# A new possible way to detect
Axion Anti-quark Nuggets

I. Lazanu, M. Parvu∗
University of Bucharest, Faculty of Physics
POBox MG-11 Bucharest-Magurele, Romania
∗ Corresponding author<EMAIL_ADDRESS>
###### Abstract

The axion anti-quark nugget (A$\bar{\mathrm{Q}}$N) model was developed to
explain in a natural way the asymmetry between matter and antimatter in
Universe. In this hypothesis, a similitude between the dark and the visible
components exists. The lack of observability of any type of dark matter up to
now, in particular A$\bar{\mathrm{Q}}$Ns, requires finding new ways of
detecting these particles, if they exist. In spite of strong interaction with
visible matter, for such objects a very small ratio of cross section to mass
is expected and thus huge detector systems are necessary. This paper presents
a new idea for the direct detection of the A$\bar{\mathrm{Q}}$Ns using
minerals as natural rock deposits acting as paleo-detectors, where the latent
signals of luminescence produced by interactions of A$\bar{\mathrm{Q}}$Ns are
registered and can be identified as an increased and symmetrical deposited
dose. The estimates were made for minerals widely distributed on Earth, for
which the TL signal is intense and if the thermal conditions are constant and
with low temperatures, the lifetime of the latent signals is kept for
geological time scales.

_K_ eywords Dark Matter $\cdot$ A$\bar{\mathrm{Q}}$N $\cdot$ TL OSL Dosimetry
$\cdot$ Minerals $\cdot$ Detectors.

##  1 Introduction

A large fraction of the matter in the Universe is not directly observable, its
origin is not yet understood, and it is generically referred to as Dark Matter
(DM). The existence of DM is established by observations from galactic to
cosmological scales, where many phenomena can be better understood if the
presence of additional (unobserved) matter is assumed. In order to understand
what DM truly is, it is necessary to unambiguously observe its possible
interactions with ordinary matter. Regarding the origins and characteristics
of DM, a wide variety of particles have been predicted in different theories
or phenomenologically, ranging from very massive macroscopic objects like
primordial black holes and MACHOs, to cold DM, warm DM, milli-charged
particles, self-interacting DM, Weakly Interacting Slim Particles (WISPs),
Ultra-light (fuzzy) Dark Matter, or Weakly Interacting Massive Particles
(WIMPs). Other examples are gravitinos, axions, neutralinos, strangelets,
Q-balls or sterile neutrinos. For a more complete review see for example
Cirelli et al. [1]. There is a wide variety of experiments searching for
axions, QCD-axions, and axion-like particles. These include ADMX [2], CAST
[3], ALPS [4], HAYSTAC [5], MADMAX [6], IAXO [7], and CASPEr [8], each
employing unique techniques to probe the existence of these elusive particles.
Notable examples of experiments that are actively searching for WIMPs include
the LUX [9] experiment, CDMS [10] and SuperCDMS [11], the XENON1T [12]
experiment, and the AMS-02 [13].

Predicting the existence of Axion (anti)Quark Nuggets as exotic constituent
particles of dark matter (DM) has a long history, beginning with the
pioneering work of Witten [14], De Rujula and Glashow [15], until their
detailed definition by Zhitnitsky [16]. The DM and Baryogenesis are two sides
of the same coin; apparently, these two phenomena are uncorrelated. In fact,
in the Big Bang model, the existence of axion quark nuggets and axion anti-
quark nuggets respectively can restore the symmetry between matter and
antimatter, if the total baryon numbers satisfy the following relations:

$B_{\text{visible}}:B_{\text{nuggets}}:B_{\text{antinuggets}}\approx 1:2:3$
(1)
$B_{\text{tot}}=0=B_{\text{nuggets}}-B_{\text{antinuggets}}+B_{\text{visible}}$
(2)

Very important, the basic available constraints from Big Bang nucleosynthesis
(BBN) and cosmic microwave background (CMB) are not in contradiction with
existence of this form of antimatter [17].

