Comments on all-loop constraints for scattering amplitudes and Feynman integrals

TL;DR

This paper investigates all-loop constraints on the symbol entries of scattering amplitudes and Feynman integrals in planar ${ m N}=4$ super-Yang-Mills theory, confirming extended Steinmann relations for various integrals and amplitudes, and proposing new constraints for higher-point MHV amplitudes.

Contribution

It demonstrates that extended Steinmann relations hold for a wide class of finite integrals and amplitudes, and introduces a list of last-two-entries for higher-point MHV amplitudes based on $ar{Q}$ equations.

Findings

Extended Steinmann relations hold for various finite integrals and two-loop NMHV amplitudes.

Cancellation observed between rational and algebraic letters in the symbol entries.

Proposed last-two-entries for $n$-point MHV amplitudes to constrain the function space.

Abstract

We comment on the status of "Steinmann-like" constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar ${\cal N}=4$ super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for $n=6,7$ to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any $n$; with suitable normalization such as minimal subtraction, they hold for $n=8$ MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for $n$-point MHV amplitudes derived from $\bar{Q}$ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.