# Exploring the detection of AQNs in large liquid detectors

I. Lazanu  and M. Parvu11footnotetext: Corresponding author.

###### Abstract

Recent work from the last years has raised the possibility that a portion of
Dark Matter could consist of exotic particles, such as axion (anti)quark
nuggets (AQN, A$\bar{\mathrm{Q}}$N). After a brief review outlining the main
features of axion antiquark nuggets, we explore potential experimental
signatures that can be leveraged to search for these stable supermassive
particles in future surface and underground experiments using large liquid
detectors. These expected signals are discussed in relation with the specific
characteristics of each detection system.

##  1 Introduction

Presently the existence of dark matter is strongly supported, but its nature
remains elusive. At the end of the 70’s, Witten [1] predicted the existence of
a new, and more stable state of matter in comparison to the normal nuclear
matter, consisting of roughly equal numbers of up, down, and strange quarks.
This idea presents many problems because the QCD phase transition is not of
the first order, but rather a crossover and fast evaporation is necessary.
These difficulties were avoided by Ariel Zhitnitsky considering the existence
of axions in the structure of nuggets. Nuggets and anti-nuggets are composite
objects with (anti)baryons in the color superconducting phase squeezed by the
axion domain wall as the shell. Ariel Zhitnitsky introduced [2] Axion
(anti)Quark Nuggets (AQNs/A$\mathrm{\bar{Q}}$Ns) as a new type of candidate
for dark matter, as an alternative to WIMPs, strangelets or nuclearites.

In the Big Bang model, it is generally assumed that the Universe began as a
symmetric state with zero global baryonic charge. Later, following the
scenarios of Sakharov [3] which is a set of three necessary conditions (1.
violation of baryon number, 2. violation of both charge conjugation symmetry
(C) and charge conjugation-parity symmetry (CP), and 3. the process must not
be in thermal equilibrium), this evolved into a state with a net positive
baryon number. Alternatively, in Zhitnitsky’s model of the AQN, the unobserved
antibaryons come to comprise the dark matter. CP-violating processes
associated with the axion $\theta$ term in QCD result in the preferential
formation of antinuggets, thus:

$B_{\text{visible}}:B_{\text{nuggets}}:B_{\text{antinuggets}}\approx 1:2:3$
(1.1)

and the net baryonic number is conserved:

$B_{\text{tot}}=0=B_{\text{nuggets}}-B_{\text{antinuggets}}+B_{\text{visible}}$
(1.2)

if
$B_{\text{DM}}=B_{\text{nugget}}+B_{\text{antinugget}}=5B_{\text{visible}}$.
The ratio $B_{\text{nugget}}/B_{\text{antinugget}}=2/3$ is determined by CP
violating parameter $\theta\sim 1$ [4].

##  2 The structure of AQN and A$\bar{\text{Q}}$N

Regarding the structure, inside the A(anti)QN exists more regions with
distinct length scales. The central core represents the size of the nugget
filled by antiquarks in the Color State (CS) phase with interactions dominated
by QCD theory. Following the predictions of this theory, it is expected that
the characteristic length scale $R_{\text{QCD}}$ is approximately $10^{-13}$
cm. This length is further extended in accord with the baryonic number as
$R=R_{\text{QCD}}B^{1/3}$. This region is surrounded by two other regions: one
represented by the presence of positrons captured by the antiquarks core
(electrosphere) extended up to distances of order of $10^{-8}$ cm, comparable
with the atom dimensions, and the second, corresponding to the axionic domain
wall (DW), much larger, with a radius of the order $R_{\text{DW}}\approx
10^{-5}$ cm. These AQNs carry a substantial (anti)baryon charge $\lvert
B\rvert\approx 10^{3}\div 10^{33}$ [5, 6]. The region with baryon numbers
between $10^{3}$ and $10^{23-24}$ is typically constrained by the expected
dark matter (DM) flux, and thus, the AQN flux. These nuggets exhibit an
approximate nuclear density, with a typical nuclear baryon number density on
the order of $10^{40}$ cm-3. With these parameters the effective interaction
is very small, $\sigma_{\text{AQN}}/M_{\text{AQN}}\approx(\pi
R^{2})/(m_{p}\lvert B\rvert)\sim 10^{-10}$ cm 2/g. In the cited article it is
supposed that the (anti)baryonic matter forms a color superconducting fluid
with a gap of the order of 100 MeV. The electric charge of the A(anti)QN core
depends on the phase of the antiquarks in the color-superconducting phase,
although there may be many different phases in the color-superconducting quark
matter. In this structure, the electric charge in the color–flavor locking
(CFL) phase is very small and can be estimated as $Z\approx-0.3\times
B^{2/3}\approx-30\div-3\times 10^{15}$ for $B$ between $10^{3}$ and $10^{24}$.
This interval of values is very different in the case when phase 2CS is
considered. In this case $Z\approx-5\times 10^{-3}\times
B\approx-5\div-5\times 10^{21}$ for an identical interval of the baryon
charge.

