The Muon Problem of UHECR Air Showers Rijksuniversiteit Groningen

Home , Air shower (physics)

The Muon Problem of UHECR Air Showers

Rijksuniversiteit Groningen

Bachelor Research Project

Jonah Stalknecht S2767163

supervised by

Professor Olaf Scholten

July 7, 2017 TABLE OF CONTENTS

SUMMARY ...... 3

I INTRODUCTION ...... 4

II HISTORY ...... 5

III THEORY ...... 8

3.1 Cosmic Rays and Air Showers ...... 8

3.2 The Heitler Model ...... 11

3.3 The Generalized Heitler Model ...... 12

3.3.1 Inelasticity ...... 14

3.3.2 Superposition Model ...... 15

3.4 Hadronic Interactions ...... 16

IV OBSERVATIONS ...... 18

V MODELS AND SIMULATIONS ...... 20

VI THE MUON PROBLEM ...... 22

VII POSSIBLE SOLUTIONS ...... 24

7.1 Enhanced Baryon Production ...... 24

7.2 Enhanced Remnant Excitation ...... 25

7.3 New Physics ...... 28

1 7.3.1 Chiral Symmetry Resotartion ...... 29

7.3.2 String Percolation ...... 30

7.3.3 Other Solutions ...... 32

VIII WHAT’S NEXT? ...... 35

IX CONCLUDING REMARKS ...... 37

2 SUMMARY

Results from the Pierre Auger Observatory have shown that more muons are being created in UHECR air showers than our current models predict. There are many proposed solutions to this muon problem, some of which are based on newly proposed physical phenomena regarding hadronic interactions, while others take a more conventional approach. Models based on new physics include Chiral Symmetry

Restoration and String Percolation, the more conventional methods are mainly based on enhanced baryon production and enhanced remnant excitation. Physics of air showers and basic hadronic interactions are used to understand and discuss these proposed solutions to the muon problem. It is found that more results from the

Auger Prime and IceCube observatory are needed before we can tell which of the proposed solutions work and which ones don’t.

3 CHAPTER I

INTRODUCTION

For almost two decades there have been hints that more muons are created in Ultra

High Energy Cosmic Ray (UHECR) air showers than our models predict, this has been conﬁrmed beyond doubt in 2016 by the Pierre Auger collaboration. The best models for simulating air showers have a discrepancy of 30-80% when it comes to predicting the number of muons that are created. The energies involved in these

UHECR air showers are far higher than the energies reached in particle accelerators here on earth, which makes them an interesting phenomenon to study how particles behave at these immensely high energies.

Many possible solutions to this muon problem have been proposed, but none have shown to be conclusive as of yet. In this bachelor research project we will look at the physics behind air showers and hadronic interactions and use this to discuss the proposed solutions to the muon problem. In the ﬁnal chapters of this bachelor research project we will focus on what future experiments can be used to narrow the broad range of possible solutions and to get closer to understanding the actual mechanisms responsible for the unexpectedly large amount of muons we observe.

4 CHAPTER II

HISTORY

All the way back in 1785, Coulomb found that electroscopes spontaneously discharge, even when properly isolated. After the discovery of spontaneous radioactivity by Bec- querel in 1896, it was found that electroscopes promptly discharge in the presence of radioactive material. At this time, the dominant opinion was that the discharge of the electroscope was due to radiation from the earth.

This idea was ﬁrst tested by Theodor Wulf.

Wulf, a Jesuit priest and physics teacher in Valken- burg, developed a very sensitive electroscope that could detect the presence of energetic charged particles. Be- cause it was thought that the main source of radiation came from the earth, Wulf predicted that the further he got from these sources, the less radiation he would detect.

He tested this idea by measuring the levels of radiation Figure 1: The electroscope at the bottom of the Eiffel Tower and at the top. But to used by Wulf around 1912 [A. his surprise, Wulf measured a slight increase of radiation De Angelis, Wikimedia] at 300 m height, instead of a decrease. This was the first hint we had for the existence of cosmic rays, but his findings were largely ignored or dismissed.

This changed when Victor Hess saw the ﬁndings of Wulf. To test the results of

Wulf further, Hess took an electroscope in a balloon ﬂight, and measured the levels of radiation at altitudes up to 5.3 km. It was found that the levels of radiation increased signiﬁcantly above an altitude of about 1 km. The conclusion was that

5 there is radiation reaching the earth from outer space. By doing the experiment during the night, Hess was able to exclude the sun as a source of this radiation. His work lead to him receiving the nobel prize in physics in 1936.

One of the ﬁrst hypotheses to explain cosmic rays was

due to the American physicist Robert Millikan. Millikan

was the son of a minister and a religious man. He believed

that cosmic rays were gamma-rays that were ’birth cries’

of atoms that were continuously being created by god to

counteract the increase of entropy in the universe as to

prevent the ’heat death’ of the universe. Millikan is also

the ﬁrst to coin the term ”cosmic rays”. The view that

cosmic rays are gamma rays did not go undisputed and

was attacked by many scientists, e.g. by Compton. By

Figure 2: The Millikan- measuring the intensity of radiation during a trip from Compton cosmic ray discus- Java to the Netherlands, Jacob Clay found that the in- sion even made it to the front- page of the New York Times tensity was dependent on latitude. It was proposed that (1932). this was because of the earth’s magnetic ﬁeld having an

inﬂuence on the cosmic rays. This would only be possible if cosmic rays consisted of charged particles, which is contradictory to Millikan’s view that cosmic rays are gamma rays. After a worldwide survey that was initiated by

Compton, it was shown conclusively that cosmic ray intensity indeed depended on latitude. The result that the intensity of radiation was dependent on latitude was conﬁrmed in 1933 by Pierre Auger and Louis Leprince-Ringuet.

It was this same Pierre Auger who, together with Roland Maze, measured time coincidence of cosmic rays in detectors 20 meters apart, meaning that they came from a common event. For a time these events, which we now call ”air-showers”, were called ”Auger showers” [Auger, 1985, page 214].

