# Are multimodal events a sign of the strangelet passage through the matter?

Ewa Gładysz-Dziaduś
e-mail<EMAIL_ADDRESS>
Institute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland
111retired

We investigate the possible connection between the multimodal events (MME)
observed in very high energy extensive air showers by the HORIZON experiment
and the so-called strongly penetrating component observed in the homogenous
lead emulsion chambers of the Pamir and Chacaltaya Experiments. We found that
both experimental observations could be connected one to the other, and could
be the manifestation of the same physical process, i.e. penetration of a
strangelet through the matter. In the first case a strangelet produces the
many-maxima long range cascades observed in the homogenous lead emulsion
chambers. In the second one the successive interactions of a strangelet in the
air are seen in the HORIZON detectors as the consecutive signals. Time
intervals between signals are between several dozen to several hundred
nanoseconds.

##  1 Introduction

The characteristics of very high energy cosmic-ray interactions over $\sim
10^{15}$ eV reveal some anomalies which are hardly explained by the physical
rules known from the hadronic interactions in lower energy accelerator
experiments. The wide spectrum of unusual events, such as for example Centauro
species [1, 2, 3], the so-called strongly penetrating component appearing in
the form of long-range many-maxima cascades [4, 5], multimodal events (MME)
[6, 7, 8], was observed above this threshold. Monte Carlo simulations show
that any kind of statistical fluctuations of ”normal” hadronic interactions
could not produce such anomalies.

In the past years some explanations based on the idea of the QGP and/or
strangelet formation have been proposed for Centauros [9, 10, 11] and for the
strongly penetrating component [12]).

In particular in the paper [12], entitled Is the strongly penetrating
component a sign of the strangelet passage through the matter? we showed that
”the many maxima cascades observed in the thick homogenous lead emulsion
chambers [4, 5] of the Pamir and Chacaltaya experiments can be produced in the
process of a strangelet penetration through the apparatus”. For description
and proposed interpretations of these unusual phenomena see the reviews [13,
14].

In this work we attempt to estimate how could manifest the strangelet passage
through the air and to consider the question of their possible existence in
the present experimental data. Our considerations indicate that the puzzling
observations of the HORIZON experiment, could be explained by the passage of
strangelets through the air. It seems that the mechanism which we proposed and
used to explain the many maxima strongly penetrating cascades observed in the
thick Pb chambers of the Pamir Experiment [12] could be also responsible for
the multimodal events found in the HORIZON detectors.

##  2 The experimental puzzles

###  2.1 The strongly penetrating many-maxima cascades

The strongly penetrating component is one of the most surprising phenomena
observed in the mountain emulsion chamber experiments [4, 5, 12, 13]. This
phenomenon manifests itself by the characteristic energy pattern revealed in
the shower development in the deep chambers (calorimeters) indicating the slow
attenuation and many maxima structures.

The very spectacular examples of such phenomena are exotic cascades which we
have found in the Centauro–like event C–K [4] detected in the deep homogenous
type Pb chamber of the Pamir experiment. It was exposed at the altitude of
4370 m a.s.l. The thickness of the chamber was 60 cm of the lead, what
corresponds to $\sim$ 3.5 interaction lengths for hadrons or $\sim$ 108
cascade unites. The cascades were detected at the X-ray films. The event,
similarly as other Centauro species is the hadron-rich family, consisting of
123 cascades. The energy deposited by the photon cascades and hadron cascades
was $\Sigma E_{\gamma}$ = 306 TeV and $\Sigma E_{h}^{\gamma}$ = 392 TeV
respectively. So, the total energy of the event was $\sim 10^{15}$ eV. The
zenithal angle of the event, calculated as the average zenithal angle of all
registered cascades is $<\theta>=27.4^{0}$. Among them three hadron cascades,
emitted in the very forward rapidity region, seem to be not ”ordinary” ones
and their transition curves exhibit surprising features:

  * •

strongly penetrative nature associated with a very slow attenuation

  * •

appearance of many maxima (11, 5 and 3 humps respectively) satisfactorily
fitted by the individual electromagnetic cascade curves

  * •

small distances between the maxima, being about two times shorter than the
calculated distance for the ”normal” hadron cascades [15])

Transition curves of the longest cascade is shown in Fig. 1.

Figure 1:  Transition curves in X-ray film darkness D (measured in three
diaphragms of a radius R = 48, 84 and 140 $\mu$) for cascade no 197.08. Energy
(in TeV units) liberated into the soft component is indicated at each hump
[4].

This exotic cascade penetrated more than 109 cascade units and 11 maxima
appeared along its transition curve. The average distance between the
neighbouring humps of many-maxima cascades detected in the ”Pamir 76/77”
chamber [4] has been found to be 7.2 cm Pb (i.e. $\sim 0.4\lambda_{int}$ for
hadrons in Pb).

The other examples of such anomalous cascades, found in the deep Pb chambers
have been described in [5]. Simulations, assuming standard models of hadron-
nucleus interactions confirmed the unusual character of these long-penetrating
cascades [16].

###  2.2 Multimodal events

HORIZON experiments are aimed to study spatial and temporal distributions of
the Extensive Air Showers (EAS) of the energy above $\sim 10^{16}$ eV. These
detector systems are located at the Tien Shan Science Station, at the altitude
$\sim$ 3340 meters above the sea level [17, 18].

A sizable number of events detected during all runs of the Horizon experiments
have features different from the ”standard” events. They revealed many maxima
structures seen in several detection stations of the Horizon system [8]. These
events called MME, i.e. multimodal events exhibit the unusual spatial and
temporal structure of pulses:

  * •

Pulses consist of several smaller peaks (called also modes or maxima) and are
separated one to the other by distances from few tens to hundreds of
nanoseconds.

  * •

Separation between the maxima increases with a distance of detectors from the
EAS core.

This phenomenon can not be obtained from simulations based on the standard
models.

Below are shown two examples of the typical unusual events that follow a
multipeak behaviour.

  1. 1.

5-peak MME event [6] registered by the Horizon-10T system on April 6, 2019.

The event is shown in Fig. 2.

Figure 2: Example of the 5-peak multimodal event detected at $R\approx 898$ m
from the axis of the event by the Horizon-10T system on April 6, 2019,
reprinted from [6].

The EAS axis of the event arrived from the Southern direction with the
zenithal angle $\theta=52^{0}$ and it was registered near the detection point
no 7 and closer to the point no 6. Fig. 2 shows pulses registered by 1 m2
scintilator located at the detection point no 9, positioned at the distance R
$\approx$ 898 m from the EAS axis. Five separate pulses can be clearly
distinguished. Tab. 1 shows their main characteristics: the delay of each
pulse from the first one [6] and the resulting time distances $\Delta\tau$
between the successive peaks.

Characteristics of 5-peak MME event
---
Peak no | 1 | 2 | 3 | 4 | 5
Delay [nsec] | 0 | 265 | 720 | 894 | 1493
$\sim\Delta\tau$ [nsec] | - | 265 | 455 | 174 | 599
Table 1:  Characteristics of the MME event from April $6^{th}$, 2018,
Horizon-10T [6].

According to [6], the characteristics of this event are totally different than
these observed as well as in ”usual” experimental and simulated by the CORSIKA
package events. In particular:

     * •

The simulated EAS produces a single pulse with a uniform structure without
breaks. It is in contrary to the multipeak structure of signals in this event.