In QCD, if one of the quarks has a non-zero mass, the Lagrangian contains a
term that violates the CP symmetry. Experimentally, QCD does not violate CP
symmetry and this aspect is solved considering a physical field, the axion,
with a potential $V(\theta)$, that goes through a phase transition at a
specific temperature $T_{c}$, that can naturally absorb the QCD CP violating
term. At $T<T_{c}$ the symmetry is broken. At $T\sim\Lambda_{\text{QCD}}$ we
have $\theta_{\text{eff}}\to 0$ (solves CP problem) and the axion field gets a
mass. The axion field interacts with photons. After the inflation, $\theta$
oscillates before the QCD phase transition at the temperature $T_{c}\sim 170$
MeV, the vacuum energy remains the same everywhere, but the phase can vary and
form topological defects because there must be continuity in the $[0,2\pi]$.
Axion domain wall topological defects start forming; quarks and anti-quarks
are trapped inside the domain walls that will form nuggets and anti-nuggets.
The process of domain wall formation stops at some temperature, suggested to
be $\sim$ 40 MeV. Domain walls shrink until they are stopped by nuclear Fermi
pressure of the Color Superconducting phase. In this model, the free parameter
is the axion mass.

The axions play a key role in the structure and stability of AQN because they
generate additional pressure to stabilize the nugget. In accord with Fischer,
Liang et. al. [7] the total energy of an AQN is minimized when the axion
contributes one-third of its total mass. The dynamics of nuggets are governed
by Equation 12 from Reference [8] which depends on the effective domain wall
tension, the pressure difference inside and outside the nugget, and the
viscosity term which describes the friction for the domain wall bubble
oscillation in plasma.

##  3 Relic abundance of A$\bar{\mathrm{Q}}$N

Cosmological production of A$\bar{\mathrm{Q}}$N in the early Universe allows
the estimation of their relic abundance. The two necessary conditions: a
nonzero dark baryon number asymmetry and a first-order phase transition can
easily be satisfied for many asymmetric dark matter models and QCD-like gauge
theories. Such models were developed by [8] and [9]. At a temperature around
40 MeV the A$\bar{\mathrm{Q}}$N is completed and the next step is to establish
the electrical neutrality. In the case of nuggets composed of matter, these
will collect electrons on the electrosphere, but this component does not
essentially modify their mass. For nuggets composed of antimatter, the
processes are more complicated and details were discussed by Ge and others in
the cited paper.

The number of AQNs hitting the earth’s surface is tiny and can be estimated in
accord with [10]:

$\varnothing_{\text{axions}}^{\text{Earth}}=\frac{\langle\dot{N}\rangle}{4\pi
R_{\text{Earth}}^{2}}=0.4\times\frac{10^{24}}{\langle
B\rangle}\frac{\rho_{\text{DM}}}{0.3\,\text{GeV/cm}^{3}}\frac{\langle
v_{\text{AQN}}\rangle}{220\,\text{km/s}}\left[\frac{1}{\text{km}^{2}\,\text{yr}}\right]$
(3.1)

which suggests that lighter AQN have a higher total flux. Considering the
local dark matter with an average velocity $<v_{\text{AQN}}>$, and the density
values ($\rho_{\text{DM}}$) in the range $\sim 0.3-0.4$ GeV/cm3 [11], the AQN
flux on Earth ($\varnothing_{\text{axions}}^{\text{Earth}}$) is estimated to
be between $4\times 10^{3}\div 0.04$ AQN/(km2 yr) for $<B>=10^{20}\div
10^{25}$. If the AQNs are stopped inside the Earth and finally annihilate in
the Earth’s core, following the formalism discussed previously, it is possible
to estimate the corresponding axion flux as a function of axion mass:

$m_{\text{axion}}\varnothing_{\text{axions}}^{\text{Earth}}=\frac{2}{3}B[\text{GeV}]\times\varnothing_{\text{axions}}^{\text{Earth}}\left[\frac{1}{\text{km}^{2}\,\text{yr}}\right]$
(3.2)

In principle, it can be supposed that the full AQN annihilates in this
process. For axions with masses of the order of $10^{-6}\mathrm{eV}\leq
m_{a}\leq 10^{-3}\mathrm{eV}$ [2] their fluxes are in the range of
$10^{35}\div 10^{38}$ [1/(km2 yr)].

##  4 Interaction rates of A$\bar{\mathrm{Q}}$N in the atmosphere and inside
Earth

In simplified hypotheses, the energy-loss mechanism occurs through atomic
collisions. When the dimensions of a nugget are larger than the atomic
dimensions, its cross section can be considered as geometric, contrary, the
effective area is determined by the electronic components. The energy losses
of the AQN/A$\bar{\mathrm{Q}}$N along the trajectories, can be calculated
following the formalism of De Rujula and Glashow [12] for nuclearites:

$\frac{dE}{dx}=\begin{cases}\sigma_{\text{AQN}}\rho(x)v_{\text{AQN}}^{2}&\text{if
}v_{\text{AQN}}(L)\geq\sqrt{\frac{\varepsilon}{\rho}}\\\
-\varepsilon\sigma_{\text{AQN}}&\text{if
}v_{\text{AQN}}(L)<\sqrt{\frac{\varepsilon}{\rho}}\end{cases}$ (4.1)

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 $L$,

$v_{\text{AQN}}(L)=v_{\text{AQN}}(0)\exp\left(-\frac{\sigma_{\text{AQN}}}{M_{\text{AQN}}}\int_{0}^{L}\rho\,dx\right)$
(4.2)

where the cross section can be modeled as $\sigma_{\text{AQN}}=\pi
R_{\text{AQN}}^{2},\quad\text{and}\quad
R_{\text{AQN}}=10^{-7}\left(\frac{B}{10^{24}}\right)^{1/3}\,\text{m}$.

