Investigating post-lhc hadronic interaction models and their predictions of cosmic ray shower observables

MSc Physics and Astronomy Gravitation and Astroparticle Physics Master Thesis Investigating post-lhc hadronic interaction models and their predictions of cosmic ray shower observables June 22, 218 6 ECTS Author: Stephan Runderkamp BSc. Student ID: 1447288 Examiners: prof. dr. ing. Bob van Eijk prof. dr. Patrick Decowski Daily Supervisor: Kasper van Dam MSc. Master Thesis Physics and Astronomy, conducted between September 216 and June 218

Abstract There are large systematic uncertainties in the interpretation of the data of cosmic ray experiments due to the unreliability of hadronic interaction models in MC simulations at high cosmic ray energies. In this work the predicted values of the number and energy of secondaries produced after the first hadronic interaction are compared for the three post-lhc models, and. This study focuses on proton-nitrogen, alpha-nitrogen and iron-nitrogen collisions with primary energies up to 1 2 ev. Large shower-to-shower fluctuations are found, but also significant differences in the predicted values between the three models. An important finding is the significant difference in the predicted number of produced baryons and mesons and the dependence of the multiplicity on primary energy and primary mass number for the three models. The position of shower maximum, the number of ground-level particles (for both photons, electrons, muons and hadrons) and the energy, arrival time and lateral distance to the shower core at ground level are compared for the three post-lhc models. Vertical proton, alpha and iron initiated showers with primary energies up to 1 16.5 ev are considered. Significant differences between the predictions of the models are already found at primary energies far below LHC energies. The three models predict the same value of the position of the shower maximum X max. Large deviations, up to 4%, are found (even below LHC energies) for the number of ground-level hadrons. The energy spectra of ground-level muons is found to be the hardest for QGSJET-II- 4, followed by, followed by. Consequently, ground-level muons arrive latest for, followed by, followed by. This is relevant for describing the lateral shower profile. Could a cosmic ray experiment constrain the models by observing these properties? 1

Contents 1 Introduction 4 2 Cosmic rays 6 2.1 Cosmic rays in the atmosphere............................ 8 2.1.1 Electromagnetic shower............................ 8 2.1.2 Hadronic shower................................ 9 2.1.3 Longitudinal profile.............................. 1 2.1.4 Lateral profile.................................. 14 3 HiSPARC 17 3.1 Charged particles in scintillators........................... 18 3.2 Signal.......................................... 22 3.3 Direction reconstruction................................ 23 3.4 Energy reconstruction................................. 27 4 Hadronic interaction models 29 4.1 Post-LHC hadronic interaction models........................ 3 4.1.1 v34............................... 3 4.1.2................................. 31 4.1.3.................................. 33 4.1.4 Summary.................................... 34 5 Analysis 34 5.1 First hadronic interaction............................... 34 5.1.1 Number of secondaries............................. 35 5.1.2 Energy of secondaries............................. 49 5.2 Cosmic ray shower observables............................ 52 5.2.1 Shower maximum............................... 53 5.2.2 Number of particles at ground level..................... 54 5.2.2.1 Photons................................ 54 5.2.2.2 Electrons............................... 56 5.2.2.3 Muons................................ 58 5.2.2.4 Hadrons................................ 6 5.2.3 Distance to shower core............................ 62 5.2.4 Arrival time................................... 63 5.2.5 Energy distribution............................... 64 6 Conclusion 66 2

7 Discussion 68 8 Outlook 68 3

1 Introduction One of the main challenges of astroparticle physics is to understand the source and propagation of cosmic rays. Cosmic rays are charged particles from extra-terrestrial origin bombarding Earth s atmosphere. The energies of these cosmic rays range from less than 1 GeV up to 1 2 ev. Cosmic rays can be measured directly by detectors in balloons and satellites. However, above 1 15 ev the flux of high-energy cosmic rays becomes too low to measure cosmic rays directly. At energies E 1 14 ev cosmic rays can be measured indirectly. Cosmic rays interact high in the atmosphere with air nuclei (mostly nitrogen and oxygen) and cosmic rays with energies E 1 14 ev initiate cascades, showers of particles, of which a sufficient amount of particles reaches the ground to be measured by ground-level cosmic ray experiments. These showers are called extensive air showers (EAS). To be able to determine the properties of the incoming cosmic ray, such as the energy, direction and particle type, measurements of cosmic ray experiments have to be compared with predictions of MC simulations of showers. Calibration of cosmic ray experiments with high-energy test beams is impossible and therefore we can only compare measured data with data from lots of simulated events. Currently, the largest source of uncertainty of cosmic rays experiments is the unreliability of these MC simulations [1]. This is because at the highest cosmic ray energies, E 1 17 ev, hadronic interactions, and especially the hadronic multiparticle production, are poorly understood, since these energies are higher than that reached by current collider experiments. There are currently three hadronic interaction models, models that describe hadronic interactions, tuned to LHC data with center-of-mass energies s =.9TeV and 7TeV, therefore called the post-lhc models: [2], [3, 4] and [5]. Proton-proton collision energy s = 7TeV corresponds to about 2.5 1 16 ev in the laboratory frame. This means that simulations of hadronic interactions with higher energies are highly uncertain. The predicted values of properties of hadronic interactions at these extrapolated energies depend on the parametrization of the chosen model and the values differ widely as will be shown in this thesis. My work in this thesis has two goals. One goal is to investigate and compare the model-dependent predicted properties of the secondaries produced in proton-nitrogen, alpha-nitrogen and ironnitrogen collisions with lab energies up to the highest cosmic ray energies (E = 1 2 ev). The focus will be on the number of produced secondaries and how the energy is distributed between the different produced particles. The other goal is to investigate and compare the predicted values of cosmic ray observables for vertical (zenith angle = ) proton, alpha and iron initiated showers with primary energies up to E = 1 16.5 ev for the three post-lhc models. Furthermore, it is discussed if and how the differences between these model-dependent predicted values of shower observables could be measured to investigate which one of the models describes the showers best. Analysis is done 4

