# Cosmic neutrino flux and spin flavor oscillations in intergalactic medium

Ashutosh Kumar Alok<EMAIL_ADDRESS>Indian Institute of Technology Jodhpur,
Jodhpur 342037, India Neetu Raj Singh Chundawat<EMAIL_ADDRESS>Indian
Institute of Technology Jodhpur, Jodhpur 342037, India Arindam Mandal
<EMAIL_ADDRESS>Indian Institute of Technology Jodhpur, Jodhpur 342037,
India

###### Abstract

The ultra high energy (UHE) cosmic neutrinos are expected to play a pivotal
role in the disquisition of physics beyond the standard model of particle
physics as well as serve as an ideal cosmic messengers. This epitomizes the
selling point of several currently running or planned neutrino telescopes. The
UHE cosmic neutrinos usually perambulate gargantuan scales in the
extragalactic universe having a magnetic field. If neutrinos have a finite
magnetic moment ($\mu_{\nu}$) owing to quantum loop corrections, this may
result in spin-flavor oscillations, which can affect the cosmic neutrino flux.
Using the current limit and assuming neutrinos to be Dirac particles, we show
that the flux of cosmic neutrinos will reduce by half if they traverse few
Mpcs through the intergalactic magnetic field, in the range of $\rm\mu G$ to
$nG$. Moreover, one can safely neglect the effect of $\mu_{\nu}$ if the
current upper bound is improved by a few orders of magnitude even if the
neutrinos travel through the size of the visible universe.

##  I Introduction

The standard model (SM) of electroweak interaction provides a compendious
description of nature up to the energy scale of $10^{12}$ eV. The discovery of
Higgs boson [1, 2] enabled the extension of the authoritativeness of the SM
far beyond the EW scale. Despite this high-handedness, SM cannot be considered
as theory of everything. The nonfeasance of the SM to assimilate gravity,
demystify dark matter and dark energy effectuates us to hunt for physics
beyond SM. The currently running Large Hadron Collider (LHC) has the potential
to search for such new physics at the multi-TeV scale.

The neutrinos of cosmic origin may have energies many order of magnitudes
larger than those accessible by any terrestrial accelerator. Therefore these
ultra high energy (UHE) neutrinos furnish access to inquest physics at a very
fundamental scale. Further, unlike photons and charged particles, they only
interact via weak interactions. Hence neutrinos can barge in on a direct line
from their source, traveling cosmological distances, passing through
interceding matter like interstellar dust and can even escape the densest
environments. Due to these accentuates, neutrinos are considered to be the
ideal cosmic messengers.

The UHE neutrino flux originates from the interaction of cosmic rays with
ambient matter and photon fields as they propagate through the universe.
Therefore these neutrinos can be affiliated with sources of cosmic rays
ranging from the Milky Way to powerful sources such as active galactic nuclei.
During their journey towards earth, UHE protons above the threshold of
photopion can interact with the cosmic microwave background or infrared
backgrounds, the Greisen-Zatsepin-Kuzmin (GZK) process [3, 4], which induces
pions and neutrons which decays to produce neutrinos [5]. These neutrinos are
known as Berezinsky-Zatspein neutrinos, or the “BZ” flux. The accretion of
such neutrinos over cosmological time is known as the BZ or the ”cosmogenic
neutrino flux” [6, 7, 8]. The length-scale of the GHZ process is 50-200 Mpc
whereas it has an energy scale limit of 50 EeV. On average, the neutrinos are
produced with $\sim$5% of the energy of protons [9]. The cosmic neutrino
fluxes can be substantial if the sources are rummaged beyond the GZK scale and
the flux of protons extends beyond the GZK cutoff [10]. Apart from protons,
cosmic rays also consist of high energy nuclei. These UHE nuclei interact with
the cosmic microwave as well as infrared backgrounds and undergo
photodisintegration [3, 4]. The cosmic neutrinos are then produced via
interaction of these disassociated nucleons with the ambient cosmic microwave
and infrared environments [11, 12].

