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

TL;DR

This work addresses the perturbative renormalisation of the non-integer dimensional $oldsymbol{Φ^4_d}$ model with $d<4$ by recoding the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) procedure into an algebra of polynomials in two variables, $X$ and $Y$, representing the fourth and second Wick powers. It constructs a linear Wick map $oldsymbol{W: ext{R}[X] o ext{R}[X,Y]}$ that makes a key diagram commute, implementing renormalisation so that $oldsymbol{W}(e^{- ext{α}X}) = e^{- ext{α}X- ext{β}Y}$ with a mass counterterm $oldsymbol{β(d, extα,N)}$ given by a finite $ extα$-expansion in divergent diagram data, and $oldsymbol{W}(X^n)$ expressed through complete Bell polynomials. The energy renormalisation is captured by a term $oldsymbol{ extγ}$ derived from a twisted antipode construction, leading to a convergent asymptotic expansion for the log-partition function as $N o obreak\infty$. The framework leverages recent multi-index Hopf-algebra results to bridge Feynman-diagram combinatorics and a polynomial representation, offering a compact algebraic approach to perturbative renormalisation in subcritical dimensions with potential implications for stochastic quantisation and SPDE renormalisation.

Abstract

We consider the perturbative renormalisation of the $Φ^4_d$ 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.