Oaknin and Zhitnitsky [18] proposed a novel scenario to explain the observed
cosmological asymmetry between matter and antimatter which relies on a
mechanism of separation of quarks and antiquarks in two coexisting phases at
the end of the cosmological QCD phase transition. Some fraction of them are
bound into heavy nuggets of quark matter in a colour superconducting phase.
Nuggets of both matter and antimatter are formed as a result of the dynamics
of the axion domain walls [19]. The observed difference between the quantities
of nuggets ($\Omega_{\mathrm{N}}$) and antinuggets
($\Omega_{\bar{\mathrm{N}}}$) generated by CP violation unequivocally suggests
that the baryon contribution ($\Omega_{\mathrm{B}}$) must be of the same order
of magnitude as $\Omega_{\bar{\mathrm{N}}}$ and $\Omega_{\mathrm{N}}$. This is
because all these components are proportional to the fundamental dimensional
parameter $\Lambda_{\mathrm{QCD}}$, as are all other dimensional parameters in
QCD, such as the color superconducting (CS) gap ($\Delta$), critical
temperature ($T_{c}$), and chemical potential ($\mu$). The remaining
antibaryons in the early universe plasma then annihilate, leaving only the
baryons. The antimatter counterparts of these baryons are bound in the excess
of antiquark nuggets and are thus unavailable for fast annihilation. Since all
asymmetry effects are of order one, this ultimately leads to similarities
among all components, both visible and dark. The observed ratio of matter to
dark matter, $\Omega_{\mathrm{DM}}\approx 5\cdot\Omega_{\mathrm{B}}$, implies
a scenario where the baryon charge concealed in antinuggets exceeds that in
nuggets by a factor of approximately
$(\Omega_{\bar{\mathrm{N}}}/\Omega_{\mathrm{N}})\approx 3/2$ at the end of
nugget formation [20]. Recently, Sebastian Baum et al. proposed a systematic
direction for investigation of neutrino interactions and searches for exotic
particles predicted in the extensions of the Standard Model or only
phenomenological, using effects in different minerals and rocks [21].

Starting from the reference work by Polymeris et al. [22], the current article
proposes the detection of axion antiquark nuggets - complex systems using the
spatial distribution of the deposited doses in various minerals from natural
deposits.

##  2 The structure of the A$\bar{\mathrm{Q}}$Ns and their interactions

Inside the A(anti)QN exists more regions with distinct structures and
different length scales; for details see for example [23] and [24]. In accord
with Gorham and Rotter [25] and Forbes and Zhitnitsky [26], these
A$\bar{\text{Q}}$Ns carry a (anti)baryon charge |B| $\approx 10^{3}\div
10^{33}$. These values of the baryon number are typically constrained by the
expected dark matter (DM) flux, and thus, the A$\bar{\text{Q}}$N flux.

The A$\bar{\text{Q}}$Ns made of antimatter are capable of releasing a
significant amount of energy when they enter Earth’s atmosphere and
annihilation processes start to occur between antimatter hidden in the form of
A$\bar{\text{Q}}$Ns and the atmospheric or other materials such as rocks. In
accord with the equation for the energy loss derived by De Rujula and Glashow
[15], and with the explicit form for the case of A$\bar{\text{Q}}$Ns:

$-\frac{dE}{dx}=\begin{cases}\sigma_{\text{AQN}}\rho(x)v_{\text{AQN}}^{2}(L)&\text{if
}v_{\text{AQN}}(L)\geq\sqrt{\frac{\varepsilon}{\rho}}\\\
\varepsilon\sigma_{\text{AQN}}&\text{if
}v_{\text{AQN}}(L)<\sqrt{\frac{\varepsilon}{\rho}}\end{cases}$ (3)

where $\sigma_{\mathrm{AQN}}$ is the cross-section associated with the area of
the nugget, and $\rho(x)$ is the density of the medium. For a well-defined
distance $L$,

$v_{\text{AQN}}(L)=v_{\text{AQN}}(0)\exp\left(-\frac{\sigma_{\text{AQN}}}{M_{\text{AQN}}}\int_{0}^{L}\rho\,dx\right).$
(4)

The equation of the energy loss breaks down at low velocity when the force
generated by the particle becomes equal or lower than the force by which the
material is confined. For velocities below $\sqrt{\varepsilon/\rho}$ the
energy loss decreases with a constant rate and is brought to zero. The
variation of the velocity of nugget with distance can be considered as
$v_{AQN}(L)$, where for the cross section can be considered geometric formula
$\sigma_{AQN}=\pi R^{2}_{AQN}$ and
$R_{AQN}=10^{-7}\left(\frac{B}{10^{24}}\right)^{1/3}$ [m].

