# Limits on the nuclearite flux using the ANTARES neutrino telescope

A. Albert  S. Alves  M. André  M. Ardid  S. Ardid  J.-J. Aubert  J. Aublin  B.
Baret  S. Basa  B. Belhorma  M. Bendahman  F. Benfenati  V. Bertin  S. Biagi
M. Bissinger  J. Boumaaza  M. Bouta  M.C. Bouwhuis  H. Brânzaş  R. Bruijn  J.
Brunner  J. Busto  B. Caiffi  D. Calvo  A. Capone  L. Caramete  J. Carr  V.
Carretero  S. Celli  M. Chabab  T. N. Chau  R. Cherkaoui El Moursli  T.
Chiarusi  M. Circella  J.A.B. Coelho  A. Coleiro  R. Coniglione  P. Coyle  A.
Creusot  A. F. Díaz  G. de Wasseige  B. De Martino  C. Distefano  I. Di Palma
A. Domi  C. Donzaud  D. Dornic  D. Drouhin  T. Eberl  T. van Eeden  D. van
Eijk  N. El Khayati  A. Enzenhöfer  P. Fermani  G. Ferrara  F. Filippini  L.
Fusco  J. García  P. Gay  H. Glotin  R. Gozzini  R. Gracia Ruiz  K. Graf  C.
Guidi  S. Hallmann  H. van Haren  A.J. Heijboer  Y. Hello  J.J. Hernández-Rey
J. Hößl  J. Hofestädt  F. Huang  G. Illuminati  C. W. James  B. Jisse-Jung  M.
de Jong  P. de Jong  M. Kadler  O. Kalekin  U. Katz  A. Kouchner  I.
Kreykenbohm  V. Kulikovskiy  R. Lahmann  M. Lamoureux  R. Le Breton  D.
Lefèvre  E. Leonora  G. Levi  S. Le Stum  D. Lopez-Coto  S. Loucatos  L.
Maderer  J. Manczak  M. Marcelin  A. Margiotta  A. Marinelli  J.A. Martínez-
Mora  K. Melis  P. Migliozzi  A. Moussa  R. Muller  L. Nauta  S. Navas  E.
Nezri  B. Ó Fearraigh  A. Păun  G.E. Păvălaş  C. Pellegrino  M. Perrin-Terrin
V. Pestel  P. Piattelli  C. Pieterse  C. Poirè  V. Popa  T. Pradier  N.
Randazzo  D. Real  S. Reck  G. Riccobene  A. Romanov  A. Sánchez-Losa  F.
Salesa Greus  D. F. E. Samtleben  M. Sanguineti  P. Sapienza  J. Schnabel  J.
Schumann  F. Schüssler  J. Seneca  M. Spurio  Th. Stolarczyk  M. Taiuti  Y.
Tayalati  S.J. Tingay  B. Vallage  V. Van Elewyck  F. Versari  S. Viola  D.
Vivolo  J. Wilms  S. Zavatarelli  A. Zegarelli  J.D. Zornoza  J. Zúñiga
(ANTARES Collaboration)  Université de Strasbourg, CNRS, IPHC UMR 7178,
F-67000 Strasbourg, France Université de Haute Alsace, F-68100 Mulhouse,
France IFIC - Instituto de Física Corpuscular (CSIC - Universitat de València)
c/ Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain Technical
University of Catalonia, Laboratory of Applied Bioacoustics, Rambla Exposició,
08800 Vilanova i la Geltrú, Barcelona, Spain Institut d’Investigació per a la
Gestió Integrada de les Zones Costaneres (IGIC) - Universitat Politècnica de
València. C/ Paranimf 1, 46730 Gandia, Spain Aix Marseille Univ, CNRS/IN2P3,
CPPM, Marseille, France Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France Aix Marseille Univ, CNRS, CNES, LAM,
Marseille, France  National Center for Energy Sciences and Nuclear Techniques,
B.P.1382, R. P.10001 Rabat, Morocco Mohammed V University in Rabat, Faculty of
Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco Institute
of Applied Physics, Mohammed VI Polytechnic University, Lot 660, Hay Moulay
Rachid Ben Guerir, 43150, Morocco. INFN - Sezione di Bologna, Viale Berti-
Pichat 6/2, 40127 Bologna, Italy Dipartimento di Fisica e Astronomia
dell’Università, Viale Berti Pichat 6/2, 40127 Bologna, Italy INFN -
Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania, Italy
Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for
Astroparticle Physics, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany University
Mohammed I, Laboratory of Physics of Matter and Radiations, B.P.717, Oujda
6000, Morocco Nikhef, Science Park, Amsterdam, The Netherlands Institute of
Space Science, RO-077125 Bucharest, Măgurele, Romania Universiteit van
Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG
Amsterdam, The Netherlands INFN - Sezione di Genova, Via Dodecaneso 33, 16146
Genova, Italy INFN - Sezione di Roma, P.le Aldo Moro 2, 00185 Roma, Italy
Dipartimento di Fisica dell’Università La Sapienza, P.le Aldo Moro 2, 00185
Roma, Italy LPHEA, Faculty of Science - Semlali, Cadi Ayyad University, P.O.B.
2390, Marrakech, Morocco. INFN - Sezione di Bari, Via E. Orabona 4, 70126
Bari, Italy Department of Computer Architecture and Technology/CITIC,
University of Granada, 18071 Granada, Spain Université Paris-Sud, 91405 Orsay
Cedex, France Università di Salerno e INFN Gruppo Collegato di Salerno,
Dipartimento di Fisica, Via Giovanni Paolo II 132, Fisciano, 84084 Italy
Laboratoire de Physique Corpusculaire, Clermont Université, Université Blaise
Pascal, CNRS/IN2P3, BP 10448, F-63000 Clermont-Ferrand, France LIS, UMR
Université de Toulon, Aix Marseille Université, CNRS, 83041 Toulon, France
Dipartimento di Fisica dell’Università, Via Dodecaneso 33, 16146 Genova, Italy
Royal Netherlands Institute for Sea Research (NIOZ), Landsdiep 4, 1797 SZ ’t
Horntje (Texel), the Netherlands Géoazur, UCA, CNRS, IRD, Observatoire de la
Côte d’Azur, Sophia Antipolis, France International Centre for Radio Astronomy
Research - Curtin University, Bentley, WA 6102, Australia Huygens-Kamerlingh
Onnes Laboratorium, Universiteit Leiden, The Netherlands Institut für
Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer Str.
31, 97074 Würzburg, Germany Dr. Remeis-Sternwarte and ECAP, Friedrich-
Alexander-Universität Erlangen-Nürnberg, Sternwartstr. 7, 96049 Bamberg,
Germany Mediterranean Institute of Oceanography (MIO), Aix-Marseille
University, 13288, Marseille, Cedex 9, France; Université du Sud Toulon-Var,
CNRS-INSU/IRD UM 110, 83957, La Garde Cedex, France INFN - Sezione di Catania,
Via S. Sofia 64, 95123 Catania, Italy Dpto. de Física Teórica y del Cosmos &
C.A.F.P.E., University of Granada, 18071 Granada, Spain IRFU, CEA, Université
Paris-Saclay, F-91191 Gif-sur-Yvette, France INFN - Sezione di Napoli, Via
Cintia 80126 Napoli, Italy Museo Storico della Fisica e Centro Studi e
Ricerche Enrico Fermi, Piazza del Viminale 1, 00184, Roma INFN - CNAF, Viale
C. Berti Pichat 6/2, 40127, Bologna Institut Universitaire de France, 75005
Paris, France Dipartimento di Fisica dell’Università Federico II di Napoli,
Via Cintia 80126, Napoli, Italy

