A Systematic Analysis of Perturbations for Hexagonal Mixing Matrix

We present a systematic analysis of perturbative Hexagonal(HG) mixing for describing recent global fit neutrino mixing data with normal and inverted hierarchy. The corrections to unperturbed mixing are parameterized in terms of small orthogonal rotations ® with modified PMNS matrix of the forms \big($R_{\alpha\beta}^l\cdot V_{HG},~V_{HG}\cdot R_{\alpha\beta}^r,~V_{HG}\cdot R_{\alpha\beta}^r \cdot R_{\gamma\delta}^r,~R_{\alpha\beta}^l \cdot R_{\gamma\delta}^l \cdot V_{HG}$,$R_{\alpha\beta}^l\cdot V_{HG}\cdot R_{\gamma\delta}^r$)\mathrm{.}Here$R_{\alpha\beta}^{l, r}$ is rotation in ij sector and $V_{HG}$ is unperturbed Hexagonal mixing matrix. The detailed numerical investigation of all possible cases is performed with scanning of parameter space using ${\chi }^2$ approach. We found that the perturbative schemes governed by single rotation are unable to fit the mixing angle data even at $3\sigma$ level. The mixing schemes which involves two rotation matrices, only \big($R_{12}^l\cdot R_{13}^l\cdot V_{HG}$, ~$R_{13}^l \cdot R_{12}^l \cdot V_{HG}$,$R_{13}^l \cdot V_{HG} \cdot R_{12}^r$,$R_{12}^l \cdot V_{HG} \cdot R_{12}^r$,$R_{13}^l \cdot V_{HG} \cdot R_{13}^r$\big ) are successful in fitting all neutrino mixing angles within $1\sigma$ range for normal hierarchy(NH). However for inverted hierarchy(IH), only $R_{13}^l \cdot V_{HG} \cdot R_{13}^r$ismostpreferableasitcanfitallmixinganglesat$1\sigma$level\mathrm{.}Theremainingperturbativecasesareeitherexcludedat3$\sigma$ level or successful in producing mixing angles only at $2-3\sigma$ level. To study the impact of phase parameter, we also looked into CP violating effects for single rotation case. The predicted value of ${\delta }_{CP}$ lies in the range ${39.0}^{\circ }({40.4}^{\circ })\le \mathrm{\mid }{\delta }_{CP}\mathrm{\mid }\le {78.7}^{\circ }({79.2}^{\circ })$ for $U_{12}^l\cdot V_{HM}$ and $U_{13}^l\cdot V_{HM}$ case with Normal(Inverted) Hierarchy.