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Diagrammatic insights into next-to-soft corrections

C. D. White

TL;DR

The paper uses diagrammatic techniques to derive and confirm next-to-soft (NE) corrections in gauge and gravity theories, showing how external and internal NE emissions combine to reproduce tree-level NE theorems for scalars, fermions, and gluons. A key result is that NE contributions organize into total angular momentum operators J^{(i)}_{ ho u}, highlighting a spin-orbit structure and a partial double-copy relationship between gauge and gravity NE factors. The approach provides a versatile, helicity-independent framework that complements existing soft-theorem analyses and offers tools for exploring loop effects and collider phenomenology. Overall, the work clarifies the diagrammatic underpinnings of NE theorems and points to practical avenues for refining infrared predictions in high-energy processes.

Abstract

We confirm recently proposed theorems for the structure of next-to-soft corrections in gauge and gravity theories using diagrammatic techniques, first developed for use in QCD phenomenology. Our aim is to provide a useful alternative insight into the next-to-soft theorems, including tools that may be useful for further study. We also shed light on a recently observed double copy relation between next-to-soft corrections in the gauge and gravity cases.

Diagrammatic insights into next-to-soft corrections

TL;DR

The paper uses diagrammatic techniques to derive and confirm next-to-soft (NE) corrections in gauge and gravity theories, showing how external and internal NE emissions combine to reproduce tree-level NE theorems for scalars, fermions, and gluons. A key result is that NE contributions organize into total angular momentum operators J^{(i)}_{ ho u}, highlighting a spin-orbit structure and a partial double-copy relationship between gauge and gravity NE factors. The approach provides a versatile, helicity-independent framework that complements existing soft-theorem analyses and offers tools for exploring loop effects and collider phenomenology. Overall, the work clarifies the diagrammatic underpinnings of NE theorems and points to practical avenues for refining infrared predictions in high-energy processes.

Abstract

We confirm recently proposed theorems for the structure of next-to-soft corrections in gauge and gravity theories using diagrammatic techniques, first developed for use in QCD phenomenology. Our aim is to provide a useful alternative insight into the next-to-soft theorems, including tools that may be useful for further study. We also shed light on a recently observed double copy relation between next-to-soft corrections in the gauge and gravity cases.

Paper Structure

This paper contains 6 sections, 42 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of necessary concepts
  3. The next-to-soft theorem for scalars and fermions
  4. The next-to-soft theorem for gluons
  5. Conclusion
  6. NE Feynman rules for graviton emission

Figures (4)

  • Figure 1: (a) A hard interaction which produces $n$ particles; (b) Emission of an eikonal gluon from an external leg.
  • Figure 2: (a) External emission of a next-to-soft gluon; (b) Internal emission of a soft gluon.
  • Figure 3: Emission of a single gluon from the $j^{\rm th}$ external leg of the hard amplitude ${\cal A}_n$.
  • Figure 4: Emission of a single graviton from the $i^{\rm th}$ external leg of the hard amplitude ${\cal M}_n$.