###### Abstract

In this work, a search for nuclearites of strange quark matter by using nine
years of ANTARES data taken in the period 2009-2017 is presented. The passage
through matter of these particles is simulated taking into account a detailed
description of the detector response to nuclearites and of the data
acquisition conditions. A down-going flux of cosmic nuclearites with Galactic
velocities ($\beta=10^{-3}$) was considered for this study. The mass threshold
for detecting these particles at the detector level is  $4\times 10^{13}$
GeV/c2. Upper limits on the nuclearite flux for masses up to $10^{17}$ GeV/c2
at the level of $\sim 5\times 10^{-17}$ cm-2 s-1 sr-1 are obtained. These are
the first upper limits on nuclearites established with a neutrino telescope
and the most stringent ever set for Galactic velocities.

##  1 Introduction

The idea of strange matter goes back to 1971 when A. Bodmer discussed
collapsed nuclei [1]. In 1984 E. Witten introduced a new form of matter named
strange quark matter (SQM) that could be more stable than ordinary matter [2,
3]. SQM is made up of roughly equal numbers of up, down and strange quarks
with color neutrality. SQM objects are confined in a hadronic structure
commonly called “bag”[4]. A bag of SQM will have a net positive charge on the
surface with an almost equal electrostatic potential of + 50 MeV. This
potential is adequate to repel positively charged nuclei at ordinary
temperatures or at reasonably low velocities, and to attract electrons to
neutralize the electric charge of SQM bags [2]. Bags can be as small as
ordinary nuclei (baryon number $A\lesssim 260$) or as large as a neutron star
(baryon number A $\simeq 2\times 10^{57}$) [2, 5]. SQM has been studied as a
part of the MIT111“MIT bag model” named after the institution (Massachusetts
Institute of Technology ) of the inventors of the model. bag model for
different strange quark masses and bag constants [4, 6, 7]. It could exist if
it is a preferred thermodynamic phase compared to the normal state of nuclear
matter consisting of protons and neutrons [8]. SQM could be formed either
during the quark-hadron phase transition in the early Universe, or during the
conversion of neutron stars into strange stars where strange quarks would be
produced through weak processes [5, 9].

SQM produced in the early Universe was originally suggested as a candidate for
dark matter [2, 7]. A. De Rújula and S. L. Glashow have discussed methods to
detect the Galactic flux of SQM objects falling on Earth and used the
available data to constrain the cosmic flux of SQM [10]. Usually in the
literature, small ($A<10^{7}$) SQM systems in the GeV-TeV range are called
strangelets. Strangelets are outside the range of neutrino telescopes: a
strangelet is searched for as an event with anomalous charge-to-mass ratio in
the cosmic radiation using balloon or space-borne spectrometers. However,
neutrino telescopes might detect nuclearites that penetrate at their depth.
The term nuclearites is used to design higher mass ($A>10^{7}$) objects.
Nuclearites are electrically neutral atom-like systems, as they would be
expected to possess an electron cloud around the core. For $A>10^{15}$,
electrons would be largely contained within the bag of nuclear matter.

Nuclearites are generally assumed to be gravitationally bound to astrophysical
objects (the Galaxy, the Galaxy cluster…) and to have a speed determined by
the virial theorem. In the case of nuclearites bound to the Milky Way, their
velocity relative to the Solar System reference frame is $\beta\sim 10^{-3}$.
Nuclearites can arrive at the Earth ground and penetrate large amounts of
material, reaching underground detectors. Usually, they are searched for by
cosmic ray detectors [11, 12]. These particles are protected against direct
interactions with the atoms constituting the traversed medium by an electronic
cloud surrounding their core. However, they would lose a large amount of
energy by elastic and quasi-elastic collisions with the atoms encountered in
their path. Hence, they leave a distinct signal during the passage in cosmic
ray detectors. In the case of a transparent medium such as water, nuclearites
signal could be detected by using the light emission from their overheated
path as a black-body radiation from an expanding cylindrical thermal shock
wave [10].

This work considers downgoing cosmic nuclearites arriving on the Earth with
velocities $\beta=10^{-3}$. In order to determine the threshold detection mass
of these particles at the detector level, nuclearites with different masses
are assumed to arrive vertically into the detector. The arrival speed of
nuclearites at the detector depth is computed after considering their energy
loss in atmosphere and sea water. Preliminary results obtained with a small
ANTARES data set equivalent to 159 days of data were presented in [13].