for the following shower observables: position of the shower maximum, the number of particles at ground level (photons, electrons, muons and hadrons) and the arrival times, energy spectra and lateral distance to the shower core of ground-level electrons and muons. In Section 2 Cosmic Rays will be described in more detail with emphasis on the properties of extensive air showers (EAS). An example of a cosmic ray experiment, the HiSPARC experiment, and its dependence on MC simulations will be described in Section 3. In Section 4 hadronic interactions and the theoretical difficulties to describe these interactions will be explained. Furthermore, in this section the different theoretical approaches to describe the hadronic interactions for the three post-lhc models, and will be described. In Section 5 the results of MC simulations for the three different hadronic interaction models are compared. First, the predicted number of produced secondaries in the first hadronic interaction and the predicted energy distribution between these secondaries are analyzed for the three post-lhc hadronic interaction models. This is done for proton-nitrogen, alpha-nitrogen and iron-nitrogen collisions with primary energies up to E = 1 2 ev. Further in Section 5 the model-dependent predicted values of the position of the shower maximum, the number of particles at ground level (photons, electrons, muons and hadrons) and the arrival times, energy spectra and lateral distance to the shower core of ground-level electrons and muons will be compared for the three models. This analysis is performed for vertical proton, alpha and iron initiated showers with energies up to E = 1 16.5 ev. The findings will be concluded in Section 6 and discussed in Section 7. Furthermore, an outlook will be given in Section 8. 5

2 Cosmic rays Cosmic rays are particles from extra-terrestrial origin bombarding Earth s atmosphere. energies of cosmic rays range from less than 1GeV up to 1 2 ev, which is orders of magnitude higher than the energy that is reached in man-made collider experiments. These cosmic rays can improve our understanding of high energy processes in both galactic and extra-galactic astrophysical sources and the propagation of these particles through the Galaxy. It is one of the main challenges of astroparticle physics to understand how these particles are accelerated to such high energies and in what sources, where these sources are (e.g. galactic or extra-galactic) and how they propagate through the insterstellar medium (or even intergalactic medium). Currently, there are different theories for this and these predict different chemical composition changes of cosmic rays over primary energy. Therefore, investigating the chemical composition of cosmic rays for different cosmic ray primary energies is crucial. From direct measurements it is known that the low primary energy cosmic rays (1.6GeV/nucleon) are mainly protons (about 93%) and alpha particles (about 6%) [6]. About one percent of the cosmic rays are heavier nuclei, such as iron, and electrons and gamma rays (high energy photons), although gamma rays are not necessarily a part of cosmic rays, since they are not charged. As already described, this particle composition of cosmic rays is expected to change over primary energy. The cosmic ray flux depends on primary energy and can be seen in Figure 1. Note that the flux is scaled with E 2.5 to make the differences in the slope visible. The cosmic ray spectrum spans multiple orders of magnitude, from several MeV s (rest masses of the particles) up to 1 2 ev, and even further. The cosmic ray spectrum follows the power law F (E) E γ with spectral index γ 2.7. This power law is associated with the acceleration processes in astrophysical sources. However, as can be seen in Figure 1, the spectral index is not constant. At an energy of 4 1 15 ev the slope steepens, this is called the knee. There are multiple explanations for this feature in the spectrum. One of the explanations is that cosmic rays leak away from our Galaxy since they are less contained by the Galactic magnetic fields at these energies. The gyromagnetic radius R for a relativistic charged particle with proton number Z is proportional to E Z. Therefore, particles with low Z of similar energies have larger gyromagnetic radii and are more likely to escape our Galaxy and not reach Earth. Another explanation is that the maximum energy to which galactic sources can accelerate particles is reached. Cosmic rays with energies up to about 1 17 ev are most likely accelerated by shock acceleration in supernova remnants. Because these shock waves have a finite life time, also a maximum energy can be reached for the particles. This maximum energy is proportional to the proton number Z. For both explanations a transition in cosmic ray chemical composition from protons to heavier nuclei (such as iron) at these energies is expected. A detailed review of the different theories to describe the knee can be found in [7]. At an energy of about 4 1 18 ev the slope flattens again, which is known as the ankle. Most theories interpret this as the transition from galactic sources to extra-galactic courses. However, The 6

the dip model [8] predicts that the slope change comes from energy losses by interactions of protons with CMB photons in extragalactic propagation, thereby producing electron-positron pairs (p + γ CMB p + e e + ), which can only be true if cosmic rays with primary energies E 1 18 ev are mostly protons from extragalactic sources. For the highest energies E 1 19 ev the cosmic ray flux is suppressed. There are multiple explanations for this flux suppression. One explanation is that it is an imprint of the interactions of protons and nuclei with CMB photons, producing pions via the resonance: γ CMB + p + p+π or γ CMB +p + n+π. This process is called the Greisen-Zatsepin-Kuzmin (GZK) [9, 1] energy loss effect and this theory only holds if the sources are extragalactic. The threshold energy for this process is.5 1 2 ev A, with A the nuclear mass of the cosmic ray. Therefore, this theory predicts extragalactic sources and a transition in cosmic ray chemical composition from protons to heavier nuclei (such as iron) at these extremely high energies. Another explanation is that the maximum energy for (extra-)galactic accelerators is reached. If the changes in cosmic ray chemical composition can not be explained by astrophysical processes, this can be a hint of new particle physics at ultrahigh energies. A detailed review of the different theories for ultrahigh energy cosmic rays can be found in [11]. Figure 1: Cosmic ray spectrum (all particles) with data from different experiments. The flux is scaled with E 2.5 to make subtle differences of the slope visible. The c.m. energies at the top-axis for proton-proton collisions correspond to the energies at the bottom-axis for cosmic ray protons. Figure from [12]. 7

