The two-loop n-point all-plus helicity amplitude

TL;DR

This work targets the explicit analytic form of the polylogarithmic part of the $n$-point, two-loop all-plus helicity amplitude in gauge theory. It introduces a compact ansatz where the finite remainder is $F_n^{(2)}=P_n^{(2)}+R_n^{(2)}$, with $P_n^{(2)}$ built from dilogarithm-based box-function building blocks $F^{2m}$ and constrained by IR/UV consistency, collinear limits, and four-dimensional unitarity. Four-dimensional unitarity and multi-cut techniques fix the polylogarithmic content while ensuring the correct singular structure, though the rational part for general $n$ remains unresolved. The result provides a compact analytic framework that reproduces known limits and hints at underlying structures that could extend to broader multi-loop amplitudes.

Abstract

We propose a compact analytic expression for the polylogarithmic part of the n-point two-loop all-plus helicity amplitude in gauge theory.

Figures (6)

• Figure 1: Diagrammatic representation of the functions $F^{2m}_{n:r,i}$. The functions are symmetric between $K_2$ and $K_4$ but their coefficients are not. The summation is over all such functions including the case when $K_2$ is a single leg ($r=1$) but leg $K_4$ must contain at least two legs (indicated by a solid disc).
• Figure 2: Pictorial representation of the identity amongst the $F$-functions..
• Figure 3: The collinear limit of these two $F$-functions.
• Figure 4: Functions whose coefficients vanish in the collinear limit
• Figure 5: The non-vanishing quadruple cut. $A$ is a MHV tree amplitude whereas $B$ is a one-loop all-plus amplitude.