6 Auger estimated that large air showers must be created by primary particles of energies of at least 1015eV . In reality, we have found primaries with energies up to

3 × 1020eV (the so called ”Oh-my-god particle”, observed in 1991). This energy is remarkable because it is even higher than the GZK-limit of 5 × 1019eV , which is a proposed upper limit for the energy of cosmic rays. These energies are millions of times larger than the highest energies achieved in particle accelerators here on earth

(the Large Hadron Collider only reaches about 1013eV ). The source of these Ultra

High Energy Cosmic Rays is still largely unknown and continues to be a hot topic of debate to this day.

7 CHAPTER III

THEORY

3.1 Cosmic Rays and Air Showers

When a cosmic ray enters the atmosphere, it will collide with the nucleus of an atom in the air. Such a cosmic ray is called a primary, which could be an electron, proton,

nucleus, photon or, rarely, a positron. There are primaries that reach the earth

constantly, with varying energies from as low as 1GeV up to 1020eV .

Low energy primaries we see quite often, while the more energetic ones are more

rare. This can be visualized in ﬁgure 3, which represents the ﬂux of cosmic rays. Here

we see that low energy primaries reach the earth often, primaries of about 1010eV

reach the earth more often than once every second per square meter of surface area,

while primaries with energies above 1020eV are only found once per century per

square kilometre. There is an approximate power law that describes this behaviour:

dN −α dE ∼ E , with α roughly equal to 3. In figure 3 we can, however, find some breaks from the power law. We talk about two main breaks we can see in the figure 3, at an energy of about 3 × 1015eV we see the so called ”knee” and at about 1018 − 1019eV

we see the so called ”ankle”. The reason we see these breaks is often explained as

a consequence of diﬀerent sources of the particles. For example, The magnetic ﬁeld

of the milky way can only contain protons with energies up to about 1015eV , even

though the shock wave of supernovae can accelerate protons up to 1017eV . So protons

with an energy above ∼ 1015eV have to be extragalactic in origin. This can be used to

explain the dip in the graph at this energy. The change at the ankle can be explained

by the transition from heavier particles like iron nuclei to protons.

The collision of a primary particle and a nucleus in the air will produce many

8 Figure 3: The ﬂux of Cosmic Rays [Olena Shmahalo, Quanta Magazine. Original data by S. Swordy, U. Chicago.] energetic hadrons, these are called secondaries. These secondaries can do either of two things, they can collide with another nucleus in the air, producing more secondaries, or they can decay before they interact again. These new secondaries undergo the same process and the process repeats at each generation. The number of particles in this extensive cascade increases with every generation, and the energy of the primary is divided among all the secondaries. At some point the average energy of the particles is low enough that more particles are stopped, than that new ones are created. At this point the number of particles in the shower starts to decrease, this point is called

9 the shower maximum.

The secondaries of an air shower mainly consist of pions. There are three diﬀerent types of pions: two charged pions, a positive π+ and a negative π−, and one neutral pion π0. The charged pions decay into muons and muon neutrinos, while the neutral pions decay into 2 photons. Whether pions will interact with a nucleus in the air or decay, depends on two things: their lifetime and their energy. The neutral π0 has a very short lifetime and will generally decay before interacting, while the longer lifetime of the two charged pions means that they will typically interact as long as their energy remains above the critical energy. This distinguishes two diﬀerent branches of the air shower, a hadronic part where π+/− interacts with the air creating more pions until the average energy dips below the critical energy, and an electromagnetic part where the photons created in π0 decay interact with the atmosphere and create electrons and positrons, which in turn will make more photons in reaction with the atmosphere and so on.

Apart from charged pions we can ﬁnd more secondaries that will contribute to the hadronic branch of the air shower. Baryons that are created will also contribute, as well as strange mesons. The reason strange mesons contribute to the hadronic branch of the air shower can be understood by looking at the ways they can decay.

By deﬁnition, a strange meson has a strange quark in it, and thus has a Strangeness that has to be conserved in all but one type of decay: weak decay. Weak decay is the only type of decay that allows for quark ﬂavor to be changed, and as the name suggests, it is quite a weak type of decay. This means that on average the decay length of strange mesons is rather large, and the chance that it will interact before decaying is also large. This means that strange mesons will contribute to the hadronic branch of the air shower.

Even though the real properties of an air shower are diﬃcult to calculate and requires sophisticated simulations in order to be able to make predictions, there are

10 some toy models which can qualitatively describe parts of the air shower and their

dependence on some basic parameters. For the electromagnetic part of the air shower

we can use the model named after Walter Heitler1. While for the hadronic part of

the air shower a generalization of the Heitler model will suﬃce.

3.2 The Heitler Model

In the Heitler model we start by considering an energetic electron moving through

the air. This electron will radiate a single photon with half its energy after having

travelled a splitting length d = λr ln 2, where λr is the radiation length. A photon travelling the same distance d will split into a e+e− pair, each with half the energy of

the photon. After this splitting has happened n times (having traversed a distance of

n x/λr x = n × d = nλr ln 2) we have a total of N = 2 = e particles. If the energy of the

E0 E0 initial electron is E0, each particle will have an energy of N = ex/λr . If the photons no longer have enough energy to undergo pair production, pair production will stop

and we will have the maximum size of the shower. Heitler takes this energy to be

the critical energy Ec of about 85MeV . The maximum size Nmax is reached when all particles have energy Ec, such that E0 = Nmax × Ec. The number of splitting lengths

(generations) after which Nmax is reached is called nc, it is easy to see that

ln ( E0 ) nc E0 Ec Nmax = 2 = → nc = . Ec ln 2

From this we ﬁnd that the depth at which the maximum occurs

E0 Xmax = ncλr ln 2 = λrln( ) . Ec

It should be noted that this model is very oversimpliﬁed but it gives us a very nice

description of two features of the electromagnetic shower: that Nmax is proportional

to E0 in a linear way, and that Xmax depends on E0 logarithmically. These predictions have been conﬁrmed experimentally.

