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In the paper Symmetries and conservation laws in non-Hermitian field theories by Jean Alexandre, Peter Millington, and Dries Seynaeve, Phys. Rev. D 96, 065027 the authors use this Lagrangian:

$$ L = \partial_\nu \phi_1^* \partial^\nu \phi_1 + \partial_\nu \phi_2^* \partial^\nu \phi_2 - m_1^2 \phi_1^* \phi_1 - m_2^2 \phi_2^* \phi_2 - \mu^2 (\phi_1^* \phi_2 - \phi_2^* \phi_1).$$

But they do not explain its origin. I can't find anywhere from where this Lagrangian has come and where in physics it's being used. Please tell about its origin and use or give any references from where I can find about it.

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  • $\begingroup$ Where did you see this Lagrangian? Probably just a typo at the end of your expression. The last-minus sign should be a plus-sign, as otherwise $\mathcal{L}$ would not be hermitian. With the correct plus-sign, you simply have to diagonalize the mass matrix to indentify the mass eigenfields. $\endgroup$
    – Hyperon
    Commented Apr 10 at 11:23
  • $\begingroup$ @Hyperon yes this is non-hermitian lagrangian and this has real energy eigenvalues even though its non-hermitian due to PT symmetry. I saw it in one of the paper by Alexander but there was nothing about its origin, they just took this system. $\endgroup$
    – Kawaljeet Kaur
    Commented Apr 10 at 14:50
  • $\begingroup$ Which paper, which page? Link? $\endgroup$
    – Hyperon
    Commented Apr 10 at 15:13
  • $\begingroup$ @Hyperon DOI: 10.1103/PhysRevD.96.065027 Symmetries and conservation laws in non-Hermitian field theories $\endgroup$
    – Kawaljeet Kaur
    Commented Apr 10 at 15:34
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    $\begingroup$ The origin is that it's a toy model. $\endgroup$
    – Connor Behan
    Commented Apr 10 at 16:32

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