On gauge amplitudes first appearing at two loops

TL;DR

This paper investigates two-loop scattering amplitudes in a specific massless non-abelian gauge theory with positive helicity gluons, revealing their non-rational nature in dimensional regularization and confirming the celestial chiral algebra bootstrap results.

Contribution

It demonstrates the non-rationality and divergence of two-loop amplitudes in this theory and extends the celestial chiral algebra bootstrap to double-trace terms at two loops.

Findings

Two-loop amplitudes are not rational functions in dimensional regularization.

Finite remainders for four gluons match Costello's formula.

Mass regulator allows exact recovery of CCA bootstrap results.

Abstract

We study scattering amplitudes in massless non-abelian gauge theory where all outgoing gluons have positive helicity. It has been argued recently by Costello that for a particular fermion representation (8 fundamentals plus one antisymmetric-tensor representation in $SU(N)$) the one-loop amplitudes vanish identically. We show that this vanishing leads to previously-observed identities among one-loop color-ordered partial amplitudes. We then turn to two loops, where Costello has computed the all-plus amplitudes for this theory, as rational functions of the kinematics for any number of gluons using the celestial chiral algebra (CCA) bootstrap. We show that in dimensional regularization, these two-loop amplitudes are not rational, and they are not even finite as $\epsilon\to0$. However, the finite remainder for four gluons agrees with the formula by Costello. In addition, we provide a mass regulator for the infrared-divergent loop integrals; with this regulator, the CCA bootstrap formula is recovered exactly. Finally, we use the CCA bootstrap to compute the double-trace terms in the theory at two loops for an arbitrary number of gluons.