Towards a Quintic Ginzburg-Landau Description of the (2,7) Minimal Model

We discuss dimensional continuation of the massless scalar field theory with the i ϕ 5 interaction term. It preserves the so-called P T symmetry, which acts by ϕ → − ϕ accompanied by i → − i . Below its upper critical dimension 10 / 3 , this theory has interacting infrared fixed points. We argue that the fixed point in d = 2 describes the nonunitary minimal conformal model M ( 2 , 7 ) . We identify the operators ϕ and ϕ 2 with the Virasoro primaries ϕ 1 , 2 and ϕ 1 , 3 , respectively, and i ϕ 3 with a quasiprimary operator, which is a Virasoro descendant of ϕ 1 , 3 . Our identifications appear to be consistent with the operator product expansions and with considerations based on integrability. Using constrained Padé extrapolations, we provide estimates of the critical exponents in d = 3 . We also comment on possible lattice descriptions of M ( 2 , 7 ) and discuss RG flows to and from this conformal field theory (CFT). Finally, we conjecture that the minimal models M ( 2 , 2 n + 1 ) are described by the massless scalar field theories with the i ϕ 2 n − 1 interaction terms.