Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory

TL;DR

This paper tests whether algebraic symbol letters are necessary in two-loop NMHV amplitudes in planar N=4 SYM by directly integrating complex topologies, revealing surprising simplifications and rational combinations.

Contribution

It demonstrates the feasibility of testing algebraic symbol letters in two-loop amplitudes through direct integration and reveals unexpected rational simplifications.

Findings

Algebraic symbol letters can combine into rational expressions in specific amplitudes.

Direct integration of complex topologies is feasible with novel techniques.

Significant reduction from millions of terms to thousands in amplitude expressions.

Abstract

It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Recent advances in loop integration technology have made this an `experimentally testable' hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is---surprisingly---rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.