# UHE neutrinos encountering decaying and non-decaying magnetic fields of
compact stars

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 Trisha Sarkar
<EMAIL_ADDRESS>Indian Institute of Technology Jodhpur, Jodhpur 342037,
India

###### Abstract

The phenomena of neutrino spin flavour precession in the presence of an
extraneous magnetic field is a repercussion of neutrino magnetic moment which
is consociated with the physics beyond the standard model of electroweak
interactions. Ultra high energy neutrinos are spawned from a number of sources
in the universe including the highly energetic astrophysical objects such as
active galactic nuclei, blazar or supermassive black holes. When such high
energy neutrinos pass through any compact stellar objects like neutron stars
or white dwarfs, their flux can significantly reduce due to the exorbitant
magnetic field provided by these compact objects. For Dirac neutrinos, such
phenomena occur due to the conversion of neutrinos to their sterile
counterparts. In this work, we consider a neutron star possessing a spatially
varying magnetic field which may or may not decay with time. We find that, for
the non-decaying magnetic field, the flux of high energy Dirac neutrinos
becomes nearly half after passing through the neutron star. The flux is
further enfeebled by $\sim 10\%$ in the presence of muons inside the neutron
star. For decaying magnetic field, the flux reduction is abated by $\sim 5\%$
as compared to the temporally static magnetic field. In the case of a white
dwarf, the depletion of flux is lesser as compared to the neutron stars.

##  I Introduction

The aberrant and largish potential of ulta high energy (UHE) neutrinos of
cosmic origin to probe the structure of the universe at the leviathan scales
has effectively propelled them to the omphalos of high energy physics. The
fact that neutrinos interact only via weak interactions, it can perambulate
colossal scales without interacting with the ambient environment which
includes compact objects as well as interstellar or intergalactic medium. This
may enable identification of sources of distant UHE cosmic ray sources as
these neutrinos are usually generated due to interaction of cosmic rays with
the cosmic microwave or infrared backgrounds. In fact, one such identification
has already been accomplished [1]. In case neutrino interactions do occur, the
nature of interactions can be revealed if this results in the change of flux
or flavor ratios as compared to that of the vacuum oscillations. Therefore UHE
neutrinos relishes unmatched potential in unsnarling remotest demesnes of our
Universe.

To capitalize on this elephantine potentiality, a profusion of experimental
facilities are either under construction or planned [2]. These can be
categorized in terms of sensitivities to energies of cosmic neutrinos. For
TeV-PeV energy range, neutrinos have high opacity [3, 4, 5, 6, 7, 8]. In
defiance of this, a smatter of neutrinos can still sally up to thousands of km
inside the Earth before interacting [9, 10, 11]. These neutrinos can be
ensnared by photonic detectors instilled in pharaonic voume of ice or
submerged in grandiose water body. In the impending future, the northern sky
will be mapped by the IceCube [12, 13] whereas KM3NeT [14], Baikal-GVD [15],
P-ONE [16] will monitor the southern sky. The experiments such as IceCube-Gen2
[17], RET-N [18], TAMBO [19] and Trinity [20, 21] are sensitive to neutrinos
in the range of (1 - 100) PeV. If neutrino energy lies in the range of EeV,
the interaction length becomes achingly diminutive [22, 23, 24, 25]. In such a
case, the experimental set up should be accoutred to dredge up neutrinos
interacting in the atmosphere, rock, ice or water outside of the detector
volume [26, 27, 28]. The experimental facilities such as Pierre Auger
Observatory [29, 30, 31], ANITA I-IV [32], ARIANNA [33], ARA [34, 35] or are
under construction RNO-G [36, 37] and PUEO [38] are bestowed with such
proficiencies.

The high energy neutrinos can beseem as an important tool to quest physics
beyond the standard model (SM) of electroweak interactions. In this context,
one of the less skirred aspects of neutrinos is its electromagnetic
properties. In particular, if a neutrino possesses a finite magnetic moment
due to quantum loop corrections [39, 40] then it may have several important
hegemonies for physics of high energy cosmic neutrinos. The neutrinos would
then be jaundiced by any external magnetic field, particularly a strong
magnetic field. For cosmic neutrinos, even a relatively small intergalactic
magnetic field of the order of $\mu$G can have sobersided ramifications. Using
the current upper bound on the magnetic moment of neutrinos, it was shown that
the flux of the cosmic neutrons can be depleted by half if they peregrinate
sufficiently large distances in the intergalactic field [41].

