Soft-gluon effective coupling and cusp anomalous dimension

Abstract

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 $\overline{\mathrm{MS}}$ coupling ${\alpha }_S$ . Although both effective couplings are well-defined in the physical four-dimensional space time, we examine their behaviour in $d=4-2ϵ$ 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 $ϵ=\beta \left({\alpha }_S\right)$ , where the d-dimensional QCD $\beta$ -function, $\beta \left({\alpha }_S\right)-ϵ$ , vanishes. We present the explicit expressions of the two soft-gluon couplings up to $O\left({\alpha }_S^2\right)$ in d dimensions. In the four-dimensional case we compute the two soft couplings up to $O\left({\alpha }_S^3\right)$ . For one of the two couplings, we confirm the $O\left({\alpha }_S^3\right)$ result previously presented by other authors. For the other coupling, we obtain the explicit relation with the cusp anomalous dimension up to $O\left({\alpha }_S^4\right)$ . We comment on Casimir scaling at $O\left({\alpha }_S^4\right)$ .