Nonperturbative strange–antistrange asymmetry and intrinsic strangeness of the nucleon

The strange and antistrange quark distributions of the proton, $s(x)$ and $\bar{s}(x)$, need not be equal: their difference is a purely nonperturbative signature of the proton's light-front Fock-state structure, it is constrained by the lattice QCD strange Dirac form factor $F_1^s(Q^2)$, and it enters neutrino and parity-violating deep-inelastic observables. A 2018 analysis combining lattice QCD, light-front holographic QCD (LFHQCD), and the meson–baryon fluctuation model found a first moment $\langle S_-\rangle=0.0011(4)$ — far below the $\sim 0.005$ once invoked to explain the NuTeV weak-mixing-angle anomaly — and showed that the two frameworks make qualitatively different small-$x$ predictions. This problem asks for a full quantitative determination of the nonperturbative strange–antistrange asymmetry and the intrinsic strange probability: the magnitude and sign pattern of $s(x)-\bar{s}(x)$ over all $x$, the intrinsic strange probability $I_s$ beyond its positivity lower bound, the separation of $s$ from $\bar{s}$ beyond the current one-parameter degeneracy, the small-$x$ behavior that distinguishes the Regge-like holographic prediction from the vanishing meson-cloud one, and the factorization-scale matching to perturbative QCD evolution. The determination is still open: global fits remain essentially insensitive to the asymmetry at current precision, lattice input reaches only intermediate $x$, and the small-$x$ region where the predictions diverge is unconstrained by any existing data.

Background

A parton distribution function (PDF) $f(x)$ gives the probability density for finding a parton of flavor $f$ carrying a fraction $x$ of a hadron's longitudinal momentum in a fast frame. In the proton, the strange quark and antiquark distributions, $s(x)$ and $\bar{s}(x)$, are sea distributions: they are not required by the valence $uud$ content. Perturbative QCD generates sea pairs radiatively through gluon splitting (the extrinsic sea); in contrast, the intrinsic sea is the nonperturbative component already present in the proton's light-front Fock-state expansion — the quantum-mechanical superposition of states such as $|uud\rangle$ and $|uud\,s\bar{s}\rangle$, where the light-front wave function describes the bound state at fixed light-front time. Because the proton carries quark number, the strange quark in the $|uud\,s\bar{s}\rangle$ component interacts differently with the valence spectators than the antistrange quark does, so $s(x)$ and $\bar{s}(x)$ need not coincide. Their difference, the strange–antistrange asymmetry $s(x)-\bar{s}(x)$, is therefore a purely nonperturbative effect (the perturbative contribution first arises at three loops and is tiny).

Two nonperturbative frameworks make predictions. In the meson–baryon fluctuation model, the proton fluctuates part of the time into a virtual kaon plus a hyperon ($K\Lambda$ or $K\Sigma$); the kaon's valence strange quark and the hyperon's antistrange spectator then acquire different momentum distributions. In light-front holographic QCD (LFHQCD) — a semiclassical approximation to QCD built by mapping light-front dynamics onto a gravitation-like theory in a higher-dimensional curved space — the strange form factor and the asymmetry follow from the Regge trajectory of the vector meson coupled to the strange current, where a Regge trajectory is the relation between a hadron's spin and its squared mass that governs its exchange behavior at high energy; its intercept controls how singular a PDF is as $x\to 0$. A third, first-principles input is lattice QCD: the strange Dirac form factor $F_1^s(Q^2)$, the strange-quark contribution to the proton's Dirac electromagnetic form factor at squared momentum transfer $Q^2$, is nonzero at $Q^2>0$ when the strange and antistrange distributions differ, $s(x)-\bar{s}(x)\neq 0$, and its $Q^2$ behavior constrains the first moment $\langle S_-\rangle=\int_0^1 dx\, x\,[s(x)-\bar{s}(x)]$ through model fits to the lattice data. Lattice computations of $F_1^s$ are difficult because the strange contributions enter through disconnected quark loops with a poor signal-to-noise ratio, and the reachable $Q^2$ is limited.

