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NeutrinoOsc3Flavor: CP Phase Dependence in Three-Flavor Neutrino Oscillations: A Numerical Study in Vacuum and Matter

Baktiar Wasir Farooq, Bipin Singh Koranga, Ansh Prasad, Imran Khan

TL;DR

NeutrinoOsc3Flavor addresses the need for a transparent, verification-oriented reference implementation of exact three-flavor neutrino oscillations in vacuum and constant-density matter. The framework solves the flavor-basis Schrödinger equation by explicitly constructing the effective Hamiltonian in the PMNS formalism, diagonalizing it numerically with NumPy, and validating the results with an independent Cardano-based analytical solution for the matter eigenvalues. A key contribution is the explicit inclusion of full CP-violating phase dependence as a diagnostic for numerical stability and correctness, enabling CP-sensitivity studies in both vacuum and matter and providing a baseline for educational use and cross-checks against larger, experimental-oriented toolkits. The code emphasizes minimal dependencies, portability across Linux/Windows/macOS, and a pedagogical, equation-level viewpoint, making it a valuable resource for theoretical verification, teaching, and benchmarking of more complex neutrino-oscillation software. The approach is particularly relevant for CP-phase studies and baseline scenarios (e.g., DUNE), where precise, traceable calculations help validate numerical solvers and support reproducible research in neutrino phenomenology.

Abstract

We present NeutrinoOsc3Flavor, a lightweight and fully transparent computational framework for exact three flavor neutrino oscillation studies in vacuum and constant density matter. The code numerically solves the Schrodinger evolution equation in the flavor basis using explicit construction and diagonalization of the effective Hamiltonian within the PMNS formalism, including full CP Violating phase dependence. In contrast to large scale oscillation toolkits optimized for experimental simulations, NeutrinoOsc3Flavor is designed as a minimal dependency reference implementation, emphasizing analytical traceability, equation level accessibility, and cross platform portability. The framework relies solely on NumPy for numerical linear algebra and runs natively on both Linux and Windows systems without external compilation or specialized libraries. As an internal consistency and validation feature, we implement an independent analytical determination of the matter modified Hamiltonian eigenvalues using the Cardano method and demonstrate excellent agreement with numerical diagonalization. CP Phase dependence is used as a sensitive diagnostic of numerical stability and correctness of the evolution operator in both vacuum and matter. NeutrinoOsc3Flavor is intended as a verification oriented and pedagogical computational tool, suitable for theoretical cross-checks, educational use, and benchmarking of more complex neutrino oscillation software, rather than as a replacement for full experimental simulation frameworks. Here, we consider the DUNE experiments baseline length in the python implementation but in general we can implement any value of baseline length.

NeutrinoOsc3Flavor: CP Phase Dependence in Three-Flavor Neutrino Oscillations: A Numerical Study in Vacuum and Matter

TL;DR

NeutrinoOsc3Flavor addresses the need for a transparent, verification-oriented reference implementation of exact three-flavor neutrino oscillations in vacuum and constant-density matter. The framework solves the flavor-basis Schrödinger equation by explicitly constructing the effective Hamiltonian in the PMNS formalism, diagonalizing it numerically with NumPy, and validating the results with an independent Cardano-based analytical solution for the matter eigenvalues. A key contribution is the explicit inclusion of full CP-violating phase dependence as a diagnostic for numerical stability and correctness, enabling CP-sensitivity studies in both vacuum and matter and providing a baseline for educational use and cross-checks against larger, experimental-oriented toolkits. The code emphasizes minimal dependencies, portability across Linux/Windows/macOS, and a pedagogical, equation-level viewpoint, making it a valuable resource for theoretical verification, teaching, and benchmarking of more complex neutrino-oscillation software. The approach is particularly relevant for CP-phase studies and baseline scenarios (e.g., DUNE), where precise, traceable calculations help validate numerical solvers and support reproducible research in neutrino phenomenology.

Abstract

We present NeutrinoOsc3Flavor, a lightweight and fully transparent computational framework for exact three flavor neutrino oscillation studies in vacuum and constant density matter. The code numerically solves the Schrodinger evolution equation in the flavor basis using explicit construction and diagonalization of the effective Hamiltonian within the PMNS formalism, including full CP Violating phase dependence. In contrast to large scale oscillation toolkits optimized for experimental simulations, NeutrinoOsc3Flavor is designed as a minimal dependency reference implementation, emphasizing analytical traceability, equation level accessibility, and cross platform portability. The framework relies solely on NumPy for numerical linear algebra and runs natively on both Linux and Windows systems without external compilation or specialized libraries. As an internal consistency and validation feature, we implement an independent analytical determination of the matter modified Hamiltonian eigenvalues using the Cardano method and demonstrate excellent agreement with numerical diagonalization. CP Phase dependence is used as a sensitive diagnostic of numerical stability and correctness of the evolution operator in both vacuum and matter. NeutrinoOsc3Flavor is intended as a verification oriented and pedagogical computational tool, suitable for theoretical cross-checks, educational use, and benchmarking of more complex neutrino oscillation software, rather than as a replacement for full experimental simulation frameworks. Here, we consider the DUNE experiments baseline length in the python implementation but in general we can implement any value of baseline length.
Paper Structure (17 sections, 29 equations, 3 figures)

This paper contains 17 sections, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Neutrino oscillations in vacuum
  4. Matter effects
  5. Diagonalizing the effective Hamiltonian using numpy:
  6. Analytical Approach to get the eigenvalues of H extasciimacronf extasciimacronm:
  7. Code implementation in NeutrinoOsc3Flavor
  8. Three Flavor Neutrinos in Vacuum:
  9. Three Flavor Neutrinos in Matter:
  10. Eigenvalues using numpy:
  11. Eigenvalues using analytical method:
  12. Plots for the Probabilities in matter:
  13. Results
  14. Comparison of the eigenvalues of the Hfm matrix
  15. Variation with CP Phase for the Vacuum case:
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Comparison of muon neutrino survival probabilities in matter.The left panel shows the result obtained using NeutrinoOsc3Flavor, while the right panel shows the corresponding result obtained using NuOscProbExact3. The orange dashed curve represents the survival probability in matter.
  • Figure 2: In vacuum, the oscillation probability will have changes but not the survival probability
  • Figure 3: In matter, the oscillation probabilities will have changes and the survival probability will also show slight changes due to matter effects.