# An analyst's take on the BPHZ theorem

## TL;DR

This paper offers a clear, self-contained mathematical formulation of the BPHZ theorem, providing a systematic way to handle divergent integrals in Euclidean space without relying on quantum field theory background.

## Contribution

It presents a new, self-contained formulation of the BPHZ theorem that clarifies the renormalization process for divergent integrals in a purely analytical setting.

## Key findings

- Provides a systematic renormalization procedure for divergent integrals
- Clarifies the role of arbitrariness in renormalization
- Appeals to an analytically minded audience without quantum field theory background

## Abstract

We provide a self-contained formulation of the BPHZ theorem in the Euclidean context, which yields a systematic procedure to "renormalise" otherwise divergent integrals appearing in generalised convolutions of functions with a singularity of prescribed order at their origin. We hope that the formulation given in this article will appeal to an analytically minded audience and that it will help to clarify to what extent such renormalisations are arbitrary (or not). In particular, we do not assume any background whatsoever in quantum field theory and we stay away from any discussion of the physical context in which such problems typically arise.