Textures of Neutrino Mass Matrix from $S_4$-flavor Symmetry

We study a texture of neutrino mass matrix characterized by two constraints consisting of one equality and another antiequality between two elements corresponding to two pairs of the matrix entries. Amidst such textures, we limit our study to three patterns which were realizable assuming an $S_4$-symmetry within type II-seesaw scenario. Three such cases were found and studied: I ($M_{\n 22}=-M_{\n 33}$ \& $M_{\n 11}=+M_{\n 23}$), II ($M_{\n 11}=-M_{\n 33}$ \& $M_{\n 22}=+M_{\n 13}$) and III ($M_{\n 11}=-M_{\n 22}$ \& $M_{\n 33}=+M_{\n 12}$). We specify the role of unphysical phases in the definition of the textures under study which were tested against experimental constraints, and were found to accommodate data with both hierarchies allowed. However, switching off the unphysical phases allows only for inverted hierarchy, except for the texture III which allows also, albeit for a very narrow parameter space region, for normal ordering. We stress that the different phenomenologies when including/excluding unphysical phases stem from the different definitions of the texture one has to adopt in order to make it insensitive to unphysical phases, rather than to any `absent' physical effects of unphysical phases. We present a complete phenomenological analysis of these three textures and justify analytically the resulting correlations. We detail the effect of the unphysical phases in diluting/deforming several correlations, which otherwise would have been ``clear". Finally, we give theoretical realizations within seesaw type II scenarios for such textures.