In this paper, a brief description of the ANTARES neutrino telescope is given
in section 2. Section 3 introduces the main nuclearite properties, in
particular their effective cross-sectional area and the light yield in a
transparent medium. The simulation of the passage of nuclearites in the
ANTARES detector and the associated event selection are discussed in sections
4 and 5, respectively. The results obtained after the analysis of 9 years of
data are presented and discussed in section 6.

##  2 The ANTARES neutrino telescope

ANTARES (Astronomy with a Neutrino Telescope and Abyss environmental RESearch)
was a neutrino telescope based on the detection of Cherenkov radiation whose
construction was completed in May 2008. The data acquisition was definitively
stopped on February, 12th 2022 and the detector decommissioning ended in June
2022\. It was made up of 12 vertical detection lines of 450 m length each,
horizontally spaced by about $60-75$ m. A floor (or a storey) was formed by
three optical modules (OMs) housing a 10” photomultiplier tube (PMT). A
detection line was an ensemble of 25 floors spaced by 14.5 m being the first
one located 100 m from the seafloor. The detector was deployed with the base
of the lines at a depth of $\sim$2475 m in the Mediterranean Sea, 42 km
offshore from Toulon in France. The lines were anchored to the seabed with a
dead weight and were held vertical by a buoy at the top. The so-called
junction box supplied the lines with electrical power and bundles the data
streams from the OMs as well as the distribution of control signals to the
various components. When photons of any origin impinge on the PMT
photocathode, a signal can be produced at the anode and converted into a
digital format by the front-end electronics boards [14], recording time and
charge, and storing information in what is called a hit. A detailed
description of the apparatus can be found in [15].

Downward going atmospheric muons and atmospheric neutrinos interacting inside
or in the vicinity of the detector produce the bulk of physics signal. In
fact, relativistic charged particles passing in or close to the telescope
induce the production of Cherenkov photons in the medium, which can be
detected by the PMTs. Nuclearites, as heavy penetrating particles, would
generate visible photons from a different mechanism (described in the next
section) while they penetrate into water, producing a characteristic and
recognizable signal in the ANTARES detector.

##  3 Interaction of nuclearites with matter

Nuclearites are assumed to be heavy strange quark nuggets. For a massive
nuclearite the energy loss is mainly due to the elastic and quasielastic
collisions with atoms constituting the traversed medium. The energy loss per
unit of length is [10]

$-\dfrac{dE}{dx}=\rho\sigma v^{2},$ (1)

where $\rho$ is the density of the traversed medium, $v$ the non-relativistic
velocity of the nuclearite and $\sigma$ the effective cross section of the
nuclearite defined as [10]

$\sigma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}[\text{cm}^{2}]}=\begin{cases}\pi\times
10^{-16}&\text{if}\quad
M_{N}<{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>10^{-9}\ \text{g}}\\\
\pi\times\bigg{(}\dfrac{3M_{N}}{4\pi\rho_{N}}\bigg{)}^{2/3}&\text{if}\quad
M_{N}\geq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>10^{-9}\ \text{g}}\end{cases},$ (2)

where $M_{N}$ is the nuclearite mass and $\rho_{N}=3.6\times 10^{14}$ g/cm3 is
the density of SQM objects [16]. In Eq. 2 the mass limit corresponds to a
nuclearite with a radius of about $10^{-10}$ m. As the chemical potential of
the s quark in SQM is slightly larger than for u and d quarks, finite SQM
objects are always positively charged and this charge is neutralized by an
electron cloud surrounding their core, thus the effective cross section for
nuclearites with $M_{N}<1.5\times 10^{-9}$ g is controlled by their electronic
cloud.

Nuclearites would travel with the typical velocity of gravitationally trapped
objects in our Galaxy. Therefore, nuclearites are assumed entering the Earth’s
atmosphere with a velocity of $\beta_{0}=10^{-3}$. Before they reach the
ANTARES detector they propagate through a large amount of material and
interact with its constituents. For nuclearites of mass $M_{N}$ penetrating a
distance $L$ in a medium of density profile $\rho(x)$, their velocity changes
as [10]

$\beta_{L}=\beta_{0}\times\exp\left({-\dfrac{\sigma}{M_{N}}\int_{0}^{L}\rho(x)dx}\right).$
(3)

To simulate the propagation of nuclearites along the Earth’s atmosphere, the
following parametrization of its density is used [17]:

$\displaystyle\rho(x)=a\times\exp\left(-\dfrac{x}{b}\right)=a\times\exp\left(-\dfrac{H-L(x)}{b}\right),$
(4)

where $a=1.2\times 10^{-3}$ g/cm3, $b=8.57\times 10^{5}$ cm, $H=50$ km is the
height of the atmosphere and $L(x)$ is the penetrating length in the
atmosphere. For the propagation in water, a constant density of $\rho(x)$ = 1
g/cm3 is assumed.

Any experimental search for nuclearites has an acceptance of the detector for
an isotropic flux of nuclearites that depends on their mass. Only nuclearites
with sufficiently large mass ($>2.5\times 10^{22}$ GeV/c2) can traverse the
Earth at typical Galactic velocities [18]. As the flux of nuclearites should
decrease when their mass increases, in this study, downward going nuclearites
with masses lower than $10^{17}$ GeV/c2 are considered. In fact, downward
going nuclearites of larger masses are expected to cause an overall saturation
of the detector, producing events with a high number of hits that could not be
processed.