2.1 Cosmic rays in the atmosphere Before galactic cosmic rays reach the Earth s atmosphere they have traveled through the interstellar medium of our Galaxy. In [13] a calculation is done that cosmic rays originating from the Galactic centre traverse a column density 1 g/cm 2. Extragalactic cosmic rays travelled through the interstellar medium of the galaxy it is produced in, the intergalactic medium and the interstellar medium of our Galaxy, thereby traversing a higher column density than galactic cosmic rays. The interaction length of protons is about 9g/cm 2, therefore some of the cosmic rays reaching Earth have already interacted. In our atmosphere the density of particles is much higher than in the interstellar medium. Our atmosphere has a column density of about 1g/cm 2, which is more than 11 interaction lengths long. Therefore, all of the cosmic rays interact with air nuclei (mostly nitrogen and oxygen) in the atmosphere before reaching sea level. We usually don t refer to column density, but to atmospheric depth X (in g/cm 2 ) at a given altitude: X = z ρ(h)dh On average the first hadronic interaction takes place at an altitude of about 2 kilometers, but this also depends on the inclination and mass number of the primary particle. When cosmic rays do not enter the atmosphere vertically (from zenith), but with a certain angle θ, the path length is longer and the particles traverse more column density and will therefore interact higher in the atmosphere. Heavier nuclei, with a higher mass number, have shorter interaction lengths. The interaction length of iron is 5g/cm 2, for instance. These particles will interact earlier and therefore higher up in the atmosphere. 2.1.1 Electromagnetic shower In the first hadronic interaction of the incoming cosmic ray with an air nucleus (mostly nitrogen and oxygen) a lot of secondary particles are produced, mostly pions. The neutral pion, π, is unstable and has a short mean lifetime of 8.52 ±.18 1 17 s and will therefore immediately decay before interacting. The π decay mode with the highest branching ratio is the decay into two photons, π γ + γ, with BR = (98.823 ±.34) %. In the vincinity of an atomic nucleus these photons (and photons of other processes) will produce an electron-positron pair, this process is called pair production: γ e + e + Charged light particles like electrons and positrons radiate photons when interacting with electromagnetic fields of nuclei, termed Bremsstrahlung: e e + γ 8

Of course there is also the positron equivalent of this process. The produced photons by Bremsstrahlung will create new electron-positron pairs by pair production, and these electrons and positrons which will later radiate photons by Bremsstrahlung, and so on. More and more particles will be produced and an electromagnetic shower with lots of electrons, positrons and photons is formed. The average energy of the particles decreases with every interaction and after some interactions the energy of the particles is so low that no new particles can be produced anymore. The low-energetic electrons and positrons will be scattered out of the shower or lose their remaining energy by ionization and eventually get absorbed by the atomic nuclei. The low energy photons will get absorbed by the photoelectric effect and Compton scattering. This process is known as an electromagnetic shower. 2.1.2 Hadronic shower In the first hadronic interaction of the incoming cosmic ray with an air nucleus also charged pions and other particles (but in lower numbers) such as kaons are produced. The charged pions, π ±, have a mean lifetime of 26ns at rest. The interaction length of pions is 12g/cm 2, which means that only charged pions with energies higher than about 3GeV will interact before decaying, due to their high Lorentz-factors. But this also depends on the altitude where these pions are produced. High in the atmosphere the density is low and particles are more likely to decay before interacting than at lower altitudes. High-energy charged pions interact with air nuclei, mostly nitrogen and oxygen (π ± + air...). This hadronic interaction produces, depending on the c.m. energy of the collision, multiple secondaries, mostly pions. With every interaction with air nuclei more and more (charged) pions are created and these form a so-called hadronic shower. The average energy of the charged pions decreases with every hadronic interaction with air and after a while the energies are too low for the charged pions to interact before decaying. The pions decay to muons and neutrinos according: π µ + ν µ π + µ + + ν µ The same mechanism is true for kaons, which have similar mean lifetimes and interaction lengths, but these can decay differently. But since kaons are produced in lower amount with respect to pions these processes are less important for understanding the overall shower. Muons are about 2 times heavier than electrons and have a mean lifetime of 2.2µs. Muons with energies of only a few GeV can, due to time dilation, travel several kilometers through the atmosphere, and even reach sea level (the ground). Since the muons are much heavier than electrons the lost energy due to Bremsstrahlung is negligible compared to that of electrons (total radiated power goes with m 4 or m 6 ). Muons on average lose about 2GeV to ionization and have a mean energy of 4GeV at sea level [6]. The muons that do not reach sea level decay to electrons/positrons 9

and neutrinos, thereby feeding the electromagnetic shower: µ + e + + ν e + ν µ µ e + ν e + ν µ 2.1.3 Longitudinal profile An important feature of cosmic ray showers that cosmic ray experiments use to determine the properties of cosmic rays, such as primary energy, particle type and inclination, is the longitudinal development of the shower: the longitudinal profile. The longitudinal profile, the evolution of the number of particles in the atmosphere (over g/cm 2 ), of showers firstly depends on the altitude of the first interaction, which depends on mass number of the primary and the inclination of the shower (as described in Section 2.1). This is the point where the growth of the number of particles starts. The longitudinal profile depends also on the properties of the first interaction itself: the number of secondaries produced in the first interaction. Therefore, the initial growth of the particles in the longitudinal profile, depends on the energy of the collision and the mass number of the incoming particle (and of the target nucleus). The more secondaries produced, the less average energy per particle, which influences the longitudinal development of the shower. In the first few interactions, the first few g/cm 2 after the first interaction, the number of particles increases almost exponentially. After every interaction the average energy of the particles decreases and after some interactions more and more low-energy electrons and positrons are absorbed by atomic nuclei or are scattered out of the shower. Also, low energy photons get absorbed by the photoelectric effect and Compton scattering. Therefore, at a certain point, at the shower maximum X max, the number of particles starts decreasing again. The shower maximum X max increases logarithmically with primary energy, therefore it is a good characteristic of a shower to determine the primary energy. An example of the longitudinal profile of electrons (e and e + ) of a vertical 1 15 ev proton shower is shown in Figure 2. The number of electrons/positrons of this particular example starts increasing from the first interaction at 1g/cm 2 ( 22km) and starts decreasing from the shower maximum X max at about 575g/cm 2 ( 5km). At the ground (sea level), corresponding to 13g/cm 2, the number of electrons/positrons is about 1 5. 1

