Calculating error bars for neutrino mixing parameters

Abstract

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 $\ensuremath{\Delta}{\ensuremath{\chi}}^{2}$ 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 $\ensuremath{\Delta}{\ensuremath{\chi}}^{2}$ 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 ${\ensuremath{\chi}}^{2}$ function; it is particularly useful for cases in which the minimum of the ${\ensuremath{\chi}}^{2}$ function is near a boundary. We point out that the question of what is the probability that a parameter is not zero is more precisely worded as what is the maximum confidence level at which the value of zero is not included. 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 $\ensuremath{\Delta}{\ensuremath{\chi}}^{2}$ from normal statistics. For example, we find that for the T2K experiment the value of $\ensuremath{\Delta}{\ensuremath{\chi}}^{2}$ corresponding to a confidence level of 90$%$ is 3.57 rather than the normal statistics value of 2.71.