# Three-flavour neutrino oscillations in a magnetic field

Alexey Lichkunov<EMAIL_ADDRESS>Department of Theoretical
Physics, Moscow State University, 119991 Moscow, Russia  Artem Popov
<EMAIL_ADDRESS>Department of Theoretical Physics, Moscow State
University, 119991 Moscow, Russia  Alexander Studenikin
<EMAIL_ADDRESS>Department of Theoretical Physics, Moscow State
University, 119991 Moscow, Russia  National Centre for Physics and
Mathematics, 607328 Satis, Nizhny Novgorod District, Russia

###### Abstract

In this paper we study Dirac neutrino oscillations in a magnetic field in the
three-flavour case. A theoretical framework developed in [1] is extended to
the case of three neutrino species. The closed expressions for neutrino
flavour and spin oscillations probabilities are derived. We show that the
probabilities exhibit a complicated interplay of oscillation on magnetic
$\mu_{\nu}B$ and vacuum $\Delta m^{2}_{ij}/2p$ frequencies. It is also shown
that neutrino oscillations in an interstellar magnetic field can modify
neutrino fluxes observed in neutrino telescopes.

## Introduction

It is well-known that massive neutrinos have nontrivial electromagnetic
properties (see [1] for a review). In particular, neutrino magnetic moments
are nonzero [2, 3]. The best terrestrial upper bounds on neutrino magnetic
moments on the level of $\mu_{\nu}<2.8\div 2.9\times 10^{-11}\mu_{B}$ were
obtained by the GEMMA reactor neutrino experiment [18] and by the Borexino
collaboration [4] from solar neutrino fluxes.

Interaction of neutrinos with nonzero magnetic moments with a transversal
magnetic field leads to a phenomenon of neutrino spin precession or neutrino
spin oscillations. Neutrino spin precession in the transversal magnetic field
${\bf B}_{\perp}$ was first considered in [5], spin-flavor precession was
discussed in [6]. The resonant amplification of neutrino spin oscillations in
${\bf B}_{\perp}$ in the presence of matter was investigated in [7, 8].

In our previous paper [9] we proposed a new approach to the problem of
neutrino mixing and oscillations in a magnetic field based on the exact
solutions of the Dirac equation for a massive neutrino state in a magnetic
field. Within the approach, a spin operator that commutes with the evolution
Hamiltonian is used to classify the massive neutrino states in a magnetic
field. We calculated the probabilities of neutrino flavour and spin
oscillations in a magnetic field for the case of two neutrino flavours. We
have shown that the probabilities exhibit an interplay of oscillations on the
vacuum $\omega_{vac}=\Delta m^{2}/4p$ and magnetic
$\omega_{B}=\mu_{\nu}B_{\perp}$ frequencies. In [10, 11] we extended some of
our results to the case of three neutrino flavours. The CP-violating effects
in oscillations of Majorana neutrinos in a supernova media has been studied
using our approach in [12].

In this paper we continue our research of neutrino oscillations in a magnetic
field and now apply our approach to the case of three neutrino flavours. The
presented derivation of the explicit expressions for the probabilities of
Dirac neutrino flavour and spin oscillations enable us to investigate the
astrophysical neutrino flavour composition in an experiment for the three
flavour case.

##  I Neutrino flavour and spin oscillations in a magnetic field

In this section we describe the approach to deriving of the neutrino
oscillations probabilities in a magnetic field developed in [9] and extend it
to the case of three neutrino flavours.

The wave function $\nu_{i}$ of a massive Dirac neutrino state that propagates
in the presence of a constant and homogeneous and arbitrary orientated
magnetic field can be found as the solution of the following system of
equations

$(\gamma_{\mu}p^{\mu}-m_{i}-{\mu_{i}}{\bm{\Sigma}\bf{B}})\nu_{i}(p)=0,$ (1)

where $\mu_{i}$ are the neutrino magnetic moments, $i={1,2,3}$. Note that in
this paper we suppose that Dirac neutrinos do not possess the transition
magnetic moments. Effects of the transition magnetic moments were studied in
[13] for the two flavour case.

