Assembling the complete strange-antistrange asymmetry of the nucleon sea: combining the intrinsic light-front contribution with the NNLO perturbative contribution under Q^2 evolution and confronting it with global data

The strange-quark and antistrange-quark parton distribution functions of the proton, $s(x)$ and $\bar{s}(x)$, need not coincide at fixed light-cone momentum fraction $x$ even though their integrals are equal by strangeness conservation. Vega, Schmidt, Gutsche and Lyubovitskij (Phys. Rev. D 93, 056001 (2016)) computed only the nonperturbative (intrinsic) piece of $s(x)-\bar{s}(x)$, using meson-baryon fluctuation amplitudes built from three light-front wave functions (one Gaussian, two motivated by light-front holographic QCD), and left the physical observable unassembled: the perturbative (extrinsic) asymmetry generated by next-to-next-to-leading order (NNLO) QCD evolution, the scale evolution of the combined asymmetry, the constraint of the model inputs by global data, the discrimination among the wave functions, and the implications for the NuTeV weak-mixing-angle anomaly all remained open. The open core survives: later theory work (chiral effective theory, lattice-QCD-constrained light-front holography, meson-baryon fluctuation models) still computes intrinsic pieces only, while modern global PDF fits (MMHT14/MSHT20) fit a weakly constrained phenomenological asymmetry without model-motivated intrinsic input, and no work has yet assembled and tested the full combined prediction.

Background

A parton distribution function (PDF) gives the probability density for finding a parton (quark, antiquark, or gluon) carrying momentum fraction $x$ of the proton's light-front momentum at a resolution scale $Q^2$. Deep-inelastic scattering (DIS) with charged leptons and neutrinos, together with $W$- and $Z$-boson production at hadron colliders, constrains these PDFs in global QCD fits. Although net strangeness conservation forces $\int_0^1 dx\,(s(x)-\bar{s}(x)) = 0$, the shapes of $s(x)$ and $\bar{s}(x)$ may differ, so the second moment $\langle x(s-\bar{s})\rangle = \int_0^1 dx\, x\,(s(x)-\bar{s}(x))$ can be nonzero. Two mechanisms generate such a strange-antistrange asymmetry (SQA). First, a nonperturbative (intrinsic) mechanism: the proton fluctuates into virtual meson-baryon states such as $\Lambda K$ (a proton splitting into a lambda hyperon and a kaon); because the $\Lambda$ (which contains the strange quark) and the $K$ (which contains the antistrange quark) share the proton momentum differently, $s(x)$ and $\bar{s}(x)$ acquire different shapes, typically differing at large $x$. Brodsky and Ma (Phys. Lett. B 381, 317 (1996)) formulated this in the light-front approach, in which the meson-baryon fluctuation amplitude is a two-body light-front wave function (LFWF) convoluted with quark LFWFs inside the hadrons. Second, a perturbative (extrinsic) mechanism: Catani, de Florian, Rodrigo and Vogelsang (Phys. Rev. Lett. 93, 152003 (2004)) showed that QCD evolution generates $s \neq \bar{s}$ starting at three loops (NNLO in the DGLAP evolution equations, the standard perturbative $Q^2$-evolution of PDFs), important mainly at small $x$, with a small negative second moment $\approx -5\times 10^{-4}$ at $Q^2 = 20$ GeV$^2$. Phenomenologically, the SQA matters because the NuTeV collaboration extracted the weak mixing angle $\sin^2\theta_W$ from neutrino and antineutrino DIS about $3\sigma$ above the standard-model value (Zeller et al., Phys. Rev. Lett. 88, 091802 (2002)); a positive second moment $\langle x(s-\bar{s})\rangle \sim 0.005$ would shift the extraction enough to attribute the anomaly to the strange asymmetry alone. Extractions from neutrino dimuon data (opposite-sign dimuon production, which tags charm production from strange quarks) are small and sign-unstable: a leading-order NuTeV analysis gave $\langle x(s-\bar{s})\rangle = (-2.7\pm 1.3)\times 10^{-3}$, the next-to-leading-order reanalysis by Mason et al. (Phys. Rev. Lett. 99, 192001 (2007)) gave $(1.96\pm 1.43)\times 10^{-3}$, and the global fit of Olness et al. (Eur. Phys. J. C 40, 145 (2005)) found $-0.001 < \langle x(s-\bar{s})\rangle < 0.004$. The MMHT 2014 global fit (Harland-Lang, Martin, Motylinski, Thorne, Eur. Phys. J. C 75, 204 (2015)) was the first in that series to parametrize $x(s-\bar{s})$ and fit it, mainly against dimuon data from the CCFR and NuTeV Fermilab neutrino experiments. Vega, Schmidt, Gutsche and Lyubovitskij (Phys. Rev. D 93, 056001 (2016)) computed the intrinsic SQA in the Brodsky-Ma approach with three LFWFs: a Gaussian and two variants motivated by light-front holographic QCD (a correspondence between light-front QCD and anti-de Sitter space, AdS/QCD, that fixes wave-function forms), where variant I matches light-front QCD at large $x$ and variant II at all $x$; they used the $N\to\Lambda K$ fluctuation probability 1.27% from Cao and Signal (Phys. Rev. D 60, 074021 (1999)), imposed a symmetrization $\phi_{BM}(x,k_\perp) = \psi_5(x,k_\perp)+\psi_5(1-x,k_\perp)$ so that variant II satisfies the fluctuation-symmetry condition $f_{BM}(x)=f_{MB}(1-x)$, and fixed the LFWF scale parameter $\kappa = 350$ MeV from other hadron data. They found $s(x)<\bar{s}(x)$ at small $x$ and $s(x)>\bar{s}(x)$ at large $x$ (opposite to earlier meson-cloud models) and agreement of variant II with the MMHT band for $x(s-\bar{s})$.

