# Antistars or antimatter cores in mirror neutron stars?

Zurab Berezhiani  E-mail<EMAIL_ADDRESS>Dipartimento di Fisica
e Chimica, Università di L’Aquila, 67100 Coppito, L’Aquila, Italy  INFN,
Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy

###### Abstract

The oscillation of the neutron $n$ into mirror neutron $n^{\prime}$, its
partner from dark mirror sector, can gradually transform an ordinary neutron
star into a mixed star consisting in part of mirror dark matter. The
implications of the reverse process taking place in the mirror neutron stars
depend on the sign of baryon asymmetry in mirror sector. Namely, if it is
negative, as predicted by certain baryogenesis scenarios, then
$\overline{n^{\prime}}-\overline{n}$ transitions create a core of our
antimatter gravitationally trapped in the mirror star interior. The
annihilation of accreted gas on such antimatter cores could explain the origin
$\gamma$-source candidates, with unusual spectrum compatible to baryon-
antibaryon annihilation, recently identified in the Fermi LAT catalog, In
addition, some part of this antimatter escaping after the mergers of mirror
neutron stars can produce the flux of cosmic antihelium and also heavier
antinuclei which are hunted in the AMS-02 experiment.

1. There may exist a hidden sector of particles which are the mirror (M) replicas of the ordinary (O) particles. Mirror parity, a discrete symmetry under the specular exchange of the O and M species (fermions, gauge bosons and Higgs fields of two sectors), implies that the two worlds should have identical microphysics. In this way, all O particles: electron $e$, proton $p$, neutron $n$, neutrinos $\nu$, photon $\gamma$ etc. must have the mass-degenerate M twins $e^{\prime}$, $p^{\prime}$, $n^{\prime}$, $\nu^{\prime}$, $\gamma^{\prime}$ etc. which are sterile to the interactions of our Standard Model (SM) but have their own SM′ interactions of exactly the same pattern. Like O matter, during cosmological evolution the M matter should form nuclei, atoms, stars and planets which, being invisible for us in terms of O photons, can represent dark matter (DM) in the Universe (for reviews see e.g. Alice ; Foot ).

A specific feature of this scenario is that any neutral particle (elementary
as e.g. neutrinos or composite as e.g. the neutron) can have maximal mixing
with its M partner, and such twin species can effectively oscillate between
each other. Namely, the active–sterile $\nu-\nu^{\prime}$ oscillations between
the O and M neutrinos, violating the lepton numbers ${\rm L}$ and ${\rm
L}^{\prime}$ of both sectors by one unit, can be experimentally observed as
the deficit of neutrinos. Analogously, active–sterile oscillation between the
neutron $n$ and M neutron $n^{\prime}$, induced by the mass mixing

$\varepsilon\,\overline{n^{\prime}}n+\text{h.c.}$ (1)

violates the baryon numbers ${\rm B}$ and ${\rm B}^{\prime}$ by one unit but
conserves the combination $\widetilde{{\rm B}}={\rm B}+{\rm B}^{\prime}$. This
phenomenon can be tested via the neutron disappearance ($n\rightarrow
n^{\prime}$) and regeneration ($n\rightarrow n^{\prime}\rightarrow n$)
experiments BB-nn' ; More ; Pokot ; HFIR .

As it was shown in Ref. BB-nn' , the phenomenological and cosmological bounds
do not exclude $nn^{\prime}$ mixing mass to be as large as $\varepsilon\sim
10^{-15}$ eV, which corresponds to few seconds for the characteristic
oscillation time $\tau=1/\varepsilon$.111In this paper we use natural units
$c=1$ and $\hbar=1$. In fact, for free neutrons $n-n^{\prime}$ oscillation is
suppressed by the medium effects as the presence of matter and magnetic fields
BB-nn' ; More . But in deep cosmos it could proceed without suppression, with
interesting implications e.g. for the cosmic rays at super-GZK energies, as
far as the neutrons produced in $p\gamma\rightarrow n\pi^{+}$ reactions on
relic photons can promptly oscillate into $n^{\prime}$ and then decay in M
sector, $n^{\prime}\rightarrow p^{\prime}e^{\prime}\overline{\nu}^{\prime}$
UHECR ; UHECR1 . Several dedicated experiments searching for $n-n^{\prime}$
oscillations still allow rather short oscillation times Ban ; Serebrov1 ;
Altarev ; Bodek ; Serebrov2 ; ILL ; Abel . Moreover, some of their results
show deviations from null hypothesis Ban ; ILL ; Nesti  and new experiments
are underway for testing these anomalies Broussard ; ESS .

It is remarkable that the nuclear stability yields no limit on $\varepsilon$
since $n\rightarrow n^{\prime}$ conversion in nuclei is forbidden by the
energy conservation BB-nn' . However, in the neutron stars (NS) $n\rightarrow
n^{\prime}$ conversion is energetically favored, and it can gradually
transform the NS into mixed stars partially consisting of M matter. Since in
super-dense nuclear medium $n-n^{\prime}$ oscillation is strongly suppressed,
for $\varepsilon\sim 10^{-15}$ eV the effective conversion time appears to be
much larger than the universe age BB-nn' . Nevertheless, it can have
observable effects for the NS which were analyzed in details in Ref. BBMT .
Various aspects of $n-n^{\prime}$ oscillation in the NS were discussed also in
Refs. INT ; Massimo ; LHEP ; Nussinov-new ; Ciancarella ; Makakin .

By the essence of the mirror matter, the neutron stars should exist also in
dark M sector. In fact, M baryons represent DM of asymmetric type, with its
abundance determined by the baryon asymmetry (BA) in M world. However, a
priori the sign of mirror BA is unknown.222Notice that by naming $n^{\prime}$
as mirror neutron, we implicitly extend the notion of our baryon charge ${\rm
B}$ to that of M species ${\rm B}^{\prime}$ classifying the latter via the
combined charge $\widetilde{{\rm B}}={\rm B}+{\rm B}^{\prime}$ which is
conserved by $nn^{\prime}$ mixing (1). In fact, the M species with
$\widetilde{{\rm B}}=1$ and $\widetilde{{\rm B}}=-1$ for us respectively are
the mirror baryons (MB) and anti-mirror baryons (AMB) (namely,
$\overline{n}^{\prime}$ is anti-mirror neutron or mirror antineutron), no
matter how they are qualified by the inhabitants of M world.

The sign of ordinary BA, ${\mathcal{B}}=\text{sign}(n_{b}-n_{\bar{b}})=1$, is
fixed by some baryogenesis mechanism which created the primordial excess of
baryons over antibaryons. At first glance, the sign of mirror BA should be the
same by M parity,
${\mathcal{B}}^{\prime}=\text{sign}(n_{b^{\prime}}-n_{\bar{b}^{\prime}})=1$.
This would be the case if the identical baryogenesis mechanisms act separately
in the O and M sectors. However, one can also envisage a co-baryogenesis
scenario e.g. via the processes which violate ${\rm B}$ and ${\rm B}^{\prime}$
but conserve $\widetilde{{\rm B}}={\rm B}+{\rm B}^{\prime}$ (which processes
can in turn be related to underlying new physics which induces $nn^{\prime}$
mixing (1)). In this case, for ${\mathcal{B}}$ being positive, the null O+M
asymmetry in the Universe would imply the negative ${\mathcal{B}}^{\prime}$.
The co-genesis models which induce BA’s in both sectors via the ${\rm
B}\\!-\\!{\rm L}$ and ${\rm B}^{\prime}\\!-\\!{\rm L}^{\prime}$ violating
cross-interactions between the O and M particle species indeed predict
${\mathcal{B}}$ and ${\mathcal{B}}^{\prime}$ of the opposite signs ADM ; ADM-
IJMP .

Hence, depending on the sign of ${\mathcal{B}}^{\prime}$, the compact stars in
M sector can be the mirror neutron stars (MNS) or anti-mirror neutron stars
(AMNS) consisting respectively of the MB or AMB. Correspondingly,
$n^{\prime}\rightarrow n$ transitions in the MNS (or
$\overline{n}^{\prime}\rightarrow\overline{n}$ in the AMNS) can create the
cores of ordinary matter (or antimatter) in their interior. In both cases
these cores can be detectable as hot compact sources emitting the photons in
the far UV and soft X-ray ranges BBMT , but this electromagnetic emission
cannot trace their composition (matter or antimatter). However, the presence
of interstellar medium makes the difference. Namely, accretion of interstellar
gas on the MNS with O matter core can only heat it and induce X-ray emission.
However, in the case of AMNS, the accreted gas will annihilate on the
antimatter core producing gamma-rays with energies up to a GeV or so.

Interestingly, the recent analysis Dupourque:2021poh  based on the 10-year
Fermi Large Area Telescope (LAT) gamma-ray source catalog Fermi-LAT:2019yla
has identified 14 point-like candidates emitting $\gamma$’s with a spectrum
compatible to baryon-antibaryon annihilation Backenstoss:1983gu , and not
associated with objects belonging to established gamma-ray source classes.

In this letter we discuss the possibility whether the unusual sources of this
type can be the AMNS with the antimatter cores which produce the annihilation
$\gamma$-rays by accretion of interstellar gas, and whether some part of this
antimatter can escape from the AMNS producing the antinuclei detected by
AMS-02 in cosmic rays.

2. Neutron stars are presumably born after the supernova (SN) explosions of massive progenitor stars followed by the neutronization of their iron cores. The NS are formed with very high rotation speed and with the surface magnetic fields as large as $10^{8}\div 10^{15}$ Gauss. This makes many of them observable as pulsars due to their electromagnetic radiation. The NS are believed to have a compact onion-like structure: the inner core of dense nuclear liquid dominantly consisting of neutrons (with less than 10 % fraction of protons and electrons), and the outer shell consisting of heavy nuclei which form a rigid crust at the surface (for reviews, see e.g. Lattimer ). The NS mass-radius ($M$–$R$) relations depend on the equation of state (EoS), i.e. the pressure–density relation in dense nuclear matter. The masses of the known pulsars range within $M=(1\div 2)\,M_{\odot}$, and the observation of $2M_{\odot}$ ones disfavors the too soft EoS. E.g. the Sly EoS Sly  allows the maximum mass $M_{\text{max}}=2.05\,M_{\odot}$, while some of the realistic EoS reviewed in Refs. Ozel  can afford somewhat larger masses, up to $2.5\,M_{\odot}$ or so. All these EoS predict the NS radii in the range $R\simeq(10\div 15)$ km.

