Neutrino Mass Matrix in Neutrino-Related Processes

Techniques are developed for constructing amplitudes of neutrino-related processes in terms of the neutrino mass matrix, with no reference to the neutrino mixing matrix. The amplitudes of neutrino oscillations in vacuum and medium, quasi-elastic neutrino scattering, $\beta$ decays and double-$\beta$ decays are considered. The proposed approach makes extensive use of Frobenius covariants within the framework of Sylvester's theorem on matrix functions. The in-medium dispersion laws are found in terms of elementary functions for three flavors of Majorana neutrinos as an application of the developed formalism. The in-medium dispersion laws for Dirac neutrinos can be determined in the general case by searching for the roots of a polynomial of degree 6. In the rest frame of baryon matter, the minimum energy of both Majorana and Dirac neutrinos is achieved at a finite neutrino momentum. In such cases, Dirac neutrinos occupy a hollow Fermi sphere at zero temperature and low densities. Fitting experimental data in terms of the neutrino mass matrix can provide better statistical accuracy in determining the neutrino mass matrix compared to methods using the neutrino mixing matrix at intermediate stages.