1The model was originally developed by Carlson and Oppenheimer

11 Figure 4: Schematic overview of the Heitler model for EM cascades (LH side) and hadronic cascades (RH side). Dashed lines are neutral π0 particles and solid lines are charged π+/− particles. [T. Pierog, 2013] [2]

3.3 The Generalized Heitler Model

For the hadronic part of the air shower we can generalize the model described above

(for a more detailed derivation of this model, see Matthews [11]). The incoming

particle is assumed to have energy E0, and after one interaction length λi every charged pion will interact again. Due to equal energy distribution, we ﬁnd that at

1 2 each interaction 3 of the pions created will be neutral, and 3 of the pions created will be charged. The neutral pions decay into photons almost instantaneously and will

contribute only to the electromagnetic part of the air shower, this energy will never

return to the hadronic part of the shower. After n generations we ﬁnd the following energy distribution for the hadronic and the electromagnetic part of the shower:

2 E = ( )nE , had 3 0 2 E = [1 − ( )n]E . EM 3 0 This means that as n increases, more and more of the energy will be in the

electromagnetic part of the shower (after 6 generations it is already ∼ 90%), therefore

the depth of shower maximum will be given by the depth of shower maximum of

12 the electromagnetic component. We have shown that for an electromagnetic shower

X = λ ln ( E0 ), however, in the present case the incoming particle will interact max r Ec after λi, producing EM particles with an energy of E0/ntot (ntot being the total amount of new particles created) and we obtain

E0 Xmax = λi + λr ln ( ) . ntotEc

Similarly as before, we assume the splitting to end when the particles reach a speciﬁc

energy, this time we’ll call it Ed, the decay energy, which is about 20GeV . If there

is a multiplicity of Nch of charged pions created in every interaction, we ﬁnd a total

n number of charged pions Nπ = (Nch) after n interaction lengths. The energy per charged pion at this point is then

E E E = had = 0 . π 3 n Nπ ( 2 Nch)

The splitting will end when the energy per charged pion Eπ is equal to the decay

energy Ed, this happens after some number of generations which we shall call nc, at this point we have:

E0 Eπ = Ed = . 3 nc ( 2 Nch)

At this point we can calculate nc:

ln [E0/Ed] nc = 3 . ln [ 2 Nch]

We now assume that all the charged pions decay into muons after nc interaction

lengths, such that Eµ = Eπ, Nµ = Nπ. We now ﬁnd that E0 = EcNmax + EdNµ,

where Nmax and Ec are from the electromagnetic part of the Heitler model. To ﬁnd

the number of electrons Ne from the total Nmax, we simply introduce some correction

Nmax factor g, such that Ne = g . We can now rewrite to ﬁnd:

Ed E0 = g × Ec[Ne + Nµ] . g × Ec

13 E0 3 nc Using that nc = ln [ ]/ ln [ Nch] and Nµ = (Nch) , we can ﬁnd Ed 2

ln Nµ = nc ln Nch = α ln (E0/Ed) with

ln Nch α = 3 . ln ( 2 Nch)

E0 α The size of the muon part of the shower can now be written as Nµ = ( ) . It is Ed important to note that this toy model does not represent the data from accelerators particularly well [2]. The parameters we use are only eﬀective quantities and are not the same as the quantities observed in accelerators.

3.3.1 Inelasticity

Another thing we need to consider in our toy model is the so called ”leading particle effect”. This effect happens when one of the particles in an interaction (the ”leading” particle) gets a significant fraction of the total energy. This energy can now not be used for the creation of more particles. The fraction of the energy taken by the leading particle is called the inelasticity, represented by the parameter κ < 1. In this inelastic toy model it is assumed that all hadron interactions will have inelastic effects and that

κ remains constant. κ is deﬁned such that (1 − κ)E is taken by the single leading particle for an interaction induced by a particle of energy E. This leaves κE energy

2 for the creation of more particles. 3 κE is taken by the Nch charged particles and the 1 1 remaining 3 κE is taken by the 2 Nch neutral pions. We will now assume that there is a total multiplicity of charged particles of (Nch + 1), which is one particle more than we have used in the model above, which will make it easier to make comparison, since we can get the simpler earlier model without inelastic considerations if we take the limit of κ → 1.

The fractions of the energy that stays in the hadronic branch of the shower and

1 can thus interact again is now (1− 3 κ). For a primary with energy E0 we ﬁnd the total 1 n energy in the hadronic branch of the air shower after n generations to be (1− 3 κ) E0,

14 n which will be distributed over (1 + Nch) particles. Again we assume the process

to stop after a critical number of generations nc, after which the particles have an average energy of 1 nc (1 − 3 κ) E0 Ec = n . (1 + Nch) c From this we ﬁnd that

ln [E0/Ec] nc = 1 . ln [(1 + Nch)/(1 − 3 κ)] 3 This formula looks similar to what we found earlier: nc = ln [E0/Ed]/ln [ 2 Nch], only 3 1 now we have substituted Nch → Nch + 1 and 2 → 1 . Using a similar trick as 1− 3 κ before to get a relation between the primary energy and the muon number, we have

α that ln (Nµ) = ln (Nπ) = nc ln [1 + Nch] = α ln [E0/Ed], so that Nµ = (E0/Ec) . This is again very similar to what we found before, but this time we ﬁnd

ln [1 + Nch] α = 1 . ln [(1 + Nch)/(1 − 3 κ)] From this we can see that the inelasticity κ has an inﬂuence on the value of α.

3.3.2 Superposition Model

In the superposition model we look at primary particles that consist of multiple nucleons, for example an iron nucleus. In this model a primary consisting of A nucleons is treated as if it were A individual, independent, nucleons. We rationalize this by noting that the binding energy of the nucleons is of the order of ∼ 5MeV , which is much lower than our interaction energies. If the primary has an energy E0 we consider each proton to have an energy Eh = E0/A. We now ﬁnd that

(A) (p) NEM,max(E0) = A × NEM,max(Eh) ,

(A) (p) Xmax(E0) = Xmax(E0/A) ,

and ﬁnally

(A) Eh 1−α (p) Nµ = A = A Nµ (E0) . Ed

15 In this formulation the superscript of (A) and (p) refer to nucleus and proton induced showers respectively. From these formulae we can see that we would expect more muons in nuclear induced showers than in proton induced showers, and the heavier the nucleus the more muons we expect. Also note that for higher numbers of nucleons we would expect a shallower depth of maximum.

