Longitudinal profiles of Extensive Air Showers with inclusion of charm and bottom particles
aa r X i v : . [ h e p - ph ] M a y Longitudinal profiles of Extensive Air Showers withinclusion of charm and bottom particles
M. A. Müller , and V. P. Gonçalves Instituto de Física e Matemática, Departamento de Física e Matemática, Universidade Federal dePelotas/UFPEL, Pelotas, RS, Brazil Instituto de Física Gleb Wataghin (IFGW), Departamento de Cronologia, Raios Cósmicos, Altas Energias eLéptons (DRCC), Universidade Estadual de Campinas, Campinas, SP - Brazil
Abstract
Charm and bottom particles are rare in Extensive Air Showers but the effect of its presence can be radical inthe development of the Extensive Air Showers (EAS). If such particles arise with a large fraction of the primaryenergy, they can reach large atmospheric depths, depositing its energy in deeper layers of the atmosphere. Asa consequence, the EAS observables ( X max , RM S and N max ) will be modified, as well as the shape of thelongitudinal profile of the energy deposited in the atmosphere. In this paper, we will modify the CORSIKAMonte Carlo by the inclusion of charm and bottom production in the first interaction of the primary cosmic ray.Results for different selections of the typical x F values of the heavy particles and distinct production models willbe presented. The description of the Extensive Air Showers (EASs) is fundamental for the Cosmic Ray Physics.Primary particles reache the Earth with energies up to eV . Such energies are well above thosereached in colliders, and, therefore, the simulations of these EASs require an extrapolation of the knownphysics.Very energetic charm and bottom heavy hadrons may be produced in the upper atmosphere when aprimary cosmic ray or a leading hadron in an EASs collide with the air. Because of its short mean life, ≈ − s ( ≈ µm ), heavy hadrons decay before interacting. At E ≈ GeV heavy hadrons reachtheir critical energy and its decay probabilities decrease rapidly. Decay lenghts grow to considerablevalues. At E ≈ GeV their decay lenght becomes of order ≈ km , implying that they tend tointeract in the air instead of decaying. Since the inelasticity in these collisions is much smaller than theone in proton and pion collisions, keeping a higher fraction of their energy after each interaction, therecould be rare events where a heavy hadron particle transports a significant amount of energy deep intothe atmosphere, giving rise to additional contributions to the EAS development. Heavy particles canbe produced at any stage of the EAS development, but it is mainly during the first interaction of theprimary collision that they are produced with a significant fraction of primary energy.The collisions of heavy hadrons with the air are very elastic. For example, a D meson after a GeV collision could keep around 55% of the initial energy, whereas a B meson will have typically 80% of theincident energy after colliding with an air nucleus. In contrast, the leading meson after a GeV pioncollision would carry in average just 22% of the energy [1]. If heavy hadrons are produced with a highfraction of the primary particle, they will interact rather than decaying. If several elastic interactionsoccur, these heavy hadrons could transport a significat amount of energy deep into the atmosphere andlikely have observable effects on the EAS development.The energy deposition of a heavy partycle near of the ground would produce muons and otherparticles that could change significantly the EASs longitudinal profile seen in fluorescence telescopes1e.g. in the Pierre Auger Observatory [2]) and/or the temporal distribution observed in the surfacedetectors. That will modify the EAS observables ( X max , RM S and N max ), besides a considerablechange of the EAS longitudinal profile shape.In this work, we will use the CORSIKA [3] (Cosmic Ray Simulations for Kaskade) Monte Carlo tosimulate the evolution of EASs in the atmosphere. We will a modified code of CORSIKA, with charmand bottom production at the cosmic ray first interaction [4] (hereafter denoted HQ CORSIKA). Inthe next section, we will explain how this version works. Moreover, the HQ CORSIKA predictionswith those derived using the standard CORSIKA (original code - without charm particle production)(hereafter denoted STD CORSIKA). In both versions (HQ CORSIKA and STD CORSIKA) we willuse the QGSJET01 MC to describe the high energy hadronic interactions and the FLUKA one for thedescription