Eq. 1 can be rewritten in the equivalent Hamiltonian form

$H_{i}\nu_{i}=E_{i}\nu_{i},$ (2)

where the Hamiltonian

$H_{i}=m_{i}\gamma_{0}+\gamma_{0}\bm{\gamma}\bm{p}+\mu_{i}\gamma_{0}\bm{\Sigma}\bf{B}.$
(3)

is introduced.

The eigenvalues of the Hamiltonian (3) are given by

$E_{i}^{s}=\sqrt{m_{i}^{2}+p^{2}+{\mu_{i}}^{2}{B}^{2}+2{\mu_{i}}s\sqrt{m_{i}^{2}{B}^{2}+p^{2}B_{\bot}^{2}}},$
(4)

where $s=\pm 1$, $p=|\bm{p}|$ and $B_{\bot}$ is a transversal (relatively to
the neutrino momentum) component of the magnetic field.

Thus, stationary solutions of Eq. 1 exist and satisfy the following relation

$H_{i}\ket{\nu_{i}^{s}(t)}=E_{i}^{s}\ket{\nu_{i}^{s}(t)}.$ (5)

To clarify the meaning of the spin number $s$ introduced in (4) and classify
the neutrino stationary states in a magnetic field we use the following spin
operator [9]

$S_{i}=\frac{m_{i}}{\sqrt{m_{i}^{2}\bm{B}^{2}+\bm{p}^{2}B^{2}_{\perp}}}\left[\bm{\Sigma}\bm{B}-\frac{i}{m_{i}}\gamma_{0}\gamma_{5}[\bm{\Sigma}\times\bm{p}]\bm{B}\right].$
(6)

One can show that the spin operator (6) commutes with the Hamiltonian

$[S_{i},H_{i}]=0,$ (7)

and satisfy the following conditions

$S_{i}^{2}=1.$ (8)

Thus, the stationary states (5) are solutions of Eq. 1 that are eigenstates of
$S_{i}$:

$S_{i}\ket{\nu_{i}^{s}}=s\ket{\nu_{i}^{s}},s=\pm 1,$ (9)
$\sum_{s}\ket{\nu_{i}^{s}}\bra{\nu_{i}^{s}}=1,$ (10)
$\braket{\nu_{i}^{s}}{\nu_{k}^{s^{\prime}}}=\delta_{ik}\delta_{ss^{\prime}}.$
(11)

In order to treat the evolution of the neutrino massive states in a magnetic
field we expand the wavefunctions $\nu_{i}$ over the stationary states

$\displaystyle\nu_{i}^{L}(t)=c_{i}^{+}\nu_{i}^{+}(t)+c_{i}^{-}\nu_{i}^{-}(t),$
(12)
$\displaystyle\nu_{i}^{R}(t)=d_{i}^{+}\nu_{i}^{+}(t)+d_{i}^{-}\nu_{i}^{-}(t),$
(13)

where $L,R$ are neutrino helicities,
$\nu_{i}^{s}(t)=\exp(-iE_{i}^{s}t)\nu_{i}^{s}(0)$, and the coefficients
$c^{\pm}_{i}$ and $d^{\pm}_{i}$ do not depend on time and can be calculated
using initial conditions.

We derive general expressions for the neutrino flavour and spin oscillations
probabilities in a magnetic field using the common defition and strightforward
expression

$P_{\nu_{\alpha}^{h}\rightarrow\nu_{\beta}^{h^{\prime}}}(t)=\big{|}\braket{\nu_{\beta}^{h^{\prime}}(0)}{\nu_{\alpha}^{h}(t)}\big{|}^{2}=\Big{|}\sum_{i}U^{*}_{\beta
i}U_{\alpha i}\braket{\nu_{i}^{h^{\prime}}(0)}{\nu_{i}^{h}(t)}\Big{|}^{2},$
(14)

where $U$ is the mixing matrix, $h,h^{\prime}=L,R$ are helicities of neutrino
initial and final states respectively, $\alpha,\beta=\\{e,\mu,\tau\\}$ are
neutrino flavours and $i=\\{1,2,3\\}$. Using (12), we derive the following
expressions for the probabilities of neutrino flavour and spin oscillations
assuming that the initial state is an electron neutrino:

