Properties of the $\epsilon$-Expansion, Lagrange Inversion and Associahedra and the $O(1)$ Model

TL;DR

This paper explores the mathematical structure of the epsilon-expansion in quantum field theory, linking it to combinatorics and geometry, and computes critical exponents for the O(1) model up to seventh order in epsilon.

Contribution

It provides exact formulas for the Wilson-Fisher fixed point coupling and anomalous dimensions using Lagrange inversion and associahedra, extending calculations to high order in epsilon.

Findings

Exact epsilon-expansion expressions for fixed point coupling.

Connection between epsilon-expansion and associahedra geometry.

Computed critical exponents for O(1) model up to epsilon^7.

Abstract

We discuss properties of the $\epsilon$-expansion in $d=4-\epsilon$ dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in $\epsilon$ in terms of the beta function coefficients. The $\epsilon$-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the $\epsilon$-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in $\epsilon$ in terms of the coefficients of the beta function and anomalous dimensions. We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar $O(1)$ symmetric model to $O(\epsilon^7)$.