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

TL;DR

This paper investigates the $c$-function and central charge of the sine-Gordon model using non-perturbative renormalization group flow, emphasizing the importance of wave-function renormalization for accurate results.

Contribution

It introduces a non-perturbative RG approach that explicitly accounts for periodicity and highlights the necessity of wave-function renormalization beyond local potential approximation.

Findings

Central charge matches expected $ riangle c=1$ in certain flow regimes.

Including higher harmonics alone is insufficient for accurate results.

Wave-function renormalization is essential for reliable RG flow analysis.

Abstract

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