     * •

The width of the pulse at R $\approx$ 898 m should be $\sim$ 600 ns. The
measured widths of peaks are between 28 ns and 35 ns [6].

     * •

The disk thickness increases and the particle density in the disk decreases
with a distance from the EAS axis. At R $\approx$ 898 m from axis, the average
particle density for simulated events of energy $E_{0}=10^{18}$ eV is below
one particle/m2 [6]. It is in contrary to $\sim$ 173 particles/m2 found in
this event.

In conclusion, the event with such characteristics can not be described by the
standard models used for description of the EAS.

  2. 2.

The MME event [8] detected by the Horizon-10T detector system [18] on Jan.
2016

This system consists of 10 detection stations which are located at different
distances from the central station named ”Center”. Their names and numbers,
and distances R[m] from the central detection point are published in [18]. The
distance of the farest station, named ”Bunker”, from the central detection
point 1 is 1000 m.

The event is shown in Fig. 3. Analysis of signals [8] indicated the following
features of the event:

     * •

The very large signal has been detected at the point 2 and the next in time
signal has been seen by detectors of the station no 5. Peak structures were
not observed in these points. It proves that EAS axis has arrived somewhere
between these points, closer to the point no 2.

     * •

The multimodal structures have been observed in detectors of all remaining
stations. The character of the observed structures strongly depends on the
distance $\Delta R_{ia}$ of the detection point $i$ from the EAS axis $a$.

     * •

The pulses observed in the detection points 1 and 7, located approximately at
the same distance from the EAS axis and further than the point 5, have
complicated structures. However, because of rather small distances of points 1
and 7 from the EAS axis it was difficult to measure the time intervals between
the particular modes [8].

     * •

The pulses from detection points 6 and 4 demonstrate apparent multimodality.
Peaks are separated by hundreds of nanosecs, see Fig. 3. As it is seen in the
Table 2 these points are located far from the point no 2, assumed to be the
EAS axis. However, some of the pulses have a complex shape, what could
indicate that they possibly consist of more than one signal. In a consequence
the true values of $\Delta\tau$ are a little shorter [8].

     * •

The apparent peak structures are observed also at the detection points 8 and
3, see the right picture of Fig. 3. They are located far from the EAS axis and
the successive modes are separated by several hundreds of nanosecs. As the
station no 3 is located at the furthest point from the EAS axis, it is
possible that some modes could be too weak to be detected. In a consequence
only 2 peaks were observed.

So, in our further considerations, we assume that the center of the EAS was
approximately in the point no 2. Taking from [8, 18] coordinates X, Y, Z of
all stations we have calculated the distances $\Delta R_{ia}$ of the detection
point $i$ from the EAS axis $a$ (see Table 2).

The values of time distances $\Delta\tau$ between the consecutive modes,
estimated from Fig. 3 are shown in the Table 2. It is seen that the signals
detected by stations located further from the event axis are more delayed. The
correlation between the length of time intervals $\Delta\tau$ and the time
arrivals of the successive signals is also observed by each station. More
delayed signals have longer $\Delta\tau$.

As it will be described in the next section, such features are in agreement
with the proposed by us explanation of the MME events.

Figure 3: The MME event seen by the HT detector, reprinted from [8]. 3 peaks
in the station no 8 and 2 peaks in the station no 3 have been detected (right
picture). 3 peaks have been detected as well as in the station no 4 and in the
station no 6 (left picture). Time delay $\tau$ of signals and time distances
$\sim\Delta\tau$ between them
---
Station no |  $R_{2i}$ [m] | Peak delay $\tau$ [nsec] |  $\sim\Delta\tau$ [nsec]
4 | 392 | 4375 | 0
4 | 392 | 4750 | 375
4 | 392 | 5100 | 350
6 | 601 | 4900 | 0
6 | 601 | 5050 | 150
6 | 601 | 5250 | 200
8 | 769 | 4800 | 0
8 | 769 | 5000 | 200
8 | 769 | 5350 | 350
3 | 848 | 5900 | 0
3 | 848 | 6300 | 400
Table 2: Delay of signals $\sim\tau$ [nsec] and time distances between the
peaks $\sim\Delta\tau$ [nsec] for detection points 4, 6, 8 and 3 in the MME
event detected on Jan.26 2016 by the HT experiment. Calculated distances of
the station $i$ from the event axis, being estimated to be close to the point
2, $R_{2i}$ are shown.

In general all MME events found by the Horizon detectors follow the same
pattern:

  * •

the large single pulse (that may have a peak structure too fine to be
resolved) seen close to the center of the event

  * •

the complicated structures seen at the intermediate distances from the axis

  * •

apparently separated peaks at the furthest points from the EAS axis

  * •

the longest time distances $\Delta\tau$ between the peaks observed for the
most delayed pulses

Summarizing, the main feature of the unusual MME events is the increase of the
time intervals between the successive peaks with both the distance of the
detection station from the center of the event (event axis) and the time delay
of the pulse relative to the first in time pulse registered by this station.

As it will be explained in the next section, we assume that the earliest in
time peak comes from the highest altitude. It is possibly produced by the EAS
particles born in the first strangelet interaction with the air nuclei. In
such scenario as well as the delay of the successive peaks and the length of
time intervals between them decreases with the increase of the altitude of a
possible strangelet interaction. On the other hand the length of the time
distances $\Delta\tau$ between the successive modes is increasing with the
distance of the detection point from the EAS axis. All these predicted by our
scenario features are observed not only in the above described events but they
are also in accordance with the experimentally observed tendency of other MME
events.

##  3 Proposed explanation

We propose and investigate the hypothesis that the mentioned in the previous
section phenomena are closely related one to the other and the same mechanism
could be responsible for their appearance. Our considerations indicate that
both phenomena could be manifestation of a strangelet passage through the
matter. As well as the strongly penetrating many-maxima cascades observed in
the thick Pb chambers and the multimodal events could be produced in the
process of the successive strangelet interactions with the lead nuclei in the
first case or with the air nuclei in the second one.

###  3.1 Many-maxima cascades as signs of a strangelet passage through the
thick lead chambers

Strangelets are hypothetical droplets of the stable strange quark matter [19,
20, 21]. To answer the question of whether a strangelet could be detectable in
the typical emulsion chambers used in the mountain experiments at Mt.
Chacaltaya and Pamirs and what should be signs of its passage through the
apparatus we have considered the several possible decay [22, 23, 24] and
interaction [25] modes of strangelets and we have simulated their travelling
through the homogenous-type thick lead chambers or tungsten calorimeters. The
method and our results are described in [12, 26]. Our simulations showed [12]
that strangelets could produce in the thick chambers/calorimeters with the
heavy absorbers the long many-maxima transition curves resembling the observed
long-lived cascades.

As we have mentioned in [26] the strong penetration power of some objects
observed in the cosmic ray experiments can be connected with both the small
interaction cross section of a strangelet and a big concentration of its
energy in a narrow region of a phase space. This energy could be liberated
into a conventional particle production in many consecutive interaction (or
evaporation) acts. As the result we proposed the energy deposition pattern in
deep calorimeters as a new strangelet signature [26]. In our opinion this idea
could be extended and applied also for interpretation of the surprising
results of the HORIZON experiments. We suppose that the consecutive
interactions of a ”stable” strangelet with the air nuclei could generate the
multimodal events, observed in the form of separated in time signals.

