# Functional reduction of Feynman integrals

## TL;DR

The paper introduces a novel method for reducing complex Feynman integrals with multiple variables to simpler forms using functional equations, enabling explicit analytic solutions in terms of hypergeometric functions.

## Contribution

It presents a new iterative functional equation approach for reducing multi-variable Feynman integrals to fewer variables with explicit analytic expressions.

## Key findings

- Reduced one-loop scalar triangle integrals to two-variable sums.
- Expressed one-loop box integrals with six variables as sums of terms with three variables.
- Derived analytic formulas for integrals using hypergeometric functions.

## Abstract

A method for reducing Feynman integrals, depending on several kinematic variables and masses, to a combination of integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle and box integrals with massless internal propagators to simpler integrals is described in detail. The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the $_2F_1$ Gauss hypergeometric function and the logarithmic function was derived. By using the functional equations, the one-loop box integral with massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the $F_1$ Appell and $_2F_1$ Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension $d$. The reduction equations for the box integral with some kinematic variables equal to zero are considered.