  Continuum spacetime is expected to emerge via phase transition in discrete
approaches to quantum gravity. A promising example is tensorial group field
theory but its 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 $\textrm{U}(1)$ tensor
fields at arbitrary rank $r$ focusing on a specific class of so-called
cyclic-melonic interactions. Projecting onto constant field configurations, we
obtain the full set of renormalization-group flow equations. At large cut-offs
we find equivalence with $r-1$ dimensional $\textrm{O}(N)$ scalar field theory
in the large-$N$ limit, modified by a tensor-specific, relatively large
anomalous dimension. However, on small length scales there is equivalence with
the corresponding scalar field theory with vanishing dimension and, 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. Thus, group field theories with non-compact
degrees of freedom will be necessary to describe a phase transition to
continuum spacetime.
