Open Path Geometric Phases in Three Flavor Neutrino Oscillations through Layered Matter

Abstract

We present a gauge invariant formulation of open path geometric phases in the context of three flavor neutrino oscillations traversing nonuniform matter. In contrast to the standard Berry phase, which is intrinsically associated with closed adiabatic cycles, neutrino propagation from source to detector typically traces an open trajectory within the parameter space of matter dependent Hamiltonians. We derive this formalism within a layered matter framework, where each discrete layer is characterized by a constant density and a local matter eigenbasis. The total evolution operator is expressed as a path ordered product of dynamical propagation factors and interface matrices that bridge adjacent matter eigenbases. This construction naturally yields geometric phase chains composed of endpoint projections and interface overlaps. We demonstrate that these chains remain invariant under arbitrary local rephasings of the instantaneous matter eigenstates, thereby resolving the gauge dependence ambiguity typically associated with open path transport. The proposed formalism elucidates the noncommutative nature of flavor evolution in layered media, illustrating that profiles with identical column densities but distinct ordering can yield different transition amplitudes. Furthermore, we analyze the influence of the Dirac CP phase, showing that it deforms the geometry of matter eigenbasis transport without being equivalent to the geometric phase itself. In the continuous limit, finite interface overlaps converge to Berry connection factors, while endpoint projections remain indispensable for ensuring open path gauge invariance. We discuss the phenomenological implications of this structure, emphasizing that geometric phases are not independent observables but contribute to transition probabilities via interference among coherent propagation chains.