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

TL;DR

This paper extends the soft-gluon effective coupling concept in QCD, providing explicit perturbative results up to third order and analyzing the large-$n_F$ limit to all orders, enhancing understanding of soft-gluon resummation.

Contribution

It introduces a generalized class of soft couplings for resummation, derives explicit perturbative expressions, and explores the large-$n_F$ limit to all orders, connecting to the cusp anomalous dimension.

Findings

Explicit second-order results for soft couplings in d dimensions.

Relation between soft couplings and cusp anomalous dimension at the conformal point.

All-order expressions for soft couplings in the large-$n_F$ limit.

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 $\beta$ 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 $\alpha_s$. In d=4 dimensions we obtain the explicit relation between the soft couplings at ${\cal O}(\alpha_s^3).$ 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.