A summary of the detection techniques and experimental results can be found in
[19, 20]. Transparent media (e.g., liquid scintillators or water) have been
used in nuclearite searches. Nuclearites, as non-relativistic objects, do not
produce Cherenkov light. They are expected to give rise to a thermal shock
through collisions with the atoms of water. The temperature of the medium
surrounding the nuclearite path rises up to the order of a few thousands of
kelvin. Thus, a hot plasma is formed that moves outward as a shock wave,
emitting blackbody radiation and producing many photons in the visible band. A
detailed description of the luminous efficiency $\eta$ defined as the fraction
of dissipated energy appearing as light is given in [10]. The authors estimate
that in pure water, a fraction of about $3\times 10^{-5}$ of the total energy
loss is provided in form of photons in the energy range [2.25-3.75] eV. This
region is smaller than the wavelengths range in which ANTARES PMTs are
sensitive (300nm-600nm). The number of visible photons radiated per unit of
path length is conservatively estimated as

$\displaystyle\dfrac{dN_{\gamma}}{dx}=\dfrac{\eta}{\pi}\times\dfrac{dE/dx}{\text{eV}}.$
(5)

The possible theoretical uncertainty on the luminous efficiency $\eta$ exactly
in the 300-600 nm band is within the systematic uncertainties arising from
water transparency and PMT detection efficiency variation as a function of
time, as discussed in sect. 6.1

At the depth of the ANTARES detector and under the previous conditions, the
number $I$ of photons reaching one photomultiplier tube is given by the
integration of the number of visible photons radiated per unit of length (Eq.
5) over the nuclearite path length into the detector. The expression is given
by

$\displaystyle
I=\int\dfrac{\Omega}{4\pi}\dfrac{dN_{\gamma}}{dx}\times\exp\bigg{(}\dfrac{-r}{\lambda_{\text{att}}}\bigg{)}dx,$
(6)

where $\Omega$ is the solid angle in which the optical module is seen from the
emission point defined as

$\displaystyle\Omega=\dfrac{A_{\text{eff}}\times\cos(\theta)}{r^{2}},$ (7)

where $r$ is the distance from the nuclearite position to the PMT and $\theta$
is the incidence angle of the photon on the PMT. The quantity
$\lambda_{\text{att}}$ is the light attenuation length in sea water.
$A_{\text{eff}}$ is the effective area of the PMT [15]. The number of
photoelectrons in the PMT is computed including the quantum efficiency of the
tubes.

##  4 Monte Carlo simulation

To simulate downward going nuclearites arriving at the ANTARES telescope, a
hemisphere of 548 m radius has been used as a generation volume filled with
water, with the value of the radius corresponding to two light attenuation
lengths in water from the closest storey. The considered medium surrounds
symmetrically the instrumented volume of the detector, as shown in Fig. 1.
Each simulated event consists of a nuclearite generated with random position
over the surface of the generation volume and with isotropic randomly
generated zenith and azimuth angles $(\theta,\varphi)$ that define the
particle direction. Since the nuclearites are assumed to arrive at the Earth
with Galactic velocities, the velocity of the particle at the generation point
is determined using the path crossed in sea water and atmosphere according to
Eq. 3. At this stage, the particle is propagated along its direction with a
time step of 2 ns. The position and the speed are evaluated at each iteration
as well as the energy loss, the number of expected visible photons and the
number of photoelectrons produced in each OM, which are calculated using the
formulas introduced in section 3. The simulation ends when the particle exits
the simulation hemisphere or when its optical energy loss (integrated over the
time step) becomes too small to produce a sufficient amount of visible photons
(i.e. it drops below 3 eV). At the end, the analysis tool simulates the
digitization of the signal on PMTs, yielding hits as for real events.

Figure 1: Simulated geometries for the event generation. Nuclearites are
generated with random position over the surface of the hemisphere, isotropic
randomly generated zenith and azimuth angles $(\theta,\varphi)$ that define
the particle direction. They are propagated and only those entering in the
detector sensitive volume are then considered in the simulation.

Atmospheric muons represent the major background source for this analysis.
They are produced in the decay of charged mesons generated by the interactions
of primary cosmic rays with nuclei present in the atmosphere. In the
simulations, they are generated in bundles and propagated down to 5 km water
equivalent with the MUPAGE event generator [21]. MUPAGE is based on parametric
formulas allowing to calculate the flux and angular distribution of underwater
or ice muon bundles. The generator takes into account the muon multiplicity
and the multi-parameter dependent energy spectrum.

The last step of the simulation chain aims at transforming the individual
light pulses into a data stream with the same format and environmental
conditions as real data. In order to meet this objective, the environmental
optical background due to bio-organisms and the decays of 40K are added to the
light produced by physics events (nuclearites or atmospheric muons). Also the
behaviour of individual optical modules can be affected by local changes of
environmental conditions. As a consequence, the time evolution of the data
acquisition is properly reproduced, as described in [22].

##  5 Events selection and analysis

In order to avoid biased results, a fraction of 10% of real data has been used
to define the event selection criteria and to compute the sensitivity of the
detector to nuclearites. In addition, as the detection of nuclearite signals
could be impacted by bioluminescence, a strict anti-bioluminescence cut is
applied. A selection is made for each data taking period of several hours,
which is called “run”. To be considered for this analysis, a “run” should not
have more than 20% of the detector elements affected by bioluminescent bursts
at any moment. After applying this filter, 839 days of livetime in the period
2009-2017 are selected.

###  5.1 Trigger and event selection

All hits with a charge above a minimum threshold of 0.3 photoelectrons (p.e.)
are denoted as L0 hits and transmitted to the shore and processed with
dedicated trigger algorithms to identify potentially interesting events that
are stored on disk [23]. L1 hits are defined as either a coincidence of two L0
hits in the same storey within a time window of 20 ns or a single hit with
high amplitude exceeding a predefined high-threshold condition (set to 3 p.e.
or 10 p.e. depending on the data acquisition conditions).

ANTARES used several trigger algorithms to filter its data. For this analysis,
the standard muon triggers T3 and 3D are used to characterise the nuclearite
signal. The 3D trigger requires at least 5 causally connected L1 hits within
2.2 $\mu$s of each other and the T3 trigger is defined as the occurrence of at
least two L1 hits in three consecutive storeys within a coincidence time
window. This coincidence time window is set to 100 ns in the case that the two
storeys are adjacent, and 200 ns in the case of next to adjacent storeys.

