It has proved convenient to define the effective lepton flavor mixing matrix $\widetilde{ U}$ and neutrino mass-squared differences $\widetilde{ \Delta}^{ }_{ ji} \equiv \widetilde{ m}^2_j - \widetilde{ m}^2_i$ (for $i,j =1,2,3$) to describe the phenomena of neutrino mixing and flavor oscillations in a medium, but the prerequisite is to establish direct and transparent relations between these effective quantities and their fundamental counterparts in vacuum. With the help of two sets of sum rules for $\widetilde{ U}$ and $\widetilde{ \Delta}^{ }_{ ji}$, we derive new and exact formulas for moduli of the nine elements of $\widetilde{ U}$ and the sides of its three Dirac unitarity triangles in the complex plane. The asymptotic behaviors of $|\widetilde{ U}^{ }_{ \alpha i}|^2$ and $\widetilde{ \Delta}^{ }_{ ji}$ (for $\alpha = e, \mu, \tau$ and $i,j =1,2,3$) in very dense matter (namely, allowing the matter parameter $A = 2\sqrt{ 2} ~ G^{ }_{ \rm F} N^{ }_e E$ to mathematically approach infinity) are analytically unraveled for the first time, and in this connection the confusion associated with the parameter redundancy of $\widetilde{ \theta}^{ }_{ 12}$, $\widetilde{ \theta}^{ }_{ 13}$, $\widetilde{ \theta}^{ }_{ 23}$ and $\widetilde{ \delta}$ in the standard parametrization of $\widetilde{ U}$ is clarified.
Categories: hep-ph hep-ex