Abstract
We propose an analysis method for the leading-twist operator product expansion based lattice QCD determinations of the valence parton distribution function (PDF). In the first step, we determine the confidence-intervals of the leading-twist $\overline{\mathrm{MS}}$ Wilson coefficients, $C_n(\mu^2 z^2)$, of the equal-time bilocal quark bilinear, given the lattice QCD matrix element of Ioffe-time distribution for a particular hadron $H$ as well as the prior knowledge of the valence PDF, $f(x,\mu)$ of the hadron $H$ determined via global fit from the experimental data. In the next step, we apply the numerically estimated $C_n$ in the lattice QCD determinations of the valence PDFs of other hadrons, and for the zero-skewness generalized parton distribution (GPD) of the same hadron $H$ at non-zero momentum transfers. Our proposal still assumes the dominance of leading-twist terms, but it offers a pragmatic alternative to the usage of perturbative Wilson coefficients and their associated higher-loop uncertainties such as the effect of all-order logarithms at larger sub-Fermi quark-antiquark separations $z$.
Highlights
The progress in determining the x-dependent hadron structures, such as the parton distribution functions (PDFs) and the generalized parton distribution functions (GPDs) [1,2,3], has been quite rapid in the recent years, owing to the perturbative matching frameworks such as the quasi-PDF [5,6], pseudo-PDF [7,8], current-current correlators [9] and the good lattice cross sections approach [10,11]
The method we advocated in this paper for lattice determination of valence PDFs consists of the following steps: (1) The leading-twist Wilson coefficients CnðμzÞ are determined directly by fitting the lattice data for a hadron, say the pion, by assuming the pion PDF is known well from phenomenological determinations and taken as an input for the lattice analysis
(2) The CnðzÞ obtained from fits is used in the analysis of lattice data to determine the PDF of another hadron, say, the proton by performing the usual reconstruction analysis methods to determine the x-dependent PDF, except that one replaces the perturbative matching coefficients with the one obtained from the pion, in this example
Summary
The progress in determining the x-dependent hadron structures, such as the parton distribution functions (PDFs) and the generalized parton distribution functions (GPDs) [1,2,3] (for a review, see [4]), has been quite rapid in the recent years, owing to the perturbative matching frameworks such as the quasi-PDF [5,6], pseudo-PDF [7,8], current-current correlators [9] and the good lattice cross sections approach [10,11]. For the valence case, xfðxÞ vanishes in the small-x limit, and enable us to neglect the associated systematic errors in the global fit determinations and in their first few Mellin moments This lets us focus only on sources of uncertainty in the lattice computations of PDF. We assume that the Wilson coefficients that enter the leading-twist OPE are hadron independent Given these two assumptions, the idea we pursue in this paper is to replace a perturbatively determined Cn with a probabilistically likely Cn that is determined directly from the lattice matrix element data, and given a trustworthy knowledge of valence PDF of a hadron from global analysis of experimental input. In the rest of the paper, we explain the method and apply it to lattice mock-data, and some real published proton and pion lattice data
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