# Combinatorics in tensor integral reduction

## TL;DR

This paper presents a rigorous combinatorial method for expressing symmetric isotropic tensors in any dimension, simplifying angular integral calculations in physics, especially in gauge theories and particle physics.

## Contribution

It introduces a systematic approach to tensor integral reduction using symmetry and combinatorics, applicable in Euclidean and Minkowski spaces, aiding complex physics calculations.

## Key findings

- Reduces angular integral computations to combinatorial counts
- Generalizes tensor reduction methods to subspace symmetries
- Applicable to regularized gauge-field theory calculations in dimensional spaces

## Abstract

We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the $n$-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use of the symmetries, one can greatly reduce the efforts to compute cumbersome angular integrals into straightforward combinatoric counts. This method is generalized into the cases in which such symmetries are present in subspaces. We further demonstrate the mechanism of the tensor-integral reduction that is widely used in various physics problems such as perturbative calculations of the gauge-field theory in which divergent integrals are regularized in $d=4-2\epsilon$ space-time dimensions. The main derivation is given in the $n$-dimensional Euclidean space. The generalization of the result to the Minkowski space is also discussed in order to provide graduate students and researchers with techniques of tensor-integral reduction for particle physics problems.