Such neutrinos can also stumble onto compact objects such as neutron stars or
white dwarf stars which have large magnetic fields. The presence of such
fields within compact stars would effectuate spin flip oscillations rendering
depletion in the neutrino flux. In this work, we examine this aspect. We
assume that high energy muon neutrinos are emitted from a point neutrino
source and after travelling some distance, it encounters a compact star with a
large internal magnetic field. The compact star is assumed to be located
sufficiently closer to the source that the initial flux ratio remains
unchanged. It would be interesting to see whether such interactions have any
observational implications or not.

The plan of our work is as follows. In section II, we describe the dynamics of
neutrino spin flip oscillations. In section III, we present a brief overview
of neutron star and then exhibit our results for UHE neutrinos traversing
through it. In the next section, we provide results for white dwarf star.
Finally, in section V, we furnish conclusions of our work.

##  II Neutrino spin flip oscillations

The phenomenon of neutrino oscillations are experimentally well-entrenched
[42, 43, 44, 45, 46, 47, 48]. Such oscillations are possible only if the
neutrino flavour state is a linear superposition of the non-degenerate mass
eigenstates. It is illustrious that a massless and chiral neutrinos cannot
have a non-zero magnetic dipole moment. However, a massive Dirac neutrino, in
general, may have a magnetic moment which emerges at the one-loop level. In
the standard electroweak theory, the Dirac neutrino magnetic moment
$\mu_{\nu}$ at the leading order in $m_{\ell}^{2}/m_{W}^{2}$ is given as [49]

$\mu_{\nu}^{\rm SM}=\frac{3eG_{F}m_{\nu}}{8\sqrt{2}\pi^{2}}\,,$ (1)

where $e$ is positive, $G_{F}$ is Fermi constant and $m_{\nu}$ ($m_{\ell}$) is
neutrino (lepton) mass. The mass of $W$ boson is denoted by $m_{W}$. It is
ostensible that at the leading order in $m_{\ell}^{2}/m_{W}^{2}$, $\mu_{\nu}$
is independent of $m_{\ell}$ and the PMNS matrix. Numerically

$\mu_{\nu}^{\rm SM}\approx\frac{3.2\times 10^{-19}m_{\ell}}{1eV}\mu_{B}\,.$
(2)

It is apparent that the neutrino magnetic moment is directly proportional to
its mass. Therefore in order to have plenteously large value of $\mu_{\nu}$ so
that it can have some observational implications, one should have sufficiently
large neutrino mass. However, we have tight constraints on the mass of
neutrinos [50]. Therefore, it might appear that the neutrino magnetic moment
cannot be enhanced enough up to a level to have any observational impact.
However, many new physics models are up to this demurral to effectuate a
nonzero magnetic moment without affecting the neutrino mass. For e.g., models
with scalar leptoquark [51, 52], vector leptoquarks in the TeV mass range with
couplings both to the left and right-handed neutrinos [53], charged scalars
[54], $R$-parity breaking SUSY [55, 56], an additional approximate horizontal
symmetry $SU(2)_{H}$ with the SM gauge group [57], non standard interaction
(NSI) [39, 58, 59, 60] can generate a non-zero neutrino magnetic moment that
may lie within the experimental reach.

On the experimental front, various reactor and accelerator experiments have
provided bounds on the neutrino magnetic moment. The reactor experiments
include the GEMMA experiment at the Kalinin Nuclear Power Plant, Russia [61]
and the TEXONO experiment at the Kuo-Sheng Reactor Neutrino Laboratory, Taiwan
[62] and ROVNO experiment at Rovno nuclear power plant, Ukraine [63]. The
accelerator experimental set ups entail LAPMF [64] and LSND [65]. The bounds
from these reactor and accelerator neutrinos are in the range
$(10^{-11}-10^{-10})~{}\mu_{B}$. Recently, the XENON1T experiment provided
upper bound on $\mu_{e\nu}$ as $(1.65-3.42)\times 10^{-11}~{}\mu_{B}$ [57].

