Reduction of General One-loop Integrals Using Auxiliary Vector

TL;DR

This paper extends an improved Passarino-Veltman reduction method with auxiliary vectors to handle general one-loop tensor integrals with arbitrary propagator powers, simplifying tensor reduction in quantum field theory calculations.

Contribution

The paper generalizes the auxiliary vector PV reduction method to encompass integrals with arbitrary tensor structures and propagator powers, enhancing analytical reduction capabilities.

Findings

Successfully generalized the reduction method to complex integrals.

Demonstrated the method with multiple examples.

Established the method as a self-contained reduction technique.

Abstract

As a key method to deal with loop integrals, Integration-By-Parts (IBP) method can be used to do reduction as well as establish the differential equations for master integrals. However, when talking about tensor reduction, the Passarino-Veltman (PV) reduction method is also widely used for one-loop integrals. Recently, we have proposed an improved PV reduction method, i.e., the PV reduction method with auxiliary vector $R$, which can easily give analytical reduction results for any tensor rank. However, our results are only for integrals with propagators with power one. In this paper, we generalize our method to one-loop integrals with general tensor structures and propagators with general powers. Our ideas are simple. We solve the generalised reduction problem by combining differentiation over masses and proper limit of reduction with power-one propagators. Finally, we demonstrate our method with several examples. With the result in this paper, we have shown that our improved PV-reduction method with auxiliary vector is a self-completed reduction method for one-loop integrals.