Soft-gluon effective coupling and cusp anomalous dimension

Abstract

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 $${\overline{\mathrm{MS}}}$$ MS¯ coupling $$\alpha _{\mathrm {S}}$$ αS . Although both effective couplings are well-defined in the physical four-dimensional space time, we examine their behaviour in $$d=4 -2\epsilon $$ 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 $$\epsilon =\beta (\alpha _{\mathrm {S}})$$ ϵ=β(αS) , where the d-dimensional QCD $$\beta $$ β -function, $$\beta (\alpha _{\mathrm {S}}) - \epsilon $$ β(αS)-ϵ , vanishes. We present the explicit expressions of the two soft-gluon couplings up to $$\mathcal{O}(\alpha _{\mathrm {S}}^2)$$ O(αS2) in d dimensions. In the four-dimensional case we compute the two soft couplings up to $$\mathcal{O}(\alpha _{\mathrm {S}}^3)$$ O(αS3) . For one of the two couplings, we confirm the $$\mathcal{O}(\alpha _{\mathrm {S}}^3)$$ O(αS3) result previously presented by other authors. For the other coupling, we obtain the explicit relation with the cusp anomalous dimension up to $$\mathcal{O}(\alpha _{\mathrm {S}}^4)$$ O(αS4) . We comment on Casimir scaling at $$\mathcal{O}(\alpha _{\mathrm {S}}^4)$$ O(αS4) .