The baryon amount $N$ is related to the NS mass as

$N=\kappa M/m\simeq\left(\frac{\kappa M}{1.5\,M_{\odot}}\right)\times 2\cdot
10^{57}$ (2)

where $m$ is the nucleon mass, and $\kappa\simeq 1$ is the EoS dependent
factor which, for a given EoS, mildly depends also on $M$. E.g. the SLy EoS
implies $\kappa\approx 1.1$ for the typical NS masses $M\simeq
1.5\,M_{\odot}$, increasing to $\kappa\approx 1.2$ for heavy NS with $M\approx
2\,M_{\odot}$ Sly . The deficit between the gravitational mass $M$ and
equivalent baryonic mass $M_{b}=mN_{b}$, $M_{b}-M=(\kappa-1)M$, corresponds to
the gravitational binding energy.

In principle, at extremal densities the energetically favored ground state can
be strange quark matter Bodmer . Therefore, the NS which mass can reach a
critical value by accretion from the companion or by the matter fall-back
after the SN explosion can be transformed BBDFL  into hybrid stars (HS) with
more onion-like structure including the quark matter core, or even entirely
quark stars (QS), depending on the amount of accreted mass and on the details
of the quark matter EoS. Interestingly, the bimodal distribution of the pulsar
masses Schwab  can indeed correspond two different types of stars, the lighter
stars being the NS and heavier stars being the HS/QS. In the following, we
shall concentrate on the NS, addressing the possibility of the HS/QS only in
proper occasions.

3. The core-collapse of O star should produce a NS entirely consisting of ordinary nuclear matter.333Modulo some tiny amount of M matter which can be accreted by the progenitor star during its lifetime, or produced via $n-n^{\prime}$ oscillation of the neutrons involved in certain chains of nuclear reactions at late stages of its evolution before the core-collapse. But once the NS is formed, then $n\rightarrow n^{\prime}$ transitions in its liquid nuclear core can effectively produce M neutrons.444The NS liquid core is dominated by the neutrons and we shall neglect subdominant fraction of protons for simplicity. In the crust the neutrons are bound in heavy nuclei and for them $n-n^{\prime}$ conversions is ineffective, but this fraction is also negligible.

The oscillation $n-n^{\prime}$ in nuclear medium is described by the
Schrödinger equation with effective Hamiltonian

$H=\left(\begin{array}[]{cc}{\Delta E}&{\varepsilon}\\\
{\varepsilon}&{0}\end{array}\right)$ (3)

where the off-diagonal term $\varepsilon$ comes from $n-n^{\prime}$ mass
mixing (1), and $\Delta E$ is the medium-induced energy splitting between $n$
and $n^{\prime}$ states. It can be taken as the difference of the in-medium
optical potentials induced by their coherent scatterings, $\Delta
E=V-V^{\prime}$. Namely, for O neutrons we have $V=2\pi an_{b}/m$, where $m$
is the neutron mass, $a=a_{3}\times 3$ fm is its scattering length, and
$n_{b}=\xi\times 0.16~{}{\rm fm}^{-3}$ is the baryon density, i.e. $\xi$ is
$n_{b}$ in units of nuclear density. If the M baryon density in the star is
small, $n^{\prime}_{b}\ll n_{b}$, $V^{\prime}$ can be neglected and we get
$\Delta E=V=a_{3}\xi\times 125\,\text{MeV}$.555Zeeman energy induced by the
neutron magnetic moment $-\mu=6\times 10^{-12}$ eV/G is also negligible since
$|\mu B|<10^{-2}$ MeV even in magnetars where the magnetic field $B$ can reach
$10^{15}$ G or so.  Hence, in a dense nuclear medium $n-n^{\prime}$ mixing
angle is $\theta=\varepsilon/V\simeq(\varepsilon_{15}/a_{3}\xi)\times 8\cdot
10^{-24}$ ($\varepsilon_{15}=\varepsilon\times 10^{15}\,{\rm eV}^{-1}$), and
the mean probability of $n-n^{\prime}$ oscillation
$P_{nn^{\prime}}=2\theta^{2}$ is extremely small.

The mirror neutrons are mainly produced via the scattering process
$nn\rightarrow n^{\prime}n$.666Also $np\rightarrow n^{\prime}p$ scatterings
work, but we neglected them since the proton fraction at the typical densities
of the liquid core is just of few per cents. Its rate can be estimated as
$\Gamma=2\theta^{2}\eta\langle\sigma v\rangle n_{b}$, where $\sigma=4\pi
a^{2}$ is the $nn$ elastic scattering cross-section and $v=p_{F}/m$ is the
mean relative velocity, while the factor $\eta\approx 0.18$ takes into account
the Pauli blocking implying that in $nn\rightarrow nn^{\prime}$ process the
final state $n$ should have the momentum above the Fermi momentum
$p_{F}=\xi^{1/3}\times 340$ MeV. Hence, dependence on the scattering length
cancels out and we obtain BBMT :

$\Gamma=\varepsilon_{15}^{2}\,\xi^{-2/3}\times 3\cdot 10^{-47}~{}\text{GeV}$
(4)

This rate depends on the baryon density $n_{b}$ which varies by an order of
magnitude from the NS centre to the outer edge of its liquid core, so that in
dense central regions $n-n^{\prime}$ transition is somewhat slower than in
peripheral regions. Hence, the factor $\xi$ in (4) should be averaged over the
baryon density profile in the star. Taking $\langle n_{b}\rangle=N/V$ where
$V$ is the NS volume and $N=\kappa M/m$ is its total baryon number, for the
effective time $\tau_{\varepsilon}=\langle\Gamma\rangle^{-1}$ of
$n-n^{\prime}$ conversion we get:

$\tau_{\varepsilon}=\frac{a_{R}}{\varepsilon_{15}^{2}}\,\left(\frac{M}{1.5\,M_{\odot}}\right)^{2/3}\times
10^{15}~{}{\rm yr}$ (5)

where factor $a_{R}=\kappa^{2/3}/R_{12}^{2}\simeq 1$, with $R_{12}=R/12\,{\rm
km}$ normalized to the typical NS radii, mildly depends on the EoS which
determines $M$–$R$ relation for the NS. As we see, $\tau_{\varepsilon}$
becomes comparable to the universe age $t_{U}=14$ Gyr for $\varepsilon=4\times
10^{-12}$ eV or so.

Thus, by integrating the Boltzmann equations

$\dot{N}^{\prime}=\langle\Gamma\rangle
N,\quad\quad\dot{N}=-\langle\Gamma\rangle N$ (6)

and observing that the overall O+M baryon number $N+N^{\prime}$ is conserved,
for the fraction of M baryons produced in the NS of the age $t<t_{U}$ we get:

$\frac{N^{\prime}}{N+N^{\prime}}=\frac{t}{\tau_{\varepsilon}}=10^{-5}\times\frac{\varepsilon_{15}^{2}}{a_{R}}\left(\frac{1.5\,M_{\odot}}{M}\right)^{\\!2/3}\\!\\!\left(\frac{t}{10\,{\rm
Gyr}}\right)$ (7)

Thus, for $\varepsilon\sim 10^{-15}$ eV the amount of M baryons in the oldest
NS are comparable to their amount in planets.

Due to the Pauli blocking, the process $nn\rightarrow nn^{\prime}$ process
takes place for the neutron momenta close to the Fermi surface, so that
$n^{\prime}$ is typically produced with the Fermi energy
$E_{F}=p_{F}^{2}/2m=\xi^{2/3}\times 60$ MeV. Hence, the energy production rate
per baryon is independent of $\xi$: $\Gamma
E_{F}\simeq\varepsilon_{15}^{2}\times 3\cdot 10^{-24}$ GeV/s. Then,
multiplying this by the baryon number in the NS (2), for the total rate of the
energy production we obtain

$\dot{\mathcal{E}}\simeq\Gamma
E_{F}N\simeq\varepsilon_{15}^{2}\left(\frac{M}{1.5\,M_{\odot}}\right)\times
10^{31}~{}\text{erg/s}$ (8)

which energy will be radiated away via the mirror photons and neutrinos.

In the case of HS/QS situation is somewhat different. Transition
$n-n^{\prime}$ should be ineffective in quark matter since there are not much
neutrons. In addition, quark matter is thought to be self-bound Bodmer , in
particular if it is in color superconducting phase Alford , and its transition
to M matter is not energetically favored. Hence in the QS, almost entirely
consisting of quark matter, $n-n^{\prime}$ transitions play practically no
role. In the hybrid stars $n-n^{\prime}$ transition should take place in its
shell consisting of the neutron liquid. Thus, in principle the HS can also
develop the M matter cores in their interior. However, the energy production
rate in the HS, with less neutrons employed, can be much lower than it was
estimated in Eq. (8) for the NS.

For the evolution time $\tau_{\varepsilon}$ being much larger than the cooling
time, Eq. (8) in fact determines the NS mass loss rate due to $n-n^{\prime}$
conversion:

$\dot{M}/M\simeq-\varepsilon_{15}^{2}\times 10^{-16}~{}{\rm yr}^{-1}$ (9)

The mass loss changes the orbiting period in the compact NS binaries as
$\dot{P}_{b}/P_{b}\simeq-2\dot{M}/M$ Nussinov-new . The observational data on
the orbital period decay in some well-studied compact binaries (as e.g. the
famous Hulse-Taylor pulsar B1913+16) yield an upper bound roughly as
$\varepsilon<10^{-13}$ eV Nussinov-new ; BBMT . Somewhat stronger bounds,
$\varepsilon<10^{-15}$ eV or so BBMT , can be obtained from the surface
temperature measurements of old pulsars, which for PSR J2144$-$3933, with
$T_{\text{surf}}<4\times 10^{4}$ K Guillot , approaches $\varepsilon<10^{-16}$
eV. But this object, being an isolated slow pulsar with characteristic age
$\tau_{c}\simeq 0.3$ Gyr and unknown mass, could in principle be quark matter
dominated. In this case the above limit would be invalid. More generally, the
observational determination of the NS surface temperature is influenced by
environmental factors as composition of the pulsar magnetosphere and
interstellar extinction, and its theoretical interpretation depends on the
cooling models. Therefore, in the following we conservatively stick to
benchmark values $\varepsilon\sim 10^{-15}$ eV.

