Study of the day-night effect for the Super-Kamiokande detector. III. The case of transitions into sterile neutrino

Using the results of a high precision calculation of the solar neutrino survival probability for Earth crossing neutrinos in the case of the Mikheyev-Smirnov-Wolfenstein (MSW) νₑ→νₛ transition solution of the solar neutrino problem, we derive predictions for the one-year averaged day-night (D-N) asymmetries in the deformed recoil-e^- spectrum and in the energy-integrated event rate due to the solar neutrinos, to be measured with the Super-Kamiokande detector. The asymmetries are calculated for three event samples, produced by the solar νₑ crossing the Earth mantle only, the core, and the mantle only + the core (the full night sample), for a large set of representative values of the MSW transition parameters Δm² and sin²2θV from the ``conservative'' νₑ→νₛ solution regions, obtained by taking into account the possible uncertainties in the predictions for the ⁸B and ⁷Be neutrino fluxes. The effects of the uncertainties in the value of the bulk matter density and in the chemical composition of the Earth core on the predictions for the D-N asymmetries are investigated. The dependence of the D-N effect related observables on the threshold recoil-e^- kinetic energy, Te,th, is studied. It is shown, in particular, that for sin²2θV<~0.030, the one year average D-N asymmetry in the sample of events due to the core-crossing neutrinos is larger than the asymmetry in the full night sample by a factor which, depending on the solution value of Δm², can be {~}(3--4) (Δm²<5×10^-6eV²) or {~}(1.5--2.5) $(5×{}{10}^{{-}6}{eV}²{}{Δ}{m}²{}8×{}{10}^{{-}6}{eV}²).$ We find, however, that at small mixing angles ${sin}²2{{θ}}V{}0.014,$ the $D{-}N$ asymmetry in the case of solar ${{ν}}ₑ{→}{{ν}}ₛ$ transitions is considerably smaller than if the transitions were into an active neutrino, ${{ν}}ₑ{→}{{ν}}_{{μ}({τ})}.$ In particular, a precision better than 1% in the measurement of any of the three one year averaged $D{-}N$ asymmetries considered by us would be required to test the small mixing angle nonadiabatic ${{ν}}ₑ{→}{{ν}}ₛ$ solution at ${sin}²2{{θ}}V{}0.01.$ For $0.0075{}{sin}²2{{θ}}V<~0.03,$ the magnitude of the $D{-}N$ asymmetry in the sample of events due to the core-crossing neutrinos is very sensitive to the value of the electron number fraction in the Earth's core, ${Y}ₑ(core):$ a change of ${Y}ₑ(core)from the standard value of 0.467 to the conservative upper limit of 0.50 can lead to an increase of the indicated asymmetry by a factor of ~(3--4). Iso-(D{-}N)$ asymmetry contours in the ${Δ}{m}²{-}{sin}²2{{θ}}V$ plane for the Super-Kamiokande detector are derived in the region ${sin}²2{{θ}}V>~{10}^{{-}4}$ for the three event samples studied for ${T}e,th=5MeV$ and 7.5 MeV, and in the case of the samples due to the core crossing and (only mantle $crossing+core$ crossing) neutrinos---for ${Y}ₑ(core)=0.467$ and 0.50. The possibility to discriminate between the ${{ν}}ₑ{→}{{ν}}ₛ$ and ${{ν}}ₑ{→}{{ν}}_{{μ}({τ})}$ solutions of the solar neutrino problem by performing high precision $D{-}N$ asymmetry measurements is also discussed.