Figure 2: Longitudinal profile of electrons (e and e + ) of a vertical 1 15 ev proton initiated shower. Figure from [14]. However, showers with the same initial conditions are not the same. There are, often large, shower-to-shower fluctuations. For instance, the column depth between two interactions is not always equal to the (average) interaction length, it is subject to probability. Also, the number of produced secondaries per interaction and the energy/momentum distributions of these secondaries are all a matter of probability (following cross-sections). Therefore, the longitudinal development of showers with the same initial conditions may differ widely. For cosmic ray experiments it is important that these shower-to-shower fluctuations are known very precisely. This is the reason why for a cosmic ray experiment lots of simulations of showers have to be done, for instance to determine the uncertainty of the experiment. An example of such shower-to-shower fluctuations is depicted in Figure 3, where the longitudinal profile of electrons (e and e + ) of 5 vertical 1 15 ev proton showers are plotted. Most of the 5 showers start developing before 2g/cm 2 and have X max at about 5g/cm 2, but there are three showers that develop slower and have X max at 8g/cm 2. The number of electrons/positrons at the ground (sea level) ranges from 5 1 4 to 5 1 5. Figure 3: Longitudinal profile of electrons (e and e + ) of 5 vertical 1 15 ev proton initiated showers. Notice the large fluctuations in position of first interaction, position of shower maximum and number of electrons at different atmospheric depths. Figure from [14]. For showers with primary energies lower than 1 14 ev almost all the particles (mostly photons and electrons/positrons) are absorbed in the atmosphere and only some of the muons can reach 11

the ground. For showers with primary energies higher than 1 14 ev the number of electrons on the ground exceeds the number of muons. For these showers enough particles reach the ground to be able to observe them with ground-level detectors. These showers are called Extensive Air Showers (EAS). An example of such an Extensive Air Shower is presented in Figure 4, where the particle tracks of a vertical 1 15 ev proton shower are plotted. Notice that the first interaction takes place at an altitude of about 2km. A high number of particles reaches sea level (z = km), therefore it is an EAS. Figure 4: Particle tracks of a vertical 1 15 ev proton initiated shower. Left: electron (e and e + ) tracks (red). Right: hadron tracks (blue) drawn over the muon (µ and µ + ) tracks (green) drawn over the electron (e and e + ) tracks (red). The number of electrons, muons and hadrons at ground level (z = ) are about 1 5, 1 4 and 2 1 3, respectively. Figure from [14]. The longitudinal profile of different particles in an EAS, in this case a vertical 1 15 ev proton shower, is shown in Figure 5. It can be seen that the most abundant particles in an EAS are photons, followed by electrons, followed by muons. The number of muons at this primary energy is quite stable for large atmospheric depths, whereas the number of photons and electrons decreases because of absorption in the atmosphere. 12

Figure 5: Longitudinal profile of photons, electrons (e and e + ) and muons (µ and µ + ) of a vertical 1 15 ev proton initiated shower. Figure from [15]. As described earlier, for non-eas showers, with primary energies lower than 1TeV (1 14 ev), most of the electrons and photons will be absorbed in the atmosphere. Therefore, the number of muons at sea level is higher than that of electrons and photons. Since the flux of these low-energy showers is much higher than that of high-energy (extensive air) showers, as can be seen again in Figure 1, the muon flux at sea level is about four times higher than the electron flux. The particle fluxes for muons, electrons and protons as a function of atmospheric depth are presented in Figure 6. The muon rate at the ground is 1cm 2 min 1. Figure 6: Particle flux for muons (both µ and µ + ), electrons (both e and e + ) and protons as a function of atmospheric depth. Figure from [15]. 13

2.1.4 Lateral profile Another important feature of cosmic ray showers that experiments use to determine properties of the cosmic ray is the lateral development of the shower: the lateral profile. With every hadronic and electromagnetic interaction the particles in the shower obtain some transverse momenta. In electromagnetic interactions transverse momentum is obtained by bremsstrahlung and pair production due to the angles between the particles. can also be deflected by Coulomb forces and Compton scattering. The (produced) particles in hadronic collisions also obtain transverse momentum, as do the particles produced by particle decay. Due to these processes the shower widens. The footprint of an EAS on the ground can be several kilometers wide. The properties of this footprint depend on the primary energy of the cosmic ray, the type of the incoming cosmic ray particle (e.g. proton, alpha, iron) and the inclination of the shower. On the ground the particle density is the highest in the shower core, which follows the axis of the incoming cosmic ray, and this particle density decreases outwards. The number of particles that can travel to the ground increases with the primary energy of the shower, therefore the particle density also increases with primary energy. Cosmic rays with higher mass numbers interact higher in the atmosphere because their interaction length is shorter (i.e. probability to interact is higher). As a result, X max is also lower (higher in the atmosphere) and particles are more attenuated, therefore decreasing the particle density on the ground. The particle densities on the ground of a proton shower is thus higher than that of an iron or alpha shower with the same primary energies. The particle densities and the form of the footprint on the ground also depend on the inclination of the shower. An inclined shower with zenith angle θ traversed 1 cos θ times more column depth than a vertical shower. Therefore particles are more attenuated, decreasing the particle densities on the ground. The transverse momenta cause the shower to develop, more or less isotropically, perpendicular to the shower axis (lateral direction). This means that for a vertical shower the footprint is circular around the shower core. But for inclined showers the footprint becomes elliptical. This effect is illustrated in Figure 7, where an inclined shower with zenith angle θ reaches the ground. (right side) at distance r from shower core arrive t later than the core of the shower (front). on the left arrived earlier than the shower core. 14

