  The renormalization of the periodic potential is investigated in the
framework of the Euclidean one-component scalar field theory by means of the
differential RG approach. Some known results about the sine-Gordon model are
recovered in an extremely simple manner. There are two phases, an ordered one
with asymptotical freedom and a disordered one where the model is
non-renormalizable and trivial. The order parameter of the periodicity, the
winding number, indicates spontaneous symmetry breaking in the ordered phase
where the fundamental group symmetry is broken and the solitons acquire
dynamical stability. It is argued that the periodicity and the convexity are so
strong constraints on the effective potential that it always becomes flat. This
flattening is reproduced by integrating out the RG equation.