M baryons produced in the NS will form a mirror core in its interior. In fact,
$n^{\prime}$ produced at the first instances after the NS formation will
undergo $\beta$-decay $n^{\prime}\rightarrow
p^{\prime}e^{\prime}\bar{\nu}^{\prime}$ forming a plasma consisting of
$p^{\prime}$ and $e^{\prime}$ components. But very soon, in hundred years or
so, the density produced M matter $n^{\prime}_{b}$ will exceed $10^{26}/{\rm
cm}^{3}$, equivalent to the baryon density in the sun’s centre. In
gravitationally captured hot M matter nuclear reactions will be ignited among
$p^{\prime}$ and $n^{\prime}$ continuously produced in the NS, forming the M
nuclei. In the very old NS where the central density of M baryons could reach
$n_{b^{\prime}}=0.07$ fm-3 or so, they can undergo a ”neutronization” and form
a degenerate liquid core dominantly consisting of $n^{\prime}$.

The energy production rate (8) can be substantial: e.g. for
$\varepsilon_{15}=10$ it would comparable to the solar luminosity
${\mathcal{L}}_{\odot}=4\times 10^{33}$ erg/s. It should be radiated away by M
photons and M neutrinos, but given the complexity of nuclear reactions in hot
core, it is difficult to precisely determine the energy fractions emitted in
two channels. In any case, the fraction $x$ emitted in thermal M photons
should be significant, and the “mirror photosphere” temperature in the NS can
be estimated by equating $x\dot{\mathcal{E}}=4\pi
R_{\text{ph}}^{2}\times\sigma T^{4}$, where $R_{\text{ph}}$ is the photosphere
radius and $\sigma$ is the Stefan-Boltzmann constant:

$T=\varepsilon_{15}^{1/2}\left(\frac{x\kappa
M}{1.5\,M_{\odot}}\right)^{1/4}\left(\frac{1~{}{\rm
km}}{R_{\text{ph}}}\right)^{1/2}\times 10^{6}~{}\text{K}$ (10)

For determining the observable temperature $T_{\infty}$, the surface redshift
effect should be taken into account.

4. Let us reverse now the situation and consider neutron stars in M sector. As we have anticipated in the introduction, the BA ${\mathcal{B}}^{\prime}$ in M world can be positive or negative, depending on the baryogenesis mechanism. Hence, if ${\mathcal{B}}^{\prime}>0$ all compact mirror stars should be the MNS, whereas if ${\mathcal{B}}^{\prime}<0$ they all should be the AMNS. As far as the O and M sectors have the same microphysics by mirror parity, the EoS describing the O and M nuclear matters should be identical, and, needless to say, it should be identical for the nuclear and antinuclear matters by C invariance of strong interactions. In other words, the NS, MNS and AMNS should have the same $M$–$R$ relations.

In the absence on $n-n^{\prime}$ mixing these will be dark compact objects,
sort of solar mass MACHOs which can be detected by microlensing, but ordinary
observer cannot distinguish between the MNS and AMNS. If $n-n^{\prime}$ mixing
is switched on, then $n^{\prime}\rightarrow n$ transitions will create O
matter in the MNS interior, while in the AMNS
$\overline{n}^{\prime}\rightarrow\overline{n}$ will take place forming the O
antimatter. These transitions with the effective time and energy production
rate given again by Eqs. (5) and (8), would form hot cores which can be
visible for us as bright compact stars with high temperatures (10). Clearly,
this will not allow to determine whether the core is composed of O matter or O
antimatter. However, the two cases can be discriminated by the accretion of
ordinary gas which, in the case of the AMNS, will annihilate with the
antibaryons in its interior. Thus, detection $\gamma$-ray sources with a
typical baryon-antibaryon annihilation spectrum Backenstoss:1983gu  can be the
way to determine the BA sign in mirror world.

Let us consider an AMNS of a typical mass $M$ and radius $R$. Transitions
$\overline{n}^{\prime}\rightarrow\overline{n}$ will produce O antimatter in
its liquid core. The production rate (antibaryon per second) can be estimated
as

$\dot{N}_{\overline{b}}=\frac{N_{\overline{b}^{\prime}}}{\tau_{\varepsilon}}\simeq\varepsilon_{15}^{2}\left(\frac{M}{M_{\odot}}\right)^{2/3}\times
3\cdot 10^{34}~{}\text{s}^{-1}$ (11)

where the amount of AMB in the AMNS $N_{\overline{b}^{\prime}}$ is given by
Eq. (2), and the conversion time $\tau_{\varepsilon}$ is given by Eq. (5). For
simplicity, we put $\kappa=1.1$ and $R=12$ km.

On the other hand, the AMNS will accrete O gas while it travels in diffuse
interstellar medium (ISM). The accretion rate (baryon per second) reads

$\dot{N}_{b}\simeq\frac{(2GM)^{2}n_{\text{is}}}{v^{3}}\simeq\frac{10^{32}}{v_{100}^{3}}\times\left(\frac{n_{\text{is}}}{1/\text{cm}^{3}}\right)\left(\frac{M}{M_{\odot}}\right)^{2}\\!\text{s}^{-1}$
(12)

where $n_{\text{is}}$ is the baryon density in the ISM and $v=v_{100}\times
100$ km/s is the star velocity relative to the ISM which is normalized taking
into account that typical kick velocities of pulsars are order 100 km/s.

The ratio between the two rates

$\frac{\dot{N}_{\overline{b}}}{\dot{N}_{b}}\simeq\varepsilon_{15}^{2}v_{100}^{3}\left(\frac{1/\text{cm}^{3}}{n_{\text{is}}}\right)\left(\frac{M}{M_{\odot}}\right)^{-4/3}\times
300$ (13)

depends, for a given $\varepsilon$, on the AMNS mass $M$ (which can range
within $M=(1\div 2)\,M_{\odot}$ as that of the normal NS) and, more
critically, on its velocity $v$ and on the ISM density. The velocity
distribution of observed pulsars seems to be bi-modal: a part of pulsars have
kick velocities $v>100$ km/s and some achieving 1000 km/s, while others
probably receive a very little kick – otherwise they would not be contained in
globular clusters which have small escape velocities. The origin of pulsar
kicks is largely unknown, but we consider that this bimodal distribution
applies also to the AMNS or MNS.

For definiteness, let us take $\varepsilon\sim 10^{-15}$ eV as a benchmark
value. For the AMNS with $v>100$ km/s we have
$\dot{N}_{\overline{b}}/\dot{N}_{b}\gg 1$, i.e. the antibaryon production rate
(11) is much larger than the baryon accretion rate (12). In this case the
antimatter core can be formed and it will emit O photons with the energy
luminosity (8), Such cores can be observed as bright point-like sources in the
far UV and soft X-ray ranges. As for the accreted baryons, they should
annihilate on the core surface. The annihilation photons will be produced with
the rate $L_{\gamma}=l_{\gamma}\dot{N}_{b}$, where $l_{\gamma}\approx 4$ is
the average multiplicity of $\gamma$’s per $p\overline{p}$ annihilation
Backenstoss:1983gu . The rate of energy production is $2m\dot{N}_{b}$, about a
half of which contributes in heating the core (in addition to (8)), and
another half will be emitted from its surface in $\gamma$-rays. Hence, the
energy flux from such a source at a distance $d$ will be $J\simeq
m\dot{N}_{b}/4\pi d^{2}$, or numerically

$J\simeq\frac{10^{-12}}{v_{100}^{3}}\left(\frac{n_{\text{is}}}{1/\text{cm}^{3}}\right)\\!\left(\frac{M}{1.5\,M_{\odot}}\right)^{2}\\!\left(\frac{50\,\text{pc}}{d}\right)^{2}\\!\frac{\rm
erg}{{\rm cm}^{2}{\rm s}}$ (14)

For the AMNS travelling with $v>100$ km/s in the Milky Way (MW) this emission
can be below the diffuse $\gamma$-background and the source may remain
unresolved unless this source is closer than 50 pc or so. However, if the AMNS
has less velocity, say $v\simeq 30$ km/s, and it incidentally crosses a high
density region as e.g. cold molecular cloud with
$n_{\text{is}}>10^{3}/\text{cm}^{3}$, the observability distance can be
increased up to several kpc.

On the other hand, for the slow AMNS moving in galactic discs with $v<10$
km/s, the antibaryons produced in its interior can be outnumbered by the
accreted baryons, i.e. $\dot{N}_{\overline{b}}<\dot{N}_{b}$. In this case the
antimatter core canot be formed and the thermal emission with temperature (10)
will be suppressed. The produced antibaryons will readily annihilate with the
already accreted baryons and the $\gamma$-ray luminosity will be
$L_{\gamma}=l_{\gamma}\dot{N}_{\overline{b}}$. In other words, the
$\gamma$-luminosity of the object will be defined by the lesser value between
$\dot{N}_{\overline{b}}$ and $\dot{N}_{b}$. Namely, if the ratio
$\dot{N}_{\overline{b}}/\dot{N}_{b}$ (13) is less than one, then instead of
Eq. (14) we would have for the energy flux

$J\simeq
10^{-12}\times\left(\frac{\varepsilon_{15}}{0.05}\right)^{2}\\!\left(\frac{M}{1.5\,M_{\odot}}\right)^{2/3}\\!\left(\frac{50\,\text{pc}}{d}\right)^{2}\\!\frac{\rm
erg}{{\rm cm}^{2}{\rm s}}$ (15)

where the $\varepsilon$-dependent factor is normalized to indicate that for
$\varepsilon<5\times 10^{-17}$ eV such a sources become too faint to be
resolved at distances larger than 50 pc or so.

The search of $\gamma$ -ray sources with a spectrum compatible with baryon-
antibaryon annihilation was recently performed in Ref. Dupourque:2021poh .
Analyzing 5787 sources included in 4FGL catalog Fermi-LAT:2019yla  based on 10
years of observations with the Fermi LAT, 14 candidates were found which were
selected by applying the following criteria:
(i) extended candidates were excluded (with angular size larger than the LAT
resolution at energies $E<1$ GeV);
(ii) sources associated with objects known from other wavelengths and
belonging to established $\gamma$-ray sources were excluded, as e.g. pulsars
and active galactic nuclei;
(iii) sources with significant higher energy tail above a GeV were excluded
since the baryon-antibaryon annihilation $\gamma$-spectrum should end up at
the nucleon mass.