Figure 7: Side view of an inclined shower with zenith angle θ at ground level. (right side) at distance r from shower core arrive t later than the core of the shower (front). on the left side of the shower core arrived earlier than the shower core. Figure from [15]. The lateral density of electrons (e and e + ), muons (µ and µ + ) and the sum of both on the ground (horizontal) of a vertical 1 16 ev proton shower and a 1 16 ev proton shower with zenith angle θ = 45 are shown in Figure 8. The density of the vertical shower on the left is dominated by the electrons. At distances larger than several hundred meters from the shower core the muon density start to become a significant fraction of the sum density and at a distance larger than 1km the muon density starts to overtake the electron density. The sum density of the θ = 45 shower on the right is also dominated by electrons, but here the muon density starts to overtake the electron density much closer to the shower core, at about 3m. The differences between the densities in the shower core of the vertical and the inclined shower is because of the 1 cos θ times higher column depth. The large decrease of the electron density with respect to the decrease of the muon density is because electrons are more attenuated than muons. The muon density starts to overtake the electron densities at some (large) distance from the shower core because at these distances the traversed column depth is higher than that to the shower core and electrons are more attenuated than muons. Another reason is the differences in expected lateral distances between electrons and muons. Electrons undergo many interactions in the shower, which is a random walk process. Therefore their expected lateral distance follows x h, with h the vertical distance. Muons travel nearly unattenuated to the ground with few (or none) interactions and their expected lateral distance follows x h. 15

Figure 8: Horizontal density at sea level of electrons (e and e + ) in red, muons (µ and µ + ) in green and the sum of both as a dashed line of a vertical (θ = ) 1 16 ev proton initiated shower (left) and a 1 16 ev proton initiated shower with zenith angle θ = 45 (right). Figure from [14]. Note that in Figure 8 the photon density is not taken into account. Photons are, however, the dominant particle at sea level. In Figure 9 the lateral profile of photons, electrons (e and e + ) and muons (µ and µ + ) at sea level is plotted of a vertical 1 15 ev proton initiated shower. The primary energy of this cosmic ray is an order of magnitude lower than that in the left panel of 8. Note that the densities of the electrons and muons (at all distances) of the vertical 1 15 ev proton initiated shower are much lower than that of the vertical 1 16 ev proton initiated shower. It can also be seen that for the 1 15 ev proton initiated shower the muon density starts to overtake the electron density much closer to the shower core. This is because of the lower X max (higher up in the atmosphere) of the 1 15 ev shower and the fact that the on average lower-energy electrons in the 1 15 ev shower are more attenuated in the atmosphere. Figure 9: Lateral profile of photons, electrons (e and e + ) and muons (µ and µ + ) at sea level of a vertical (θ = ) 1 15 ev proton initiated shower. Figure from [15]. 16

The total number of particles on the ground, the shape of the footprint and the lateral densities can be used by cosmic ray experiments to investigate cosmic rays. An example of such a cosmic ray experiment, the HiSPARC experiment, will be described in the next chapter. Also will be explained how this experiment uses the properties of the particles at ground level to investigate cosmic rays and how the experiment relies on simulations of showers. 3 HiSPARC The HiSPARC Cosmic Ray Experiment aims to investigate cosmic ray air showers and reconstruct the primary energy and direction of showers. The scientific purpose of HiSPARC is to study large-scale correlated effects like measuring the Gerasimova-Zatsepin effect. The Gerasimova-Zatsepin effect is when a solar photon interacts with a cosmic ray nucleus, emitting a particle, mostly a proton. A detailed description of this effect can be found in [16]. This effect can be measured by detecting both the shower from the residual cosmic ray nucleus and the shower from the emitted proton. HiSPARC is suitable for measuring this effect because it has a relatively large surface area and (some of) the detectors are close together, much closer than that of the Pierre Auger observatory in Argentina [17], for instance. Because the detectors are closer together lower energy showers can be detected, from which there are many, and the possibility of measuring both the showers is higher. Apart from the scientific questions, another purpose of HiSPARC is to reach out to High School students (and teachers) and introduce them to research in modern physics. Currently, HiSPARC has over 1 detection stations spread out over the roofs of high schools and (scientific) institutions in the Netherlands, Denmark and England, with the vast majority in the Netherlands (about 9%). A HiSPARC detector is composed of a 2cm thick 1cm 5cm plastic scintillator, a plastic light guide and a photo multiplier tube (PMT) and is illustrated in Figure 1. The detector stations can have two or four detectors and are put in separate ski-boxes on the roofs of the participating high schools and institutions. There are two configurations for four-detector stations: an equilateral triangle with a fourth detector in the barycenter or a 6 diamond configuration. These two configurations are depicted in Figure 11. In Figure 12 the locations of the HiSPARC detector stations in the Netherlands can be seen. Figure 1: Illustration of HiSPARC detector with scintillator in white, light guide in gray and PMT in black. Figure from [14]. 17

