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 $$ coupling $$ . Although both effective couplings are well-defined in the physical four-dimensional space time, we examine their behaviour in $$ space time dimensions. We uncover an all-order perturbative relation with the cusp anomalous dimension: the (four dimensional) cusp anomalous dimension is equal to the d-dimensional soft-gluon effective coupling at the conformal point $$ , where the d-dimensional QCD $$ -function, $$ , vanishes. We present the explicit expressions of the two soft-gluon couplings up to $$ in d dimensions. In the four-dimensional case we compute the two soft couplings up to $$ . For one of the two couplings, we confirm the $$ result previously presented by other authors. For the other coupling, we obtain the explicit relation with the cusp anomalous dimension up to $$ . We comment on Casimir scaling at $$ .