Pancharatnam-Berry phase in neutrino mixing

The Pancharatnam - Berry phase (PBP) of purely geometric origin appears as a reparametrization invariant quantity of ray Space. In this article, we investigate the properties exhibited by PBP in neutrino mixing. We map the neutrino flavour modes to independent flavour vacuum states and compute PBP using Bargmann invariant. We derive the exact formula for PBP in two flavour approximation using the kinematic approach. Our result reproduces previous results of Blasone et al. under cyclic condition. Inspired by the work of Mukunda and Simon, we investigate the total and dynamical phases separately. This method leads us to identify the existence of nodal points in the mixing parameter space. At nodal points, PBP changes by a value $\pi$, and it originates from the total phase. We report the direct relation between nodal points and MSW resonance, giving physical meaning to nodal points. Our analysis shows the ability of PBP to differentiate between different mass hierarchies and set numerical bounds to $\Delta m^2$ by changing total energy. We extend our studies to three flavour model and found that PBP is sensitive to the Dirac $CP$ phase ($\delta_{CP}$). Using our $N$-qubit architecture of the $N$-flavour neutrino model, one can immediately study the dynamical characteristics like mode entanglement between neutrino flavour modes.