∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Theoretical Physics, Faculty of Physics, Moscow
State University, 119991 Moscow, Russia

# Resonance enhancement of neutrino oscillations due to transition magnetic
moments

A.V. Chukhnova e1,addr1  A.E. Lobanov e2,addr1

(Received: date / Accepted: date)

###### Abstract

We prove that a resonance enhancement of neutrino oscillations in magnetic
field is possible due to transition magnetic moments and demonstrate that this
resonance is strictly connected to the neutrino polarization. To study the
main properties of this resonance, we obtain the probabilities of transitions
between neutrino states with definite flavor and helicity in inhomogeneous
electromagnetic field in the adiabatic approximation. Since the resonance is
present only when the adiabaticity condition is fulfilled, we also obtain and
discuss this condition.

###### Keywords:

neutrino oscillations, neutrino magnetic moments

††journal: Eur. Phys. J. C

The neutrino oscillations are usually described using the phenomenological
theory based on the ideas by B. Pontecorvo (see, e.g. Bilenky1978 ). Within
this approach the interaction with matter can be taken into account with the
help of an effective potential Wolfenstein1978 , which modifies the neutrino
dispersion law. As a result, the resonance behavior of flavor oscillations,
i.e. the Mikheev–Smirnov–Wolfenstein (MSW) effect MS_en , can be observed
Bethe . However, since neutrino is a massive particle, it is necessary to take
into account not only the neutrino flavor oscillations, but also the spin
rotation effect to construct a complete description of neutrino evolution.

When neutrino propagates in vacuum, in matter at rest or parallel to the
direction of magnetic field AK1988 , the helicity does not change. In this
case it is possible to consider the evolution of a neutrino state with a
definite helicity. Nevertheless, the presence of electromagnetic field
Fujikawa1980 ; Shrock1982 , moving or polarized matter Lobanov2001  (see also
Studenikin2004 ) in general case can modify neutrino spin orientation. Spin
rotation of neutrino in electromagnetic field was widely discussed about 40
years ago (see, e.g., OVV1986 ; Voloshin1986 ). In particular, spin rotation
was considered as a possible explanation of the solar neutrino problem. For
the first time the external medium as a factor, which results in an actual
spin precession of the neutrino, was considered in Lobanov2001 . However, the
neutrino propagation in moving matter has been studied for a long time. The
dispersion law for neutrino in moving matter was considered in Pal1989 ;
Nieves1989_1 . In orsesmo1986_en ; Semikoz1987_en ; Semikoz1989_en  the
polarization of the background matter was taken into account using the concept
of induced neutrino magnetic moment. The value of the induced magnetic moment
should be calculated for any preset composition of the background medium. In
particular, in Nunokawa1997  it was calculated for the medium composed of
electrons only.

Since there are correlations between flavor oscillations and spin rotation,
while describing neutrino evolution in the general case it is necessary to
take into account these processes simultaneously. The results of such
description may be significant for astrophysics Akhmedov1988 ; Volpe2015 ;
Kartavtsev2015 ; Dobrynina2016 ; Vlasenko2014 ; Ternov2016 ; Kurashvili2017 ;
Ternov2019 .

A rigorous quantum field theoretical description of Dirac neutrino propagation
taking into account both flavor oscillations and spin rotation can be
constructed within the Standard Model modification tmf2017_en ; lobanov2019 ,
where both the mass states and their arbitrary superpositions are considered
as different quantum states of an $SU(3)$ neutrino multiplet. This description
can be generalized for the case of neutrino propagating in dense matter
izvu2016_en ; VMU2017_en  and electromagnetic field PR2020 . Then the neutrino
evolution can be studied on the base of neutrino wave equation, which has the
same meaning as the Dirac–Schwinger equation of quantum electrodynamics (see,
e.g. B_SH_en ). As a result of this approach in PR2020  we obtain that the
cosine of the effective mixing angle for neutrino, moving in electromagnetic
field with constant characteristics, changes its sign as a function of the
magnetic induction, and therefore the resonance behavior of probabilities due
to neutrino transition magnetic moments is possible PR2020 . As it is well-
known, the resonance can be observed when the background characteristics vary
slowly in space. In the present paper we obtain explicit formulas for neutrino
spin-flavor transition probabilities in the case of inhomogeneous
electromagnetic field within the adiabatic approximation, which is usually
used to describe the MSW effect for neutrino in matter. We study the
properties of the new resonance taking into account the restrictions on the
magnetic induction, for which the adiabatic approximation is valid.

The wave equation for neutrino in electromagnetic field, which takes into
account the direct interaction of neutrino multiplet with the field due to the
anomalous magnetic moments and the transition magnetic and electric moments,
is as follows PR2020

$\bigg{(}\mathrm{i}\gamma^{\mu}{\partial}_{\mu}{\mathds{I}}-{\mathds{M}}-\frac{\mathrm{i}}{2}\mu_{0}F^{\mu\nu}\sigma_{\mu\nu}{\mathds{M}}-\frac{\mathrm{i}}{2}F^{\mu\nu}\sigma_{\mu\nu}{\mathds{M}}_{h}\\\
-\frac{\mathrm{i}}{2}{}^{\star\\!\\!}F^{\mu\nu}\sigma_{\mu\nu}{\mathds{M}}_{ah}\\!\bigg{)}\,{\varPsi}(x)=0.$
(1)

The wave functions $\varPsi(x)$ describe the $SU(3)$ neutrino multiplet as a
whole. Here $\mathds{I}$ is the identity matrix, $\mathds{M}$ is the neutrino
mass matrix,
${}^{\star\\!}F^{\mu\nu}=-\frac{1}{2}e^{\mu\nu\rho\lambda}F_{\rho\lambda}$ is
the dual tensor to the electromagnetic field tensor $F^{\mu\nu}$. The
interaction with the electromagnetic field via transition magnetic and
transition electric moments is taken into account by introducing the Hermitian
matrix of transition magnetic moments ${\mathds{M}}_{h}$ and the matrix of the
transition electric moments ${\mathds{M}}_{ah}$. In the first approximation
the diagonal magnetic moments are proportional to the neutrino masses, i.e.
the matrix of the diagonal magnetic moments is defined as $\mu_{0}\mathds{M}$,
where