Figure 11: Two configurations of four-detector stations. Triangle station on the left and 6 diamond station on the right. Figure from [14]. Figure 12: HiSPARC detector station locations in the Netherlands. Figure from [14]. 3.1 Charged particles in scintillators High-energy charged particles in matter can lose energy by collisions with atoms in the medium (ionization and excitation) and by radiation (Bremsstrahlung). In the HiSPARC detector this medium is a plastic scintillator. It consists of polyvinyltoluene, a solid plastic solvent, with a small amount of the fluor anthracene dissolved into it. The electrons in the atoms of the polyvinyltoluene absorb energy from the charged particles, get excited and emit photons in their re-excitation which will excite the fluor (anthracene). π-molecular orbital free valence electrons in the fluor can get excited to singlet or triplet excited states, see Figure 13. After internal degradation, which is decay without radiation, the molecule de-excites from the S 1 state to the ground state or one of the vibrational sublevels of the ground state S. This process is called fluorescence and in this process photons are emitted. In the indirect de-excitation of the lowest excited triplet state (T ) to the ground state also photons are emitted. However, this process, which is called phosphorescence, is slower than the singlet de-excitation (fluorescence). The emitted scintillation photons will not be absorbed in the scintillator and can travel through the scintillator and the plastic light guide to the PMT (transmission). The PMT signal from 18

the photoelectrons is converted to a digital value (ADC counts) and this is the detector signal. Figure 13: π-orbital energy levels, both singlet S and triplet T states of a fluor molecule. In the de-excitation from S 1 to the ground state (or its vibrational sublevels S 1, S 2 etc.) fluorescence takes place and photons are emitted. In the indirect de-excitation from T to the ground state (or its vibrational sublevels) phosphorescence takes place and photons are emitted, although slower. There are no photons emitted by internal degradation. Figure from [15]. The detector signal depends on the amount of scintillation photons produced and therefore the amount of energy lost by the particle in the medium by ionization/excitation. The mean energy loss of a particle in a medium, the stopping power, can be calculated with the Bethe-Bloch equation. de = K dx 2 ( Z 1 A β 2 ln 2m ec 2 β 2 γ 2 T max I 2 + F δ C ), (1) where de dx is the mean energy loss per thickness of the medium, the stopping power, in MeV g 1 cm 2. K is a constant depending on the electron mass and classical electron radius, Z is the proton number and A is the mass number of the medium. β is the relative velocity, the velocity of the particle relative to the speed of light, v c, and is close to 1 for most of the incoming particles. γ = 1 β 1 is the Lorentz factor and can become very high for particles with velocities 2 close to the speed of light. T max is the maximum kinetic energy transfer in one collision and depends on the mass of the incoming particle. Therefore, this parameter is different for muons and electrons/positrons, because of the mass difference. I is the mean excitation energy of the medium. F is another parameter that is different for different types of particles and depends on the velocities of the incoming particles. δ is the density correction and is different for different regimes of the value βγ. C is a so-called shell correction and depends on βγ of the particles, the mean excitation energy I of the medium and proton number Z of the medium. In [14] 19

the stopping power is calculated with the values of Z, A, δ, I and C for polyvinyltoluene, the scintillator material. The stopping power of the scintillator for muons, electrons and positrons is shown in Figure 14 as a function of βγ. However, in this figure only the stopping power due to collisions (ionization) is shown. The contribution of radiation (Bremsstrahlung) to the total stopping power, which becomes important for values βγ > 1, is not shown because the produced photons in this process do not interact with the medium, therefore leave the scintillator, and will not be detected. Figure 14: Stopping power from ionization for electrons, positrons and muons in the HiSPARC scintillator detector (polyvinyltoluene). Figure from [14]. The mean energy loss de dx can be calculated by the Bethe-Bloch equation, but in reality the energy loss follows a distribution because of a process called energy straggling. This distribution is called the Landau energy loss distribution and follows: f( ) = 1 φ(λ), (2) ξ with the energy loss in MeV and ξ equal to: ξ = KZ 2Aβ 2 x, (3) where x is the thickness of the medium in g cm 2, which for our 2cm thick scintillator with density 1.3 g cm 3 equals to 2.6g cm 2. However, this is only true for vertically incoming particles. incoming under an angle θ have traveled a thickness φ(λ) is the Landau probability density function with λ given by: λ = ξ x cos θ in the medium. ( ) ξ ln β 2 1 + C T E, (4) max where C E is the Euler-Mascheroni constant and is the mean energy loss (from stopping power). In [14] the energy loss distributions for vertically incoming high-energy electrons, positrons 2

and muons are calculated for HiSPARC s 2cm thick polyvinyltoluene-based scintillator. These distributions can be seen in Figure 15. The most probable energy losses, corresponding to the location of peaks in the distributions, are lower than the mean energy losses calculated from the Bethe-Bloch equation. According to [14], the Landau distribution is different for different particles. However, there is disagreement, in [18] only very small differences are found between the Landau distribution of different particles. Figure 15: Landau energy loss distribution for vertically incoming high-energy electrons, positrons (βγ > 68) and muons (βγ > 4.3) for the HiSPARC scintillator detector. Figure from [14]. As described earlier, particles coming in with zenith angle θ have traveled a larger distance in the medium, following x cos θ. Consequently, the Landau energy loss distribution of the particles changes with θ. In Figure 16 the energy loss distributions for high-energy muons (βγ 1) for three different zenith angles is presented. It can be observed that the most probable value of energy loss (location of the peak) increases with θ and also the distributions broaden. Figure 16: Landau energy loss distribution for high-energy muons (βγ 1) for zenith angles, 22.5 and 45. Notice that the most probable value of energy loss (location of peak) increases with θ and also the distributions broaden. Figure from [14]. 21

3.2 Signal In the following subsections a summary will be given of the HiSPARC signal and how the direction and energy of cosmic rays are reconstructed. A more detailed and more actual description can be found in [18]. For an event to be stored at least two or more detectors of the same (four-detector) station have to have had a signal within a 1.5µs timeframe. The idea is that if the event is an EAS more than one detector will have detected a particle. Events where only one detector detects a signal are associated with background muons from low-energy cosmic rays (e.g. E < 1 14 ev) and have to be rejected. The threshold for a signal is if two detectors of the same station have a signal larger than 7mV or three detectors detect a signal larger than 3mV. An example of an event where all four detectors in a four-detector station have a signal can be seen in Figure 17. It is clear that there can be a time difference of several nanoseconds between signals of the same event and that the ADC counts peak (pulse height) is at different heights for the same event in the four different detectors. The peaks of the signal (in ADC counts) is called the pulse height and the integral of the signals (in ADC counts ns) is the pulse integral. The pulse height and pulse integral of a full week of detecting events of a four-detector station are shown in Figure 18 on the left and right, respectively. The peak of the pulse integral is associated with the peak of the Landau distribution. It is the most probable energy loss, and the minimum of the stopping power distribution. Therefore, the value at the peak of the pulse integral is taken as the value for 1 MIP (minimum ionizing particle) instead of in ACD ns. This is, however, not entirely accurate because it depends on the angle of incidence where this MIP peak should exactly be. Figure 17: Signals (in ADC counts) of an event with a signal in all four detectors of a four-plate station. Figure from [14]. 22