3.4 Hadronic Interactions

The study of hadronic interactions is a complex ﬁeld of study. Two of the more important ﬁelds to be able to understand hadronic interactions are perturbative Quantum

Chromo Dynamics and Gribov-Regge Theory, I will not go into depth on these here, instead I will give a short introduction to the basic concepts we will use later. We consider two hadrons (let’s say protons) colliding at high speed. The protons are made up from smaller parts called partons (quarks and gluons). Partons carry what is called a color charge, a charge that is related to the strong interaction. There are three diﬀerent colors labeled as red, green and blue (or sometimes red, yellow and blue), as well as their anti-color. The term ’color’ is used because of the similari- ties with the mixing of coloured light: if you mix red, green and blue light you will

ﬁnd that the resulting light will be white. Also a color and its anti-color will add up to white, for example red and anti-red will produce white, in analogy to colours anti-red can be seen as the negative colour of red, which is cyan. In reality we ﬁnd that only particles with a total white color charge (colorless) can be observed. Now let’s go back to our colliding protons. When two protons collide at high speed, they will interact. This means that there is a color charge exchange between two partons by the transmission of a gluon. Often one parton of each proton will carry a large fraction of the total momentum and will be shot away at high speeds while the other partons remain largely unchanged. The partons have interacted through exchange of a gluon, and the original set of partons can thus not be colorless all by themselves.

16 Now in order for the observable particles to have a total color charge of white, color

connections between the partons that were shot away have to be created, these color

connections are called color strings, this newly created color string stores the kinetic energy of the partons. When the partons are far enough apart, the energy in the color string can become large enough for there to be pair production. The new partons that are created in this process will attach them to the partons at the end of the color strings in such a way that the new combination of partons is colorless. This is a process called string fragmentation.

Figure 5: Schematic illustration of string fragmentation. In the top picture we see partons that are connected by a color string. In the bottom picture a new particle pair is created when the partons get far enough away from one another. [F. Tanedo, Quantum Diaries]

17 CHAPTER IV

OBSERVATIONS

We have come a long way since the ﬁrst detection of cosmic rays by Wolf, and, a little later, Hess. To directly observe cosmic rays you would need to do so above the earth’s atmosphere, but high energy cosmic rays are rare enough that the observatory would need to be impractically large. Therefore we only observe high energy cosmic rays at ground level, through the air shower they produce when interacting with the earth’s atmosphere. There are many observatories around the world that try to detect cosmic ray air showers around the clock, some of these are huge, like the Pierre Auger

Observatory, which spans an area of about 3000km2, about 30 times the size of Paris!

Modern observatories use a number of methods to detect cosmic ray air showers.

One of the ways to detect air showers is by measuring the number of electrons and muons that reach the ground. Giant tanks of pure water are used to detect charged particles that reach the earth’s surface. These charged particles will have large speeds due to the high energies involved, and will thus emit Cherenkov light, which is then detected. Another method to detect air showers is the so called ”Fly’s Eye” method.

Finely tuned light sensors point in every direction measuring the ﬂuorescence of particles in the air shower when interacting with air molecules. The amount of light that is observed in this way is directly proportional to the number of particles in the air shower, and thus to the energy of the primary cosmic ray.

Now there is a diﬃculty in observing the diﬀerence between electrons and muons.

Electrons and muons behave almost exactly the same, they are almost exactly the same. The one big diﬀerence between the two is their mass, the muon has a mass of

106MeV , about 207 times heavier than the electron. This diﬀerence in mass is also

18 Figure 6: Schematic illustration of the Auger Observatory. The white dots are Cherenkov detectors. The coloured semi-circles are fluorescence detectors, the lines coming from the light blue fluorescence detector traces the energy loss of the shower as it travels through the atmosphere. The red line is the shower trajectory [T. Gaisser, 2016] what is used to be able to distinguish between the two as secondaries reaching the earth. To keep the discussion simple we can introduce two scaling parameters, RE and Rhad. RE allows for a possible shift in the energy calibration (by the fluorescence detectors) and Rhad rescales the hadronic component of the shower. The fact that the muon is so much more massive than the electron has as a consequence that muons will be attenuated less over the same path length. The path length (in grams/cm2 that a particle travels is directly responsible for the attenuation, the longer the path length, the more attenuation. The atmospheric path length is a function of the zenith angle, by looking at sufficiently many observations over a range of zenith angles the

Auger Collaboration were able to untangle RE from Rhad[1], the results of this are discussed in chapter VI.

19 CHAPTER V

MODELS AND SIMULATIONS

To be able to make actual predictions about air showers, toy models like described above won’t do. More advanced models have been developed to be able to make accurate predictions. These models have their roots in Quantum Chromo Dynamics

(QCD) which is the theory of the strong interaction. Interactions where a large part of the momentum is transferred are called hard interactions, for these hard interactions we can use perturbative QCD (pQCD) to make predictions. For interactions with a small momentum transfer, called soft interactions, this is, however, not an option.

Some interactions in the more advanced models can have various outcomes. To simulate these we make use of pseudo-random numbers following a known probability distribution, this process is called a Monte Carlo (MC) simulation. Truly random numbers won’t do because the simulation has to be fully reproducible. Because

Monte Carlo simulations will produce certain statistical ﬂuctuations, they can be very realistic in producing a model of air showers [13]. The energies involved in UHECR air showers are much higher than what we can achieve in accelerators here on earth.

There is currently no theory on which we can base an veriﬁable extrapolation to these high energies. The models that exist to simulate air showers have to make due, and are based on phenomenology, but they are tuned to the accelerator data.