A precise estimation of cosmic neutrino flux relies upon the origin as well as
cosmic ray source models [13]. Ever since the first observation of cosmic rays
in 1960, we have observed cosmic rays with energies up to a few times
$10^{20}$ eV which is about 40 million times the energy of particles
accelerated by the LHC. However, the origin of cosmic rays is still a mystery.
In 2018, IceCube Collaboration announced the observation of neutrinos emerging
from high energy blazar TXS 0506+056 which is estimated to be located at a
distance of about 1.75 Gpc from the earth [14]. This blazar is also slated to
produce cosmic rays. Before this observation, the farthest observed neutrino
source was supernova SN 1987A. About 20 neutrinos in the 10-40 MeV energy
range were detected from this source. Further, there exist a number of
possibilities for the cosmic ray source models, see for e.g., [15, 16, 17, 18,
19]. Due to these reasons, the predicted neutrino flux suffers from
significant uncertainties. The measurement of UHE cosmic neutrino fluxes would
not only allow determination of composition of cosmic rays at all distant
scales [20, 21] and hence screening out cosmic ray source models but would
also empower determination of the source of cosmic rays [9, 6, 22, 23, 7, 21,
16, 24, 25, 26, 27, 28, 29, 19, 30, 31, 32]. In fact, the existing
experimental results have already ostracized a few cosmogenic models [33, 10,
34, 35].

Currently, the IceCube observatory is the largest neutrino telescope covering
a 1 ${\rm km^{3}}$ detection volume [36]. It is an in-ice Cherenkov neutrino
observatory located at the Amundsen–Scott South Pole Station in Antarctica.
Over the last decade, IceCube collaboration has measured neutrinos in the TeV
to PeV range. IceCube-Gen2 is the planned extension of IceCube where the
instrumented volume is expected to be around 7.9 ${\rm km^{3}}$ along with an
effective area which would be $\sim$ 8 times that of IceCube between 100 TeV
and 1 PeV [37, 38]. Located in the Mediterranean Sea, 100 km off the coast of
Sicily, KM3NeT is a planned experiment [39] which is the successor to ANTARES,
This project is anticipated to be completed by 2024 [40]. Located in lake
Baikal in Siberia, Baikal-GVD is a detector having a volume of gigatons [41].
It develops on the extant NT-200 detector. With an effective volume of 0.35
${\rm km^{3}}$ , this is operational from 2018. The completion is expected in
2025 with an effective volume of 1.5 ${\rm km^{3}}$. Baikal-GVD has already
observed at least one candidate neutrino cascade event with reconstructed
energy of 91 TeV [42]. The Pacific Ocean Neutrino Experiment (P-ONE) is a
planned water Cherenkov experiment which will be ensconced in the Cascadia
basin of Vancouver Island for which the infrastructure is already in place
[43]. P-ONE is expected to be complete in 2030. These new telescopes KM3NeT,
Baikal-GVD and P-ONE will improve our sensitivity to TeV to PeV neutrinos to
the Southern Sky.

In the next two decades, there are several planned experiments based on
different strategies [44, 45]. These include radar based detector RET-N [46],
Tau Air-Shower Mountain Based Observatory (TAMBO) located in a deep canyon in
Peru [47], satellite based NASA Astrophysics Probe-class mission POEMMA which
will observe air fluorescence produced by extensive air showers from UHE
neutrinos [48] and Ashra Neutrino Telescope Array (Ashra NTA) which will form
a 25 km triangle watching the total air mass surrounded by Mauna Loa, Mauna
Kea, and Hualalai, and a single site station at the center with full-sky
coverage [49]. These detectors are expected to improve our sensitivity to
neutrino energies above the energy range where IceCube becomes too small to
detect a significant flux.