The limits on $\mu_{\nu}$ can also be obtained from astrophysics and cosmology
through the evolution of stars [66], plasmon decay in the stellar environment
[67], supernova events [68] and ${}^{4}_{2}He$ nucleosynthesis [69]. In fact,
the most stringent limits on the neutrino magnetic moment comes from the red
giant branch (RGB) of globular cluster which is $\mu_{\nu}<4.5\times
10^{-12}~{}\mu_{B}$ [70]. This value is about one order of magnitude smaller
than that obtained from several terrene experiments .

Since neutrinos are deprived of electric charge, they can partake in
electromagnetic interactions only by coupling with photons through quantum
corrections. The Hamiltonian for the neutrino-photon interaction is given by
[39]

$\displaystyle\mathcal{H}_{EM}=\bar{\nu}(x)\Lambda^{\mu}\nu(x)A_{\mu}\sim
J_{\mu}^{EM}A_{\mu},$ (3)

where $A_{\mu}$ is the electromagnetic field and $\Lambda_{\mu}$ represents
the vertex function subsuming the electromagnetic properties of neutrino which
is a $4\times 4$ matrix in the spinor space and containes all the form factors
corresponding to the four electromagnetic properties, given by (a) electric
moment, (b) magnetic moment, (c) neutrino charge and (d) anapole moment. In
Dirac picture, the form factor corresponding to electric moment vanishes due
to the assumption of $CP$ invariance and the hermiticity of the
electromagnetic current, $J_{\mu}^{EM}$. The hermiticty of the current,
$J_{\mu}^{EM}$ also restricts the form factors corresponding to the magnetic
moment, neutrino charge and anapole moment to be real.

The magnetic moment spawns the neutrino spin to precess in the contiguity of
an external magnetic field which ushers to the mixing between the left handed
and right handed neutrino current. In the two flavour framework, the
Hamiltonian for the neutrino state evolution is expressed as $4\times 4$
Hermitian matrix. In our work we consider only Dirac neutrinos, for which the
basis vector is $(\nu_{eL},\nu_{\mu L},\nu_{eR},\nu_{\mu R})$ and the
Hamiltonian in this case manifests as [39, 40]

$H_{D}=\begin{pmatrix}\frac{-\Delta
m^{2}}{4E_{\nu}}cos2\theta+V_{e}&\frac{\Delta
m^{2}}{4E_{\nu}}sin2\theta&\mu_{ee}B_{\perp}&\mu_{e\mu}B_{\perp}\\\
\frac{\Delta m^{2}}{4E_{\nu}}sin2\theta&\frac{\Delta
m^{2}}{4E_{\nu}}cos2\theta+V_{\mu}&\mu_{e\mu}B_{\perp}&\mu_{\mu\mu}B_{\perp}\\\
\mu^{*}_{ee}B_{\perp}&\mu^{*}_{\mu e}B_{\perp}&\frac{-\Delta
m^{2}}{4E_{\nu}}cos2\theta&\frac{\Delta m^{2}}{4E_{\nu}}sin2\theta\\\
\mu^{*}_{e\mu}B_{\perp}&\mu^{*}_{\mu\mu}B_{\perp}&\frac{\Delta
m^{2}}{4E_{\nu}}sin2\theta&\frac{\Delta m^{2}}{4E_{\nu}}cos2\theta\\\
\end{pmatrix}.$ (4)

A spin rotation from negative helicity state towards positive state will
reduce the resultant effective weak neutral and charged-current scattering
cross sections for neutrino which emerge, in the relativistic case,
predominantly from the negative helicity state for neutrinos.

