Double soft current at one-loop in QCD
Yu Jiao Zhu
TL;DR
This work derives compact one-loop expressions for double soft gluon and double soft quark emissions in QCD with multiple Wilson lines, revealing predominantly dipole-type color correlations and an Abelian factorization piece. The kinematic dependence is captured by polylogarithmic functions, with explicit time-like results and a careful analytic continuation to other kinematic regions, uncovering crossing-induced phases that do not affect the squared amplitude. A key result is the crossing invariance of the squared amplitude, which implies universality of the TMD soft function up to three loops across Drell-Yan, SIDIS, and e^+e^- annihilation. By combining these results with existing two-loop single-soft and tree-level triple-soft amplitudes, the paper lays a foundation for understanding NNNLO soft correlations involving multiple hard directions and supports a universal soft-sector description in high-precision QCD phenomenology.
Abstract
We investigate the soft behavior of QCD amplitudes involving multiple Wilson lines and derive compact analytic expressions for double soft gluon and double soft quark emissions at one loop. The color correlations of the soft current exhibit a predominantly dipole structure, coupling to two hard legs at a time, apart from an abelian contribution that factorizes into products of one-loop and tree-level single soft currents, which may involve up to three hard legs. The kinematic dependence of the one-loop soft currents is expressed in terms of polylogarithmic functions, with explicit results presented for time-like kinematics. We further discuss the analytic continuation to other kinematic configurations and identify non-trivial crossing effects when continuing into incoming states. The squared amplitude is found to be invariant under crossing, which implies that the fully differential soft function and in particular the TMD soft function, remains universal up to three loops.