Techniques of Distributions in Perturbative Quantum Field Theory (I) Euclidean asymptotic operation for products of singular functions

TL;DR

This paper introduces a new systematic mathematical technique based on distribution theory for analyzing multiloop Feynman diagrams, providing a more powerful alternative to existing methods like BPHZ.

Contribution

It formalizes the asymptotic expansion of products of singular functions in Euclidean Feynman diagrams, offering a complete solution for multiloop diagram analysis.

Findings

Developed a systematic Euclidean asymptotic operation for singular function products.

Provided a complete solution for asymptotic expansions in Euclidean Feynman diagrams.

Enhanced the mathematical toolkit for multiloop quantum field theory calculations.

Abstract

We present a systematic description of the mathematical techniques for studying multiloop Feynman diagrams which constitutes a full-fledged and inherently more powerful alternative to the BPHZ theory. The new techniques emerged as a formalization of the reasoning behind a recent series of record multiloop calculations in perturbative quantum field theory. It is based on a systematic use of the ideas and notions of the distribution theory. We identify the problem of asymptotic expansion of products of singular functions in the sense of distributions as a key problem of the theory of asymptotic expansions of multiloop Feynman diagrams. Its complete solution for the case of Euclidean Feynman diagrams (the so-called Euclidean asymptotic operation for products of singular functions) is explicitly constructed and studied.