We call ”stable” strangelets [12, 26] the long-lived objects capable to reach
and pass through the apparatus without decay. As it has been estimated in [23]
the hyperstrange multiquark droplets, having the strange to baryon ratio
$f_{s}$ = 2.2 - 2.6 can be the subject of only the weak leptonic decays and
their lifetime is estimated to be longer than $\tau_{0}\sim 10^{-4}$ sec. Such
long-lived objects should pass without decay not only through the thick Pb
chambers but also, before evaporation into the bundle of neutrons, could
travel the long distances in the atmosphere. While traveling, the ”stable”
strangelet collides and interacts with the matter. Following the
phenomenological indications of [25] we have simulated and described their
possible interactions in the lead chambers [12] and in the thick tungsten
calorimeters [26].

The strangelet was considered as the object of the radius

$R_{str}=r_{0}A^{1/3}_{str}$ (1)

where the rescaled radius

$r_{0}=({\frac{3\pi}{2(1-\frac{2\alpha_{s}}{\pi})[\mu^{3}+(\mu^{2}-m_{s}^{2})^{3/2}]}})^{1/3}$
(2)

$\mu$ and $m_{s}$ are the chemical potential and the mass of the strange quark
respectively and $\alpha_{s}$ is the QCD coupling constant. The mean
interaction path of strangelets in the lead absorber

$\lambda_{s-Pb}=\frac{A_{Pb}\cdot
m_{N}}{\pi(1.12A^{1/3}_{Pb}+r_{0}A^{1/3}_{str})^{2}}$ (3)

Penetrating through the Pb chamber a strangelet collides with the lead nuclei.
In each act of a collision, the ”spectator” part of a strangelet survives (a
strangelet is much more strongly bound than a normal nucleus) continuing the
passage through the chamber. The ”wounded” part is destroyed. The process is
ended, when in the successive interaction the whole strangelet is destroyed.
Particles, generated in the consecutive collision points, interact with the
absorber nuclei in a ”normal” way, resulting in the electromagnetic-nuclear
cascade, developing in the chamber matter.

The general conclusion from our studies was that ”stable” strangelets can
produce in the thick Pb chambers or tungsten calorimeters long range many
maxima cascades. The distances between the successive peaks depend both on the
value of the mean interaction path of the strangelet in the absorber matter
$\lambda_{str}$ and on the value of the mean interaction paths of usual
particles.

As we will show in the next sections the puzzling results of the HORIZON
experiments also support the picture of a ”stable” strangelet passage through
the air layer above the EAS detectors.

##  4 Are many-maxima MME events observed by Horizon experiments signs of a
strangelet passage through the air?

The possibility of appearance of many maxima structures and their character
depends on several factors. The most important are both the absorber
properties and the type of interacting objects. In particular the mean
interaction path of objects interacting with the absorber nuclei should be
long enough to resolve the successive interactions. From the other hand it
must be short enough to give opportunity for several projectile interactions
in the absorber layer. It is the reason that the direct comparison of many
maxima structures observed in the Pamir and in the Horizon experiments is not
possible. In the first case we deal with the heavy absorber (Pb) and in the
second one with the light nuclei of the air. So, at first we should look at
and compare the values of the mean interaction paths of interacting objects in
both kind of the experiments.

###  4.1 Strangelet mean interaction path in the Pb absorber

It is obvious that the mean distances between peaks, observed in the Pamir Pb
chambers, should depend on the interaction length $\lambda_{str}$ of
strangelets passing through the absorber.

Fig. 4 shows $\lambda_{str}$ in the Pb absorber as a function of a strangelet
mass number $A_{str}$. These calculations have been performed for $\alpha_{s}$
= 0, for two different values of the geometrical factor (GEO = 0 or GEO = 1.5)
and for three values of a strangelet quark-chemical potential ($\mu$ = 300,
600 and 1000 MeV). We used the procedure and equations described in the
section 3.1. The curves obtained for $\alpha_{s}$ = 0.3 are very close to
these shown in Fig. 4, so they are not presented here. The mean values of the
experimentally observed distances between the consecutive peaks of many maxima
cascades [4] are marked by the blue straight lines. As it is illustrated in
Fig. 4 the blue lines lie between the calculated $\lambda_{str}$ curves. It
indicates the close relations between the calculated values of $\lambda_{str}$
and the measured distances between the successive peaks of the Pamir many-
maxima cascades. As it was mentioned in the previous section, this statement
has been confirmed in our simulations. Simulating the passage of strangelets
through the thick Pb chambers we have obtained the long many- maxima cascades
resembling the experimentally found unusual events [12].

The observed many maxima structures could arise in the result of a passage of
rather low mass ($Astr<100$) strangelets through the thick Pb chambers [12,
26]. The high mass strangelets could be rather responsible for the phenomenon
called ”halo”. These very high energy events have been observed at the X-ray
films in the form of large area black spots, penetrating through the whole
chamber. In this case it was impossible to separate the individual peaks. For
description of various forms of the observed strongly penetrating component
see our review [13].

So, the assumed scenario seems to be the good direction for understanding of
the origin and the properties of the unusual many-maxima events. Complementing
the conclusions of our earlier works [12, 26] we can say:

  * •

The calculated values of a strangelet mean interaction paths $\lambda_{str}$
in Pb turned out to be appropriate for explanation of the observed values of
distances between the successive peaks of many-maxima cascades.

  * •

The calculated distances between the successive peaks produced by ”standard”
particles are about two times longer [15]. They would be located not only far
from the experimental blue line but also far from the calculated
$\lambda_{str}$ values.

Figure 4: Calculated strangelet mean interaction path $\lambda_{str}$ [cm] in
Pb absorber as a function of a strangelet mass number $A_{str}$, for
$\alpha_{s}$ = 0 and GEO = 1 (left picture) and GEO = 1.5 (right picture). The
mean distance between experimentally observed peaks [4] is shown by the blue
line.

###  4.2 Strangelet mean interaction path in the air

Similarly as in the case of the thick Pb chambers, the idea of a possible
strangelet detection by the EAS stations is based on the expectation that
strangelets should have geometrical radii much smaller than the ordinary
nuclei of the same mass number, and correspondingly, much smaller interaction
cross sections.

There are, however, essential differences between both kind of experiments.
The main difference between the Pamir and the Horizon experiments is that they
detect nuclear electromagnetic cascades developing in different absorbers. In
the Pamir chambers we detect cascades developing in Pb, being very heavy
absorber. The signals detected by the Horizon experiment are produced by the
nuclear and electromagnetic cascades developing in the air. The air is not
only much lighter absorber than Pb but additionally its density changes with
the altitude. Fig. 5 shows $\lambda_{str}$ in the air as a function of a
strangelet mass number $Astr$, calculated for $\alpha_{s}$ = 0.3, GEO = 1 and
1.5, three values of $\mu$ = 300, 600 and 1000 MeV, for Pamir and Tien-Shan
altitudes. Interaction lengths of strangelets $\lambda_{str}$ have been
expressed in centimeters (left pictures) and in nanoseconds (right pictures),
by assuming that strangelets are flying with the time velocity $c$. The mean
distance between the peaks observed in the 5-peak MME event [6] is indicated
by the blue straight line.

