# The Method of Arbitrarily Large Moments to Calculate Single Scale Processes in Quantum Field Theory

## TL;DR

This paper introduces a novel method leveraging differential and difference equations to efficiently compute Mellin moments of single scale quantities in quantum field theory, enabling precise numerical and potential analytic reconstructions.

## Contribution

The paper presents a new approach to calculate large sets of Mellin moments using differential/difference equations derived from integration-by-parts identities, simplifying complex quantum calculations.

## Key findings

- Allows calculation of many Mellin moments efficiently
- Enables analytic reconstruction of physical quantities
- Provides highly precise numerical representations

## Abstract

We device a new method to calculate a large number of Mellin moments of single scale quantities using the systems of differential and/or difference equations obtained by integration-by-parts identities between the corresponding Feynman integrals of loop corrections to physical quantities. These scalar quantities have a much simpler mathematical structure than the complete quantity. A sufficiently large set of moments may even allow the analytic reconstruction of the whole quantity considered, holding in case of first order factorizing systems. In any case, one may derive highly precise numerical representations in general using this method, which is otherwise completely analytic.