A$\bar{\mathrm{Q}}$Ns are highly dense objects with a radius on the order of
µm. Thus, when such an object travels through matter the temperature should
locally increase. In the lithosphere, the integrity of rocks persists up to
the energy density $\varepsilon=6.2102$ eV/cm3. Because most of the major
rocks and minerals have very similar densities, around 2.6 to 3.0 g/cm3, for
velocities of nuggets below the value of 188 m/s, the energy loss is
independent of their motion. At rest, nuggets will accumulate in the Earth’s
crust and annihilate.

In the atmosphere, the parameters of interest vary strongly, depending on a
multitude of factors. The atmosphere composition can be approximated
considering only molecular nitrogen which represents nearly 78% of the
atmosphere concentration and oxygen 21%. Molecular nitrogen has a triple bond
between the two atoms, one sigma bond, and two pi bonds. This bond is very
strong and requires 941 kJ/mol of energy to break and is only 495 kJ/mol for
molecular oxygen. Considering an average density for the atmosphere around
0.657 kg/m3 (as a mean value between sea level and and altitude of $2\times
10^{4}$m), the energy loss is proportional with $v^{2}$ up to approximately
6.4 m/s. The average variation of air pressure, temperature and density with
altitude (the standard atmosphere and the mean values of the parameters) are
available in [27].

Initially, the A$\bar{\mathrm{Q}}$N is neutral electric. The
A$\bar{\mathrm{Q}}$Ns made of antimatter are capable to release a significant
amount of energy when they enter in the atmosphere and the Earth’s crust. In
accord with [20], the binding energy of positrons is $E_{\mathrm{bound}}\sim$
keV. A large number of weakly bound positrons get excited and can leave the
system. If an electric field is produced in the vicinity of
A$\bar{\mathrm{Q}}$N through polarization effects, then positrons can be
accelerated up to 10 MeV. As a result of these processes, A$\bar{\mathrm{Q}}$N
acquires a negative electric charge and supplementary will ionize the medium
as a projectile with a mass $M\approx m_{p}B$.

In the work cited above, Ariel Zhitnitski invokes arguments related to the
emission by A$\bar{\mathrm{Q}}$N of axions and their expected characteristics.
The total energy of an A$\bar{\mathrm{Q}}$N finds its equilibrium minimum when
the axion domain wall contributes about 1/3 of its total mass and does not
emit axions. When the annihilation processes start, the equilibrium state
breaks because the mass of the A$\bar{\mathrm{Q}}$N decreases, thus reducing
its size. The surrounding domain wall starts to oscillate, generating
excitation modes and emitting axions with a typical velocity of
$v_{\text{axion}}\approx 0.6c$. Unfortunately, the details of the annihilation
process of baryons or nuclei with antiquark matter in the colour
superconducting phase (2CS or CLF state) are not known. As a simple
approximation, in order to estimate the particles generated after the
annihilation process and their energy distributions, we supposed that the
probability for the interaction of the antiquark core with nucleons is similar
to the antiproton cross-section on nuclear matter. The mechanism in which
antiprotons annihilate in interaction with nuclei was explained by Egidy [28],
Richard [29] or Amsler and Myhrer [30]. In a standard scenario a primary
annihilation produces mesons, and some of them penetrate the nucleus, giving
rise to a variety of other phenomena: pion production, nucleon emission,
internal excitation, etc. Complete and detailed experimental data exist in the
classical paper of Chamberlain et al. [31].

