Renormalization Group Flows for the Second Z N Parafermionic Field Theory for N Even

Extending the results obtained in the case N odd, the effect of slightly relevant perturbations of the second parafermionic field theory with the symmetry ZN , for N even, are studied. The renormalization group equations, and their infra red fixed points exhibit the same structure in both cases. In addition to the standard flow from the p-th to the (p − 2)-th model, another fixed point corresponding to the (p − 1)-th model is found. Conformal field theories describe critical points of some statistical system, i.e. the fixed point of the renomalization group. In order to get some insights into the neighborhood of this critical point, one can study the effects of slightly relevant perturbations of the correponding conformal field theory, using a renormalization group approach. Parafermionic conformal field theories describe systems enjoying an extended conformal symmetry : they possess an additional cyclic symmetry ZN . While the first series of parafermionic conformal field theories [1] are well studied and applied in various domains [2, 3, 4], far less is known about the second parafermionic series [5, 6, 7, 8]. These conformal theories have a richer structure than the first parafermions. For a given N ≥ 5, there are infinitetely many second parafermionic theories Z (2) N (p), labeled by the integer p ≥ N − 1. These theories are unitary and correspond to degenerate representations of the corresponding parafermionic chiral algebra. The presence of the parameter p, for a given ZN , opens a way to reliable perturbative studies. It allows in particular to study the renormalisation group flows in the space of these conformal theory models, under various perturbations. This problem has already been adressed for the case N odd [17, 18], where the effect of two slightly relevant perturbations were studied. In this letter these results are generalized to the case N even. A particular case of this problem has already been treated. The second parafermionic theory Z (2) N (p) with N even is a particular exemple of symmetric coset on a simply laced Lie algebra, and as such its perturbation by one particular relevant operator has been studied in [15]. It is already known that the corresponding renormalization group equations describe a flow from Z (2) N (p) to Z (2) N (p− 2). Guided by the results obtained for the case N odd, we consider a more general perturbation, and an additional fixed point is found. The theory Z (2) N (p) perturbed by two appropriate fields will exhibit two different infrared (IR) fixed points : • one of these fixed point is described by the expected Z N (p− 2)