Analytic all-plus-helicity gluon amplitudes in QCD

TL;DR

This work tackles the challenge of obtaining analytic, all-plus leading-color two-loop gluon amplitudes in pure Yang-Mills theory for six- and seven-point external legs. It combines four-dimensional unitarity to construct the cut-constructible piece P^{(2)}_n with augmented recursion, including axial-gauge analysis and Risager shifts, to determine the purely rational piece R^{(2)}_n. The authors present compact analytic expressions for R_6^{(2)} and R_7^{(2)}, built from carefully organized factorisations and pole-residue structures, and validate them through cross-checks and symmetry properties. The results extend the frontier of analytic two-loop amplitudes beyond five points for this helicity configuration and establish a robust framework for tackling other helicity sectors in non-supersymmetric theories.

Abstract

We detail the calculation of two-loop all-plus helicity amplitudes for pure Yang-Mills theory. The four dimensional unitarity methods and augmented recursion techniques we have developed, together with a knowledge of the singular structure of the amplitudes allow us to compute these in compact analytic forms. Specifically we present the computation and analytic results for the six- and seven-gluon leading colour two-loop amplitudes these being the first QCD two-loop amplitudes beyond five-points.

Figures (4)

• Figure 1: The non-vanishing quadruple cut. $A$ is a MHV tree amplitude whereas $B$ is a one-loop all-plus amplitude. The set $K_2$ may consist of a single leg but the set $K_4$ must contain at least two legs. The integral function depends upon $S\equiv (k_{i-1}+K_2)^2$ and $T\equiv (K_2+k_{i+r})^2$.
• Figure 2: The origin of the double pole. The double pole corresponds to the coincidence of the singularity arising in the 3-pt integral with the factorisation corresponding to $K^2=s_{ab}\to 0$.
• Figure 3: Diagram containing the leading and sub-leading poles as $s_{ab}\to 0$. The axial gauge construction permits the off-shell continuation of the internal legs. The two internal helicity configurations must be summed over to obtain the complete contribution to the $1/s_{ab}$ residue. The one-loop current $\tau_n^{(1)}$ can be built from the on-shell $n$-point one-loop single minus amplitude.
• Figure 4: The factorisations that $\tau_n^{(1)}$ must reproduce.