The fact that cosmic rays have been observed up to $10^{20}$ eV energy range,
neutrino flux up to EeV energies can be engendered due to interaction of
cosmic rays with the ambient environment. However, till date, no such events
have been reported. Even higher energy neutrinos, in the ZeV range, can be
spawned from the kinks of cosmic string loops which move at the speed of light
on strings emitting moduli which eventually decay into pions and neutrinos via
hadronic cascades [50]. There are several next generation planned or under
construction EeV–ZeV range detectors. These include IceCube-Gen2 [37, 38],
EUSO-SPB2 [51], AugerPrime [52], BEACON [53], and TAROGE [54].

Neutrinos travelling through the Universe on its peregrination towards earth
may encounter strong magnetic fields of compact objects or may simply expedite
through the intergalactic space which may also have relatively weak magnetic
fields. The detection of distant static magnetic fields is a congenitally
arduous task. Howbeit several measurements over the last few decades show that
such fields do exist and that too at revelatory strengths and spanning over
astonishingly large scales in the extragalactic universe. These fields are
mainly due to contributions coming from individual galaxies, the disencumbered
magnetic energy is being stored in demesnes as colossal as intergalactic
separations. Typically this field is of the order of $\rm\mu G$ or less. For
e.g., the interstellar magnetic field of Milky way is measured to be 2.93
$\rm\mu G$ within a small uncertainty [55]. Further, the first direct
measurement of the magnetic field strength in the Coma Cluster is about 2
$\rm\mu G$ spanning over a scale of 13-40 kpc [56]. The magnetic field has
also been measured in extrement remote locations of the universe outside the
galaxy clusters. The observed value of magnetic field in Coma–A1367
supercluster is $\sim$ (0.3-0.6) $\rm\mu G$ [57].

Neutrino oscillation has substantiated the evidence for non-zero neutrino
mass. Massive neutrinos can also have a non-zero magnetic moment. Although
they cannot directly couple to photons, a magnetic moment of neutrinos can be
generated via quantum loops. The minimally extended SM (MESM) value of a
neutrino magnetic moment is $\mu_{\nu}=10^{-20}\mu_{B}$ [58]. Various beyond
SM scenarios can enhance the value of $\mu_{\nu}$ [59]. The current
experimental upper limit on $\mu_{\nu}$ is $(2.8-2.9)\times 10^{-11}\mu_{B}$
[60, 61]. Due to this non-zero magnetic moment, neutrinos can be affected
while passing through a magnetic field. In particular, a plausible possibility
is that the spin of neutrinos would be flipped while moving through the
magnetic field. This spin flip is, of course, caused by an external field.
However, neutrino oscillation is an intrinsic property of ultra-relativistic
neutrinos. Thus flavor oscillations subsist during the passage through an
external magnetic field. As a result, both spin and flavor oscillations can
proliferate during their propagation. This is known as spin-flavor oscillation
of neutrinos [62, 63, 64, 65, 66, 67].

This phenomenon has been a topic of interest in the context of different
compact objects having a very high magnetic field [68, 69]. This is because a
high magnetic field is required to compensate for the smallness of the
magnetic moment. However neutrinos traversing cosmic distances pass through
the intergalactic magnetic field. Although this field is small ($O(\rm\mu G)$
or less), this can effectuate observable spin-flavor oscillations since the
neutrinos can travel a very large distance in the interstellar and
intergalactic field. This possibility was studied in [70] in the context of
the interstellar magnetic field of the Milky Way. In ref. [69], a formalism
involving stationary eigenstates was developed. Further, in [71], it was shown
that even after traversing through the intergalactic magnetic field, the
neutrino flavor oscillation will mimic the vacuum flavor ratio for vacuum
oscillations. In this work, we show that for Dirac neutrinos, although this
flavor ratio is identical to the flavor oscillations in vacuum, the flux will
be reduced to half of the value expected from the flavor oscillations. This
would be of extreme importance for experiments related to the high energy
neutrinos of cosmic origin. We also analyze the conditions for which the
effects of neutrino magnetic moment can be safely neglected.