A study has been performed in order to check if low speed particles such as
nuclearities are able to trigger ANTARES filters. In Fig. 2, it is presented
the expected number of visible photons per centimeter, as expected from Eq. 5,
as a function of the nuclearite mass. Three positions at different levels of
the ANTARES detector (on the top, middle and bottom of the detector string)
are considered. In that figure, nuclearites are assumed to be penetrating
vertically to the detector with an initial velocity at the top of the
atmosphere of
$\beta{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}_{0}}=10^{-3}$.
Nuclearites with a mass of $10^{13}$ GeV/c2 generate less than $10^{3}$
photons per cm. For reference, the number of Cherenkov photons in the
sensitivity range of the PMTs emitted by a muon is $\sim$ 350/cm. However, the
atmospheric muon events detected by ANTARES are mostly muon bundles which emit
considerably more light per cm than single muons. For example, muon bundles
with multiplicity of about 80 produce about 30000 photons/cm [21], and few
events of this type are expected in the considered ANTARES livetime.
Therefore, a mass of $4\times 10^{13}$ GeV/c2 is taken as a threshold mass in
the analysis for nuclearite detection since it provides a light yield well
separated from atmospheric events.

Figure 2: Number of visible [300-600 nm] photons per centimeter generated by
vertically incident nuclearites at different detector levels. For masses
higher than $8.4\times 10^{14}$ GeV/c2 all electrons must be inside the quark
bag. The cross section starts to increase with the nuclearite mass due to the
change in the cross section, resulting in more generated visible photons.

When an event is triggered, all PMT pulses that happened within 2 $\mu$s
before the first triggered hit and 2 $\mu$s after the last one are recorded.
This collection of hits is referred to as a snapshot. In the case in which two
or more events have some overlapping hits, a merging of the events is done and
a larger snapshot results. As nuclearites would typically have low velocities
in the detector, both T3 and 3D triggers are expected to record a series of
overlapped snapshots for these particles.

The combined efficiency of ANTARES triggers depends on the nuclearite mass and
its arrival direction. When averaged over an isotropic flux from the upper
hemisphere, the fraction of events with mass $>10^{14}$ GeV/c2 recorded by the
T3 or the 3D triggers ranges from 60% to almost 100% depending on the mass.
The efficiency decreases for lower masses. Table 1 presents the overall
efficiency for each nuclearite mass. It is defined as the number of triggered
events divided by the number of generated events. The trigger efficiency
increases from 25% of generated events for nuclearites of $4\times 10^{13}$
GeV/c2 mass to about 97% at the highest masses. The low trigger efficiency for
low masses includes the effect of the energy loss of these particles from the
generation surface of Fig. 1 before they reach the sensitive volume of the
detector, and the fact that clipping particles are not luminous enough. These
factors become negligible for high nuclearite masses.

###  5.2 Preliminary quality cut on L0

From the set of preselected runs, only events satisfying the predefined T3 or
the 3D trigger conditions are included in this analysis. Two discriminating
variables are used to isolate nuclearites signals. The first one is a
dimensionless variable denoted as log${{}_{10}(nhits3)/nfloor}$, quantity that
is proportional to the total amount of photons reaching the OMs of the
detector. For a given event, $nhits3$ is the number of hits with charge of at
least three photoelectrons, and $nfloor$ is the number of floors in the
detector recording at least one hit for this particular event. The second
discrimination variable is the event duration, $dt$, which represents the
transit time of the event in the detector and refers to the time between the
first and the last hit in the event. Nuclearites, as they are slow heavy
particles, are expected to have a much larger transit time in the detector
than background relativistic particles. It is important to mention that by
combining these variables, the efficiency of discriminating nuclearites events
increases significantly compared to usage of a single variable, as for
instance the number of large intensity hits (nhits3) or the number of involved
floors (nfloor).

Due to their low speed and high masses, nuclearite events would generate a
high number of visible photons around the detector in a relatively long period
(reaching milliseconds). They are expected to have high values for the
log${}_{10}(nhits3)/nfloor$ variable as $nfloor$ is bound by the total number
of detector floors. However, atmospheric muon events with low number of L0
hits affect our signal region in the log${}_{10}(nhits3)/nfloor$ variable and
they must be removed. Fig. 3 illustrates log${}_{10}(nhits3)/nfloor$ versus
log10(L0) for simulated atmospheric muon events (left) and the same quantities
are presented for nuclearite with a mass of $10^{16}$ GeV/c2 (right).

A first quality cut requiring at least 300 L0 hits for each event has been
applied to clean our sample. This cut removes events firing with high
amplitude the few PMTs present in a restricted region of the detector because
of (for instance) a cascade of secondary particles induced by an atmospheric
muon. Table 1 summarises the efficiency of the L0 cut for atmospheric muons
and different masses of nuclearites. The efficiency is defined as the number
of events after applying the L0 cut to the total number of triggered events.
The L0 cut removes 85% of the simulated atmospheric muon events.

Figure 3: Distribution of log${}_{10}(nhits3)/nfloor$ versus
log${}_{10}(\text{L0})$ for simulated atmospheric muons (left), and for
nuclearites with a mass of $10^{16}$ GeV/c2 (right). The palette of colors for
the left panel represents the expected number of events in 839 days of
livetime, while for the right panel represents the number of simulated events.
The cut at 300 L0 hits (red dashed line) allows to reject background events
with high values of log${}_{10}(nhits3)/nfloor$ characterized by a low number
of L0 hits. Only 16% of the simulated atmospheric muons survive the L0 cut.

###  5.3 Selection efficiency

In order to discriminate nuclearite events from the background, variables that
reflect the behavior of nuclearites in the detector were used. Nuclearites are
expected to generate a high number of visible photons at the vicinity of the
detector. This could result in a high number of fired floors with a high
amplitude during a relatively long period. Fig. 4 illustrates the
log${}_{10}(nhits3)/nfloor$ variable and shows the L0 cut effect on both data
and atmospheric muons, for 839 days of livetime. From the right plot, most
nuclearite events for different masses have a high value of this variable
compared to muon events whose distribution stop at around $\sim$$0.015$. The
signal of these particles is also characterised by a long snapshot duration.