Figure 18: Pulse height and pulse integral of a full week of events for four detectors in a station. Figure from [14]. Due to the uncertainty in quantum efficiency of the PMT and the uncertainty of how many of the produced photons travel to the PMT (transmission efficiency) the real signal (the pulse integral) is a convolution of the Landau energy loss distribution and the detector resolution, which roughly follows a Gaussian distribution. Therefore, the shape of the pulse integral is somewhat broader than the Landau energy loss distribution. 3.3 Direction reconstruction If at least three or more detectors of a four-detector station detect signals from one shower, the direction of the incoming cosmic ray can be reconstructed from the arrival times of the particles in the detectors. This is also the case if three or more stations from the same cluster, for instance the cluster at the Science Park in Amsterdam, have detected a signal. A detailed description of the direction reconstruction of the HiSPARC experiment can be found in [15]. The direction reconstruction is more accurate for larger arrival time differences between the detectors. Then, the timing uncertainties due to the transport in the detector and the shower front shape are relatively smaller. The arrival time differences are larger if the distances between the detectors are larger. Therefore, for the four-detector station in the form an equilateral triangle with a fourth detector in the barycenter, on the left in Figure 11, the direction reconstruction is most accurate if the three corner detectors detect a signal. In Figure 19 the fraction of simulated events for which all three corner detectors detect at least one charged particle is plotted as a function of core distance for 1PeV proton initiated showers for zenith angles θ =, 22.5 and 35. The detection efficiency is larger for smaller zenith angles. This is because the number of ground-level particles is larger for showers with smaller zenith due to the shorter distance traversed in the atmosphere. Therefore, the probability to measure particles is larger for these showers. The detection efficiency decreases over distance from the shower core. This is because near the shower core the particle density is higher and it is therefore more probable to measure particles. If the arrival times between two detectors is 23

larger than the light time between two detectors ( 33ns) the detected event is rejected and no direction reconstruction is done because the outcome would be unphysical. Especially events at large distances from the shower core, where the number of detected particles is small and the shape of the shower front is curved more of the detected events will be rejected. This can be seen in Figure 2, where the fraction of detected events is shown for which a direction reconstruction can be done as a function of distance to the shower core for 1PeV initiated proton showers for three zenith angles. The reconstruction efficiency is larger for smaller zenith angles, because of the higher number of ground-level particles for such showers. Figure 19: Fraction of simulated events for which three HiSPARC detectors at the corners of a 1m-sided equilateral triangle detect at least one charged particle as a function of core distance for 1PeV initiated proton showers for three zenith angles. The detection efficiency is larger for smaller zenith angles and decreases over distance from the shower core. Figure from [15]. 24

Figure 2: Fraction of detected events for which a direction reconstruction can be done as a function of the distance to the core for 1PeV initiated proton showers for three zenith angles. The reconstruction efficiency is larger for smaller zenith angles and decreases over distance from the shower core. Figure from [15]. In Figure 21 the precision of the azimuth reconstruction is shown as a function of simulated azimuth angle. This is done for 1PeV initiated proton showers with zenith angle θ = 22.5. It is found that the azimuth reconstruction is more precise if at least two charged particles have to be detected in all corner detectors instead of one. Note that the uncertainty range in terms of standard deviations is larger than these 5% regions. The precision of the zenith reconstruction as a function of simulated zenith angle is shown for 1 PeV initiated proton showers in Figure 22. The zenith reconstruction is also more precise for the stronger condition that at least two charged particles have to be detected in the three corner detectors. The precision of the zenith reconstruction increases with zenith angle. This is because for showers with large zenith angles the arrival time difference between the detector is larger. Therefore the relative timing uncertainties due to the transport in the detector and the shower front shape are relatively smaller, improving the precision of the zenith reconstruction. Note that the uncertainty range in terms of standard deviations is larger than these 5% regions. 25

Figure 21: The precision of the azimuth reconstruction as a function of the simulated azimuth angle for 1PeV initiated proton showers with zenith angle θ = 22.5. The white dots are the median values and the grey range contains 25% of the events both above and below the median (5%). For the figure on the right at least two charged particles (instead of one) have to be detected in all corner detectors, which leads to a more precise azimuth reconstruction. Figure from [15]. Figure 22: The precision of the zenith reconstruction as a function of the simulated zenith angle for 1PeV initiated proton showers. The white dots are the median values and the grey range contains 25% of the events both above and below the median (5%). For the figure on the right at least two charged particles (instead of one) have to be detected in all corner detectors, which leads to a more precise zenith reconstruction. The precision of the zenith reconstruction increases with zenith angle. Figure from [15]. 26

