# Neutrino eigenstates and flavour, spin and spin-flavour oscillations in a constant magnetic field

## Abstract

We further develop a recently proposed new approach to the description of the relativistic neutrino flavour \(\nu _e^L \leftrightarrow \nu _{\mu }^L\), spin \(\nu _e^L \leftrightarrow \nu _{e}^R\) and spin-flavour \(\nu _e^L \leftrightarrow \nu _{\mu }^R\) oscillations in a constant magnetic field that is based on the use of the exact neutrino stationary states in the magnetic field. The neutrino flavour, spin and spin-flavour oscillations probabilities are calculated accounting for the whole set of possible conversions between four neutrino states. In general, the obtained expressions for the neutrino oscillations probabilities exhibit new inherent features in the oscillation patterns. It is shown, in particular, that: (1) in the presence of the transversal magnetic field for a given choice of parameters (the energy and magnetic moments of neutrinos and the strength of the magnetic field) the amplitude of the flavour oscillations \(\nu _e^L \leftrightarrow \nu _{\mu }^L\) at the vacuum frequency is modulated by the magnetic field frequency, (2) the neutrino spin oscillation probability (without change of the neutrino flavour) exhibits the dependence on the mass square difference \(\varDelta m^2\). It is shown that the discussed interplay of neutrino oscillations in magnetic fields on different frequencies can have important consequences in astrophysical environments, in particular in those peculiar for magnetars.

## 1 Introduction

The neutrino magnetic moment precession in the transversal magnetic field \(\mathbf{B}_{\perp }\) was first considered in [3] (this possibility was also mentioned in [10]), then the spin-flavor precession in vacuum was discussed in [11], the importance of the matter effect was emphasized in [12]. The effect of the resonant amplification of neutrino spin oscillations in \(\mathbf{B}_{\perp }\) in the presence of matter was proposed in [13, 14], the magnetic field critical strength the presence of which makes spin oscillations significant was introduced [15, 16], the impact of the longitudinal magnetic field \(\mathbf{B}_{||}\) was discussed in [17] and just recently in [18]. In a series of papers [19, 20, 21, 22] the solution of the solar neutrino problem was discussed on the basis of neutrino oscillations with a subdominant effect from the neutrino transition magnetic moments conversion in the solar magnetic field (the spin-flavour precession).

Following to the general idea first implemented in [23, 24], we further develop a new approach to the description of the relativistic neutrino flavour \(\nu _e^L \leftrightarrow \nu _{\mu }^L\), spin \(\nu _e^L \leftrightarrow \nu _{e}^R\) and spin-flavour \(\nu _e^L \leftrightarrow \nu _{\mu }^R\) oscillations in the presence of an arbitrary constant magnetic field. Our approach is based on the use of the exact stationary states in the magnetic field for the classification of neutrino spin states, contrary to the customary approach when the neutrino helicity states are used for this purpose.

Within this customary approach the helicity operator is used for the classification of a neutrino spin states in a magnetic field. The helicity operator does not commute with the neutrino evolution Hamiltonian in an arbitrary constant magnetic field and the helicity states are not stationary in this case. This resembles situation of the flavour neutrino oscillations in the presence of matter when the neutrino mass states are also not stationary. In the presence of matter the neutrino flavour states are considered as superpositions of stationary states in matter. These stationary states are characterized by “masses” \({\widetilde{m}}_{i}(n_{eff})\) that are dependent on the matter density \(n_{eff}\) and the effective neutrino mixing angle \(\tilde{\theta }_{eff}\) is also a function of the matter density.

The proposed alternative approach to the problem of neutrino oscillations in a magnetic field is based on the use of the exact solutions of the corresponding Dirac equation for a massive neutrino wave function in the presence of a magnetic field that stipulates the description of the neutrino spin states with the corresponding spin operator that commutes with the neutrino dynamic Hamiltonian in the magnetic field. In what follows, we also account for the complete set of conversions between four neutrino states.