The plan of this work is as follows. In section II, neutrino spin-flavor
oscillations are introduced. And in section III, the effect of spin-flavor
oscillations on neutrino flux from a very large distance and interplay between
various parameters governing the effect, is shown. We conclude our results in
section IV.

##  II Neutrino spin flavor oscillations

As discussed in section I, neutrinos can posses magnetic moment via quantum
corrections. This gives rise to the spin-flavor oscillations. This phenomenon
is quantified by the spin-flavor oscillation probabilities, as in the case of
flavor oscillations. The flavor and spin-flavor oscillation probabilities can
be obtained from the stationary states of neutrinos in a magnetic field 111In
the case of Majorana neutrinos, in general, the neutrinos are converted to the
antineutrinos. For Majorana neutrinos, only the off diagonal magnetic matrix
elements contribute. The analysis of off-diagonal magnetic moments calls for a
more rigorous analysis and requires setting up a new framework [72] which is
beyond the scope of the current work. For this reason, even for the Dirac
neutrinos, only diagonal magnetic moments are considered in the present work.
[69]. The Dirac equation obeyed by neutrinos in the presence of a magnetic
field, is given as,

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

where $\nu_{i}^{s}$ is the wave function of neutrino of $i$-th mass eigenstate
and $s\,(=\pm 1)$ being the eigenvalues of the spin operator $\hat{S}_{i}$ and
$\mu_{i}$ stands for magnetic moment of neutrino. This $\hat{S}_{i}$, commutes
with the Hamiltonian ($\hat{H}_{i}$) in the presence of magnetic field.
$\hat{H}_{i}$ and $\hat{S}_{i}$ are given by,

$\hat{H}_{i}=\gamma_{0}\boldsymbol{\gamma
p}+\mu_{i}\gamma_{0}\boldsymbol{\Sigma B}+m_{i}\gamma_{0},$ (2)

and

$\hat{S}_{i}=\frac{m_{i}}{\sqrt{m_{i}^{2}\boldsymbol{B}^{2}+\boldsymbol{p}^{2}\boldsymbol{B}_{\perp}^{2}}}\left[\boldsymbol{\Sigma
B}-\frac{i}{m_{i}}\gamma_{0}\gamma_{5}[\boldsymbol{\Sigma\times\boldsymbol{p}}]\boldsymbol{B}\right]\,,$
(3)

respectively. Neutrinos are assumed to propagate along the positive $z$-axis,
thus the momentum of neutrino is $\boldsymbol{p}=p_{z}$ and the magnetic field
is given by, $\boldsymbol{B}=(B_{\perp},0,B_{\parallel})$. The energy of
neutrino is given as,

$E_{i}^{s}=\sqrt{m_{i}^{2}+p^{2}+\mu_{i}^{2}\boldsymbol{B}^{2}+2\mu_{i}s\sqrt{m_{i}^{2}\boldsymbol{B}^{2}+\boldsymbol{p}^{2}\boldsymbol{B}_{\perp}^{2}}}\,.$
(4)

In absence of magnetic field, the energy of neutrinos reduces to the energy of
free neutrinos $\sqrt{m_{i}^{2}+p^{2}}$, as expected. It is quite practical to
consider $p\gg m$ and $p\gg\mu_{\nu}B$. For these approximations,

$E_{i}^{s}\approx
p+\frac{m_{i}^{2}}{2p}+\frac{\mu_{i}^{2}B^{2}}{2p}+\mu_{i}sB_{\perp}.$ (5)

The mixing between neutrinos states is given by,

$|\nu_{\alpha}^{h}\rangle=\sum_{i}U^{*}_{\alpha i}|{\nu_{i}^{h}}\rangle,$ (6)

where $U$ is the PMNS matrix and $h$ denotes the helicity of the neutrino.
This analysis follows [73, 61] for PMNS matrix elements and mass-square
differences. The states $|\nu_{\alpha}^{h}\rangle$ and $|{\nu_{i}^{h}}\rangle$
refer to the flavor and mass states, respectively. Note that in the case of
standard neutrino oscillations, neutrinos of particular handedness transform
from one flavor to another. But in the case of spin-flavor oscillations,
neutrinos of both helicities are coupled to each other.Thus it is important to
consider left and right-handed neutrinos.

