An alternative: two-mixing texture for three neutrinos or three-mixing texture for four neutrinos

Abstract

The alternative formulated in the title has a chance to be settled, when the existence of the LSND effect is experimentally excluded or confirmed. The first option, much discussed in literature, works in the case of three active neutrinos $\nu_e, \nu_\mu, \nu_\tau$, when among their massive states $\nu_1, \nu_2, \nu_3$ there is no direct mixing between $\nu_1$ and $\nu_3$, and the mass hierarchy $m^2_1 < {\sim} m^2_2 \ll m^2_3$ holds. This option is consistent with the observed deficits of solar $\nu_e$'s and atmospheric $\nu_\mu $'s, if $\Delta m^2_{21} \leftrightarrow \Delta m^2_{\rm sol}$ and $ \Delta m^2_{32} \leftrightarrow \Delta m^2_{\rm atm}$. On the other hand, the second option is an extension of the idea of the former to the case of four neutrinos $\nu_s, \nu_e, \nu_\mu, \nu_\tau$ (including one sterile neutrino $\nu_s$), when among their massive states $\nu_0, \nu_1, \nu_2, \nu_3$ there are no direct mixings between $\nu_0$ and $nu_2$, $\nu_0$ and $\nu_3$, $\nu_1$ and $\nu_3$, and the mass hierarchy $m^2_0 < {\sim} m^2_1 \ll m^2_2 < {\sim} m^2_3$ is now valid. Such an option, belonging to a class of textures widely discussed in literature, may be consistent with the observed deficits of solar $\nu_e$'s and atmospheric $\nu_\mu$'s as well as with the LSND appearance of $\nu_e$'s in the beam of accelerator $\nu_\mu$'s, if now $\Delta m^2_{10} \leftrightarrow \Delta m^2_{\rm sol}$, $\Delta m^2_{32} \leftrightarrow \Delta m^2_{\rm atm}$ and $\Delta m^2_{21} \leftrightarrow \Delta m^2_{\rm LSND}$.