$\displaystyle
P_{\nu_{e}^{L}\rightarrow\nu_{\alpha}^{L}}(t)=\Bigl{|}\left(|c_{1}^{+}|^{2}e^{-iE_{1}^{+}t}+|c_{1}^{-}|^{2}e^{-iE_{1}^{-}t}\right)U_{11}U_{\alpha
1}^{*}+$
$\displaystyle\left(|c_{2}^{+}|^{2}e^{-iE_{2}^{+}t}+|c_{2}^{-}|^{2}e^{-iE_{2}^{-}t}\right)U_{12}U_{\alpha
2}^{*}+$ (15)
$\displaystyle\left(|c_{3}^{+}|^{2}e^{-iE_{3}^{+}t}+|c_{3}^{-}|^{2}e^{-iE_{3}^{-}t}\right)U_{13}U_{\alpha
3}^{*}\Bigr{|}^{2},$

$\displaystyle
P_{\nu_{e}^{L}\rightarrow\nu_{\alpha}^{R}}(t)=\Bigl{|}\left((d_{1}^{+})^{*}c_{1}^{+}e^{-iE_{1}^{+}t}+(d_{1}^{-})^{*}c_{1}^{-}e^{-iE_{1}^{-}t}\right)U_{11}U_{\alpha
1}^{*}+$
$\displaystyle\left((d_{2}^{+})^{*}c_{2}^{+}e^{-iE_{2}^{+}t}+(d_{2}^{+})^{*}c_{2}^{+}e^{-iE_{2}^{-}t}\right)U_{12}U_{\alpha
2}^{*}+$ (16)
$\displaystyle\left((d_{3}^{+})^{*}c_{3}^{+}e^{-iE_{3}^{+}t}+(d_{3}^{-})^{*}c_{3}^{-}e^{-iE_{3}^{-}t}\right)U_{13}U_{\alpha
3}^{*}\Bigr{|}^{2}.$

Below we calculate the coefficients $c_{i}^{s}$ and $d_{i}^{s}$ and generalise
(I) and (I) to the case of an arbitrary flavour of initial neutrino.

Using (9), (10) and (11), one can show that

$\braket{\nu_{i}^{h^{\prime}}(0)}{\nu_{i}^{h}(t)}=\sum_{s}e^{-iE_{i}^{s}t}\braket{\nu_{i}^{h^{\prime}}(0)}{\nu_{i}^{s}}\braket{\nu_{i}^{s}}{\nu_{i}^{h}(0)}=\sum_{s}e^{-iE_{i}^{s}t}\bra{\nu_{i}^{h^{\prime}}(0)}\hat{P}_{i}^{s}\ket{\nu_{i}^{h}(0)},$
(17)

where the projection operators are introduced

$P_{i}^{\pm}=\ket{\nu_{i}^{\pm}}\bra{\nu_{i}^{\pm}}=\frac{1\pm S_{i}}{2}.$
(18)

Thus, the amplitudes of the transitions between massive neutrino helicity
states are given by the plane wave expansion of the following form

$\braket{\nu_{i}^{h^{\prime}}(0)}{\nu_{i}^{h}(t)}=\sum_{s}C_{is}^{h^{\prime}h}e^{-iE_{i}^{s}t},$
(19)

where

$C_{is}^{h^{\prime}h}=\bra{\nu_{i}^{h^{\prime}}(0)}P_{i}^{s}\ket{\nu_{i}^{h}(0)}.$
(20)

The coefficients $C_{is}^{h^{\prime}h}$ are related with $c_{i}^{s}$ and
$d_{i}^{s}$ introduced in (12) by the following expressions

$|c_{i}^{s}|^{2}=C_{is}^{LL},\;|d_{i}^{s}|^{2}=C_{is}^{RR}.$ (21)

Using the plane-wave expansion (19) one can derive the general expression for
the probabilities of neutrino oscillations for the case of three neutrinos as

$P_{\nu_{\alpha}^{h}\rightarrow\nu_{\beta}^{h^{\prime}}}(t)=\sum_{i,j}\sum_{s,s^{\prime}}U^{*}_{\beta
i}U_{\alpha i}U_{\beta j}U_{\alpha
j}^{*}C_{is}^{h^{\prime}h}\left(C_{js^{\prime}}^{h^{\prime}h}\right)^{*}e^{-i(E_{i}^{s}-E_{j}^{s^{\prime}})t},$
(22)

where $s,s^{\prime}=\pm 1$.

