The rational parts of one-loop QCD amplitudes III: The six-gluon case

TL;DR

This paper develops and applies a direct Feynman-integral method to compute the rational parts of six-gluon one-loop QCD amplitudes with scalars in the loop. By combining tensor-reduction techniques with symmetry-based organization, it provides explicit analytic results for two MHV and two NMHV helicity configurations, independently cross-checking against BDKE bootstrap results and factorization tests. The completed rational parts, together with the previously known cut-constructible pieces, yield the full partial helicity amplitudes for the 6-gluon one-loop QCD process and pave the way for automated, higher-multiplicity calculations in NLO phenomenology. The work demonstrates a practical, verifiable pipeline from Feynman integrals to compact analytic expressions suitable for numerical implementation at scale.

Abstract

The rational parts of 6-gluon one-loop amplitudes with scalars circulating in the loop are computed by using the newly developed method for computing the rational parts directly from Feynman integrals. We present the analytic results for the two MHV helicity configurations: $(1^-2^+3^+4^-5^+6^+)$ and $(1^-2^+3^-4^+5^+6^+)$, and the two NMHV helicity configurations: $(1^-2^-3^+4^-5^+6^+)$ and $(1^-2^+3^-4^+5^-6^+)$. Combined with the previously computed results for the cut-constructible part, our results are the last missing pieces for the complete partial helicity amplitudes of the 6-gluon one-loop QCD amplitude.

Figures (38)

• Figure 1: The Feynman rules for sewing trees to loop. The blob denotes an expansion of tree amplitude.
• Figure 2: All the possible one-loop Feynman diagrams for six gluons. The index $i$ can run from 1 to 3 in (d) and (l), and 1 to 6 in the rests if there is an index $i$.
• Figure 3: For two adjacent same helicities, the tensor reduction for the combination of two diagrams is even simpler.
• Figure 4: For three adjacent same helicities, the tensor reduction for the combination of these four diagrams is also quite simple.
• Figure 5: These 4 diagrams are tensor reduced simply with the two pairs of external lines $k_{2,3}$ and $k_{5,6}$ with the same helicities.