Threshold resummation to any order in (1-x)

TL;DR

The paper develops a momentum-space threshold-resummation framework for physical evolution kernels near x→1, proposing that each order in the expansion is governed by a RG-invariant jet or soft Sudakov anomalous dimension depending on a single scale. The approach is grounded in a large-$\beta_0$ dispersive representation and extended to finite $\beta_0$ via dispersive effective charges, yielding explicit representations for both jet and soft dimensions and connecting them to the quark form factor. It provides parallel treatments for DIS and Drell–Yan, including a systematic $N\to\infty$ (moment-space) expansion, and highlights the distinct, process-dependent expansion parameters required in each case. The results offer a transparent way to capture both logarithmic and constant threshold terms and pave the way for checks against fixed-order calculations and potential universality in Sudakov charges across processes.

Abstract

A simple ansatz is suggested for the structure of threshold resummation of the momentum space physical evolution kernels (`physical anomalous dimensions') at all orders in (1-x), taking as examples Deep Inelastic Scattering (F_2(x, Q^2) and F_L(x, Q^2)) and the Drell-Yan process. Each term in the expansion is associated to a distinct renormalization group and scheme invariant perturbative object (`physical Sudakov anomalous dimension') depending on a single momentum scale variable. Both logarithmically enhanced terms and constant terms are captured by the ansatz at any order in the expansion. The ansatz is motivated by a large--beta_0 dispersive calculation. A dispersive representation at finite beta_0 of the physical Sudakov anomalous dimensions is also obtained, associated to a set of `Sudakov effective charges' which encapsulate the non-Abelian nature of the interaction. It is found that the dispersive representation requires a non-trivial, and process-dependent, choice of variables in the (x,Q^2) plane. Some interesting properties of the physical Sudakov anomalous dimensions are pointed out. The ensuing 1/N expansion in moment space is straightforwardly derived from the momentum space expansion.