Direct Extraction Of One Loop Rational Terms

TL;DR

The work develops a D-dimensional unitarity framework for one-loop amplitudes that directly extracts rational terms missed by four-dimensional cuts. By treating the extra dimensions as a uniform mass, it derives box, triangle, and bubble coefficients from large-mass limits of massive-tree cuts, omitting independent pentagon contributions. The authors provide analytic results for gluon amplitudes up to six external legs and extend the method to amplitudes with external fermions, validating against known results and demonstrating a viable numerical approach via Fourier projections. This approach promises compact, explicit formulas and potential speed gains for computing rational parts in gauge theories, with broad applicability to massive external states and loop fermions.

Abstract

We present a method for the direct extraction of rational contributions to one-loop scattering amplitudes, missed by standard four-dimensional unitarity techniques. We use generalised unitarity in $D=4-2\e$ dimensions to write the loop amplitudes in terms of products of massive tree amplitudes. We find that the rational terms in $4-2\e$ dimensions can be determined from quadruple, triple and double cuts without the need for independent pentagon contributions using a massive integral basis. The additional mass-dependent integral coefficients may then be extracted from the large mass limit which can be performed analytically or numerically. We check the method by computing the rational parts of all gluon helicity amplitudes with up to six external legs. We also present a simple application to amplitudes with external massless fermions.

Figures (6)

• Figure 1: A general quadruple cut with loop momentum flowing clockwise and all external momenta outgoing.
• Figure 2: Momentum conventions for the triple cut; all momenta are outgoing and the loop momentum flows clockwise.
• Figure 3: Pure Bubble and Triangle terms contributing to the bubble coefficients.
• Figure 4: Cut diagrams for the integral coefficients contributing to the rational parts of the $n$-gluon amplitudes.
• Figure 5: Integral basis for the rational part of the leading colour contribution to the $gg\to q\bar{q}$ amplitude, $R_4^{[L]}$.