Figure 4: Left: Distribution of the discrimination variable
log${}_{10}(nhits3)/nfloor$ without and with the cut L0 $\geq$ 300 on the
total number of PMT hits using 839 days of livetime. The black histogram
corresponds to data, while the blue one refers to simulated atmospheric muons.
Right: The same distribution with the L0 $\geq$ 300 cut for data, simulated
atmospheric muons and simulated and triggered nuclearites with different
masses (other colors).

In order to reduce the background, events are selected based on both
parameters characterizing the signal. Fig. 5 shows log${}_{10}(nhits3)/nfloor$
versus $dt$ for each nuclearite mass. An important fraction of nuclearite
events show a long duration, $dt$, typically higher than $3\times 10^{3}$ ns,
combined with high values of log${}_{10}(nhits3)/nfloor$. The scatter plots in
Fig. 5 show that both variables are well suited to discriminate between
nuclearites and background from cosmic ray muons. By applying the appropriate
cuts, the region dominated by the noise could easily be discriminated from the
region relevant for our signal. However, in order to maximise the performance
of these cuts, an optimisation is required. The methods used to optimise the
cuts are discussed in the next subsection. As a result, the signal to
background discrimination is maximized by removing events according to the
conditions:

$\displaystyle\text{log}_{10}(dt/1\text{ns})<4.125\ .AND.\
\text{log}_{10}(nhits3)/nfloor<0.025$ (8)

Table 1 shows the selection efficiency of the optimised cuts for each
nuclearite mass.

Nuclearite mass (GeV/c2) | $10^{16}$ | $10^{15}$ | $10^{14}$ | $4\times 10^{13}$ | Atm. muons
---|---|---|---|---|---
Trigger efficiency (%) | 97 | 88 | 61 | 25 | 9.5
L0 cut efficiency (%) | 40 | 42 | 31 | 17 | 15
Selection efficiency (%) | 18 | 19 | 25 | 68 | 0
Final efficiency (%) | 6.9 | 7.0 | 4.7 | 2.8 | 0
Table 1: Selection efficiencies as a function of the nuclearite masses at the
different steps discussed in the text. The trigger efficiency in the first row
is computed with respect to the number of simulated events, while the
efficiencies in the following rows are computed with respect to the previous
one. The selection efficiency of atmospheric muons are also reported in the
last column. Figure 5: Scatter plot of log${}_{10}(nhits3)/nfloor$ versus
log${}_{10}(dt{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\text{/1ns}})$
for events with at least 300 L0 hits. Real data in black, atmospheric muons in
blue, and for different nuclearite masses. The red dotted lines represent the
optimised cuts on both discrimination variables.

###  5.4 Optimization of the two variables

In order to achieve the best sensitivity of ANTARES to nuclearites without any
bias, the Model Rejection Factor (MRF) is used [24] to optimise the cuts on
the discrimination variables taking into account the statistical fluctuations
in the simulated atmospheric muons distributions. The MRF technique consists
of varying the cuts with small steps until the minimum of MRF is reached, this
minimum corresponds to the values that give the best sensitivity.

The sensitivity at 90% confidence level (CL), denoted as $S_{90}$, is computed
using the Feldman-Cousins method [25], assuming events with a Poissonian
distribution:

$\bar{\mu}_{\text{90}}=\sum_{n_{\text{obs}}=0}^{\infty}\mu_{\text{90}}(n_{\text{obs}},n_{\text{b}})\times\dfrac{n_{\text{b}}^{n_{\text{obs}}}}{n_{\text{obs}}!}\times
e^{-n_{\text{b}}},$ (9)
$S_{\text{eff}}=\dfrac{n_{\text{Nuc}}}{\Phi_{\text{Nuc}}},$ (10)
$S_{90}=\dfrac{\bar{\mu}_{90}(n_{\text{b}})}{S_{\text{eff}}\times T},$ (11)

where $n_{\text{obs}}$ is the number of observed events and $n_{\text{b}}$ the
number of expected background events from the full dataset. $T$ is the
duration of data taking corresponding to the 2009-2017 period and
$\mu_{\text{90}}(n_{\text{obs}},n_{\text{b}})$ represents the upper limit for
$n_{\text{obs}}$ events and $n_{\text{b}}$ expected backgroung events.
$n_{\text{Nuc}}$ denotes the number of nuclearites remaining after applying
the optimized cuts, and $\Phi_{\text{Nuc}}$ the corresponding flux of
generated nuclearites.

The best values on $dt$ and log${}_{10}(nhits3)/nfloor$ are those minimising
the MRF for each nuclearite mass. Fig. 6 shows an example of the MRF for the
highest nuclearite mass as a function of the selection parameters. The values
that minimise the MRF for the four values of the simulated nuclearite masses
are reported in the first two rows of Table 2. The fraction of event passing
all the selection criteria is reported in the third row, while the number of
remaining background events after the selection cuts and the value of the MRF
are in the rows 4 and 5, respectively.

Figure 6: MRF as function of log${}_{10}(nhits3)/nfloor$ and $dt$ for the
highest nuclearite mass considered ($10^{16}$ GeV/c2).

##  6  Systematic effects and experimental results

###  6.1 Systematic uncertainties

Experimental searches of nuclearites rely on the crossing time of the
particles in the detector and the light yield per unit length of the track, as
given in the De Rújula and Glashow paper [10]. Following [10], in the present
paper the luminous efficiency $\eta$ as derived for transparent water in the
[550-330] nm wavelength range was used. This wavelength region is well within
sensitivity of ANTARES PMTs. The energy distribution of photons follows the
black-body distribution, which for nuclearites arriving in the detector with
masses above the threshold of $4\times 10^{13}$ GeV/c2 has a peak at
wavelengths shorted than 300 nm. The intensity of Cherenkov emission produced
by relativistic particles is proportional to $1/\lambda^{2}$, and also in this
case short wavelengths dominate. When the two different spectra are folded in
our simulations with the water transmission [27] and the quantum efficiency
the PMTs as a function of the wavelength [26], the differences between the two
different spectra in the photoelectron yields on the PMTs are within 30-40%.
The minimum nuclearite mass ($4\times 10^{13}$ GeV/c2) considered in the
analysis ensures that the number of emitted photons per cm are of the order of
106 at each detector depth: as a consequence, the trigger and selection
efficiencies are not affected by the variation in the emission spectra of
visible photons.

