# Direct reduction of multiloop multiscale scattering amplitudes

## TL;DR

This paper introduces a series representation method for directly reducing complex multi-loop, multi-scale scattering amplitudes into master integrals, simplifying calculations by avoiding inverse matrix and dimension shift complexities.

## Contribution

The proposed approach simplifies multi-loop amplitude reduction by using series representation and Feynman parameterization, bypassing traditional complex tensor reduction steps.

## Key findings

- Successfully applied to a two-loop W boson production amplitude
- Avoids complicated inverse matrix and dimension shift calculations
- Provides a new efficient framework for multi-loop amplitude reduction

## Abstract

We propose an alternative approach based on series representation to directly reduce multi-loop multi-scale scattering amplitude into set of freely chosen master integrals. And this approach avoid complicated calculations of inverse matrix and dimension shift for tensor reduction calculation. During this procedure we further utilize the Feynman parameterization to calculate the coefficients of series representation and obtain the form factors. Conventional methodologies are used only for scalar vacuum bubble integrals to finalize the result in series representation form. Finally, we elaborate our approach by presenting the reduction of a typical two-loop amplitude for W boson production.