arXiv : Oscillation probabilities for a $\rho$$\tau$-symmetric non-Hermitian two-state system

Alexandre, Jean; Mason, Robert; Millington, Peter; Ellis, John; Dale, Madeleine

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Preprint
Report number	arXiv:2302.11666 ; KCL-PH-TH/2023-17 ; CERN-TH-2023-032 ; LTH 1334
Title	Oscillation probabilities for a $\rho$$\tau$-symmetric non-Hermitian two-state system
Related title	Oscillation probabilities for a PT-symmetric non-Hermitian two-state system
Author(s)	Alexandre, Jean (King's Coll. London) ; Dale, Madeleine (Rome U., Tor Vergata ; INFN, Rome ; Cyprus U. ; Humboldt U., Berlin) ; Ellis, John (King's Coll. London ; CERN) ; Mason, Robert (Liverpool U.) ; Millington, Peter (Manchester U.)
Imprint	2023-02-22
Number of pages	5
Subject category	hep-th ; Particle Physics - Theory ; hep-ph ; Particle Physics - Phenomenology ; quant-ph ; General Theoretical Physics
Abstract	There is growing interest in viable quantum theories with$\rho$$\tau$-symmetric non-Hermitian Hamiltonians, but a formulation of transition matrix elements consistent with positivity and perturbative unitarity has so far proved elusive. This Letter provides such a formulation, which relies crucially on the ability to span the state space in such a way that the interaction and energy eigenstates are orthonormal with respect to the same positive-definite inner product. We mention possible applications to the oscillations of mesons and neutrinos.
Other source	Inspire
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0)

$: Oscillation probabilities for the Hermitian (topmost two planes, \smash{$\mathbb{P}^{\rm Herm}_{i\to j}$}) and non-Hermitian (bottommost two planes, \smash{$\mathbb{P}^{\mathcal{PT}}_{i\to j}$}) models as a function of $\eta$ and the phase $\vartheta\equiv\Delta \omega \Delta t/2$. The legend is ordered from topmost to bottommost planes. : Squared eigenmasses of the Hermitian ($m^2_{\pm\rm{Herm}}$) and non-Hermitian ($m^2_{\pm\mathcal{PT}}$) models divided by $m_1^2+m_2^2$ versus $\eta$ for $(m_1^2-m_2^2)/(m_1^2+m_2^2)=0.5$.$ $: Comparison of the transition and survival probabilities (a) and squared eigenmasses (b) for the Hermitian and non-Hermitian models as a function of the parameter $\eta$. For the Hermitian case, the mass eigenvalues diverge for large $\eta$, with the lower eigenvalue crossing zero and becoming negative at a value of $\eta^2=(m_1^2+m_2^2)^2/(m_1^2-m_2^2)^2-1$. This occurs before the transition and survival probabilities saturate. For the non-Hermitian case, the mass eigenvalues merge at the exceptional point $\eta=1$, at which the probabilities saturate. : Caption not extracted$ $Polar plot of the Dirac norm $r(\vartheta)/r(\pi)$ for different values of the parameter $\eta$:\ that for $\eta=0.1$ (dashed, blue) is close to a unit circle, centered on the origin; that for $\eta=0.5$ (dot-dashed, yellow) clearly deviates from the unit circle; and that for $\eta=0.9$ (solid, green) has a distinctive cardioid shape.$ Show more plots

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Record created 2023-02-24, last modified 2023-09-14

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