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All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology

A. Prygarin, Marcus Spradlin, C. Vergu, Anastasia Volovich

TL;DR

This paper uses symbol-based methods to derive a compact, universal expression for the leading-logarithmic Mandelstam-cut contribution to all two-loop MHV amplitudes in multi-Regge kinematics for any number of particles. By parameterizing MRK in momentum-twistor space and isolating the relevant first-entry terms in the symbol, the authors obtain a simple recursive structure that matches a parallel BFKL computation, strengthening the connection between SYM amplitudes and high-energy QCD. The results demonstrate the power of symbology to access otherwise intractable multi-loop limits and provide predictions applicable to QCD through the planar N=4 SYM correspondence. They also discuss consistency checks and potential beyond-the-symbol contributions, concluding that the primary result captures the essential transcendental content in this regime.

Abstract

Recent progress on scattering amplitudes has benefited from the mathematical technology of symbols for efficiently handling the types of polylogarithm functions which frequently appear in multi-loop computations. The symbol for all two-loop MHV amplitudes in planar SYM theory is known, but explicit analytic formulas for the amplitudes are hard to come by except in special limits where things simplify, such as multi-Regge kinematics. By applying symbology we obtain a formula for the leading behavior of the imaginary part (the Mandelstam cut contribution) of this amplitude in multi-Regge kinematics for any number of gluons. Our result predicts a simple recursive structure which agrees with a direct BFKL computation carried out in a parallel publication.

All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology

TL;DR

This paper uses symbol-based methods to derive a compact, universal expression for the leading-logarithmic Mandelstam-cut contribution to all two-loop MHV amplitudes in multi-Regge kinematics for any number of particles. By parameterizing MRK in momentum-twistor space and isolating the relevant first-entry terms in the symbol, the authors obtain a simple recursive structure that matches a parallel BFKL computation, strengthening the connection between SYM amplitudes and high-energy QCD. The results demonstrate the power of symbology to access otherwise intractable multi-loop limits and provide predictions applicable to QCD through the planar N=4 SYM correspondence. They also discuss consistency checks and potential beyond-the-symbol contributions, concluding that the primary result captures the essential transcendental content in this regime.

Abstract

Recent progress on scattering amplitudes has benefited from the mathematical technology of symbols for efficiently handling the types of polylogarithm functions which frequently appear in multi-loop computations. The symbol for all two-loop MHV amplitudes in planar SYM theory is known, but explicit analytic formulas for the amplitudes are hard to come by except in special limits where things simplify, such as multi-Regge kinematics. By applying symbology we obtain a formula for the leading behavior of the imaginary part (the Mandelstam cut contribution) of this amplitude in multi-Regge kinematics for any number of gluons. Our result predicts a simple recursive structure which agrees with a direct BFKL computation carried out in a parallel publication.

Paper Structure

This paper contains 10 sections, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Invitation: The Six-Particle Amplitude
  3. Outline of the Calculation
  4. A Momentum Twistor Parameterization of Multi-Regge Kinematics
  5. Mandelstam Regions
  6. Results
  7. Consistency Checks
  8. The Main Formula
  9. Beyond-the-Symbol Terms
  10. Expressing Composite Four-Brackets in Terms of $u_{ij}$ Cross-Ratios

Figures (2)

  • Figure 1: A heptagonal light-like Wilson loop projected onto the $(+,-)$-plane. The $j$-th edge vector is the momentum $p_j$ of particle $j$ in the corresponding scattering amplitude. In this region the remainder function vanishes in multi-Regge kinematics.
  • Figure 2: The Wilson loop of Fig. \ref{['fig:Wilson-Regge']} with the edges corresponding to particles $4$, $5$ and $6$ being flipped to a positive energy region. In this region the Mandelstam cut gives a logarithmically divergent contribution to the remainder function in multi-Regge kinematics.