A new and trivial CP symmetry for extended A₄ flavor

The combination of $\nu_\mu$-$\nu_\tau$ exchange together with CP conjugation in the neutrino sector (known as $\mathrm{CP}^{\mu\tau}$ symmetry or $\mu\tau$-reflection) is known to predict the viable pattern: $\theta_{23}=45^\circ$, maximal Dirac CP phase and trivial Majorana phases. We implement such a CP symmetry as a new CP symmetry in theories with $A_4$ flavor. The implementation in a complete renormalizable model leads to a new form for the neutrino mass matrix that leads to further predictions: normal hierarchical spectrum with lightest mass and $m_{\beta\beta}$ ($0\nu\beta\beta$) of only few meV, and either $\nu_1$ or $\nu_2$ has opposite CP parity. An approximate $L_\mu-L_\tau$ symmetry arises naturally and controls the flavor structure of the model. The light neutrino masses are generated by the extended seesaw mechanism with 6 right-handed neutrinos (RHNs). The requirement of negligible one-loop corrections to light neutrino masses, validity of the extended seesaw approximation and not too long-lived BSM states to comply with BBN essentially restricts the parameters of the model to a small region: three relatively light right-handed neutrinos at the GeV scale, heavier neutrinos at the electroweak scale and Yukawa couplings smaller than the electron Yukawa. Such a small Yukawa couplings render these RHNs unobservable in terrestrial experiments.