# Strange fireball as an explanation of the muon excess in Auger data

###### Abstract

We argue that ultrahigh energy cosmic ray collisions in the Earth’s atmosphere can probe the strange quark density of the nucleon. These collisions have center-of-mass energies , where is the nuclear baryon number. We hypothesize the formation of a deconfined thermal fireball which undergoes a sudden hadronization. At production the fireball has a very high matter density and consists of gluons and two flavors of light quarks (). Because the fireball is formed in the baryon-rich projectile fragmentation region, the high baryochemical potential damps the production of and pairs, resulting in gluon fragmentation mainly into . The strange quarks then become much more abundant and upon hadronization the relative density of strange hadrons is significantly enhanced over that resulting from a hadron gas. Assuming the momentum distribution functions can be approximated by Fermi-Dirac and Bose-Einstein statistics, we estimate a kaon-to-pion ratio of about and expect a similar (total) baryon-to-pion ratio. We show that, if this were the case, the excess of strange hadrons would suppress the fraction of energy which is transferred to decaying ’s by about 20%, yielding a enhancement of the muon content in atmospheric cascades, in agreement with recent data reported by the Pierre Auger Collaboration.

###### pacs:

96.50sd, 13.85.Tp, 24.85.+p## I Introduction

Ultrahigh-energy () cosmic rays provide a formidable beam to study particle collisions at center-of-mass energies and kinematic regimes not accessible at terrestrial accelerators. The incident cosmic radiation interacts with the atomic nuclei of air molecules and produces air showers which spread out over large areas. If the primary cosmic ray is a baryon, hundreds to thousands of secondary particles are usually produced at the interaction vertex, many of which also have energies above the highest accelerator energies Anchordoqui:1998nq (). These secondary products are of course intrinsically hadrons. Generally speaking, by extrapolating final states observed at collider experiments, we can infer that, for collisions at center-of-mass energy , the jet of hadrons contains about 75% pions (including 25% ’s, in accord with isospin invariance), 15% kaons, and 10% nucleons GarciaCanal:2009xq ().

During the shower evolution, the hadrons propagate through a medium
with an increasing density as the altitude decreases and the
hadron-air cross section rises slowly with energy. Therefore, the
probability for interacting with air before decay increases with
rising energy. Moreover, the relativistic time dilation increases the
decay length by a factor , where and are the
energy and mass of the produced hadron. When the ’s (with a
lifetime of ) do decay promptly to
two photons, they feed the electromagnetic component of the
shower. For other longer-lived mesons, it is instructive to estimate
the critical energy at which the chances for interaction and decay are
equal. For a vertical transversal of the atmosphere, such a critical
energy is found to be: ,
, , Gondolo:1995fq (). The
dominant branching ratios are to , to
, to , and to , whereas those of the are to , to , and for we have , , , Olive:2016xmw (). With these figures in mind, to a first
approximation it seems reasonable to assume that in each generation of
particles about 25% of the energy is transferred to the
electromagnetic shower, and all hadrons with
energy interact rather than decay, continuing
to produce the hadronic shower.^{1}^{1}1The electromagnetic
shower fraction from pions only is less than 25%, but simulations show that inclusion of
other hadronic resonances brings the electromagnetic shower fraction
up to about 25% Aab:2016hkv (). We take 25% as a reasonable
estimate of the energy transfer to the electromagnetic shower.
Eventually, the electromagnetic cascade dissipates around 90% of the
primary particle’s energy and the remaining 10% is carried by muons
and neutrinos.

As the cascade process develops in the atmosphere, the number of particles in the shower increases until the energy of the secondary particles is degraded to the level where ionization losses dominate. At this point the density of particles starts to decline. The number of particles as a function of the amount of atmosphere penetrated by the cascade ( in ) is known as the longitudinal profile. A well-defined peak in the longitudinal development, , occurs where the number of in the electromagnetic shower is at its maximum. increases with primary energy, as more cascade generations are required to degrade the secondary particle energies. Evaluating is a fundamental part of many of the composition analyses done when studying air showers. The generic shower properties can be qualitatively well understood using the superposition principle, which states that a shower initiated by a nucleus with nucleons and energy behaves to a good approximation as the superposition of proton showers with initial energy Anchordoqui:2004xb (). This phenomenological assumption relies on the fact that the effect of nuclear binding must be negligible at extremely high energies. Thus, for a given total energy , showers initiated by a heavy nucleus have smaller than proton induced showers.

The integrated longitudinal profile provides a calorimetric measurement of the energy of the primary cosmic ray, with a relatively small uncertainty due to the correction for energy lost to neutrinos and particles hitting the ground. The characteristics of the cascade depend dominantly on the elasticity (fraction of incoming energy carried by the leading secondary particle) and the multiplicity of secondary particles in the early, high-energy interactions. Modeling the development of a cosmic-ray air shower requires extrapolation of hadronic interaction models tuned to accommodate LHC data d'Enterria:2011kw (). Not surprisingly, such extrapolation usually leads to discrepancies between measured and simulated shower properties. The hadronic interaction models are further constrained by independent measurements of and the density of muons at from the shower core Ulrich:2010rg (). The mean is primarily sensitive to the cross-section, elasticity, multiplicity, and primary mass. The mean is primarily sensitive to the multiplicity, the energy fraction (the fraction of incident energy carried by ’s in hadronic interactions), and the primary mass.