The mass states can be constructed from the stationary states $(\nu_{i}^{s})$
obtained from the Dirac equation, as

$|\nu_{i}^{L}(t)\rangle=c_{i}^{+}|\nu_{i}^{+}(t)\rangle+c_{i}^{-}(t)|\nu_{i}^{-}(t)\rangle,\quad|\nu_{i}^{R}(t)\rangle=d_{i}^{+}|\nu_{i}^{+}(t)\rangle+d_{i}^{-}|\nu_{i}^{-}(t)\rangle,$
(7)

where $c_{i}^{s}$ and $d_{i}^{s}$ are time independent coefficients (in
general, complex numbers). From Eq. 6 and Eq. 7 the mixing relation becomes,

$|\nu_{\alpha}^{L}(t)\rangle=\sum_{i}U^{*}_{\alpha
i}(c_{i}^{+}|\nu_{i}^{+}(t)\rangle+c_{i}^{-}(t)|\nu_{i}^{-}(t)\rangle),$ (8)
$|\nu_{\alpha}^{R}(t)\rangle=\sum_{i}U^{*}_{\alpha
i}(d_{i}^{+}|\nu_{i}^{+}(t)\rangle+d_{i}^{-}(t)|\nu_{i}^{-}(t)\rangle).$ (9)

Thus the mixing of neutrinos is now defined in terms of the stationary
eigenstates. The spin-flavor oscillation probability can be written in a
generic form as [71],

$P_{\alpha\beta}^{hh^{\prime}}(x)=|\langle\nu_{\beta}^{h^{\prime}}(0)|\nu_{\alpha}^{h}(x)\rangle|^{2}\,,$
(10)

which will lead us to the explicit oscillation probability,

$\begin{split}P_{\alpha\beta}^{hh^{\prime}}(x)=\delta_{\alpha\beta}\delta_{hh^{\prime}}-4\sum_{\\{i,j,s,s^{\prime}\\}}{\rm{Re}}([A_{\alpha\beta}^{hh^{\prime}}]_{i,j,s,s^{\prime}})\sin^{2}\left(\frac{E_{i}^{s}-E_{j}^{s^{\prime}}}{2}\right)x\\\
+2\sum_{\\{i,j,s,s^{\prime}\\}}{\rm{Im}}([A_{\alpha\beta}^{hh^{\prime}}]_{i,j,s,s^{\prime}})\sin\left(E_{i}^{s}-E_{j}^{s^{\prime}}\right)x,\end{split}$
(11)

where $[A_{\alpha\beta}^{hh^{\prime}}]_{i,j,s,s^{\prime}}=U^{*}_{\beta
i}U_{\alpha i}U_{\beta j}U^{*}_{\alpha
j}(C_{is}^{h^{\prime}h})(C_{is^{\prime}}^{h^{\prime}h})^{*}$ and
$C_{is}^{h^{\prime}h}=\langle\nu_{i}^{h^{\prime}}|\hat{P}_{i}^{s}|\nu_{i}^{h}\rangle$.
The projection operator is defined as,
$\hat{P}_{i}^{\pm}=\frac{1\pm\hat{S}_{i}}{2}$ and

$\sum_{\\{i,j,s,s^{\prime}\\}}=\sum_{i>j;s,s^{\prime}}+\sum_{s>s^{\prime};i=j}.$
(12)

For a neutrino flux traversing a very large distance, the probability 11 is
averaged out to,