A realistic model will need to take a number of things into account, such as the interaction of the projectile particle (primaries and secondaries) with nuclei in the air,

their decay properties, the layering of the atmosphere, Cherenkov light production,

multiple scattering and many more. Most of these are understood quite well and

can be modelled realistically in Monte Carlo simulations. But there are still great

20 inaccuracies in the way we model hadronic and nuclear interactions [17]. Two of the leading models for the high energy hadronic interactions are EPOS and QGSJET.

The most recent versions of these Hadronic Event Generators (HEGs), EPOS-LHC and QGSJET-II-04, have been tuned to the recent LHC data. These models are based on Gribov-Regge theory of multiple pomeron exchange. A more in depth overview about how these models work can be found in [9] for EPOS and [10] for QGSJET.

Models for lower energies are generally more accurate in their predictions because the physics is understood better and can generally be veriﬁed in accelerator experiments.

Popular models for the lower energies are GHEISHA, FLUKA, and UrQMD.

Figure 7: Comparison for the muon number prediction of models pre-LHC (QGSJET-II-03 and EPOS 1.99) and LHC tuned models (QGSJET-II-04 and EPOS LHC) for proton induced air showers (solid line) and iron-nucleus induced air showers (dotted line) [Pierog, 2013] [2]

21 CHAPTER VI

THE MUON PROBLEM

The number of muons that we observe in Ultra High Energy Cosmic Ray air showers is larger than what the best hadronic interaction models predict. Using the energy rescaling method discussed in the Observations chapter, the Pierre Auger Observatory has found a discrepancy for UHECR air showers from 1019eV primaries of 30-80% between the observed data and EPOS-LHC and QGSJET-II-04 [1]. The fact that there is a discrepancy between our best models and the number of muons we detect has been seen before, in the year 2000, HiRes-MIA Hybrid Array in Utah also found a larger muon fraction than was predicted by the best models at the time.

Figure 8: The Auger collaboration found no signiﬁcant rescaling is needed for RE (RE = 1.00±0.10 for EPOS-LHC and RE = 1.00 ± 0.14 for QGSJET-II-04) while the hadronic component requires rescaling (Rhad = 1.33±0.16 for EPOS-LHC and RE = 1.61±0.21 for QGSJET-II-04) [Pierre Auger Collaboration, 2016] [1]

22 The diﬃculty with trying to explain this problem is in the ﬁtting of two pieces

of data. On the one hand the number of muons created in UHECR air showers Nµ

should increase, and on the other hand the mean depth of shower maximum Xmax should still be in accordance with the data. This is not easy, a lot of the ’simple’

solutions to the muon problems fail to do this. For example, it is well know that

heavier primaries, like iron nuclei, produce more muons than lighter primaries. We

saw this in the section on the superposition model, where we also saw that this would

change the value of Xmax. It turns out that Xmax is changed too much if we tune this to the observed number of muons. We have also seen in our toy models that decreasing the number of generations increases the number of muons. This is because muons are created at the end of the hadronic branch of the air shower, and with every generation more energy is lost to the electromagnetic branch of the spectrum, leaving less energy for the muons. But we also saw that the total number of generations inﬂuences Xmax, and ﬁtting the number of generations to the number of muons observed would also

change Xmax too much. There are many other ways to increase the number of muons created, for example

by altering the strangeness or charm creation, multiplicity or forward scattering. But

this would only be a qualitative approach because of inherently diﬀerent Monte Carlo

event generators. If one model predicts more muons than another, we cannot say for

certain how important the eﬀect we considered is; it is very possible that other eﬀects

which we have ignored are also important.

23 CHAPTER VII

POSSIBLE SOLUTIONS

There are numerous proposed solutions that would explain the high number of muons

we detect. The proposed solutions can be divided in three categories:

1. Higher number of created baryons/anti-baryons

2. Enhanced Remnant Excitation

3. New physics

In the following sections I will explain what the proposed solutions are and how they

work.

7.1 Enhanced Baryon Production

In 2008 T. Pierog and K. Werner showed that by artiﬁcially changing the number of

(anti)-baryons produced in the EPOS model it is possible to inﬂuence the number of

muons created without substantially changing the depth of shower maximum [8]. We

can understand why this is by looking at our generalized Heitler model described in

E0 α 3 chapter 3. From our toy model we know that Nµ = ( Ed ) with α = ln Nch/ln ( 2 Nch). 3 Looking back to our derivation of α we can see that the 2 term in α comes from the fact that we have only considered pions as secondary particles where the ratio of

the number of hadrons produced in an interaction to the total number of particles

created in the interaction R = Nhad = 2 . For an arbitrary R we can rewrite α = Ntot 3

Nch ln Nch/ ln ( R ). Increasing R would increase the value of α, and thus also the total number of muons. Baryons as secondaries will typically interact again, so if we now consider more baryons as secondaries we can see that this would increase this fraction

24 R, since more energy stays in the hadronic branch. Thus we would ﬁnd an increase in the total number of muons created if we consider an increase in the total number of

(anti)-baryons created. This becomes especially dominant if we consider the leading particle eﬀect.

7.2 Enhanced Remnant Excitation

It is not just baryons that can have a signiﬁcant inﬂuence on the ratio R between the number of hadronic particles and all the particles produced in an interaction. Every particle that does not decay into the electromagnetic branch before interacting can play a similar role, since the energy is in this case kept within the hadronic branch of the air shower. One of the proposed explanations is that a higher production of vector mesons due to excited remnants contributes in a similar fashion as the (anti)-baryons before.

In the process of string fragmentation which we have brieﬂy discussed in the section on Hadronic Interactions, we can distinguish the two string ends of a color string that are attached to the partons of the interacting particles, and the newly created particles. The newly created particles we will call the remnant. Now there is a chance that this remnant is in an excited state, let’s say the probability of this happening is pex. The value of this pex is very important for modelling air showers [3]. Let’s look at diﬀerent reactions that undergo string fragmentation.

If the collision is induced by a baryon, the remnant will always be another baryon.

As we know, baryons will keep the energy in the hadronic branch of the air shower.

