⚛️ high-energy theory

Wave packet description of Majorana neutrino oscillations in a magnetic field

This paper analytically solves the modified Dirac equation for Majorana neutrinos with transition magnetic moments in a magnetic field using a wave packet formalism to derive oscillation probabilities and demonstrate that decoherence effects, which depend on the relative strengths of vacuum and magnetic frequencies, can occur during propagation in supernova magnetic fields.

Original authors: Artem Popov, Alexander Studenikin, Alexander Tcvirov

Published 2026-02-04

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CC BY 4.0

Original authors: Artem Popov, Alexander Studenikin, Alexander Tcvirov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Neutrinos as "Ghostly Runners"

Imagine neutrinos as tiny, ghostly runners on a track. They are so light and interact so weakly with matter that they can pass through entire planets without stopping. In this paper, the authors are studying what happens to these runners when they encounter a very strong magnetic field, like the ones found inside exploding stars (supernovae).

Specifically, they are looking at a special type of neutrino called a Majorana neutrino. Think of a Majorana neutrino as a "chameleon" that is its own antiparticle. Unlike other particles that have a distinct "mirror image" (antiparticle), a Majorana neutrino is its own mirror. Because of this unique nature, it can only change its "magnetic personality" (called a transition magnetic moment) when it interacts with a magnetic field.

The Problem: The "Wave Packet" and the "Split"

To understand the paper, you need to understand two concepts: Oscillations and Decoherence.

Oscillations (The Dance): Neutrinos come in different "flavors" (like electron, muon, and tau). As they travel, they don't stay in one flavor; they dance back and forth between them. This is called oscillation.
Wave Packets (The Cloud): In quantum physics, a particle isn't just a single point; it's a fuzzy cloud of probability called a "wave packet." Imagine a runner not as a single dot, but as a cloud of mist.
Decoherence (The Clouds Drifting Apart): The paper focuses on what happens when the magnetic field causes these clouds to split. If a neutrino is a mix of two different "states" (like two different running speeds), the magnetic field might make one part of the cloud run slightly faster than the other.

If the two parts of the cloud run at different speeds, they will eventually drift so far apart that they stop "talking" to each other. When they stop talking, the neat "dance" of oscillation stops. The runner loses its rhythm. This stopping of the rhythm is called decoherence.

What the Authors Did

The authors used advanced math (solving a modified version of the famous Dirac equation) to track these "ghostly runners" through a magnetic field. They treated the neutrinos not as simple points, but as these fuzzy "wave packet" clouds.

They calculated two main things:

How likely is the neutrino to change flavor? (e.g., turning from an electron-neutrino to a muon-neutrino).
How far can the neutrino travel before the "clouds" drift apart and the oscillation stops? This distance is called the Coherence Length.

The Two Scenarios: A Tale of Two Speeds

The paper finds that the behavior of these neutrinos depends on a "tug-of-war" between two forces:

Vacuum Frequency ( $\omega_{vac}$ ): The natural rhythm of the neutrino changing flavors just because it has mass (even without a magnetic field).
Magnetic Frequency ( $\omega_B$ ): The rhythm forced upon the neutrino by the external magnetic field.

The authors discovered two distinct regimes:

1. The "Quiet" Field ( $\omega_{vac} \gg \omega_B$ ):
If the magnetic field is weak compared to the neutrino's natural rhythm, the magnetic field barely matters. The neutrino behaves just like it does in empty space. The distance it travels before the clouds drift apart (coherence length) is the same as it would be in a vacuum.

2. The "Stormy" Field ( $\omega_{vac} \ll \omega_B$ ):
If the magnetic field is incredibly strong (like in a supernova), it dominates the neutrino's behavior. Here is the paper's big discovery:

The distance the neutrino can travel before losing its rhythm (coherence length) becomes massively sensitive to its speed.
Specifically, the distance grows with the cube of the neutrino's energy.
The Analogy: Imagine a runner. In a normal field, if you double their speed, they might run twice as far before getting tired. But in this "stormy" magnetic field, if you double their speed, they can run eight times (2 cubed) as far before their rhythm breaks.

The Supernova Connection

The authors applied this math to a real-world scenario: Supernovae (exploding stars).

Supernovae have incredibly strong magnetic fields (trillions of times stronger than Earth's).
They produce huge numbers of neutrinos.
The authors calculated that for the neutrinos coming from a supernova, the magnetic field is strong enough to trigger this "stormy" regime.

The Result: In a supernova, the "clouds" of the neutrinos might drift apart much faster or much slower than expected, depending on their energy. This means the "flavor dance" of the neutrinos could be damped or stopped entirely before they even leave the star. This is a crucial detail for understanding what we see when we detect neutrinos from exploding stars.

Summary of Findings

New Physics: They successfully described how Majorana neutrinos behave in magnetic fields using the "wave packet" method, which accounts for the fuzziness of quantum particles.
The Cube Law: In strong magnetic fields, the distance a neutrino travels before losing its quantum rhythm is proportional to the cube of its energy. This is a unique signature of Majorana neutrinos in these conditions.
Supernova Impact: This effect is likely happening right now in supernovae. The strong magnetic fields there could cause neutrinos to "forget" their oscillation patterns due to decoherence, changing how we interpret the signals from these cosmic explosions.

