Effective Field calculations of the Energy Spectrum of the $\mathcal{PT}% $-Symmetric ($-x^{4}$) Potential

TL;DR

This paper demonstrates that effective field theory methods can be applied to the $ ext{PT}$-symmetric $-x^4$ potential, providing a stable vacuum description and spectrum approximation, with potential extensions to quantum field theory.

Contribution

It introduces a novel application of effective field theory to $ ext{PT}$-symmetric $-x^4$ potentials, showing bounded effective potentials and calculable metric operators.

Findings

Effective potential is bounded from below, indicating vacuum stability.

Spectrum predictions closely match exact results.

Perturbative calculation of the metric operator is feasible and extendable.

Abstract

In this work, we show that the traditional effective field approach can be applied to the $\mathcal{PT}$-symmetric wrong sign ($-x^{4}$) quartic potential. The importance of this work lies in the possibility of its extension to the more important $\mathcal{PT}$-symmetric quantum field theory while the other approaches which use complex contours are not willing to be applicable. We calculated the effective potential of the massless $-x^{4}$ theory as well as the full spectrum of the theory. Although the calculations are carried out up to first order in the coupling, the predicted spectrum is very close to the exact one taken from other works. The most important result of this work is that the effective potential obtained, which is equivalent to the Gaussian effective potential, is bounded from below while the classical potential is bounded from above. This explains the stability of the vacuum of the theory. The obtained quasi-particle Hamiltonian is non-Hermitian but $\mathcal{PT}$-symmetric and we showed that the calculation of the metric operator can go perturbatively. In fact, the calculation of the metric operator can be done even for higher dimensions (quantum field theory) which, up till now, can not be calculated in the other approaches either perturbatively or in a closed form due to the possible appearance of field radicals. Moreover, we argued that the effective theory is perturbative for the whole range of the coupling constant and the perturbation series is expected to converge rapidly (the effective coupling $g_{eff}={1/6}$).