# Scalar one-loop four-point Feynman integrals with complex internal masses

## TL;DR

This paper derives analytic formulas for scalar one-loop four-point Feynman integrals with complex internal masses, extending previous methods and enabling direct tensor integral evaluation, with implementation and validation in a software package.

## Contribution

It introduces a new method for calculating scalar and tensor one-loop four-point integrals with complex masses, overcoming limitations of traditional approaches.

## Key findings

- Analytic formulas valid for complex and real masses.
- Successful implementation in the ONELOOP4PT.CPP package.
- Perfect agreement with LoopTools in numerical tests.

## Abstract

Based on the method in Refs.~{\tt [D.~Kreimer, Z.\ Phys.\ C {\bf 54} (1992) 667} and {\tt Int.\ J.\ Mod.\ Phys.\ A {\bf 8} (1993) 1797]}, we present analytic results for scalar one-loop four-point Feynman integrals with complex internal masses. The results are not only valid for complex internal masses, but also for real internal mass cases. Different from the traditional approach proposed by G. 't Hooft and M. Veltman in the paper {\tt[Nucl.\ Phys.\ B {\bf 153} (1979) 365]}, this method can be extended to evaluate tensor integrals directly. Therefore, it may open a new approach to cure the inverse Gram determinant problem analytically. We then implement the results into a computer package which is {\tt ONELOOP4PT.CPP}. In numerical checks, one compares the program to {\tt LoopTools version} $2.12$ in both real and complex mass cases. We find a perfect agreement between the results generated from this work and {\tt LoopTools}.