Analytic resummation for the quark form factor in QCD

TL;DR

The paper develops an analytic, RG-invariant resummation framework for the quark form factor in QCD using dimensional regularization. It derives explicit all-orders structures for the counterterm function K and provides concrete one- and two-loop resummed expressions for the logarithm of the form factor, demonstrating that infrared and collinear poles exponentiate into a controlled ε-pole with computable residues. Three equivalent methods for one-loop resummation are presented, and the analysis is extended to two loops, preserving the simple leading singularity while revealing richer logarithmic behavior at higher orders. The results offer a scalable approach to precision perturbative QCD amplitudes and potential insights into nonperturbative aspects and broader applications such as Drell–Yan processes.

Abstract

The quark form factor is known to exponentiate within the framework of dimensionally regularized perturbative QCD. The logarithm of the form factor is expressed in terms of integrals over the scale of the running coupling. I show that these integrals can be evaluated explicitly and expressed in terms of renormalization group invariant analytic functions of the coupling and of the space--time dimension, to any order in renormalized perturbation theory. Explicit expressions are given up two loops. To this order, all the infrared and collinear singularities in the logarithm of the form factor resum to a single pole in epsilon, whose residue is determined at one loop, plus powers of logarithms of epsilon. This behavior is conjectured to extend to all loops.