In equation (4), $B_{\perp}$ denote transverse magnetic field, while $\mu$ is
the neutrino magnetic moment. Here, $\Delta m^{2}=m_{2}^{2}-m_{1}^{2}$.
$V_{e}$ and $V_{\mu}$ are the potentials experienced by $\nu_{e}$ and
$\nu_{\mu}$, respectively which are dependent on the matter composition of the
medium through which the neutrino travels. If the matter is consisted of only
nucleons and electron ($npe$), the matter potentials take the form of,
$V_{e}=\sqrt{2}G_{F}(n_{e}-n_{n}/2)$ and $V_{\mu}=-\sqrt{2}G_{F}n_{n}/2$,
while if muons are added with it ($npe\mu$), $V_{\mu}$ is altered to be,
$V_{\mu}=-\sqrt{2}G_{F}(n_{\mu}-n_{n}/2)$ due to the occurence of $\nu_{\mu}$
charged current (CC) interaction in matter in addition to that of the
electrons also. Here, $n_{e}$, $n_{n}$ and $n_{\mu}$ are the number densities
of electron, neutron and muon respectively.

##  III Cosmic neutrinos passing through neutron star

Neutron stars (NS) are unique compact stars engendered when the core nuclear
fuel of a main sequence star of intermediate mass ($M\geq 8M_{\odot}$) is
completely exhausted with the formation of the most stable iron nuclei. At
this stage, the hydrodynamic equilibrium in the stellar structure is sustained
by the degeneracy pressure of excess neutrons, against the inward
gravitational collapse, which are the byproducts of photodisintegration and
the following process of neutron drip from the iron nuclei. With further
infalling matter from the outer layer towards the center, the density of the
system may even exceed nuclear saturation density and the system detonates
with type-II supernova (SN) explosion. As a consequence, NSs are born [72].
The typical mass and radius of a NS are of the order of $\sim 2M_{\odot}$ and
$\sim 10$ km respectively, with its central density being a few times of the
nuclear saturation density. The exact estimation of these quantities are
obtained from its matter configuration and the equation of state (EOS). The
simplest NS models consist of matter containing neutrons, protons and
electrons ($npe$). However, muons start to appear in the matter when the
chemical potential of the electrons exceeds the rest mass of muons ($105.6$
MeV) [74]. NSs also possess extremely high magnetic fields, also known as
magnetars, which can even reach as high values as $10^{18}$ G at its center
[75]. The central and surface values of magnetic fields may differ by one or
two orders of magnitude.

NSs are the natural sources of neutrinos having energy in the $\sim$MeV
($10^{6}$ eV) range which are initially trapped for a few seconds during the
birth of the star. Later during the different stages of its thermal evolution,
neutrinos also continue to be emitted from the star as they are the most
efficient heat carriers and help to reduce the temperature of the star. These
$\nu_{e}$ neutrinos can also undergo spin flip precession, see for e.g., [77,
78]. However, in this work we are not considering the neutrinos which are
generated inside the NS. Rather, we consider UHE neutrinos generated from
existing sources nearby the NS [76], such as active galactic nuclei (AGN) [79,
80] or a blazar [81, 82]. These objects are able to produce mainly $\nu_{\mu
L}$ 111For convenience, from now on, we write $\nu_{\mu}$ for $\nu_{\mu L}$.
with energies which can transcend the $\sim$EeV ($10^{19}$ eV) range. We
contemplate these UHE neutrinos peregrinating through the NS which are
expected to be affected by the star’s magnetic field and hence exhibit the
reduction in their flux due to the SFP phenomena.

In our analysis, we consider a massive NS dovetailed of normal matter and
having mass $2.3M_{\odot}$ and radius $11.8$ km, i.e., we consider the
composition of $npe$ and $npe\mu$ with large magnetic field at its core
$10^{18}$ G. The matter configuration of the star is described by a nonlinear
Walecka model containing $\sigma$, $\omega$ and $\rho$-mesons [83, 84], and is
governed by the GM1 equation of state (EOS) [85]. The neutrinos are assumed to
be Dirac in nature and have a finite magnetic moment of $\sim
10^{-11}~{}\mu_{B}$.