$\displaystyle\mu_{0}=\frac{3e\mathrm{G_{F}}}{8\sqrt{2}\pi^{2}}.$ (2)

The evolution of ultra-relativistic neutrinos can be studied in the quasi-
classical approximation. In this case we can assume that the neutrino
4-velocity $u^{\mu}$, which is proportional to the neutrino kinetic momentum,
is constant arlomur . Then the evolution equation takes the form

$\mathrm{i}\dot{\varPsi}(\tau)={\mathcal{F}}\varPsi(\tau),$ (3)

where

${\mathcal{F}}={\mathds{M}}-\mu_{0}\mathds{M}\gamma^{5}\gamma^{\mu}{\,}^{\star\\!\\!}F_{\mu\nu}u^{\nu}\\\
-\mathds{M}_{h}\gamma^{5}\gamma^{\mu}{\,}^{\star\\!\\!}F_{\mu\nu}u^{\nu}+\mathds{M}_{ah}\gamma^{5}\gamma^{\mu}F_{\mu\nu}u^{\nu}.$
(4)

The dot denotes differentiation with respect to the proper time $\tau$.

For simplicity we will consider the two-flavor model. In the two-flavor model
the mass matrix $\mathds{M}$ and the matrices of transition moments
$\mathds{M}_{h}$, $\mathds{M}_{ah}$ are $2\times 2$ matrices and may be
expressed in the terms of the Pauli matrices. The corresponding wave functions
$\varPsi(\tau)$ are $8$-component objects. In the mass representation

$\displaystyle\begin{array}[]{l}\displaystyle\mathds{M}=\frac{1}{2}(\sigma_{0}(m_{1}+m_{2})-\sigma_{3}(m_{2}-m_{1})),\\\\[6.0pt]
\displaystyle\mathds{M}_{h}=\frac{1}{2}(m_{1}+m_{2})\mu_{1}\sigma_{1},\\\\[6.0pt]
\displaystyle\mathds{M}_{ah}=\frac{1}{2}{(m_{1}-m_{2})}\varepsilon_{1}\sigma_{2},\end{array}$
(5)

where $\sigma_{i},i=1,2,3$ are the Pauli matrices, $\sigma_{0}$ is the
identity $2\times 2$ matrix. For the Standard Model neutrinos the values of
the coefficients $\mu_{1}$ and $\varepsilon_{1}$, which characterize the
neutrino transition magnetic and electric moments, can be found in Shrock1982
(see also Giunti2015 ). Within the Standard Model the transition moments are
suppressed in comparison to the diagonal ones due to GIM-mechanism GIM_1970 ,
and the neutrino transition electric moments are smaller than the transition
magnetic moments. In this work we assume that the transition electric moments
are small, and so we neglect them. In this case we are able to obtain an
analytical solution of the evolution equation in the two flavor model.
Possible effects of the transition electric moment are discussed further in
the paper.

Then matrix (4) in the mass representation looks like

${\mathcal{F}}\rightarrow\frac{1}{2}\Big{\\{}(\sigma_{0}(m_{1}+m_{2})\\\
-\sigma_{3}(m_{2}-m_{1}))(1-\mu_{0}\gamma^{5}\gamma^{\mu}{\,}^{\star\\!\\!}F_{\mu\nu}u^{\nu})\\\
-\sigma_{1}(m_{1}+m_{2})\mu_{1}\gamma^{5}\gamma^{\mu}{\,}^{\star\\!\\!}F_{\mu\nu}u^{\nu}\Big{\\}}.$
(6)

When the direction of the magnetic induction is constant, the operator

${\cal S}=\gamma^{5}\gamma^{\mu}{\,}^{\star\\!\\!}F_{\mu\nu}u^{\nu}/N,\quad
N=\displaystyle\sqrt{\displaystyle
u_{\mu}{\,}^{\star\\!\\!}F^{\mu\alpha}{\,}^{\star\\!\\!}F_{\alpha\nu}u^{\nu}}.$
(7)

is an integral of motion for the evolution equation (6). Note that for
neutrino propagating orthogonally to the purely magnetic field with the
induction $\bf{B}$ we have $N=u_{0}|\bf{B}|$, and for neutrino propagating
parallel to the magnetic field $N=|{\bf B}|$, i.e. $N$ is the absolute value
of the magnetic induction in the neutrino rest frame. The operator ${\cal S}$
defines the projection of the spin on the direction of the magnetic field in
the neutrino rest frame. The corresponding neutrino polarization vector is
defined as follows

$\bar{s}^{\mu}=-{\,}^{\star\\!\\!}F_{\mu\nu}u^{\nu}/N.$ (8)

If the magnetic field varies slowly, the operator ${\cal S}$ can be considered
as an approximate integral of motion, when the energy of the neutrino
interaction with the field due to the magnetic moment is much larger than the
inverse characteristic time of the field variation. To put it in a more formal
way, the following conditions must be satisfied Lobanov2006

$\frac{\kappa}{2\mu_{0}N}\ll 1,$ (9) $\frac{\varkappa}{\kappa}\ll 1,$ (10)

where

$\begin{array}[]{l}\displaystyle\kappa=\sqrt{H^{2}\dot{H}^{2}-(H\dot{H})^{2}}/{N^{2}},\\\
\displaystyle\varkappa=\frac{N}{H^{2}\dot{H}^{2}-(H\dot{H})^{2}}e^{\alpha\beta\gamma\delta}\ddot{H}_{\alpha}\dot{H}_{\beta}H_{\gamma}u_{\delta}.\end{array}$
(11)

Here $H^{\mu}={}^{\star\\!}F^{\mu\nu}u_{\nu}$.

However, the neutrino Lorentz factor is rather large $u_{0}\gg 1$. Under the
Lorentz transformations to the neutrino rest frame the longitudinal component
of the magnetic field remains the same, while the orthogonal component
increases proportionally to $u_{0}$. Hence, the direction of the magnetic
induction in the neutrino rest frame is almost orthogonal to the velocity of
this reference frame, except for the case of the neutrino propagating
precisely in the direction of the magnetic field or against it. Except for
this case, the directions of the first and second derivatives of the magnetic
induction in the neutrino rest frame are also almost orthogonal to the
direction of the neutrino velocity, and condition (10) is satisfied.