of the low energy hadronic interactions. We will demonstrate that the charm and bottomproduction rates are only relevant for the highest energies. The modified MC allows us to analyse theeffects that the production and propagation of heavy hadrons has in the EAS development.Our goal is to verify the impact on the EAS observables of different selections of for the minumumvalue of the x F variable , as well of different models to describe the heavy hadron production - CGC(Color Glass Condensate) and IQM (Intrinsic Quark Model). As mentioned earlier, we have included in the CORSIKA MC the production of heavy hadron (mesonsand/or baryons) in the first interaction of primary cosmic ray with an incident energy > × eV . Weconsider that Λ s, Ξ s, Σ s, Ω s, Ds and Bs are produced in this interaction. These particles are producedaccording to the probability for the quark charm to be present in different species of hadrons after p − Air collision [6].The CORSIKA MC doesn’t explicitly include the charm and bottom heavy interactions. It is nec-essary that in the extraction of the source code the option CHARM is “switch on”. Not all packages canhandle production and propagation of heavy particles. In the CORSIKA, the DPMJET and QGSJETpackages of high energy hadronic interactions consider the interaction of these particles. The D + s (mainsource of tau leptons) particle (PDG code 431) is not included by the QGSJET01.We assume that the heavy particles can be produced via two models: • Color Glass Condensate - CGC. • Intrinsic Quark Model - IQM.The choice of the production model and the kind of heavy particle (charm or bottom) generatedin the first interaction is made via CORSIKA INPUT - key COLLDR. From the first interaction, thepackage of hadronic interaction PYTHIA [5] makes the decay and interaction of the heavy particles.The key PROPAQ determines whether the propagation of heavy particles will be handled by PYTHIA,or by standard routine (e.g. QGSJET01). Through key SIGMAQ, the cross sections for the charm andbottom mesons and baryons are determined [4].
In this model, a heavy flavor quark-antiquark pair is created through the fluctuaction of a gluon in theprojectile particle. Charmed and bottom hadrons are formed from the hadronization of these heavyquarks with sea quarks, in a mechanism called Uncorrelated Fragmentation. More information of themodel in [7] and [8].When a proton of energy E p (in GeV) collides with a nucleus in the atmosphere, the probability toproduce a heavy hadron carrying a fraction of energy is given by: X max is the depth of the maximum energy deposited in the atmospheric (starting at the top of the atmosphere) bythe EAS, RMS is the fluctuation of the X max (Standard Deviation) and N max is the maximum number of particles atthe EAS. Probability of production of a heavy hadron carrying values larger than a certain fraction of primary energy. Above ( x F > . ) [6]: P ( E p ) = 0 . − . ∗ ln ( E p ) + 0 . ∗ ln ( E p ) (1) • Above ( x F > . ): P ( E p ) = − . . ∗ ln ( E p ) + 0 . ∗ ln ( E p ) (2)The energy distribution of charms produced has the general form [6]: dPdx F = a ∗ (1 − x . F ) b x cF (3)where: a = 0 . . × − ) ∗ ln (10 ∗ E p ) , (4) b = 8 . − . ∗ ln (10 ∗ E p ) , c = 1 . . ∗ ln (10 ∗ E p ) At leading order in QCD, heavy quarks are produced by the processes qq → QQ and gg → QQ . Whenthese heavy quarks arise from fluctuation of the initial state, its wave function can be represented as asuperposition of Fock state fluctuations: | h > = c | n v > + c | n v g > + c | n v qq > + c | n v QQ > ... (5)where | n v > is the hadron ground state, composed only by its valence quarks.When the projectile scatters on the target the coherence of the Fock components is broken andthe fluctuations can hadronize, either with sea quarks or with spectator valence quarks. The lattermechanism is called Coalescence. For instance, the production of Λ + c in p − N collisions comes fromfluctuation of the Fock state of the proton to | uudcc > state. To obtain a Λ − c in the same collisiona fluctuation to | uuduuddcc > would be required. Thus, since the probability of a five quarks stateis larger than that of a 9 quarks state, Λ + c production is favored over Λ − c in proton reactions. Theco-moving heavy and valence quarks have the same rapidity in these states but the larger mass of theheavy quarks implies they carries most of the projectile momentum. Heavy hadrons form from thesestates can have a large longitudinal