But if the remnant gets excited and then breaks up, it will typically also produce a number of neutral pions, and this will transfer some of the energy into the electromagnetic branch of the air shower [3]. So break-up of excited remnants from baryon induced collisions will produce π0 particles as well, and will thus reduce the energy in the hadronic branch of the air shower, and this will reduce the number of muons.

25 If we look at pion induced collisions we need to distinguish between charged pions

(π+/−) and neutral pions (π0). Collisions induced by charged pions will have remnants that can either be one of the charged pions, or a neutral pion. So charged pion induced reactions will increase the production of neutral pions, and thus lower the number of muons created.

For neutral pions we see the opposite of what we saw for the charged pions. Again the remnant can be of either one of the charged pions, or a neutral pion. So in total this will increase the number of charged pions and thus increase the amount of energy that goes into the hadronic branch of the air shower.

Now the question arises: which of these processes is more dominant? Are there more neutral pions created due to charged muon remnant break-up, or will we ﬁnd more charged pions due to the break-up of the remnants of neutral pions? We can answer this question by looking at two distinct cases. First we look at the case where we have low multiplicity, that is, where the decaying remnant only produces two particles. The remnant will be neutral or charged with equal probability. This can be understood by looking at the following diagram. Here we look at an ud particle

where we insert a qq pair, which we will underline for clarity.

projectile → [remnant] + stringends

ud → [u d] + d d (1)

ud → [u u] + u d (2)

ud → [d u] + u u (3)

ud → [d d] + d u (4)

We can see that (1) and (3) will give a charged remnant and (2) and (4) will give

a neutral remnant. So we have charged and neutral remnants both with a probability

of 50%. Now the remnant will try to produce an equal amount of charged and neutral

26 particles when decaying. Since we have said that only two particles are created in the

remnant decay, we ﬁnd two main possible decays: a (π+, π−)pair or a (π0, η(0)) pair.

Now the η(0) has a mass of about 958MeV , while the pions have a mass of about

135MeV. Due to the large mass of the η(0) particle, its production will be suppressed,

resulting in the creation of more (π+, π−) pairs. For larger multiplicities we know

1 2 0 that we get 3 neutral and 3 charged particles. So the π fraction will be reduced strongly.

Figure 9: The inﬂuence enhanced remnant excitation has on muon production. On the y-axis we see the ratio of muons predicted by the Picco model [3] (pex = 1.0) and the QGSJET II model (pex = 0.6) [Drescher, 2007] [3]

So all in all we ﬁnd that the break-up of excited remnants will increase the energy

fraction that goes into the hadronic branch of the air shower, and this process will

thus increase the amount of muons created. In 2007 Hans-Joachim Drescher combined

this idea with higher baryon production which we have discussed before. Using EPOS

Drescher has found that the combination of the two possible solutions can together

contribute to a 40% higher number of muons created in the air shower. In this paper

[3], Drescher explains the motivation for enhanced remnant excitation (higher pex) by

27 pointing out that ”at higher energies the projectile probes smaller gluon momenta in the target and therefore encounters a higher gluon density”

7.3 New Physics

The energies involved in UHECR air showers are far above the energies reached in the most powerful accelerators here on earth, and even some of the secondaries have energies that exceed the upper limit of the LHC experiments. This justifies a decent amount of freedom to use new physics in the explanations of the muon problem. The reason there is a lot of freedom for new theories comes from the idea that there are only a few constraints that have to be satisfied in order for it to qualify as a valid explanation. One of the things that must be embedded in theories of new physics is the idea that the newly proposed phenomenon must affect most air showers, instead of being a rare occurrence, this because the variation in observed muon numbers is relatively small [4]. Note that this doesn’t mean that every ultra high energy interaction has to manifest this new physics, as long as the effect responsible for the high muon numbers has a threshold of about 1017eV and manifests at the majority of events at these energies. Furthermore, the new physics has to influence a large portion of the energy distribution in the final states of UHECR air showers, as to be able to explain the high number of muons we observe.

Using these requirements described above, G.R. Farrar and J.D. Allen argued in

2013 against some ideas of possible new physics as an explanation for the high number of muons. They argue that many proposed exotic particles will in fact decrease the number of muons since they take energy away from the hadronic branch of the air shower, particles predicted by supersymmetry would simply have too small an impact, the creation of a Quark-Gluon plasma wouldn’t signiﬁcantly alter the particle composition at the end of the shower, and also the creation of black holes would not alter the number of muons we would observe (by altering the multiplicity strongly

28 you could ﬁnd more muons with this method, but that would alter Xmax too much to be an acceptable explanation). In this same 2013 paper Farrar and Allen also give a possible solution that would work given the above constraints. They proposed Chiral

Symmetry Restoration (CSR) as a possible mechanism.

7.3.1 Chiral Symmetry Resotartion

Chiral symmetry is a symmetry that applies to massless fermions. This means that the left- and right handed components can be rotated independently without diﬀer- ence to the theory. In our case chiral symmetry is only an approximate symmetry.

This is because the masses of normal quarks (up and down) have very small masses compared to the QCD scale. At normal temperatures this symmetry can be broken spontaneously, because in presence of light quark-antiquark pair condensates, QCD interactions can lower the vacuum energy. However at higher temperatures this is not the case, and Chiral Symmetry is restored. Whenever Chiral Symmetry is broken, a so called Goldstone boson has to be produced, in the case of chiral symmetry this usu- ally is a pion. At the energies of UHECR’s the concept of temperature sort of breaks down, because there is no equilibrium. But we can talk about the energy density instead of temperature. We can consider a shock front from the initial collision where the inside of the bubble has a very high energy density, and the outside has a normal

(low) atmospheric energy density. Now there are a few ways that Chiral Symmetry

Restoration (CSR) can inﬂuence baryon production relative to pion production.

One of the ways CSR can increase baryon production relative to pion production is because pions cannot be created as Goldstone bosons when chiral symmetry is restored, pions can only be created after the energy density has gotten low enough for Chiral Symmetry to be broken again. So in this initial high energy density state where CSR is present, only baryons can be produced. This will lead to more energy in the hadronic branch of the air shower and thus more muons.