The main experimental characteristics of the particles in the final states
are: i) The number of pions (as average at rest and in flight): $\langle
N_{\pi}\rangle=5.36\pm 0.3$, with average total energy (at rest and in
flight): $\langle E_{\pi}\rangle=350\pm 18\mathrm{MeV}/\pi$; ii) Out of these
pions, 1.3 and 1.9 interact with the nucleus at rest and in flight
respectively, giving rise to nuclear excitation and nucleon emission; iii) 0.4
of interacting pions are inelastically scattered and the effect is a
degradation of the primary pion energy to $\langle E_{\pi}\rangle=339\pm
18\mathrm{MeV}$; iv) An average number of $1.6\pm 0.1$ of the pions produced
in the annihilation interacts with the nucleus in which the annihilation
occurs, with the effect of nuclear excitation and nucleon emission. The
average number of protons emitted per annihilation is $\langle
N_{B}\rangle=4.1\pm 0.3$ and the corresponding total average energy release in
protons and neutrons is $\langle\Sigma E_{B}\rangle=490\pm 40\mathrm{MeV}$.

In the laboratory frame, from $\pi_{0}\to\gamma+\gamma$ the energies of
photons are in the energy range $0\leq E_{\gamma}\leq E_{\pi^{0}}$ with a flat
distribution. From the decay of charged pions, $\pi\to\mu+\nu$, the energy of
muon is $0.58E_{\pi}\leq E_{\mu}\leq E_{\pi}$. Since the annihilation
processes occur within nuclei, nuclear effects may alter these results.
However, the similarity between values at rest and in flight suggests minimal
differences.

If the annihilation takes place in the nucleus, in accord with the description
of Plendl and co-workers [32], the principal steps that are assumed to take
place just before, during and after the intranuclear cascade (INC) caused by
annihilation are the following: a) capture of antimatter structure into high-n
atomic orbit; b) cascade to low-n atomic orbit as intermediate steps from
annihilation; c) Mesons: $K$, $\eta$, $\rho$, $\omega$, $\pi$, between 2 up to
8, with energies: 20 to 600 MeV/$\pi$ are produced; d) Direct emission of
kaons, pions from annihilation; e) Other successive processes: multi-pion
induced INC of $\pi N$, $\pi NN$, $NN$, energetic $n$, $p$ from INC
coalescence and emission of $d$, $t$, 3He, 4He or from pre-equilibrium,
multifragmentation, and others take place. The time scale for all these
processes is between 0 s (direct annihilation) up to $10^{-18}\div 10^{-17}$
s.

Processes initiated in A$\bar{\mathrm{Q}}$N present critical differences
compared with nucleon – antinucleon annihilation. Depending on the parameters,
the inside of the A$\bar{\mathrm{Q}}$N will exist in a color superconducting
phase (CS or CFL) with a gap parameter $\Delta\approx 100$ MeV, while the
outside will consist of a free Fermi gas of positrons to maintain neutrality.
In the CFL phase, colored quark matter and gluons at high densities can be in
excited states. They are strongly interacting quasi-particles and the energy
will ultimately be transferred to lower energy degrees of freedom which are
the Nambu-Goldstone (NG) bosons. The NG bosons are collective excitations of a
diquark condensate. More details from the theoretical point of view and
concrete equations for masses calculations and the dispersion relations are
given in the papers of Alford and collaborators [33] and Lawson and Zhitnisky
[34]. For these states, peculiarities include an inverted mass spectrum, with
kaons being lighter than pions.

In this article we are interested in determining the energy deposited by the
A$\bar{\mathrm{Q}}$N following its annihilation in rocks in order to find a
direct detection method through absorbed dose measurements. Zhitnitsky [20]
and Budker et al. [35] estimated the number of frontal collisions of the
environmental molecules with an A$\bar{\mathrm{Q}}$N per unit time, the energy
rate deposited by A$\bar{\mathrm{Q}}$Ns as a result of annihilation processes
and the energy transfer per unit distance in the material of the environment.
Unfortunately, in the absence of relevant experimental results, the energy
transfer following the annihilation process from the nuclear state of colored
superconductivity to that of normal nuclear matter, along with the kinematic
properties of the resulting particles are not fully understood.

In order to estimate the annihilation probability for the incident protons on
the A$\bar{\mathrm{Q}}$N core, it is assumed that the typical cross section is
of the order of antiproton cross section on the nuclear matter and a value of
$\sigma\approx$ 0.4 b was considered [36, 37, 38]. This annihilation process
is considered near the surface of the A$\bar{\mathrm{Q}}$N and different
estimations of the energies of different particles produced after
annihilations are given. In a recent paper, the same authors [39] introduced a
series of constraints in the case of annihilations, considering the absorption
of energy after these processes through strong and electromagnetic
interactions that prevent weak disintegrations with neutrino emission. The
authors estimated that a suppression factor $<3.3\times 10^{-4}$ must be
considered for the emitted mesons. The numerical results that we will present
below will use these assumptions.

