Power-counting theorem for non-local matrix models and renormalisation

TL;DR

This paper develops a power-counting theorem for non-local matrix models using Wilson-Polchinski renormalisation, aiding the understanding of renormalisability in noncommutative field theories.

Contribution

It introduces a power-counting theorem applicable to general matrix models with non-local propagators, linking topological graph data and scaling dimensions to renormalisability.

Findings

Power-counting degree depends on propagator scaling and graph topology.

Large enough scaling dimensions are necessary for renormalisability.

Additional locality conditions from orthogonal polynomials are required for finite couplings.

Abstract

Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation.