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MHV amplitude for 3->3 gluon scattering in Regge limit

J. Bartels, L. N. Lipatov, A. Prygarin

TL;DR

The paper tackles the breakdown of the BDS ansatz for the six-gluon planar MHV amplitude in the Mandelstam channel for the 3→3 transition in the multi-Regge limit. It analytically continues the two-loop remainder $R_6^{(2)}$ (GSVV) to the Mandelstam region and uses all-loop dispersion relations to extract leading and subleading terms, including an all-loop prediction for the real part of the remainder multiplied by the BDS phase. At three loops, it provides the leading-logarithmic imaginary part $R_6^{(3)\; LLA}$ and the real part at next-to-leading order via $\Re(R_6^{(3)\; NLLA}) = (i\pi \delta / a) R_6^{(2)\; LLA}$, and shows that an all-loop relation $\Re(R_{6} e^{-i\pi \delta}) = \cos \pi \omega_{ab}$ holds, with $\delta$ and $\omega_{ab}$ tied to the cusp anomalous dimension. The results reinforce Regge-factorization between the $2\to4$ and $3\to3$ channels and yield concrete all-loop predictions, potentially testable at strong coupling, while highlighting subtleties from oscillatory behavior at large coupling. Overall, the work deepens understanding of the analytic structure of six-gluon amplitudes in the Regge regime and clarifies the role of Mandelstam cuts in the remainder function.

Abstract

We calculate corrections to the BDS formula for the six-particle planar MHV amplitude for the gluon transition 3->3 in the multi-Regge kinematics for the physical region, in which the Regge pole ansatz is not valid. The remainder function at two loops is obtained by an analytic continuation of the expression derived by Goncharov, Spradlin, Vergu and Volovich to the kinematic region described by the Mandelstam singularity exchange in the crossing channel. It contains both the imaginary and real contributions being in agreement with the BFKL predictions. The real part of the three loop expression is found from a dispersion-like all-loop formula for the remainder function in the multi-Regge kinematics derived by one of the authors. We also make a prediction for the all-loop real part of the remainder function multiplied by the BDS phase, which can be accessible through calculations in the regime of the strong coupling constant.

MHV amplitude for 3->3 gluon scattering in Regge limit

TL;DR

The paper tackles the breakdown of the BDS ansatz for the six-gluon planar MHV amplitude in the Mandelstam channel for the 3→3 transition in the multi-Regge limit. It analytically continues the two-loop remainder (GSVV) to the Mandelstam region and uses all-loop dispersion relations to extract leading and subleading terms, including an all-loop prediction for the real part of the remainder multiplied by the BDS phase. At three loops, it provides the leading-logarithmic imaginary part and the real part at next-to-leading order via , and shows that an all-loop relation holds, with and tied to the cusp anomalous dimension. The results reinforce Regge-factorization between the and channels and yield concrete all-loop predictions, potentially testable at strong coupling, while highlighting subtleties from oscillatory behavior at large coupling. Overall, the work deepens understanding of the analytic structure of six-gluon amplitudes in the Regge regime and clarifies the role of Mandelstam cuts in the remainder function.

Abstract

We calculate corrections to the BDS formula for the six-particle planar MHV amplitude for the gluon transition 3->3 in the multi-Regge kinematics for the physical region, in which the Regge pole ansatz is not valid. The remainder function at two loops is obtained by an analytic continuation of the expression derived by Goncharov, Spradlin, Vergu and Volovich to the kinematic region described by the Mandelstam singularity exchange in the crossing channel. It contains both the imaginary and real contributions being in agreement with the BFKL predictions. The real part of the three loop expression is found from a dispersion-like all-loop formula for the remainder function in the multi-Regge kinematics derived by one of the authors. We also make a prediction for the all-loop real part of the remainder function multiplied by the BDS phase, which can be accessible through calculations in the regime of the strong coupling constant.

Paper Structure

This paper contains 4 sections, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Analytic continuation
  3. $3 \to 3$ remainder function at three loops
  4. Discussion

Figures (4)

  • Figure 1: The $3 \to 3$ gluon scattering amplitude.
  • Figure 2: The $3 \to 3$ gluon scattering amplitude in the Mandelstam channel given by $s_1, s_3, s_{13}, s_{02} <0\;\;\; \text{and} \;\;\; s, t^{'}_2>0$.
  • Figure 3: The $3 \to 3$ gluon scattering amplitude with discontinuity in $t^{'}_2$ channel.
  • Figure 4: The $3 \to 3$ gluon scattering amplitude for the Mandelstam channel, where $t_2', s, s_1, s_3 < 0$ and $s_{13}, s_{02} >0$.