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The six-point remainder function to all loop orders in the multi-Regge limit

Jeffrey Pennington

TL;DR

The paper derives an all-orders leading-logarithmic formula for the six-point remainder function in multi-Regge kinematics in planar ${ m N}=4$ SYM, expressed in the basis of single-valued harmonic polylogarithms. It confirms consistency with independent Lipatov-Prygarin integral results up to at least 14 loops and establishes a differential equation linking MHV and NMHV configurations, all within SVHPL space. The authors also analyze the near-collinear limit, deriving all-orders expansions organized by log|w| with leading DLLA given by modified Bessel functions, and provide explicit subleading terms. Although a formal proof is not given, extensive cross-checks and the absence of unwanted multiple zeta values bolster the validity of the all-orders construction and suggest rich connections to OPE and strong-coupling analyses.

Abstract

We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric N=4 Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem---the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w*, which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w|->0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log|w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log|w| is given in terms of modified Bessel functions.

The six-point remainder function to all loop orders in the multi-Regge limit

TL;DR

The paper derives an all-orders leading-logarithmic formula for the six-point remainder function in multi-Regge kinematics in planar SYM, expressed in the basis of single-valued harmonic polylogarithms. It confirms consistency with independent Lipatov-Prygarin integral results up to at least 14 loops and establishes a differential equation linking MHV and NMHV configurations, all within SVHPL space. The authors also analyze the near-collinear limit, deriving all-orders expansions organized by log|w| with leading DLLA given by modified Bessel functions, and provide explicit subleading terms. Although a formal proof is not given, extensive cross-checks and the absence of unwanted multiple zeta values bolster the validity of the all-orders construction and suggest rich connections to OPE and strong-coupling analyses.

Abstract

We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric N=4 Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem---the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w*, which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w|->0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log|w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log|w| is given in terms of modified Bessel functions.

Paper Structure

This paper contains 11 sections, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The six-point remainder function in multi-Regge kinematics
  3. Review of single-valued harmonic polylogarithms
  4. Six-point remainder function in MRK and LLA
  5. The all-orders formula
  6. Consistency of the MHV and NMHV formulas
  7. Collinear limit
  8. MHV
  9. NMHV
  10. The real part of the MHV remainder function in NLLA
  11. Conclusions

Figures (1)

  • Figure 1: The MHV remainder function in the near-collinear limit of the LL approximation of MRK. It has been rescaled by an exponential damping factor. See eq. \ref{['eq:damping']}.