  The phase structure of a higher derivative sine-Gordon model in four
dimensions is analysed. It is shown that the inclusion of a relevant
two-derivative term in the action significantly modifies some of the results
obtained by neglecting this operator, and the final picture is substantially
different from the one describing the phase diagram associated with the
two-dimensional Berezinskii-Kosterlitz-Thouless (BKT) transition. The study is
carried out with the help of the Renormalization Group (RG) flow equations,
determined for a set of three parameters, and numerically solved both for a
truncated series expansion approximation, and for the complete set of
equations. In both cases, a continuous line of fixed points, terminating at a
particular point presenting universal properties, is found, together with a
manifold that separates two phases, roughly characterized by the sign of the
coupling $\widetilde z_k$ associated with this newly included operator. While
the phase corresponding to $\widetilde z_k>0$ shows some pathologies, the one
with $\widetilde z_k<0$ has a well-behaved infrared limit, where the system
reduces to a Gaussian-like model. We also briefly comment about the possibility
that our model could capture some of the qualitative features of the
ultraviolet (UV) critical manifold of conformally reduced gravity.
