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Multi-Soft gluon limits and extended current algebras at null-infinity

Tristan McLoughlin, Dhritiman Nandan

TL;DR

The paper develops a two-dimensional current-algebra framework for four-dimensional Yang-Mills soft limits. By using double-soft limits, it constructs a holomorphic Sugawara energy-momentum tensor from the leading $ rak{su}(N)$ currents and verifies its OPEs, linking soft theorems to conformal Ward identities and Knizhnik-Zamolodchikov equations for positive-helicity, MHV amplitudes. It then elevates the discussion to sub-leading soft terms, introducing a one-parameter family of sub-leading currents and deriving their algebras with leading and among themselves, while also accommodating an anti-holomorphic sector with nontrivial holomorphic–anti-holomorphic OPEs. The work further explores the emergence of a sub-leading stress sector and a conjugation operator within a CFT-like structure, and discusses holographic interpretations and connections to BCJ relations, indicating a richer symmetry structure at null infinity beyond the leading current algebra. Overall, the results suggest a multi-layered 2D symmetry description of YM amplitudes at null infinity, with potential implications for holography and flat-space scattering.

Abstract

In this note we consider aspects of the current algebra interpretation of multi-soft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities correspond to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic currents and study some of its implications.

Multi-Soft gluon limits and extended current algebras at null-infinity

TL;DR

The paper develops a two-dimensional current-algebra framework for four-dimensional Yang-Mills soft limits. By using double-soft limits, it constructs a holomorphic Sugawara energy-momentum tensor from the leading currents and verifies its OPEs, linking soft theorems to conformal Ward identities and Knizhnik-Zamolodchikov equations for positive-helicity, MHV amplitudes. It then elevates the discussion to sub-leading soft terms, introducing a one-parameter family of sub-leading currents and deriving their algebras with leading and among themselves, while also accommodating an anti-holomorphic sector with nontrivial holomorphic–anti-holomorphic OPEs. The work further explores the emergence of a sub-leading stress sector and a conjugation operator within a CFT-like structure, and discusses holographic interpretations and connections to BCJ relations, indicating a richer symmetry structure at null infinity beyond the leading current algebra. Overall, the results suggest a multi-layered 2D symmetry description of YM amplitudes at null infinity, with potential implications for holography and flat-space scattering.

Abstract

In this note we consider aspects of the current algebra interpretation of multi-soft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities correspond to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic currents and study some of its implications.

Paper Structure

This paper contains 20 sections, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Soft limits for gluon amplitudes
  3. Single-soft limits
  4. Multi-soft limits
  5. Current algebra interpretation of Yang-Mills soft limits
  6. An $\mathfrak{su}(N)$ current algebra
  7. Sugawara construction for amplitudes
  8. Energy-momentum tensor/current OPE
  9. Energy-momentum OPE
  10. Comments on a Knizhnik-Zamolodchikov equation
  11. Sub-leading currents
  12. Sub-leading algebra
  13. $J_{\rm sub} J$ :
  14. $J_{\rm sub} J_{\rm sub}$ :
  15. Anti-holomorphic currents
  16. ...and 5 more sections

Figures (1)

  • Figure 1: The gluons emerging from the amplitude travel on the light-cone and intersect with the sphere at null infinity, or any finite time representation, at points which can be given complex coordinates $z$, for the soft gluon, and $u_i$ for the hard gluons. The soft gluon is then interpreted as the insertion of a current operator at the point $z$ while the $u_i$ denote the insertion points of additional fields.