3.4 Energy reconstruction For an Extensive Air Shower (EAS) the lateral profile on the ground, such as the ones in Figure 8 is often approximated by the Nishimura-Kamata-Greisen (NKG) function [19, 2], which depends on the distance from the shower core r and is given by: ( ) r s 2 ( ρ(r) = kn 1 + r ) s 4.5, (5) R M R M with ρ(r) the particle density at a distance r from the shower core, k a constant, N the number of particles, R M the Molière radius and s a paramater for the age of the shower. In reality the position of the shower core and the total number of particles N in the shower are not known. These have to be determined by minimizing the following function: χ 2 = n ( ) wi ρ 2 i, (6) i=1 with w i the real signal in detector i, ρ i the expected signal in detector i from the NKG-function and σ i = ρ i. If we now write the NKG-function as ρ(r) = Nv(r) the χ 2 function changes to: σ i χ 2 = n i=1 (w i Nv i ) 2 Nv i. (7) The value N for which χ 2 is minimized, in other words for which δχ2 δn =, then equals: ( w2 i v ) N = i (8) vi With this condition for N the χ 2 function becomes: χ 2 ( w2 i v ) = 2 i vi 2 w i (9) vi The real difficulty is then finding the right shower core position for which χ 2 is minimized. After trying lots of different shower core positions the optimal shower core position is found for which χ 2 is minimized. From this the corresponding number of particles, or shower size, N can be determined. With the expected shower size N and the expected zenith angle θ from the direction reconstruction the cosmic ray energy can be determined. This is done with the help of a large number of simulated cosmic ray air showers of different energies and with different inclinations. The 27

dependence of the shower size N e+µ on cosmic ray energy for three zenith angles, 45 and 6 is depicted in Figure 23. Figure 23: Shower size N e+µ (with errors) as a function of cosmic ray energy for three zenith angles, 45 and 6 from cosmic ray simulations for proton primaries. Figure from [14]. The error bars describe the shower-to-shower fluctuations between the showers with the same initial conditions. It is important to emphasize that this energy reconstruction can only be done by comparing with lots of simulated events. However, these simulations, and therefore also the link between shower size and the estimated cosmic ray energy for different inclinations, depend on the chosen hadronic interaction model in these simulations. In the next section will be explained what an hadronic interaction model is and why it is so difficult to model high-energy hadronic interactions. Also, three of the main hadronic interaction models will be described in depth, what their differences are and what the advantages and disadvantages of these models are. 28

4 Hadronic interaction models Hadrons are particles consisting of quarks and gluons. These quarks and gluons have color charge. When these color charges are exchanged between two separate hadrons a hadronic interaction takes place. Inside a hadron there are both valence quarks and an indefinite amount of virtual (sea) quarks. Valence quarks determine the quantum number of the hadron and sea quarks do not. There are two types of hadrons: baryons and mesons. Baryons contain three valence quarks, mesons contain two. Protons and neutrons are examples of baryons. Protons consist of two up quarks and one down quark (uud) and neutrons consist of two down quarks and one up quark (udd). Pions are examples of mesons. π +, for instance, consist of one up quark and one down antiquark (u d). Hadronic interactions, like proton-proton collisions in hadron colliders, can be described by quantum chromodynamics (QCD). This theory describes strong interactions, the interactions between quarks and gluons. However, perturbative QCD can only be used for hard interactions: processes with large momentum transfers. It is known that valence quarks carry most of the momentum of the hadron, often over 5% [21]. In hadronic interactions we can distinguish two kinds of produced particles: leading particles, the particles which contain one or more valence quarks, and the large majority of particles which do not contain any of these valence quarks. Most of the produced particles without valence quarks have a low momentum fraction of the initial hadron. Therefore a lot of these can be produced. On the contrary, leading particles have a flat distribution in momentum. These particles often have large momentum fraction of the initial hadron and only a few of them will be produced. QCD can well describe the production of hard (often leading) particles, but inaccurately describes the production of soft particles. Therefore, to understand multiparticle production, we need to combine QCD with phenomenological models. At high energies other processes that cannot be described by QCD, for instance, parton saturation, take place. At high energies the parton densities, the density of the gluons and quarks in the hadrons, can become so high that the wave functions of the partons start to overlap. This process is called parton saturation [22]. This process decreases the growth of individual partons and therefore decreases the multiplicity, the amount of secondary particles produced, over energy. This process becomes more important with increasing center-of-mass (c.m.) energy. To predict hadronic interactions at extrapolated energies, at center-of-mass energies much higher than that of LHC, a flexible model that describes the processes above for proton-proton, proton-nucleus and nucleus-nucleus interactions is necessary. In the next section three hadronic interaction models are described. These models are called post-lhc models, since they are tuned to the newest LHC data. 29

4.1 Post-LHC hadronic interaction models 4.1.1 v34 The Monte Carlo event generator EPOS [23, 2, 24] was originally used to describe heavy-ion collisions at RHIC, a heavy-ion collider experiment in the United States. is a much faster version of the original EPOS model, but has more parameters and is therefore less precise than the original one for heavy-ion collisions. For instance, it does not take into account the 3D hydrodynamic flow calculations [24] of the original EPOS model for heavy-ion collisions. But, since these precise calculations are less relevant (less collective hydrodynamic flow) for hadronhadron and hadron-nucleus collisions, is a very good model to describe minimum bias proton-proton (hadron-hadron) and proton-nucleus (hadron-nucleus) collisions. The parameters of the model are tuned to large sets of hadronic interaction data of both hadron-hadron, hadronnucleus and nucleus-nucleus collisions. EPOS is based on the Gribov-Regge effective field theory [25, 26, 27, 28]. This describes a hadronic interaction with multigluon diagrams, Pomerons, for both soft and (semi)hard and hard interactions. In a hadronic interaction many interactions are taking place in parallel. An example of such an elementary scattering is depicted in Figure 24 and is called the parton ladder or the cut Pomeron. If the first parton of the projectile/target (or nucleon in Figure 24) is a sea quark or a gluon and the momentum transfer q 2 is lower than Q 2 (few GeV 2 ), the cutoff above which pqcd is applicable, a soft Pomeron is exchanged. These soft Pomeron emissions lead to a soft preevolution; a nonperturbative parton cascade takes place. This gives rise to the large number of low transverse momentum hadrons. The semi-hard scatterings with q 2 > Q 2 can be calculated perturbatively and the parton evolution follows the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations. For hard interactions the ladder partons eventually form a flux tube, see Figure 24, where fragmentation takes place. Quark-antiquark pairs are formed, which later form the hadrons. Figure 24: Elementary interaction in the EPOS model. Figure from [24]. In the non-linear effects are parametrized. At higher energy the gluon densities 3