A Two-Loop Four-Point Form Factor at Function Level

TL;DR

This paper lifts a two-loop four-point form factor in planar N=4 super Yang-Mills from symbol to function level, providing explicit iterated representations and verifying symmetries and limits.

Contribution

It introduces a method to obtain the full function from the symbol for the two-loop form factor, including symmetries and physical limits, extending previous symbol-level results.

Findings

Successfully lifted the two-loop form factor to function level.

Confirmed antipodal self-duality at the function level.

Validated the result against known physical limits and symmetries.

Abstract

Recently, the maximally-helicity-violating four-point form factor for the chiral stress-energy tensor in planar $\mathcal{N}=4$ super Yang-Mills was computed to three loops at the level of the symbol associated with multiple polylogarithms. It exhibits {\it antipodal self-duality}, or invariance under the combined action of a kinematic map and reversing the ordering of letters in the symbol. Here we lift the two-loop form factor from symbol level to function level. We provide an iterated representation of the function's derivatives (coproducts). In order to do so, we find a three-parameter limit of the five-parameter phase space where the symbol's letters are all rational. We also use function-level information about dihedral symmetries and the soft, collinear, and factorization limits, as well as limits governed by the form-factor operator product expansion (FFOPE). We provide plots of the remainder function on several kinematic slices, and show that the result is compatible with the FFOPE data. We further verify that antipodal self-duality is valid at two loops beyond the level of the symbol.