Interestingly, the distribution of the sources in the sky shown in Fig. 1 of
Ref. Dupourque:2021poh  very much resembles the distribution of observed
pulsars. Only two candidates have galactic coordinates compatible with the MW
disc, while the 11 candidates having galactic latitudes $|b|>10^{\circ}$
(among which 7 candidates with $|b|>30^{\circ}$) can be assigned to the MW
halo. Interestingly, the sources belonging to the disc are the brightest, with
the energy fluxes $J\geq 10^{-11}$ erg cm-2 s-1, while the ones with higher
galactic latitudes become increasingly fading, and the source J2330-2445
($b=-71,7^{\circ}$) is the taintest, with $J<2\times 10^{-12}$ erg cm-2 s-1.
In view of Eq. (12), this may well explained by correlation of the accretion
rate with the distribution of the ISM densities, though the velocity
distribution of the stars remains the key issue.

Probably it is too early to claim the discovery. These sources are all faint,
close to the Fermi LAT detectability threshold, and they may well belong to a
known $\gamma$-ray source classes, or maybe mimicked by imperfections of the
background interstellar emission. In fact, the authors of Ref.
Dupourque:2021poh  take a conservative attitude and translate their findings
into an upper limit on the local fraction of such objects with respect to
normal stars. The clear identification of these sources is very challenging,
and requires serious multiwavelength search.

The possibility of mirror NS (or HS) being the engines of our antimatter can
have interesting implications since they can produce antinuclei in the ISM.
Namely, for $\varepsilon\sim 10^{-15}$ eV, transitions
$\overline{n}^{\prime}\rightarrow\overline{n}$ transition in the AMNS produces
about $10^{52}$ antibaryons forming the hot and dense core in which nuclear
reactions should form the antinuclei. However, these antinuclei are
gravitationally trapped and the question is how they can escape from the star.
This possibility can be provided by the mirror neutron star mergers. In the
coalescence of the two AMNS, their small antimatter cores do not merge at the
same instant but continue the orbiting and then explode due to the
decompression producing a hot cloud of the neutron rich antinuclei. Most of
these antinuclei, being stable only at the extreme densities, will decay into
the lighter ones which are stable in normal conditions. Therefore, the
antinuclei produced by such “sling” effect can leave the coalescence site and
propagate in the outer space. In addition, they can have some additional
acceleration if reasonably large magnetic fields are formed the rotating
ordinary anti-core during its evolution before the merger. The AMS-02
experiment hunting for the antinuclei in the cosmic rays has reported, as the
preliminary results, the detection of eight antihelium events (among which two
are compatible with antihelium-4 and the rest with antihelium-3), The fraction
$\sim 10^{-8}$ of antihelium with respect to measured fluxes of the helium is
too high to be explained by the conventional production mechanisms.
Interestingly, the rate of the NS mergers $\sim 10^{3}$ Gpc-3 yr-1, with $\sim
10^{52}$ antibaryons produced per a merger, is nicely compatible with this
fraction of antihelium. In addition, our mechanism should produce also the
heavier antinuclei, and thus AMS-02 can be the place where to find fantastic
animals as anticarbon or antioxygen which would be a key for many mysteries.

5. Let us discuss our scenario in wider context, and have a pleasant walk and pleasant talk with the Walrus and the Carpenter, viewing and reviewing panorama of two parallel worlds. This picture is based on the direct product $G\times G^{\prime}$ of two identical gauge groups represented by the SM (or some its extension), with the total Lagrangian

${\mathcal{L}}_{\rm
tot}={\mathcal{L}}+{\mathcal{L}}^{\prime}+{\mathcal{L}}_{\rm mix}$ (16)

where ${\mathcal{L}}$ describes the O matter and ${\mathcal{L}}^{\prime}$ the
dark M matter, while the mixed Lagrangian ${\mathcal{L}}_{\rm mix}$ describes
the possible cross-interactions between the particles of two sectors. The
identical forms of ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ is ensured by
a discrete symmetry $G\leftrightarrow G^{\prime}$ interchanging all O species:
the fermion, gauge and Higgs fields of $G$ sector, with their M partners: the
fermion, gauge and Higgs fields of $G^{\prime}$ sector. In the context of
extra dimensions, it can be viewed as a geometric symmetry between two
parallel 3-branes on which the O and M particle species are localized.

M baryons as cosmological DM have specific cosmological implications Khlopov ;
BDM ; BCV ; IV ; BCCV ; BCCP . Although O and M components have identical
microphysics, their cosmological realizations cannot be quite different.
Namely, the viability of M sector requires the following conditions BDM ; BCV
:
– Asymmetric reheating: after inflation the O and M sectors are reheated
asymmetrically, with $T>T^{\prime}$, which can naturally occur in certain
models;
– Out-of-equilibrium: interactions between O and M particles are feeble enough
in order to maintain the initial temperature asymmetry in subsequent epochs.
In other words, all cross-interactions in ${\mathcal{L}}_{\text{mix}}$ should
remain out-of-equilibrium at any stage after inflation, at least before the
Big Bang Nucleosynthesis (BBN);
– No extra heating: after inflation both sectors evolve almost adiabatically
and the temperature difference $T^{\prime}<T$ is not erased by entropy
production in M sector due to possible 1st order phase transitions.

Namely, the BBN bounds demand $T^{\prime}/T<0.5$ BCV  while the post-
recombination cosmology is yet more restrictive, requiring $T^{\prime}/T<0.2$
or so BCCV . This has interesting consequences for the primordial chemical
content in M world: while O world is dominated by hydrogen, with it primordial
mass fraction being 75 % and that of helium being 25 %, M world should be
helium dominated, with the mass fractions of M hydrogen and M helium being
respectively 25 % and 75 % or so BCV .

Along with the ordinary stars also dark M stars can be formed the Galaxy.
However, since mirror world is colder, the first M stars should start to form
somewhat earlier than the first (population III) stars in O sector. M stars,
being helium dominated, should have somewhat different initial mass function,
and their evolution should be more rapid as compared to O stars BCCP . Since
helium is less opaque than hydrogen, the massive M stars should suffer less
mass losses due to stellar winds end up their life collapsing directly into
black holes (BH). The intermediate mass M stars can explode as SN and form
mirror neutrons stars, while the solar mass M stars as white dwarfs can
survive until present times. All these objects can constitute a dominant
fraction of dark matter in the Galaxy. Namely, the galactic halo can be viewed
as elliptical mirror galaxy of M stars and BH in which O matter forms the disc
BCV . M matter could contribute also to the disc, but the density of M stars
in the disc should not exceed the density of O stars Roux . All these objects
can be observed via microlensing as the Machos in different mass ranges. The
present limits from EROS-MACHO observations do not exclude the possibility of
the galactic halo dominated by dark objects as BH with masses $M>(10\div
100)\,M_{\odot}$ while the fraction of dark stars with $M\sim M_{\odot}$ can
be $\sim 10\%$ or so. This proportion can correspond to the abundance of the
LIGO gravitational wave (GW) signals LIGO  from the BH mergers with typical
masses $M\sim(10\div 50)\,M_{\odot}$ or so Merab . In addition, some of the
peculiar LIGO events with one or both light components and no optical
counterpart can be viewed as the BH-MNS or MNS-MNS mergers Merab1.

Now the time has come to talk of many things: of shoes and ships and sealing-
wax, of cabbages and kings…. there is a subtlety related to the chiral
character of the fermion representations in the SM: in our weak interactions
fermions are left-handed (LH) while the antifermions are right-handed (RH),
and two systems would be symmetric if not CP-violating effects. The value of
the BA in the Universe, and in particular its sign
${\mathcal{B}}=\text{sign}(n_{b}-n_{\bar{b}})=1$, is fixed by (unknown)
baryogenesis mechanism a lá Sakharov Sakh  that created primordial excess of
baryons over antibaryons due to CP-violation in some out-of-equilibrium
processes violating ${\rm B}$ (or ${\rm B}-{\rm L}$ KRS ). In parallel sector
the situation is same, apart of an ambiguity in the CP-violation pattern
distinguishing between the M fermions and M antifermions, and determiing the
sign of mirror BA
${\mathcal{B}}^{\prime}=\text{sign}(n_{b}-n_{\bar{b}^{\prime}})$. This
ambiguity is related to the fact that $G\leftrightarrow G^{\prime}$ symmetry
can be realized in two ways: with or without chirality change between the O
and M species Alice .

Namely, for three families of O fermions $f_{L,R}$ described by $SU(3)\times
SU(2)\times U(1)$, the left-handed (LH) quarks $q_{L}=(u_{L},d_{L})$ and
leptons $\ell_{L}=(\nu_{L},e_{L})$ are weak doublets while the right-handed
(RH) ones $u_{R},d_{R}$ and $e_{R}$ are singlets. The antifermion fields are
obtained by complex-conjugation, $\bar{f}_{R,L}=C\gamma_{0}f_{L,R}^{\ast}$,
and they have opposite chiralities: $\bar{q}_{R}=(\bar{u}_{R},\bar{d}_{R})$
and $\bar{u}_{L},\bar{d}_{L}$ are antiquarks, and
$\bar{\ell}_{R}=(\bar{\nu}_{R},\bar{e}_{R})$ and $\bar{e}_{L}$ are
antileptons. The invariance under the transformation CP:
$f_{L,R}\rightarrow\bar{f}_{R,L}$, i.e. symmetry between the fermions and
antifermions, is violated by the complex Yukawa couplings with the Higgs
doublet $\phi$.

As for three analogous families of $SU(3)^{\prime}\times SU(2)^{\prime}\times
U(1)^{\prime}$ in M sector, we invert definitions and denote the species
$\overline{f}^{\prime}_{L,R}$ with the LH weak interactions as M antiquarks:
$\bar{q}^{\prime}_{L}=(\bar{u}^{\prime}_{L},\bar{d}^{\prime}_{L})$ and
$\bar{u}^{\prime}_{R},\bar{d}^{\prime}_{R}$, and M antileptons:
$\bar{\ell}^{\prime}_{L}=(\bar{\nu}^{\prime}_{L},\bar{e}^{\prime}_{L})$ and
$\bar{e}^{\prime}_{R}$. Correspondingly, the respective anti-species
$f^{\prime}_{R,L}$ with the RH weak interactions we call M quarks:
$q^{\prime}_{R}=(u^{\prime}_{R},d^{\prime}_{R})$ and
$u^{\prime}_{L},d^{\prime}_{L}$, and M leptons:
$\ell^{\prime}_{R}=(\nu^{\prime}_{R},e^{\prime}_{R})$ and $e^{\prime}_{L}$.
This is just a convention: we could name them in the opposite way. The M
fermions and antifermions are equivalent modulo CP violating phases in the
Yukawa couplings of the mirror Higgs doublet $\phi^{\prime}$.

