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IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N=4 SYM Theory

Stephen G. Naculich, Howard J. Schnitzer

TL;DR

The paper derives an all-orders IR-divergent expression for the four-gluon amplitude in ${\cal N}=4$ SYM under the hypothesis that higher-loop soft anomalous dimension matrices are proportional to the one-loop matrix, enabling a compact all-orders description and validating prior conjectures for leading IR poles. It then analyzes the Regge limit, finding planar amplitudes exhibit leading logarithmic growth $\sim [\log(-s/t)]^L$ while subleading-color contributions grow as $\sim [\log(-s/t)]^{L-1}$, and shows 1/$N^2$ corrections alter Regge residues but not the leading planar trajectory at two loops. The study also reveals Regge-type behavior for double-trace amplitudes, including a Regge cut at three loops, and provides an inductive argument that IR-divergent pieces of higher-loop subleading-color amplitudes cannot exceed $\log^{L-1}(-s/t)$ growth. Overall, the work demonstrates an iterative IR structure across color sectors and clarifies nonplanar Regge phenomena in ${\cal N}=4$ SYM, with implications for the understanding of nonplanar amplitudes and their high-energy behavior.

Abstract

We derive a compact all-loop-order expression for the IR-divergent part of the N=4 SYM four-gluon amplitude, which includes both planar and all subleading-color contributions, based on the assumption that the higher-loop soft anomalous dimension matrices are proportional to the one-loop soft anomalous dimension matrix, as has been recently conjectured. We also consider the Regge limit of the four-gluon amplitude, and we present evidence that the leading logarithmic growth of the subleading-color amplitudes is less severe than that of the planar amplitudes. We examine possible 1/N^2 corrections to the gluon Regge trajectory, previously obtained in the planar limit from the BDS ansatz. The double-trace amplitudes have Regge behavior as well, with a nonsense-choosing Regge trajectory and a Regge cut which first emerges at three loops.

IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N=4 SYM Theory

TL;DR

The paper derives an all-orders IR-divergent expression for the four-gluon amplitude in SYM under the hypothesis that higher-loop soft anomalous dimension matrices are proportional to the one-loop matrix, enabling a compact all-orders description and validating prior conjectures for leading IR poles. It then analyzes the Regge limit, finding planar amplitudes exhibit leading logarithmic growth while subleading-color contributions grow as , and shows 1/ corrections alter Regge residues but not the leading planar trajectory at two loops. The study also reveals Regge-type behavior for double-trace amplitudes, including a Regge cut at three loops, and provides an inductive argument that IR-divergent pieces of higher-loop subleading-color amplitudes cannot exceed growth. Overall, the work demonstrates an iterative IR structure across color sectors and clarifies nonplanar Regge phenomena in SYM, with implications for the understanding of nonplanar amplitudes and their high-energy behavior.

Abstract

We derive a compact all-loop-order expression for the IR-divergent part of the N=4 SYM four-gluon amplitude, which includes both planar and all subleading-color contributions, based on the assumption that the higher-loop soft anomalous dimension matrices are proportional to the one-loop soft anomalous dimension matrix, as has been recently conjectured. We also consider the Regge limit of the four-gluon amplitude, and we present evidence that the leading logarithmic growth of the subleading-color amplitudes is less severe than that of the planar amplitudes. We examine possible 1/N^2 corrections to the gluon Regge trajectory, previously obtained in the planar limit from the BDS ansatz. The double-trace amplitudes have Regge behavior as well, with a nonsense-choosing Regge trajectory and a Regge cut which first emerges at three loops.

Paper Structure

This paper contains 19 sections, 113 equations.

Table of Contents

  1. Introduction
  2. ${\cal N} =4$ SYM IR divergences to all loops
  3. IR divergences in the $1/N$ expansion
  4. Leading IR divergence of $A^{(L,k)}$
  5. IR divergences of $A^{(L,L)}$
  6. IR divergences of $A^{(L,1)}$
  7. Regge limit of ${\cal N} =4$ SYM four-gluon amplitudes
  8. Expectations from transcendentality
  9. Regge limit of planar amplitudes
  10. Regge limit of $A^{(1,1)}$
  11. Regge limits of $A^{(2,1)}$ and $A^{(2,2)}$
  12. Regge limit of IR-divergences of higher-loop amplitudes
  13. Regge trajectories
  14. Planar gluon Regge trajectory
  15. $1/N^2$ corrections
  16. ...and 4 more sections