The rational parts of one-loop QCD amplitudes II: The five-gluon case

TL;DR

The paper addresses the challenge of computing the rational parts of five-gluon one-loop QCD amplitudes. It advances a direct-from-Feynman-integrals approach, employing tensor reduction with spinor-helicity techniques and supersymmetric decomposition as a framework. The authors derive explicit rational parts for the two independent MHV helicity configurations and show agreement with the Bern, Dixon and Kosower results, up to conventions between real versus complex scalar loops, via the relation $R = \tfrac{1}{2} \tilde{R}$. These results demonstrate the method’s efficiency and lay groundwork for completing the remaining six-gluon rational parts (xyziii).

Abstract

The rational parts of 5-gluon one-loop amplitudes are computed by using the newly developed method for computing the rational parts directly from Feynman integrals. We found complete agreement with the previously well-known results of Bern, Dixon and Kosower obtained by using the string theory method. Intermediate results for some combinations of Feynman diagrams are presented in order to show the efficiency of the method and the local cancellation between different contributions.

Figures (16)

• Figure 1: The Feynman rules for sewing trees to loop. The blob denotes an expansion of the tree amplitude.
• Figure 2: All the possible one-loop Feynman diagrams for 5 gluons. The index $i$ runs from 1 to 5 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: A combination of 2 diagrams with the same adjacent helicity.
• Figure 5: The first diagram is the only 5-point diagram. Its combination with the pinched $k_{1,2} \to k_{12}$ 4-point diagram leads to triangle diagrams only by making use of the reduction formula (\ref{['eqreductiona']}).