Hence, one can impose a symmetry ${\mathcal{Z}}_{2}:$
$f_{L,R}\leftrightarrow\overline{f}^{\prime}_{L,R}$ interchanging the twin
species of the same chirality, i.e. each O fermion with the corresponding M
antifermion. Alternatively, we can employ
${\mathcal{Z}}^{LR}_{2}={\mathcal{Z}}_{2}\times\text{C}P$ under exchange
$f_{L,R}\leftrightarrow f^{\prime}_{R,L}$ between the O and M fermions which,
in our definition, have the opposite chiralities. (Clearly, both of these
transformations should be also complemented by a proper exchange between the
gauge and Higgs fields of two sectors.) In the former case the M antimatter
should have exactly the same CP-violating physics as our matter. In the latter
case the equivalence holds between the ‘left-handed’ O matter and the ‘right-
handed’ M matter which means that P parity, maximally violated in weak
interactions of each sector, is in some sense restored between two sectors. In
fact, this was the original motivation for introducing mirror fermions Mirror
; FLV  (for a historical overview see Ref. Okun ). But the real difference is
related to CP-violation which was not yet discovered at the time of original
works Mirror . In the absence of CP-violating phases ${\mathcal{Z}}_{2}$ and
${\mathcal{Z}}_{2}^{LR}$ would be equivalent.

Both of these discrete symmetries ensure the identical form of the O and M
Lagrangians ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ in (16), modulo the
issues of CP-violation. Nevertheless, in the absence of cross-interaction
terms ${\mathcal{L}}_{\text{mix}}$ in (16), with the O and M particles
interacting only gravitationally, no experiment can discriminate between the
two possibilities.

On the other hand, there are no fundamental reasons for neglecting the cross-
interactions in ${\mathcal{L}}_{\text{mix}}$ which in fact are the portals for
the mirror DM detection and identification. For example, the simplest
possibility is a kinetic mixing term between the O and M photons, $\varepsilon
F^{\mu\nu}F^{\prime}_{\mu\nu}$ Holdom . The particles of two sectors can
interact also through the gauge bosons of e.g. the common family symmetry
$SU(3)_{H}$ PLB  or common $U(1)_{{\rm B}-{\rm L}}$ symmetry ABK . They can
also share the Peccei-Quinn symmetry and their cross-interaction can be
mediated by the axion BGG . The M gas (dominated by helium, and probably
containing some M nuclei of carbon-neutrogen-oxygen) can be subject of direct
detection via the kinetic mixing of the O–M photons or via other portals in
${\mathcal{L}}_{\text{mix}}$ Cerulli .

The most interesting interactions in ${\mathcal{L}}_{\text{mix}}$ are the ones
that violate baryon and lepton numbers of both sectors.

In fact, the conservation of ${\rm B}$ and ${\rm L}$ in the SM is related to
accidental global symmetries of the Lagrangian at the level of the
renormalizable terms. However, these global symmetries can be broken by higher
order operators cutoff emerging from new physics at some large energy scales.
Namely, ${\rm L}$ should be violated if the neutrino have the Majorana masses,
while ${\rm B}$ (or ${\rm B}-{\rm L}$) violation is needed for the
baryogenesis.

In particular, the lowest dimension effective operators which violate the
lepton numbers are of dimension D=5:

$\frac{A}{M}\big{(}\ell_{L}\phi\big{)}^{2}+\frac{A^{\prime}}{M}\big{(}\bar{\ell}^{\prime}_{L}\phi^{\prime}\big{)}^{2}+\frac{\mathcal{A}}{M}\big{(}\bar{\ell}^{\prime}_{L}\phi^{\prime}\big{)}\big{(}\ell_{L}\phi\big{)}~{}+~{}\text{h.c.}$
(17)

where $M$ is the relevant mass scale, and $A\\!=\\!A^{T}$,
$A^{\prime}\\!=\\!A^{\prime T}$ and ${\mathcal{A}}$ are the matrices of
generically complex Wilson coefficients in family space (the family indices as
well as $C$-matrix are suppressed). The first term, violating ${\rm L}$ by two
units ($\Delta{\rm L}=2$), after substituting the Higgs VEV
$\langle\phi\rangle=v\sim 10^{2}$ GeV, induces the small Majorana masses to
the neutrinos, $m_{\nu}\sim v^{2}/M$ Weinberg . The second term ($\Delta{\rm
L}^{\prime}=2$) works analogously in M sector and induces the Majorana masses
of M neutrinos. As for the third term, it violates both ${\rm L}$ and ${\rm
L}^{\prime}$ by one unit conserving the combination $\widetilde{{\rm L}}={\rm
L}+{\rm L}^{\prime}$, and induces the O–M neutrino (active-sterile) mixing ABS
.

The first operator in (17) can be induced by seesaw mechanism, by introducing
the heavy Majorana fermions $N$ in weak singlet and triplet representations
which are coupled to the leptons via the Yukawa terms
$Y\ell_{L}N\phi+\text{h.c.}$, with $Y$ being the matrix of respective coupling
constants. Then the second operator is induced via the Yukawa terms
$Y^{\prime}\bar{\ell}^{\prime}_{L}{N}^{\prime}\phi^{\prime}+\text{h.c.}$ with
the analogous heavy fermions $N^{\prime}$ of M sector. However, there can
exist also singlet heavy fermions ${\mathcal{N}}$ which are coupled with both
O and M leptons:
${\mathcal{Y}}\ell_{L}{\mathcal{N}}\phi+{\mathcal{Y}}^{\prime}\bar{\ell}^{\prime}_{L}{\mathcal{N}}\phi^{\prime}+\text{h.c.}$,
and thus are messengers between two sectors. In this way, all three operators
in (17) are induced by the seesaw mechanism, and for their coefficients we
get:

$\displaystyle
A=YZY^{T}+{\mathcal{Y}}{\mathcal{Z}}{\mathcal{Y}}^{T},~{}~{}A^{\prime}=Y^{\prime}Z^{\prime}Y^{\prime
T}+{\mathcal{Y}}^{\prime}{\mathcal{Z}}{\mathcal{Y}}^{\prime T},$
$\displaystyle\qquad\qquad\qquad\widetilde{A}={\mathcal{Y}^{\prime}}{\mathcal{Z}}{\mathcal{Y}}^{T}$
(18)

where the coefficient matrices $Z$ etc. parametrize the inverse mass matrices
of $N$, $N^{\prime}$ and ${\mathcal{N}}$ fermions respectively as $Z/M$,
$Z^{\prime}/M$ and ${\mathcal{Z}}/M$. Without losing generality, these
matrices can be taken to be diagonal and real, while the symmetry under
$N\leftrightarrow N^{\prime}$ implies $Z^{\prime}=Z$.

Now the question comes to the sign of BA in M sector. Clearly, this depends on
the baryogenesis models, for which two possible realizations of the discrete
inter-sector symmetry, ${\mathcal{Z}}_{2}$ and ${\mathcal{Z}}_{2}^{LR}$, have
different implications.

Let us consider the case of a baryogenesis mechanisms acting separately in O
and M sectors in identical manner. For example, the BA’s can be induced by
means of electroweak baryogenesis (EWB) in both sectors BCV .777Certainly, in
the SM context the EWB cannot work but one can consider e.g. two Higgs doublet
or supersymmetric extension. Then, in the case of ${\mathcal{Z}}_{2}$, which
yields the CP-violation pattern for the M antimatter identical to that of the
O matter, the BA’s ${\mathcal{B}}$ and ${\mathcal{B}}^{\prime}$ induced in two
sectors should have the opposite signs. Alternatively, we can consider a
leptogenesis scenario related to the seesaw mechanism, due to the CP-violation
in the decays of the heavy Majorana fermions $N\rightarrow\ell_{L}\phi$ and
$N^{\prime}\rightarrow\bar{\ell}^{\prime}_{L}\phi^{\prime}$ (and analogous
decays of ${\mathcal{N}}$-fermions coupled to both sectors). Then
${\mathcal{Z}}_{2}$ ($\ell_{L}\leftrightarrow\overline{\ell}^{\prime}_{L}$)
implies $Y^{\prime}=Y$ and ${\mathcal{Y}}^{\prime}={\mathcal{Y}}$ for the
Yukawa couplings in (Antistars or antimatter cores in mirror neutron stars?),
and so for ${\mathcal{B}}$ being positive, ${\mathcal{B}}^{\prime}$ should be
negative.

As for ${\mathcal{Z}}^{LR}_{2}$ ($\ell_{L}\leftrightarrow\ell^{\prime}_{R}$),
it implies the equivalent CP-violation between the O and M fermions, and the
above mechanisms of separate baryogenesis mechanisms in this case predict the
same signs of ${\mathcal{B}}$ and ${\mathcal{B}}^{\prime}$.

However, the BA of opposite signs between the O and M sectors can be generated
also in the case of ${\mathcal{Z}}^{LR}_{2}$ symmetry. This occurs e.g. in the
co-leptogenesis models discussed in BB-PRL ; EPJ-ST  which assumes that after
inflation the O and M sectors are reheated asymmetrically, with $T\gg
T^{\prime}$, and masses of messenger ${\mathcal{N}}$ fermions between two
sectors are larger than the reheating temperature. Nevertheless, the operators
in (17) mediate scattering processes as
$\ell_{L}\phi\rightarrow\bar{\ell}^{\prime}_{L}\phi^{\prime}$,
$\ell_{L}\phi\rightarrow\ell^{\prime}_{R}\bar{\phi}^{\prime}$ etc. which
violate both ${\rm L}$ and ${\rm L}^{\prime}$, and they are out-of-
equilibrium. The invariance under ${\mathcal{Z}}^{LR}_{2}$
($\ell_{L}\leftrightarrow\ell^{\prime}_{R}$) for the Yukawa couplings in
(Antistars or antimatter cores in mirror neutron stars?) implies
$Y^{\prime}=Y^{\ast}$ and ${\mathcal{Y}}^{\prime}={\mathcal{Y}}^{\ast}$, in
which case the CP-violating factors in the above scattering processes are non-
zero, and they induce ${\mathcal{B}}$ and ${\mathcal{B}}^{\prime}$ of the
opposite signs ADM ; ADM-IJMP . (Interestingly, this mechanism is ineffective
in the case of ${\mathcal{Z}}_{2}$ symmetry yielding
${\mathcal{Y}}^{\prime}={\mathcal{Y}}^{\dagger}$ since CP-violating factors
appear to be vanishing ADM ; ADM-IJMP .) Let us remark that this mechanism
implies $\Omega_{b^{\prime}}>\Omega_{b}$, which is related to the fact that M
sector is colder and the produced ${\rm B}^{\prime}-{\rm L}^{\prime}$ suffers
less damping EPJ-ST . Hence, it can naturally explain the observed
cosmological fractions of the baryons and DM,
$\Omega_{b^{\prime}}/\Omega_{b}\simeq 5$, which also makes clear who has eaten
more oysters, the Walrus of the Carpenter.

