Wave packet description of Majorana neutrino oscillations in a magnetic field

Majorana neutrino oscillations in a magnetic field are considered using the wave packets formalism. The modified Dirac equation for Majorana neutrinos with non-zero transition magnetic moments propagating in a magnetic field is solved analytically in the two flavour case. The expressions for the oscillations probabilities are derived accounting for the decoherence effect emerging at distances exceeding the coherence length. It is shown that for Majorana neutrinos propagating in a magnetic field the coherence length coincides with the coherence length for neutrino oscillations in vacuum when the vacuum frequency is much greater than the magnetic frequency ($ω_{vac} \gg ω_B$), while it is proportional to the cube of the average neutrino momentum if ($ω_{vac} \ll ω_B$). We show that the decoherence effect may appear during neutrino propagation in a magnetic field of supernova.

💡 Research Summary

The paper investigates how Majorana neutrinos with non‑zero transition magnetic moments behave when they travel through a strong magnetic field, using the wave‑packet formalism. Starting from the effective Lagrangian that couples the antisymmetric magnetic‑moment matrix µij to the external field, the authors construct one‑dimensional Gaussian wave packets for the mass eigenstates and derive a modified Dirac equation that includes the magnetic interaction term. By diagonalising the resulting 2 × 2 Hamiltonian for the two‑flavour system, they obtain analytic expressions for the energy eigenvalues, which contain both the usual vacuum oscillation frequency ωvac = Δm²/(4k) and the magnetic frequency ωB = µB⊥.

Assuming ultra‑relativistic neutrinos (p0 ≫ mi) and a magnetic interaction much weaker than the masses (µB⊥ ≪ mi), the dispersion relation is expanded to first order in 1/k. The group‑velocity difference Δv between the two mass eigenstates is then calculated. Two asymptotic regimes emerge:

1. Vacuum‑dominated regime (ωvac ≫ ωB). Here Δv ≈ Δm²/(2k²), identical to the standard vacuum case.

2. Magnetic‑dominated regime (ωvac ≪ ωB). In this case Δv ≈ (Δm²)²/(ωB k³) + 2 ωB (m1 + m2)²/k³, showing a k⁻³ dependence.

The group‑velocity difference controls the separation of the two wave packets and therefore the decoherence (or loss of coherence) of the oscillation pattern. The coherence length is defined as Lcoh = 2√2 σx / Δv, where σx = 1/(2σp) is the spatial width of the packet. Consequently, the coherence length behaves as

• Lcoh ≈ 4√2 σx p0²/Δm² for ωvac ≫ ωB (the usual vacuum result), and

• Lcoh ≈ 2√2 σx p0³