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Soft Theorems in Chern-Simons Matter Theories

Avi Wadhwa

TL;DR

This work analyzes tree-level soft theorems in 4+1 dimensional gauge theories augmented by Chern-Simons terms. By combining Sen's covariantisation framework with explicit Chern-Simons vertices for both Abelian and non-Abelian cases, it shows that the leading soft photon and soft gluon theorems remain unchanged, while the subleading terms receive CS-induced corrections. The corrections arise from the CS 3-point (and higher) vertices and are computed for single soft emissions and generalized to multiple soft emissions, where CS effects remain subleading due to extra momentum factors. The results illustrate that gauge invariance is not strictly necessary for leading soft universality and highlight the distinct role of CS terms in modifying subleading soft behavior, providing explicit expressions in both QED and QCD. The findings have implications for the interplay between gauge invariance, soft theorems, and topological terms in higher-dimensional gauge theories.

Abstract

We describe tree level soft theorems for Chern-Simons QED and QCD in 4+1 dimensions. Soft theorems have been studied in various context in the past. The universal behavior of the leading and subleading soft theorems is understood as a consequence of the local on-shell invariance of the scattering amplitudes. Chern-Simons terms are gauge invariant only up to a boundary term. Hence, they make for an interesting case to study the relationship between gauge invariance and soft theorems. In this paper, we show that the subleading soft theorems in QED and QCD get corrected when Chern-Simons terms are introduced and also derive those corrections. We start with a discussion on soft theorems as a consequence of gauge invariance. Then we move on to derive the Chern-Simons vertices. Finally, we calculate the correction to soft theorems due to the addition of Chern-Simons terms for both QED and QCD.

Soft Theorems in Chern-Simons Matter Theories

TL;DR

This work analyzes tree-level soft theorems in 4+1 dimensional gauge theories augmented by Chern-Simons terms. By combining Sen's covariantisation framework with explicit Chern-Simons vertices for both Abelian and non-Abelian cases, it shows that the leading soft photon and soft gluon theorems remain unchanged, while the subleading terms receive CS-induced corrections. The corrections arise from the CS 3-point (and higher) vertices and are computed for single soft emissions and generalized to multiple soft emissions, where CS effects remain subleading due to extra momentum factors. The results illustrate that gauge invariance is not strictly necessary for leading soft universality and highlight the distinct role of CS terms in modifying subleading soft behavior, providing explicit expressions in both QED and QCD. The findings have implications for the interplay between gauge invariance, soft theorems, and topological terms in higher-dimensional gauge theories.

Abstract

We describe tree level soft theorems for Chern-Simons QED and QCD in 4+1 dimensions. Soft theorems have been studied in various context in the past. The universal behavior of the leading and subleading soft theorems is understood as a consequence of the local on-shell invariance of the scattering amplitudes. Chern-Simons terms are gauge invariant only up to a boundary term. Hence, they make for an interesting case to study the relationship between gauge invariance and soft theorems. In this paper, we show that the subleading soft theorems in QED and QCD get corrected when Chern-Simons terms are introduced and also derive those corrections. We start with a discussion on soft theorems as a consequence of gauge invariance. Then we move on to derive the Chern-Simons vertices. Finally, we calculate the correction to soft theorems due to the addition of Chern-Simons terms for both QED and QCD.

Paper Structure

This paper contains 9 sections, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Subleading Soft Theorem from Sen's Covariantisation Procedure
  3. Chern-Simons Vertices in D=4+1
  4. Abelian Case - Chern-Simons QED
  5. Non-Abelian Case - Chern-Simons QCD
  6. Chern-Simons Correction to Single Soft Photon Theorem in QED
  7. Chern-Simons Correction to Single Soft Gluon Theorem in QCD
  8. Chern-Simons Correction to Soft Theorems with Multiple Simultaneous Soft Emissions
  9. Conclusion

Figures (3)

  • Figure 1: $A1$: Soft boson attaches to the external leg of a finite energy particle via a 1PI vertex $\Gamma^{\left(3\right)}$.
  • Figure 2: $A2$: Soft boson attaches to an internal line.
  • Figure 3: Diagram containing a 5-point Chern-Simons vertex contributing to the subleading soft theorem with four soft emissions