The uncertainties for the signal produced by atmospheric muons are derived
from the statistical fluctuations in the signal region for both discrimination
variables ($d$t and log${}_{10}(nhits3)/nfloor$); this is recovered by
extrapolating the distributions using a Landau type function (see ref. [28]
for further details). Systematic uncertainties on the effect of atmospheric
muon events are mainly due to the detector description and to the knowledge on
the environmental parameters. These are mainly related to the uncertainties on
the angular acceptance of the optical modules and on the light absorption and
scattering lengths in the sea water [26, 27]. Hence, an overall of +35% -30%
effect on the expected atmospheric muon rate results from $\pm$15% as a
maximum of uncertainty on the optical module acceptance and $\pm$10% on the
light absorption in water along the entire wavelength spectrum [29]. The
computation of the number of observed events $n_{obs}$ and the number of
expected background events $n_{b}$ given in equation 9 incorporate systematic
uncertainties using the method described in [30].

###  6.2 Unblinding results

After unblinding the remaining 90% of data not used for testing the
optimisation procedure, no event is present in the region beyond the optimised
cuts. Since no event is found, the upper limit on the flux of nuclearites at
90% CL is computed as

$\phi_{\text{90}}=\dfrac{{\mu}_{\text{90}}(n_{\text{obs}},n_{\text{b}})}{S_{\text{eff}}\times
T},$ (12)

where the confidence interval at 90% CL
$\mu_{90}(n_{\text{obs}},n_{\text{b}})$ is computed from the unified approach
of Feldman-Cousins [25]. $S_{eff}$ is defined in equation 10 and $T$ is the
livetime of the analysis.

Nuclearite mass (GeV/c2) | $10^{16}$ | $10^{15}$ | $10^{14}$ | $4\times 10^{13}$
---|---|---|---|---
Best cut on log10(dt/1ns) |  $\geq$ 4.27 |  $\geq$ 4.27 |  $\geq$ 4.27 |  $\geq$ 4.35
Best cut on log${}_{10}(nhits3)/nfloor$ |  $\geq$ 0.029 |  $\geq$ 0.024 |  $\geq$ 0.024 |  $\geq$ 0.024
Remaining nuclearites (%) | 6.9 | 7.0 | 4.7 | 2.8
Remaining background | 1.05 | 0.35 | 0.35 | 0.31
MRF | 1.6$\times 10^{-5}$ | 2$\times 10^{-5}$ | 2.7$\times 10^{-5}$ | 4.12$\times 10^{-5}$
Flux upper limit 90 %CL (cm${}^{-2}\times$ sr${}^{-1}\times$ s-1) | 6.6$\times 10^{-18}$ | 8.1$\times 10^{-17}$ | 1.1$\times 10^{-17}$ | 1.6$\times 10^{-17}$

Table 2: Values of the optimised parameters and the results of the MRF and
upper limits for the different nuclearite masses for 839 days of livetime.

The 90% CL flux upper limit values reported in Table 2 include both
statistical and systematic uncertainties.

Figure 7: Upper limit on the flux of nuclearites with Galactic velocities
($\beta=10^{-3}$) in red line, using 839 days of livetime in the period
2009-2017 with the ANTARES detector. The green line corresponds to a previous
ANTARES result obtained with a smaller data sample [13]. Results from other
experiments, MACRO [19] and SLIM [20], are also shown.

The obtained upper limit (UL) on the nuclearite flux for $T=839$ days of
livetime of the ANTARES detector is shown in Fig. 7. For masses higher than
$10^{16}$ GeV/c2, nuclearites emit more light. Therefore, the limit of the
last test point can be taken as a conservative limit also for higher
nuclearite masses, up to the value of $\sim 10^{17}$ GeV/c2 where detector
saturation effects start to occur.

##  7 Conclusion

A search for nuclearites in the mass range from $4\times 10^{13}$ GeV/c2 to
$10^{16}$ GeV/c2 reaching the ANTARES neutrino telescope using nine years of
data (for an equivalent livetime of 839 days) is presented. Typical Galactic
velocities ($\beta=10^{-3}$) at the surface of the Earth atmosphere have been
considered for this study. After propagation down to the sea level, a large
fraction of events trigger the apparatus and they are discriminated against
the background formed by atmospheric muon bundles. The selection efficiency
increases with increasing nuclearite masses. No events survive the selection
cuts and 90% CL upper limits are derived as a function of the nuclearite mass.
These are the most stringent limits ever set for nuclearites with Galactic
velocities and the first ones set by a neutrino telescope.

## Acknowledgments

The authors acknowledge the financial support of the funding agencies: Centre
National de la Recherche Scientifique (CNRS), Commissariat à l’énergie
atomique et aux énergies alternatives (CEA), Commission Européenne (FEDER fund
and Marie Curie Program), Institut Universitaire de France (IUF), LabEx
UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), Région Île-de-France (DIM-
ACAV), Région Alsace (contrat CPER), Région Provence-Alpes-Côte d’Azur,
Département du Var and Ville de La Seyne-sur-Mer, France; Bundesministerium
für Bildung und Forschung (BMBF), Germany; Istituto Nazionale di Fisica
Nucleare (INFN), Italy; Nederlandse organisatie voor Wetenschappelijk
Onderzoek (NWO), the Netherlands; Executive Unit for Financing Higher
Education, Research, Development and Innovation (UEFISCDI), Romania;
Ministerio de Ciencia, Innovación, Investigación y Universidades (MCIU):
Programa Estatal de Generación de Conocimiento (refs. PGC2018-096663-B-C41,
-A-C42, -B-C43, -B-C44 and refs. PID2021-124591NB-C41, -C42, -C43)
(MCIU/FEDER), Generalitat Valenciana: Prometeo (PROMETEO/2020/019), Grisolía
(refs. GRISOLIA/2018/119, /2021/192) and GenT (refs. CIDEGENT/2018/034,
/2019/043, /2020/049, /2021/023) programs, Junta de Andalucía (ref.
A-FQM-053-UGR18), La Caixa Foundation (ref. LCF/BQ/IN17/11620019), EU: MSC
program (ref. 101025085), Spain; Ministry of Higher Education, Scientific
Research and Innovation, Morocco, and the Arab Fund for Economic and Social
Development, Kuwait. We also acknowledge the technical support of Ifremer, AIM
and Foselev Marine for the sea operation and the CC-IN2P3 for the computing
facilities.

