Renormalization of the bilocal sine-Gordon model

Abstract

The functional renormalization group treatment is presented for the twodimensional sine-Gordon model by including a bilocal term in the potential, which contributes to the flow at tree level. It is shown that the flow of the bilocal term can substitute the evolution of the wave function renormalization constant, and then the Kosterlitz-Thouless type phase transition can be recovered. I. INTRODUCTIONThe quantum field theoretical models suffer from ultraviolet (UV) divergences, and we should remove the infinities by regularizations. They imply that we should consider an effective dynamics of the models, where certain degrees of freedom or modes of the physical system are not followed, they do not belong to the system anymore, they are pushed to the environment. However, if we integrate out certain degrees of freedom, then the remaining effective model of the system modes becomes nonlocal [1]. Earlier works investigated nonlocal field theories at perturbative level [2,3]. Nevertheless, the theories including nonlocal interactions have to face with the violation of causality [4-6], instability problems [7,8], or the spoiling of gauge invariance [9][10][11].The introduction of a momentum cutoff, as a simple regularization, also leads to nonlocal interactions. The same problem takes place, when we use the functional renormalization group (RG) method [12][13][14][15][16]. Recently it has been pointed out, that RG blocking step introduces nonlocal contributions to the evolution [17][18][19]. During the blocking some ultraviolet (UV) modes in the momentum shell are integrated out, transforming system degrees of freedom into the environmental ones. The nonlocality has not been followed during the traditional the RG method, because we usually use local actions and do not let the nonlocal terms to evolve. However there is a nontrivial saddle point for the integrated UV modes, which introduces a nonlocal term, moreover it contributes to the RG evolution at tree level.