In this analysis, we consider the simplification of zero vacuum mixing
$(\theta_{12}=0)$ which yields the Hamiltonian to have a form of $2\times 2$
matrix, given by [71]

$H=\begin{pmatrix}-\Delta m_{12}^{2}/4E+\Delta V/2&\mu_{e\mu}B\\\
\mu_{e\mu}B&\Delta m_{12}^{2}/4E-\Delta V/2\end{pmatrix}.$ (5)

Here $\Delta V=\sqrt{2}G_{F}\rho Y_{e}^{eff}/m_{N}$ with $\rho$ being the
matter density. $Y_{e}$ and $m_{n}$ denote the effective electron fraction and
nucleonic mass, respectively. Further, $Y_{e}^{eff}=(3Y_{e}-1)/2$ for Dirac
basis. Following eqn. (5), the neutrino spin flavour evolution equation in the
presence of an external magnetic field is expressed as [71]

$\frac{d^{2}\nu_{\mu L}}{dr^{2}}-\left(\frac{\mu\dot{B}}{\mu
B}+i\zeta\right)\frac{d\nu_{\mu L}}{dr}+\left[\phi^{2}+i\frac{d\phi}{dr}+(\mu
B)^{2}-i\phi\frac{\mu\dot{B}}{\mu B}+\phi\zeta\right]\nu_{\mu L}=0,$ (6)

where

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\frac{-\Delta
m^{2}}{4E}+\frac{1}{\sqrt{2}}G_{F}n_{e},$ $\displaystyle\zeta$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}G_{F}n_{n},~{}\nu_{\mu
L}\rightarrow\nu_{\mu R}\,.$ (7)

The transition probability of left handed electron neutrino can be obtained
after solving (6) along with (III).

###  III.1 Non-decaying magnetic field

We consider the following universal profile of the internal magnetic field of
the NS

$B(x)=B_{c}(1-1.6x^{2}-x^{4}+4.2x^{6}-2.4x^{8}),$ (8)

where $x=r/R$, with $r$ is the radial distance from the center of the star and
$R$ is the radius of the star. $B_{c}$ is the magnetic field at the center
which is assumed to be $B_{c}=10^{18}$ G. The profile is independent of any
free parameter which works at its advantage. The magnetic field at the surface
of the star is obtained to be $\sim 10^{17}$ G, following the profile given in
eqn. (8).

Figure 1: Variation of survival probability $P(\nu_{\mu}\rightarrow\nu_{\mu})$
with the radial distance inside the star for two different energy sets of
incoming $\nu_{\mu}$. The star contains nuclear matter with $npe$
configuration.

Fig. 1 demystifies the survival probability of the incoming $\nu_{\mu}$ beam.
In the left and right panels of the figure, we consider the neutrino energies
to be $E=10^{15}$ eV and $E=10^{19}$ eV, respectively. We contemplate the
magnetar as a sphere in which the neutrino beam enters at the point $r=-R$ and
leaves the star from the point $r=R$. As stated before, $R$ is the radius of
the star with value $11.8$ km. We assume that the incoming UHE neutrinos
travel along the diameter of the star as being the largest traversed distance,
the effect of SFP will be maximal. It can be observed from the figure that the
survival probability of $\nu_{\mu}$ at its point of emission from the star at
$r=R$ is $\sim 0.5$ which implies that almost half of the incoming neutrinos
are converted into their sterile counterparts. The result does not vary
significantly with increase in incoming neutrino energy, as observed comparing
left and right panels of the figure. The survival probability of $\nu_{\mu}$
is obviously unity at the point of entry to the star, $r=-R$, as we have
considered zero vacuum mixing and hence the neutrino beam incident on the
magnetar only consists of $\nu_{\mu}$.

Figure 2: Same as fig. 1, comparing $npe\mu$ and $npe$ matter.

In fig. 2, we portray the plot for $\nu_{\mu}$ survival probability after
crossing the NS for two kinds of matter composition, $npe$ and $npe\mu$. In
the left and right panels of the plot, the incoming energy of $\nu_{\mu}$ beam
is considered to be $10^{15}$ eV and $10^{19}$ eV, respectively. The plots
clearly show that starting from the point of incidence ($r=-R$) till the
center of the star, the survival probability is nearly similar for the two
kinds of matter configuration. However, the difference starts to be pronounced
once it crosses the stellar center and becomes greater near the surface
region. It is to be noted that in the presence of muon in NS matter, at the
point of emission ($r=R$), the survival probability of $\nu_{\mu}$ is further
lowered to a value of $\sim 0.45$ from $\sim 0.5$ which is the muon neutrino
survival probability for only $npe$ matter at $r=R$. This implies that the
presence of muon affects the SFP phenomena to a greater extent and thus aids
to generate a larger flux of sterile neutrino. The flux of incoming
$\nu_{\mu}$ is diminished by $50\%$ after they drift through the $npe$ matter
of NS whereas the reduction is enhanced to $55\%$ in the presence of muons.
The result does not exhibit any significant variation for the case of lower
and higher energies as observed from assimilating the two panels of the
figure. The results are summarized in table 1.

