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

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 three different scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. The main application is the renormalisation problem of field theories on noncommutative R^D written in matrix formulation. It turns out that the propagator for the real scalar field has anomalous scaling dimensions, which for D>2 result in arbitrarily high power-counting degrees of divergence. This feature is known as UV/IR-mixing, which we conclude to emerge in any non-local matrix model with anomalous scaling dimensions of the propagator. Models in which the propagator has regular scaling dimensions are for D=2,4 power-counting renormalisable but acquire due to non-locality an infinite number of relevant or marginal interactions. By a reduction-of-couplings mechanism it is possible to express all interactions in terms of a finite number of base interactions. In this way, a regular propagator with respect to a given base determines via the renormalisation flow the renormalisable interaction, which will rarely coincide with the expression obtained by inserting *'s into classical formulae