  We continue the series of articles on the application of Landau-Ginzburg
mean-field theory to unveil the basic phase structure of tensorial field
theories which are characterized by combinatorially non-local interactions.
Among others, this class covers tensor field theories (TFT) which lead to a new
class of conformal field theories highly relevant for investigations on the
AdS/CFT conjecture. Moreover, it also encompasses models within the tensorial
group field theory (TGFT) approach to quantum gravity. Crucially, we find in
the infrared that the effective mass of such theories generically vanishes not
only at criticality but also throughout the entire phase of non-vanishing
vacuum expectation value due to the non-locality of the interactions. As a
consequence, one encounters there the emergence of scale invariance which is
potentially enhanced to conformal invariance on configuration space.
