Neutrinos are produced in weak interactions as states with definite flavor—electron, muon, or tau—and these flavor states are superpositions of states of different mass. As a neutrino propagates through space, the different mass eigenstates interfere, resulting in time-dependent flavor oscillation. Though matter is transparent to neutrinos, the flavor oscillation probability is modified when neutrinos travel through matter. Herein, we present an introduction to neutrino propagation through matter in a manner accessible to advanced undergraduate students. As an interesting application, we consider neutrino propagation through matter with a piecewise-constant density profile. This scenario has relevance in neutrino tomography, in which the density profile of matter, like the Earth's interior, can be probed via a broad-spectrum neutrino beam. We provide an idealized example to demonstrate the principle of neutrino tomography.
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For a purely two-neutrino system, the mixing matrix can always be taken as real since complex phases can be absorbed into the definitions of the fields. For a three-neutrino system, this is not the case. Three complex phases survive, though only one of these, the Dirac CP phase, is observable in neutrino oscillations.
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