The original experimental motivation was the NuTeV anomaly: the NuTeV collaboration extracted the weak mixing angle $\sin^2\theta_W$ from the Paschos–Wolfenstein ratio of neutral-current to charged-current neutrino and antineutrino deep-inelastic scattering cross sections on an iron target and found a value about $3\sigma$ above the Standard Model prediction. Since $\langle S_-\rangle$ enters this ratio, a positive strangeness asymmetry of order $\langle S_-\rangle\sim 0.005$ was one candidate explanation. In the 2018 analysis that defines this problem, the meson–baryon fluctuation model and LFHQCD, normalized to the lattice strange form factor, both give $\langle S_- \rangle = 0.0011(4)$ — an order of magnitude too small — and they disagree qualitatively at small $x$: LFHQCD predicts Regge behavior $s(x)-\bar{s}(x)\simeq -0.044\,x^{-0.01}$ as $x\to 0$, while the fluctuation model predicts the asymmetry vanishes. The holographic positivity bound gives a lower limit $I_s \ge 0.92\%$ on the intrinsic strange probability $I_s=\int_0^1 dx\, s(x)=\int_0^1 dx\, \bar{s}(x)$, and the separation of $s(x)$ from $\bar{s}(x)$ is fixed only up to a free parameter $\alpha$ constrained by the positivity of both distributions but not uniquely determined. Finally, comparing these hadronic-scale model distributions with the perturbatively evolved distributions used in global PDF fits requires choosing a factorization scale that neither framework specifies (the 2018 comparison assumed $\mu = 1$ GeV).

Problem Statement

Determine the nonperturbative strange–antistrange asymmetry of the nucleon quantitatively: obtain $s(x)$ and $\bar{s}(x)$ — equivalently the asymmetry $s(x)-\bar{s}(x)$ together with the intrinsic strange probability $I_s=\int_0^1 dx\, s(x)=\int_0^1 dx\, \bar{s}(x)$ — for the intrinsic (nonperturbative) strange sea of the proton, with uncertainties, including:

1. The magnitude and sign of the first moment $\langle S_-\rangle=\int_0^1 dx\, x\,[s(x)-\bar{s}(x)]$ and the sign pattern of $s(x)-\bar{s}(x)$ across the full $x$ range (the lattice-implied pattern is negative at small $x$ and positive at large $x$).
2. The value of the intrinsic strange probability $I_s$ as a determination rather than the positivity lower bound $I_s\ge 0.92\%$, including resolution of the one-parameter ($\alpha$) degeneracy that currently fixes only the separation between $s(x)$ and $\bar{s}(x)$.
3. The small-$x$ behavior of $s(x)-\bar{s}(x)$ — the Regge-like $s(x)-\bar{s}(x)\sim x^{-0.01}$ predicted by light-front holographic QCD versus the vanishing behavior of the meson–baryon fluctuation model.
4. The factorization (hadronic) scale at which the nonperturbative model distributions are matched to perturbative QCD evolution, so that the comparison with global-fit PDFs is unambiguous.

These components jointly serve one overarching objective: a single, constraint-consistent determination of the nonperturbative strange sea, validated against lattice QCD input for the strange form factor $F_1^s(Q^2)$ (with controlled continuum and physical-pion-mass systematics, extended to larger $Q^2$ than presently available) and against experimental constraints such as neutrino dimuon production, $W/Z$ production at hadron colliders, and parity-violating deep-inelastic scattering (scattering asymmetries generated by the interference of the electromagnetic and weak neutral currents, which are sensitive to individual sea-quark flavors). Resolving whether $\langle S_-\rangle$ can account for the NuTeV weak-mixing-angle discrepancy is part of the lineage of this problem but is not by itself a solution: the determination itself — in particular at small $x$, where current data and lattice calculations are blind — is the target.

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