Over the past few decades, it has been suspected that the number of
registered muons at the surface of the Earth is by some tens of
percentage points higher than expected with extrapolations of existing
hadronic interaction
models AbuZayyad:1999xa (); Aab:2014pza ().^{2}^{2}2See, however,
Fomin:2016kul (). Very recently, a study from the Pierre Auger
Collaboration ThePierreAuger:2015rma () has strengthened this
suspicion, using a novel technique to mitigate some of the measurement
uncertainties of earlier methods Aab:2016hkv (). The new analysis
of Auger data suggests that the hadronic component of showers (with
primary energy ) contains about
to more muons than expected. The significance of the
discrepancy between data and model prediction is somewhat above .

Changing the energy fraction or suppressing decay are
the only modifications which can be used to increase without
coming into conflict with the
observations Farrar:2013sfa ().^{3}^{3}3We note in passing that
muons may be able to escape the shower core before reaching the
Earth surface if their production angle is increased by boosting the
distribution. Since the muon lateral distribution function is
a steeply falling function of the radius, a larger production angle
would increase . However, as shown in Farrar:2013sfa (),
the measured zenith angle dependence of the ground signal vetoes the
correlated flattening of the muon lateral distribution function
necessary to accommodate Auger data. Several new physics models
have been proposed exploring these two
possibilities Farrar:2013sfa (); Allen:2013hfa (); AlvarezMuniz:2012dd (). In
this paper we adopt a purely phenomenological approach to develop an
alternative scheme. In sharp contrast to previous models, our proposal
is based on the assumption that ultrahigh energy cosmic rays are heavy
(or medium mass) nuclei. Our work builds upon some established
concepts, yet contravenes others.

We conceive the production and separation of strangeness in a baryon-rich Centauro-like fireball before its spontaneous explosive decay Halzen:1985eh (); Panagiotou:1989zz (); Panagiotou:1991kv (); Asprouli:1994ch (); Angelis:2003zn (). At production the fireball has a very high matter density and consists of gluons and two flavors of light quarks (). Because the fireball is formed in the baryon-rich projectile fragmentation region, the high baryochemical potential slows down the creation of and pairs, resulting in gluon fragmentation dominated by . The larger amount of and with respect to and gives a higher probability for to find or and form or than for to form the antiparticle counterparts. Prompt hard kaon emission then carries away all strange antiquarks and positive charge, lowering somewhat the initial temperature and entropy. The late-stage hadronization is characterized by production of , , nucleons, pions, and strange baryons. Overall, after hadronization is complete, the relative density of strange hadrons is significantly enhanced over that resulting from a hadron gas alone, damping the energy fraction.

The layout of the paper is as follows. We begin in Sec. II with an overview of the fireball paradigm and make a critical review of the available experimental data from colliders. After that, in Sec. III we discuss the particulars of air shower evolution and present our results from Monte Carlo simulations. We show that the formation of a plasma, with gluons and massive quarks, could play a key role in the hadronization process, modifying shower observables. In particular, we demonstrate that air showers triggered by a fireball explosion tend to increase , and under some reasonable assumptions can accommodate Auger data. Finally, we summarize our results and draw our conclusions in Sec. III.

## Ii Fireball Phenomenology

It has long been suspected that, for systems of high energy density, the elementary excitations can be safely approximated by an ensemble of free quarks and gluons at finite temperature and baryon number density Collins:1974ky (); Iachello:1974vx (); Cabibbo:1975ig (). This is because when the energy density is extremely high, the expected average particle separation is so small that the effective strength of interactions is weak (asymptotic freedom) Gross:1973id (); Politzer:1973fx (). For many purposes, the order of the energy density of matter inside a heavy nucleus is immense,

(1) |

where is the proton mass, is the nuclear baryon number, and is the nuclear radius. However, at typical energy densities inside of nuclei, quarks and gluons are very probably confined, on the average, inside of hadrons such as protons, neutrons, or pions. For all of these hadrons could be squeezed so tightly together that on the average they will all overlap and the system would become an unconfined plasma of quarks and gluons, which are free to roam the system. The energy scale at which hadrons begin to overlap is above the energy density of matter inside the proton, , where we have taken the radius of the proton as measured in electron scattering . Indeed, using phenomenological considerations it is straightforward to estimate that the critical energy density to form a non-hadronic medium is around Bjorken:1982qr (). This result is supported by high statistics lattice-QCD calculations, which yield , where we have taken Bazavov:2009zn ().