##  3 Basic concepts for detection of the A$\bar{\mathrm{Q}}$Ns

Because the fluxes of A$\bar{\mathrm{Q}}$Ns are unknown and not ambiguously
observed until now, the conservative assumption is that the interaction rates,
if they exist, are small. Therefore, massive detectors sensitive to these
particles and long exposure times are required, or it is necessary to use
nonstandard techniques for detection. Large classes of geological minerals
present sensitivity to energies deposed in different interaction processes
with projectile particles. An additional requirement is that the signals
produced in the form of latent signals should be stable for as long as
possible, depending on the ambient conditions. For different materials, due to
the link of the luminescence process with the deposited energy, this is an
excitation process used for dosimetric purposes. The predicted lifetimes [22],
for storage at 15 ∘C is of the order of $10^{8}$ yr, and decrease
exponentially with increasing temperature.

Fluorite (CaF2) is commercially known as fluorspar. The deposits of fluorite
occur in a variety of geologic environments throughout the globe - see for
example Figure 1 from reference [40]. Silicate minerals are rock-forming
minerals made up of silicate groups. These minerals represent up to
approximately 90 percent of Earth’s crust. For the present study especially
SiO2 is important. The U.S. Mineral Resources Data System (MRDS) [41] offers
comprehensive maps and precise location data of silicon dioxide (SiO2) and
calcium fluoride (CaF2) deposits.

Sedimentary quartz (SiO2), natural calcium fluoride (CaF2) and feldspars
(x[AlSi3O8] with x= K, Na, Ca) present luminescence properties and are large
distributed in the Earth. Minerals, exposed to different types of radiation
since the time of their formation, have accumulated over time doses of
radiation in the form of latent signals, which can be extracted by appropriate
physical methods. In the case of luminescence signals, the "reading" of this
information can be extracted through the thermoluminescence (TL) or Optically
Stimulated Luminescence (OSL) techniques. Conventionally, TL measurements are
made by recording light emission during heating sample. Much more information
may be gained by monitoring the details of the emission spectrum during
thermoluminescence. TL spectra of minerals exhibit changes as a result of
crystal purity, radiation dose, dose rate and thermal history. [42].

If the dose accumulation is proportional to the deposited energy and there are
no fading effects, then the read dose will be proportional to the total
energy. In the case of rare events, in principle, the total integrated dose is
the sum of contributions due to the cosmic background or from other sources
and singular contributions, with random spatial locations associated with
axion antiquark nuggets type events. Different components relevant for the
calculation of dose rate for luminescence include: cosmic rays from space
(protons, heavy ions, electrons, muons, photons, etc.), which decrease with
depth; external radiation flux from neighboring grains, such as 40K, 232Th,
and 238U chains (including $\alpha$, $\beta$, and $\gamma$ radiation); and
internal radiation, potentially from 40K and other sources.

This background, as paleodose $D_{b}$ [Gy], is calculated as the integrated
luminescence [yr] multiplied by the dose rate [Gy/yr]. If a singular
annihilation event of an A$\bar{\mathrm{Q}}$N occurs in a region with mineral
deposits of interest (such as feldspars, quartz, calcite, etc.) at a certain
depth, an overdose (D${}_{\text{AQN}}$) with a certain spatial distribution
and approximately spherical symmetry will be generated, so that:

$D_{AQN}=D-D_{b}.$ (5)

##  4 Numerical results and predicted doses produced by the annihilation of
A$\bar{\mathrm{Q}}$Ns

###  4.1 Energy loss of the A$\bar{\mathrm{Q}}$N in air and rocks

The energy produced by annihilation inside the A$\bar{\mathrm{Q}}$N when it
travels the atmosphere or underground per unit length can be expressed as
follows [35]:

