Renormalization of an Abelian Tensor Group Field Theory: Solution at Leading Order

TL;DR

This paper analyzes a rank-six Abelian tensor group field theory, establishing its renormalization, deriving flow equations, and proving the existence of a unique solution at leading order.

Contribution

It introduces a formalism for the intermediate field, characterizes leading order graphs, and solves the model at leading order with rigorous bounds.

Findings

Renormalization group flow computed for the model

Closed equations derived for two and four point functions

Existence of a unique solution at small coupling proven

Abstract

We study a just renormalizable tensorial group field theory of rank six with quartic melonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and write its expansion in terms of effective couplings. We then establish closed equations for the two point and four point functions at leading (melonic) order. Using the effective expansion and its uniform exponential bounds we prove that these equations admit a unique solution at small renormalized coupling.