Two flavor neutrino oscillations in presence of non-Hermitian dynamics

TL;DR

This paper develops a mathematical framework for two-flavor neutrino oscillations with non-Hermitian dynamics, comparing two approaches and highlighting the limitations of one in $ ext{PT}$-symmetric cases.

Contribution

It introduces a consistent density matrix approach for non-Hermitian neutrino oscillations, addressing issues with probability conservation in the $ ext{PT}$-symmetric regime.

Findings

The $ ext{G}$ metric approach fails to conserve probabilities in $ ext{PT}$-symmetric regimes.

The density matrix approach by Brody and Graefe provides a positive semi-definite map.

Steady state probabilities can deviate from 1/2, indicating non-Markovian behavior.

Abstract

We develop a consistent mathematical framework for studying two flavor neutrino oscillations in presence of non-Hermitian dynamics. We consider two approaches : (a) bi-orthonormal inner product defined by a positive-definite metric operator $\mathcal{G}$ and (b) the density matrix prescription by Brody and Graefe [Phys. Rev. Lett. 109, 230405 (2012)]. For the $\mathcal{PT}$-symmetric case, we show that the $\mathcal{G}$ metric approach does not work well (probabilities are not conserved) both in $\mathcal{PT}$-unbroken as well as $\mathcal{PT}$-broken regime. Hence, we adopt the density matrix prescription by Brody and Graefe which is a positive semi-definite map. In the density matrix prescription, we note that probability in the steady state limit is not necessarily $1/2$ thereby indicating non-Markovian behavior.