Radiative neutrino mass matrix for three active plus one sterile species

A simple unifying mass matrix is presented for the three active and one sterile neutrinos $\nu_e$, $\nu_\mu$, $\nu_\tau$ and $\nu_s$, using an extension of the radiative mechanism proposed some time ago by Zee. The total neutrino-oscillation data are explained by the scheme $\nu_e \leftrightarrow \nu_s$ (solar), $\nu_\mu \leftrightarrow \nu_\tau$ (atmospheric) and $\nu_e \leftrightarrow \nu_\mu$ (LSND). We obtain the interesting approximate relationship $(\Delta m^2)_{atm} \simeq 2[(\Delta m^2)_{solar} (\Delta m^2)_{LSND}]^{1/2}$ which is well satisfied by the data.