29 Figure 10: Schematic illustration of a CSR mechanism that may suppres meson prodcuion. A qq (meson) wavefunction has very little overlap between the low exteriour energy density and the high interiour energy density. The wavefunction will reflect off the boundery of the shock wave ’bubble’ which leads to the q and q being available to be bound in other states. A qqq or qqq (baryon or anti-baryon) state is expected to have more overlap between the wavefunction inside and outside the bubble, and can thus escape. From this we find more (anti)-baryons being created in this process than mesons (pions). [Farrar and Allen, 2013] [4]

7.3.2 String Percolation

The study of percolation is the study of connected vertices or edges in a lattice. A graph where adjacent vertices are connected randomly with some probability will display a critical phenomena, that means that the system can drastically change when some parameter is adjusted. This can be seen as a phase transition. In the case of percolation this can be understood when we look at the connectedness of the boundaries of our lattice, for a low probability of adjacent vertices to be connected it will be very unlikely for the boundaries to be connected through some path of connected vertices, while for higher probabilities of connected vertices this becomes very likely.

The String Percolation Model (SPD) looks at the percolation of color strings that

30 Figure 11: The percolation of a 9x9 square grid. Adjacent vertices are connected with a probability 1 2 1 of 3 (a) and 3 (b). We see that the boundaries of the grid are not connected for a probability of 3 , 2 but they are connected for a probability of 3 [Hunt, 2014] [18]

are created in a hadronic interaction at ultra high energies. The main idea is that

connected color strings can cluster and fuse together, inﬂuencing the nature of the

strings, which determines what and how many particles are created when the string

fragmentates. In SPD it is predicted that at the energy range of 20-30 TeV in the

center of mass frame of reference, we could expect enough color strings to be created

for such a phase transition to occur, and it is expected to have a signiﬁcant inﬂuence

on hadronic interactions at energies around 30-40 TeV. This is larger than the energies

reached at the LHC but can be found in secondary interactions of UHECR air showers.

SPD would increase the number of strange mesons produced as well as increase the

overall number of baryons produced [6]. Both of which contribute to the hadronic

branch of the air shower. If parameters in this model are ﬁxed to the LHC data, Monte

Carlo simulations estimate about 50% more muons due to SPD for a proton induced

air shower with an energy of 1019eV . Another prediction that SPD makes is that

the muonic and the electromagnetic shower maxima evolve diﬀerently. This can be

tested experimentally by comparing data from the Fluoresence Detectors, which can

31 mesure the Xmax for the electromagnetic component, to the data from the Cherenkov

Detectors, which are sensitive to the Xmax of the muonic component[6]. This can be used to test this model [6].

7.3.3 Other Solutions

Other theories of new physics have to satisfy the constraints discussed in the beginning of this section. Allen and Farrar have discussed four possible models in their paper [5], and have argued for observable diﬀerences between the models. The four models they considered are: CSR (which we have discussed in the section on Chiral Symmetry

Restoration), Pion Decay Suppression (PDS), Pion Production Suppression (PPS) and a variant on PPS that only appears at High Energies (PPS-HE).

The simplest way to satisfy the constraints discussed in the beginning of this section is by ﬁnding a mechanism that reduces the π0 energy fraction. One way to

reduce the π0 energy fraction is through PDS [5]. One way of rationalizing this is by

pointing out that most Monte Carlo simulations of pion decay treat the process as

if it were happening in a vacuum. But pions in UHECR air showers travel at very

high speeds, which means that for them the atmosphere is actually very dense, so the

approximation that it is happening in a vacuum does not hold. PDS suggests that

this suppresses the pion decay at high energies. This will have the same consequences

as simply lowering the pion production, which has as a consequence that π0s do not

decay and this decreases the energy in the electromagnetic branch of the air shower.

A more straightforward way to get to the same result as PDS is by directly lowering

pion production, which we see in the PPS model. There are two diﬀerent variants

of PPS, PPS at all energies, and PPS only at High Energies (PPS-HE) [5]. Both

have the same core idea of suppressing the pion production, the diﬀerence is that for

PPS-HE the phenomenon only happens for energies above 1017eV .

One of the ways to discriminate between these four models is by looking at the

32 Nµ − Xmax plane. The correlation between Nµ and Xmax is largely dependent on the mass composition and the threshold of the energy where the pion production/decay

suppression is present. This can be understood by pointing out that the high energy

secondary particles that are created in the ﬁrst few generations of the air shower are

mainly responsible for the depth of shower maximum, while muons are created only

as a product of pion decay, and that only happens at low energies. All four theories

have enough freedom in them to be able to predict both Nµ and Xmax in accordance

with observation. But the way Nµ and Xmax change with respect to one another is diﬀerent for each theory.

Figure 12: The predicted muon density at 1000 meter from the shower core versus Xmax for the diﬀerent models. [Allen, 2013] [5]

The way we expect the Nµ and Xmax correlation for the diﬀerent models can be seen in Figure 12. We can see in the graph that PPS and PPS-HE both have a negative correlation between Nµ and Xmax, CSR shows almost no correlation and

PDS shows a positive correlation. This shows that the Nµ vs Xmax dependency is an important new observable that can help to distinguish between diﬀerent explanations,

and already this comparison between Nµ and Xmax has been used to put constraints

33 on interaction models [19].

34 CHAPTER VIII

WHAT’S NEXT?

There are plenty of questions left to answer before we can start to explain what causes the excess of muons from UHECR air showers we observe. Luckily for us, there are a multitude of experiments currently running that can shed new light on the matter.