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 the value
$v_{\text{AQN}}(L)<\sqrt{\frac{\varepsilon}{\rho}}$ the energy loss decreases
with a constant rate and is brought to zero. The problem of A(anti)QN’s
impacting the Earth’s atmosphere and lithosphere was investigated by
Zhitnitsky and Gorham [13] and [14].

The parameters of interest for the atmosphere vary strongly depending on a
multitude of factors. The average variation of air pressure (P), temperature
(T), and density ($\rho$) with altitude is called the standard atmosphere and
the mean values of the parameters are available [15]. 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 value between sea level
and $2\times 10^{4}$ m), the energy loss is proportional with $\nu^{2}$ up to
approximately 6.4 m/s. In the lithosphere, the integrity of rocks persists up
to the energy density of $\epsilon$=0.1 eV per molecular bond (or equivalent
$\epsilon=6.2\texttimes 10^{20}$ eV/cm3). For the processes considered inside
Earth, most of the major rocks and minerals have very similar densities,
around 2.6 to 3.0 g/cm2 (for depths at which detection systems are usually
placed). In this case, for the velocities of nuggets below the value 188 m/s
the energy loss is independent of their motion. At rest, nuggets will
accumulate in the Earth’s crust and annihilate. If the annihilation occurs in
the vicinity of a detection system, then the effects may be detectable. A
detailed discussion of these aspects will be given further from a perspective
other than Gorham’s [14].

##  5 Processes generated by the structure of A$\bar{\mathrm{Q}}$N and the
signals of interest for direct detection

In this section, we investigate the possibility of dark matter nuggets
producing measurable signals inside the detector that are distinguishable from
bias and other sources.

The spectral emissivity of photon energies below the electron mass of a nugget
at effective temperature $T$, in the frame of the Thomas-Fermi model, was
previously calculated [6, 16, 14] and particular results are given for three
values of the effective surface temperature (10 keV, 30 keV, 80 keV) in Figure
1 of Gorham’s paper [14].

In the present discussion, we will consider only the case of A(anti)QN
composed by a core with antiquarks structure in a color superconductivity
phase (2CS or CLF state), an axion domain wall, and an external electrosphere
with positrons bounded to establish a neutral system. These A(anti)QN will
strongly interact with other nuggets or ordinary nuclear matter. The
constituent subsystems interact. A simplified hypothesis assumes a partially
decoupled approach between the different processes of interaction with the
external medium.

a. Ionization processes. For an isolated nugget, the system is neutral
electric. A large number of weakly bound positrons from the electrosphere get
excited and can easily leave the system. Thus, A(anti)QN acquires a negative
electric charge $\approx|e|Q$, with

$Q\approx 1.5\times
10^{6}\left(\frac{T}{\text{eV}}\right)^{\frac{5}{4}}\left(\frac{R}{2.25\times
10^{-5}\,\text{cm}}\right)^{2}$ (5.1)

and can ionize the medium as a large object with mass $M\approx m_{p}B$.

b. Interaction atom-antiatom, with one in a neutral state and the other
ionized. In general, the atom-antiatom interactions are important in the
treatment of annihilation processes. In this case, the problem consists in the
fact that the anti-nugget structure is not in reality an antiatom, but in an
approximate approach it can be considered as such. In any matter-antimatter
mixture in which the atoms and antiatoms are not fully ionized, the
rearrangement collisions can lead to bound states of the nucleus and
antinucleus from which the annihilation can proceed. Two distinct classes of
processes exist. The first corresponds to the annihilations between electrons
and positrons; the second is associated with the processes between nuclei and
the antinuclear core. Since the cross sections for such collisions at low
energies are considerably higher than the direct particle-antiparticle
annihilation cross sections, atom-antiatom interactions may play a dominant
role in such a mixture. At energies lower than 100 eV, the rearrangement
collisions are important, the annihilation follows the formation of the
nucleus-antinucleus bound state. For the energies above this value, the direct
particle-antiparticle annihilation cross-sections apply.

