Soft-gluon effective coupling and cusp anomalous dimension

Abstract

We consider the extension of the CMW soft-gluon effective coupling in the context of soft-gluon resummation for QCD hard-scattering observables beyond the next-to-leading logarithmic accuracy. We present two proposals of a soft-gluon effective coupling that extend the CMW coupling to all perturbative orders in the MS ¯ coupling αS. Although both effective couplings are well-defined in the physical four-dimensional space time, we examine their behaviour in d= 4 - 2 ϵ space time dimensions. We uncover an all-order perturbative relation with the cusp anomalous dimension: the (four dimensional) cusp anomalous dimension is equal to the d-dimensional soft-gluon effective coupling at the conformal point ϵ= β(αS) , where the d-dimensional QCD β-function, β(αS) - ϵ, vanishes. We present the explicit expressions of the two soft-gluon couplings up to O(αS2) in d dimensions. In the four-dimensional case we compute the two soft couplings up to O(αS3). For one of the two couplings, we confirm the O(αS3) result previously presented by other authors. For the other coupling, we obtain the explicit relation with the cusp anomalous dimension up to O(αS4). We comment on Casimir scaling at O(αS4).