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

The Ising limit is a correlated limit in which two bare Lagrangian paramete rs, the coupling constant g and the negative mass squared -m(2), both appro ach infinity with the ratio -m(2)/g=alpha >0 held fixed. In a conventional Hermitian parity-symmetric scalar quantum field theory, with interaction te rm g\phi\(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 te rm -g(i phi)(N)/N, the renormalized mass diverges in the same correlated li mit. Nevertheless, the asymptotic theory still has interesting properties. In particular, the one-point Green's function approaches the value -i alpha (1/(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 ins tanton 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 proportion al to an inverse power of g. We thus observe a weak-coupling/strong-couplin g duality: the Ising limit itself is a strong-coupling limit, but the expan sion about this limit takes the form of a conventional weak-coupling expans ion. A possible generalization to dimensions D <4 is briefly discussed. (C) 2001 American Institute of Physics.