The rearrangement cross section at low energies for which the interatomic
potential energy is negative and proportional to the distance at a negative
power, applicable for the cases in which one member is neutral and the second
ionized was investigated by Morgan et. al. [17]. For these cases, the cross-
section can be approximated as

$\sigma=\left(\frac{2\alpha}{E}\right)^{1/2}\pi a_{0}^{2},$ (5.2)

where $\alpha$ is the polarizability of the neutral member (medium in this
case), $a_{0}$ is the Bohr radius ($0.529\times 10^{-10}$ m), with $\alpha$
and $E$ in atomic units (a.u.). As an example, for Ar the polarizability is
11.083 a.u., and for Xenon 27.32 a.u. respectively [18]. With these values,
considering an energy around 1 eV, the corresponding cross sections are
$\sigma=24.57\pi a_{0}^{2}$ and $\sigma=38.5\pi a_{0}^{2}$ respectively.

c. Annihilation of positrons. In accord with Flambaum et. al. [19] the
effective charge of the electron in the positron cloud varies from 1/3 in the
ultrarelativistic case to 2/3 in the non-relativistic case. Taking into
account the screening of the electron charge in the positron cloud, the
probability of direct annihilation is $4\times 10^{-9}$ and $2\times 10^{-8}$
for positronium formation probability in the 3-body collision, considering the
following parameters: interaction energy 0.25 eV, and the positron density at
this point: $n_{e^{+}}\approx 1.5\times 10^{-4}a_{0}^{-3}$.

d. Domain wall, its properties and interactions. In models like the one
proposed by Huang and Sikivie [20], the axionic domain walls appear with
finite size on the order of the horizon correlated with a temperature
$T_{\text{QCD}}\approx 100$ MeV and the probability of a larger size being
exponentially suppressed. Sikivie [21] showed that the domain wall (DW) does
not carry any electromagnetic field of its own - see Equations 4 and 5 from
the cited paper, but the coupling is permitted. The strength of the coupling
hold in GUT is:

$-\frac{1}{137}\frac{1}{8\pi}\frac{Na}{\langle\Phi\rangle}F_{\mu\nu}\overline{F}^{\mu\nu},$
(5.3)

where the unrenormalized value of the electroweak angle is
$\sin^{2}\theta_{W}^{0}=\frac{3}{8}$ and $a$ is the axion field before mixing
with the pion. For a static wall in the plane xy and a static magnetic or
electric field, supposed to be oriented in the z direction, the domain wall
becomes electrically charged with surface charge density:
$\frac{2e^{2}}{3\pi}\overrightarrow{\rm{n}}\cdot\overrightarrow{\rm{B}}$ or
$-\frac{2e^{2}}{3\pi}\overrightarrow{\rm{n}}\times\overrightarrow{\rm{E}}$
respectively. Here, $\overrightarrow{\rm{n}}$ is the direction of increasing
$\theta$. Because the environmental medium is supposed to be ionized, the
conditions to generate a charge density on the surface of the DW are
satisfied. The domain walls can oscillate, and dissipate energy in the form of
gravitational radiation, and this process permits to estimate the lifetime
independent of its size [22, 20, 23]. Thus,

$\tau\approx\left(G\omega\right)^{-1}\approx\left(\frac{m_{P}}{\eta}\right)^{2}\frac{1}{m_{a}}\geq
6\times
10^{5}\,\text{sec}\left(\frac{10^{10}\,\text{GeV}}{\langle\Phi\rangle}\right)\approx
0.2\times 10^{9}\,\text{y}.$ (5.4)

Here, $G$ is Newton’s constant, $\omega$ is the energy per unit area of the
axion domain, $\eta$ is the field variable and $m_{P}$ is the Planck mass with
a numerical value of $1.22\times 10^{22}$ eV. $\langle\Phi\rangle$ is the
magnitude of the vacuum expectation value (VEV) that spontaneously breaks the
$U_{PQ}$(1) symmetry of Peccei and Quinn. At temperatures around 100 MeV, QCD
effects break $U_{PQ}$(1) symmetry. For numerical estimations, we considered
$\langle\Phi\rangle\approx$ 1 GeV. Here, the aspect of interest is whether the
presence of axions can generate an observable signal as an effect of the
interaction.

Axion-photon conversion. In accord with Di Luzio et. al. [24], for the
coupling axion–photon $g_{a\gamma}$ and effective axion–nucleon coupling
$g_{aN}^{eff}$, we are going to assume that the axion is massless or very
light. If the massless case is considered, the conversion probability of an
axion to a photon $P_{a\to\gamma}$ is energy independent and
$P_{a\to\gamma}=\frac{g_{a\gamma}^{2}B^{2}L^{2}}{4}$, where $B$ is the
magnetic field and $L$ is the length of the conversion volume. When the mass
of the axion is considered, the probability must be supplemented by
multiplication with the term $\frac{2(1-\cos(qL))}{(qL)^{2}}$, where $q$ is
the transferred momentum given by $m_{a}^{2}/(2\omega)$ and $\omega$ is the
energy of the axion. This probability is negligible or zero.