momentum and carry a large fraction of the primary energy, whichis crucial for their propagation [4]. At the figure 1 we can see the differential energy fraction distributionfor some charmed and bottom hadrons.Figure 1: Feynman - x distribution for some charmed (left) and bottom (right) hadrons derived usingthe Intrinsic Quark production model[4]. | uudcc > in the proton is [9]: P ( p → uudcc ) ≈ [ m p − X i =1 m ⊥ i x i ] − (6)where the transverse mass is m ⊥ i and we take i = 4 , for c, c . A detailed description of this model canbe found in [9] and [10]. In this section we will analyse the longitudinal profiles of total energy deposited in the atmosphere,considering ≈ EAS. We will assume that the primary cosmic ray is × eV , the zenithal angleis ◦ and primary particle is a proton. Production of charm or bottom heavy particles can occur viatwo models - CGC and IQM. We will use two selections for the minimum of x F (via CGC) - x F > . and x F > . . The comparisons will be made between CORSIKA HQ and STD CORSIKA. For thelongitudinal profile of energy deposited, the curves will be separated according to the energy fraction( F E ) of the heavy particles produced in the first interaction - F E < . and F E ≥ . for the CGCmodel and F E < . , F E ≥ . and F E ≥ . for the IQM production model.For this analysis, we will restrict the heavy hadron production to Λ + c , D , D + , D + s , B + and B andtheir anti-particles .In Fig. 2 we show the evolution of production of secondary charms in the EAS. The charm particlesare “written” as they are produced during the EAS development, from the first interaction of cosmicray down to the sea level. We assume bins of g/cm of atmospheric depths and half decadeof energy. The charm particles are dominantly produced with low energy (below GeV ) and areproduced between and g/cm . On the other hand, the energetic charms are produced in thefirst interactions, i.e. when the depth of the atmosphere is less than g/cm . We have that ≈ is the average number of charm produced above GeV , which will decay and produce high energy µ and ν that reach the ground. Regarding the total number of charms produced in all bins (energyhigher than GeV ), we have an average of ≈ charms, being ≈ D , ≈ D + , ≈ D s and ≈
240 Λ c . From this total we have that ≈ charm are produced with energy higher than GeV . ) Atmospheric depth (g/cm
Log ( E ( G e V )) Figure 2:
Evolution in the production of secondary charms in the EAS for x F > . . We assume thatthe energy of the primary cosmic ray is × eV . In Fig. 3, we present the energy distribution for charms produced in the first interaction obtainedusing the HQ CORSIKA. Comparing the CGC predictions for x F > . and x F > . , we havethat the secondary charms have a higher average energy for x F > . . However, for x F > . the Not all high hadronic interactions packages can handle this kind of heavy particles. D + s , B + B and their anti-particlesfor example is not considered by QGSJET01. x F > . some particles reach ≈ × eV . For thisenergy the charm decay lenght becomes of order of ≈ km , what could make such particle reach theground with reasonable energy. Comparing now different heavy production models, CGC and IQM, thesecondary generated via Intrinsic Quark Model reaches a much larger energy than produced via ColorGlass Condensate. Via IQM, we have secondary particles been produced at first interaction with largerfractions of primary energy. Such particles can carry almost all primary energy. Energy (eV) × P r obab ili t y > 0.01 F HQ CORSIKA, charm via CGC - x > 0.05 F HQ CORSIKA, charm via CGC - xHQ CORSIKA, charm via IQM
Figure 3:
Energy distribution of the charm hadrons generated at the first interaction of the HQ COR-SIKA. The curves are separated according to x F values ( x F > . and x F > . ) and the modelused to describe the heavy quark production. We assume that the energy of the primary cosmic ray is × eV . Considering now the energy distribution of the charmed particles that hits the ground. Independentlyof the selection for the values of x F and the production model(CGC and IQM), the number of particlesis negligible. Consequently, the most part of charm particles that are produced in the EAS decay orinteract before hit the ground.Muons are a key prediction in EAS simulations. Although the presence of heavy hadrons will notintroduce significant differences in the total number of muons at the ground level, there are otherobservables that may be more sensitive