First Constraints on the Complete Neutrino Mixing Matrix with a Sterile Neutrino

Neutrino oscillation models involving one extra mass eigenstate beyond the standard three ($3+1$) are fit to global short baseline experimental data and the recent IceCube $\nu_\mu+\bar\nu_\mu$ disappearance search result. We find a best fit of $\Delta m^2_{41} = 1.75\; \text{eV}^2$ with $\Delta \chi^2_{null-min}$ (dof) of 50.61 (4). We find that the combined IceCube and short baseline data constrain $\theta_{34}$ to $< 80^\circ (< 6^\circ) $ at 90\% C.L. for $\Delta m^2_{41} \approx 2 (6)~\text{eV}^2$, which is improved over present limits. Incorporating the IceCube information provides the first constraints on all entries of the 3+1 mixing matrix.