The Last of the Finite Loop Amplitudes in QCD

TL;DR

This work computes the last unknown finite one-loop QCD amplitudes with a massless qq̄ pair and n−2 positive-helicity gluons using on-shell recursion, addressing unreal poles unique to complex momenta. It derives two independent primitive amplitudes, A_n^{L-s} and A_n^s, and provides compact all-n formulas for both, validated by factorization checks and cross-checks with QED and Mahlon’s all-plus gluon results. The study also yields a new compact representation for Mahlon’s one-loop n-gluon amplitudes with a single negative helicity, linking quark-based and all-gluon results. These results complete the set of finite loop amplitudes in massless QCD and offer robust consistency checks for recursion-based loop calculations, highlighting unreal poles as a key feature in loop-level analytic structure.

Abstract

We use on-shell recursion relations to determine the one-loop QCD scattering amplitudes with a massless external quark pair and an arbitrary number (n-2) of positive-helicity gluons. These amplitudes are the last of the unknown infrared- and ultraviolet-finite loop amplitudes of QCD. The recursion relations are similar to ones applied at tree level, but contain new non-trivial features corresponding to poles present for complex momentum arguments but absent for real momenta. We present the relations and the compact solutions to them, valid for all n. We also present compact forms for the previously-computed one-loop n-gluon amplitudes with a single negative helicity and the rest positive helicity.

Figures (7)

• Figure 1: In diagram (a) the fermion line (following the arrow) turns left on entering the loop and is an $L$ type primitive amplitude. In diagram (b) the fermion line turns right and is an $R$ type.
• Figure 2: In diagram (a) the external fermion line passes to the "left" of the loop, following the fermion arrow, and is designated an $L$ type. In (b) it passes to the "right" and is an $R$ type. A gluon, fermion or scalar can circulate in the loop. The same decomposition also holds even if we emit additional gluons off the external fermion lines.
• Figure 3: The real-pole diagram in the recursion relation for $A_5^{L-s}(1_{\! f}^+, 2_{\! f}^+,3^+,4^+,5^+)$. The vertices labeled by a $T$ are trees, and the ones labeled by an $L$ are loops.
• Figure 4: The extra unreal pole contribution in the recursion relation for $A_5^{L-s}(1_{\! f}^+, 2_{\! f}^+,3^+,4^+,5^+)$.
• Figure 5: The diagrams corresponding to the terms in the recursion relation in eq. (\ref{['SFullRecurrence']}). In diagram (a) $l$ runs over $\{j+1,j+2, \ldots, n-2\}$. Diagram (c) contains a double pole as well as an unreal pole underneath it.