⟨x⟩u−dfrom lattice QCD at nearly physical quark masses

Abstract

We determine the second Mellin moment of the isovector quark parton distribution function x u−d from lattice QCD with N f = 2 sea quark flavours, employing the non-perturbatively improved Wilson-Sheikholeslami-Wohlert action at a pseudoscalar mass mπ = 157(6) MeV. The result is converted non-perturbatively to the RI'-MOM scheme and then perturbatively to the MS scheme at a scale µ = 2 GeV. As the quark mass is reduced we find the lattice prediction to approach the value extracted from experiments. PACS numbers: 12.38.Gc,14.20.Dh Almost all visible matter is composed of protons and neutrons. Analyses of the scattering of cosmic ray particles off nuclei or of results from fixed target and colliding hadron beam experiments require a quantitative understanding of the partonic structure of nucleons. The theoretical framework for this is known [1, 2] since the inception of quantum chromodynamics (QCD); see also [3,4].Of particular importance for the experimental programmes at the Large Hadron Collider are unpolarized parton distribution functions (PDFs). These, in the lightcone frame, parameterize the likelihood of a parton to carry the Bjorken momentum fraction x at a renormalization scale µ. While these PDFs have been mapped out very well from fits to experimental data, ideally one would wish to evaluate them directly from the underlying fundamental theory, QCD.The present method of choice is lattice QCD, where in principle all approximations can be removed and systematic uncertainties controlled by taking the limits of infinite volume, of vanishing lattice spacing (a → 0) and of physical quark masses. However, in this Monte Carlo simulation approach to QCD, the statistical errors and the reliability of the extrapolation to the physical point are limited by the power of available computers and the efficiency of numerical algorithms. Moreover, only Mellin moments of the PDFs can be accessed. Thus, present-day lattice simulation cannot compete in terms of precision with determinations of isovector unpolarized PDFs from fits to experimental photon-nucleon scattering data that have been collected over decades of dedicated effort. For a summary of the present status of PDF parametrizations, see [5].The possibility to predict averages over the momentum fraction, however, is complementary to experimental measurements that can only cover a limited range of x values. In particular, the strangeness and gluonic PDFs are determined rather indirectly from experiment, with large uncertainties [6]. Consequently, lattice QCD is already on the verge of becoming essential to constrain these and other less well known quantities, e.g., the pion nucleon σ term [7][8][9][10][11], the strangeness fraction of the mass of the proton f Ts [8][9][10][11][12][13], the strange quark contribution to the spin of the proton ∆s + ∆s [13,14] or individual (valence and sea) quark contributions to the proton's momentum x u , x d and x s [15]. Naturally, for lattice predictions of such quantities to be trusted, lattice QCD needs to demonstrate its ab...

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