$P_{\alpha\beta}^{hh^{\prime}}(x)=\delta_{\alpha\beta}\delta_{hh^{\prime}}-2\sum_{\\{i,j,s,s^{\prime}\\}}{\rm{Re}}([A_{\alpha\beta}^{hh^{\prime}}]_{i,j,s,s^{\prime}})\,.$
(13)

##  III Cosmic neutrinos flux reduction

Ultrahigh energetic neutrinos can be produced in various astrophysical
environments, as discussed in section I. The flux of this UHE neutrinos passes
through omnipresent intergalactic magnetic field. The value of magnetic field
is very small, $B\sim nG-{\mu}G$. But since neutrinos traverse cosmic
distances to reach earth, they may undergo spin-flavor oscillations driven by
the intergalactic magnetic field. Since under this phenomenon, active
neutrinos convert to right handed and vice versa, some crucial effect on
active neutrino flux is expected. In this section we will discuss, in details,
about these effects and conditions of feasibility of these effects.

The averaged probability given in Eq.13 describes the probability of
converting from one flavor to another after traversing through a very large
distance. Thus the active neutrino flux on earth from a very distant source
can be obtained as,

$\Phi_{\beta}^{\oplus}=\sum_{\alpha}P_{\alpha\beta}^{LL}\Phi_{\alpha}^{s},$
(14)

where $\Phi^{\oplus}$ and $\Phi^{s}$ are the flux on earth and at the source,
respectively.

In [72], it was shown that the flux ratio of neutrinos does not alter form
that of flavor oscillations, by spin-flavor oscillation for a neutrino flux
arriving from large distance. However, we find that even if the neutrino
flavor ratio remains unchanged under such circumstances, the flux of the
neutrinos will become half of the expected value from flavor oscillations. In
Eq. 13, $C_{is}^{LL}=|c_{i}^{s}|^{2}\approx\frac{1}{2}$. Thus expanding the
Eq. 13 we find that $\sum_{\alpha}P_{\alpha\beta}^{LL}=0.5$, unlike the case
of flavor oscillations. This is due to the fact that half of the active
neutrinos becomes sterile neutrinos while traversing large distance which
leads to $\Phi_{\beta}^{\oplus SF}=\frac{\Phi_{\beta}^{\oplus F}}{2}$, for
each of the flavor.

But this process of averaging is subject to the number of cycles completed by
the oscillation wave forms given in Eq. 11. This factor depends on the energy
differences, $(E_{i}^{s}-E_{j}^{s^{\prime}})$. From Eq. 5,

$E_{i}^{s}-E_{j}^{s^{\prime}}=\frac{\Delta
m_{ij}^{2}}{2p}+\mu_{\nu}(s-s^{\prime})B_{\perp},$ (15)

where $\Delta m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}$ and the magnetic moment of
neutrinos are assumed to be equal for all the neutrino states. One can thus
define the phase of oscillations as,

$\phi=\left[\frac{\Delta
m_{ij}^{2}}{2p}+\mu_{\nu}(s-s^{\prime})B_{\perp}\right]x.$ (16)

As one can observe from Eq. 16, if the energy difference is very small due to
very high energy of neutrinos and very small magnetic moment, then huge $x$ is
required for the phase to be significant $(\phi\sim\pi)$. The phase $\phi$ is
composed of two different phases: one of which is the phase for vacuum
oscillations, $\phi_{v}=\frac{\Delta m_{ij}^{2}}{2p}x$ and another due to
magnetic field is $\phi_{B}=\mu_{\nu}(s-s^{\prime})B_{\perp}x$.