Conditions |  $P(\nu_{\mu}\rightarrow\nu_{\mu})$ at $r=R$ |  $\%$ decrease in flux at $r=R$
---|---|---
|  $10^{15}$ eV |  $10^{19}$ eV |
Without muon | $0.5$ | $0.5$ | $50\%$
With muon | $0.45$ | $0.45$ | $55\%$
Table 1: Survival probability and the $\nu_{\mu}$ flux reduction at the point
of emission from the star, i.e., at $r=R$, $R=11.8$ km.

###  III.2 Decaying magnetic field

In NS, the magnetic field starts to deplete over time which may undergo via
three mechanisms: [87] (a) Ohmic decay, in case of which the magnetic field
diffuses with respect to the charged particles [88], (b) ambipolar diffusion,
which is a dissipative effect where the diffusion of electrons and protons in
opposite directions in a neutron background takes place [89] and (c) Hall
drift [90]. In our analysis, we consider the dissipation of magnetic field by
the process of ambipolar diffusion which is the most important mechanism
occurring inside the core of the manetar and affects the magnetic field most
significantly at the lower core temperature ($\sim 10^{8}$ K) [89]. To
incorporate the effect of magnetic field dissipation in the magnetar, we
consider only the occurrence of ambipolar diffusion for the transport of
magnetic flux and follow the formalism mentioned in the ref. [91, 89],
according to which the expression for time varying magnetic field in $npe$
matter, is given by

$B(r,t)=\frac{B_{0}(r)}{1+t/\tau}\,.$ (9)

Here $B_{0}$ is the constant magnetic field without taking the decaying effect
into account and is obtained from eqn. (8). Further, $\tau$ is the
characteristic decay time for the magnetic field due to the occurrence of
ambipolar diffusion and can be expressed as,

$\tau\approx
25\,L_{5}^{2}\,B_{17}^{-2}\,T_{9}^{2}\,(\rho/\rho_{0})^{2/3}\,\,{\rm yrs},$
(10)

where $L_{5}=L/10^{5}$ cm, $L$ being the distance over which the variation of
magnetic field is considered, $B_{17}=B/10^{17}$ G and $T_{9}=T/10^{9}$ K.
$\rho_{0}$ is the saturation nuclear density which is dependent on the choice
of EOS of NS matter. For GM1 EOS, $\rho_{0}=0.153$ fm-3 [92].

The surface temperature of our model star varies within a range of $\sim
10^{7}-10^{4}$ K over a temporal span of $10^{6}$ years, starting after the
birth of the NS and the corresponding surface magnetic field varies from $\sim
10^{18}$ G to $10^{4}$ G which are depicted in the left and right panels of
fig. 3, respectively. For a massive NS like our model star, the initial
thermal relaxation period of the stellar thermal evolution lasts upto $\sim
100$ years, during which the surface magnetic field also shows drastic
reduction in its value from $\sim 10^{14}$ G to $10^{11}$ G, as observed from
fig. 3.

Figure 3: Variation of surface temperature (left panel) and magnetic field
(right panel) with time of a magnetar of mass $2.3~{}M_{\odot}$ and radius
$11.8$ km.

Figure 4: Variation of survival probability $P(\nu_{\mu}\rightarrow\nu_{\mu})$
with radial distance for decaying magnetic field in $npe$ matter of NS.