In paper PR2020  we obtain solutions of the evolution equation in the case of
constant background conditions with the help of the resolvent $U(\tau)$

$\varPsi(\tau)=\frac{1}{\sqrt{2u_{0}}}\,U(\tau)\varPsi_{0}.$ (12)

Here $\varPsi_{0}$ is a constant object, which defines the neutrino initial
state and in the two flavor model has $8$ components. For a neutrino pure
state with a definite initial polarization it can be presented in the form

$\begin{array}[]{l}\displaystyle\varPsi_{0}=\frac{1}{2}(1-\gamma^{5}\gamma_{\mu}{s}_{0}^{\mu})(\gamma_{\mu}u^{\mu}+1)\left(\psi^{0}\otimes
e_{j}\right),\\\\[6.0pt] \bar{\varPsi}_{0}\varPsi_{0}=2.\end{array}$ (13)

Here $\psi^{0}$ is a constant bispinor, $e_{j}$ is an arbitrary unit vector in
the two-dimensional vector space over the field of complex numbers, and
${s}_{0}^{\mu}$ is a $4$-vector of neutrino polarization such that
$(u{s}_{0})=0$.

When the external conditions vary slowly, it is also possible to write the
resolvent $U(\tau)$. Since the operator $\cal{S}$ with the eigenvalues $\zeta$
can be considered as an integral of motion, we can study the evolution of
neutrino states with definite $\zeta$ independently. Note that the states with
definite $\zeta$ in the general case are not the states with definite
helicity.

The matrix, which determines the evolution equation for the states with
definite $\zeta$, can be diagonalized in the mass representation at a given
point $\tau$ using the matrix $\mathds{U}^{(eff)}_{\zeta}(\tau)$

$\mathds{U}^{(eff)}_{\zeta}(\tau)=\left(\begin{matrix}\cos{\theta^{\prime}}_{\zeta}(\tau)&\sin{\theta^{\prime}}_{\zeta}(\tau)\\\
-\sin{\theta^{\prime}}_{\zeta}(\tau)&\cos{\theta^{\prime}}_{\zeta}(\tau)\end{matrix}\right).$
(14)

Here ${\theta}^{\prime}_{\zeta}(\tau)$ is an effective mixing angle in
electromagnetic field in the mass representation for the states with definite
$\zeta$ (for more detail see PR2020 ). The value of the angle
${\theta}^{\prime}_{\zeta}(\tau)$ is determined by the relations

$\begin{array}[]{l}\displaystyle
X^{\prime}_{\zeta}(\tau)=\sin{2\theta^{\prime}_{\zeta}(\tau)},\\\\[6.0pt]
Y^{\prime}_{\zeta}(\tau)=\cos{2\theta^{\prime}_{\zeta}(\tau)},\end{array}$
(15)

where

$\begin{array}[]{l}Y^{\prime}_{\zeta}(\tau)=\displaystyle\frac{1}{Z_{\zeta}(\tau)}\Big{(}\big{(}m_{2}-m_{1}\big{)}\big{(}1-\zeta\mu_{0}N(\tau)\big{)}\Big{)},\\\\[8.0pt]
X^{\prime}_{\zeta}(\tau)=\displaystyle\frac{1}{Z_{\zeta}(\tau)}\Big{(}-\zeta\mu_{1}N(\tau)\big{(}m_{2}+m_{1}\big{)}\Big{)},\\\\[8.0pt]
Z_{\zeta}(\tau)=\Big{\\{}\Big{(}\big{(}m_{2}-m_{1}\big{)}\big{(}1-\zeta\mu_{0}N(\tau)\big{)}\Big{)}^{2}\\\
\phantom{Z_{\zeta}(\tau)======}+\Big{(}\big{(}m_{2}+m_{1}\big{)}\mu_{1}N(\tau)\Big{)}^{2}\Big{\\}}^{1/2},\end{array}$
(16)

Then for $\sin{\theta^{\prime}_{\zeta}(\tau)}$ and
$\cos{\theta^{\prime}_{\zeta}(\tau)}$ we have

$\begin{array}[]{l}\displaystyle\sin{\theta^{\prime}_{\zeta}(\tau)}=\mathrm{sgn}X^{\prime}_{\zeta}(\tau)\sqrt{(1-Y^{\prime}_{\zeta}(\tau))/2},\\\\[6.0pt]
\displaystyle\cos{\theta^{\prime}_{\zeta}(\tau)}=\sqrt{(1+Y^{\prime}_{\zeta}(\tau))/2}.\end{array}$
(17)

The adiabatic approximation is valid when $|\dot{\theta}_{\zeta}|\ll
Z_{\zeta}$. We can introduce the adiabaticity parameter $\Gamma$ for neutrino
in electromagnetic field by analogy with what is usually done for neutrino in
matter (see, e.g., Giunti2007 ). The adiabaticity condition takes the form

$\Gamma=\frac{2Z_{\zeta}^{3}}{\mu_{1}|\dot{N}(\tau)|(m_{2}^{2}-m_{1}^{2})}\gg
1.$ (18)

If the external conditions vary slowly, then the Hamiltonian of the system is
almost diagonal at every point. Then to write down the solution with rather
high accuracy, it is enough to diagonalize the matrix of the equation at the
current point $\tau$ and to perform the inverse transformation at the initial
point $\tau=0$ Fedoryuk_en . The method is absolutely similar to what is
usually done to describe MSW resonance. Using this approach, we can obtain the
resolvent $U^{\prime}(\tau)$ in the mass representation in the adiabatic
approximation.

