# Reduction of Feynman Integrals in the Parametric Representation

## TL;DR

This paper presents a parametric method for reducing Feynman integrals that is more efficient than traditional IBP methods, simplifies tensor integral parametrization, and enhances symmetry transparency for automated calculations.

## Contribution

It introduces a parametric reduction technique for Feynman integrals that improves efficiency and simplifies tensor and symmetry analysis compared to existing methods.

## Key findings

- Parametric method outperforms IBP in efficiency.
- Tensor integrals can be parametrized directly without tensor reduction.
- Symmetries are more transparent in the parametric representation.

## Abstract

In this paper, the reduction of Feynman integrals in the parametric representation is considered. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum space. Tensor integrals can directly be parametrized without performing tensor reductions. The integrands of parametric integrals are functions of Lorentz scalars, instead of four momenta. The complexity of a calculation is determined by the number of propagators that are present rather than the number of all the linearly independent propagators. Furthermore, the symmetries of Feynman integrals under permutations of indices are transparent in the parametric representation. Since all the indices of the propagators are nonnegative, an algorithm to solve those identities can easily be developed, which can be used for automatic calculations.