(No) phase transition in tensorial group field theory

Pithis AG, Thürigen J

Abstract

Continuum spacetime is expected to emerge via phase transition in discrete approaches to quantum gravity. A promising example are tensorial and group field theories but their phase diagram remains an open issue. The results of recent attempts in terms of the functional renormalization group method remain inconclusive since they are restricted to truncations of low order. We overcome this barrier with a local-potential approximation for U(1) tensor fields at arbitrary rank r focusing on a specific class of so-called cyclic-melonic interactions. Projecting on constant field configurations, we obtain the full set of renormalization-group flow equations. At large scales we find equivalence with r-1 dimensional O(N) scalar field theory in the large-N limit, modified by a tensor-specific, relatively large anomalous dimension. As a consequence, in the large-volume limit there is a non-Gaussian fixed point with special non-integer critical exponents for rank r=4 but not else. However, on small scales we find equivalence with the corresponding scalar field theory with vanishing dimension, thus no phase transition. This is confirmed by numerical analysis of the full non-autonomous equations where we always find symmetry restoration. The essential reason for this effect are isolated zero modes. This result should therefore be true for tensor field theories on any compact domain and including any tensor-invariant interactions. As a consequence, we expect that group field theory on a non-compact group will be necessary to describe gravitational degrees of freedom in a quantum regime together with a phase transition to continuum spacetime.