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Single-valued harmonic polylogarithms and the multi-Regge limit

Lance J. Dixon, Claude Duhr, Jeffrey Pennington

TL;DR

This work identifies Brown's single-valued harmonic polylogarithms as the natural function space for the multi-Regge limit of six-gluon scattering in planar ${\mathcal{N}}=4$ SYM and develops a Fourier-Mellin dictionary to translate between the $(w,w^*)$ and $(\nu,n)$ representations. It achieves all-order LL and NLL predictions for the six-point MHV remainder in MRK up to weight 10, and extends to NMHV via an operator, leveraging the SVHPL basis to simplify the extraction of BFKL eigenvalues and impact factors. By combining SVHPL techniques with symbol constraints from the four-loop remainder, the authors determine NNLLA and N$^3$LLA corrections, including explicit analytic forms in $(\nu,n)$ space with a small set of beyond-the-symbol constants, and they show these results can be expressed solely through polygamma derivatives and simple building blocks. The methodology promises broader applicability to higher-loop amplitudes, NMHV extensions, and potential QCD adaptations, while highlighting open questions about symbol-only ambiguities and exact fixed constants.

Abstract

We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar N=4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w,w*). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν,n) which are related to (w,w*) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν,n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to NNNLLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

Single-valued harmonic polylogarithms and the multi-Regge limit

TL;DR

This work identifies Brown's single-valued harmonic polylogarithms as the natural function space for the multi-Regge limit of six-gluon scattering in planar SYM and develops a Fourier-Mellin dictionary to translate between the and representations. It achieves all-order LL and NLL predictions for the six-point MHV remainder in MRK up to weight 10, and extends to NMHV via an operator, leveraging the SVHPL basis to simplify the extraction of BFKL eigenvalues and impact factors. By combining SVHPL techniques with symbol constraints from the four-loop remainder, the authors determine NNLLA and NLLA corrections, including explicit analytic forms in space with a small set of beyond-the-symbol constants, and they show these results can be expressed solely through polygamma derivatives and simple building blocks. The methodology promises broader applicability to higher-loop amplitudes, NMHV extensions, and potential QCD adaptations, while highlighting open questions about symbol-only ambiguities and exact fixed constants.

Abstract

We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar N=4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w,w*). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν,n) which are related to (w,w*) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν,n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to NNNLLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

Paper Structure

This paper contains 45 sections, 195 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The six-point remainder function in the multi-Regge limit
  3. Harmonic polylogarithms and their single-valued analogues
  4. Review of harmonic polylogarithms
  5. Single-valued harmonic polylogarithms
  6. Explicit construction
  7. The six-point remainder function in LLA and NLLA
  8. The six-point NMHV amplitude in MRK
  9. Single-valued HPLs and Fourier-Mellin transforms
  10. The multi-Regge limit in $(\nu,n)$ space
  11. Symmetries in $(\nu,n)$ space
  12. General construction
  13. Examples
  14. Applications in $(\nu,n)$ space: the BFKL eigenvalues and impact factor
  15. The impact factor at NNLLA
  16. ...and 30 more sections

Figures (2)

  • Figure 1: Imaginary parts $g_{\ell-1}^{(\ell)}$ of the MHV remainder function in MRK and LLA through 10 loops, on the line segment with $w={w^\ast}$ running from 0 to 1. The functions have been rescaled by powers of 4 so that they are all roughly the same size.
  • Figure 2: Imaginary parts $g_{\ell-2}^{(\ell)}$ of the MHV remainder function in MRK and NLLA through 9 loops.