Study on perturbation schemes for achieving the real PMNS matrix from various symmetric textures

The PMNS matrix displays an obvious symmetry, but not exact. There are several textures proposed in literature, which possess various symmetry patterns and seem to originate from different physics scenarios at high energy scales. To be consistent with the experimental measurement, all of the regularities slightly decline, i.e. the symmetry must be broken. Following the schemes given in literature, we modify the matrices (9 in total) to gain the real PMNS matrix by perturbative rotations. The transformations may provide hints about the underlying physics at high energies and the breaking mechanisms which apply during the evolution to the low energy scale, especially the results may be useful for the future model builders.