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

TL;DR

This paper explores the reduction of one- and two-loop quantum field theory amplitudes at the integrand level, improving computational efficiency by analyzing tensor structures and coefficients, applicable to complex diagrams.

Contribution

It introduces a method to write integrand-level relations for one- and two-loop amplitudes, clarifying their structure and connection to spurious terms, applicable to higher loops.

Findings

Counted tensor structures and independent coefficients for amplitude relations.

Clarified the connection between integrand relations and spurious terms.

Applicable to both planar and non-planar diagrams at multiple loops.

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.