The rational parts of one-loop QCD amplitudes I: The general formalism

TL;DR

The paper develops a direct, Feynman-integral–based formalism to compute the rational parts of one-loop QCD amplitudes, exploiting tensor reduction, spinor-helicity techniques, and the BDDK theorem. By deriving recursive relations and closed-form expressions for the rational contributions of bubble, triangle, and box integrals (up to two-mass-hard boxes), the authors provide a practical framework to obtain the elusive rational terms without full tensor reductions. This approach complements unitarity-based methods and lays groundwork for efficient, automated calculations of 5- and 6-gluon amplitudes, with extensions to massive loops and potential synergy with other modern formalisms. The results offer concrete, checkable formulas and correction terms essential for accurate NLO predictions in high-multiplicity QCD processes relevant to the LHC and similar colliders.

Abstract

A general formalism for computing only the rational parts of oneloop QCD amplitudes is developed. Starting from the Feynman integral representation of the one-loop amplitude, we use tensor reduction and recursive relations to compute the rational parts directly. Explicit formulas for the rational parts are given for all bubble and triangle integrals. Formulas are also given for box integrals up to two-masshard boxes which are the needed ingredients to compute up to 6-gluon QCD amplitudes. We use this method to compute explicitly the rational parts of the 5- and 6-gluon QCD amplitudes in two accompanying papers.

Figures (8)

• Figure 1: The composition of 3 external particles in tree amplitudes.
• Figure 2: The composition of 4 external particles in tree amplitudes. The blob denotes an expansion as given in Fig. \ref{['One']}. The explicit expression of $\epsilon_{i (i+1)(i+2)(i+3)}$ will not be given.
• Figure 3: The tensor reduction for two adjacent same helicities generally gives three terms.
• Figure 4: For two adjacent same helicities, the tensor reduction for the combination of two diagrams is even simpler by a judicious choice of the reference momenta.
• Figure 5: For three adjacent same helicities, the tensor reduction for the combination of these four diagrams is also quite simple if we choose the reference momenta appropriately.