Neutrino oscillations: what do we know about $\theta_{13}$

The phenomenon of neutrino oscillations is reviewed. A new analysis tool for the recent, more finely binned Super-K atmospheric data is outlined. This analysis incorporates the full three-neutrino oscillation probabilities, including the mixing angle $\theta_{13}$ to all orders, and a full three- neutrino treatment of the Earth's MSW effect. Combined with the K2K, MINOS, and CHOOZ data, the upper bound on $\theta_{13}$ is found to arise from the Super-K atmospheric data, while the lower bound arises from CHOOZ. This is caused by the linear in $\theta_{13}$ terms which are of particualr importance in the region $L/E>10^4$ m/MeV where the sub-dominant expansion is not convergent. In addition, the enhancement of $\theta_{12}$ by the Earth MSW effect is found to be important for this result. The best fit value of $\theta_{13}$ is found to be (statistically insignificantly) negative and given by $\theta_{13}=-0.07^{+0.18}_{-0.11}$. In collaboration with Jesus Escamilla, Vanderbilt University and David Latimer, University of Kentucky.