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Some analytic results for two-loop scattering amplitudes

Luis F. Alday

TL;DR

The paper establishes analytic results for the finite double-pentagon contributions to the two-loop eight-point MHV amplitude in planar N=4 SYM within restricted kinematics governed by two cross-ratios χ^±. By leveraging momentum-twistor Grassmannian integrals, differential operators, and the symbol/motive framework, it derives compact, transcendentality-four expressions that factorize into products f(χ^+)g(χ^-) after summing over cyclic permutations. This approach tightly constrains the functional form, avoids large intermediate expressions, and reveals deep structures akin to Wilson-loop duality. The work suggests powerful avenues for extending to the full amplitude and for connecting perturbative results with integrable-system ideas and OPE constraints.

Abstract

We present analytic results for the finite diagrams contributing to the two-loop eight-point MHV scattering amplitude of planar N=4 SYM. We use a recently proposed representation for the integrand of the amplitude in terms of (momentum) twistors and focus on a restricted kinematics in which the answer depends only on two independent cross-ratios. The theory of motives can be used to vastly simplify the results, which can be expressed as simple combinations of classical polylogarithms.

Some analytic results for two-loop scattering amplitudes

TL;DR

The paper establishes analytic results for the finite double-pentagon contributions to the two-loop eight-point MHV amplitude in planar N=4 SYM within restricted kinematics governed by two cross-ratios χ^±. By leveraging momentum-twistor Grassmannian integrals, differential operators, and the symbol/motive framework, it derives compact, transcendentality-four expressions that factorize into products f(χ^+)g(χ^-) after summing over cyclic permutations. This approach tightly constrains the functional form, avoids large intermediate expressions, and reveals deep structures akin to Wilson-loop duality. The work suggests powerful avenues for extending to the full amplitude and for connecting perturbative results with integrable-system ideas and OPE constraints.

Abstract

We present analytic results for the finite diagrams contributing to the two-loop eight-point MHV scattering amplitude of planar N=4 SYM. We use a recently proposed representation for the integrand of the amplitude in terms of (momentum) twistors and focus on a restricted kinematics in which the answer depends only on two independent cross-ratios. The theory of motives can be used to vastly simplify the results, which can be expressed as simple combinations of classical polylogarithms.

Paper Structure

This paper contains 14 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Amplitudes as a Grasmannian integral and restricted kinematics
  3. Restricted kinematics
  4. Scattering of eight-particles
  5. Double-pentagon contribution to the two-loop eight point amplitude in restricted kinematics
  6. Diagram of type $(b)$
  7. Diagram of type $(c)$
  8. Combining everything
  9. Conclusions
  10. Some details on the explicit computations
  11. One loop example
  12. Two-loop double-pentagon example
  13. Symbol of transcendental functions
  14. Result for $\partial_{\chi^+}I^{(c)}$

Figures (2)

  • Figure 1: Different kind of double pentagon diagrams contributing to the scattering of eight particles in the restricted kinematics. There are other two contributions that are the reflections of diagrams $(a)$ and $(c)$.
  • Figure :