Axion-electron interactions, inverse Primakoff conversion on nuclei and the
axioelectric effect [25]. An axion can scatter off an electron to produce a
photon in the Compton-like process $a+e\to\gamma+e$. The energy spectrum of
the photons depends on the axion mass, while the spectra of electrons can be
determined from $E_{e}=E_{a}-E_{\gamma}$. The last two effects are
$a+Z\to\gamma+Z$ and $a+e+Z\to e+Z$. In the considered hypothesis regarding
the mass of the axion, these mechanisms are not capable of producing
observable signals from axion interactions.

e. Annihilation of antiquarks from the structure of anti-nuggets and nucleons.
Quark matter occurs in various forms, depending on the temperature $T$ and
quark chemical potential. At sufficiently high density and low temperature, it
is possible to imagine that quarks form a degenerate Fermi liquid. Because QCD
is asymptotically free, the quarks near the Fermi surface are almost free. The
quark-quark interaction is certainly attractive in some channels since we know
that quarks bind together to form baryons. These conditions are sufficient to
guarantee color superconductivity at sufficiently high density. At zero
temperature, the thermodynamic potential which we loosely refer to as the
”free energy” is $E-\mu N$, where $E$ is the total energy of the system, $\mu$
is the chemical potential, and $N$ is the number of fermions. If there were no
interactions, then the energy required to add or subtract particles or holes
near the Fermi surface would cost zero energy. With a weak attractive
interaction in any channel, if we add a pair of particles or holes with the
quantum numbers of the attractive channel, the free energy is lowered by the
potential energy of their attraction. Numerous pairs will be generated in the
vicinity of the Fermi surface, and as they are bosonic, they will combine to
form a condensate. The ground state will consist of a superposition of states
with varying pair numbers, leading to the breaking of the fermion number
symmetry [26]. This argument, originally developed by Bardeen, Cooper, and
Schrieffer (BCS theory) is completely general in QCD: the “color Coulomb”
interaction is attractive between quarks whose color wave function is
antisymmetric, meaning that superconductivity arises as a direct consequence
of the primary interaction in the theory [27]. The pairs of quarks cannot be
color singlets and the Cooper pair condensate in quark matter will break the
local color symmetry. Jiang and Kuo [28] investigated and evaluated the
binding energy per nucleon for superconducting nuclear matter and compared the
results with the values corresponding to normal nuclear matter. In Table 1 the
ratio for ground-state energy per nucleon, expressed in MeV, for
superconducting (s) and normal (n) nuclear matter are given for different
values of Fermi’s momentum in the frame of Skyrme interaction.

$k_{F}$ [fm-1] | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.1 | 1.2
---|---|---|---|---|---|---|---
$\frac{-\left(E_{0}^{S}/N\right)}{-\left(E_{0}^{n}/N\right)}$ | 1.056 | 1.028 | 1.018 | 1.010 | 1.006 | 1.002 | 1.002
Table 1:  Ratio for ground-state energy per nucleon, in MeV, for
superconducting and normal nuclear matter for different values of Fermi’s
momentum.

Because the charge $Q$ can be large, it is possible that a nugget will capture
protons from the surrounding medium considering these as quasi-free particles.
Capture radius depends on both temperatures: (i) internal ($T$) corresponding
to the electric charge of the AQN because of $Q(T)$ and (ii) external gas
temperature – essentially determined by the hypothesis of ionized plasma as an
environmental medium where the temperature ($T_{\text{gas}}$) is generated by
the typical velocities of protons in plasma. The parameterization is:

$R_{\text{capture}}(T)\approx
0.2\,\text{cm}\times\left(\frac{T}{\text{eV}}\right)^{5/4}\left(\frac{\text{eV}}{T_{\text{gas}}}\right)\gg
R_{\text{geom}}.$ (5.5)

When the AQN enters in the region of relevant baryon density, the annihilation
processes start and the internal temperature increases. In these conditions,
the number of annihilation events per time per unit volume is:

$\frac{dN_{\text{ann}}}{dtdV}\approx\left[\pi
R_{\text{capture}}^{2}(T)nn_{\text{AQN}}v_{\text{AQN}}\right]x_{e}(2\,\text{GeV}).$
(5.6)

Here $n$ represents the baryon number density of the surrounding material;
$n_{\text{AQN}}\approx 0.3\times 10^{-25}$ cm-3 (10${}^{25}/B$) and usually it
is supposed that $v_{\text{AQN}}\approx 0.3c$. In the case of proton captures,
$n$ is the proton number density. In this equation, the square bracket is the
number of protons captured per time and unit volume and the total rate of
annihilation is the product between this number and the fraction of
ionization. If $x_{e}(T_{\text{gas}})$ is the ionization fraction of the gas
as a function of temperature, for highly ionized gas it is possible to
consider $x_{e}\approx 1$; for other cases $x_{e}\leq 1$. The temperature will
be estimated using the following equation:

$T\approx
0.4\,\text{eV}\left(\frac{n}{\text{cm}^{-3}}\right)^{4/17}\left(\frac{v_{\text{AQN}}}{10^{-3}c}\right)^{4/17}\kappa^{4/17},$
(5.7)

where $\kappa$ is a factor that considers the theoretical uncertainties in the
annihilation process [29] and $\kappa\leq 1$. A plausible value is
$\kappa\approx 0.25$ [30].

