Researchers computed one-loop corrections for unpolarized quark and gluon quasi- and pseudo-distributions in the MS-bar (modified minimal subtraction) scheme and matched them to the usual light-cone parton distribution functions (PDFs). Quasi- and pseudo-distributions are versions of the PDFs tailored for lattice quantum chromodynamics (QCD), where operators are separated in space rather than along the light cone. The paper gives explicit formulas that let lattice results be converted into the standard PDFs used in high-energy physics.

For gluons the authors organized the calculation by decomposing the gluon correlator into six covariant form factors. From this decomposition they derived a 6×6 renormalization mixing matrix. The matrix’s eigenvectors identify combinations of operators that renormalize multiplicatively (that is, without mixing into other combinations). This means that, for any choice of the extended gluon operator on the lattice, one can project onto the authors’ results and obtain the one-loop matching without doing a new loop calculation. The amplitudes were checked in both a covariant gauge and an axial gauge and were generated with an automated pipeline from QGRAF to master integrals.

A second major result concerns the region |x|>1 in quasi-distributions. Here x is the parton momentum fraction that PDFs depend on, and the Ioffe time ν is a variable related to the operator separation used on the lattice. The authors evaluated all Fourier transforms in general dimension and found that the equal-time correlator contains a short-distance logarithm with a branch cut at ν=0. By the Paley–Wiener theorem (a mathematical statement about how analytic functions transform), the analytic part of the correlator produces only the usual interior region |x|≤1. The exterior region |x|>1 therefore comes entirely from the branch cut in the short-distance logarithm. At leading power in the large-momentum expansion, that exterior region carries no independent non-perturbative information: it is fixed pointwise by the light-cone PDF through the renormalization group.