From (22) it can be shown that

$\sum_{\beta,h^{\prime}}P_{\nu_{\alpha}^{h}\rightarrow\nu_{\beta}^{h^{\prime}}}(t)=\sum_{\alpha,h}P_{\nu_{\alpha}^{h}\rightarrow\nu_{\beta}^{h^{\prime}}}(t)=1$
(23)

and

$P_{\nu_{\alpha}^{h}\rightarrow\nu_{\beta}^{h^{\prime}}}(0)=\delta_{\alpha\beta}\delta_{hh^{\prime}}.$
(24)

##  II Neutrino oscillations probabilities in the ultrarelativistic limit

The expression (22) for the probabilities of neutrino oscillations in a
magnetic field can be significantly simplified for the ultrarelativistic case.

For the neutrino energy spectrum in the ultrarelativistic limit $m_{i}\ll p$
we get

$E^{s}_{i}\approx
p+\frac{m^{2}_{i}}{2p}+\frac{\mu_{i}^{2}B^{2}}{2p}+\mu_{i}sB_{\perp}.$ (25)

Given that neutrino magnetic moments are of order of $10^{-11}\mu_{B}$ or
smaller, it is also reasonable to assume that $\mu_{i}B_{\perp}\ll m_{i}$ (see
[9] for a detailed discussion). Thus we can approximately write neutrino
energy spectrum in a magnetic field as

$E^{s}_{i}\approx p+\frac{m^{2}_{i}}{2p}+\mu_{i}sB_{\perp}.$ (26)

Since the differences of neutrino energies and the corresponding frequencies
of neutrino oscillations are

$E^{s}_{i}-E^{s^{\prime}}_{j}\approx\frac{\Delta
m^{2}_{ij}}{2p}+(\mu_{i}s-\mu_{j}s^{\prime})B_{\perp},$ (27)

we expect that the probabilities of neutrino oscillations in a magnetic field
exhibit the inherent interplay of the oscillations on the vacuum frequencies
$\Delta m^{2}_{ik}/4p$ and magnetic frequencies $\mu_{i}B_{\perp}$.

The plane-wave expansion coefficients $C^{hh^{\prime}}_{is}$ also can be
simplified if we account for the fact that in the ultrarelativistic limit
neutrino helicity states are given by the corresponding limit of the Dirac
equation solutions:

$\ket{\nu_{i}^{L}(0)}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ -1\\\ 0\\\
1\end{pmatrix},\ \ \
\ket{\nu_{i}^{R}(0)}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ 0\\\ 1\\\
0\end{pmatrix}.$ (28)

Then from (20) we have

$C^{LL}_{is}\approx\frac{1}{2}\left(1+s\frac{m_{i}B_{\parallel}}{{\sqrt{m_{i}^{2}B^{2}_{\parallel}+(m_{i}^{2}+p^{2})B^{2}_{\perp}}}}\right),$
(29)
$C^{RL}_{is}\approx-\frac{s}{2}\frac{pB_{\perp}}{\sqrt{m_{i}^{2}B^{2}_{\parallel}+(m_{i}^{2}+p^{2})B^{2}_{\perp}}}.$
(30)

In the derivation above we use the fixed reference frame for which
$B_{\parallel}=B_{z}$, $\bm{B_{\bot}}=(B_{x},0,0)$. Note that in the limit
$B_{\bot}=0$ the coefficients $C^{RL}_{is}=0$, and the probabilities of
neutrino spin oscillations $\nu_{\alpha}^{L}\to\nu_{\beta}^{R}$ are zeros.
Thus, neutrino spin oscillations are generated by the transversal magnetic
field, as it should be.

Finally, in the ultrarelativistic limit we assume that $t\approx x$, where $x$
is the distance travelled by neutrino.

