^{1}and David C. Latimer

^{1}

### Abstract

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.

The authors are grateful for useful comments from Greg Elliott, Joel Franklin, Mary James, and Jerry Shurman.

I. INTRODUCTION

II. VACUUM OSCILLATION

III. MATTER INTERACTIONS

IV. PIECEWISE-CONSTANT DENSITY PROFILE

V. THE INVERSION PROBLEM: LOW DENSITY LIMIT

VI. CONCLUDING REMARKS

### Key Topics

- Neutrinos
- 95.0
- Neutrino oscillations
- 40.0
- Forward scattering
- 12.0
- Tomography
- 10.0
- Weak interactions
- 10.0

## Figures

Forward scattering of an electron neutrino on an electron mediated by the charged-current interaction.

Forward scattering of an electron neutrino on an electron mediated by the charged-current interaction.

The solid line is a plot of the ratio as a function of energy in units of the resonance energy. The dashed line is a plot of the ratio as a function of energy. Both curves use the vacuum mixing angle .

The solid line is a plot of the ratio as a function of energy in units of the resonance energy. The dashed line is a plot of the ratio as a function of energy. Both curves use the vacuum mixing angle .

Oscillation probability in vacuum or constant-density matter with fixed vacuum mixing angle and energy E = 100 MeV. Thebaseline is expressed in terms of the vacuum oscillation wavelength km. The solid line depicts vacuum oscillations. For the dotted line, the density is g/cm3 so that . For the dashed line, the density is g/cm3 so that .

Oscillation probability in vacuum or constant-density matter with fixed vacuum mixing angle and energy E = 100 MeV. Thebaseline is expressed in terms of the vacuum oscillation wavelength km. The solid line depicts vacuum oscillations. For the dotted line, the density is g/cm3 so that . For the dashed line, the density is g/cm3 so that .

The solid curves depict the oscillation probability for 200 MeV neutrinos traveling through the density profile in Eq. (22) with g/cm3 and g/cm3. The white areas indicate the regions of density , and the shaded areas represent the regions of density . The dashed lines represent the oscillation probability were the higher density region not present. The vacuum mixing angle is , km, and km. For (a), and . For (b), and .

The solid curves depict the oscillation probability for 200 MeV neutrinos traveling through the density profile in Eq. (22) with g/cm3 and g/cm3. The white areas indicate the regions of density , and the shaded areas represent the regions of density . The dashed lines represent the oscillation probability were the higher density region not present. The vacuum mixing angle is , km, and km. For (a), and . For (b), and .

The solid curves depict the oscillation probability for 200 MeV neutrinos traveling through the density profile in Eq. (22) with g/cm3 and g/cm3. The white areas indicate the regions of density , and the shaded areas represent the regions of density . The dashed lines represent the oscillation probability were the higher density region not present. The vacuum mixing angle is , km, and km. For (a), and . For (b), and .

The difference in oscillation probabilities, , for km, km, and km. The vacuum mixing angle is , and the matter densities are g/cm3 and g/cm3. The anti-neutrino energy ranges from 1 to 10 MeV. The solid curve is computed without approximation from Eq. (23) . The dashed curve is computed from the approximate expression in Eq. (29) .

The difference in oscillation probabilities, , for km, km, and km. The vacuum mixing angle is , and the matter densities are g/cm3 and g/cm3. The anti-neutrino energy ranges from 1 to 10 MeV. The solid curve is computed without approximation from Eq. (23) . The dashed curve is computed from the approximate expression in Eq. (29) .

The magnitude of the Fourier transform of . Only positive values of are shown.

The magnitude of the Fourier transform of . Only positive values of are shown.

A plane wave impinges upon a slab of thickness d at normal incidence. The primed coordinates refer to positions within the slab.

A plane wave impinges upon a slab of thickness d at normal incidence. The primed coordinates refer to positions within the slab.

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