The matrices in (5), (6) and the resolvent $U(\tau)$ are presented in the mass
representation. To obtain the same matrices in the flavor representation one
should use the transformation

$U(\tau)=\mathds{U}U^{\prime}(\tau)\mathds{U}^{\dagger}.$ (19)

Here $\mathds{U}$ is the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix, which
in the two-flavor model is defined by the vacuum mixing angle $\theta$ as
follows

$\mathds{U}=\left(\begin{matrix}\cos{\theta}&\sin{\theta}\\\
-\sin{\theta}&\cos{\theta}\end{matrix}\right),$ (20)

To obtain the resolvent in the flavor representation one needs to take into
account relation (19). So, in adiabatical approximation we derive the
following expression for the resolvent of the wave equation, which describes
the neutrino multiplet in inhomogeneous magnetic field

$U(\tau)=\sum\limits_{\zeta=\pm
1}e^{-\mathrm{i}\int\limits_{0}^{\tau}(T_{\zeta}(\tilde{\tau})/2)d\tilde{\tau}}\times\Big{(}C_{\zeta}\cos(\theta_{\zeta}(\tau)-\theta_{\zeta}(0))\\\
-\mathrm{i}S_{\zeta}\sigma_{1}\sin(\theta_{\zeta}(\tau)+\theta_{\zeta}(0))+\mathrm{i}C_{\zeta}\sigma_{2}\sin(\theta_{\zeta}(\tau)-\theta_{\zeta}(0))\\\
+\mathrm{i}S_{\zeta}\sigma_{3}\cos(\theta_{\zeta}(\tau)+\theta_{\zeta}(0))\Big{)}\Lambda_{\zeta}.$
(21)

The values of the mixing angles in the flavor representation are given by the
relation

$\theta_{\zeta}(\tau)=\theta+\theta^{\prime}_{\zeta}(\tau).$ (22)

In Eq. (21) the following notations are used

$\begin{array}[]{l}\displaystyle
S_{\zeta}=\sin\\!\\!{\int\limits_{0}^{\tau}\\!\\!{\big{(}Z_{\zeta}(\tilde{\tau})}/{2}\big{)}\,d\tilde{\tau}},C_{\zeta}=\cos\\!\\!{\int\limits_{0}^{\tau}\\!\\!{\big{(}Z_{\zeta}(\tilde{\tau})}/2\big{)}\,d\tilde{\tau}},\\\\[12.0pt]
\displaystyle
T_{\zeta}(\tau)=\displaystyle\big{(}m_{2}+m_{1}\big{)}\big{(}1-\zeta\mu_{0}N(\tau)\big{)},\end{array}$
(23)

which generalize the notations used in paper PR2020  for constant
electromagnetic field. The projection operators on the eigenstates of operator
${\cal S}$ are defined by the expressions

$\Lambda_{\zeta}=\frac{1}{2}\left(1+\zeta{\cal
S}\right),\quad[\gamma^{\mu}u_{\mu},\Lambda_{\zeta}]=0,\quad\zeta=\pm 1.$ (24)

To calculate the probabilities of transitions between states with definite
flavor and polarization, we use quasi-classical spin-flavor density matrices
introduced similarly to the quasi-classical spin density matrices (see
Lobanov2006 )

$\rho_{\alpha}(\tau)=\frac{1}{4u^{0}}U(\tau)\big{(}\gamma^{\mu}u_{\mu}+1\big{)}\left(1-\gamma^{5}\gamma_{\mu}{s}^{\mu}_{0}\right){\mathds{P}}_{0}^{(\alpha)}\bar{U}(\tau)$
(25)

In this formula ${s}^{\mu}_{0}$ defines the initial polarization state of the
neutrino, the projection operator ${\mathds{P}}_{0}^{(\alpha)}$ defines its
initial flavor state and the resolvent $U(\tau)$ is given by Eq. (21). The
probability of a transition from the state $\alpha$ to the state $\beta$ in
the time $\tau$ is determined by the following relation

$W_{\alpha\rightarrow\beta}=\mathrm{Tr}\left\\{\rho_{\alpha}(\tau)\rho^{{\dagger}}_{\beta}(\tau=0)\right\\}.$
(26)

Note that since in our model these states of the neutrino multiplet are pure
states, all the final formulas may also be obtained using the neutrino
multiplet wave functions.

In the flavor representation the projection operators on states with the
definite flavor take the form

$\begin{array}[]{l}\displaystyle\mathds{P}^{(\alpha)}_{0}=\frac{1}{2}(1+\xi_{\alpha}\sigma_{3}),\displaystyle\mathds{P}^{(\beta)}_{0}=\frac{1}{2}(1+\xi_{\beta}\sigma_{3}),\\\\[6.0pt]
\displaystyle\xi_{\alpha},\xi_{\beta}=\pm 1.\end{array}$ (27)

To obtain the projection operators on the initial and final state with the
electron flavor one should choose $\xi_{\alpha},\xi_{\beta}=1$, otherwise
$\xi_{\alpha},\xi_{\beta}=-1$. We assume that in the initial and final states
neutrino has a definite helicity, i.e.

$\begin{array}[]{l}\displaystyle{s}_{0}^{(\alpha)\mu}=\zeta_{\alpha}{s}^{\mu}_{sp},\quad{s}_{0}^{(\beta)\mu}=\zeta_{\beta}{s}^{\mu}_{sp},\\\\[4.0pt]
\displaystyle{s}^{\mu}_{sp}=\\{|{\bf u}|,u^{0}{\bf u}/|{\bf
u}|\\},\quad\zeta_{\alpha},\zeta_{\beta}=\pm 1,\end{array}$ (28)

where the values $\zeta_{\alpha},\zeta_{\beta}=1$ correspond to the right-
handed neutrino and $\zeta_{\alpha},\zeta_{\beta}=-1$ correspond to the left-
handed neutrino. The spin-flavor transition probabilities can be presented in
the form

$W_{\alpha\rightarrow\beta}=\frac{1+\xi_{\alpha}\xi_{\beta}}{2}\frac{1+\zeta_{\alpha}\zeta_{\beta}}{2}W_{1}+\frac{1+\xi_{\alpha}\xi_{\beta}}{2}\frac{1-\zeta_{\alpha}\zeta_{\beta}}{2}W_{2}\\\
\\!\\!\\!+\frac{1-\xi_{\alpha}\xi_{\beta}}{2}\frac{1+\zeta_{\alpha}\zeta_{\beta}}{2}W_{3}+\frac{1-\xi_{\alpha}\xi_{\beta}}{2}\frac{1-\zeta_{\alpha}\zeta_{\beta}}{2}W_{4},$
(29)

