QFT with tensorial and local degrees of freedom: Phase structure from functional renormalization

Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the r-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on Rd. We find that the effective r − 1 dimensions of the torus part dynamically vanish along the renormalization group flow while the d local dimensions persist up to small momentum scales. Consequently, for d > 2 one can find a phase structure allowing also for phase transitions.