Besides the early universe, the conditions of extremely high temperature and density necessary for the appearance of unconfined quark and gluons could occur in at least two other physical phenomena: (i) the interiors of neutron stars Ivanenko:1969gs (); Itoh:1970uw (); Chapline:1976gq (); Freedman:1977gz (); Chapline:1977rn (); Alcock:1986hz (); Olinto:1986je () and (ii) high-energy nucleus-nucleus collisions, whether artificially produced at accelerators or naturally occurring interactions of cosmic rays with particles in the Earth’s atmosphere Anishetty:1980zp (); Halzen:1981zx (); Cleymans:1982cc (); Halzen:1983gx (). We estimate the energy density in nucleus-nucleus collisions of cosmic rays following Bjorken:1982qr (). We assume there exists a “central-plateau” structure in the inclusive particle productions as function of the pseudorapidity variable. The energy per particle should be of the order of the typical transverse momentum per particle . More precisely, in the fireball frame each isotropically emitted particle has an energy given by

(2) |

where is the particle’s mass Panagiotou:1991kv (). The energy content is approximately , where is the charged-hadron multiplicity per collision, and the number of nucleon-nucleon interactions in the fireball during the collision. Consider two nuclei of transverse radius which collide in the center-of-mass frame. The longitudinal size of the nuclei is Lorentz contracted forming a transverse thin slab at mid-pseudorapidity. The initial fireball volume is , where is the typical time scale for the formation and decay of a central fireball and hence is the longitudinal size, with the pseudorapidity width at . The time scale can be estimated as the time to traverse at light speed the fireball diameter, i.e., . The energy density during the cosmic ray (CR) collision is then

(3) |

The average transverse-momentum of charged hadrons produced in collisions can be parametrized as a function of the squared center-of-mass energy,

(4) |

with in appropriate units of Khachatryan:2010us ().
On the other hand, for a cosmic-ray nitrogen nucleus
(for simplicity, we choose the beam nucleus to be that
of the dominant element in air^{4}^{4}4In air, nitrogen is a dimer, N, with total
However, the electronic binding is eV, whereas the nuclear binding is MeV,
so the energy scales and length scales differ by a million,
and the dimerization is not expected to survive the cosmic-ray production process.
The nuclear binding of the single nucleus apparently does survive this process.
)
of colliding with an air
nucleus, we have
, yielding and d'Enterria:2011jc (). Throughout, we take as fiducial in
our calculations and assume that half of these number of nucleons interact
per collision, producing the fireball. Therefore, taking an effective mass and , the energy density in such a scattering
process is

(5) |

well above , complying with the requirement for the formation of a deconfined thermal fireball.

We envision the fireball as a plasma of massive quarks and gluons maintained in both kinetic and chemical equilibrium. Because the total number of particles is allowed to fluctuate, we adopt the viewpoint of the grand canonical ensemble. In this representation, which allows exchange of particles among the system and the reservoir, the control variables are the baryochemical potential and the temperature . In the limit and the system becomes the vacuum.

The momentum distribution functions can be approximated by Fermi-Dirac and Bose-Einstein statistics,

(6) |

yielding following equilibrium number densities

(7) |

where the index runs over , because of the total strangeness neutrality of the initial state, , and are the particle’s momentum and mass, and is the spin-color degeneracy factor. The plus sign is to be used for quarks and minus sign for gluons, with . The density of the strange quarks is found to be (two spins and three colors)

(8) | |||||

where is the second order modified Bessel function Rafelski:1983hg (); Glendenning:1984ta (). In addition, there is a certain light antiquark density ( stands for either or ):

(9) | |||||

Note that the baryonic chemical potential exponentially suppresses the pair production. This reflects the chemical equilibrium between and the presence of a light quark density associated with the net baryon number. Now, since the chemical potentials satisfy , it follows that Kolb:1990vq ()

(10) | |||||

Note that this result is exact and not a truncated series. The gluon density also follows from (7) and is given by

(11) |

where is the Riemann function. (See Appendix A for details.) By comparing (8) and (9), it is straightforward to see that there are often more than anti-quarks of each light flavor,

(12) |

For and , there are about as many and quarks as there are quarks.

For , many of the properties of the quark-gluon plasma can be calculated in the framework of thermal perturbation theory. Neglecting quark masses in first-order perturbation theory, the energy density of the ideal quark gluon plasma is found to be

(13) | |||||

where and are the gluon and quark degrees of freedom, is the QCD coupling constant, and is the difference between the energy density of the perturbative and the nonperturbative QCD vacuum (the bag constant) Muller:1994rb (); Asprouli:1995zk (); GladyszDziadus:2001cq (). One observes that (13) is essentially the equation of state of a gas of massless particles with corrections due to QCD perturbative interactions, which are always negative and thus reduce the energy density at given temperature .