where

$\begin{array}[]{l}\displaystyle
W_{1}=\frac{1}{8}\Big{(}(1\\!-\\!\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1+C_{+1}^{2}\cos{2\Delta_{+1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{1}}+S_{+1}^{2}\\!\cos{2\Theta_{+1}(\tau)}\big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{1}}\displaystyle\\!+\\!(1\\!+\\!\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1+C_{-1}^{2}\\!\cos{2\Delta_{-1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{1}}\\!+\\!S_{-1}^{2}\\!\cos{2\Theta_{-1}(\tau)}\big{)}\Big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{1}}\displaystyle\\!+\\!\frac{1}{2}(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}\xi_{\alpha}F_{1}(\tau)\sin{\Phi(\tau)}+D_{1}(\tau)\cos{\Phi(\tau)}\big{)},\\\\[8.0pt]
\displaystyle
W_{2}\\!=\\!\frac{1}{8}\Big{(}(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}1+C_{+1}^{2}\cos{2\Delta_{+1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{2}}+S_{+1}^{2}\cos{2\Theta_{+1}(\tau)}\big{)}\\\\[4.0pt]
\phantom{W_{2}}\displaystyle+(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}1+C_{-1}^{2}\cos{2\Delta_{-1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{2}}+S_{-1}^{2}\\!\cos{2\Theta_{-1}(\tau)}\big{)}\Big{)}\\\
\displaystyle\phantom{W_{4}}-\\!\frac{1}{2}(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}\xi_{\alpha}F_{1}(\tau)\sin{\Phi(\tau)}+D_{1}(\tau)\cos{\Phi(\tau)}\big{)},\\\\[8.0pt]
\displaystyle
W_{3}\\!=\\!\frac{1}{8}\Big{(}(1\\!-\\!\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1-C_{+1}^{2}\\!\cos{2\Delta_{+1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{3}}\\!-\\!S_{+1}^{2}\\!\cos{2\Theta_{+1}(\tau)}\big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{3}}\displaystyle\\!+\\!(1\\!+\\!\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1-C_{-1}^{2}\\!\cos{2\Delta_{-1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{3}}-S_{-1}^{2}\\!\cos{2\Theta_{-1}(\tau)}\big{)}\Big{)}\\\
\displaystyle\phantom{W_{3}}+\\!\frac{1}{2}(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}\xi_{\alpha}F_{2}(\tau)\sin{\Phi(\tau)}\\!+\\!D_{2}(\tau)\cos{\Phi(\tau)}\big{)},\\\\[8.0pt]
\displaystyle
W_{4}\\!=\\!\frac{1}{8}\Big{(}(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}1-C_{+1}^{2}\\!\cos{2\Delta_{+1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{4}}-S_{+1}^{2}\\!\cos{2\Theta_{+1}(\tau)}\big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{4}}\displaystyle\\!+\\!(1\\!-\\!(\bar{s}s_{sp})^{2})\big{(}1-C_{-1}^{2}\\!\cos{2\Delta_{-1}(\tau)}\\\\[4.0pt]
\displaystyle\phantom{W_{4}}\\!-\\!S_{-1}^{2}\\!\cos{2\Theta_{-1}(\tau)}\big{)}\Big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{4}}\\!-\\!\frac{1}{2}(1\\!-\\!(\bar{s}s_{sp}))^{2})\big{(}\xi_{\alpha}F_{2}(\tau)\sin{\Phi(\tau)}\\!+\\!D_{2}(\tau)\cos{\Phi(\tau)}\big{)}.\end{array}$
(30)

Here we use the notations

$\begin{array}[]{l}\displaystyle\Phi(\tau)=\mu_{0}(m_{1}+m_{2})\int\limits_{0}^{\tau}N(\tilde{\tau})d\tilde{\tau},\\\\[4.0pt]
F_{1}(\tau)=C_{+1}S_{-1}\cos\Delta_{+1}(\tau)\cos\Theta_{-1}(\tau)\\\
\phantom{F_{1}(\tau)}-S_{+1}C_{-1}\cos\Theta_{+1}(\tau)\cos\Delta_{-1}(\tau),\\\
D_{1}(\tau)=C_{+1}C_{-1}\cos\Delta_{+1}(\tau)\cos\Delta_{-1}(\tau)\\\
\phantom{D_{1}(\tau)}\,+S_{+1}S_{-1}\cos\Theta_{+1}(\tau)\cos\Theta_{-1}(\tau),\\\
F_{2}(\tau)=C_{+1}S_{-1}\sin\Delta_{+1}(\tau)\sin\Theta_{-1}(\tau)\\\
\phantom{F_{2}(\tau)}\,-S_{+1}C_{-1}\sin\Theta_{+1}(\tau)\sin\Delta_{-1}(\tau),\\\
D_{2}(\tau)=C_{+1}C_{-1}\sin\Delta_{+1}(\tau)\sin\Delta_{-1}(\tau)\\\
\phantom{D_{2}(\tau)}\,+S_{+1}S_{-1}\sin\Theta_{+1}(\tau)\sin\Theta_{-1}(\tau),\end{array}$
(31)

where

$\Delta_{\pm 1}(\tau)=\theta_{\pm 1}(\tau)-\theta_{\pm 1}(0),\quad\Theta_{\pm
1}(\tau)=\theta_{\pm 1}(\tau)+\theta_{\pm 1}(0).$ (32)

In the general case the formulas are rather complicated even in the two-flavor
model. The spin-flavor transition probabilities can be simplified for a
neutrino, which is generated in the region with high values of magnetic
induction and detected in vacuum. Then the final value of the mixing angle in
the flavor representation is $\theta_{\zeta}({\tau})=\theta$, and the initial
value we denote as $\theta_{\zeta}^{0}=\theta_{\zeta}(0)$. As we usually have
no information concerning the exact location of the neutrino generation
process, we should average the probabilities over the proper time $\tau$.
Thus, the averaged values of the probabilities are

$\begin{array}[]{l}\displaystyle
W_{1}=\frac{1}{8}\Big{(}(1-\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1+\cos{2\theta}\,\cos{2\theta}_{+1}^{0}\big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{1}=}+(1+\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1+\cos{2\theta}\,\cos{2\theta}_{-1}^{0}\big{)}\Big{)},\\\\[4.0pt]
\displaystyle
W_{2}=\frac{1}{8}(1-(\bar{s}s_{sp})^{2})\big{(}2+\cos{2\theta}(\cos{2\theta}_{+1}^{0}+\cos{2\theta}_{-1}^{0})\big{)},\\\\[6.0pt]
\displaystyle
W_{3}=\frac{1}{8}\Big{(}(1-\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1-\cos{2\theta}\,\cos{2\theta}_{+1}^{0}\big{)}\\\\[4.0pt]
\displaystyle\phantom{W_{3}=}+(1+\zeta_{\alpha}(\bar{s}s_{sp}))^{2}\big{(}1-\cos{2\theta}\,\cos{2\theta}_{-1}^{0}\big{)}\Big{)},\\\\[4.0pt]
\displaystyle
W_{4}=\frac{1}{8}(1-(\bar{s}s_{sp})^{2})\big{(}2-\cos{2\theta}(\cos{2\theta}_{+1}^{0}+\cos{2\theta}_{-1}^{0})\big{)}.\end{array}$
(33)

