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

Using the renormalization group approach, the Coulomb gas and the coset techniques, the effect of slightly relevant perturbations is studied for the second parafermionic field theory with the symmetry ZN , for N odd. New fixed points are found and classified. ∗ Unité Mixte de Recherche UMR 7589 Parafermionic conformal field theories describe systems enjoying conformal symmetry and a cyclic symmetry ZN . The first series of parafermionic conformal field theories appeared in 1985 [1]. Since then they have been well studied and applied in various domains [2, 3, 4]. The second parafermionic series Z (2) N has been developed fairly recently [5-8] and it still awaits its applications. In the case of the first series Z (1) N , to a given N is associated a single conformal theory. This is different for the second series: for a given N , there exist an infinity of unitary conformal theories Z (2) N (p), with p = N − 1, N − 2, .... These theories correspond to degenerate representations of the corresponding parafermionic chiral algebra. They are much more rich in their content of physical fields, as compared to the theories of the first series. They are also much more complicated. But, on the other hand, the presence of the parameter p, for a given ZN , opens a way to reliable perturbative studies. It allows in particular to study the renormalization group flows in the space of these conformal theory models, under various perturbations. In the theory of second order phase transition, it is widely accepted that fixed points of the renormalization group should be described by conformal field theories. Saying it differently, a CFT describes the critical point of some statistical system. In order to investigate the behavior of the renormalization group in the vicinity of a fixed point, it is useful to study the effects of slightly relevant perturbations of the correponding conformal field theory. In this paper we shall present results for the renormalization group flows of the Z (2) N (p) theories with N odd, being perturbed by two slightly relevant fields. In a previous letter [9] we have studied the case of the Z (2) 5 (p) theories. These results are here generalized to any odd integer N , and more details are given. The details of the Z (2) N parafermionic theory with N odd could be found in [6]. The q charge of ZN takes values in ZN , so that in the Kac table of this theory one finds the Z (2) N neutral fields, of q = 0, the q = 1 . . . N − 1 doublets, and the Z2 disorder fields. The symmetry of the theory is actually DN , which is made of ZN rotations and the