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In this paper we study the $c$-function of the sine-Gordon model takingexplicitly into account the periodicity of the interaction potential. Theintegration of the $c$-function along trajectories of the non-perturbativerenormalization group flow gives access to the central charges of the model inthe fixed points. The results at vanishing frequency $\beta^2$, where theperiodicity does not play a role, are retrieved and the independence on thecutoff regulator for small frequencies is discussed. Our findings show that thecentral charge obtained integrating the trajectories starting from therepulsive low-frequencies fixed points ($\beta^2 <8\pi$) to the infrared limitis in good quantitative agreement with the expected $\Delta c=1$ result. Thebehavior of the $c$-function in the other parts of the flow diagram is alsodiscussed. Finally, we point out that also including higher harmonics in therenormalization group treatment at the level of local potential approximationis not sufficient to give reasonable results, even if the periodicity is takeninto account. Rather, incorporating the wave-function renormalization (i. e.going beyond local potential approximation) is crucial to get sensible resultseven when a single frequency is used.

$c$-function and central charge of the sine-Gordon model from the non-perturbative renormalization group flow