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We discuss the viability of the $mu$--$tau$ interchange symmetry imposed on the neutrino mass matrix in the flavor space. Whereas the exact symmetry is shown to lead to textures of completely degenerate spectrum which is incompatible with the neutrino oscillation data, introducing small perturbations into the preceding textures, inserted in a minimal way, lead however to four deformed textures representing an approximate $mu$--$tau$ symmetry. We motivate the form of these `minimal textures, which disentangle the effects of the perturbations, and present some concrete realizations assuming exact $mu$--$tau$ at the Lagrangian level but at the expense of adding new symmetries and matter fields. We find that all these deformed textures are capable to accommodate the experimental data, and in all types of neutrino mass hierarchies, in particular the non-vanishing value for the smallest mixing angle.
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Motivated by the recent results from Daya Bay, Reno and Double Chooz Collaborations, we study the consequences of small departures from exact $mu-tau$ symmetry in the neutrino sector, to accommodate a non-vanishing value of the element $V_{e3}$ from the leptonic mixing matrix. Within the see-saw framework, we identify simple patterns of Dirac mass matrices that lead to approximate $mu-tau$ symmetric neutrino mass matrices, which are consistent with the neutrino oscillation data and lead to non-vanishing mixing angle $V_{e3}$ as well as precise predictions for the CP violating phases. We also show that there is a transparent link between neutrino mixing angles and see-saw parameters, which we further explore within the context of leptogenesis as well as double beta decay phenomenology.
Inspired by the neutrino oscillations data, we consider the exact $mu-tau$ symmetry, implemented at the level of the neutrino mass matrix, as a good initial framework around which to study and describe neutrino phenomenology. Working in the diagonal basis for the charged leptons, we deviate from $mu-tau$ symmetry by just modifying the phases of the neutrino mass matrix elements. This deviation is enough to allow for a non-vanishing neutrino mixing entry $|V_{e3}|$ (i.e. $theta_{13}$) but it also gives a very stringent (and eventually falsifiable) prediction for the atmospheric neutrino mixing element $|V_{mu3}|$ as a function of $|V_{e3}|$. The breaking by phases is characterized by a single phase and is shown to lead to interesting lower bounds on the allowed mass of the lightest neutrino depending on the ordering of neutrino masses (normal or inverted) and on the value of the Dirac ${cal CP}$ violating phase $delta_{CP}$. The allowed parameter space for the effective Majorana neutrino mass $m_{ee}$ is also shown to be non-trivially constrained.
The $mu$-$tau$ exchange symmetry in the neutrino mass matrix and its breaking as a perturbation are discussed. The exact $mu$-$tau$ symmetry restricts the 2-3 and 1-3 neutrino mixing angles as $theta_{23} = pi/4$ and $theta_{13} = 0$ at a zeroth order level. We claim that the $mu$-$tau$ symmetry breaking prefers a large CP violation to realize the observed value of $theta_{13}$ and to keep $theta_{23}$ nearly maximal, though an artificial choice of the $mu$-$tau$ breaking can tune $theta_{23}$, irrespective of the CP phase. We exhibit several relations among the deviation of $theta_{23}$ from $pi/4$, $theta_{13}$ and Dirac CP phase $delta$, which are useful to test the $mu$-$tau$ breaking models in the near future experiments. We also propose a concrete model to break the $mu$-$tau$ exchange symmetry spontaneously and its breaking is mediated by the gauge interactions radiatively in the framework of the extended gauge model with $B-L$ and $L_mu - L_tau$ symmetries. As a result of the gauge mediated $mu$-$tau$ breaking in the neutrino mass matrix, the artificial choice is unlikely, and a large Dirac CP phase is preferable.
Nonstandard interactions (NSIs), possible subleading effects originating from new physics beyond the Standard Model, may affect the propagation of neutrinos and eventually contribute to measurements of neutrino oscillations. Besides this, $ mu-tau $ reflection symmetry, naturally predicted by non-Abelian discrete flavor symmetries, has been very successful in explaining the observed leptonic mixing patterns. In this work, we study the combined effect of both. We present an $S_4$ flavor model with $mu-tau$ reflection symmetry realized in both neutrino masses and NSIs. Under this formalism, we perform a detailed study for the upcoming neutrino experiments DUNE and T2HK. Our simulation results show that under the $mu-tau $ reflection symmetry, NSI parameters are further constrained and the mass ordering sensitivity is less affected by the presence of NSIs.
We study the consequences of the $Z_2$-symmetry behind the $mu$--$tau$ universality in neutrino mass matrix. We then implement this symmetry in the type-I seesaw mechanism and show how it can accommodate all sorts of lepton mass hierarchies and generate enough lepton asymmetry to interpret the observed baryon asymmetry in the universe. We also show how a specific form of a high-scale perturbation is kept when translated via the seesaw into the low scale domain, where it can accommodate the neutrino mixing data. We finally present a realization of the high scale perturbed texture through addition of matter and extra exact symmetries.
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