Hence, depending on the baryogenesis model and the type of discrete exchange
symmetry, M sector can have positive or negative BA which sign should be also
correlated to the chirality of M baryon in their weak interactions. In
principle, two situations could be distinguished by the neutrino oscillations
between two sectors. Namely, the ordinary SN explosions are believed to be
accompanied by the short neutronization burst. Some part of them can oscillate
into M neutrinos creating the deficit in the expected neutrino flux but this
can be difficult to identify, mainly due to uncertainties in the SN core-
collapse modelling. However, the neutronization bursts from the mirror SN
explosions, depending on the case, could be observed (via the O and M neutrino
oscillation) in terms of our neutrinos or antineutrinos, without being
accompanied by the optical SN. In particular, the observation of the
antineutronization burst can be a smoking gun signal which could shade light
on the mirror nature of DM and sterile neutrinos.

6. Analogously to leptons, the mixed Lagrangian ${\mathcal{L}}_{\text{mix}}$ can include the following D=9 operators with quarks:

$\frac{1}{{\mathcal{M}}^{5}}\overline{(u^{\prime}d^{\prime}d^{\prime})}_{L}(udd)_{R}~{}+~{}{\rm
h.c.}$ (19)

where the parentheses contain the possible gauge invariant spin 1/2 chiral
combinations of three O quarks and analogous combinations of three M quarks
(the gauge and Lorentz indices are omitted). These operators violate both
${\rm B}$ and ${\rm B}^{\prime}$ by one unit but conserve $\widetilde{{\rm
B}}={\rm B}+{\rm B}^{\prime}$, and they induce $n-n^{\prime}$ mass mixing (1)
with BB-nn' :888We single out $n-n^{\prime}$ mixing though generically
operators (19) can induce also other mixings as e.g.
$\Lambda-\Lambda^{\prime}$ between the O and M hyperons etc.

$\varepsilon\simeq\frac{\Lambda_{\rm
QCD}^{6}}{{\mathcal{M}}^{5}}\simeq\left(\frac{10~{}{\rm
TeV}}{{\mathcal{M}}}\right)^{5}\times 10^{-15}~{}{\rm eV}$ (20)

In an UV complete theory they can be induced via a seesaw like mechanism
involving new heavy particles, as color-triplet scalars $S$ and $S^{\prime}$
and a neutral Dirac fermion $F$, so that we have ${\mathcal{M}}^{5}\sim
M_{S}^{4}M_{F}$ modulo the Yukawa coupling constants BB-nn' ; B-M . Hence, for
color scalars at few TeV, the underlying theories can be testable at the LHC
and future accelerators B-M . Interestingly, if ${\mathcal{F}}$ fermions are
allowed to have also a small Majorana mass term, $\mu\ll M_{F}$, then the same
seesaw mechanism would induce also $\Delta{\rm B}=2$ operators
$\sim(udd)_{R}^{2}$ (and their M counterparts) leading to the neutron-
antineutron ($n-\overline{n}$) with mixing
$\varepsilon_{n\bar{n}}=(\mu/M_{F})\varepsilon$ BB-nn' ; B-M . Thus, the
origin of both $n-n^{\prime}$ and $n-\overline{n}$ oscillation phenomena can
be related to the same new physics.

However, $n-\bar{n}$ oscillation Kuzmin  is strongly restricted by the direct
experimental limit $\varepsilon_{n\bar{n}}<7.7\times 10^{-24}$ eV, while the
nuclear stability bounds are yet stronger yielding
$\varepsilon_{n\bar{n}}<2.5\times 10^{-24}$ eV (for a review, see Ref.
Phillips ). By this limits, $n-\overline{n}$ mixing cannot have a great
effects on the neutron stars. The effective time of $n-\overline{n}$
conversion in the NS, which can be roughly estimated by replacing
$\varepsilon$ into $\varepsilon_{n\bar{n}}$ in (5), is larger than $10^{33}$
yr or so, while the energy production rate due to $n-\overline{n}$
annihilation can be no more than $10^{14}$ erg/s.

As for $n-n^{\prime}$ oscillation, it has no restrictions from the nuclear
stability BB-nn'  while the experimental limits Ban ; Serebrov1 ; Altarev ;
Bodek ; Serebrov2 ; ILL ; Abel  still allow $n-n^{\prime}$ mixing mass as
large as $\varepsilon\sim(10^{-15}\div 10^{-16})$ eV taking into account that
the $n-n^{\prime}$ oscillation probability can be suppressed by the
environmental effects as e.g. mirror magnetic fields which can be comparable
to the normal magnetic field at the Earth More . As we have discussed in this
paper, in this case the (anti)mirror NS can get antimatter cores in their
interior which can be rendered visible, via the baryon-antibaryon annihilation
$\gamma$-rays, by accretion of ordinary matter from the ISM.

We have discussed these effects in the minimal situation which assumes that
$n-n^{\prime}$ mixing occurs only due to mass mixing (1), $n-\bar{n}^{\prime}$
mixing is absent, and $n$ and $n^{\prime}$ are exactly degenerate in mass.
However, the concept permits more variations which we briefly mention below:

Transitional magnetic moment. In difference from the $n-\overline{n}$ system
where transitional magnetic moment between $n$ and $\overline{n}$ is forbidden
by Lorentz invariance, non-diagonal magnetic moment $\mu_{nn^{\prime}}$ (or
dipole electric moment) is allowed between $n$ and $n^{\prime}$ Arkady ; MDPI
and they can be effectively induced in certain models of $n-n^{\prime}$ mixing
LHEP . In this case the $n-n^{\prime}$ transition time will depend on the
magnetic field in the NS, and it can be simply estimated by replacing
$\varepsilon$ into $|\mu_{nn^{\prime}}B|$ in Eq. (5), or more concretely

$\varepsilon_{15}~{}\longrightarrow~{}\varepsilon_{15}^{B}=\left(\frac{\mu_{nn^{\prime}}}{10^{-27}~{}\text{eV/G}}\right)\left(\frac{B}{10^{12}~{}G}\right)$
(21)

Taking e.g. $\mu_{nn^{\prime}}\sim 10^{-27}$ eV/G, which is 16 orders of
magnitude smaller than the neutron magnetic moment itself, and making
replacement (21) in Eq. (11), we see that for a mirror magnetar ($B\sim
10^{15}$ G) the antimatter production rate will be $\sim 10^{40}/\text{s}$
while for an old recycled AMNS with $B\sim 10^{8}$ G it will be $\sim
10^{26}/\text{s}$. Thus, the former objects should be very bright while the
latter can be practically invisible in annihilation $\gamma$-rays. Therefore,
in this case the analysis similar to that of Ref. Dupourque:2021poh , would
require a specific selection of the source samples which would take into
account the distribution of magnetic field values in the NS.

$n-\bar{n}^{\prime}$ mixing. For a simplicity, we have considered the
situation with only $n-n^{\prime}$ mixing (1), induced via effective D=9
operators (19), which conserves $\widetilde{{\rm B}}={\rm B}+{\rm
B}^{\prime}$. However, there can exist also $n-\bar{n}^{\prime}$ mixing: there
is no fundamental reason to forbid it. However, the latter mixing, due to the
SM structure, emerges from D=10 operators shortcut , and the depending on the
model parameters, $n-\bar{n}^{\prime}$ mixing mass
$\varepsilon_{n\bar{n}^{\prime}}$ can be much smaller than $n-n^{\prime}$
mixing mass $\varepsilon$, but can be also comparable to it. In the latter
case, with $\varepsilon_{n\bar{n}^{\prime}}\sim\varepsilon_{nn^{\prime}}$,
both the MNS or AMNS could produce the baryon-antibaryon annihilation
$\gamma$-rays, without ‘help’ of the ordinary gas accretion. Interestingly, in
the presence of both $\varepsilon_{n\bar{n}^{\prime}}$ and
$\varepsilon_{nn^{\prime}}$ with the comparable values is not conflict with
the nuclear stability limits, while for the free neutron case it could allow
to effectively induce $n-\bar{n}$ oscillation with pretty large rates provided
that experimental conditions are properly tuned shortcut .

$n-n^{\prime}$ mass splitting. We considered the minimal situation when $n$
and $n^{\prime}$ have exactly the same masses in which case the experimental
bounds Ban ; Serebrov1 ; Altarev ; Bodek ; Serebrov2 ; ILL ; Abel  imply
$\varepsilon<10^{-15}$ eV or so. In this case the time of $n-n^{\prime}$
transition (5) is much larger than the Universe age, and thus it should be an
ongoing process in the existing neutron stars (or M neutron stars). However,
much larger values of $\varepsilon$ are allowed by the experiment if $n$ and
$n^{\prime}$ are not degenerate in mass. In particular, $n-n^{\prime}$
oscillation e.g. with $\varepsilon\sim 10^{-10}$ eV or so can solve the
neutron lifetime problem, the $4\sigma$ discrepancy between the neutron
lifetimes measured via the bottle and beam experiments, provided that $n$ and
$n^{\prime}$ have a mass splitting $m_{n^{\prime}}-m_{n}\sim 100$ neV lifetime
. In fact, mass splitting will emerge in models in which M parity is
spontaneously broken BDM  but with a rather small difference between the O and
M Higgs VEVs $\langle\phi\rangle$ and $\langle\phi^{\prime}\rangle$ Nussinov .
In this case $n-n^{\prime}$ conversion time will be much smaller,
$\tau_{\varepsilon}\sim 10^{6}$ yr or so, so that the most of existing NS
should be already transformed in maximally mixed stars with equal amounts and
equal radii of the O and M components. Hence, half of the AMNS in this case
will be our antinuclear matter, and the $\gamma$-ray emission rate due to
accretion will be given by Eq. (14).