to these heavy hadrons: Events with late energy depositionfrom the decay of a heavy meson or a τ lepton. The fraction of these events is low [1]. Events withleptons of PeV energies, coming from charm decays. d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) HQ CORSIKA, charm via CGC (x F > E ≥ Figure 4: Longitudinal profiles for the total energy deposited in the atmosphere for charm productionvia CGC ( x F > . ) and F E ≥ . .In the Figs. 4, 5, and 6 we present our predictions for the longitudinal profiles. The longitudinalprofiles are separated according to the fraction of the energy of the heavy particles generated in the5rst interaction, F E < . and F E ≥ . , which allows to highlight the profiles. Concerning to F E < . ( x F > . ), for both charm and bottom production (via CGC), the longitudinal profiles followapproximately the same behaviour in comparison with the standard profiles (STD CORSIKA).Concerning to F E ≥ . ( x F > . , via CGC), presented in the Fig. 4 they represent less than 6%of total . Looking for the profiles with a fraction < . we can’t see significant changes in the profile. Infact, they follow the same general shape of profiles with no heavy hadron production (STD CORSIKA).If we look now at the profiles with larger fractions, the effect is more pronounced. We have a slightdifference in the maximum of energy deposited, i.e, we have a shift to deeper layers of the atmosphere.These profiles have a smaller value for the peak of energy deposited. d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) HQ CORSIKA, charm via IQM − F E ≥ Figure 5: Longitudinal profiles of the total energy deposited in the atmosphere for charm productionvia IQM, F E ≥ . . d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) HQ CORSIKA, bottom via IQM − F E ≥ Figure 6: Longitudinal profiles of the total energy deposited in the atmosphere for bottom productionvia IQM, F E ≥ . .Once again we separated the profiles for F E < . , F E ≥ . and F E ≥ . for the IQM heavyproduction model, considering the charm and bottom production separately. For these cases, we havea large number of profiles with high energy fraction . For the highlighted profiles with F E ≥ . wepredict a non- negligible change of the longitudinal profiles, for both particles, charm and bottom. Inparticular, for F E ≥ . , we have the most important results. In some profiles the maximum of totalenergy deposit is largely shifted to larger depths. We also see in some profiles a big change in the profileshape, i.e, the peak of energy deposited is much smaller and the profiles are quite elongated (See Figs. The highest value of the fraction of primary energy reached is ≈ . (CGC production model), ie, ≈ × eV . As mentioned before, heavy hadrons produced via IQM model can have a large longitudinal momentum and carry alarge fraction of the primary energy. In some cases the peak of total energy deposited reaches just
P eV /g/cm . The standard profile reaches about P eV /g/cm . F E ≥ . ,we see significant changes, mainly for bottom production. In particular, we can see some double coreprofiles. Our purpose in this analysis was to study how the presence of a heavy hadrons could modify thefundamental parameters of the cascade development, such as the shape of longitudinal profile, thenumber of particles reaching the ground, the position and deviation of the shower maximum. Heavyhadrons propagating with an energy above its critical value will travel long paths. In particular, showerswith zenithal angle of degrees have an atmosphere slant depth of ≈ g/cm . After several elasticinteractions, we expect heavy particles to deposit its remaining energy deep in the atmosphere and somecould reach the ground carrying substancial energy fraction. If the heavy hadron carries a significantfraction of the primary’s energy, we can expect a large impact on the EAS observables. As the heavyparticle energy fraction increases, more accentuated these effects will be. d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavy − Xmax=(802.0 ± HQ CORSIKA, charm via CGC − x F > ± HQ CORSIKA, charm via CGC − x F > ± HQ CORSIKA, bottom via CGC − x F > ± d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavy − Xmax=(802.0 ± HQ CORSIKA, charm via CGC − x F > ± Figure 7: Longitudinal profiles of the energy deposited in the atmosphere considering the HQ CORSIKA(CGC, to x F > . and x F > . ) and the STD CORSIKA. Upper panel: Predictions for F E < . .Lower panel: F E ≥ . .In what follows