Neutrino Flavor Changes in the DRUMS Framework

1. Neutrino as Superfluid Excitations

In DRUMS, neutrinos are excitations of the coherent superfluid medium coupled to the cubic magnetic substrate:

\[ \Psi_\nu(\mathbf{x},t) = \sum_i \psi_i(\mathbf{x},t) e^{i\theta_i(\mathbf{x},t)} \]

Each \(\psi_i\) corresponds to a flavor state \(\nu_e, \nu_\mu, \nu_\tau\).

2. Flavor Mixing via Phase Coupling

Flavor changes arise from differential phase evolution:

\[ \frac{d}{dt} \begin{pmatrix} \psi_e \\ \psi_\mu \\ \psi_\tau \end{pmatrix} = - i \begin{pmatrix} \phi_e & \kappa_{e\mu} & \kappa_{e\tau} \\ \kappa_{e\mu} & \phi_\mu & \kappa_{\mu\tau} \\ \kappa_{e\tau} & \kappa_{\mu\tau} & \phi_\tau \end{pmatrix} \begin{pmatrix} \psi_e \\ \psi_\mu \\ \psi_\tau \end{pmatrix} \]

Where \(\phi_i\) are phase potentials from local superfluid density and \(\kappa_{ij}\) are coupling terms induced by cubic substrate interactions.

3. Oscillation Probability

The probability for a neutrino to change flavor over distance \(L\) is:

\[ P_{\nu_\alpha \to \nu_\beta} = | \langle \psi_\beta(L) | \psi_\alpha(0) \rangle |^2 = \sin^2(2\theta) \sin^2\left( \frac{\Delta \phi \, L}{2} \right) \]

Here \(\Delta \phi = \phi_2 - \phi_1\) represents phase difference between eigenstates; \(\theta\) is effective mixing angle from substrate-mediated couplings.

4. Matter Effects

Superfluid density variations in stellar interiors or Earth modify \(\phi_i\), producing MSW-like resonance:

\[ \phi_i = \phi_i^0 + V_i(\rho_{sf}) \]

Where \(V_i\) is the effective potential from local superfluid density \(\rho_{sf}\).

5. Energy Dependence

Phase accumulation depends on neutrino energy \(E\):

\[ \Delta \phi(E) = \frac{\Delta m_{eff}^2}{2 E} \]

Where \(\Delta m_{eff}^2\) emerges from superfluid-mediated interactions, reproducing observed energy-dependent oscillation patterns.

6. Coherence and Decoherence

Superfluid coherence length sets the maximum distance for flavor oscillations:

\[ L_{coh} \sim \frac{2 \pi}{|\nabla \theta_1 - \nabla \theta_2|} \]

Beyond \(L_{coh}\), oscillations average out, consistent with observed long-baseline neutrino behavior.

7. Final Interpretation

Within the DRUMS framework, neutrino flavor changes are fully explained as:

  • Phase-dependent oscillations of neutrino excitations in the superfluid medium
  • Coupling induced by the cubic magnetic substrate determines effective mixing angles
  • Energy and matter dependence naturally emerge from superfluid density variations
  • Coherence length and phase accumulation reproduce observed oscillation probabilities

No ad hoc neutrino mass assumptions are required; the flavor change is a direct consequence of superfluid dynamics and substrate interactions.