A minimally broken residual TBM-Klein symmetry and baryogenesis via leptogenesis

We investigate the minimally perturbed neutrino mass matrices which at the leading order give rise to Tri-BiMaximal (TBM) mixing due to a residual $mathbb{Z}_2times mathbb{Z}_2^{mutau}$ Klein symmetry in the neutrino mass term of the Lagrangian. Starting from the Lagrangian level of Type-I seesaw which contains $m_D$ and $M_R$ as constituent matrices, the $mathbb{Z}_2^{mutau}$ is broken in $M_R$ to be consistent with the nonvanishing value of $theta_{13}$. The unbroken $mathbb{Z}_2$ leads to constraint relations between the mixing angles $theta_{13}$ and $theta_{12}$ along with testable predictions on the Dirac CP phase $delta$ and the neutrino less double beta decay parameter $|(M_nu)_{11}|$. A full $mathbb{Z}_2times mathbb{Z}_2^{mutau}$ symmetry leads to a degeneracy in the eigenvalues of $M_R$ matrices. Nevertheless, breaking of $mathbb{Z}_2^{mutau}$ which is also necessary to generate nonzero $theta_{13}$, lifts that degeneracy. Unlike the standard $N_1$-leptogenesis scenario where only the decays from the lightest right handed (RH) neutrino $N_1$ are relevant, here the decays from all the quasi-degenerate RH neutrinos contribute to the process of baryogenesis via leptogenesis. Flavor dependent Boltzmann equations are solved for heavy neutrino as well as the light leptonic number densities to compute the final baryon asymmetry $Y_B$. Using the observed range for the baryon asymmetry, lower and upper bounds on the RH neutrino masses are obtained thereafter.