Sum rules and asymptotic behaviors of neutrino mixing in dense matter

It has proved convenient to define the effective lepton flavor mixing matrix $\tilde{U}$ and neutrino mass-squared differences ${\tilde{\Delta }}_{ji}\equiv {\tilde{m}}_j^2-{\tilde{m}}_i^2$ (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 $\tilde{U}$ and ${\tilde{\Delta }}_{ji}$, we derive new and exact formulas for moduli of the nine elements of $\tilde{U}$ and the sides of its three Dirac unitarity triangles in the complex plane. The asymptotic behaviors of $|{\tilde{U}}_{\alpha i}{|}^2$ and ${\tilde{\Delta }}_{ji}$ (for $\alpha =e,\mu ,\tau$ and $i,j=1,2,3$) in very dense matter (namely, allowing the matter parameter $A=2\sqrt{2}{G}_F{N}_{e}E$ to mathematically approach infinity) are analytically unraveled for the first time, and in this connection the confusion associated with the parameter redundancy of ${\tilde{\theta }}_{12}$, ${\tilde{\theta }}_{13}$, ${\tilde{\theta }}_{23}$ and $\tilde{\delta }$ in the standard parametrization of $\tilde{U}$ is clarified.