Since the and flavors are almost massless even in the fireball phase, we can get an estimate of the baryochemical potential by considering only the contribution from the nearly massless (even in the fireball phase) two quarks at . For two quark flavors, and so (13) simplifies to

(14) |

Substituting (5) into (14), with Kaczmarek:2005ui () and the MIT bag constant Asprouli:1995zk (), we obtain

(15) |

The temperature of the plasma can be approximated by Kolb:1990vq ()

(16) |

A point worth noting at this juncture is that the shapes of the spectra are expected to be determined by an interplay between two effects: the thermal motion of the particles in the fireball and a pressure-driven radial flow, induced by the fireball expansion. To disentangle the two contributions, namely thermal motion and transverse flow, one has to rely on model calculations which seem to indicate that the observed temperature in the particle spectra is close if not exactly equal to the temperature value that would be present in the chemically equilibrated fireball Rafelski:1996hf ().

Substituting (15) and (16) into (12), with , we obtain . The pion-to-nucleon density ratio is found to be

(17) |

where the factor 3/2 comes from the number of the particle species, is the pion mass, is the average baryon mass, and is a normalization constant fixed by the choice of the boundary values Panagiotou:1991kv ().

It is worth commenting on an aspect of this analysis which may seem discrepant at first blush. From relativistic heavy ion experiments one infers a temperature falling with the chemical potential, from . However, to accommodate the muon excess in Auger data one needs . Details of this discrepancy and eventual accommodation are given in Appendix B.

In our analysis we will adopt a pragmatic approach and avoid the details of theoretical modeling of the hadronization process. Which of the two points of view one may find more convincing, it seems most conservative at this point to depend on experiment to resolve the issue. The multiplicity ratio has been eyeball-fitted to reproduce the anomalous muon signal observed in Auger data. We show the success of this protocol in the next section. Here we simply note that these ratios are in partial agreement with (12) and (17) for .

## Iii Air Shower Evolution

We now make contact with the shower evolution. As a first-order approximation we adopt a basic phenomenological approach. Namely, we assume that the hadronic shower carries a fraction of the total energy of the primary cosmic ray , which scales as

(18) |

where is the number of generations required for most pions
to have energies below and is the
average fraction in electromagnetic particles per
generation Matthews:2005sd (). In the canonical framework
hadronic interaction models transfer about 25% of the energy to the
electromagnetic shower. Conspicuously, the production of light hadrons
in this canonical framework is virtually local in rapidity and,
therefore, since the interaction models are tuned to fit collider data,
would remain approximately constant with energy. We have
found that for the considerations in the present work we can safely
approximate that each interaction diverts about 75% of the available
energy into pions, 15% into kaons, and 10% continues as
nucleons. Roughly speaking, this is consistent with a multiplicity ratio . Now, taking then and we conclude that .^{5}^{5}5The average
neutrino energy from the direct pion decay is and that of the muon is , where
is the
ratio of muon to the pion mass squared. Thus, we can safely neglect
the missing energy in neutrinos. This is in agreement with the
estimates of Matthews:2005sd () which indicate that the
number of interactions needed to reach is for , respectively.

A comprehensive study of the uncertainties associated with the modeling of hadronic interactions indicates that a simple reduction of to increase correlates with , which becomes too shallow before is sufficiently increased to accommodate Auger data Farrar:2013sfa ().

Now, if the shower is initiated in a fireball explosion in the first generation, then we have seen that we can approximate the multiplicity ratios by . Moreover, this is a completely inelastic process, but differs from the usual inelastic processes in that a fireball is also produced. We assume this fireball creates a higher multiplicity of particles and to a first approximation equally partitions energy among the secondaries (thereby negating a large leading particle effect). The fireball production thus accelerates the cooling of the cascade and could reduce the number of generations. We denote with the numbers of generations for a shower initiated by a fireball. We may then assume that or 8 would be enough to reach the critical shower energy . To include the fireball effects we rewrite (18) as

(19) |

where is the fraction of electromagnetic energy emitted by the fireball. We arrived at from considering the fraction, equal to 1/3 the total fraction, as before. By substituting our fiducial numbers in (19) it is straightforward to see that the hadronic shower is increased by about 30% if and by about 70% if , in agreement with recent Auger results Aab:2016hkv ().

Of course, (19) is a dreadful simplification of the shower evolution. As we have noted before, the shower evolution depends on primary energy, as well as the elasticity and particle multiplicity which also depend on . Note that all these restrictions have not been specified explicitly as separate parameters in (19), but rather as a combined constant . Moreover, heavy meson production must also be taken into consideration when modeling the shower evolution (distinctively, and contribute about 4% to the electromagnetic cascade). Unfortunately, it is difficult to estimate accurate parameter values of the shower evolution in a simple analytical fashion. For such a situation, a full-blown simulation may be the only practical approach.