Fig. 4 illustrates the muon neutrino survival probability for the case of a
decaying magnetic field of a NS for $npe$ matter. For comparison, the plot for
a non-decaying field is also shown. We consider a time span of $10^{3}$ years
in which the magnetic field of the magnetar is reduced at least by five orders
of magnitude. Further, it should be mentioned that at $t\approx 10^{3}$ years,
the core temperature is $\sim 10^{8}$ K which implies that the ambipolar
diffusion could be the most significant process for the dissipation of the
magnetic field in the magnetar. It can be observed from the figure that at
$t=10^{3}$ years, around the central region of the star ($R\approx 0$), the
decaying magnetic field enhances the survival probability whereas at the point
of $\nu_{\mu}$ emission from the NS $i.e.$ at $r=R$, the decaying magnetic
field induces slightly higher survival probability ($\sim 0.53$) as compared
to that of no decay ($\sim 0.5$). Thus the decaying magnetic field favours
$\sim 5\%$ enhancement in the $\nu_{\mu}$ flux as compared to the temporally
static magnetic field. The results remain almost the same at $t=10^{5}$ years,
as can be seen from the figure. Thus, it can be concluded that there can only
be a marginal alteration in the outgoing neutrino flux from the NS for a
decaying magnetic field as compared to the static field. Further, these
conclusions are almost independent of the age of the NS. The results are
further illustrated in table 2.

Quantities | Without decay | With decay
---|---|---
$P(\nu_{\mu}\rightarrow\nu_{\mu})$ at $r=R$ | $0.5$ | $0.53$
$\%$ reduction in $\nu_{\mu}$ flux at $r=R$ | $50\%$ | $47\%$
Table 2: Survival probability and the $\%$ reduction of the incoming
$\nu_{\mu}$ flux, at the point of emission from the star at $r=R$, $R=11.8$
km.

##  IV Cosmic neutrinos passing through white dwarf

White dwarfs (WD) are another branch of compact stars the formation of which
originated from the progenitor stars having lower mass ($1M_{\odot}\leq M\leq
8M_{\odot}$). When there is sufficient nuclear fusion in the core of such a
low mass main sequence star which generates helium or some successive heavier
nuclei, the outer layer of the star expands to form a red giant (RG). Due to
rapid expansion and cooling of the outer layer it becomes unstable and is
expelled as a planetary nebulae. The remnant forms WD in which the inward
gravitational pull is balanced by the electron degeneracy pressure [93]. The
maximum mass of a WD is about $1.475~{}M_{\odot}$ which is the Chandrashekhar
limit, having its central density $\sim 10^{6}$ gm cm-3. The core of a low
mass WD usually consists of helium, while heavier ones may contain carbon or
oxygen. WDs are smaller compared to NSs and are less massive. Considering non-
relativistic degenerate electron gas, the mass radius relation ($M-R$) and the
central density are expressed as [94]

$\displaystyle R$ $\displaystyle=$ $\displaystyle(10,500~{}{\rm
km})~{}\left(\frac{0.6M_{\odot}}{M}\right)^{1/3}(2Y_{e})^{5/3}$
$\displaystyle\rho_{central}$ $\displaystyle=$ $\displaystyle 1.46\times
10^{6}~{}{\rm
gm~{}cm^{-3}}\left(\frac{M}{0.6M_{\odot}}\right)^{2}(2Y_{e})^{-5},$ (11)

where $0.6M_{\odot}$ is the canonical mass of a WD. $Y_{e}$ is the electron
number density per baryon, $Y_{e}=n_{e}/(n_{p}+n_{n})$ with $n_{e}$, $n_{p}$
and $n_{n}$ being the number density of electron, proton and neutron,
respectively. WDs usually consist of $npe$ matter. Muons cannot be present
inside WDs, as the Fermi energy of muons is very small inside the star.
Isolated WDs may contain large internal magnetic fields, although the surface
value of the magnetic field may be vanishing. In a recent work, the upper
bound of the WD magnetic field is estimated to be as [94]

$B_{max}=8.8\times 10^{11}G\left(\frac{M}{0.6M_{\odot}}\right)^{1/3}$ (12)

WDs are also capable of producing MeV neutrinos at their core. However,
similar to the previous analysis in case of NS, we proceed with UHE Dirac
neutrinos coming from a local source which pass through the WD. Due to large
magnetic field of WD, it is expected that a fraction of the incoming UHE is
converted to their sterile counterpart.