The probability in Eq. 11, is basically sum of terms like
$(\sin^{2}\phi_{v})(\sin^{2}\phi_{B})$, which forms a modulated waveform. The
frequency of oscillation for the two sinusoidal waves individually is given as
$\omega_{v}=\frac{\Delta m_{ij}^{2}}{2p}$ and
$\omega_{B}=\mu_{\nu}(s-s^{\prime})B_{\perp}$, respectively. Whichever of them
have smaller frequency, will form the envelope of of the wave form. The other
one will oscillate inside the envelope. Thus to derive the condition for
averaging out the probability, Eq. 11 to get Eq.13, one needs to identify the
frequencies. The phase $\phi_{v}$ (and hence frequency $\omega_{v}$) depends
on the energy of the neutrinos, whereas $\omega_{B}$ depends on the magnetic
moment of neutrinos and the magnetic field. The oscillation length
corresponding to each of them is given by,

$l_{v}=\frac{\pi}{\omega_{v}},\quad l_{B}=\frac{\pi}{\omega_{B}}.$ (17)

Obviously, smaller frequency corresponds to larger oscillation length. So,
whichever have larger oscillation length, will form the envelope. And thus we
need to find the condition of averaging out for larger oscillation length.
(Note that, both $\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$ appear in the
probability. However, the larger oscillation length corresponds to $\Delta
m_{21}^{2}$). In any case, if $n=\frac{x}{l_{v/B}}\gg 1$, then only we can
average out the probability. We consider $n=100$ (at least) as the condition
to average out the probability.

Figure 1: The upper panel shows the variation of $l_{B}$ with energy. The
black dashed line denotes the distance of TXS blazar from earth and the blue
line corresponds to the distance to the center of Milky Way from earth. The
energy dependence of $l_{v}$ is shown in the lower panel. The purple line
shows the minimum possible value of $l_{B}$ which is attainable at
$\mu_{\nu}B=5\times 10^{-18}\mu_{B}G$. Here we have considered a more
stringent value, $B=1\times 10^{-6}$ G and $\mu_{\nu}=5\times
10^{-12}\,\mu_{B}$, as compared to the current experimental upper limits.

As shown in fig.1, up to 1 ZeV neutrino energy, $l_{B}>l_{v}$. This is the
minimum possible value of $l_{B}$ with the observational upper limits of
intergalactic magnetic field and the magnetic moment of neutrinos. For energy
above that, however, it will depend on the $\mu_{\nu}$ and energy of the
neutrinos. Thus there are two regions of interest: (i) $E<1$ ZeV and (ii)
$E>1$ ZeV.

Let us first consider scenario (i). For neutrino flux having energy $E<1$ ZeV,
to average out the spin-flavor oscillation probability, we need
$\frac{d}{l_{B}}>100$ as per our condition, where $d$ is the distance to the
source of neutrinos. Thus the source must be situated at least at a distance
$d_{min}=100\,l_{B}$. We can see from fig.1, operating an average over the
oscillation probability for a neutrino flux sourced within the Milky Way, is
only possible if $\mu B>10^{-17}$ $\rm\mu_{B}G$ (since $d_{min}>100\,l_{B}$).
But this value is not achievable with the considered upper limits on the
neutrino magnetic moment and the intergalactic magnetic field as shown in Fig.
2. On the other hand, let us take the example of neutrino flux sourced at the
TXS blazar, located at a distance of 5.7 billion light years from earth. This
neutrino flux can complete 100 cycles of spin-flavor oscillations, for large
region of allowed parameter space. In particular, even near $nG$ order of the
intergalactic magnetic field, probability averaging is possible. Fig.2 also
illustrates the parameter space between the magnetic field and the magnetic
moment for the minimum distance of the source of neutrino flux for which the
averaging out condition is satisfied.

Figure 2: The allowed parameter space of $\mu_{\nu}$ and intergalactic
magnetic field satisfying the condition for averaging out the spin-flavor
oscillation for different values of the distance of neutrino sources.

Let us now turn to the scenario (ii). In this energy region, both
$l_{v}>l_{B}$ and $l_{v}<l_{B}$ is possible. For $l_{v}<l_{B}$, the conditions
are identical to the scenario (i). In this region, $l_{v}>l_{B}$ iff

$p>\frac{6.3\times 10^{3}}{\mu_{\nu}B}\,\rm{eV}$ (18)

This is the condition for the flavor oscillations to form the envelope. For
$\mu B=10^{-22}$ $\rm\mu_{B}G$, the required energy of the neutrino flux to
follow Eq. 18 is $\sim 10^{25}$ eV. This is the GUT scale energy. Since the
energy of neutrinos usually lies below this value, thus the condition required
for scenario (i) is sufficient to study the depletion of neutrinos.