As a second order approximation, we estimate the fireball spectrum and propagate the particles in the atmosphere using the algorithms of aires (version 2.1.1) Sciutto:1999jh (). Most of the large multiplicity of observable quanta emitted in the fireball is expected to come through hadronic jets produced by the quarks. We assume that nucleons (each with initial energy ) produce the fireball and that the remaining nucleons scatter inelastically at the collision vertex. We further assume that for the nucleons producing the fireball, each of the valence quarks interact to give dijet final states without any leading particle. We calculate the total energy of each jet in the rest frame of the fireball from the momentum fraction carried by the up and down quarks, using CTEQ6L parton distribution functions Pumplin:2002vw ().

The precise nature of the fragmentation process is unknown. We adopt the quark hadron fragmentation spectrum originally suggested by Hill

(20) | |||||

where , with the energy of any hadron in the jet. This is consistent with the so-called “leading-log QCD” behavior and seems to reproduce quite well the multiplicity growth as seen in colliders experiments Hill:1982iq (). provides a reasonable parametrization of (20) for . And so we set the infrared cutoff to . The main features of the jet fragmentation process derived from this simplified parameterization are listed in Table 1. Using the multiplicity ratios derived in the previous section and the fractional equivalent energies given in Table 1, we construct the fireball particle spectra. More specifically, for each jet, we start first from and integrate down in until three leading hadron particles are obtained. The resulting interval in is , as shown in column three, with and values listed in columns two and one, respectively. We assign one of three species-types to each hadron using the weights. The energy fraction of the jet contained in these three hadrons is given in column four, and the average energy fraction per each of these three hadrons, denoted , is given in column five. Next, we duplicate the procedure for the remaining hadron species following the splitting of fractional energies, , as given in column 3, and assigning to each hadron in the interval the corresponding from column 5. Note that for each subsequent interval, the fractional hadron energy is significantly smaller; this feature allows us to sensibly truncate the process after five intervals at our cutoff value . The mean energy of each of the fifth and final batch of 30 hadrons produced by wee partons is of the energy of each hadron produced by large partons in first batch. The average particle multiplicity per jet is the sum of column three entries, approximately 54. Charge and strangeness conservation are separately imposed by hand.

0.0750 | 1.0000 | 3 | 0.546 | 0.182 |

0.0350 | 0.0750 | 3 | 0.155 | 0.052 |

0.0100 | 0.0350 | 9 | 0.167 | 0.018 |

0.0047 | 0.0100 | 9 | 0.062 | 0.007 |

0.0010 | 0.0047 | 30 | 0.069 | 0.002 |

All secondary particles are boosted to the laboratory frame. The particles are tightly beamed due to their very high boost. The boosted secondaries are then injected into aires as primaries of an air shower initiated at the collision vertex. The vertex of the primary interaction is determined using the mean free path of a N nucleus with . We set the observation point at 1.5 km above sea level, which is the altitude of Auger. All shower particles with energies above the following thresholds were tracked: 750 keV for gammas, 900 keV for electrons and positrons, 10 MeV for muons, 60 MeV for mesons, and 120 MeV for nucleons. The geomagnetic field was set to reproduce that prevailing upon the Auger experiment. For further details, see Appendix C. Secondary particles of different types in individual showers were sorted according to their distance to the shower axis. Our results are encapsulated in Fig. 1, where we compare the density distribution of at ground level for a typical nitrogen shower processed by the aires kernel with that of a nitrogen fireball explosion. The vertical axis is given in arbitrary units to indicate the large systematic uncertainty in the normalization of the lateral distribution, which is induced by the different predictions of high-energy hadronic event generators. Importantly, the comparison of hadron and fireball fluxes is not arbitrary. It is easily seen that the number of muons at is about 40% higher in the fireball-induced shower.

The third order approximation should include a precise determination of the fireball particle spectra using scaling hydrodynamic equations, which contain the probability amplitudes for production and annihilation both in the fireball phase and in the hadron gas phase Rafelski:1982pu (); Rafelski:1982ii (); Kapusta:1986cb (); Matsui:1985eu (); Matsui:1986dx (); Koch:1986ud (). It should also contain a thorough high-level competitive analysis of theoretical systematics emanating from hadronic interaction models. This rather ambitious project is beyond the scope of this paper, but will be the topic of a future publication.

