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Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach

Eric Laenen, Lorenzo Magnea, Gerben Stavenga, Chris D. White

TL;DR

This work extends soft-gluon exponentiation from the eikonal to next-to-eikonal order using a diagrammatic framework. It derives effective NE Feynman rules, proves NE exponentiation for factorizable contributions, and introduces a remainder that encodes inter-group correlations, including a novel two-gluon NE vertex. The authors validate the approach by matching to path-integral results and applying the rules to Drell–Yan production, reproducing known threshold logarithms up to NNLO in abelian-like sectors. The results lay groundwork for threshold resummation beyond leading power and outline future steps to incorporate non-factorizable pieces and hard-collinear effects. Overall, the paper provides a concrete, diagrammatic method to organize NE soft radiation and demonstrates its consistency with alternative formalisms and with explicit cross-section calculations.

Abstract

We consider the problem of soft gluon resummation for gauge theory amplitudes and cross sections, at next-to-eikonal order, using a Feynman diagram approach. At the amplitude level, we prove exponentiation for the set of factorizable contributions, and construct effective Feynman rules which can be used to compute next-to-eikonal emissions directly in the logarithm of the amplitude, finding agreement with earlier results obtained using path-integral methods. For cross sections, we also consider sub-eikonal corrections to the phase space for multiple soft-gluon emissions, which contribute to next-to-eikonal logarithms. To clarify the discussion, we examine a class of log(1 - x) terms in the Drell-Yan cross-section up to two loops. Our results are the first steps towards a systematic generalization of threshold resummations to next-to-leading power in the threshold expansion.

Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach

TL;DR

This work extends soft-gluon exponentiation from the eikonal to next-to-eikonal order using a diagrammatic framework. It derives effective NE Feynman rules, proves NE exponentiation for factorizable contributions, and introduces a remainder that encodes inter-group correlations, including a novel two-gluon NE vertex. The authors validate the approach by matching to path-integral results and applying the rules to Drell–Yan production, reproducing known threshold logarithms up to NNLO in abelian-like sectors. The results lay groundwork for threshold resummation beyond leading power and outline future steps to incorporate non-factorizable pieces and hard-collinear effects. Overall, the paper provides a concrete, diagrammatic method to organize NE soft radiation and demonstrates its consistency with alternative formalisms and with explicit cross-section calculations.

Abstract

We consider the problem of soft gluon resummation for gauge theory amplitudes and cross sections, at next-to-eikonal order, using a Feynman diagram approach. At the amplitude level, we prove exponentiation for the set of factorizable contributions, and construct effective Feynman rules which can be used to compute next-to-eikonal emissions directly in the logarithm of the amplitude, finding agreement with earlier results obtained using path-integral methods. For cross sections, we also consider sub-eikonal corrections to the phase space for multiple soft-gluon emissions, which contribute to next-to-eikonal logarithms. To clarify the discussion, we examine a class of log(1 - x) terms in the Drell-Yan cross-section up to two loops. Our results are the first steps towards a systematic generalization of threshold resummations to next-to-leading power in the threshold expansion.

Paper Structure

This paper contains 21 sections, 207 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Introduction to eikonal exponentiation
  3. Abelian eikonal exponentiation
  4. Non-abelian eikonal exponentiation
  5. NE corrections to Feynman diagrams
  6. NE emission from a scalar particle
  7. NE emission from a spin-$\frac{1}{2}$ particle
  8. Factorization of next-to-eikonal diagrams
  9. The NE propagator
  10. The NE vertex
  11. The remainder term
  12. Exponentiation for NE matrix elements
  13. Next-to-eikonal Feynman rules
  14. Relation to the path integral formalism
  15. An illustrative example: the Drell-Yan process
  16. ...and 6 more sections

Figures (19)

  • Figure 1: Soft photon emission from an energetic line.
  • Figure 2: A process involving two eikonal lines A and B, interacting through the exchange of soft gluons forming diagram $G$.
  • Figure 3: Decomposition of a soft photon graph into connected subdiagrams $G_c^{(i)}$, each of which occurs $N_i$ times.
  • Figure 4: Decomposition of a higher order non-Abelian diagram in terms of webs.
  • Figure 5: Gluon groups obtained from the lower eikonal line of fig. \ref{['webdecomp']}, with the same color coding as in that figure.
  • ...and 14 more figures