Soft-gluon effective coupling: perturbative results and the large-nF limit to all orders

Abstract

We consider extensions of the soft-gluon effective coupling that generalize the Catani-Marchesini-Webber (CMW) coupling in the context of soft-gluon resummation beyond the next-to-leading logarithmic accuracy. Starting from the probability density of correlated soft emission in d dimensions we introduce a class of soft couplings relevant for resummed QCD calculations of hard-scattering observables. We show that at the conformal point, where the d-dimensional QCD β function vanishes, all these effective couplings are equal and they are also equal to the cusp anomalous dimension. We present explicit results in d dimensions for the soft-emission probability density and the soft couplings at the second-order in the QCD coupling α$_{S}$. In d = 4 dimensions we obtain the explicit relation between the soft couplings at $$ \mathcal{O} $$($$ {\alpha}_{\textrm{S}}^3 $$). Finally, we study the structure of the soft coupling in the large-n$_{F}$ limit and we present explicit expressions to all orders in perturbation theory. We also check that, at the conformal point, our large-n$_{F}$ results agree with the known result of the cusp anomalous dimension.