Hidden sterile neutrino and the 2+2 sum rule

We discuss the oscillations of atmospheric and solar neutrinos into sterile neutrinos in the 2+2 scheme. A zeroth order sum rule requires equal probabilities for oscillation into ${\nu }_s$ and ${\nu }_{\tau }$ in the solar+atmospheric data sample. Data do not favor this claim. Here we use scatter plots to assess the corrections of the zeroth order sum rule when (i) the $4\times 4$ neutrino mixing matrix assumes its full range of allowed values, and (ii) matter effects are included. We also introduce a related “product rule.” We find that the sum rule is significantly relaxed, due to both the inclusion of the small mixing angles (which provide a short-baseline contribution) and to matter effects. The product rule is also dramatically altered. The observed relaxation of the sum rule weakens the case against the 2+2 model and the sterile neutrino. To invalidate the 2+2 model, the daunting task of fitting global data with the small mixing angles included seems to be required.