Non-perturbative calculations for the effective potential of the $PT$ symmetric and non-Hermitian $(-g\phi^{4})$ field theoretic model

TL;DR

This paper explores both perturbative and non-perturbative methods to calculate the effective potential of the PT-symmetric, non-Hermitian (-gφ^4) field theory, introducing new resummation techniques and analytical results.

Contribution

It develops a novel, unified resummation method for the effective potential across all coupling regions, providing new analytic forms and insights into the theory's amplitudes and bound states.

Findings

Effective potential vanishes exponentially as G approaches G+

New form of the effective potential up to G^3 showing perturbative behavior

A unique resummation formula interpolates between weak and strong coupling regimes

Abstract

We investigate the effective potential of the $PT$ symmetric $(-g\phi^{4}) $ field theory, perturbatively as well as non-perturbatively. For the perturbative calculations, we first use normal ordering to obtain the first order effective potential from which the predicted vacuum condensate vanishes exponentially as $G\to G^+$ in agreement with previous calculations. For the higher orders, we employed the invariance of the bare parameters under the change of the mass scale $t$ to fix the transformed form totally equivalent to the original theory. The form so obtained up to $G^3$ is new and shows that all the 1PI amplitudes are perurbative for both $G\ll 1$ and $G\gg 1$ regions. For the intermediate region, we modified the fractal self-similar resummation method to have a unique resummation formula for all $G$ values. This unique formula is necessary because the effective potential is the generating functional for all the 1PI amplitudes which can be obtained via $\partial^n E/\partial b^n$ and thus we can obtain an analytic calculation for the 1PI amplitudes. Again, the resummed from of the effective potential is new and interpolates the effective potential between the perturbative regions. Moreover, the resummed effective potential agrees in spirit of previous calculation concerning bound states.