The Singular Behaviour of QCD Amplitudes at Two-loop Order

TL;DR

This paper establishes a universal colour-space factorization framework for infrared singularities in on-shell QCD amplitudes at two-loop order. It derives a two-loop insertion operator structure, $I^{(1)}$ and $I^{(2)}_{\mathrm{RS}}$, plus a remainder $H^{(2)}$ that yields all $\mathcal{O}(1/\epsilon)$ terms, and provides explicit coefficients for the leading poles $1/\epsilon^4$, $1/\epsilon^3$, and $1/\epsilon^2$ that are RS-independent. The results are demonstrated in concrete cases: amplitudes with two partons, with a $q\bar{q}$ pair in CDR, and with three partons including $q\bar{q}g$, with explicit expressions for the two-loop $qq$ and $qqg$ singular structures and for $e^+e^- \to 3$ jets at NNLO. These findings offer a framework to check two-loop calculations and to organize NNLO jet computations by isolating divergent, analytically integrable components from finite remainders.

Abstract

We discuss the structure of infrared singularities in on-shell QCD amplitudes at two-loop order. We present a general factorization formula that controls all the $\ep$-poles of the dimensionally regularized amplitudes. The dependence on the regularization scheme is considered and the coefficients of the $1/\ep^4, 1/\ep^3$ and $1/\ep^2$ poles are explicitly given in the most general case. The remaining single-pole contributions are also explicitly evaluated in the case of amplitudes with a $q{\bar q}$ pair.