Abstract
Many present lattice QCD approaches to calculate the parton distribution functions (PDFs) rely on a factorization formula or effective theory expansion of certain Euclidean matrix elements in boosted hadron states. In the quasi- and pseudo-PDF methods, the matching coefficient in the factorization or expansion formula includes large logarithms near the threshold, which arise from the subtle interplay of collinear and soft divergences of an underlying 3D momentum distribution. We use the standard prescription to resum such logarithms in the Mellin-moment space at next-to-leading logarithmic accuracy, which also accounts for the DGLAP evolution, and we show that it can suppress the PDF at large $x$. Unlike the deep inelastic scattering and Drell-Yan cross sections, the resummation formula is away from the Landau pole. We then apply our formulation to reanalyze the recent lattice results for the pion valence PDF, and find that within the current data sensitivity, the effect of threshold resummation is marginal for the accessible moments and the PDF at large $x$.
Highlights
Parton distribution functions (PDFs) are among the most basic quantities that describe the inner structure of the proton in high-energy experiments such as the deep inelastic scattering (DIS) and Drell-Yan (DY) processes
We studied the threshold resummation in the qPDF, the pPDF and the corresponding spatial correlator, which have been used for the lattice calculation of PDFs in the large-momentum effective theory (LaMET) and pPDF approaches
We find that the threshold logarithms in them originate from the interplay between the collinear and soft divergences in a 3D momentum distribution, and they are sensitive to longrange correlations of the spatial correlator in coordinate space
Summary
Parton distribution functions (PDFs) are among the most basic quantities that describe the inner structure of the proton in high-energy experiments such as the deep inelastic scattering (DIS) and Drell-Yan (DY) processes. The LaMET approach has attracted wide interest and led to much progress towards systematic calculations of the x-dependence of the PDF over the past few years Within this approach, the PDF can be obtained from the qPDF, which is the Fourier transform of a spatial correlator in a boosted hadron state, through an effective theory expansion in powers of large hadron momentum [31]. Similar large logarithms are observed in the matching coefficient for the short-distance expansion of the current-current correlator [40], which, is not the focus of this work The forms of such logarithms are very similar to the threshold logarithms in the DIS and DY cross section formulas, but their origins are not yet understood, especially for the pPDF or spatial correlator, since there is no clear correspondence to a soft-momentum limit in its variables.
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