Concluding, we have discussed a possibility of M world having a negative BA,
so that the M neutron stars are the AMNS, and $\bar{n^{\prime}}-\bar{n}$
transition in their interior can create antimatter cores. The ordinary gas
accreted from the ISM annihilating on the surface of these cores give rise to
$\gamma$-rays with the typical spectrum reducible to the baryon-antibaryon
annihilation.

The alternative our mechanism can be the existence of antimatter stars
(antistars) Steigman:1976ev . The commonly accepted baryogenesis mechanisms
fix the value as well as the sign of the BA universally in the whole Universe.
In addition, the observations rule out the existence of significant amount of
antimatter on scales ranging from the solar system to galaxy and galaxy
clusters, and even at very large scales comparable to the present horizon
Cohen ; Steigman . However, more exotic baryogenesis mechanisms (for a review
see Dolgov:1991fr ) can in principle allow the existence of small domains at
well-tempered scales in which antimatter could survive in the form of anistars
Khlopov-AS ; Dolgov-AS . In particular, the Affleck-Dine mechanism Affleck
can be extended by the coupling of the Affleck-Dine ${\rm B}$-charged scalar
field to the inflaton Dolgov-Silk . This modification, with properly tuned
parameters, can give rise to large baryon overdensities at needed scales in
which the stars of specific pattern (or the baryon-dense objects (BDO) as they
were named in Ref. Dolgov-BD ) can be formed. In addition, in these
overdensities the difference between the baryon and antibaryon amounts can be
non-vanishing, and it could be positive as well as negative. Provided that
part of the BDO consisting of antibaryons survive in the Milky Way (MW) halo
up to present days, they can be observed as the emitters of the
$p\overline{p}$ annihilation $\gamma$-rays.

In principle, the BDO and AMNS mechanisms can be distinguished by the spectral
shape of the annihilation $\gamma$-rays. Namely, the proton annihilation on
the surfaces of the BDO should produce $\gamma$-rays with typical spectrum
peaked at 70 MeV or so Backenstoss:1983gu . In the case of the AMNS, the
spectral shape will be deformed by the surface redshift effect, by a factor
$\exp[\phi]$, where $\phi=\phi(r)$ is the gravitational potential at the
surface of antimatter core inside the AMNS. This will rescale down the
spectral shape by $(15-30)\,\%$ depending on the AMNS mass, the EoS specifics
and the radius of antimatter core $r<R$. In addition, one has to take into
account the energy blueshift of the accreted protons: in fact, at the core
surface they will be semi-relativistic, with the speeds nearly approaching the
speed of light .

In addition, let us recall that the AMNS can radiate substantial energy (8)
via the photons in the far UV/soft X-ray ranges (provided that the ratio (13)
is much larger than 1) which can be an additional tracer for their
identification. In addition, in Ref. Dupourque:2021poh  the sources possibly
associated with pulsars were excluded from the possible antistar candidates.
On the other hand, it is plausible that the AMNS are also observable as
ordinary pulsars, if large ordinary magnetic fields are somehow developed in
their antibaryon cores. This could be realized, for example, if along with
$nn^{\prime}$ mixing, there is also a kinetic mixing between the O and M
photons Holdom  which effectively renders the mirror electrons and protons
mini-charged (with tiny ordinary electric charges). The value of these
electric mini-charges are severely restricted by the the cosmological Lepidi
and experimental Vigo  bounds. Nevertheless, their existence can be effective.
Since the antimatter core in the AMNS should consist of the heavy antinuclei
and positrons, the AMNS rotation can induce circular electric currentsin its
antimatter core by the drag mechanism BDT  which can be sufficient for these
cores to acquire significant magnetic field, as it was discussed in Ref. BBMT
. Therefore, the AMNS could mimic ordinary pulsars, perhaps with some unusual
properties. Having this in mind, maybe the pulsars should not be excluded from
the candidate selection provided that their $\gamma$-emission has no high
energy tail above 1 GeV or so.

Acknowledgements

I thank Igor Tkachev for paying my attention to Ref. Dupourque:2021poh  and
useful discussions. The work was supported in part by the research grant “The
Dark Universe: A Synergic Multimessenger Approach” No. 2017X7X85K under the
program PRIN 2017 funded by the Ministero dell’Istruzione, Università e della
Ricerca (MIUR), and in part by Shota Rustaveli National Science Foundation
(SRNSF) of Georgia, grant DI-18-335/New Theoretical Models for Dark Matter
Exploration.