Fig. 6 shows the ratios of calculated $\lambda_{str}$ for strangelets passing
through the air to these for strangelets passing through the Pb absorber.

Our calculations indicate that:

  * •

The general behaviour of $\lambda_{str}$ as a function of $Astr$ is similar in
both cases. As well as for strangelets passing through the air and through the
Pb absorbers, $\lambda_{str}$ decreases with an increase of $Astr$ and
increases with $\mu$.

  * •

The calculated $\lambda_{str}$ are longer for GEO = 1 than for GEO = 1.5.
Dependence of $\lambda_{str}$ on $\alpha_{s}$ is very weak.

  * •

The mean interaction paths of strangelets in the air are much longer than
these in Pb, i.e. $\lambda_{str-air}>>\lambda_{str-Pb}$. For $0<A_{str}<1000$
their ratios are in the range $\sim$ 1500-4500. As well as the ratios of cross
sections $\sigma_{str-air}/\sigma_{str-Pb}$ and the ratios of $\lambda_{str-
air}/\lambda_{str-Pb}$ do not depend on the geometrical factor GEO. They also
depend very weakly on the $\alpha_{s}$ value.

  * •

The curves obtained for the Pamir and for the Tien-Shan altitudes are close
one to the other.

  * •

As it is seen from the position of the blue straight lines (Fig. 5) the
experimentally found time distances $\Delta\tau$ between arrivals of the
successive signals are within the calculated values of $\lambda_{str}$ (with
exception of very low mass strangelets). It indicates that the calculated
$\lambda_{str-air}$ are related to the observed values of time distances
between the successive pulses of many-maxima signals detected by the HORIZON
experiment. It is the same conclusion as that drawn from our earlier analysis
of exotic many-maxima cascades registered in the thick Pb chambers.

Figure 5: Calculated $\lambda_{str}$ in the air as a function of a strangelet
mass number $A_{str}$, expressed in centimeters (left pictures) and in
nanoseconds (right pictures). $\alpha_{s}$ = 0.3 and GEO = 1 (upper pictures)
or GEO = 1.5 (lower pictures) were assumed. Comparison of $\lambda_{str}$
calculated for Pamir (solid line) and for Tien - Shan (dotted line) is shown.
The mean time distance $<\Delta\tau>$ between the peaks observed in the 5-peak
MME event [6] is indicated by the blue straight line.

Figure 6: Calculated ratios of $\lambda_{str}$ [cm] in the air to these in the
Pb absorber, at Tien-Shan (left picture) and Pamir (right picture). GEO = 1
has been assumed in both cases.

###  4.3 Comparison of the Pamir Pb chambers and the Horizon Tien Shan
detectors

As it has been already mentioned in the previous subsection the main
difference between both detectors is the kind of absorbers. In the Pamir Pb
chambers the mean interaction path of a strangelet $\lambda_{str-Pb}$ is much
shorter than $\lambda_{str-air}$ in the air being the absorber for the Horizon
detectors.

In a consequence we are able to observe directly each point of a strangelet
interaction with the Pb nucleus and directly watch the strangelet passage
through the Pb chamber. The nuclear electromagnetic cascades, developing in
Pb, after each strangelet interaction are visible at X-ray film detection
layers in the form of black spots, of the radius of only several hundreds of
microns. The distances between the starting points of the successive cascades
are directly connected with the length of strangelet interaction paths
$\lambda_{str-Pb}$. The 60 cm Pb thickness of the chamber corresponds to $\sim
3.5\lambda_{int}$ for nucleons. In the case of strangelets, characterized by
the shorter interaction paths, the chamber is a suitable tool for detection
and for watching of many consecutive strangelet interactions.

In the case of strangelets passing through the air layer the situation is
different. The long $\lambda_{str-air}$ causes that the air cascades
developing between the consecutive strangelet interactions create the EAS with
large as well as longitudinal and lateral sizes. The direct measurements of
distances between the successive strangelet interactions are impossible
because the developing EAS is detected at the only one level - at the Tien-
Shan detection station altitude. However, time distances $\Delta\tau$ between
arrivals of signals from the subsequent interactions are connected with the
mean interaction paths of the objects responsible for their production. The
knowledge of $\lambda_{str-air}$ enables to estimate a time of strangelet
interactions with the air nuclei.

As it is seen from Fig. 5 the experimental $<\Delta\tau>$ value is located
between the calculated curves, what indicates a consistency of assumed by us
scenario with the experimental observations. However, to draw the more
detailed conclusions, a dependence of $\Delta\tau$ on a distance $R$ should be
studied. This question will be discussed in the next section.

At the moment the next essential point should be mentioned. The considered
detectors are located at two different altitudes. The difference of their
altitudes $\Delta H_{Pamir-TS}\approx 100g/cm^{2}$. Expressing $\Delta H$ in a
number of strangelet mean interaction paths $N[\lambda_{str-air}]$ we found
that for $\mu\approx$ 300 - 600 MeV and for strangelets with $Astr\geq 100$
this number is close to the thickness of 60 cm Pb chamber $L_{Pb}$, i.e.
$\Delta H_{Pamir-TS}[\lambda_{str-air}]\approx L_{Pb}[\lambda_{str-Pb}]$. So,
we can expect the similar mean number of the consecutive strangelet
interactions in both detectors. For example, for strangelets with $Astr$ = 100
(500), $\mu$ = 300 MeV and $\alpha_{s}$ = 0.3 the thickness of the Pamir Pb
chamber $L_{Pb}\approx$ 8 (13) $\lambda_{str-Pb}$ and the thickness of the air
layer between Pamir and Tien-Shan stations $\Delta H_{Pamir-TS}\approx$ 6 (13)
$\lambda_{str-air}$.

For details see Figs. 7 and 8.

Fig. 7 shows thicknesses of the 60 cm Pb chamber $L_{Pb}[\lambda_{str}]$
(solid lines) and the 100 g/cm2 air layer between the Pamir and Tien-Shan
stations $L_{air}$[$\lambda_{str}$] (dotted lines), as a function of $Astr$.
The curves calculated for 3 values of a strangelet quark-chemical potentials
are marked by different colours. The solid and the dotted curves of the same
colour are close one to the other what proves a similar thickness of both
detectors.

The ratios of the both absorber thicknesses are shown in Fig. 8. The positions
of the blue straight lines, drawn at
$L_{Pb}[\lambda_{str}]/L_{air}[\lambda_{str}]=1$, show that for reasonable
values of strangelet parameters ($Astr>100$, $\mu\sim 300-600$ MeV) the
thickness of the Pb chamber is close or only a little greater than the
thickness of the air layer between the Pamir and Tien-Shan stations.

For illustration see the sketch drawn in Fig.9.

Figure 7: Thicknesses L [$\lambda_{str}$] of the 60 cm Pb chamber and 100
g/cm2 of the air layer between the Pamir and Tien-Shan stations, expressed in
a number of strangelet mean interaction paths $\lambda_{str}$.

Figure 8: Ratio of thicknesses: 60 cm Pb chamber to 100 g/cm2 of the air layer
between the Pamir and Tien-Shan stations, expressed in $\lambda_{str}$.