## References

  * [1] A. Bodmer, Collapsed Nuclei, Phys. Rev. D 4 (1971) 1601.
  * [2] E. Witten, Cosmic separation of phases, Phys. Rev. D 30 (1984) 272-285.
  * [3] H. Terazawa, Super-Hypernuclei in the Quark-Shell Model. II , J. Phys. Soc. Japan, 58 (1989), 4388-4393.
  * [4] E. Farhi and R. L. Jaffe, Strange matter, Phys. Rev. D 30 (1984), 2379.
  * [5] A. Bauswein, H.-T. Janka, R. Oechslin, G. Pagliara, I. Sagert, J. Schaffner-Bielich, M. M. Hohle, and R. Neuhäuser, Mass Ejection by Strange Star Mergers and Observational Implications, Phys. Rev. Lett. 103 (2009), 011101.
  * [6] A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn, Baryon structure in the bag theory, Phys. Rev. D 10 ( 1974), 2599.
  * [7] T. DeGrand, R. L. Jaffe, K. Johnson, and J. Kiskis, Masses and other parameters of the light hadrons, Phys. Rev. D 12 (1975), 2060.
  * [8] C. Greiner, A. Diener, J. Schaffner, H. Stoecker, Strange matter — a new domain of nuclear physics, Nucl. Phys. A 566 (1994), 157-165.
  * [9] J. D. Bjorken and L. D. McLerran, Explosive quark matter and the ”Centauro” event, Phys. Rev. D 20 (1979), 2353.
  * [10] A. De Rújula and S. L. Glashow, Nuclearites - a novel form of cosmic radiation, Nature 312 (1984), 734–737.
  * [11] J. Madsen, Strangelet propagation and cosmic ray flux, Phys. Rev. D 71 (2005), 014026.
  * [12] M. Spurio, Searches for Magnetic Monopoles and other Stable Massive Particles, arXiv: 1906.02039 [hep-ph].
  * [13] G. E. Păvălaş [ANTARES Collaboration], Search for nuclearites with the ANTARES neutrino telescope, Proceedings of The 34th International Cosmic Ray Conference — PoS(ICRC2015), 236 (2015), 1060.
  * [14] J. A. Aguilar [ANTARES Collaboration], Performance of the front-end electronics of the ANTARES neutrino telescope, Nucl. Instrum. Meth. A 622 (2010) 59 [arXiv:1007.2549].
  * [15] M. Ageron [ANTARES Collaboration], ANTARES: the first undersea neutrino telescope, Nucl. Instrum. Meth. A 656 (2011), Issue 1, 11-38.
  * [16] S. A. Chin and A. K. Kerman, Possible Long-Lived Hyperstrange Multiquark Droplets, Phys. Rev. Lett. 43, 1292.
  * [17] T. Shibata, Study of Shower Phenomena in the Atmosphere. I: Analytical Derivation of Lateral Structure Function of Hadronic Components, Prog. Theor. Phys. 57 (1977), Issue 3, 882–900.
  * [18] Wan-Lei Guo, Cheng-Jun Xia, Tao Lin, and Zhi-Min Wang, Exploring detection of nuclearites in a large liquid scintillator neutrino detector, Phys. Rev. D 95 (2017), 015010.
  * [19] M. Ambrosio [MACRO Collaboration], Nuclearite search with the MACRO detector at Gran Sasso, Eur. Phys. J. C 13 (2000), 453–458 .
  * [20] S. Balestra et al, Results of the search for strange quark matter and Qballs with the SLIM experiment, Eur. Phys. Jour. C 57 (2008), 525.
  * [21] G. Carminati, M. Bazzotti, A. Margiotta and M. Spurio, Atmospheric MUons from PArametric formulas: a fast GEnerator for neutrino telescopes (MUPAGE), Comput. Phys. Commun. 179 (2008), Issue 12, 915-923.
  * [22] A. Albert [ANTARES Collaboration], Monte Carlo simulations for the ANTARES underwater neutrino telescope, J. Cosmol. Astropart. Phys. (2021).
  * [23] J. A. Aguilar [ANTARES Collaboration], The data acquisition system for the ANTARES Neutrino Telescope, Nucl. Instrum. Meth. A 570 (2007) 107 [astro-ph/0610029].
  * [24] G. C. Hill and K. Rawlins, Unbiased cut selection for optimal upper limits in neutrino detectors: the model rejection potential technique, Astropart. Phys. 19 (2003), Issue 3, 393-402.
  * [25] Gary J. Feldman and Robert D. Cousins, Unified approach to the classical statistical analysis of small signals, Phys. Rev. D 57 (1998), 3873 .
  * [26] P. Amram [ANATRES Collaboration], The ANTARES optical module, Nucl. Instrum. Methods Phys. Res. A: Accel. Spectrom. Detect. Assoc. Equip. 484 (2002), 1–3, 369-383.
  * [27] J. A. Aguilar [ANTARES Collaboration], Transmission of light in deep sea water at the site of the ANTARES, Astropart. Phys. 23 (2005) 131-155.
  * [28] A. Albert et al. (ANTARES Collaboration), Search for magnetic monopoles with ten years of the ANTARES neutrino telescope, JHEAp 34 (2022) 1-8.
  * [29] J. A. Aguilar [ANTARES Collaboration], Zenith distribution and flux of atmospheric muons measured with the 5-line ANTARES detector, Astropart. Phys. 34 (2010) 179-184.
  * [30] J. Conrad, O. Botner, A. Hallgren, and C. Pérez de los Heros Including systematic uncertainties in confidence interval construction for Poisson statistics, Phys. Rev. D 67, 012002 (2003).