For our analysis, we consider an isolated WD of mass $0.6~{}M_{\odot}$ and
radius $10,500$ km having constant internal magnetic field of $8.8\times
10^{11}$ G with $npe$ matter configuration having $Y_{e}=0.5$. We assume that
the local UHE neutrino source gives away $\nu_{\mu}$ beam which crosses the
white dwarf along its diameter, entering at the point $r=-10,500$ km
considering at its center at $r=0$ and exiting from $r=10,500$ km. We
determine the survival probability of $\nu_{\mu}$ after they pass through the
star at the point of emission, $r=10,500$ km.

Figure 5: Variation of survival probability
$P({\nu_{\mu}}\rightarrow{\nu_{\mu}})$ with the radial distance inside the WD
star for two different energies.

In fig. 5, we represent $\nu_{\mu}$ survival probability varying with the
radial distance inside the WD star. Similar to fig. 1, in the case of WD, we
also present our results considering the energy values of $10^{15}$ eV and
$10^{19}$ eV in the left and right panels of the plots, respectively. It is
observed that the survival probability follows an oscillatory pattern
throughout the star and retains a value of $\sim 0.75$ at the exit point from
the star at $r=10,500$ km implying a reduction of incoming $\nu_{\mu}$ flux by
$25\%$. The result does not vary significantly in case of lower and higher
energy values. Compared to NS, UHE Dirac $\nu_{\mu}$s are affected to a lesser
extent by the SFP phenomena inside the WD which can be accredited to a smaller
value of magnetic field. Thus, in the case of a WD, the conversion of Dirac
neutrinos into the sterile form occurs in a lower amount as compared to the
NS. This difference can be determined from the proper estimation of the flux
by a suitable terrestrial neutrino detector with high efficiency.

Unlike NS, the decay of magnetic fields in a magnetic WD is not realistic as
the decay time is very large $\sim(2-6)\times 10^{11}$ yrs. Therefore, we
curtail our analysis to the estimation of flux reduction of UHE neutrinos
passing through WD stars containing $npe$ matter and constant magnetic field
without delving into the concept of field decay.

##  V Conclusion

We appraise the reduction of flux of ultra high energy Dirac neutrinos passing
through two of the most compact stellar objects in the universe, neutron star
and white draft, after being generated from a point source residing close
enough to the compact star. As a model star, we consider a neutron star with
mass $2.3~{}M_{\odot}$ and radius $11.8$ km having a mammoth central magnetic
field of $10^{18}$ G. Using the current experimental bound on the magnetic
moment of neutrinos and assuming GM1 parametrization, we evaluate the survival
probability of the incoming UHE $\nu_{\mu}$ beam at the point of emission for
$npe$ as well as $npe\mu$ matter. We observe the following:

  * •

For a non-decaying magnetic field, the flux of a $\nu_{\mu}$ beam can be
reduced to half at the emission point for $npe$ matter.

  * •

For muons in the neutron star matter, the survival probability of $\nu_{\mu}$
decreases from 0.50 to 0.45.

  * •

These results remain unaltered in the $\nu_{\mu}$ energy range of
($10^{15}-10^{19}$) eV.

  * •

For a decaying magnetic field in $npe$ matter of neutron star, the survival
probability of $\nu_{\mu}$ increases by $\sim 5\%$ as compared to the
temporally static field. Although the decaying and non-decaying scenarios show
a significant difference in the values of neutrino survival probability around
the central region of the neutron star, this may not have any observational
implications.

We then consider a white dwarf star consisting of only $npe$ matter and having
mass $0.6M_{\odot}$ and radius $10,500$ km with a constant magnetic field of
$8.8\times 10^{11}$ G. This leads to $\nu_{\mu}$ survival probability of $\sim
0.8$ at the point of emission after they traverse along its diameter. Like
neutron stars, the conclusions are almost the same in the energy range of
($10^{15}-10^{19}$) eV. We do not consider the case of magnetic field
evolution in white dwarf due to its unusually large decaying time scale.

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