On the Effective Mass of the Electron Neutrino in Beta Decay

In the presence of mixing between massive neutrino states, the distortion of the electron spectrum in beta decay is, in general, a function of several masses and mixing angles. For $3\nu$-schemes which describe the solar and atmospheric neutrino data, this distortion can be described by a single effective mass, under certain conditions. In the literature, two different definitions for the effective mass have been suggested. We show that for quasi-degenerate mass schemes (with an overall mass scale $m$ and splitting $\Delta m^2$) the two definitions coincide up to $(\Delta m^2)^2/m^4$ corrections. We consider the impact of different effective masses on the integral energy spectrum. We show that the spectrum with a single mass can be used also to fit the data in the case of $4\nu$-schemes motivated, in particular, by the LSND results. In this case the accuracy of the mass determination turns out to be better than $(10 - 15)%$.