The above analysis was performed using a more stringent values of $\mu_{\nu}$
and $B$ as compared to their current experimental upper limits. However, if we
use the current upper limits, $B\approx 3\times 10^{-6}$ G and
$\mu_{\nu}\approx 3\times 10^{-11}\,\mu_{B}$, then the $l_{B}$ values reduces
by an order of magnitude which allows the conditions for averaging out to be
satisfied by the sources of neutrino flux lying within the Milky Way as shown
in fig. 3. It is evident from the figure that the averaging out conditions
will be satisfied only for $\mu_{\nu}$ above $\sim 1.2\times
10^{-11}\,\mu_{B}$ (dashed-line).

Figure 3: The magenta region shows the parameter space for which the averaging
out is possible for neutrino sources located within the Milky Way galaxy. The
black shaded region is excluded by the upper limits on the magnetic field. The
dashed-line represents the cut-off on $\mu_{\nu}$ for which the averaging out
conditions will be satisfied.

A neutrino flux originated from a distance objects, can potentially be halved
by the spin flavor oscillation, if they meet the above conditions. Future
neutrino observatories will expectantly measure neutrino flux from sources
located at very large distances. If the neutrino flux measured by these future
experiments turns out to be half of the expected flux (from sources situated
at a minimum distance specified above), then that would be a signature of
large neutrino magnetic moment. On the other hand if the neutrino magnetic
moment is smaller than $10^{-13}\mu_{B}$, then neutrino flux in the future
experiments will be shielded from the effect of neutrino spin-flavor
oscillations.

##  IV Conclusions

The UHE neutrinos are expected to play a crucial role in determining the
origin of cosmic rays as well as testing several cosmic ray source models. The
fact that these cosmic neutrinos can have energies million times the energy of
particles generated by any of the currently running planned or terrestrial
accelerators, it has immense potential to probe physics beyond Standard Model
extending even up to Planck scale. The IceCube Collaboration has already
observed the neutrinos emerging from blazar which is located at a distance of
about 1.75 Gpc from the earth which is approximately one tenth of the size of
the visible universe. There are cornucopia of under construction or planned
neutrino telescopes to observe such UHE cosmogenic neutrino fluxes. The UHE
neutrino flux traversing humongous distances from its source to the earth,
encounters the omnipresent intergalactic magnetic field. These fields are
relatively weaker compared to the magnetic field of the compact objects. If
neutrinos have magnetic moment due to the quantum loop corrections, the
phenomenon of spin flavour oscillations may be induced. Therefore it is a
legitimate question to ask whether this magnetic moment, which is expected to
be small, would affect the propagation of neutrinos through the intergalactic
medium or not. We investigate this question in the current work.

Using the current limit on $\mu_{\nu}$ and magnetic field ranging from $\mu G$
to $nG$, we find that the condition for averaging out the probability, i.e.,
the reduction of flux by a factor of half, is satisfied for the neutrino
sources lying at a distance of a several kpc from the earth for energy up to 1
ZeV. For energies beyond 1 ZeV, the minimum required distance is about few
Mpc. If the current bound on $\mu_{\nu}$ improves, the condition for averaging
out will be satisfied for larger distances. We find that the reduction of
cosmic neutrino flux is not possible if the current upper limit of $\mu_{\nu}$
is improved up to $\sim 10^{-13}\,\rm\mu_{B}$ even if the neutrinos travel
through the entire length of the visible universe. Therefore, for magnetic
moment in this range, the effect of spin flavor oscillations can be safely
neglected.

Note added: Few hours before submitting this manuscript, ref. [74] appeared on
the arXiv on similar lines.

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