Neglecting term of order of $m_{i}^{2}/p^{2}$ and smaller, we arrive to the
expressions for the neutrino flavour $\nu^{L}_{\alpha}\to\nu^{L}_{\beta}$ and
spin $\nu^{L}_{\alpha}\to\nu^{R}_{\beta}$ oscillations probabilities:

$P_{\nu_{\alpha}^{L}\to\nu_{\beta}^{L}}(x)=\frac{1}{4}\sum_{i,j}U^{*}_{\beta
i}U_{\alpha i}U_{\beta j}U_{\alpha
j}^{*}\sum_{s,s^{\prime}}e^{-i(E_{i}^{s}-E_{j}^{s^{\prime}})x},$ (31)
$P_{\nu_{\alpha}^{L}\to\nu^{R}_{\beta}}(x)=\frac{1}{4}\sum_{i,j}U^{*}_{\beta
i}U_{\alpha i}U_{\beta j}U_{\alpha
j}^{*}\sum_{s,s^{\prime}}ss^{\prime}e^{-i(E_{i}^{s}-E_{j}^{s^{\prime}})x}.$
(32)

These expressions can be further simplified. For the probabilities of neutrino
three-flavour oscillations we get

$\displaystyle P_{\nu_{\alpha}^{L}\to\nu_{\beta}^{L}}(x)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}|U_{\alpha i}|^{2}|U_{\beta
i}|^{2}\cos^{2}(\mu_{i}B_{\perp}x)$ $\displaystyle+$
$\displaystyle\sum_{i>j}2\cos(\mu_{i}B_{\perp}x)\cos(\mu_{j}B_{\perp}x)\left[\text{Re}(A_{ij}^{\alpha\beta})\cos\Big{(}\frac{\Delta
m^{2}_{ij}}{2p}x\Big{)}+\text{Im}(A_{ij}^{\alpha\beta})\sin\Big{(}\frac{\Delta
m^{2}_{ij}}{2p}x\Big{)}\right],$

where $A_{ij}^{\alpha\beta}=U^{*}_{\beta i}U_{\alpha i}U_{\beta j}U_{\alpha
j}^{*}$. Note that
$\text{Im}A^{\alpha\beta}_{ij}=J\sin\delta\sum_{k,\gamma}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}$,
where $\delta$ is CP-violating phase and $J\approx 0.034$ is the leptonic
Jarlskog invariant (see [15]). Thus, the last term in (II) describes CP-
violating effects.

It makes sense to introduce a new observable that describes the total
probability of conversion into a sterile neutrino state
$P_{\nu_{\alpha}^{L}\to\nu^{R}}=P_{\nu_{\alpha}^{L}\to\nu^{R}_{e}}+P_{\nu_{\alpha}^{L}\to\nu^{R}_{\mu}}+P_{\nu_{\alpha}^{L}\to\nu^{R}_{\tau}}$:

$P_{\nu_{\alpha}^{L}\to\nu^{R}}(x)=\frac{1}{4}\sum_{\beta}\sum_{i,j}U^{*}_{\beta
i}U_{\alpha i}U_{\beta j}U_{\alpha
j}^{*}\sum_{s,s^{\prime}}ss^{\prime}e^{-i(E_{i}^{s}-E_{j}^{s^{\prime}})x}.$
(34)

This expression can be further simplified. Using the unitarity condition of
the mixing matrix $\sum_{\beta}U^{*}_{\beta i}U_{\beta j}=\delta_{ij}$ and the
fact that $E_{i}^{L}-E_{i}^{R}=2\mu_{i}B_{\perp}$, we arrive to the final
expression for the probability of neutrino spin oscillations

$P_{\nu_{\alpha}^{L}\to\nu^{R}}(x)=\sum_{i=1}^{3}|U_{\alpha
i}|^{2}\sin^{2}(\mu_{i}B_{\perp}x)$ (35)

Note that the probability of spin oscillations (35) does not depend on the
value of the CP-violating phase since it depends only on absolute values of
the mixing matrix entries.