For high energy neutrinos the assumption $(\bar{s}s_{sp})=0$ is valid with
high accuracy, except for a narrow region of angles when neutrino velocity and
the vector of magnetic induction are almost parallel (see PR2020 ). Therefore,
the probabilities take the form

$\begin{array}[]{l}\displaystyle
W_{1,2}=\frac{1}{8}\left(2+\cos{2\theta}(\cos{2\theta}_{+1}^{0}+\cos{2\theta}_{-1}^{0})\right),\\\\[6.0pt]
\displaystyle
W_{3,4}=\frac{1}{8}\left(2-\cos{2\theta}(\cos{2\theta}_{+1}^{0}+\cos{2\theta}_{-1}^{0})\right).\end{array}$
(34)

Taking into account Eq.(15) and Eq.(22), we have

$\cos
2\theta_{\zeta}^{0}=Y^{\prime}_{\zeta}\cos{2\theta}-X^{\prime}_{\zeta}\sin{2\theta}.$
(35)

In the explicit form (22) can be presented as follows

$\cos
2\theta_{\zeta}^{0}=\Big{(}(m_{2}-m_{1})(1-\zeta\mu_{0}N(0))\cos{2\theta}\\\
+\zeta(m_{2}+m_{1})\mu_{1}N(0)\sin{2\theta}\Big{)}\\\
\times\Big{(}\\!\big{(}(m_{2}-m_{1})(1-\zeta\mu_{0}N(0))\big{)}^{2}\\!\\\
+\big{(}(m_{2}+m_{1})\mu_{1}N(0)\big{)}^{2}\\!\Big{)}^{\\!-1/2}\\!\\!.$ (36)

Note that when the neutrino state can be described as a superposition of the
mass eigenstates, the effective mixing angles $\theta_{\zeta}$ are equal to
their vacuum values.

The theoretical predictions for the Standard Model neutrinos give
$\mu_{1}/\mu_{0}\sim 10^{-4}$ (see Shrock1982 ), and so it can be expected
that

$r=\frac{\mu_{1}(m_{1}+m_{2})}{\mu_{0}(m_{2}-m_{1})}\ll 1.$ (37)

In this case in the first approximation the transition probabilities take the
form

$\begin{array}[]{l}W_{1,2}\\!=\\!\displaystyle\frac{1}{8}\\!\bigg{(}2\\!+\\!\Big{(}1\\!+\\!\mathrm{sgn}\big{(}1\\!-\\!\frac{\\!\mu_{0}(m_{2}\\!-\\!m_{1})\\!}{m_{2}-m_{1}}N(0)\big{)}\Big{)}\\!\cos^{2}{2\theta\\!}\bigg{)},\\\\[10.0pt]
W_{3,4}\\!=\\!\displaystyle\frac{1}{8}\\!\bigg{(}2\\!-\\!\Big{(}1\\!+\\!\mathrm{sgn}\big{(}1\\!-\\!\displaystyle\frac{\\!\mu_{0}(m_{2}\\!-\\!m_{1})\\!}{m_{2}-m_{1}}N(0)\big{)}\Big{)}\\!\cos^{2}{2\theta\\!}\bigg{)}.\end{array}$
(38)

Obviously, when $\mu_{0}N(0)\approx 1$, expression (38) is not valid, and the
exact formula (34) is necessary.

However, if the adiabaticity condition is not fulfilled, even for initial
fields, which exceed the resonance value, the resonance behavior of transition
probabilities will not be observed. Since in the resonance region
$Z_{\zeta}\approx\mu_{1}N(m_{1}+m_{2})$, the adiabaticity parameter $\Gamma$
in (18) becomes proportional to $\mu_{1}^{2}$. Hence, though $\mu_{1}$ is
absent in Eq. (38), if we put $\mu_{1}=0$ from the start, the adiabaticity
condition can not be fulfilled in the resonance region and we will not obtain
the resonance behavior of the transition probabilities.

The coefficient $r$ defined by Eq. (37) actually determines the region of the
fields, where $\mu_{1}$ can not be neglected. That means, the greater is the
parameter $r$, the wider is the resonance region. Note that for smaller values
of $r$ the adiabaticity condition (18) results in stronger restrictions on the
possible value of the field gradient.

In Fig. 2 the behavior of the neutrino flavor-survival probability
$W_{1}+W_{2}$ for different values of $r$ is demonstrated in the case when the
neutrino velocity is orthogonal to the magnetic induction vector. Here we
choose the value of the vacuum mixing angle such that $\sin^{2}\theta=0.307$.

In the present paper we neglect the transition electric moment, since no
explicit analytical solution can be found for arbitrary values of the
transition electric moment. However, the resonance behavior of the flavor
transition probabilities will still be present, since it is determined by the
diagonal elements of the matrices in the wave equation expressed in the mass
representation. For the Standard Model neutrinos the effect of the transition
magnetic and electric moments becomes significant only for the values of the
initial magnetic field, which are in the resonance region $\mu_{0}N\sim 1$.
That is, the resonance denominator of the effective mixing angle contains the
value $Z_{\zeta}$, which can be approximated as

$Z_{\zeta}\approx\Big{(}\big{(}(m_{2}-m_{1})(1-\zeta\mu_{0}N)\big{)}^{2}+\big{(}(m_{1}+m_{2})\mu_{1}N\big{)}^{2}\\\
+\big{(}\alpha(m_{2}-m_{1})\varepsilon_{1}\tilde{N}\big{)}^{2}\Big{)}^{1/2},$
(39)

where $|\alpha|\leq 1$. Indeed, the interaction with the electric transition
moment is determined by the spin operator

$\tilde{\cal
S}=-\gamma^{5}\gamma^{\mu}F_{\mu\nu}u^{\nu}/\tilde{N},\quad\tilde{N}=\displaystyle\sqrt{\displaystyle
u_{\mu}F^{\mu\alpha}F_{\alpha\nu}u^{\nu}}.$ (40)

The square of this operator, as well as the square of any spin operator, is
equal to unity, and therefore this approximation can be obtained. Obviously,
the electric moment will not make the resonance disappear. When the transition
electric moment is taken into account, no spin integral of motion exists for
Eq. (4). Thus, the neutrino can not propagate in any spin eigenstate. This may
lead to the following consequences. Firstly, the value of the resonance field
may be slightly changed. Secondly, the probabilities $W_{1}$ and $W_{2}$ may
differ from each other, as well as $W_{3}$ may differ from $W_{4}$ (see (34)).

