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 $ν_e, ν_μ, ν_τ$, when among their massive states $ν_1, ν_2, ν_3$ there is no direct mixing between $ν_1$ and $ν_3$, 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 $ν_e$'s and atmospheric $ν_μ$'s, if $Δm^2_{21} \leftrightarrow Δm^2_{\rm sol}$ and $ Δm^2_{32} \leftrightarrow Δm^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 $ν_s, ν_e, ν_μ, ν_τ$ (including one sterile neutrino $ν_s$), when among their massive states $ν_0, ν_1, ν_2, ν_3$ there are no direct mixings between $ν_0$ and $nu_2$, $ν_0$ and $ν_3$, $ν_1$ and $ν_3$, 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 $ν_e$'s and atmospheric $ν_μ$'s as well as with the LSND appearance of $ν_e$'s in the beam of accelerator $ν_μ$'s, if now $Δm^2_{10} \leftrightarrow Δm^2_{\rm sol}$, $Δm^2_{32} \leftrightarrow Δm^2_{\rm atm}$ and $Δm^2_{21} \leftrightarrow Δm^2_{\rm LSND}$.