Reconciling TM$_2$ Mixing with LMA and Dark-LMA Data based on Minimal Corrections from Charged-Lepton Sector

Motivated by the increasing precision of neutrino oscillation data, we study the corrections to the TM$_2$ neutrino mixing framework, emanating from $(1,2)$ sector of the charged lepton, for both the standard LMA and dark-LMA solutions. We employ the Wolfenstein parameterization of the charged-lepton mixing matrix, characterized by two additional parameters $(\lambda,\delta)$, which effectively reconciles the TM$_2$ neutrino-mixing predictions with current oscillation data. For the LMA solution, the allowed ranges are $0.1 \lesssim \lambda \lesssim 0.33$ and $\delta \in (20^\circ\!-\!90^\circ)\oplus(270^\circ\!-\!340^\circ)$, while the dark-LMA case requires $\lambda>0.24$ and $125^\circ<\delta<235^\circ$. Interestingly, for LMA case, the upper bound $\lambda \le 0.33$ is found to be dictated by the atmospheric mixing angle $\theta_{23}$. The model predicts sizeable CP violation, with $|J_{CP}|$ reaching values as large as $0.13$. We, also, analyze the effective Majorana mass parameter $m_{ee}$ relevant for neutrinoless double beta decay. The inverted hierarchy region lies within the sensitivity of future experiments for both solutions, whereas only part of the normal hierarchy region can be tested.