Unfortunately, the details of the annihilation process of baryons with
antiquark matter in the color superconducting phase are not known. In order to
estimate the annihilation probability for the incident antiquark core with
nucleons, the single possibility is to assume that the typical cross-section
of the nucleon on the antiquark core is similar to the antiproton cross-
section on nuclear matter.

There have been several studies regarding the annihilation of antinucleons on
nuclei. The mechanism in which antiprotons annihilate in interaction with
nuclei was clearly explained by Egidy [31]. In accord with the review of J-M
Richard [32] a typical scenario is: a primary annihilation produces mesons,
and some of them penetrate the nucleus, giving rise to a variety of phenomena:
pion production, nucleon emission, internal excitation, etc. Some detailed
properties have been studied. In accord with recent results [33], for the
annihilation cross sections, if we exclude effects that may regard specific
nuclei, general arguments suggest some systematic trends at large and small
energies: $\sigma_{\text{ann}}$ is almost energy independent near its black-
sphere value $4\pi R_{\text{eff}}^{2}$ (where $R_{\text{eff}}$ is not far from
the nuclear radius). In particular, one expects
$\sigma_{\text{ann}}(p)\approx\sigma_{\text{ann}}(n)$, and both proportional
to $A^{2/3}$. For $p\to 0$, in the absence of resonances,
$\sigma_{\text{ann}}\to 1/p(n)$ and $1/p^{2}(p)$. For a comparison with the
experimental data, we used results from one classical article on
nucleon–antinucleon processes [34], which are in accord with the usual models
and have been used for different interpretations of simulated data. The main
experimental data is given in Table 2:

Particles produced in annihilation in complex nuclei can be divided into: |  At rest |  In flight |  Dominant decay mode
---|---|---|---
Charged pions |  $48\pm 6\%$ |  $45\pm 7\%$ |  $\mu\nu$ $(\approx 100\%)$
Neutral particles (other than $n$, $K^{0}$), in particular $\pi^{0}$ |  $28\pm 7\%$ |  $22\pm 7\%$ |  $\gamma\gamma$ $(\approx 98\%)$
$K$ mesons |  $3\pm 1.5\%$ |  $3\pm 1.5\%$ |  $\mu\nu(63.6\%)+\pi^{-/+}\pi^{0}+3\pi$ (and thus $\mu\nu+\gamma\gamma$)
Cascade of nucleons and nuclear excitation |  $21\pm 2\%$ |  $30\pm 2\%$ |  $\gamma$
Table 2:  The classes of particles produced in annihilation in complex nuclei,
at rest and in flight

The primary antiproton annihilation gives rise to the number of pions (as
average at rest and in flight): $<N_{\pi}>=5.36\pm 0.3$, with average total
energy (at rest and in flight): $<E_{\pi}>=350\pm 18\mathrm{MeV}/\pi$. 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. 0.4 of
interacting pions are inelastically scattered and the effect is a degradation
of the primary pion energy to $<E_{\pi}>=339\pm 18$ MeV. 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 $<N_{B}>=4.1\pm 0.3$ and the corresponding total average
energy release in protons and neutrons is $<\Sigma E_{B}>=490\pm 40$ MeV.
Thus, in the laboratory frame, from $\pi^{0}\to\gamma+\gamma$ the energies of
photons are in the energy range $0\leq E_{\gamma}\leq E_{\pi^{0}}$. The
distribution is flat. For 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 process is produced in the
environmental region, the only particles of interest for the identification of
nuggets are the photons produced directly from the decay of neutral pions and
as secondary particles from other decays. These are the only primary signals
directly generated by the nuggets that can penetrate the active medium of the
detection systems. The distribution of the opening angle ($\alpha$) between
the two photons from $\pi^{0}$ in the lab frame is:

$\frac{dN}{d\alpha}=\frac{1}{4\beta\gamma}\cdot\frac{1}{\sqrt{\gamma^{2}\sin^{2}\left(\frac{\alpha}{2}\right)}-1}\cdot\frac{\cot\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)},$
(5.8)

peaked at the value $\alpha_{\text{min}}$ where
$\sin\left(\frac{\alpha_{\text{min}}}{2}\right)=\frac{1}{\gamma}$ and vanishes
at $\alpha_{\text{max}}=\pi$.

##  6 Numerical results

In the following, we consider different types of signals that can be produced
inside the active medium of the detector.

(i) For a simplified analysis, we can consider that the annihilation of the
core region of the nuggets takes place directly in the liquid active medium of
the considered detection systems. The positive and neutral pions, resulting
from annihilation, are the dominant particles. As charged pions predominantly
decay into muons and neutrinos, the photons produced from the decay of neutral
pions serve as the primary source of measurable signals, allowing for the
detection of the A$\bar{\mathrm{Q}}$Ns. With a high probability, the signals
resulting from photon interactions in the considered active media (liquid
argon or liquid xenon) appear as electromagnetic showers and will be observed
in coincidence. In the laboratory system, the opening angle between the
primary photons is described by Equation 5.8.