## Iv Conclusions

Teasing out the physics of ultrahigh-energy cosmic rays has proven to be extraordinarily challenging. The Pierre Auger Observatory employs several detection methods to extract complementary information about the extensive air showers produced by ultrahigh-energy cosmic rays ThePierreAuger:2015rma (). Two types of instruments are employed: Cherenkov particle detectors on the ground sample air shower fronts as they arrive at the Earth’s surface, whereas fluorescence telescopes measure the light produced by air-shower particles exciting atmospheric nitrogen. These two detector systems provide complementary information, as the surface detector (SD) measures the lateral distribution and time structure of shower particles arriving at the ground, and the fluorescence detector (FD) measures the longitudinal development of the shower in the atmosphere. A subset of hybrid showers is observed simultaneously by the SD and FD. These are very precisely measured and provide an invaluable tool for energy calibration, minimizing systematic uncertainties and studying composition by simultaneously using SD and FD information. Very recently, the Pierre Auger Collaboration exploited the information in individual hybrid events initiated by cosmic rays with to study hadronic interactions at ultrahigh energies Aab:2016hkv (). The analysis indicates that the observed hadronic signal of these showers is significantly larger (30 to 60%) than predicted by the leading LHC-tuned models, with a corresponding excess of muons. The significance of the discrepancy between data (411 hybrid events) and model prediction is above about . A deployment of a scintillator on top of each SD is foreseen as a part of the AugerPrime upgrade of the Observatory to measure the muon and electromagnetic contributions to the ground signal separately Aab:2016vlz (). This will provide additional information to reduce systematic uncertainties and perhaps increase the significance of the muon excess.

Even though the excess is not statistically significant yet, it is interesting to entertain the possibility that it corresponds to a real signal of QCD dynamics flagging the onset of deconfinement. In this paper, we have proposed a model that can explain the observed excess in the muon signal. We have assumed that ultrarelativistic nuclei () that collide in the upper atmosphere could create a deconfined thermal fireball which undergoes a sudden hadronization. At production, the fireball has a very high matter density and consists of gluons and two flavors of light quarks (). Because the fireball is formed in the baryon-rich projectile fragmentation region, the high baryochemical potential damps the production of and pairs, resulting in gluon fragmentation mainly into . The strange quarks then become much more abundant and upon hadronization the relative density of strange hadrons is significantly enhanced over that resulting from a hadron gas. We have shown that the augmented production of strange hadrons by the fireball, over that resulting from a hadron gas alone, provides a mechanism to increase the muon content in atmospheric cascades by about 40%, in agreement with the data of the Auger facility.

Contrary to previous proposals Farrar:2013sfa (); Allen:2013hfa (); AlvarezMuniz:2012dd ()
to explain the muon excess in Auger data our model relies on the assumption
that ultrahigh-energy cosmic rays are heavy (or medium mass)
nuclei. As noted elsewhere Ahlers:2009rf (), upper limits on the
cosmic diffuse neutrino flux provide a constraint on the proton
fraction in ultrahigh-energy cosmic rays, and therefore can be used to
set indirect constraints on the model proposed herein. In particular,
the nearly guaranteed flux of cosmogenic neutrinos is a decay product
from the generated pions in interactions of ultrahigh energy cosmic
rays with the cosmic microwave
background and related radiation Beresinsky:1969qj ().
The spectral shape and intensity
of this flux depend on whether the cosmic-ray particles are protons or
heavy nuclei. For proton primaries, the energy-squared-weighted flux
peaks between and , and the intensity is
around 1 in Waxman-Bahcall (WB)
units Stecker:1978ah (); Hill:1983xs (); Engel:2001hd (); Fodor:2003ph (); Anchordoqui:2007fi (); Ahlers:2010fw (); Kotera:2010yn (); Kampert:2012mx ().^{6}^{6}61 WB
= Waxman:1998yy (). For heavy nuclei, the peak is at
much lower energy (around Hooper:2004jc (); Ave:2004uj ()) and the intensity is about
0.1 to 0.01 WB, depending on source
evolution Kotera:2010yn (); Kampert:2012mx (). The sensitivity of
existing neutrino-detection facilities has a reach to 1 WB, challenging
cosmic-ray models for which the highest energies are
proton-dominated Ahlers:2012rz (); Aab:2015kma (); Unger:2015laa (); Aloisio:2015ega (); Heinze:2015hhp (); Supanitsky:2016gke (); Aartsen:2016ngq (); Yoshida:2016hba (). Next-generation neutrino detectors will systematically probe the
entire range of the parameter space of cosmogenic
neutrinos Allison:2011wk (); Aartsen:2014njl (); Martineau-Huynh:2015hae (); Adrian-Martinez:2016fdl (); Neronov:2016zou (); Connolly:2016pqr (). Observation
of the cosmogenic neutrino flux with intensity of 0.1 to 0.01 WB could
become the smoking gun for the ideas discussed in this
paper. Complementary information can be obtained with accompanying
cosmogenic photons Hooper:2010ze (). As a matter of fact, the
Pierre Auger Observatory has begun to probe the region of the
parameter space relevant for proton primaries Aab:2016agp ().
A third probe of proton-dominated UHECR models is the extragalactic
gamma-ray background seen by Fermi-LAT (and, in the future, CTA) Supanitsky:2016gke (); Berezinsky:2016jys (). For
heavy nuclei, however, the cosmogenic photon intensity is almost
negligible and therefore cannot be used (for the time being) as a
harbinger signal Anchordoqui:2006pd ().

