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CHIC: Caley-Hamilton, Invariants and Constants for Neutrino Oscillation Probabilities and Gradients

Pablo Fernández-Menéndez

TL;DR

This work introduces an analytic framework for three-flavor neutrino oscillations in constant-density matter using the Cayley–Hamilton theorem, yielding a clean separation between energy and baseline dependencies through Hamiltonian invariants. It derives explicit expressions for the oscillation amplitude $\Psi$ as a second-order polynomial in the reduced Hamiltonian and provides analytic derivatives $\partial_\zeta P$ with respect to mixing parameters, enabling gradient-based analyses. The authors implement these results in the CHIC software (C++ and Python), achieving fast computation comparable to state-of-the-art codes while offering derivatives and the novel oscillograds visualization to probe local features of neutrino mixing. These capabilities support efficient data analyses, interpolation, and sensitivity studies for current and upcoming experiments like DUNE, Hyper-K, and JUNO. The work lays a foundation for extensions to varying-density media and beyond-Standard-Model scenarios within an efficient analytic framework.

Abstract

We use the Caley-Hamilton theorem to derive analytical solutions for the three-flavor neutrino propagation amplitude in a constant-density medium and their derivatives with respect to the mixing parameters. This approach avoids the diagonalization of the Hamiltonian and exploits precomputed matrix invariants to separate the dependence of oscillation probabilities on neutrino energy and propagation baseline. The results are implemented in the CHIC software, which provides simple, fast and efficient computation of oscillation probabilities and their derivatives. Finally, we demonstrate the value of probability gradients for neutrino data analyses and introduce a complementary visualization, the oscillograds, to probe underlying features of neutrino mixing.

CHIC: Caley-Hamilton, Invariants and Constants for Neutrino Oscillation Probabilities and Gradients

TL;DR

This work introduces an analytic framework for three-flavor neutrino oscillations in constant-density matter using the Cayley–Hamilton theorem, yielding a clean separation between energy and baseline dependencies through Hamiltonian invariants. It derives explicit expressions for the oscillation amplitude as a second-order polynomial in the reduced Hamiltonian and provides analytic derivatives with respect to mixing parameters, enabling gradient-based analyses. The authors implement these results in the CHIC software (C++ and Python), achieving fast computation comparable to state-of-the-art codes while offering derivatives and the novel oscillograds visualization to probe local features of neutrino mixing. These capabilities support efficient data analyses, interpolation, and sensitivity studies for current and upcoming experiments like DUNE, Hyper-K, and JUNO. The work lays a foundation for extensions to varying-density media and beyond-Standard-Model scenarios within an efficient analytic framework.

Abstract

We use the Caley-Hamilton theorem to derive analytical solutions for the three-flavor neutrino propagation amplitude in a constant-density medium and their derivatives with respect to the mixing parameters. This approach avoids the diagonalization of the Hamiltonian and exploits precomputed matrix invariants to separate the dependence of oscillation probabilities on neutrino energy and propagation baseline. The results are implemented in the CHIC software, which provides simple, fast and efficient computation of oscillation probabilities and their derivatives. Finally, we demonstrate the value of probability gradients for neutrino data analyses and introduce a complementary visualization, the oscillograds, to probe underlying features of neutrino mixing.

Paper Structure

This paper contains 18 sections, 39 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Neutrino Oscillations
  3. Three-Flavor Neutrino Propagation
  4. Gradient of the Neutrino propagation
  5. CHIC Library
  6. Results and Applications
  7. Conclusions
  8. Acknowledgements
  9. Derivatives of the Hamiltonian
  10. Electron Density, ne
  11. dm21
  12. dm31
  13. th23
  14. th12
  15. th13
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Comparison of cubic spline interpolation with (Hermite) and without the derivative information for the oscillation probability of $\nu_\mu$ to $\nu_e$ assuming an energy of 0.6 GeV and a baseline of 295 km.
  • Figure 2: Two-dimensional plots showing the transition probability (oscillogram, top) and its derivative with respect to $\delta_{CP}$ (oscillograd, bottom) of $\nu_\mu$ to $\nu_e$ assuming an energy of 0.6 GeV and a baseline of 295 km, and as a function of $\Delta m^2_{31}$ (vertical axis) and $\delta_{CP}$ (horizontal axis).
  • Figure 3: Two-dimensional plots showing the transition probability (oscillogram, top) and its derivative with respect to $\delta_{CP}$ (oscillograd, bottom) of $\nu_\mu$ to $\nu_e$ assuming a baseline of 1,300 km, and as a function of $\Delta m^2_{31}$ (vertical axis) and $\delta_{CP}$ (horizontal axis).