Double soft current at one-loop in QCD

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.