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

We discuss dimensional continuation of the massless scalar field theory with the iϕ^{5} interaction term. It preserves the so-called PT 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 iϕ^{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(2,2n+1) are described by the massless scalar field theories with the iϕ^{2n-1} interaction terms.