To simulate the expected signals, we have used the FLUKA 4-3.4 code [35, 36]
alongside the Flair graphical interface [37]. The output has been written
using the USDRAW entry of the MGDRAW subroutine and analyzed in the ROOT
framework [38]. The spatial evolution of the showers induced by photons from
pion decay is presented in Figures 1 and 2 for LAr and LXe. In both cases, the
projections in two planes are represented. As expected, the showers develop
over significantly different distances in these cases. Figure 3 presents a
comparison between the time necessary in every case to complete the
development of the shower.

Figure 1: The spatial evolution of the electromagnetic cascade generated by
$\pi^{0}$ decay in LAr considering a uniform energy distribution up to 500
MeV.

Figure 2: The spatial evolution of the electromagnetic cascade generated by
$\pi^{0}$ decay in LXe considering a uniform energy distribution up to 500
MeV. Figure 3: A comparison between the time necessary to develop the complete
electromagnetic shower in LAr and LXe considering a uniform energy
distribution up to 500 MeV for the decaying $\pi^{0}$.

A distinct problem is the capability of these detectors to discriminate
between the signals of interest and the background and the identification of
other sources that mimic the expected processes. Table 3 presents the main
properties of interest for the present analysis for Ar and Xe.

Target |  LAr |  LXe
---|---|---
Atomic number |  18 |  54
Atomic mass |  40 |  131.3
Boiling point Tb [K] |  87.3 |  165
Liquid density at Tb [g/cm3] |  1.40 |  2.94
Scintillation wavelength [nm] |  128 |  178
Ionization energy [eV] |  23.6 |  15.6
Table 3: Noble Liquid Targets Parameters

In ton-scale to multi-kton scale liquid detectors such as ProtoDUNE and DUNE
(LAr) or LUX-ZEPLIN (LXe), numerous radioactive materials are present inside
the detector as well as in the surrounding environment, such as the rock in
the underground cavern walls. Internal radioactivity is associated not only
with dust deposition and radon daughters but also with the components within
the detector. Muons, neutrons, and neutrinos represent the main sources of
background from cosmogenic sources for this kind of experiment. In LAr
detectors [39], the intrinsic radioactive background is dominated by the
effect of the content of 39Ar in natural atmospheric argon, with a superior
limit of $(8.0\pm 0.6)\times 10^{-16}$ g/g, or $(1.01\pm 0.08)$ Bq/kg. This
undesirable isotope with the $\beta$ end-point energy of 565 keV leads to very
short tracks ($\sim$ 1 mm). The background value for the energy losses is
$dE/dx\approx$2.1 MeV/cm. DUNE requires an average light yield of $>20$
photoelectrons/MeV with a minimum of 0.5 photoelectron/MeV, corresponding to a
photon detection efficiency (PDE) of 2.6% and 1.3% respectively. Caratelli and
co-workers [40] extended the discussion for the next generation of
experiments. For detectors placed at the surface, particularly in the case of
ProtoDUNE Vertical Drift, there exists a supplementary source of background
associated with its position with respect to the beam pipe [41]. In the LUX
Zeplin experiment, a comprehensive analysis of the backgrounds is published in
[42]. The observed background rate after WIMP search criteria were applied was
$(6.3\pm 0.5)\times 10^{-5}$ events/keVee/kg/day in the low-energy region
after analysis of the contributions from alpha spectra of 222Rn, 218Po, 216Po,
214Po, and 212Po, activated 125Xe, 127Xe, ${}^{131\text{m}}$Xe,
${}^{129\text{m}}$Xe, and 133Xe. Cavern gamma-ray measurements consider
contributions from 40K, 238U and 232Th. Background contributions from beta and
gamma-ray sources produce flat spectra in the detector.

Showers initiated by photons up to approximately 350 MeV can have various
sources: (i) due to primary cosmic photons, (ii) produced by $\pi^{0}$ decay
(for example in a reaction such as $\bar{\nu_{\mu}}N\to\mu^{+}\pi^{0}N$, with
threshold energy 740 MeV), or (iii) photons resulting directly from the
bremsstrahlung of very energetic muons. The values of the muon antineutrino
flux at the threshold, in the Kamioka location, are around $2\times 10^{2}$
m-2 s-1 sr-1 GeV and decrease at higher values [43].