The paper concludes that to truly understand the neutrinos coming from exploding stars, we cannot ignore the fact that their "wave clouds" might be drifting apart due to the intense magnetic fields they are passing through.

Technical Summary: Wave Packet Description of Majorana Neutrino Oscillations in a Magnetic Field

Problem Statement
While it is established that massive neutrinos possess non-zero electromagnetic form factors, including anomalous magnetic moments, the description of their propagation in astrophysical environments with extreme magnetic fields (e.g., supernovae) requires a rigorous treatment of quantum coherence. Previous studies have applied the wave packet formalism to vacuum oscillations, matter effects, and Dirac neutrino oscillations in magnetic fields. However, a comprehensive wave packet description for Majorana neutrinos propagating in strong magnetic fields, specifically accounting for the decoherence effects arising from the separation of wave packets with different group velocities, has not been fully developed. This paper addresses the need to analytically solve the modified Dirac equation for Majorana neutrinos with transition magnetic moments and derive oscillation probabilities that include decoherence damping.

Methodology
The authors employ the wave packet formalism to describe the propagation of Majorana neutrinos. The methodology involves the following steps:

Theoretical Framework: The interaction is described via an effective Lagrangian involving a hermitian and antisymmetric matrix of transition magnetic moments ( $\mu_{ik}$ ) in the massive neutrino basis. The authors utilize the modified Dirac equation for Majorana neutrinos in a magnetic field.
Wave Packet Construction: Massive neutrino states are modeled as one-dimensional Gaussian wave packets with an average momentum $p_0$ and momentum spread $\sigma_p$ . The authors distinguish between left-handed (neutrino) and right-handed (antineutrino) states, noting that for Majorana particles, right-handed states behave similarly to Dirac antineutrinos in creation/annihilation processes.
Analytical Solution: The modified Dirac equation is solved analytically for the two-flavor case. The Hamiltonian is constructed as a block matrix, and its positive eigenvalues (dispersion relations) are derived.
Approximations: The analysis assumes ultra-relativistic neutrinos ( $p_0 \gg m_i$ ) and that the neutrino mass dominates the magnetic interaction energy ( $m_i \gg \mu B_\perp$ ), a condition justified by current experimental bounds even for strong astrophysical fields.
Decoherence Calculation: The group velocity difference ( $\Delta v$ ) between the stationary states is calculated. This difference drives the spatial separation of wave packets, leading to decoherence. The authors derive expressions for oscillation probabilities between flavors ( $\nu_\alpha \to \nu_\beta$ ) and between neutrinos and antineutrinos ( $\nu_\alpha \to \bar{\nu}_\beta$ ), incorporating an exponential damping term dependent on the coherence length ( $L_{coh}$ ).

Key Contributions and Results
The paper provides the first analytical derivation of oscillation probabilities for Majorana neutrinos in a magnetic field that explicitly accounts for wave packet decoherence. The primary results include:

Modified Oscillation Probabilities: The authors derive expressions for transition probabilities (Eqs. 18–20) that include a damping factor $\exp(-L^2/L_{coh}^2)$ . These expressions generalize previous plane-wave results to the wave packet regime.
Coherence Length Regimes: A critical finding is the behavior of the coherence length ( $L_{coh}$ $L_{co h}$ ) in two asymptotic regimes defined by the ratio of the vacuum oscillation frequency ( $\omega_{vac}$ $ω_{v a c}$ ) to the magnetic oscillation frequency ( $\omega_B$ $ω_{B}$ ):
- Weak Magnetic Field ( $\omega_{vac} \gg \omega_B$ ): The coherence length coincides with that of vacuum neutrino oscillations, scaling as $L_{coh} \propto p_0^2$ . In this regime, neutrino-antineutrino transitions are strongly suppressed.
- Strong Magnetic Field ( $\omega_{vac} \ll \omega_B$ ): The coherence length becomes proportional to the cube of the average neutrino momentum ( $L_{coh} \propto p_0^3$ ). This cubic dependence is identified as a distinctive signature of Majorana neutrinos in strong magnetic fields.
Supernova Application: The authors apply their results to supernova environments with magnetic fields of $10^{11}$ to $10^{12}$ Gauss. Assuming a transition magnetic moment of $\mu = 10^{-12}\mu_B$ and neutrino energies of $\sim 10$ MeV, they calculate coherence lengths of $\sim 10–100$ km. This scale is comparable to the size of the supernova region where strong magnetic fields are observed.

Significance and Claims
The paper claims that for Majorana neutrinos propagating in the strong magnetic fields of supernovae, the regime $\omega_B \gg \omega_{vac}$ is likely realized. In this regime, the decoherence effect due to wave packet separation is significant and may occur within the supernova region itself. Consequently, this effect could lead to the damping of both flavor transitions and neutrino-antineutrino transitions, potentially influencing the flavor evolution of supernova neutrinos.

The authors note that while their numerical estimations suggest decoherence is important, a more detailed analysis would require incorporating neutrino interactions with matter and realistic magnetic field models, which they identify as a topic for future research. The work is supported by the Russian Science Foundation and the National Center for Physics and Mathematics.