Effective Action study of $\mathcal{PT}$-Symmetry Breaking for the non-Hermitian $\left( i\phi^{3}\right) _{6-\epsilon}$ Theory and The Yang-Lee Edge Singularity

TL;DR

This paper investigates the $ ext{PT}$-symmetry breaking in a non-Hermitian $i\,\phi^3$ field theory near six dimensions, revealing critical exponents, stability issues, and connections to Yang-Lee edge singularity through effective potential analysis.

Contribution

It provides a detailed effective potential analysis of $ ext{PT}$-symmetry breaking in non-Hermitian field theory, including stability, unitarity, and critical exponent calculations at non-integer dimensions.

Findings

Critical exponents match exact values from literature.

Vacuum amplitude is zero at $ ext{PT}$-symmetry breaking point.

Connection established between $ ext{PT}$-symmetry breaking and Yang-Lee edge singularity.

Abstract

We use the effective potential method to study the $\mathcal{PT}$-symmetry breaking of the non-Hermitian $i\phi^{3}$ field theory in $6-\epsilon$ space-time dimensions. The critical exponents so obtained coincide with the exact values listed in the literature. We showed that at the point of $\mathcal{PT}$-symmetry breaking, the vacuum-vacuum amplitude is certainly zero and the fugacity is one which mimics a Yang-Lee edge singularity in magnetic systems. What makes this work interesting is that it takes into account problems which are always overlooked in the literature for the Yang-Lee model like stability, unitarity and generation of Stokes wedges at space-time dimensions for which divergences occur in the theory . Besides, here we make direct calculation of critical exponents from the dependance of the order parameter on external magnetic field not from the density of zeros of the partition function.