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Threshold Resummation in Momentum Space from Effective Field Theory

Thomas Becher, Matthias Neubert

TL;DR

The paper develops a soft-collinear effective theory (SCET) framework to perform threshold resummation of Sudakov logarithms for the DIS structure function F_2 near x→1 directly in momentum space, avoiding Landau-pole issues of Mellin-space methods. It derives an explicit all-order factorization into hard, jet, and PDF components with exact RG evolution, expressing the resummed F_2^{ns} in terms of a Sudakov-exponentiated C_V, a jet-function transform, and endpoint PDFs. A key result is an exact momentum-space formula that remains free of Landau-pole integrals and is readily extendable to other hard QCD processes, achieving high perturbative control (NNLO/N^3LL). The approach provides a transparent scale separation and a simpler alternative to conventional moment-space resummation, with broad practical impact for precision QCD predictions.

Abstract

Methods from soft-collinear effective theory are used to perform the threshold resummation of Sudakov logarithms for the deep-inelastic structure function F_2(x,Q^2) in the endpoint region x->1 directly in momentum space. An explicit all-order formula is derived, which expresses the short-distance coefficient function C in the convolution F_2=C*phi_q in terms of Wilson coefficients and anomalous dimensions defined in the effective theory. Contributions associated with the physical scales Q^2 and Q^2(1-x) are separated from non-perturbative hadronic physics in a transparent way. A crucial ingredient to the momentum-space resummation is the exact solution to the integro-differential evolution equation of the jet function, which is derived. The methods developed in this Letter can be applied to many other hard QCD processes.

Threshold Resummation in Momentum Space from Effective Field Theory

TL;DR

The paper develops a soft-collinear effective theory (SCET) framework to perform threshold resummation of Sudakov logarithms for the DIS structure function F_2 near x→1 directly in momentum space, avoiding Landau-pole issues of Mellin-space methods. It derives an explicit all-order factorization into hard, jet, and PDF components with exact RG evolution, expressing the resummed F_2^{ns} in terms of a Sudakov-exponentiated C_V, a jet-function transform, and endpoint PDFs. A key result is an exact momentum-space formula that remains free of Landau-pole integrals and is readily extendable to other hard QCD processes, achieving high perturbative control (NNLO/N^3LL). The approach provides a transparent scale separation and a simpler alternative to conventional moment-space resummation, with broad practical impact for precision QCD predictions.

Abstract

Methods from soft-collinear effective theory are used to perform the threshold resummation of Sudakov logarithms for the deep-inelastic structure function F_2(x,Q^2) in the endpoint region x->1 directly in momentum space. An explicit all-order formula is derived, which expresses the short-distance coefficient function C in the convolution F_2=C*phi_q in terms of Wilson coefficients and anomalous dimensions defined in the effective theory. Contributions associated with the physical scales Q^2 and Q^2(1-x) are separated from non-perturbative hadronic physics in a transparent way. A crucial ingredient to the momentum-space resummation is the exact solution to the integro-differential evolution equation of the jet function, which is derived. The methods developed in this Letter can be applied to many other hard QCD processes.

Paper Structure

This paper contains 5 sections, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Factorization formula
  3. Solutions of the renormalization group equations
  4. Momentum-space resummation
  5. Conclusions

Figures (1)

  • Figure 1: Dependence of the resummed result for $F_2^{\rm ns}(x,Q^2)$ on the hard (left) and intermediate (right) scales, at different orders in RG-improved perturbation theory: LO (dotted), NLO (dashed), and NNLO (solid). We use $Q=30$ GeV, $x=0.9$, $\mu_f=5$ GeV, and $b(\mu_f)=4$.