Double-scaling limit of a broken symmetry quantum field theory in dimensions D<2

Abstract

The Ising limit is a correlated limit in which two bare Lagrangian parameters, the coupling constant g and the negative mass squared −m2, both approach infinity with the ratio −m2/g=α>0 held fixed. In a conventional Hermitian parity-symmetric scalar quantum field theory, with interaction term g|φ|N/N, the renormalized mass of the asymptotic theory is finite in this limit, and the limiting theory exhibits universality in N. For a non-Hermitian 𝒫𝒯-symmetric but parity-violating Lagrangian, with interaction term −g(iφ)N/N, the renormalized mass diverges in the same correlated limit. Nevertheless, the asymptotic theory still has interesting properties. In particular, the one-point Green’s function approaches the value −iα1/(N−2) independently of the space–time dimension D for D<2. Moreover, while the Ising limit of a conventional theory is dominated by a dilute instanton gas, the corresponding correlated limit of this 𝒫𝒯-symmetric theory is dominated by a constant-field configuration with corrections determined by a weak-coupling expansion in which the expansion parameter is proportional to an inverse power of g. We thus observe a weak-coupling/strong-coupling duality: the Ising limit itself is a strong-coupling limit, but the expansion about this limit takes the form of a conventional weak-coupling expansion. A possible generalization to dimensions D<4 is briefly discussed.