we will present our predictions for the average longitudinal profiles. In Fig. 7, we7an’t observe significant differences between the average profiles. Small changes of the observable occuronly for x F > . ( F E ≥ . ). In this case the X max is more deeper and the RM S larger in comparisonwith EAS simulated by the STD CORSIKA. We have a relative difference of 9% of to X max and 28% to RM S . Comparing now the x F assumed, we have small differences in the shape of longitudinal profiles.For x F > . , the X max and RM S is slightly larger.Regarding the average profiles obtained using IQM production model, presented in Figs. 8 and 9, wecan see significant changes in the longitudinal profiles. We have a deeper X max and a higher RM S whenthe energy fraction of the heavy particles is increased. For charm production with F E ≥ . we predictthe values of . g/cm and . g/cm , respectively, for X max and RM S . For bottom productionwith F E ≥ . we obtain . g/cm and . g/cm . We can also observe a significant change in theshape of the profile. Depositing less energy according to energy fraction. Bottom and charm particleinteraction are more elastic than other particles, therefore charms and bottoms produced with highprimary fraction will deposit energy more slowly in the atmosphere and can carry large energies deeperin the atmosphere. Such effect is larger in bottom particles. At higher fractions of energy ( F E ≥ . ),the impact on the average longitudinal profile is larger. In the case of charm production we have a X max shifted to deeper layers of the atmosphere in relation to standard CORSIKA, with the relative differencebeing about 12%. For the RM S , we have a relative difference of 40%. Regarding bottom productionwe have a more radical effect, being 22% for X max (shift to deeper layers) and 100% to RM S . Forthe RMS, we have larger values, which means that the fluctuation of depth of the maximum of EAS( X max ) is larger. This happens because the heavy particle interaction is more elastic. All these EASeffects ( X max shift, larger RM S and more elongated shape of the EAS profiles) can change significantlythe EASs longitudinal profile seen in fluorescence telescopes and/or the temporal distribution observedin the surface detectors. In fact, we could see some double core profiles. The global effect of all thesechanges in the longitudinal profile is the lower energy reconstructed of the EAS and higher uncertainties.For the energy of the primary cosmic rays considered ( ≈ eV ), the values of X max and RM S foundin experiments such as the Pierre Auger Observatory for example are respectively ≈ g/cm and ≈ g/cm [11]. The X max and RM S are directly linked to the mass composition of the primarycosmic ray. The appearance of charm and bottom in the EAS makes it more difficult to make such aconnection, because of the deeper X max and larger RM S . d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavy − Xmax=(802.0 ± HQ CORSIKA, charm via IQM − < ± HQ CORSIKA, charm via IQM − ≥ ± HQ CORSIKA, charm via IQM − ≥ ± Figure 8: Longitudinal profiles of the energy deposited in the atmosphere considering HQ CORSIKA(IQM - charm production) and the STD CORSIKA. Predictions for F E < . , F E ≥ . and F E ≥ . .In Figs. 8 and 9 we can also see a more elongated shape of the EAS, ie, a slower energy deposit.In these figures we analyze when the energy deposition in the shower is shifted to large depths. The Here it is taken into account that we have a mixture of several mass compositions (H, He, C, Fe, etc.) X max is affected. The number of particles at maximum decrease, whilethe number of particles that reach the ground increase. In Fig. 10, we show the ratio between thenumber of particles in the maximum of the shower and the number of particles at the ground level( E max /E ground ) according to the fraction of energy carried by the heavy hadron. Such ratio is sensitiveto the change in the profile’s shape. The EAS energy deposited amplitude decrease as the X max is shiftedto higher depths. The effect is higher to bottom particle production because of its higher elasticity. d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavy − Xmax=(802.0 ± HQ VCORSIKA, bottom via IQM − < ± HQ CORSIKA, bottom via IQM − ≥ ± HQ CORSIKA, bottom via IQM − ≥ ± Figure 9: Longitudinal profiles of the energy deposited in the atmosphere considering the HQ CORSIKA(IQM - bottom production) and the STD CORSIKA. Predictions for F E < . , F E ≥ . and F E ≥ . .
