Abstract
The off-diagonal parton-scattering channels g + γ* and q + ϕ* in deep-inelastic scattering are power-suppressed near threshold x → 1. We address the next-to-leading power (NLP) resummation of large double logarithms of 1 − x to all orders in the strong coupling, which are present even in the off-diagonal DGLAP splitting kernels. The appearance of divergent convolutions prevents the application of factorization methods known from leading power resummation. Employing d-dimensional consistency relations from requiring 1/ϵ pole cancellations in dimensional regularization between momentum regions, we show that the resummation of the off-diagonal parton-scattering channels at the leading logarithmic order can be bootstrapped from the recently conjectured exponentiation of NLP soft-quark Sudakov logarithms. In particular, we derive a result for the DGLAP kernel in terms of the series of Bernoulli numbers found previously by Vogt directly from algebraic all-order expressions. We identify the off-diagonal DGLAP splitting functions and soft-quark Sudakov logarithms as inherent two-scale quantities in the large-x limit. We use a refactorization of these scales and renormalization group methods inspired by soft-collinear effective theory to derive the conjectured soft-quark Sudakov exponentiation formula.
Highlights
Resummations of logarithmically enhanced loop corrections are a powerful and often essential tool to enlarge the predictivity of QCD perturbation theory
Where the first expression is the unfactorized partonic cross section, and the second the unfactorized parton distribution function (PDF). The latter shows that the gluon PDF in the x → 1 limit must be considered as a two-scale object already at leading power (LP), since fgLP,LL depends on the softcollinear in addition to the collinear virtuality
This goes beyond the standard paradigm of soft-collinear effective theory (SCET), where the large component of collinear momenta are assumed to be of the order of the hard scale, the momentum fraction z appearing in a B-type current is treated as an order one parameter z = O(1). z is integrated from 0 to 1
Summary
Resummations of logarithmically enhanced loop corrections are a powerful and often essential tool to enlarge the predictivity of QCD perturbation theory. Resummation is necessary when a ratio of kinematic invariants, λ, becomes small such that αs lnk λ, where αs is the strong coupling and k = 1 or 2, is no longer a good expansion parameter Recent interest in this subject has focused on understanding the structure of such logarithmic terms at next-to-leading power (NLP) in λ with the aim of summing them to all orders in αs. A number of methods has been used, but it has become evident that a generalization to the next-to-leadinglogarithmic (NLL) order is not straightforward This is to be compared to the situation at leading power (LP), where resummation is often understood to any logarithmic order, even though one faces technical challenges of high-order loop calculations in practice. Vogt and collaborators [10,11,12] found that the all-order quark-gluon splitting function with LL accuracy is given in moment space by
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