##  III Cosmogenic neutrino oscillations in interstellar media

As an example of neutrino evolution in a magnetic field we study propagation
of cosmogenic neutrinos in the interstellar media (the case of two neutrino
flavours was studied in [13]). Cosmogenic neutrinos are neutrinos that
originate from interaction of ultra-high energy cosmic rays (UHECR) with the
cosmic microwave background [16]. Cosmogenic neutrino fluxes typically have
energies of order of $1$ EeV (i.e. $10^{18}$ eV) [17].

Consider neutrinos with the energy $p=1$ EeV and , $B=2.8$ $\mu$ G for the
interstellar magnetic field strength (see [14]). We also set the value of CP-
violating phase $\delta$ to zero. For the sake of simplicity we start our
analysis with the case of equal neutrino magnetic moments:
$\mu_{1}=\mu_{2}=\mu_{3}=10^{-11}\mu_{B}$. The probabilities of neutrino
flavour oscillations $\nu_{e}\to\nu_{e}$ , $\nu_{e}\to\nu_{\mu}$ and
$\nu_{e}\to\nu_{\tau}$ as functions of distance travelled by neutrino in
parsec are shown in Fig. 1a and Fig. 1b. The probabilities indeed exhibit a
complicated interplay of oscillations on the following vacuum and magnetic
lengths:

$L^{B}_{i}=\frac{\pi}{\mu_{i}B_{\perp}}=2.17\cdot\left(\frac{B}{\text{G}}\right)^{-1}\left(\frac{\mu_{i}}{10^{-11}\mu_{B}}\right)^{-1}10^{-3}\;\text{pc},$
(36)

$L^{vac}_{ij}=\frac{4\pi p}{\Delta m^{2}_{ij}}=5.02\cdot\left(\frac{\Delta
m_{ij}^{2}}{\text{eV}^{2}}\right)^{-1}\left(\frac{p}{\text{GeV}}\right)\cdot
10^{-13}\;\text{pc}.$ (37)

For the choice of parameters given above, oscillations on the vacuum lengths
$L^{vac}_{12}\approx 7$ pc and $L^{vac}_{13}\approx 0.2$ pc that can be
observed at smaller scales (Fig. 1a) are modulated by oscillations on the
magnetic length $L^{B}\approx 370$ pc (Fig. 2a). This behaviour reproduces
phenomenon observed in [13, 9] for the case of two neutrino flavours. For the
case of $\mu_{1}=\mu_{2}=\mu_{3}=10^{-11}\mu_{B}$, Eq. (II) reduces to

$P_{\nu_{\alpha}^{L}\to\nu_{\beta}^{L}}(x)=\cos^{2}(\mu
B_{\perp}x)\left[\sum_{i=1}^{3}|U_{\alpha i}|^{2}|U_{\beta
i}|^{2}+2\sum_{i>j}A_{ij}^{\alpha\beta}\cos\Big{(}\frac{\Delta
m^{2}_{ij}}{2p}x\Big{)}\right],$ (38)

Here the expression in the square brackets is the vacuum oscillations
probability and it is modulated by oscillations on magnetic frequency $\mu
B_{\perp}$.

The interplay of oscillations on the vacuum and magnetic frequencies becomes
even more pronounced if neutrino magnetic moments are not equal to each other.
In Fig. 2a we show the probabilities of neutrino flavour oscillations for this
particular case

Finally, in Fig. 2b we plot the neutrino spin oscillations probability
$\nu_{e}\to\nu^{R}$ for different values of the neutrino magnetic moment
$\mu_{2}$. The magnetic moment $\mu_{1}$ value is fixed as $10^{-11}\mu_{B}$.
Since the last term in the sum in (35) for the case of initial electron is
proportional to a quite small value $\sin^{2}\theta_{13}\approx 0.022$, we can
neglect the contribution of the magnetic moment $\mu_{3}$. In the case of
equal magnetic moments $\mu_{1}=\mu_{2}=\mu$ neutrino spin oscillations
probability is

$P_{\nu_{e}\to\nu^{R}}(x)=\sin^{2}(\pi x/L_{B}),$ (39)

where the oscillations length $L_{B}\approx 370$ pc. For different choices of
the neutrino magnetic moments oscillations probability exhibit an interplay of
oscillations on two magnetic frequencies $\omega_{1}=\mu_{1}B_{\perp}$ and
$\omega_{2}=\mu_{2}B_{\perp}$. Note that distance-averaged spin oscillations
probability equals $1/2$ regardless of choice of neutrino magnetic moments.