## 2 Massive neutrino in a magnetic field

*L*is the normalization length. Thus, for the quadratic combinations of the coefficients we get

## 3 Neutrino flavour, spin and spin-flavour oscillations in a magnetic field

From the obtained expression (32) a new phenomenon in the neutrino flavour oscillation in a magnetic field can be seen. It follows that the neutrino flavour oscillations in general can be modified by the neutrino magnetic moment interactions with the transversal magnetic field \(B_{\perp }\). In the case of zeroth magnetic moment and/or vanishing magnetic field Eq. (32) reduces to the well known probability of the flavour neutrino oscillations in vacuum.

Consider the mass square difference \(\varDelta m^2=7\times 10^{-5} \ {\text {eV}} ^2\) and the magnetic moments \(\mu _1 = \mu _2=\mu \sim 10^{-20}\mu _B\) that corresponds the Standard Model prediction (1) for neutrino masses of the order \(m \sim 0.1 \ {\text {eV}}\). In Fig. 1 we show the probability (32) of the neutrino flavour oscillations \(\nu ^L_e \rightarrow \nu _{\mu }^L\) in the transversal magnetic field for this particular choice of parameters and the neutrino energy \(p = 1 \ {\text {MeV}}\). It is clearly seen that the amplitude of oscillations at the vacuum frequency \(\omega _{vac}=\frac{\varDelta m^2}{4p}\) is modulated by the magnetic field frequency \(\omega _{B}=\mu B_{\perp }\). The corresponding oscillation length is \(L=1/\mu B \sim 50 \ {\text {km}} \). This value indeed exceeds the typical dimensions of magnetars \(R_{mgt} \sim 20 - 30 \ {\text {km}}\) [26], but the effect of the oscillation amplitude modulation, as it is clearly illustrated by the Fig. 1, is still sufficient.

A similar phenomenon of the neutrino spin and flavour oscillations modulation by the magnetic field frequency is discussed also in [29], where the case \(\mu _{11}=\mu _{22}\) is considered.

## 4 Conclusions

We have developed a new approach to description of different types of neutrino oscillations (flavour \(\nu _e^L \leftrightarrow \nu _{\mu }^L\), spin \(\nu _e^L \leftrightarrow \nu _{e}^R\) and spin-flavour \(\nu _e^L \leftrightarrow \nu _{\mu }^R\) oscillations) in the presence of a constant magnetic field. Our treatment of neutrino oscillations is based on the use of the exact neutrino stationary states in the magnetic field and also accounts for four neutrino states (two different mass neutrinos each in two spin states).

- 1.
the amplitude modulation of the probability of flavour oscillations \(\nu _e^L \rightarrow \nu _{\mu }^L\) in the transversal magnetic field with the magnetic frequency \(\omega _{B}=\mu B_{\perp }\) (in the case \(\mu _1 = \mu _2\)) and more complicated dependence on harmonic functions with \(\omega _{B}\) for \(\mu _1 \ne \mu _2\);

- 2.
the dependence of the spin oscillation probability \(P_{\nu _{e}^L \rightarrow \nu _{e}^R}\) on the mass square difference \(\varDelta m^2\);

- 3.
the appearance of the spin-flavour oscillations in the case \(\mu _1=\mu _2\) and \(\mu _{12}=0\), the transition goes through the two-step processes \(\nu _e^L \rightarrow \nu _{\mu }^L \rightarrow \nu _{\mu }^R\) and \(\nu _e^L \rightarrow \nu _{e}^R \rightarrow \nu _{\mu }^R\).

## Notes

### Acknowledgements

The authors are thankful to Anatoly Borisov, Alexander Grigoriev, Konstantin Kouzakov, Alexey Lokhov, Pavel Pustoshny, Konstantin Stankevich and Alexey Ternov for useful discussions. This work is supported by the Russian Basic Research Foundation Grants nos. 16-02-01023 and 17-52-53-133.

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