## References

  * (1) Z. Berezhiani, Int. J. Mod. Phys. A 19, 3775 (2004) [arXiv:hep-ph/0312335]; Z. Berezhiani, “Through the looking-glass: Alice’s adventures in mirror world,” In From Fields to Strings, Circumnavigating Theoretical Physics, Eds. M. Shifman et al., vol. 3, pp. 2147-2195 [arXiv:hep-ph/0508233]
  * (2) R. Foot, Int. J. Mod. Phys. A 29, 1430013 (2014) [arXiv:1401.3965 [astro-ph.CO]]
  * (3) Z. Berezhiani and L. Bento, Phys. Rev. Lett. 96, 081801 (2006) [arXiv:hep-ph/0507031]
  * (4) Z. Berezhiani, Eur. Phys. J. C 64, 421 (2009) [arXiv:0804.2088 [hep-ph]]
  * (5) Y. N. Pokotilovski, Phys. Lett. B 639, 214 (2006) [arXiv:nucl-ex/0601017].
  * (6) Z. Berezhiani, M. Frost, Y. Kamyshkov, B. Rybolt and L. Varriano, Phys. Rev. D 96, no. 3, 035039 (2017) [arXiv:1703.06735 [hep-ex]]
  * (7) Z. Berezhiani and L. Bento, Phys. Lett. B 635, 253 (2006) [arXiv:hep-ph/0602227]
  * (8) Z. Berezhiani and A. Gazizov, Eur. Phys. J. C 72, 2111 (2012) [arXiv:1109.3725 [astro-ph.HE]]
  * (9) G. Ban et al., Phys. Rev. Lett. 99, 161603 (2007) [arXiv:0705.2336 [nucl-ex]].
  * (10) A. Serebrov et al., Phys. Lett. B 663, 181 (2008) [arXiv:0706.3600 [nucl-ex]].
  * (11) I. Altarev et al., Phys. Rev. D 80, 032003 (2009) [arXiv:0905.4208 [nucl-ex]].
  * (12) K. Bodek et al., Nucl. Instrum. Meth. A 611, 141 (2009).
  * (13) A. Serebrov et al., Nucl. Instrum. Meth. A 611, 137 (2009) [arXiv:0809.4902 [nucl-ex]].
  * (14) Z. Berezhiani et al., Eur. Phys. J. C 78, 717 (2018) [arXiv:1712.05761 [hep-ex]]
  * (15) C. Abel et al., Phys. Lett. B 812, 135993 (2021) [arXiv:2009.11046 [hep-ph]]
  * (16) Z. Berezhiani and F. Nesti, Eur. Phys. J. C 72, 1974 (2012) [arXiv:1203.1035 [hep-ph]]
  * (17) L. J. Broussard et al., EPJ Web Conf. 219, 07002 (2019) [arXiv:1912.08264 [physics.ins-det]].
  * (18) A. Addazi et al., J. Phys. G 48, no.7, 070501 (2021) [arXiv:2006.04907 [physics.ins-det]]
  * (19) Z. Berezhiani, R. Biondi, M. Mannarelli and F. Tonelli, arXiv:2012.15233 [astro-ph.HE]
  * (20) Z. Berezhiani, ”Unusual effects in $n-n^{\prime}$ conversion”, talk at the Workshop INT-17-69W, Seattle, 23-27 Oct. 2017, `http://www.int.washington.edu/talks/WorkShops/` `int_17_69W/People/Berezhiani_Z/Berezhiani3.pdf`
  * (21) M. Mannarelli, Z. Berezhiani, R. Biondi and F. Tonelli, “nn′ conversion and neutron stars”, talk at NORDITA Workshop “Particle Physics with Neutrons at the ESS”, Stockholm, Sweden, 10–14 Dec. 2018, `https://indico.fysik.su.se/event/6570/timetable/` `#20181213`
  * (22) Z. Berezhiani, LHEP 2, no. 1, 118 (2019) [arXiv:1812.11089 [hep-ph]]
  * (23) I. Goldman, R. N. Mohapatra and S. Nussinov, Phys. Rev. D 100, no. 12, 123021 (2019) [arXiv:1901.07077 [hep-ph]]
  * (24) R. Ciancarella, F. Pannarale, A. Addazi and A. Marciano, Phys. Dark Univ. 32, 100796 (2021) [arXiv:2010.12904 [astro-ph.HE]]
  * (25) D. McKeen, M. Pospelov and N. Raj, [arXiv:2105.09951 [hep-ph]].
  * (26) Z. Berezhiani, arXiv:1602.08599 [astro-ph.CO]
  * (27) Z. Berezhiani, Int. J. Mod. Phys. A 33, no. 31, 1844034 (2018)
  * (28) S. Dupourqué, L. Tibaldo and P. Von Ballmoos, Phys. Rev. D 103, no.8, 083016 (2021) [arXiv:2103.10073 [astro-ph.HE]]
  * (29) S. Abdollahi et al. [Fermi-LAT], Astrophys. J. Suppl. 247, no.1, 33 (2020) [arXiv:1902.10045 [astro-ph.HE]]
  * (30) G. Backenstoss et al., Nucl. Phys. B 228, 424-438 (1983)
  * (31) J. M. Lattimer, Ann. Rev. Nucl. Part. Sci. 62, 485 (2012) [arXiv:1305.3510 [nucl-th]]; I. Vidana, Eur. Phys. J. Plus 133, no. 10, 445 (2018) [arXiv:1805.00837 [nucl-th]]
  * (32) F. Douchin and P. Haensel, Astron. Astrophys. 380, 151 (2001) [arXiv:astro-ph/0111092].
  * (33) F. Özel and P. Freire, Ann. Rev. Astron. Astrophys. 54, 401 (2016) [arXiv:1603.02698 [astro-ph.HE]]; G. F. Burgio, H. J. Schulze, I. Vidana and J. B. Wei, Symmetry 13, no. 3, 400 (2021)
  * (34) A. R. Bodmer, Phys. Rev. D 4, 1601 (1971); E. Witten, Phys. Rev. D 30, 272 (1984)
  * (35) Z. Berezhiani, I. Bombaci, A. Drago, F. Frontera and A. Lavagno, Astrophys. J. 586, 1250 (2003) [arXiv:astro-ph/0209257]
  * (36) J. Schwab, P. Podsiadlowski and S. Rappaport, Astrophys. J. 719, 722 (2010) [arXiv:1006.4584 [astro-ph.HE]].
  * (37) M. G. Alford, A. Schmitt, K. Rajagopal and T. Schäfer, Rev. Mod. Phys. 80, 1455-1515 (2008) [arXiv:0709.4635 [hep-ph]]; R. Anglani at al., Rev. Mod. Phys. 86, 509-561 (2014) [arXiv:1302.4264 [hep-ph]]
  * (38) S. Guillot et al., Astrophys. J. 874, no. 2, 175 (2019) [arXiv:1901.07998 [astro-ph.HE]]
  * (39) S. I. Blinnikov and M. Y. Khlopov, Sov. Astron. 27, 371 (1983); M. Y. Khlopov et al., Sov. Astron. 35, 21 (1991); H. M. Hodges, Phys. Rev. D 47, 456 (1993)
  * (40) Z. Berezhiani, A. D. Dolgov and R. N. Mohapatra, Phys. Lett. B 375, 26 (1996) [arXiv:hep-ph/9511221]; Z.. Berezhiani, Acta Phys. Polon. B 27, 1503 (1996) [arXiv:hep-ph/9602326]; R. N. Mohapatra and V. L. Teplitz, Astrophys. J. 478, 29-38 (1997) [arXiv:astro-ph/9603049]
  * (41) Z. Berezhiani, D. Comelli and F. Villante, Phys. Lett. B 503, 362 (2001) [arXiv:hep-ph/0008105]
  * (42) A. Y. Ignatiev and R. R. Volkas, Phys. Rev. D 68, 023518 (2003) [arXiv:hep-ph/0304260]
  * (43) Z. Berezhiani, P. Ciarcelluti, D. Comelli and F. Villante, Int. J. Mod. Phys. D 14, 107 (2005) [arXiv:astro-ph/0312605]
  * (44) Z. Berezhiani, S. Cassisi, P. Ciarcelluti and A. Pietrinferni, Astropart. Phys. 24, 495 (2006) [astro-ph/0507153]
  * (45) J. S. Roux and J. M. Cline, Phys. Rev. D 102, no.6, 063518 (2020) [arXiv:2001.11504 [astro-ph.CO]].
  * (46) R. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. X 11, 021053 (2021) [arXiv:2010.14527 [gr-qc]].
  * (47) R. Beradze and M. Gogberashvili, Mon. Not. Roy. Astron. Soc. 487, no.1, 650-652 (2019) [arXiv:1902.05425 [gr-qc]]; MDPI Physics 1, no.1, 67-75 (2019) [arXiv:1905.02787 [gr-qc]]
  * (48) A. Addazi and A. Marcianò, Int. J. Mod. Phys. A 33, no. 29, 1850167 (2018) [arXiv:1710.08822 [hep-ph]]; R. Beradze, M. Gogberashvili and A. S. Sakharov, Phys. Lett. B 804, 135402 (2020) [arXiv:1910.04567 [astro-ph.HE]]; R. Beradze and M. Gogberashvili, Mon. Not. Roy. Astron. Soc. 503, 2882 (2021) [arXiv:2101.12532 [astro-ph.CO]]
  * (49) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967)
  * (50) V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985)
  * (51) T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956); I. Y. Kobzarev, L. B. Okun and I. Y. Pomeranchuk, Sov. J. Nucl. Phys. 3, no. 6, 837 (1966)
  * (52) R. Foot, H. Lew and R. R. Volkas, Phys. Lett. B 272 67 (1991)
  * (53) L. B. Okun, Phys. Usp. 50, 380 (2007) [arXiv:hep-ph/0606202]
  * (54) B. Holdom, Phys. Lett. B 166, 196 (1986)
  * (55) Z. Berezhiani, Phys. Lett. B 417, 287-296 (1998) [arXiv:hep-ph/9609342]
  * (56) B. Belfatto and Z. Berezhiani, Eur. Phys. J. C 79, no.3, 202 (2019) [arXiv:1812.05414 [hep-ph]]; B. Belfatto, R. Beradze and Z. Berezhiani, Eur. Phys. J. C 80, no.2, 149 (2020) [arXiv:1906.02714 [hep-ph]]
  * (57) A. Addazi, Z. Berezhiani and Y. Kamyshkov, Eur. Phys. J. C 77, no. 5, 301 (2017) [arXiv:1607.00348 [hep-ph]]
  * (58) Z. Berezhiani, L. Gianfagna and M. Giannotti, Phys. Lett. B 500, 286-296 (2001) [arXiv:hep-ph/0009290]
  * (59) R. Cerulli et al., Eur. Phys. J. C 77, no. 2, 83 (2017) [arXiv:1701.08590 [hep-ex]]; A. Addazi et al., Eur. Phys. J. C 75, no. 8, 400 (2015) [arXiv:1507.04317 [hep-ex]].
  * (60) S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979)
  * (61) E. K. Akhmedov, Z. G. Berezhiani and G. Senjanovic, Phys. Rev. Lett. 69, 3013 (1992) [arXiv:hep-ph/9205230]; R. Foot, H. Lew and R. Volkas, Mod. Phys. Lett. A 7, 2567 (1992); R. Foot and R. Volkas, Phys. Rev. D 52, 6595 (1995) [arXiv:hep-ph/9505359]; Z. G. Berezhiani and R. N. Mohapatra, Phys. Rev. D 52, 6607 (1995) [arXiv:hep-ph/9505385]
  * (62) L. Bento and Z. Berezhiani, Phys. Rev. Lett. 87, 231304 (2001) [arXiv:hep-ph/0107281]; L. Bento and Z. Berezhiani, Fortsch. Phys. 50, 489 (2002); arXiv:hep-ph/0111116
  * (63) Z. Berezhiani, Eur. Phys. J. ST 163, 271 (2008)
  * (64) Z. Berezhiani, Eur. Phys. J. C 76, no. 12, 705 (2016) [arXiv:1507.05478 [hep-ph]]
  * (65) V. Kuzmin, Pisma Zh. Eksp. Teor. Fiz. 12, 335 (1970); R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980)
  * (66) D. G. Phillips et al., Phys. Rept. 612, 1 (2016) [arXiv:1410.1100 [hep-ex]]; K. S. Babu et al., arXiv:1310.8593 [hep-ex]; arXiv:1311.5285 [hep-ph]
  * (67) Z. Berezhiani and A. Vainshtein, Int. J. Mod. Phys. A 33, no. 31, 1844016 (2018); Phys. Lett. B 788, 58 (2019) [arXiv:1809.00997 [hep-ph]]
  * (68) Z. Berezhiani, R. Biondi, Y. Kamyshkov and L. Varriano, MDPI Physics 1, no. 2, 271 (2019) [arXiv:1812.11141 [nucl-th]].
  * (69) Z. Berezhiani, Eur. Phys. J. C 81, no. 1, 33 (2021) [arXiv:2002.05609 [hep-ph]].
  * (70) Z. Berezhiani, Eur. Phys. J. C 79, no. 6, 484 (2019) [arXiv:1807.07906 [hep-ph]].
  * (71) R. N. Mohapatra and S. Nussinov, Phys. Lett. B 776, 22 (2018) [arXiv:1709.01637 [hep-ph]].
  * (72) G. Steigman, Ann. Rev. Astron. Astrophys. 14, 339-372 (1976)
  * (73) A. G. Cohen, A. De Rujula and S. L. Glashow, Astrophys. J. 495, 539-549 (1998) [arXiv:astro-ph/9707087 [astro-ph]]
  * (74) G. Steigman, JCAP 10, 001 (2008) [arXiv:0808.1122 [astro-ph]]
  * (75) I. Affleck and M. Dine, Nucl. Phys. B 249, 361-380 (1985)
  * (76) A. Dolgov and J. Silk, Phys. Rev. D 47, 4244-4255 (1993)
  * (77) A. D. Dolgov, Phys. Rept. 222, 309-386 (1992)
  * (78) M. Y. Khlopov at al., Astropart. Phys. 12, 367-372 (2000) [arXiv:astro-ph/9810228 [astro-ph]]; M. Y. Khlopov, S. G. Rubin and A. S. Sakharov, Phys. Rev. D 62, 083505 (2000) [arXiv:hep-ph/0003285 [hep-ph]]; K. M. Belotsky at al., Phys. Atom. Nucl. 63, 233-239 (2000)
  * (79) C. Bambi and A. D. Dolgov, Nucl. Phys. B 784, 132-150 (2007) [arXiv:astro-ph/0702350 [astro-ph]]; A. D. Dolgov, M. Kawasaki and N. Kevlishvili, Nucl. Phys. B 807, 229-250 (2009) [arXiv:0806.2986 [hep-ph]]; A. D. Dolgov, S. I. Godunov, A. S. Rudenko and I. I. Tkachev, JCAP 10, 027 (2015) [arXiv:1506.08671 [astro-ph.CO]]
  * (80) A. D. Dolgov, V. A. Novikov and M. I. Vysotsky, JETP Lett. 98, 519-522 (2013) [arXiv:1309.2746 [hep-ph]]
  * (81) S. I. Blinnikov, A. D. Dolgov and K. A. Postnov, Phys. Rev. D 92, no.2, 023516 (2015) [arXiv:1409.5736 [astro-ph.HE]].
  * (82) Z. Berezhiani and A. Lepidi, Phys. Lett. B 681, 276 (2009) [arXiv:0810.1317 [hep-ph]]
  * (83) C. Vigo et al., Phys. Rev. Lett. 124, no. 10, 101803 (2020) [arXiv:1905.09128 [physics.atom-ph]].
  * (84) Z. Berezhiani, A. D. Dolgov and I. I. Tkachev, Eur. Phys. J. C 73, 2620 (2013) [arXiv:1307.6953 [astro-ph.CO]].