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We present a systematic study on minimally perturbed neutrino mass matriceswhich at the leading order give rise to Tri-BiMaximal (TBM) mixing due to aresidual $\mathbb{Z}_2\times \mathbb{Z}_2^{\mu\tau}$ Klein symmetry in theneutrino mass term of the low energy effective seesaw Lagrangian. Consideringonly the breaking of $\mathbb{Z}_2^{\mu\tau}$ with two relevant breakingparameters ($\epsilon_{4,6}^\prime$), after a comprehensive numerical analysis,we show that the phenomenologically viable case in this scenario is a specialcase of TM1 mixing. For this class of models, from the phenomenologicalperspective, one always needs large breaking (more than $ 45\%$) in one of thebreaking parameters. However, to be consistent the maximal mixing of$\theta_{23}$, while more than $ 35\%$ breaking is needed in the other, a range$49.4^\circ-53^\circ$ and $38^\circ-40^\circ$ could be probed allowing breakingup to $ 25\%$ in the same parameter. Thus though this model cannot distinguishthe octant of $\theta_{23}$, non-maximal mixing is preferred from the viewpointof small breaking. The model is also interesting from leptogenesis perspective.Unlike the standard $N_1$-leptogenesis scenario, here all the RH neutrinoscontribute to lepton asymmetry due to the small mass splitting controlled bythe $\mathbb{Z}_2^{\mu\tau}$ breaking parameters. Inclusion of flavour couplingeffects (In general, which have been partially included in all the leptogenesisstudies in perturbed TBM framework) makes our analysis and results pertainingto a successful leptogenesis more accurate than any other studies in existingliterature. Finally, in the context of recent discovery of the ultra highenergy (UHE) neutrino events at IceCube, assuming UHE neutrinos originate frompurely astrophysical sources, we obtain prediction on the neutrino flux ratiosat neutrino telescopes.

A study on a minimally broken residual TBM-Klein symmetry with its implications on flavoured leptogenesis and ultra high energy neutrino flux ratios