So, we can expect that:

  * •

Almost the same kind and the same number of cosmic ray species which is
falling and interacting in the Pamir chambers is reaching the Horizon
detectors

  * •

In the case of the strangelet projectiles with the same characteristics we can
expect almost the same mean number of the consecutive strangelet interactions
in both detectors.

Figure 9: Illustration and comparison of Pamir and Horizon detector abilities
to a cosmic ray strangelet detection

It should be, however, mentioned that the Pb chambers are more suitable for
observation and investigation of low mass strangelets in contrary to the
Horizon detectors which seem to be more appropriate for detection of high mass
strangelets.

###  4.4 Proposed explanation and its correspondence to the experimental data

####  4.4.1 General idea

For explanation of the multimodal events we propose the simplified picture of
”stable” strangelet interactions in the air. The mechanism is the same as that
we used and described in [12] for explanation of the strongly penetrating many
maxima cascades detected in the Pamir Experiment. Fig. 10 illustrates the
idea.

Figure 10: Scheme of successive strangelet interactions in the air responsible
for the appearance of many peak structures in detectors. Only 3 interactions
are drawn for illustration.

In our simplified picture we assume that:

  * •

Each of $n$ signals seen by the Horizon detector located at the same distance
$R$ from the EAS axis comes from the successive strangelet interaction with
the air nuclei. The most delayed peak, marked here by no 1 comes from a
strangelet interaction at the altitude $\lambda_{1}$ above the detection
level, peak no 2 from a strangelet interaction at the altitude
$\lambda_{1}+\lambda_{2}$, peak no 3 from a strangelet interaction at the
altitude $\lambda_{1}+\lambda_{2}+\lambda_{3}$ etc.

  * •

The distances between the successive strangelet interactions are equal to the
values of the corresponding strangelet interaction paths.

$\lambda_{1}\approx\lambda_{2}\approx\lambda_{3}...\approx\lambda_{n}\equiv\lambda\equiv\lambda_{str}$,
where $\lambda$ is the mean value of the strangelet interaction paths for the
detected collisions.

  * •

As well as the strangelet and the resulting nuclear-electromagnetic cascades
fly with the light velocity $c$.

Expressing $\lambda_{str}$ in the time units we can calculate the time
distances $\Delta\tau_{n(n-1)}$ between arrival of signals from the successive
strangelet interactions. For example, if $n$ = 3 we have:

  1. 1.

time distance between interactions at the height $H_{2}=2\lambda$ and
$H_{1}=1\lambda$:
$\Delta\tau_{21}\approx\lambda+\sqrt{(\lambda^{2}+R^{2})}-\sqrt{(2\lambda)^{2}+R^{2}}$

  2. 2.

time distance between interactions at the height $H_{3}=3\lambda_{3}$ and
$H_{2}=2\lambda$:
$\Delta\tau_{32}\approx\lambda+\sqrt{(2\lambda)^{2}+R^{2}}-\sqrt{(3\lambda)^{2}+R^{2}}$

We see that $\Delta\tau$ depends as well as on the interaction height $H$, the
values of the strangelet interaction paths and on the distance $R$ of the
detector from the EAS axis.

The following limit values of $\Delta\tau$ can be expected:

  1. 1.

for $R=0$ or $R<<\lambda$:
$\Delta\tau_{21}\rightarrow 0$, $\Delta\tau_{32}\rightarrow 0$,
$\Delta\tau_{43}\rightarrow 0$ … $\Delta\tau_{n(n-1)}\rightarrow 0$

  2. 2.

for $R>>n\lambda$:
$\Delta\tau_{21}\rightarrow\lambda_{2}$,
$\Delta\tau_{32}\rightarrow\lambda_{3}$,
$\Delta\tau_{43}\rightarrow\lambda_{4}$ …
$\Delta\tau_{n(n-1)}\rightarrow\lambda_{n}\equiv\lambda$

  3. 3.

for $R\approx\lambda$:
$\Delta\tau_{21}\approx\lambda(1+\sqrt{2}-\sqrt{5})\simeq 0.178\lambda$
$\Delta\tau_{32}\approx\lambda(1+\sqrt{5}-\sqrt{10})\simeq 0.073\lambda$
$\Delta\tau_{43}\approx\lambda(1+\sqrt{10}-\sqrt{17})\simeq 0.030\lambda$
….
$\Delta\tau_{n(n-1)}\rightarrow 0$ when $n\rightarrow\infty$
i.e. $\Delta\tau$ is decreasing with the increase of the successive strangelet
interaction altitude.

In this picture the first detected signal comes from the first strangelet
interaction, i.e. the interaction at the highest altitude. The last signal
comes from the strangelet interaction at the height $H\approx\lambda_{n}$ from
the detection level. In accordance with the experimental observations, time
distances $\Delta\tau$ between the successive signals are decreasing with the
increase of the strangelet interaction altitude (i.e. $\Delta\tau$ are the
longest for the most delay signals, coming from interactions close to the
detector level). This effect is additionally enhanced by the increase of the
$\lambda_{str}$ value with the decrease of $Astr$ in each of the successive
interaction.

The experimental MME events, shown in the section 2.2 have been detected at
rather long distances $R$ from the EAS axis. $R$ in the range of $\sim$
300-1000 m from the EAS axis means that we are somewhere between situations
described in the points 2 and 3. For the 5-peak MME event (Fig. 2) detected at
the very long distance ($R\simeq 898$ m) from the EAS axis we can expect that
the criterion 2 ($R>>\lambda$) is fulfilled, so the observed $\Delta\tau$ are
close to the $\lambda_{str}$ values (see the blue line at Fig. 5). For
explanation of the events detected closer to the EAS axis the situation
described in the point 3 ($R\simeq\lambda$) is more appropriate.

The assumed picture allows to understand the strange behaviour of MME events.
The most important statement is that the multipeak structure of signals could
be the result of the successive strangelet interactions in the air. The
observed increase of the time intervals $\Delta\tau_{n(n-1)}$ between two
successive signals is caused by two factors: the above described geometry and
additionally by a dependence of $\lambda_{str}$ on $A_{str}$. As it has been
shown in the section 4.2, $\lambda_{str}$ is longer for the low mass
strangelets. As the mass number of a strangelet $A_{str}$ is decreasing in
each act of interaction, $\lambda_{str}$ is increasing with the number of
collisions. It directly results in elongation of the measured time distances
between two successive signals $\Delta\tau_{n(n-1)}$ with the time of their
arrival.

Figs. 11 and 12 summarized the predicted behaviour of $\Delta\tau_{n(n-1)}$
distances.

Fig. 11 shows the time distances between signals coming from strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$,
3$\lambda_{str}$ and 2$\lambda_{str}$, 4$\lambda_{str}$ and 3$\lambda_{str}$,
5$\lambda_{str}$ and 4$\lambda_{str}$, 10$\lambda_{str}$ and 9$\lambda_{str}$
above the detector level, as a function of $A_{str}$. Calculations have been
performed for $\alpha_{s}$ = 0.3, GEO = 1 and $R$ = 1000 m from the EAS axis.
Two upper pictures have been performed for $\mu$ = 300 MeV and 600 MeV
respectively and the lower one for $\mu$ = 1000 MeV. For all $\mu$ the
increase of $\Delta\tau$ with a decrease of as well as $Astr$ and
$\Delta\tau_{n(n-1)}$ indices is predicted. Keeping in mind that $\Delta\tau$
indices mean the height of the possible strangelet interactions (expressed in
a number $\lambda_{str}$), it directly proofs the increase of $\Delta\tau$
with the time arrival of signals to the detector.

