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 R-d. 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.