Double-Scaling Limit of a Broken Symmetry Quantum Field Theory

TL;DR

This paper explores the double-scaling limit of a non-Hermitian $ ext{PT}$-symmetric quantum field theory, revealing universal properties and a weak-coupling/strong-coupling duality in the asymptotic behavior of the theory.

Contribution

It introduces a novel double-scaling limit for $ ext{PT}$-symmetric quantum field theories, showing universal properties and a duality between weak and strong coupling regimes.

Findings

The renormalized mass diverges in the $ ext{PT}$-symmetric limit.

The one-point Green's function approaches a fixed value independent of space-time dimension.

A weak-coupling expansion describes the asymptotic theory despite the divergence of the mass.

Abstract

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.