ar X iv : h ep - t h / 06 10 24 2 v 1 2 3 O ct 2 00 6 October 2006 . Renormalization group flows for the second Z 5 parafermionic field theory

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 Z5. New fixed points are found and classified. ∗ Unité Mixte de Recherche UMR 7589 While the first series of parafermionic conformal field theories [1] is well studied and applied in various domains [2,3,4], the second parafermionic series, with the symmetry ZN , has been developed fairly recently [5-8] and it still awaits its applications. In the case of the first series, to a given N (of ZN) is associated a single conformal theory. This is different for the second series: for a given ZN , there exist an infinity of unitary conformal theories ZN,p, with p = N−2+k, k = 1, 2, 3, ...∞. 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 renormalisation group flows in the space of these conformal theory models, under various perturbations. In this paper we shall present results for a particular case of this problem: for the renormalization group flows of the Z5,p theories, being perturbed by two slightly relevant fields. The details of the Z5 parafermionic theory, the second one, could be found in [5]. The q charge of Z5 takes values q = 0,±1,±2, so that in the Kac table of this theory one finds the Z5 neutral fields, of q = 0, the q = ±1 and the q = ±2 doublets, and the Z2 disorder fields. The symmetry of the theory is actually D5, which is made of Z5 rotations and the Z2 reflections in 5 different axises. These last symmetry elements amount to the charge conjugation symmetry: q → −q. We want to perturbe by the Z5 (D5 in fact) neutral fields, in order to preserve this symmetry. Perturbatively well controlled domain of Z5,p theories is that of p ≫ 1, giving a small parameter ǫ ∼ 1/p. This is similar to the original perturbative renormalization group treatment of minimal models for Virasoro algebra based conformal theory [9,10]. In this domain, i.e. for p ≫ 1, one finds, in the lower part of the Kac table of Z5 parafermionic theory, two Z5 neutral fields which are slightly relevant and which close