For , the mean and dispersion of inferred from fluorescence Auger data point to a light composition (protons and helium) towards the low end of this energy bin and to a large light-nuclei content (around helium) towards the high end (see Fig. 3 in Aab:2016zth ()). However, when the signal in the water Cherenkov stations (with sensitivity to both the electromagnetic and muonic components) is correlated with the fluorescence data, a light composition made up of only proton and helium becomes inconsistent with observations Aab:2016htd (). The hybrid data indicate that intermediate nuclei, with , must contribute to the energy spectrum in this energy bin. Moreover, a potential iron contribution cannot be discarded.

At this stage, it is worthwhile to point out that the production of a fireball may modify the shower evolution. For example, after emitting and , the fireball has a finite excess of quarks, and because of the quarks’ stabilizing effects, the fireball could form heavy multiquark droplets , with large strangeness Witten:1984rs (); Liu:1984ta (); Farhi:1984qu (); Halzen:1984qc (); Greiner:1987tg (). The energy lost by the particles during collision with nucleons is primarily through hard scattering, and so the fractional energy loss per collision is Anchordoqui:2004bd (); Anchordoqui:2007pn (). production may thus slow down the shower evolution. Although this effect has not been included in our simulations, one may wonder whether the structure observed in the elongation rate above about Aab:2014kda () could be ascribed to the onset of production. Of course one would not expect a fireball to be created when nuclei just slide along each other. The admixture of peripheral and fireball collisions would then produce large fluctuations in the number of muons at ground level. However, since the critical energy for charged pions and kaons is roughly the same, the elongation rate of the muon channel would be almost unaltered and so the muon shower maximum would have small fluctuations. Moreover, if the fireball indeed modifies the elongation rate of the electromagnetic shower, the peripheral collisions would also tend to increase the dispersion of , mimicking what is expected for a light composition in the canonical framework where no fireballs are being produced in this energy range.

In closing, we comment on the differences between our model and the proposal by Farrar and Allen (FA) Farrar:2013sfa (), which also relies on QCD dynamics at high temperature. To better understand the differences between these models we first note that at low energies, QCD exhibits two interesting phenomena: confinement and (approximate) spontaneous chiral-symmetry breaking. These two phenomena are strong-coupling effects, invisible to perturbation theory. The confinement force couples quarks to form hadrons and the chiral force binds the collective excitations to Goldstone bosons.

As a matter of course, there is no relation a priori between these two phenomena; in thermodynamics, the associated scales are characterized by two distinct temperatures, and . For , hadrons dissolve into quarks and gluons, whereas for , the chiral symmetry is fully restored and quarks become massless, forming an ideal quark-gluon plasma. The phase diagram of hadronic matter thus contains a confined phase consisting of an interacting gas of hadrons (a resonance gas) and a deconfined phase comprising a (non-ideal) gas of quarks and gluons Rapp:1999ej (). The phase boundary reflects the present uncertainties from lattice QCD extrapolated to finite baryochemical potential. The intermediate region in-between the hadron gas and the ideal quark-gluon plasma is the domain of the thermal fireball. The existence of this intermediate region with deconfined massive quarks and gluons is also conjectured from high-statistics lattice-QCD calculations, which indicate that, for , the energy density is about 85% of the Stefan-Boltzmann energy density for the ideal quark-gluon plasma Bazavov:2009zn (). Note that this temperature is not inconsistent with our estimate in (16). In the FA model, the pion suppression is a direct consequence of massless quarks living above the chiral symmetry restoration temperature, i.e., . In our model, however, the pion suppression is the result of the large baryochemical potential which forbids the creation of light and pairs, allowing abundant production of massive via gluon fragmentation. This process naturally occurs in the fireball boundary phase (where ), and is a consequence of the high nucleon density of Lorentz-boosted nuclei. In principle, it is possible that the muon excess observed in Auger data originates in a combination of these two high-temperature QCD phenomena. Note that if ultrahigh-energy cosmic rays are heavy (or medium mass) nuclei and the observed muon excess is not the result of a large baryochemical potential suppressing the production of and pairs, but rather an effect of chiral symmetry restoration, then the excess should also be visible at , which corresponds to the center-of-mass energy in collisions of projectile cosmic ray protons with . This disparity could be use to discriminate among dominance between these two pion-suppression mechanisms.

On the one hand, a model consistent with all data requiring heavy nuclei at the high-energy end of the spectrum is generally considered a bit disappointing Aloisio:2009sj (), especially for neutrino and cosmic-ray astronomy. On the other hand, we have shown that even if heavy nuclei dominate at the highest energies, upon scattering in the Earth’s atmosphere these nuclei could become compelling probes of QCD dynamics at high temperatures, particularly in the not-so-well-understood fireball boundary phase. Should this be the case, future AugerPrime data would provide relevant information on the strange quark density of the nucleon, complementing measurements at heavy-ion colliders.

