Restricting the LSND and MiniBooNE sterile neutrinos with the IceCube atmospheric neutrino data

We study oscillations of the high energy atmospheric neutrinos in the Earth into sterile neutrinos with the eV-scale mass. The MSW resonance and parametric enhancement of the $\bar{\nu}_\mu\to\bar{\nu}_s$ oscillations lead to distortion of the zenith angle distribution of the muon-track events which can be observed by IceCube. Due to matter effect, the IceCube signal depends not only on the mixing element $U_{\mu 4}$ relevant for LSND and MiniBooNE but also on $U_{\tau 4}$ and the CP-violating phase $\delta_{24}$. We show that the case with $U_{\tau 4} = \delta_{24} = 0$ leads to the weakest IceCube signal and therefore should be used to bound $U_{\mu 4}$. We compute the zenith angle distributions of the $\nu_\mu-$events for different energy intervals in the range (0.1 - 10) TeV and find that inclusion of the energy information (binning in energy) improves the sensitivity to $\nu_s$ drastically. We estimate that with already collected (during 3 - 4 years) IceCube statistics the bound $|U_{\mu 4}|^2 < 0.01$ ($99\%$ C.L.) can be established and the mixing required by LSND and MiniBooNE can be excluded at $(4 - 6) \sigma$ confidence level.