$\mathcal{PT}$-symmetry of Particle mixing theories and the equation of motion matrix

TL;DR

This paper explores the $ ext{PT}$-symmetry in a non-Hermitian scalar field model, analyzing the equations of motion and Hamiltonian through similarity transformations to resolve discrepancies in eigenvalues and equations.

Contribution

It introduces a method using similarity transformations to address $ ext{PT}$-symmetry issues in particle mixing theories, ensuring consistent eigenvalues and equations of motion.

Findings

Discrepancies in equations of motion are resolved by similarity transformations.

Eigenvalues' reality depends on the mass terms in the model.

Similarity-transformed Hamiltonian aligns with the equations of motion.

Abstract

A non-Hermitian complex scalar field model is considered from its $\mc{PT}$ symmetric aspect. A matrix constructed from the Euler-Lagrange equations of motion is utilized to analyze the states of the model. The model has two mass terms which determine the real or complex nature of the eigen values. A mismatch is found in the Lagrange equations of motion of the fields as the equations do not agree with the other after complex conjugation of the either. This is resolved by exploiting a preferred similarity transformation of the Lagrangian. The discrepancy even at the Hamiltonian level is found to have vanished once we consider the similarity transformed Hamiltonian.