For high values of magnetic field all the spin-flavor transition probabilities
become equal

$W_{1}=W_{2}=W_{3}=W_{4}=\frac{1}{4}.$ (41)

This means that all the information about neutrino initial state is lost.
Expression (41) follows directly from Eq. (34) not only when condition (37) is
satisfied.

Although for most situations the condition $(\bar{s}s_{sp})=0$ is satisfied,
there is a region of angles, for which neutrino velocity and the vector of
magnetic induction are almost parallel and this condition is not fulfilled. As
it is already mentioned, for ultra-relativistic neutrinos this region of
angles is very narrow, and even a small angular deviation makes neutrino
behave almost like in the case of orthogonal propagation PR2020 . Because of
this instability, from the phenomenological point of view the situation is
hardly of any practical value.

Figure 1: The dependence of the flavor survival probability on the initial
magnetic induction for a neutrino propagating orthogonally to the magnetic
field for $r=0.005$ (the solid line) and for $r=0.1$ (the dashed line).

Figure 2:  The dependence of the flavor survival probability on the initial
magnetic induction for a left-handed neutrino propagating against the
direction of the magnetic field (or a right-handed neutrino propagating in the
direction of the field) for $r=0.005$ (the solid line) and for $r=0.1$ (the
dashed line).

However, understanding this case is very important since it helps us to
explain, why in the general case the number of the neutrinos of the second
flavor does not predominate the number of the neutrinos of initial flavor even
for the magnetic fields higher than the resonance field (see Fig. 2). Let us
consider the limiting case, when the neutrino moves either in the direction of
the magnetic field or against it. In these cases the helicity operator becomes
an integral of motion, and the spin-flip transitions are absent since
$(\bar{s}s_{sp})=\pm 1$, i.e. $W_{2}=W_{4}=0$.

For neutrino moving against the direction of the magnetic field we have
$(\bar{s}s_{sp})=1$. Hence,

$\begin{array}[]{l}\displaystyle
W_{1}=\frac{1-\zeta_{\alpha}}{4}(1+\cos{2\theta}\cos{2\theta}_{+1}^{0})\\\\[6.0pt]
\displaystyle\phantom{W_{1}=}+\frac{1+\zeta_{\alpha}}{4}(1+\cos{2\theta}\cos{2\theta}_{-1}^{0}),\\\\[6.0pt]
\displaystyle
W_{3}=\frac{1-\zeta_{\alpha}}{4}(1-\cos{2\theta}\cos{2\theta}_{+1}^{0})\\\\[6.0pt]
\displaystyle\phantom{W_{3}=}+\frac{1+\zeta_{\alpha}}{4}(1-\cos{2\theta}\cos{2\theta}_{-1}^{0}).\end{array}$
(42)

If inequality (37) holds, we can write the approximate expressions for the
transition probabilities

$\begin{array}[]{l}\displaystyle
W_{1}=\frac{1\\!-\\!\zeta_{\alpha}}{4}\bigg{(}1\\!+\\!\mathrm{sgn}\Big{(}1\\!-\\!\frac{\\!\mu_{0}(m_{2}\\!-\\!m_{1})\\!}{m_{2}-m_{1}}N(0)\Big{)}\cos^{2}{\\!2\theta}\bigg{)}\\\
\displaystyle\phantom{W_{1}=}+\frac{1\\!+\\!\zeta_{\alpha}}{4}(1\\!+\\!\cos^{2}{2\theta}),\\\\[6.0pt]
\displaystyle
W_{3}=\frac{1\\!-\\!\zeta_{\alpha}}{4}\bigg{(}1\\!-\\!\mathrm{sgn}\Big{(}1\\!-\\!\frac{\\!\mu_{0}(m_{2}\\!-\\!m_{1})\\!}{m_{2}-m_{1}}N(0)\Big{)}\cos^{2}{\\!2\theta}\bigg{)}\\\
\displaystyle\phantom{W_{3}=}+\frac{1\\!+\\!\zeta_{\alpha}}{4}(1\\!-\\!\cos^{2}{2\theta}).\end{array}$
(43)

According to these formulas, the resonance is present for left-handed
neutrinos, while for right-handed neutrinos it is absent.

For neutrino moving in the direction of the magnetic field we have
$(\bar{s}s_{sp})=-1$. The probabilities in this case can be obtained if we
change $\zeta_{\alpha}\rightarrow-\zeta_{\alpha}$ in (42), (43). So the
resonance is present only for right-handed neutrinos, which are not observed
in experiments. The dependence of the flavor-survival probability
$W_{1}+W_{2}$ on the value of the magnetic induction is demonstrated in Fig.
2.

Therefore, an important conclusion can be derived. The presence of the
resonance depends on the neutrino polarization. Let us look back at formula
(34). For a left-handed particle propagating orthogonally to the magnetic
field the probabilities to observe the spin projection in the direction of the
field and opposite to it are both equal to $1/2$, and the neutrino spin-flavor
transition probabilities are the sum of the resonant and non-resonant terms.
It is a well-known fact that the MSW resonance is observed for the left-handed
neutrinos only. Since for the neutrinos in matter at rest the helicity is
conserved, the problem of the correlation between the possibility of the
resonance behavior and the actual polarization of the particle did not arise
in the studies of the resonance in matter. In this sense the description of
resonance in the magnetic field is a more complicated problem, than of the MSW
resonance for neutrino propagation in matter at rest. In this paper we
consider in detail the case of neutrinos with definite initial and final
helicities. However, our approach enables one to study the states with any
polarization. It can be seen from (33) that the transition probabilities
exhibit full resonance when the initial neutrino spin is directed along the
magnetic induction vector, and the resonance is absent when the neutrino spin
is directed opposite the magnetic induction no matter what the direction of
neutrino velocity is. This may be important for the case of low neutrino
energies, when the neutrino chiral states differ significantly from the
neutrino helicity states.