Figure 11:  $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$,
3$\lambda_{str}$ and 2$\lambda_{str}$, 4$\lambda_{str}$ and 3$\lambda_{str}$,
5$\lambda_{str}$ and 4$\lambda_{str}$, 10$\lambda_{str}$ and 9$\lambda_{str}$
above the detector level. Calculations have been performed for $\alpha_{s}$ =
0.3, GEO = 1 and $R$ = 1000 m from the EAS axis, as a function of a strangelet
mass number $Astr$. Two upper pictures have been performed for $\mu$ = 300 MeV
and 600 MeV respectively and the lower one by assuming $\mu$ = 1000 MeV.

Fig. 12 shows $\Delta\tau$ distances between signals coming from strangelet
interactions at several different heights above the detector level, calculated
for $\alpha_{s}$ = 0, GEO = 1.5, three $\mu$ values (300, 600 and 1000 MeV)
and $R$ = 1000 m from the EAS axis. Comparison with the calculated
$\lambda_{str}$ is also presented. Also this figure supports the statement
about the increase of $\Delta\tau$ values with the arrival time of the
successive signals to the detector and confirms the weak dependence of this
effect on the assumed values of parameters. It is also important to note that
for such long distances of the detector from the EAS axis (R = 1000 m)
$\Delta\tau$ values lie very close to (or between) the calculated
$\lambda_{str}$ curves.

Figure 12:  $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$,
3$\lambda_{str}$ and 2$\lambda_{str}$, 4$\lambda_{str}$ and 3$\lambda_{str}$,
5$\lambda_{str}$ and 4$\lambda_{str}$, 10$\lambda_{str}$ and 9$\lambda_{str}$
above the detector level (dotted lines). Calculations have been performed for
$\alpha_{s}$ = 0, GEO = 1.5 and $R$ = 1000 m from the EAS axis, as a function
of a strangelet mass number $Astr$. Comparison with the calculated
$\lambda_{str}$ (solid lines) is shown.

Summarizing our investigation of $\Delta\tau$ behaviour we can conclude that
the proposed scenario predicts the experimentally observed behaviour of
unusual MME events. $\Delta\tau$ are longer for the signals coming from
interactions closer to the detector level than these coming from the high
altitudes, i.e.
$\Delta\tau_{21}>\Delta\tau_{32}>\Delta\tau_{43}>...\Delta\tau_{89}$ etc.

####  4.4.2 Correspondence to the detected events

We have checked the correspondence of the proposed explanation to the
experimental data comparing the characteristics of two MME events described in
the section 2.2 with the predictions of our model. The comparison is
illustrated in Figures 13-17.

Our purpose was as well as to investigate the general correspondence of the
experimental data to the theoretical predictions and, in particular, to study
a possible connection between the multimodal pictures obtained from different
detection stations.

5-peak MME event published in [6]

The event has been already described in the section 2.2. It is shown in Fig. 2
and its characteristics are summarized in the Table 1.

We have compared the experimental $\Delta\tau$ values with the calculated ones
for two series of parameters.

Fig. 13 shows $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$,
3$\lambda_{str}$ and 2$\lambda_{str}$, 4$\lambda_{str}$ and 3$\lambda_{str}$,
5$\lambda_{str}$ and 4$\lambda_{str}$ above the detector level, calculated for
three values of $\mu$ (300, 600 and 1000 MeV) and for $R=898$ m from the EAS
axis. Calculated $\lambda_{str}$ are also presented. $\alpha_{s}$ = 0 and GEO
= 1.5 have been assumed. Experimental values of $\Delta\tau$ for 5-peak MME
event [6], have been shown by the violet straight lines. Because of
fluctuations we can not, of course, expect the direct correspondence between
the features of single experimental events and the theoretical predictions. It
is seen, however, that for this event the experimental $\Delta\tau$ values are
within the calculated $\lambda_{str}$ curves. Even more, intersections of the
violet lines with the calculated $\Delta\tau_{21}$ at $Astr\approx 20$ and
$\Delta\tau_{32}$ at $Astr\sim 230$ (marked by stars) suggest that the event
could arise from the interaction of a strangelet with $Astr\geq 230$ (and
$\mu\geq 300$ MeV) at the altitude of $\sim 3\lambda_{str}$ resulting into a
strangelet characterized by $Astr\leq 20$ and $\mu\leq$ 600 MeV which
interacted at the altitude of $\sim 2\lambda_{str}$ above the detector level.
The observed increase of the strangelet mass number $Astr$ with the altitude,
i.e. a decrease of $Astr$ with the number of the successive collisions is in
accordance with the proposed scenario.

The experimental $\Delta\tau_{43}$ and $\Delta\tau_{54}$ values are, however,
far from the calculated $\Delta\tau$ values. There are several possible
explanations of such observation. High $\Delta\tau_{43}$ and $\Delta\tau_{54}$
values could be caused simply by fluctuatios, by a very massive ($Astr>1000$)
strangelet interacting at the altitudes of $\sim 4\lambda_{str}$ and $\sim
5\lambda_{str}$ above the detector level or by the additional, very weak
interactions, with the signal below the detection threshold, at the altitude
$\sim(3-5)\lambda_{str}$.

Figure 13:  $\Delta\tau$ distances between signals coming from strangelet
interactions at the altitudes of 2$\lambda_{str}$ and 1$\lambda_{str}$ ,
3$\lambda_{str}$ and 2$\lambda_{str}$ , 4$\lambda_{str}$ and 3$\lambda_{str}$
, 5$\lambda_{str}$ and 4$\lambda_{str}$ above the detector level and
calculated for $R$ = 898 m from the EAS axis, as a function of a strangelet
mass number $Astr$ are shown by the dotted lines. Calculated $\lambda_{str}$
are shown by the solid lines. Colours: black, red and green correspond to
$\mu$ = 1000, 600 and 300 MeV respectively. $\alpha_{s}$ = 0 and GEO = 1.5
have been assumed. Experimental values of $\Delta\tau$ for 5-peak MME event
[6], have been shown by the violet straight lines. Intersections of the
experimental $\Delta\tau_{21}$ and $\Delta\tau_{32}$ values with the
calculated $\Delta\tau$ curves are marked by stars.

Fig. 13 illustrates the possible explanations of the 5-peak MME event by
comparison of the experimental $\Delta\tau$ values with these calculated for
GEO = 1.5 and $\alpha_{s}$ = 0. The general interpretation of this figure is
consistent with that resulting from Fig. 14, obtained by assuming GEO = 1 and
$\alpha_{s}$ = 0.3. In both cases the experimental $\Delta\tau_{21}$ and
$\Delta\tau_{32}$ values are in agreement with the theoretical expectations.
Both $\Delta\tau_{21}$ values could be explained by the successive strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$ above the
detector level, and $\Delta\tau_{32}$ values by the strangelet interactions at
the heights of 3$\lambda_{str}$ and 2$\lambda_{str}$. Independently on the
values of parameters which we used in calculations, the estimated $Astr$
corresponding to $\Delta\tau_{32}$ is greater than $Astr$ for
$\Delta\tau_{21}$. The possible estimated $Astr$ values are: $Astr\sim 20-120$
for $\Delta\tau_{21}$ and $\mu\approx$ 600-1000 MeV, $Astr\approx 230-920$ for
$\Delta\tau_{32}$ and $\mu\approx$ 300-600 MeV.