The Pierre Auger Observatory was the ﬁrst group to establish without doubt that there is a muon problem. They did this by analysing the huge amount of data that they acquired using their observatory. New data from an upgrade of the Pierre Auger

Observatory by the name of Auger Prime, expected to run until the end of 2024, is expected to bring forth many new results that can help us understand UHECR air showers. Among other things, the upgrade will be using plastic scintillators on top of Cherenkov detectors to better determine the energy fraction of muonic and electromagnetic particles separately. The expected data sample (double the size of the data currently available) will be used to determine what the composition of the primary particles are. With more data on the energy fraction of the muonic and electromagnetic secondaries, as well as a better construction of the primary particles we will be able to signiﬁcantly improve the simulations used to model UHECR air showers. This will lead to more certainty in our simulations and a more accurate comparison to simulations of proposed solutions of the muon problem. As discussed in the section on Other Solutions, the Xmax−Nµ dependency can help us narrow down the possibilities of new physics as a solution to the muon problem. The data from

Auger Prime can be used for an event-by-event analysis of the correlation between

Xmax and Nµ. This can be used to rule out some of the explanations for the muon problem [19]. This same data can also be used to check the String Percolation model,

35 which predicts that shower maxima of the muonic and electromagnetic particles evolve diﬀerently as a function of energy for a ﬁxed primary composition.

Another experiment that can greatly increase our knowledge of the muonic component of air showers is the Ice Top experiment. IceTop is the top part of the IceCube

Observatory, a cubic kilometer large observatory located underneath the Antarctic ice. The combination of the IceTop and IceCube observations will better our knowledge of the energies of the muons that reach the earth. More speciﬁcally, the IceTop

- IceCube combination will allow us to look at the specific energies of the muons that reach the earth’s surface [14]. This can be used to discriminate between the different proposed solutions, as different models predict different spectra of muon energies.

This is not a method that is currently talked about in current assessment of the muon problem, and a multitude of simulations would be needed to understand the diﬀerence in muon spectrum from the diﬀerent models.

Figure 13: The size and elements of the IceCube observatory visualized [Upssala University]

36 CHAPTER IX

CONCLUDING REMARKS

In this bachelor research project it is shown that our current models of UHECR air showers fail when it comes to predicting the muonic component of the air shower.

The muon problem has proven itself diﬃcult to resolve, in that almost all simple solutions fail, and it gives plenty of freedom for theorists to come up with new ideas.

The proposed solutions that appear to work on paper have a wide variety of diﬀerent mechanisms that is used to explain the muon excess. Currently we are not able to evaluate the validity of these solutions, but we have been able to narrow the chunk of solutions to a manageable slice that is mainly based around decreasing the electromagnetic energy fraction of the air shower.

More data is needed before we can get closer to ﬁnding the actual mechanism at work. We eagerly await the data from Auger Prime, whose data on primary composition and Xmax and Nµ correlation is expected to greatly improve our understanding of the mechanisms underlying muon production in UHECR air showers.

37 REFERENCES

[1] A. Aab et al.(Pierre Auger Collaboration), Testing Hadronic Interactions at Ul-

trahigh Energies with Air Showers Measured by the Pierre Auger Observatory,

Phys. Rev. Lett. 117, 192001 (2016)

[2] T. Pierog, LHC data and extensive air showers, Eur. Phys. J. Web Conf. 52, 03001

(2013)

[3] H.J. Drescher, Remnant Break-up and Muon Production in Cosmic Ray Air Show-

ers, Phys. Rev. D 77, 056003 (2008)

[4] G.R. Farrar and J.D. Allen, A new physical phenomenon in ultra-high energy

collisions, Eur. Phys. J. Web Conf. 53, 07007 (2013)

[5] J. Allen and G. Farrar, Testing models of new physics with UHE air shower obser-

vations, Proceedings of the 33rd International Cosmic Ray Conference (ICRC),

Rio de Janeiro, Brazil, 2013, paper 1182; arXiv:1307.7131 (2013)

[6] J. Alvarez-Muiz, L. Cazon, R. Conceio, J.D. de Deus, C. Pajares, and M. Pimenta,

Muon production and string percolation eﬀects in cosmic rays at the highest en-

ergies, arXiv:1209.6474 (2012)

[7] R. Engel, D. Heck and T. Pierog, Extensive Air Showers and Hadronic Interactions

at High Energy, Annu. Rev. Nucl. Part. Sci. 2011. 61:46789 (2011)

[8] T. Pierog, K. Werner, New facts about muon production in Extended Air Shower

simulations, Phys. Rev. Lett. 101, 171101 (2008)

[9] T. Pierog et al. , EPOS LHC : test of collective hadronization with LHC data,

arXiv:1306.0121v2 (2013)

38 [10] S. Ostapchenko, QGSJET-II: physics, recent improvements, and results for air

showers, EPJ Web of Conferences 52, 02001 (2013)

[11] J. Matthews, A Heitler model of extensive air showers, Astropart. Phys. 22, 387

(2005)

[12] J. D. Haverhoek, Ultra High Energy Cosmic Ray Extensive Air Shower simula-

tions using CORSIKA, University of Leiden (2006)

[13] M. Alania, I. J. Araya, A. V. Chamorro Gomez, H. M. Huerta, A. P. Flores and

J. Knapp, Air Shower Simulations, AIP Conference Proceedings 1123, 150 (2009)

[14] S. R. Klein, Studying High pT Muons in Cosmic-Ray Air Showers, arXiv:astro-

ph/0612051 (2006)

[15] K. H. Kampert and A. A. Watson, Extensive Air Showers and Ultra High-Energy

Cosmic Rays: A Historical Review , arXiv:1207.4827 (2012)

[16] T. Gaisser, Viewpoint: Cosmic-Ray Showers Reveal Muon Mystery, Physics 9,

125 (2016)

[17] J.G. Gonzales, Measurement of the Muon Content of Air Showers with IceTop,

Journal of Physics: Conference Series 718, 052017 (2016)

[18] A. G. Hunt, B. Ghanbarian and R. P. Ewing, Saturation Dependence of Solute

Diﬀusion in Porous Media: Universal Scaling Compared with Experiments, Vadose

Zone Journal, Vol. 13 No. 4 (2014)

[19] A. Aab et al.(Pierre Auger Collaboration), The Pierre Auger Observatory Up-

grade ”AugerPrime”, arXiv:1604.03637v1 (2016)