###### Acknowledgements.

We would like to acknowledge many useful discussions with our colleagues of the Pierre Auger Collaboration. L.A.A. is supported by U.S. National Science Foundation (NSF) Grant No. PHY-1620661 and by the National Aeronautics and Space Administration (NASA) Grant No. NNX13AH52G. T.J.W. is supported in part by the Department of Energy (DoE) Grant No. DE-SC0011981.## Appendix A Fermi–Dirac and Bose–Einstein integrals

For the sake of completeness, in this appendix we provide all of the formulae used for computing the number densities of , , and .

The Fermi-Dirac and Bose-Einstein distributions (6) can be rewritten as infinite sums of Boltzmann distributions,

(21) | |||||

Following Chaudhuri:2012yt () we introduce the dimensionless variables, and :

In terms of and , the number density for the Boltzmann distribution can be written as,

(22) | |||||

A closed form expression can be found for in terms of the modified Bessel function

(23) |

Note that has another representation, which follows from (23) by partial integration,

(24) |

The modified Bessel function has a recurrence relation,

(25) |

such that if the expressions for and are known, all the others can be easily obtained. From (24) it is straightforward to obtain

(26) |

and so (22) becomes

(27) |

For the Fermi-Dirac distribution, the number density is found to be

(28) |

For the Bose-Einstein distribution, the number density is given by

(29) |

In the limit and we immediately obtain

(30) |

where is the Riemann function. Finally, using (28) and (30), it is straightforward to obtain (8), (9) and (11).

## Appendix B connection

Our best understanding of the thermodynamical properties of QCD at vanishing baryon density is rooted on high statistics lattice QCD numerical simulations. However, if the baryon density is non-zero, these simulations break down. The only physically relevant analyses for which the obstacles of lattice QCD can be circumvented are those dealing with small baryon density or, more accurately, small . Namely, if we are interested in the observable , we can expand it in powers of as

(31) |

and try to determine the series coefficients Bonati:2016fbi (). Which values of can be considered small enough to give reliable results with this procedure is something that can only be determined a posteriori by the convergence property of the series, limited by the accuracy in the evaluation of the expansion coefficients. One such observable is the quark-hadron crossover line. Data from heavy-ion colliders suggest that the energy dependence of this line can be parametrized as

(32) |

with , and , and that the baryochemical potential can be parametrized as

(33) |

with and Cleymans:2005xv (). Interestingly, the lattice-QCD calculation converges towards the quark-hadron crossover line as Becattini:2012xb (); Becattini:2016xct (), but it appears to depart from this line at large values of Tawfik:2004sw (). This a widely discussed feature Letessier:2005qe (); McLerran:2008nn (); Andronic:2009gj (); Li:2016wzh (); McLerran:2016ozl () which has, however, not been conclusively understood. (Several hadronization schemes have been proposed, see e.g. Torrieri:2002jp (). They differ in the geometry and in the flow velocity profile.) This perplexing region may well be relevant to cosmic-ray observations. Our results in the region shown in Fig. 2, suggest that this is so.

## Appendix C Monte Carlo simulation of air showers

The aires simulation engine Sciutto:1999jh () provides full space-time particle propagation in a realistic environment, taking into account the characteristics of the atmospheric density profile (using the standard US atmosphere), the Earth’s curvature, and the geomagnetic field (calculated for the location of Auger with an uncertainty of a few percent Cillis:1999gk ()). The following particles are tracked in the Monte Carlo simulation: photons, electrons, positrons, muons, pions, kaons, eta mesons, lambda baryons, nucleons, antinucleons, and nuclei up to . The high-energy collisions are processed invoking external hadronic event generators, whereas the low-energy ones are processed using an extension of the Hillas splitting algorithm Knapp:2002vs ().

The aires program consists of various interacting procedures that operate on a data set with a variable number of records. Several data arrays (or stacks) are defined. Every record within any of these stacks is a particle entry and represents a physical particle. The data contained in every record are related to the characteristics of the corresponding particle. The particles can move inside a volume within the atmosphere where the shower takes place. This volume is limited by the ground, the injection surfaces, and by vertical planes which limit the region of interest. Before starting the simulation, all the stacks are empty. The first action is to add the first stack entry, which corresponds to the primary particle. Then the stack processing loop begins. The primary is initially located at the injection surface, and its downwards direction of motion defines the shower axis. After the primary’s fate has been decided, the corresponding interaction begins to be processed. The latter generally involves the creation of new particles which are stored in the empty stacks and remain waiting to be processed. Particles entries are removed when one of the following events happen: (i) the energy of the particle is below the selected cut energy; (ii) the particle reaches ground level; (iii) a particle going upwards reaches the injection surface; (iv ) a particle with quasi-horizontal motion exists the region of interest. After having scanned all the stacks, it is checked whether or not there are new particle entries pending further processing. If the answer is positive, then all the stacks are scanned once more; otherwise the simulation of the shower is complete.

For the present analysis, we use the aires module for special and/or multiple primary particles. This useful feature allows to dynamically call a user-defined module that tracks the interactions of a bundle of particles returning a handy list of secondaries, which can be conveniently controlled by the propagating engine for further processing.

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