The Daya Bay, RENO, and Double Chooz experiments have discovered a large non-zero value for θ13. We present a global analysis that includes these three experiments, Chooz, the Super-K atmospheric data, and the νµ → νe T2K and MINOS experiments that are sensitive to the hierarchy and the sign of θ13. We report preliminary results in which we fix the mixing parameters other than θ13 to those from a recent global analysis. Given there is no evidence for a non-zero CP violation, we assume δ = 0. T2K and MINOS lie in a region of L/E where there is a hierarchy degeneracy in the limit of θ13 → 0 and no matter interaction. For nonzero θ13, the symmetry is partially broken, but a degeneracy under the simultaneous exchange of both hierarchy and the sign of θ13 remains. Matter effects break this symmetry such that the positions of the peaks in the oscillation probabilities maintain the two-fold symmetry, while the magnitude of the oscillations is sensitive to the hierarchy. This renders T2K and NOνA, with different baselines and different matter effects, better able in combination to distinguish the hierarchy and the sign of θ13. The present T2K and MINOS data do not distinguish between hierarchies or the sign of θ13, but the large value of θ13 yields effects from atmospheric data that do. We find for normal hierarchy, positive θ13, sin 2 2θ13 = 0.090 ± 0.020 and is 0.2% probable it is the correct combination; for normal hierarchy, negative θ13, sin 2 2θ13 = 0.108 ± 0.023 and is 2.2% probable; for inverse hierarchy, positive θ13, sin 2 2θ13 = 0.110±0.022 and is 7.1% probable; for inverse hierarchy, negative θ13, sin 2 2θ13 = 0.113 ± 0.022 and is 90.5% probable, results that are inconsistent with two similar analyses.

One goal of contemporary particle physics is to determine the mixing angles and mass-squared differences that constitute the phenomenological constants that describe neutrino oscillations. Of great interest are not only the best fit values of these constants but also their errors. Some of the neutrino oscillation data is statistically poor and cannot be treated by normal (Gaussian) statistics. To extract confidence intervals when the statistics are not normal, one should not utilize the value for chisquare versus confidence level taken from normal statistics. Instead, we propose that one should use the normalized likelihood function as a probability distribution; the relationship between the correct chisquare and a given confidence level can be computed by integrating over the likelihood function. This allows for a definition of confidence level independent of the functional form of the !2 function; it is particularly useful for cases in which the minimum of the !2 function is near a boundary. We present two pedagogic examples and find that the proposed method yields confidence intervals that can differ significantly from those obtained by using the value of chisquare from normal statistics. For example, we find that for the first data release of the T2K experiment the probability that chisquare is not zero, as defined by the maximum confidence level at which the value of zero is not allowed, is 92%. Using the value of chisquare at zero and assigning a confidence level from normal statistics, a common practice, gives the over estimation of 99.5%.Comment: 9 pages, 6 figure

The only experimentally observed phenomenon that lies outside the standard model of the electroweak interaction is neutrino oscillations. A way to try to unify the extensive neutrino oscillation data is to add a phenomenological mass term to the Lagrangian that is not diagonal in the flavor basis. The goal is then to understand the world's data in terms of the parameters of the mixing matrix and the differences between the squares of the masses of the neutrinos. An outstanding question is what is the correct ordering of the masses, the hierarchy question. We point out a broken symmetry relevant to this question, the symmetry of the simultaneous interchange of hierarchy and the sign of θ 13 . We first present the results of an analysis of data that well determine the phenomenological parameters but are not sensitive to the hierarchy. We find θ 13 = 0.152±0.014, θ 23 = 0.25 +0.03 −0.05 π and ∆ 32 = 2.45±0.14×10 −3 eV 2 , results consistent with others. We then include data that are sensitive to the hierarchy and the sign of θ 13 . We find, unlike others, four isolated minimum in the χ 2space as predicted by the symmetry. Now that Daya Bay and RENO have determined θ 13 to be surprisingly large, the Super-K atmospheric data produce meaningful symmetry breaking such that the inverse hierarchy is preferred at the 97.2 % level.