Boxing with neutrino oscillations

Abstract

We develop a characterization of neutrino oscillations based on the coefficients of the oscillating terms. These coefficients are individually observable; although they are quartic in the elements of the unitary mixing matrix, they are independent of the conventions chosen for the angle and phase parametrization of the mixing matrix. We call these reparametrization-invariant observables ``boxes'' because of their geometric relation to the mixing matrix, and because of their association with the Feynman box diagram that describes oscillations in field theory. The real parts of the boxes are the coefficients for the CP- or T-even oscillation modes, while the imaginary parts are the coefficients for the CP- or T-odd oscillation modes. Oscillation probabilities are linear in the boxes, so measurements can straightforwardly determine values for the boxes (which can then be manipulated to yield magnitudes of mixing matrix elements). We examine the effects of unitarity on the boxes and discuss the reduction of the number of boxes to a minimum basis set. For the three-generation case, we explicitly construct the basis. Using the box algebra, we show that CP violation may be inferred from measurements of neutrino flavor mixing even when the oscillatory factors have averaged. The framework presented here will facilitate general analyses of neutrino oscillations among $n>~3$ flavors.