Electromagnetic showers in the ICARUS TPC for $E>200$ MeV with $\gamma$
initiated from $\pi^{0}$ decay were reported by Behera [44]. These showers
represent a very important background source. Following Behera’s
recommendations, overshielding of the detector is deemed necessary.
Additionally, an anticoincidence measurement between external muons, acting as
cosmogenic background after suppression, and signals originating from the
shower generation inside the active medium is required. This measurement aims
to put in evidence only the signals associated with axion nugget annihilation.
Of course, similar problems appear in LXe detectors. To help mitigate the
cosmic ray flux, supplementary shielding is necessary for experiments at the
ground level to be considered at least above the detector to stop particles
before they reach the detector. For example, for the ICARUS detector, a 3 m
thick concrete overburden was installed [44] and the results of the
simulations are presented in the cited paper. Additional triggers for
cosmogenic background studies are necessary to reject the contribution of
muons, especially for very high-energy muons, able to produce energetic
photons from bremsstrahlung inside the active volume. A simple system for
cosmic-ray background rejection using an external muon counter stack, able to
tag $\approx 80$% of the cosmic rays, in a surface neutrino experiment, the
MicroBooNE detector, is described in two successive papers [45, 46]. For
example, in the process $\bar{\nu_{\mu}}N\to\mu^{+}\pi^{0}N$ the muon is
produced inside the detector and is not counted; in this case, the neutral
pion is a background for the nuggets searches. To reject these processes, a
supplementary solution is necessary and a Cosmic Ray Tagging (CRT) system must
be used. The idea of CRT is not new – see for example [47, 48]. The ICARUS
collaboration also considered this method [49] where the system surrounds the
detector. An alternative solution was suggested for DUNE, presenting a module
with extremely low background levels [50]. Managing radioactive backgrounds
from detector components involves enhancements in material selection, opting
for materials with low radioactive concentrations. Additionally, measures such
as increased neutron shielding and minimizing radon levels within the detector
contribute to effective background control. The cryostat filled with
atmospheric LAr contains a second cryostat enclosing underground LAr. The
installation of appropriate electronics in both cryostats can allow a
rejection of the background and simultaneously a coincident detection of the
signal of interest.

Using timing information from the CRT hits and detected light signals, it is
possible to discriminate between particle tracks either entering or exiting
the detector or produced inside. A supplementary analysis is necessary to
identify the particle. In particular, if the muon identification is possible,
the rejection of $\bar{\nu_{\mu}}N\to\mu^{+}\pi^{0}N$ reaction is possible.
Details of the time analysis of signals are presented by the ICARUS
Collaboration in the cited paper.

(ii) During the annihilation processes, additional emissions may include
weakly coupled axions, neutrinos, and X-rays. X-rays can also be generated in
the de-excitation processes of the excimer states of LAr or LXe. The most
important feature of the X-ray spectrum from AQN is that it is peaked in the
(10–50) keV energy range, as claimed by Liang and co-workers [51]. The
interaction of photons with matter involves three competing processes: the
photoelectric effect, Compton scattering, and pair production. In all these
processes, electrons are produced, which lose their energy through ionization
or excitation, leading to exciton production. Using photon cross sections
database for each interaction process and attenuation coefficients for
$\gamma$-rays in LAr [52], it is observed that photoelectric absorption is the
predominant process below 80 keV, while above this energy, Compton scattering
becomes the dominating interaction. Assuming for LAr a density of 1.396 g/cm3
at 1 atm, the interaction length for $\gamma$-rays at 60 keV is $\sim$1.5 cm
and $\sim$12.4 cm at 1 MeV [53]. For example, for the measurements of the low
energy gamma rays, the next project GRAMS [54], using a large LArTPC propose
methods to register and reconstruct these signals. For energies above the
threshold for electron-positron production, will be possible to reconstruct
the momentum of the pair and to determine the incident gamma ray. For lower
energies, where photo-absorption and Compton scattering are dominant, the
detector relies on accurately determining the position and energy of the
electron.

(iii) The existence of the coupling of the axion to the electromagnetic field
opens the window to the direct detection of axions. All rare gas atoms form
fairly stable positive molecule-ions. By the addition of an electron in one or
another Rydberg molecular orbital, any rare-gas molecule-ion should give rise
to very numerous stable excimeric states [55]. In argon, the basic aspects are
the following: light yield $\sim$ a few 10000’s of photons per MeV (depending
on the electric field, particle type, and purity); the dominant wavelength of
emission is 128 nm and other transitions are discussed in [56]; the light with
two characteristic time constants: fast component (6 ns) and slow component
(1500 ns); the argon is highly transparent to its scintillation light. There
are two low-lying excited states: a singlet state $1\Sigma u^{+}$ and a
triplet state $3\Sigma u^{+}$. The singlet and triplet refer to how the spin
of the electron and argon dimer couple in the Rydberg “atom”. The mechanisms
of scintillation in LAr are self-trapped exciton luminescence and
recombination luminescence [55]. For xenon, most of the details are discussed
by Mulliken [55].

The ratios between the intensities of different light transitions are known.
The existence of the axion-photon coupling allows the conversion of axions and
thus the excitation of higher Rydberg states. If the de-excitations are
spectrometrically recorded with satisfactory precision, the change in the
ratios between their intensities will be correlated with the processes induced
by the axions.

##  7 Summary

This paper explores various types of signals expected to be produced as an
effect of the interactions of the Axion (anti)Quark Nuggets in large liquid
detectors. The expected signals are discussed in correlation with the
potential for discrimination against various background sources within the
active medium, along with methods to reject signals originating from cosmic
rays. The primary signal types include electromagnetic showers resulting from
the annihilation of the core of antiquark nuggets, X-ray spectra, and the
potential for direct detection of axions through the de-excitations of the
excimeric states of argon and xenon.

## Acknowledgments

We thank Professor Konstantin Zioutas for the suggestion to address this
topic. MP would also like to thank Nicusor Arsene (ISS) for the useful
discussions regarding this subject. 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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