$-\frac{dE}{dx}=\begin{cases}\approx
10^{4}\cdot\kappa\left(\frac{B}{10^{25}}\right)^{2/3}\left(\frac{n_{\mathrm{air}}}{10^{21}\mathrm{cm}^{-3}}\right)\frac{\mathrm{J}}{{\mathrm{m}}}\text{
for atmosphere},\\\ \approx
10^{7}\cdot\kappa\left(\frac{B}{10^{25}}\right)^{2/3}\left(\frac{n_{\mathrm{rock}}}{10^{24}\mathrm{cm}^{-3}}\right)\frac{\mathrm{J}}{{\mathrm{m}}}\text{
for rocks},\end{cases}$ (6)

where $n_{\mathrm{air}}$ is the total number of nucleons in atoms such that
$\rho_{\mathrm{air}}=n_{\mathrm{air}}m_{p}$, thus $n_{\mathrm{air}}=7.7\times
10^{21}\mathrm{cm}^{-3}$ and $n_{\mathrm{rock}}$ is the total number of
nucleons in atoms such that $\rho_{\mathrm{rock}}=n_{\mathrm{rock}}m_{p}$ ,
thus $n_{\mathrm{rock}}=1.8\times 10^{24}\mathrm{cm}^{-3}$. $\kappa$
represents a parameter that acknowledges the phenomenon where not all matter
striking the nugget will undergo annihilation, and not all of the energy
released by an annihilation event will be fully thermalized within the
nuggets. For instance, a portion of the energy will be emitted in the form of
axions and neutrinos.

Considering $\kappa\approx 1$, the energy released in the atmosphere is $\sim
7.7$ kJ/m and the energy released in rocks is $\sim 18$ MJ/m. Increasing the
density of the surrounding material significantly amplifies the released
annihilation energy. In the underground case, if one removes the low-energy
positrons from the electrosphere, an additional suppression factor $\xi\sim
10^{-2}$ and even much smaller has to be introduced in the estimation of the
energy loss via annihilation processes.

The internal temperature of the A$\bar{\mathrm{Q}}$N propagating in the
Earth’s atmosphere can be expressed as [43]:

$T_{\mathrm{atm}}\approx
40\mathrm{keV}\cdot\left(\frac{n_{\mathrm{atm}}}{10^{21}\mathrm{cm}^{-3}}\right)^{4/17}\kappa^{4/17},$
(7)

and the electric charge of the A$\bar{\mathrm{Q}}$N can be written as a
function of the temperature $T$ as [20]:

$Q\approx\frac{4\pi R^{2}}{\sqrt{2\pi\alpha}}(m_{e}T)\frac{T}{m_{e}}^{1/4}$
(8)

In case of solids the temperature is $T\approx(100-200)$ keV, thus the
electric charge of the A$\bar{\mathrm{Q}}$N becomes $\approx(7.8\times
10^{8}\div 1.86\times 10^{9})|\mathrm{e}|$, thus the energy loss via
ionization is $\approx 10^{16}$ GeV/cm. The collision of A$\bar{\mathrm{Q}}$Ns
with dense media such as rocks should release a huge amount of energy, on the
order of hundreds of MJ/m, equivalent to several kg of TNT in accord with
Table 1 from Ref. [39].

The rate at which antimatter quark nuggets (A$\bar{\mathrm{Q}}$Ns) hit Earth’s
surface is very low, and can be calculated as [44]:

$\varnothing_{\text{AQN}}^{\text{Earth}}=\frac{\langle\dot{N}\rangle}{4\pi
R_{\text{Earth}}^{2}}=0.4\times\frac{10^{24}}{\langle
B\rangle}\frac{\rho_{\text{DM}}}{0.3\,\text{GeV/cm}^{3}}\frac{\langle
v_{\text{AQN}}\rangle}{220\,\text{km/s}}\left[\frac{1}{\text{km}^{2}\,\text{yr}}\right]$
(9)

This indicates that lighter A$\bar{\mathrm{Q}}$Ns tend to have a higher
overall flux. Given that the average speed of local dark matter is $\langle
v_{\text{AQN}}\rangle$ and the density $\rho_{\text{DM}}$ is approximately
0.3-0.4 GeV/cm3 [45], the estimated flux of AQNs hitting Earth
($\varnothing_{\text{AQN}}^{\text{Earth}}$) is:

$\varnothing_{\text{AQN}}^{\text{Earth}}=\begin{cases}2\times 10^{12}\text{
AQNs/year,}&\text{for }\langle B\rangle=10^{20},\\\ 2\times 10^{7}\text{
~{}~{}AQNs/year,}&\text{for }\langle B\rangle=10^{25}.\end{cases}$ (10)

To simulate the expected dose, we have used the FLUKA 4-3.4 code [46, 47]
alongside the Flair graphical interface [48]. FLUKA code can simulate nuclear
reactions, produce secondary particles and model advanced geometries in order
to ensure accurate and reliable assessment of the deposited doses in different
types of environments. The transport of charged particles is carried out using
the Multiple Coulomb scattering algorithm, with an optional single scattering
method also available. The electromagnetic physics models in FLUKA describe
continuous energy losses of charged particles, energy loss straggling, delta-
ray production, and multiple Coulomb scattering.

The spatial distribution of particles produced in $\bar{\mathrm{p}}$
annihilation in CaF2 and SiO2 is spherically symmetric. Figure 1 represents
the dose deposited by $e^{+}/e^{-},p,\pi^{+}/\pi^{-},d,\alpha$ in one
annihilation process in CaF2 and SiO2 respectively with respect to the
distance from the annihilation point. The difference between the total
deposited dose and the contribution of these particles is labeled as "Other
particles", i.e. $n,\mu,t,K$ etc. The ratio between the deposited dose by each
type of particle and the total dose is represented in Figure 2. It can be seen
that the main contribution is from electrons and positrons, along with protons
and charged pions.

Figure 1: The dose deposited by $e^{+}/e^{-}$, $p$, $\pi^{+}/\pi^{-}$, $d$,
$\alpha$ in one annihilation process in CaF2 (left) and SiO2 (right) as a
function of the distance from the annihilation point. The difference between
the total deposited dose and the contribution of these particles is labeled as
"Other particles", i.e., $n$, $\mu$, $\tau$, $K$, etc. The suppression of the
emission of some particles, particularly pions, was not taken into account.
Figure 2: Fraction of dose generated by each type of particle in CaF2 and SiO2
for each type of particle produced in the antiproton annihilation event. The
main contributions are from electrons and positrons, along with protons and
charged pions. The suppression of the emission of some particles, particularly
pions, was not taken into account.

A comparison between the total doses generated by particles from a single
annihilation process in each medium, as a function of the z-axis, is shown in
Figure 3.

In order to get the total deposited dose, these values should be multiplied by
the number of annihilation processes. Luminescence dosimetry has a high level
of sensitivity, with remarkably low detection limits for radiation doses. The
smallest dose recorded using a natural dosimeter with TL technology is 1 µGy,
achieved with calcium fluoride (CaF2:N). In contrast, the Lowest Detectable
Dose Limit (LDDL) for sedimentary quartz dosimeters is 1 mGy, although in some
cases, this limit can be reduced by one order of magnitude [49, 50, 22].

Figure 3: A comparison between the total dose generated by a single antiproton
annihilation in CaF2 and SiO2, as a function of the z-axis. In order to obtain
the total deposited dose, these values should be multiplied by the number of
annihilation processes.

Analysing the energies deposited in the two minerals by all types particles
following the annihilation, we reached the following conclusions: a) Over 50%
of the deposited energy is due to electrons/positrons, over 20% is due to
pions, and the remainder is due to other particles. b) Working under
conservative assumptions, and including the suppression of the emission of
some particles, particularly pions, the doses contributed by
electrons/positrons remain unaffected. c) If we consider only the doses due to
electrons and positrons, then, at a distance of 1 meter from the place of
annihilation, the value obtained will exceed the Lowest Detectable Dose Limit
in CaF2 if the number of annihilated protons is greater than $10^{10}$ and in
SiO2 if more than $10^{12}$ annihilations occur.