For both series of parameters the straight lines showing the experimental
$\Delta\tau_{43}$ and $\Delta\tau_{54}$ values are above the calculated ones.
This observation could be explained by fluctuations or the appearance of the
additional undetected strangelet interactions.

Figure 14: $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$,
3$\lambda_{str}$ and 2$\lambda_{str}$, 4$\lambda_{str}$ and 3$\lambda_{str}$
or 5$\lambda_{str}$ and 4$\lambda_{str}$ above the detector level and
calculated for $R$ = 898 m from the EAS axis, as a function of a strangelet
mass number $Astr$. $\alpha_{s}$ = 0.3, GEO = 1 have been assumed. In the
upper figure only curves for $\mu$ = 300 MeV and 600 MeV have been shown.
Experimental $\Delta\tau$ values for 5-peak event [6], have been shown by the
straight lines in the same colour as the corresponding $\Delta\tau$.

Summarizing, we can say that the experimental characteristics of this event
support the proposed scenario. The theoretically predicted decrease of
$\Delta\tau$ with the increase of as well as $Astr$ and the height of a
strangelet interaction is observed.

MME event published in [8]

As it was described in the section 2.2 (see Fig. 3), the multimodal pattern of
this event was observed in several stations. The most apparently seen peak
structures have been detected in stations: 8, 6 and 4.

The possible explanation of peaks detected by the station 8, located at the
distance $R\approx$ 769 m from the event axis is shown in Fig. 15. Calculated
$\Delta\tau$ distances between signals coming from strangelet interactions at
several different heights above the detector level, as a function of a
strangelet mass number $Astr$ are shown. Calculations have been performed for
$\alpha_{s}$ = 0, GEO = 1, $\mu$ = 300, 600 and 1000 MeV, and $R$ = 769 m from
the EAS axis. Experimental $\Delta\tau$ values are shown by the straight lines
of the same colour as the corresponding $\Delta\tau$ curves. Intersections of
the experimental lines with the calculated curves are marked by stars.

Assuming the values of distances between peaks $\Delta\tau_{21}\simeq$ 200 ns
and $\Delta\tau_{32}\simeq$ 350 ns we find that this event could be explained
by the following scenario: a strangelet with Astr $\geq$ 570 and $\mu\geq$ 300
MeV interacts at the altitude $H\approx 3\lambda_{str}$ and the new born
strangelet with Astr $\geq$ 180 and $\mu\sim$ 300 MeV interacts at the
altitude $H\approx 2\lambda_{str}$.

Figure 15:  $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 2$\lambda_{str}$ and 1$\lambda_{str}$ (black),
3$\lambda_{str}$ and 2$\lambda_{str}$ (violet), 4$\lambda_{str}$ and
3$\lambda_{str}$ (red), 5$\lambda_{str}$ and 4$\lambda_{str}$ (blue), 10
$\lambda_{str}$ and 9$\lambda_{str}$ (green) above the detector level, as a
function of a strangelet mass number $Astr$. Calculations have been performed
for $\alpha_{s}$ = 0, GEO = 1, $\mu$ = 300, 600 and 1000 MeV, and $R$ = 769 m
from the EAS axis. Experimental $\Delta\tau$ values are shown by the straight
lines with te same colour as the corresponding $\Delta\tau$ curve.
Intersections of curves are marked by stars.

Fig. 16 shows possible explanation of the MME signals detected by the station
no 6. As the amplitude of the signals is very low it is possible that the last
weak signal was lost [8]. In that case the observed time distances between
peaks correspond to $\Delta\tau_{32}\approx$ 200 MeV and
$\Delta\tau_{43}\approx$ 150 MeV.

As it is seen from Fig. 16 the red straight line showing the experimental
value of $\Delta\tau_{32}$ lies between the corresponding theoretical curves
(for $\mu$ = 300, 600 and 1000 MeV) in the total investigated range (0 $<$
Astr $<$ 1000). So, we can only say that the experimental $\Delta\tau_{32}$
value is in agreement with the theoretical predictions but it is impossible to
precise the value of the interacting strangelet mass number $Astr$. It should
be noted, however, that the theoretical $\Delta\tau_{32}$ line, corresponding
to $\mu$ = 1000 MeV, is the closest to the experimental one. So, assuming that
it gives the best description of the data and knowing the intersection point
of both lines we can expect that the mass number of the strangelet interacting
at the altitude of $\sim 3\lambda_{str}$ above the detector level $Astr\approx
580$.

The same problem concerns also the interpretation of $\Delta\tau_{43}$ line.
The violet straight line showing the experimental value of $\Delta\tau_{43}$
lies between the corresponding theoretical curves (for $\mu$ = 300, 600 and
1000 MeV) in the total investigated range (0 $<$ Astr $<$ 1000). In this case,
we are only able to say that the experimental $\Delta\tau_{43}$ values are in
agreement with the theoretical predictions and we are not able to estimate how
big the strangelet interacting at the altitude of $4\lambda_{str}$ could be.

Figure 16:  $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 5$\lambda_{str}$ and 4$\lambda_{str}$,
4$\lambda_{str}$ and 3$\lambda_{str}$ 3$\lambda_{str}$ and 2$\lambda_{str}$,
2$\lambda_{str}$ and 1$\lambda_{str}$, above the detector level, as a function
of a strangelet mass number $Astr$. Calculation have been performed for
$\alpha_{s}$ = 0, GEO = 1, $\mu$ = 300, 600 and 1000 MeV, and $R$ = 601 m from
the EAS axis. Experimental $\Delta\tau$ values are shown by the straight lines
with te same colour as the corresponding $\Delta\tau$ curves. Intersections of
curves are marked by stars.

Summarizing, the results obtained in the stations 8 and 6, could be explained
by the following picture. A big strangelet interacts at the altitude of $\sim
4\lambda{str}$ above the detector level. The successive interactions of the
new born strangelets with Astr $\geq$ 580 and Astr $\geq$ 200 occur at the
altitude $H\sim 3\lambda_{str}$ and $\sim 2\lambda_{str}$ respectively (see
Fig. 16).

Fig. 17 shows calculated $\Delta\tau$ predicted for the station no 4. As this
station is close to the event axis (R = 392 m) all predicted $\Delta\tau$
values are shorter than 300 nanosecs. Three apparent peaks detected at the
station 4 are separated by much longer distances (see Fig. 3). However, as it
has been already mentioned by authors of [8] the pulses have a complex shape,
what indicates that they consist of more than one signal. As their shapes are
too complicated to separate individual pulses it is impossible to compare this
data with the theoretical predictions.

