTeV anomaly, a roughly $3\sigma$ discrepancy between the weak mixing angle extracted from neutrino–antineutrino deep-inelastic scattering (DIS) and the world average, to which a nonzero $\langle x(s-\bar{s})\rangle$ contributes; the 2016 calculation reduces it by at most $0.5\sigma$. Empirically the asymmetry is barely constrained: the NuTeV dimuon data (neutrino-induced events containing two oppositely charged muons, produced by charm creation off the strange sea) give $S^-=(1.96\pm1.43)\times10^{-3}$ at $Q^2=16$ GeV$^2$ in NLO analysis (Mason et al.), while a phenomenological reanalysis including nuclear-correction and parametrization uncertainties yields $(0\pm2)\times10^{-3}$ (Bentz et al.), and global PDF fits have mostly assumed $s=\bar{s}$. The source work closed by identifying what is needed: better theory (regulator and SU(3)-flavor-breaking systematics, decuplet hyperons — the spin-3/2 SU(3) baryon multiplet $\Delta$, $\Sigma^*$, $\Xi^*$, $\Omega$ — higher-order chiral and multi-meson corrections, strange polarization), phenomenological implementation of the singular $x=0$ component in global analysis, and better data (high-precision neutrino/antineutrino DIS on hydrogen or deuterium; associated charm–weak-boson production at the LHC).

Problem Statement

Determine quantitatively the strange–antistrange asymmetry $s(x)-\bar{s}(x)$ of the proton at a hadronic scale ($Q^2\sim1$ GeV$^2$) — where $s(x)$ and $\bar{s}(x)$ are the strange-quark and strange-antiquark parton distribution functions (number densities as functions of the fraction $x$ of the proton's longitudinal momentum carried by the parton, defined at four-momentum transfer $Q^2$) — establishing its sign, shape, and moments together with a defensible uncertainty budget. As posed by the source work, this single objective comprises three jointly necessary, mutually dependent components: (1) Theory — complete the chiral-effective-theory prediction of $s-\bar{s}$ beyond the lowest-order, octet-baryon-only kaon-loop calculation, by including decuplet hyperon intermediate states (the spin-3/2 SU(3) baryon multiplet $\Delta$, $\Sigma^*$, $\Xi^*$, $\Omega$; incorporated so far only within one-loop nonlocal formulations), higher-order chiral corrections, and multi-meson exchange, and by quantifying the residual regulator dependence and the SU(3)-flavor-breaking corrections (at the 10%–20% level in the spin-flavor input relations), yielding the predicted shape, sign, and moments such as $\langle x(s-\bar{s})\rangle$ with quantified systematics; the decuplet, higher-order, and multi-meson contributions are leading sources of systematic uncertainty to be addressed with the best available treatment of each and bounded, not pursued to unbounded chiral order or meson multiplicity. (2) Structure at $x=0$ — establish the phenomenological status of the $\delta(x)$-type contribution to $\bar{s}(x)$ at $x=0$ (a Dirac-$\delta$ term localized at $x=0$ that carries strange number but no momentum) and the associated valencelike component of $s(x)$ (strange quarks at larger $x$ unaccompanied by $\bar{s}$, as for valence quarks): whether $\int dx\,[s(x)-\bar{s}(x)]\neq0$, how such a singular component should be parametrized in global parton-distribution fits, and what empirical handles can constrain it. (3) Empirical determination — constrain $s-\bar{s}$ from data by incorporating a nonvanishing strangeness-asymmetry PDF in global QCD analyses and by high-precision neutrino/antineutrino deep-inelastic scattering (DIS) on hydrogen or deuterium and associated $W$+charm production at colliders, sufficient to decide whether the asymmetry is nonzero and, if so, its magnitude and shape. The components serve one common objective — a reliable quantitative determination of $s-\bar{s}$ — with the theory fixing the expected shape and singular structure that fits must parametrize, and the data adjudicating among allowed regulator and parametrization choices; a completed answer states the asymmetry (or a bound on it) with uncertainties and reconciles theory and phenomenology. A bound with an uncertainty budget — in particular, a data-driven bound on $s-\bar{s}$ that leaves the beyond-leading-order theory extensions bounded but not computed — discharges components (1) and (3), which may each be satisfied by a quantitative result or by such a bound rather than by an unbounded expansion of the underlying calculation. Component (2) is not discharged by a bound on $s-\bar{s}$ and remains mandatory: it requires an explicit statement of whether $\int dx\,[s(x)-\bar{s}(x)]\neq0$ and of how the $\delta(x)$-type component and the associated valencelike $s(x)$ are to be parametrized.

References

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