###  4.2 Numerical values for the contribution of the radioactive background
to dose

The cosmic rays interact with the nuclei of atmospheric constituents,
producing a cascade of interactions and secondary reaction products that
contribute to cosmic ray exposure that decreases in intensity with depth in
the atmosphere, from the stratosphere to the ground level. The primary types
of radiation that originate in outer space and impinge on the top of the
earth’s atmosphere consist of 87% protons, 11% $\alpha$ particles, about 1%
nuclei of atomic number between 4 (beryllium) and 26 (iron), and about 1%
electrons of very high energy. They originate outside the solar system, and
only a small fraction is normally of solar origin; however, the solar
component becomes significant during solar flares, which follows an 11-year
cycle. The interactions of the primary particles with atmospheric nuclei
produce electrons, $\gamma$ rays, neutrons and mesons. At the ground level,
the dominant component of the cosmic ray field consists of neutrinos and muons
with energy mainly between 1 and 20 GeV [51].

A world-wide contour map of the cosmic-ray dose rates at the ground level is
available at [52], for an interval of dose rate between 20 to 600 nSv/h (or
equivalent 0.18 to 5.26 mSv/yr). The values are calculated in the frame of the
model PARMA coupled with the global relief data ETOPO2v2 which is provided by
National Geophysical Data Center.

In the absence of direct information related to the dose rates produced by the
abundances of uranium, thorium, potassium and their descendants in minerals
and rocks, we will investigate the abundances of these elements in the
different classes of materials, predominantly those with low contents of
radioactive elements. Most of the information presented here are from the Ref.
[53]. For uranium, igneous rocks as basalts and other mafic rocks or
ultramafic are characterized by abundances between 0.001–1 ppm, intermediate
rocks 1–6 ppm, Granites and rhyolites 2–50 ppm or syenites and phonolites
0.1–26 ppm. Between sedimentary rocks shales, clays, mudrocks are 1–5 ppm,
sandstones 0.5–4 ppm, limestones, dolomites <0.1–9 ppm. The metamorphic rocks
are characterized by abundances <1–5 ppm. The thorium is a relatively rare
element. For basalts and other mafic rocks, ultramafic rocks and limestones,
dolomites the abundances are in the range <0.05–3 - 4 ppm, the lower
abundances.

Potassium abundance in igneous rocks are the lower values in ultramafic rocks
(<10 ppm–1%) and basalts and other mafic rocks (1–2)%. In sedimentary rocks in
pure gypsum potassium not exist, and rocks of interest are shales, clays,
mudrocks, limestones and dolomites with abundances <0.01–7.1%. In order to
look for singular events of interaction of A$\bar{\mathrm{Q}}$N with these
rocks, first of all the complete or partial annihilation of the
A$\bar{\mathrm{Q}}$N core, the rocks/minerals in which these processes must be
searched for, must be characterized by the lowest abundance of uranium,
thorium and potassium, the doses accumulated from cosmic radiation should be
as low as possible, and the temperature of the environment constant and as low
as possible throughout the year.

##  5 Summary and conclusions

The paper investigates the possibility that A$\bar{\mathrm{Q}}$Ns, potential
exotic particles associated with dark matter, could be detected using natural
mineral deposits (CaF2 and SiO2) by exploiting TL/OSL techniques. The proposed
approach involves identifying regions with excess dose in these materials.
Because A$\bar{\mathrm{Q}}$Ns are currently hypothetical particles, a series
of simplifying hypotheses have been considered for their interactions with
matter. If this region is characterized by spherical symmetry and has a
spatial extension predicted by calculations, it is plausible to attribute
these signals to A$\bar{\mathrm{Q}}$N annihilations. Given a radioactive
background kept as low as possible, and a minimum detectable dose limit for
TL/OSL in both materials in accordance with values reported in the literature,
A$\bar{\mathrm{Q}}$Ns can be detected if at least $10^{10}$ protons annihilate
in CaF2 and $10^{12}$ in SiO2 respectively.

## Acknowledgments

We express our deep gratitude to Professor Konstantin Zioutas for inviting and
encouraging us to publish this article, as well as for the relevant scientific
discussions related to the components of dark matter and its interactions in
the universe. His comments and new ideas were extremely stimulating for our
work. We also express our gratitude to Professor Ariel Zhitnitsky for his
comments on certain aspects of the mechanisms of the interactions of nuggets
with matter. This work was performed with the financial support of the
Romanian Program PNCDI III, Programme 5, Module 5.2 CERN-RO, under contract
no. 04/2022.

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