  The Ising limit of a conventional Hermitian parity-symmetric scalar quantum
field theory is a correlated limit in which two bare Lagrangian parameters, the
coupling constant $g$ and the {\it negative} mass squared $-m^2$, both approach
infinity with the ratio $-m^2/g=\alpha>0$ held fixed. In this limit the
renormalized mass of the asymptotic theory is finite. Moreover, the limiting
theory exhibits universal properties. For a non-Hermitian $\cal PT$-symmetric
Lagrangian lacking parity symmetry, whose interaction term has the form
$-g(i\phi)^N/N$, the renormalized mass diverges in this correlated limit.
Nevertheless, the asymptotic theory still has interesting properties. For
example, 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 parity-symmetric quantum field theory is
dominated by a dilute instanton gas, the corresponding correlated limit of a
$\cal PT$-symmetric quantum field theory without parity symmetry is dominated
by a constant-field configuration with corrections determined by a
weak-coupling expansion in which the expansion parameter (the amplitude of the
vertices of the graphs in this expansion) is proportional to an inverse power
of $g$. We thus observe a weak-coupling/strong-coupling duality in that while
the Ising limit is a strong-coupling limit of the quantum field theory, the
expansion about this limit takes the form of a conventional weak-coupling
expansion. A possible generalization of the Ising limit to dimensions $D<4$ is
briefly discussed.
