Perturbative renormalisation of the $\Phi^4_{4-\varepsilon}$ model via generalized Wick maps

TL;DR

This paper introduces a simplified algebraic approach to perturbative renormalisation of the $\,\Phi^4_{d}$ model in non-integer dimensions, replacing complex diagram combinatorics with polynomial algebra and Bell polynomials.

Contribution

It presents a novel algebraic encoding of BPHZ renormalisation using polynomials in two variables, simplifying the combinatorial complexity of the process.

Findings

Renormalisation expressed as a Wick map on polynomials.

Uses Bell polynomials to encode renormalisation operations.

Applicable to non-integer dimensions $d<4$.

Abstract

We consider the perturbative renormalisation of the $\Phi^4_d$ model from Euclidean Quantum Field Theory for any, possibly non-integer dimension $d<4$. The so-called BPHZ renormalisation, named after Bogoliubov, Parasiuk, Hepp and Zimmermann, is usually encoded into extraction-contraction operations on Feynman diagrams, which have a complicated combinatorics. We show that the same procedure can be encoded in the much simpler algebra of polynomials in two unknowns $X$ and $Y$, which represent the fourth and second Wick power of the field. In this setting, renormalisation takes the form of a \lq\lq Wick map\rq\rq\ which maps monomials into Bell polynomials. The construction makes use of recent results by Bruned and Hou on multiindices, which are algebraic objects of intermediate complexity between Feynman diagrams and polynomials.