The resonance condition is $\mu_{0}N\approx 1$, and therefore it is determined
by the value of the magnetic induction in the neutrino rest frame. Since the
value of $\mu_{0}$ given by the Standard Model is very small, extremely high
values of the magnetic field are required. The estimates of the values of
neutrino energy and the magnetic field, which are necessary for the resonance
to take place, are discussed in detail in PR2020 . For neutrino propagating
orthogonally to the magnetic field the resonance is reached when
$u_{0}B/B_{0}\approx 1.3\times 10^{13}$, where $B_{0}=4.41\times 10^{13}$
Gauss is the Schwinger magnetic field. That is, $B\approx 5.8\times
10^{26}(m_{\nu}/\mathcal{E}_{\nu})$ Gauss, where $m_{\nu}$ is the average
neutrino mass and $\mathcal{E}_{\nu}$ is the neutrino energy. For neutrino
with the mass $m_{\nu}=0.033$ eV in the magnetic field $B=10^{16}$ Gauss the
energy about $1.9$ GeV is needed for the resonance to take place. These
estimates indicate that not only the resonance, but even the spin oscillations
are very unlikely to be observed in currently known magnetars, since the
characteristic length of spin oscillations seems to be much greater than the
size of the corresponding astrophysical objects. It is also very important
that since for the Standard Model neutrinos $r\ll 1$, the adiabaticity
condition (18)

$\frac{r^{2}(m_{2}^{2}-m_{1}^{2})}{{\cal E}_{\nu}}\gg\frac{|dN/dx|}{N}.$ (44)

seems to give a very strict restriction on the value of the field gradient
$dN/dx$. Here ${\cal E}_{\nu}$ is the neutrino energy, and $x$ is the spacial
coordinate along the neutrino trajectory. Even when the magnetic field and the
neutrino energy are high enough for the resonance to take place, if relation
(44) is violated, no resonance will be present. The resonance might become
observable for Standard Model neutrinos if some exotic compact objects with
higher values of magnetic fields or larger size than the currently known
magnetars are discovered. In our opinion taking into account this resonance as
well as neutrino spin rotation might also be interesting in the studies of the
Early Universe, since these effects change the flavor and helicity
characteristics of the neutrino flux and therefore might influence the
particle composition of the background matter.

However, there are models of New Physics, which predict greater values of
neutrino magnetic moments than the Standard Model. Since the Standard Model
theoretical prediction is $\displaystyle\mu_{\nu}=\mu_{0}m_{\nu}\sim 3\cdot
10^{-19}\left(\frac{m_{\nu}}{1\text{eV}}\right)\mu_{B}$ and the current
experimental restriction on the neutrino magnetic moment is
$\mu_{\nu}<2.9\cdot 10^{-11}\mu_{B}$ Agostini2017 ; Beda2013 , these models
can not be excluded. If such New Physics exists, then near magnetars the spin-
flip effect and even the resonance behavior of neutrino transition
probabilities may be observed (see Fig. 2). For such models the mass matrix
and the matrix of the diagonal magnetic moments may be not proportional.
Obviously, our results are valid within the models of New Physics with the
values of the transition moments, which are significantly smaller than the
diagonal magnetic moments. For this case our results are also applicable. To
obtain the expressions for transition probabilities within such models one
should only replace in all our formulas

$\begin{array}[]{l}\mu_{0}m_{1}\rightarrow\mu_{11},\qquad\mu_{0}m_{2}\rightarrow\mu_{22},\\\\[6.0pt]
\mu_{1}(m_{1}+m_{2})\rightarrow 2\mu_{12}.\end{array}$ (45)

Here $\mu_{11}$, $\mu_{22}$ are the diagonal magnetic moments, and $\mu_{12}$
is the transition magnetic moment. The resonance condition in this case takes
the form

$N\frac{|\mu_{22}-\mu_{11}|}{|{m_{2}-m_{1}}|}=|1-r\tan{2\theta}|^{-1},$ (46)

where $r=2\mu_{12}/|\mu_{22}-\mu_{11}|$. For neutrinos propagating parallel to
the magnetic field this condition was obtained in AK1988 . If $r\gtrsim 1$ the
adiabaticity condition (18) takes the form

$\frac{(m_{2}^{2}-m_{1}^{2})}{{\cal E}_{\nu}}\gg\frac{|dN/dx|}{N}$ (47)

and the following effects might become observable. The spin rotation effect
might become significant for neutrinos produced inside the magnetars or other
compact objects with the high values of the magnetic field. Therefore, the
total flux of observable neutrinos of all flavors might become less than the
initial flux, since only left-handed neutrinos interact with a terrestrial
detector. For solar neutrinos the possibility of this effect was discussed in
OVV1986 . The flavor composition of the flux of the neutrinos produced inside
some compact object might differ significantly from the the same flux measured
in terrestrial conditions due to the resonance studied above. However, what
effects will really be observed depends on the definite value of $\mu_{\nu}$.
It should be emphasized that in the models of New Physics with large neutrino
transition moments for the high values of the magnetic field all the averaged
spin-flavor transition probabilities become approximately equal to $1/4$
similar to the case of Standard Model neutrinos (see (41)).

In this paper we generalize the approach used in PR2020 , where the problem
was studied in the case of constant external conditions. We find analytical
expressions for solutions of the neutrino wave equation in magnetic field in
adiabatic approximation in two-flavor model taking into account transition
magnetic moments. We derive the formulas for the transition probabilities and
indeed obtain a resonance enhancement of neutrino oscillations due to
transition magnetic moments, which was predicted in paper PR2020 . We show
that the type of the resonance is determined by the neutrino polarization and
in the case of extra-high value of magnetic fields the averaged values of all
spin-flavor transition probabilities are equal to $1/4$.

###### Acknowledgements.

The authors are grateful to A.V. Borisov, E.M. Murchikova, I.P. Volobuev, and
V.Ch. Zhukovsky for fruitful discussions. A.V.C. acknowledges support from the
Foundation for the advancement of theoretical physics and mathematics “BASIS”
(Grant No. 19-2-6-100-1).

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