Unpolarized Quark and Gluon TMD PDFs and FFs at N$^3$LO

TL;DR

This paper analytically computes the N$^3$LO perturbative matching coefficients for unpolarized quark and gluon TMD PDFs and FFs, advancing precision in QCD phenomenology.

Contribution

It provides the first complete N$^3$LO calculations of TMD PDFs and FFs, including resummation of small-$z$ logarithms, crucial for high-precision QCD analyses.

Findings

N$^3$LO matching coefficients for TMD PDFs and FFs derived.

Resummation of double logarithmic terms at small $z$ performed.

Results enhance the accuracy of TMD-based phenomenology.

Abstract

In this paper we calculate analytically the perturbative matching coefficients for unpolarized quark and gluon Transverse-Momentum-Dependent (TMD) Parton Distribution Functions (PDFs) and Fragmentation Functions (FFs) through Next-to-Next-to-Next-to-Leading Order (N$^3$LO) in QCD. The N$^3$LO TMD PDFs are calculated by solving a system of differential equation of Feynman and phase space integrals. The TMD FFs are obtained by analytic continuation from space-like quantities to time-like quantities, taking into account the probability interpretation of TMD PDFs and FFs properly. The coefficient functions for TMD FFs exhibit double logarithmic enhancement at small momentum fraction $z$. We resum such logarithmic terms to the third order in the expansion of $\alpha_s$. Our results constitute important ingredients for precision determination of TMD PDFs and FFs in current and future experiments.