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

Abstract

We consider the perturbative renormalisation of the ${\Phi }_d^4$ 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 “Wick map” that maps monomials into Bell polynomials. The construction makes use of recent results by Bruned and Hou on multi-indices, which are algebraic objects of intermediate complexity between Feynman diagrams and polynomials.