Counting to one: reducibility of one- and two-loop amplitudes at the integrand level

TL;DR

The paper investigates whether perturbative amplitudes can be reduced at the integrand level for both one- and two-loop diagrams by expressing numerators as linear combinations of denominators. It shows that at one loop, linear-in-loop-momentum coefficients suffice to decompose many topologies in $d=4$, linking these linear terms to the OPP spurious terms. At two loops, decompositions require higher-order terms (quadratic or cubic, depending on dimension) to reach a unitarity-like basis, and the authors provide a systematic, numerical approach to count independent tensor structures and coefficients. The work lays groundwork for a two-loop OPP-like framework and discusses connections to IBP reductions, emphasizing that the resulting integrand basis is not necessarily minimal but can guide efficient reduction strategies across planar and non-planar diagrams.

Abstract

Calculation of amplitudes in perturbative quantum field theory involve large loop integrals. The complexity of those integrals, in combination with the large number of Feynman diagrams, make the calculations very difficult. Reduction methods proved to be very helpful, lowering the number of integrals that need to be actually calculated. Especially, the reduction at the integrand level technique, improves the speed and set-up of these calculations. In this article we demonstrate, by counting the numbers of tensor structures and independent coefficients, how to write such relations at the integrand level for one- and two-loop amplitudes. We clarify their connection to the so-called spurious terms at one loop and discuss their structure in the two-loop case. This method is also applicable to higher loops, and the results obtained apply to both planar and non-planar diagrams.