The method of regions and next-to-soft corrections in Drell–Yan production

We perform a case study of the behaviour of gluon radiation beyond the soft approximation, using as an example the Drell–Yan production cross section at NNLO. We draw a careful distinction between the eikonal expansion, which is in powers of the soft gluon energies, and the expansion in powers of the threshold variable 1−z1−z, which involves important hard-collinear effects. Focusing on the contribution to the NNLO Drell–Yan K-factor arising from real–virtual interference, we use the method of regions to classify all relevant contributions up to next-to-leading power in the threshold expansion. With this method, we reproduce the exact two-loop result to the required accuracy, including z-independent non-logarithmic contributions, and we precisely identify the origin of the soft-collinear interference which breaks simple soft-gluon factorisation at next-to-eikonal level. Our results pave the way for the development of a general factorisation formula for next-to-leading-power threshold logarithms, and clarify the nature of loop corrections to a set of recently proposed next-to-soft theorems.