Figure 1: Neutrino flavour oscillations probabilities for neutrino energy
$E=1$ EeV and magnetic moments $\mu_{1}=\mu_{2}=\mu_{3}=10^{-11}\mu_{B}$.

Figure 2: a) Neutrino flavour oscillations probabilities for neutrino energy
$E=1$ EeV and magnetic moments $\mu_{1}=10^{-11}\mu_{B}$, $\mu_{2}=2\cdot
10^{-11}\mu_{B}$ and $\mu_{3}=3\cdot 10^{-11}\mu_{B}$; b) Neutrino spin
oscillations probability $\nu_{e}^{L}\to\nu^{R}$ for different values of
magnetic moments.

##  IV Conclusion

Our previous results on neutrino oscillations in magnetic fields [9] were
extended to the case of three neutrino flavours. We have shown that the
probabilities of neutrino flavour oscillations in a magnetic field exhibit a
complicated interplay of oscillations on both vacuum $\omega^{vac}_{ij}=\Delta
m^{2}_{ij}/4p$ and magnetic $\omega^{B}_{i}=\mu_{i}B_{\perp}$ frequencies,
while spin oscillations probability depends only on the magnetic frequencies
$\omega^{B}_{i}$. As an example of neutrino oscillations in a magnetic field
we have considered oscillations of cosmogenic neutrinos in the interstellar
magnetic field.

Using our findings on three flavour neutrino oscillations, we have shown that
the presence of interstellar magnetic field indeed can modify neutrino fluxes
observed by terrestrial neutrino telescopes. In particular, we find that in
the considered Dirac case in average half of neutrinos convert into sterile
neutrino state. Using three flavour oscillations probabilities one can also
compute fluxes of active neutrinos at the distance $x$ from neutrino source as

$\Phi_{\alpha}(x)=\sum_{\beta}\Phi^{0}_{\beta}P_{\nu^{L}_{\beta}\to\nu^{L}_{\alpha}}(x),$
(40)

where $\Phi^{0}_{\beta}$ are the fluxes at the neutrino source, and calculate
neutrino flavour composition observed by a neutrino telescope:

$r_{\alpha}(x)=\frac{\Phi_{\alpha}(x)}{\sum_{\beta}\Phi_{\beta}(x)}.$ (41)

It is predicted that UHE neutrino flavour composition follows pattern
$\Phi_{e}^{0}:\Phi_{\mu}^{0}:\Phi_{\tau}^{0}\approx 1:2:0$ [19]. Then, for the
case of vacuum oscillations, neutrino flavour composition at the terrestrial
neutrino telescope is
$(r_{e}^{vac},r_{\mu}^{vac},r_{\tau}^{vac})\approx(1/3,1/3,1/3)$ [20]. It is
likely that in next 20 years neutrino telescopes such as Baikal-GVD, KM3NeT,
P-ONE, TAMBO and IceCube-Gen2 will collect enough data to determine the
flavour composition of high-energy cosmic neutrino fluxes [21]. If the
measured composition contradicts the predicted pattern (1/3, 1/3, 1/3), this
might be a signal of new physics, including neutrino magnetic moments. Using
the probabilities of neutrino oscillations in interstellar magnetic field we
can study these possible effects of nonzero neutrino magnetic moments on UHE
neutrino fluxes flavour composition. However, to study neutrino propagation
through cosmic scales we have to account for the effects of coherence in
neutrino oscillations and generalize the probabilities (II). We are going to
continue our research on neutrino oscillations in a magnetic field and
consider effects of wave packet separation in a forthcoming paper.

## Acknowledgements

The work is done within the programme of the Interdisciplinary Scientific and
Educational School of Moscow University “Fundamental and Applied Space
Research” and is supported by the Russian Science Foundation under grant
No.22-22-00384. The work of A.P. has been supported by the Foundation for the
Advancement of Theoretical Physics and Mathematics “BASIS” under Grant No.
21-2-2-26-1.

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