Problem Statement

Construct and test the complete strange-antistrange asymmetry of the proton sea. Concretely, building on the intrinsic calculation of Vega et al. (Phys. Rev. D 93, 056001 (2016)): (i) combine the nonperturbative (intrinsic) contribution to $s(x)-\bar{s}(x)$ from the meson-baryon fluctuation light-front calculation — with its candidate light-front wave functions (Gaussian, holographic variants I and II), fluctuation probability, symmetrization treatment, and scale parameter $\kappa$ — with the perturbative (extrinsic) contribution generated by NNLO QCD evolution, without double counting; (ii) evolve the combined asymmetry in $Q^2$ with DGLAP evolution to the scales of the constraining data, obtaining the net $x$-dependent $s(x,Q^2)-\bar{s}(x,Q^2)$ and its scale dependence over the full $x$ range; (iii) confront the combined prediction with global data — neutrino dimuon (opposite-sign dimuon) DIS, collider $W/Z$ production, and other PDF-sensitive observables — so as to determine the quantitative uncertainties of the model inputs (the $N\to\Lambda K$ fluctuation probability, the symmetrization treatment, and $\kappa$) and to establish which light-front wave function, with which parameter set, is preferred by a comprehensive fit including perturbative evolution; and (iv) quantify, with a careful accounting of all contributions and their scale dependence, how much of the NuTeV weak-mixing-angle anomaly the combined intrinsic-plus-perturbative asymmetry — in particular its second moment $\langle x(s-\bar{s})\rangle$, compared against the magnitude required to shift the extracted $\sin^2\theta_W$ — can account for, and whether other sources are required. The single scientific objective uniting these four components — all of which are required — is a complete, data-constrained physical prediction of the observable $s(x,Q^2)-\bar{s}(x,Q^2)$, in which the model inputs are fixed by global data rather than by hadron-spectroscopy conventions alone, together with its phenomenological verdict on the NuTeV attribution.

References

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