100 1000 0 0.2 0.4 0.6 0.8 1 E m a x / E g r ound x F HQ CORSIKA, charm via IQMHQ CORSIKA, bottom via IQM
STD CORSIKA, no heavy
Figure 10: Ratio between the number of particles at EAS maximum and the number of particles atground level according to fraction of energy carried by the heavy hadron. The red point it related toratio predicted by the STD CORSIKA.In Fig. 11 we present the average longitudinal profiles for the energy deposited by muons andneutrinos. The shape of curves is shifted to IQM bottom production, both for muons an neutrinos.Again, this effect is more significant when we consider higher energy primary fractions.9 d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavyHQ CORSIKA, charm via CGC − x F > d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavyHQ CORSIKA, charm via CGC − x F > d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavyHQ CORSIKA, bottom via IQM − < ≥ ≥ d E / d X ( P e V / ( g / c m )) Atmospheric depth (g/cm ) STD CORSIKA, no heavyHQ CORSIKA, bottom via IQM − < ≥ ≥ Figure 11: Longitudinal profiles of the energy deposited in the atmosphere by muons (left) and neutrinos(right) considering the HQ CORSIKA (CGC, to x F > . , and IQM) and the STD CORSIKA. Above, F E ≥ . . Below, we have F E < . , F E ≥ . and F E ≥ . . Regarding average longitudinal profile via CGC, we observe small changes in the X max , RM S and N max observables. We can see small effects only for individual profiles. The energy of the heavy secondariesproduced via CGC reaches up to ≈ × eV , what could make such particles reach the groundwith reasonable energy. However, the fraction of energy carried by the particles is very small to causeconsiderable effects in the EAS development.Regarding average longitudinal profiles via IQM, the heavy particles reach higher energy fraction,causing larger changes in the observables. We observed a considerable change in X max , RM S and N max . We highlight some longitudinal profiles with high larger modification in its shape. In fact we canobserve some double core profiles. This behavior certainly will cause important effects in EAS detection.Charm and bottom particles are very rare in EAS, but we shown that its effects are radical in the EASdevelopment. All these EAS effects ( X max shifted, larger RM S and more elongated shape of the EASprofiles) can change significantly the EASs longitudinal profiles seen in fluorescence telescopes and/orthe temporal distribution observed in the surface detectors. The global effect of all these changes inlongitudinal profile is a smaller value of the energy reconstructed of the EAS and higher uncertainties.As for the energy of the primary cosmic rays in question ( ≈ eV ), the values of X max and RM S found in experiments such as the Pierre Auger are smaller. The X max and RM S are directly linked tothe mass composition of the primary cosmic ray. The appearance of charms and bottoms in the EASmakes it more difficult to make such a connection, because of the deeper X max and larger RM S . Thediscussion of whether the detection of EAS with heavy particles is feasible in Fluorescence telescopes orSurface Detectors needs to be analyzed in more detail and will be done in a future publication.The inclusion of heavy hadrons in CORSIKA opens the possibility to test new possibilities of theories10o heavy hadron production and propagation.
Acknowledgments
We would like to thank Alberto Gascón for useful discussions. This work was partially financed by theBrazilian funding agencies CNPq, CAPES, FAPERGS and INCT-FNA (process number 464898/2014-5).
References [1] C. A. García, et al,
Production and propagation of heavy hadrons in air-shower simulators
Corsika implementation of heavy quark production and propagation inExtensive Air Showers , Computer Physics Communications, 185, 638-650, 2014.[5] http://home.thep.lu.se/ ∼ torbjorn/Pythia.html[6] A. Gascón and A. Bueno, Charm production and identification in EAS , Gap Note (Internal notesof Auger Collaboration), 2011-019, 2011.[7] V. P. Gonçalves and M. V. T. Machado,
Saturation physics in ultra high energy cosmic rays: heavyquark production , Journal of High Energy Physics, 04:028, 2007.[8] E. R. Cazaroto, V. P. Gonçalves and F. S. Navarra,
Heavy quark production at LHC in the colordipole formalism , Nuclear Physics A, 872(1):196-209, 2011.[9] N. Sakai, P. Hoyer, C. Peterson and S. J. Brodsky,
The intrinsic charm of the proton , Phys. Lett.B, 93:451-455, 1980.[10] R. Vogt,
Charm Production in Hadronic Collision , Nuclear Physics A, 553:791-798, 1993.[11] Pierre Auger Collaboration,
Depth of Maximum of Air-Shower Profiles at the Pierre Auger Obser-vatory: Measurements at Energies above . eVeV