Soft-gluon effective coupling and cusp anomalous dimension

We consider the extension of the CMW soft-gluon effective coupling [1] 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 over bar coupling alpha S. Although both effective couplings are well-defined in the physical four-dimensional space time, we examine their behaviour in d=4-2 epsilon 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 epsilon=beta(alpha S), where the d-dimensional QCD beta-function, beta(alpha S)-epsilon, vanishes. We present the explicit expressions of the two soft-gluon couplings up to O(alpha S2)in d dimensions. In the four-dimensional case we compute the two soft couplings up to O(alpha S3). For one of the two couplings, we confirm the O(alpha S3)result previously presented by other authors. For the other coupling, we obtain the explicit relation with the cusp anomalous dimension up to O(alpha S4). We comment on Casimir scaling at O(alpha S4).