Figure 17:  $\Delta\tau$ distances between signals coming from strangelet
interactions at the heights of 10$\lambda_{str}$ and 9$\lambda_{str}$,
5$\lambda_{str}$ and 4$\lambda_{str}$, 4$\lambda_{str}$ and 3$\lambda_{str}$
3$\lambda_{str}$ and 2$\lambda_{str}$, 2$\lambda_{str}$ and 1$\lambda_{str}$,
above the detector level, as a function of a strangelet mass number $Astr$.
Calculation have been performed for $\alpha_{s}$ = 0.3, GEO = 1, $\mu$ = 300,
600 and 1000 MeV, and $R$ = 392 m from the EAS axis.

Summarizing our attempt to explain these two MME events by means of the
strangelet scenario we can say:

  * •

Because of fluctuations it is of course difficult to draw the final conclusion
from analysis of the individual events.

  * •

However, the features of the analysed individual events do not contradict the
assumed strangelet scenario. They are consistent with such explanation.

  * •

The average behaviour of other detected MME events is in agreement with the
proposed model. In particular such MME features as the appareance of multipeak
structures and their dependence on the distance from the detected stations are
expected by our strangelet scenario.

  * •

Proposed explanation of the MME events does not depend in essential way on the
values of parameters assumed in the calculations.

  * •

Further research and simulations are desirable.

##  5 Summary

The present work is continuation of our earlier investigations [11] concerning
the possible connection between the strongly penetrating cascades and the
strangelets. As it has been shown in our previous studies the long-range many-
maxima cascades observed in the lead chambers of the Pamir and Chacaltaya
Experiments could be born during a strangelet passage through the apparatus.
The data coming from the mountain experiments, however, are not sufficient to
decide, if the exotic cascades are produced by stable or unstable strangelets.
In both cases it was possible to get by simulations the transition curves
resembling the experimental ones.

In this work we checked the hypothesis that the delay signals detected by the
Horizon experiments are produced by the successive strangelet interactions in
the atmosphere. Our results indicate that the extraterrestial strangelets
penetrating deeply into the atmosphere and degradated in the successive
interactions with the air nuclei could be responsible for the observed
multimodal events. Further studies and simulations are required.

Acknowledgments

I would like to thank Dr Dmitriy Beznosko for sending some papers concerning
the unusual events found by the HORIZON-T experiments and for giving me the
permission to use the published there data in my work. Thank you for careful
reading of my manuscript and for many essential remarks allowing for better
understanding of the HORIZON-T experimental results. I am especially grateful
for the helpful discussions, for pointing out many problems with the data
interpretation and for comments and questions, which hopefully, will help to
solve the mystery of the Chacaltaya, Pamir and Horizon-T exotic events,
possibly by means of the same scenario.

## References

  * [1] G.M.G. Lattes, Y. Fujimoto and S. Hasegawa, Phys. Rep.65 (1980) 151;
S. Hesegawa, ”Centauro Species in Cosmic-Ray Observation”, talk at FNAL CDF
Seminar (1987);

  * [2] A. S. Borisov et al.,Phys. Lett. B190 (1987) 226;
  * [3] S. Hasegawa and M. Tamada, Nucl. Phys. B474(1996) 141;
  * [4] Z. Buja, E. Gładysz, S. Mikocki, M. Szarska and L. Zawiejski, Pamir Collaboration Workshop, Cedzyna (Poland), 1980, Uniwersytet Łódzki;
Z. Buja, E. Gładysz, S. Mikocki, M. Szarska and L. Zawiejski, Proc. 17th ICRC
(Paris 1981) Vol.11, p.104;

  * [5] T. Arisawa et al., Nucl. Phys. B424 (1994) 241;
  * [6] R. Beisembaev et al., Intern. Conf. of Cos. Rays (ICRC 2019), PoS (2020) 195;
  * [7] D. Beznosko for HORIZON-T Coll., J. Phys. Conf. Ser. 1342 (2020) 1, 012007, contribution to TAUP-2017;
  * [8] D. Beznosko et al., EPJ Web Conf. 145 (2017) 14001, contribution to ISVHECRI 2016;
  * [9] J. D. Bjorken and L. D. McLerran, Phys. Rev. D20 (1979) 2353;
K. Goulianos, Commun. Nucl. Part. Phys. 17 (1987) 195;
J. D. Bjorken, K. L. Kowalski and C. C. Taylor, SLAC - PUB - 6109 (1993);

  * [10] A. D. Panagiotou et al., Z. Phys. A333 (1989) 335;
A. D. Panagiotou et al., Phys. Rev. D45 (1992) 3134;

  * [11] M. N. Asprouli, A. D. Panagiotou and E. Gładysz-Dziaduś, Astroparticle Phys. 2 (1994) 167;
E. Gładysz-Dziaduś and A. D. Panagiotou, Intern. Symposium on Strangenss and
Quark Matter, Krete (1994), World. Scientific, ed’s G. Vassiliadis et al.,
p.265;

  * [12] E. Gładysz-Dziaduś and Z. Włodarczyk, J. Phys. G: Nucl. Part. Phys. 23 (1997) 2057-2067;
  * [13] E. Gładysz-Dziaduś, Physics of Particles and Nuclei: 34 no 3 (2003) 565-678;
  * [14] J. Kempa, B. Pattison, E. Gładysz-Dziaduś, L.W. Jones, R. Mukhamedshin M. Tamada and Z. Wlodarczyk, Central Eur.J.Phys. 10 (2012) 723-741;
  * [15] A.A. Bielyayev, F.P. Iwanienko et al., it Electromagnetic Cascades in Cosmic Rays of Ultra High Energy (in Russian), Moscow, 1980;
  * [16] M. Tamada and V.V. Kopenkin Nucl. Phys. B494 (1997) 3;
M. Tamada and A. Ohsawa, Nucl. Phys., Proc. Suppl. 97B (2001) 146;

  * [17] R. Beisembaev et al., 39th Inter. Conf. on High Energy Physics, PoS (ICHEP2018) 208;
  * [18] R. Beznosko et al., arxiv:physics.ins-det/1803.08309;
  * [19] A. R. Bodmer, Phys. Rev. D4 (1971) 1601;
  * [20] H. Terazawa, INS-Report-336 (INS, Univ. of Tokyo) May, 1979;
H. Terazawa, J. Phys. Soc. Jpn. 58 (1989) 3555;
H. Terazawa, J. Phys. Soc. Jpn. 58 (1989) 4388;
H. Terazawa, J. Phys. Soc. Jpn. 59 (1990) 1199;

  * [21] E. Witten, Phys. Rev. D30 (1984) 272;
  * [22] M. S. Berger and R. L. Jaffe, Phys. Rev. C35 (1987) 213;
  * [23] S. A. Chin and A. K. Kerman, Phys. Rev. Lett. 43 (1979) 1292;
  * [24] C. Greiner and H. Stocker, Phys. Rev. D44 (1991) 3517;
J.Madsen, Intern. Symposium on Strangeness and Quark Matter, Krete (1994),
World Scientific, ed’s G. Vassiliadis et al., p.191;

  * [25] G. Wilk and Z. Włodarczyk, J. Phys. G22 (1996) L105;
G. Wilk and Z. Włodarczyk, Heavy Ion Physics 4 (1996) 395;

  * [26] A.L.S. Angelis, J. Bartke, E. Gładysz-Dziaduś, Z. Wlodarczyk, Eur.Phys.J.direct 1 (1999) 9.