Universal Treatment of Reduction for One-Loop Integrals in Projective Space

TL;DR

This paper introduces a universal method combining Feynman parametrization and embedding formalism in projective space to simplify the reduction of general one-loop integrals, including degenerated cases.

Contribution

It presents a novel, unified approach for reducing one-loop integrals with tensor structures and higher propagator powers using projective space techniques.

Findings

Method provides compact, symmetric reduction formulas.

Applicable to degenerated cases like vanishing Gram determinants.

Enhances understanding of one-loop integral structures.

Abstract

Recently a nice work about the understanding of one-loop integrals has been done in [1] using the tricks of the projective space language associated to their Feynman parametrization. We find this language is also very suitable to deal with the reduction problem of one-loop integrals with general tensor structures as well as propagators with arbitrary higher powers. In this paper, we show that how to combine Feynman parametrization and embedding formalism to give a universal treatment of reductions for general one-loop integrals, even including the degenerated cases, such as the vanishing Gram determinant. Results from this method can be written in a compact and symmetric form.