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C O N T E N T S

List of Figures v

List of Tables vii

introduction ix

I Astroparticle Physics & Hadron Collisions 1

1 astrophysics of cosmic rays & neutrinos 3

1.1 Cosmic radiation . . . 3

1.1.1 Composition & origin . . . 4

1.1.2 UHECR: the GZK limit . . . 6

1.1.3 GZK paradox . . . 9

1.2 Neutrino . . . 13

1.2.1 Expected flux & origin . . . 14

1.3 Search for astrophysical neutrinos: IceCube . . . 16

1.3.1 Detector & working principle . . . 16

1.3.2 IceCube latest results . . . 20

1.4 Summary of the chapter . . . 23

2 hadronic interactions & forwards physics: an overview 25 2.1 Parton model in hadron collisions . . . 25

2.1.1 Elastic electron-proton scattering . . . 26

2.1.2 Deep Inelastic Scattering & Bjorken scaling . . . 28

2.1.3 Parton Distribution Functions . . . 32

2.1.4 Small-x Bj behavior of the structure functions: scaling violation . 36 2.2 Forward physics at LHC . . . 42

2.2.1 LHC layout . . . 42

2.2.2 Forward program . . . 44

2.3 Summary of the chapter . . . 48

2.A Appendix A to this Chapter . . . 49

2.B Appendix B to this Chapter . . . 50

II Experimental proposal 53

3 ip5-based preliminary work 55

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ii Contents

3.1 Monte Carlo . . . 56

3.1.1 Settings . . . 56

3.1.2 Results . . . 58

3.1.3 Background . . . 62

3.2 Summary of the chapter . . . 66

4 conceptual detecting strategy 67 4.1 Set-up . . . 67

4.1.1 Channels explored & apparatus . . . 68

4.1.2 Impact of the magnetic field on a charge . . . 69

4.1.3 Measurement proposal . . . 72

4.2 Monte Carlo . . . 75

4.2.1 Preliminary: production at IP . . . 76

4.2.2 General background . . . 78

4.2.3 Comparison with experimental data . . . 80

4.3 µ+µ− production from light unflavored mesons . . . 84

4.3.1 Simulation results . . . 84

4.3.2 Exit angles as discriminant key for background . . . 85

4.3.3 un f lavored→µ+µ−signal at the detector . . . 87

4.4 Charm decay channels . . . 91

4.4.1 Charm production rate at IP . . . 91

4.4.2 Geometrical cutoffs . . . 93

4.4.3 Cutoff from un f lavored→µ+µ−measurement . . . 95

4.4.4 Signal from the complete detector . . . 98

4.4.5 Approximate size of the detector . . . 99

4.5 Summary of the chapter . . . 101

conclusions 103 appendix 107 1 Standard Model overview . . . 107

1.1 Electromagnetic interactions . . . 113

1.2 Strong interactions . . . 113

1.3 Electroweak interactions . . . 115

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L I S T O F F I G U R E S

Figure 1.1.1 All-particle-spectrum of cosmic rays. . . 5

Figure 1.1.2 Feynman diagram for the process qfγCMB →qfγCMB. . . 7

Figure 1.1.3 Differential spectrum of the UHECR. . . 10

Figure 1.1.4 Total cross-section for the inclusive process γp. . . . 11

Figure 1.1.5 Hillas Plot. . . 12

Figure 1.2.1 Measured and expected fluxes of natural and artificially-generated neutrinos. . . 15

Figure 1.3.1 Sketch of the IceCube detector. . . 17

Figure 1.3.2 Feynman diagrams of the processes exploited by IceCube for neutrino detection. . . 18

Figure 1.3.3 Neutrino signatures from three different lepton-flavors. . . 20

Figure 1.3.4 IceCube spectrum of the reconstructed muon-energy proxy for νµ events. . . 21

Figure 1.3.5 Atmospheric conventional and prompt νe, νµ flux. . . 22

Figure 2.1.1 Elastic scattering of an electron off a proton at rest and related Feynman diagram. . . 26

Figure 2.1.2 Feynman diagram for inelastic electron-proton scattering. . . . 28

Figure 2.1.3 Differential cross-section of E1=10GeV electrons against hydro-gen against the final hadronic-state energy. . . 31

Figure 2.1.4 Bjorken scaling at xBj =0.25. . . 32

Figure 2.1.5 Experimental proof of the Callan-Gross relation. . . 32

Figure 2.1.6 Proton structure functions measured at HERA. . . 37

Figure 2.1.7 Parton Distribution Functions plotted against xBj for two values of the transferred momentum. . . 38

Figure 2.1.8 Kinematic domains in (xBj, Q2) probed by several particle accel-erators. . . 41

Figure 2.2.1 A sketch of the LHC layout. . . 42

Figure 2.2.2 Partonic representation of a pp event. . . 45

Figure 2.2.3 Muon density from a proton primary cosmic ray of E1017eV against its lateral distribution at ground. . . 46

Figure 3.1.1 Primary proton beam, as shown on MARS GUI. . . 57

Figure 3.1.2 Momentum distribution at the IP for both positive and negative particles. . . 58

Figure 3.1.3 Neutral particles at the three stations. . . 59

Figure 3.1.4 Low-energy particles at the three stations. . . 59

Figure 3.1.5 Middle-energy particles at the three stations. . . 60

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iv List of Figures

Figure 3.1.6 High-energy particles at the three stations. . . 61

Figure 3.1.7 Color key for the hadron production. . . 61

Figure 3.1.8 Scatter plots comparison at the second station, to evaluate back-ground coming from shower production inside the pipe. . . 63

Figure 3.1.9 Momentum distribution of the K0at IP . . . . 63

Figure 3.1.10 Scatter plots comparison at the second station, to evaluate back-ground coming from decay inside the pipe. . . 64

Figure 3.1.11 200simulated pp events at√s=13TeV, seen on MARS GUI. . . 65

Figure 3.2.1 pT-distribution at IP: positives. . . 66

Figure 3.2.2 pT-distribution at IP: negatives. . . 66

Figure 4.1.1 Schematic representation of the IP set-up. . . 69

Figure 4.1.2 Impact of a magnetic dipole on a moving charge. . . 70

Figure 4.1.3 Momentum-kick of a dipole to a moving charge. . . 72

Figure 4.1.4 Geometric explanation for the measurement proposal. . . 73

Figure 4.2.1 Examples of LO Feynman diagrams contributing to the charm production. . . 76

Figure 4.2.2 Spectrum of all the muons generated at the IP. . . 77

Figure 4.2.3 ID of the µ±’s mother hadrons. . . 77

Figure 4.2.4 Muon spectrum at IP, separating the charm production from the rest. . . 78

Figure 4.2.5 Spectrum of the muons coming from the process un f lavored→ µ+µ−. . . 79

Figure 4.2.6 π/K spectrum at the IP. . . 79

Figure 4.2.7 Spectrum of the simulated charms going to µ± in the LHCb kinematical region. . . 83

Figure 4.3.1 µ+ and µ− spectrum from unflavored: results from different cutoffs applied. . . 85

Figure 4.3.2 Exit angles of µ+ and µ−. . . 86

Figure 4.3.3 Exit angles of µ+ and µ, background to µ+µ final state. . . . 87

Figure 4.3.4 Spectrum of µ+ and µ−from unflavored, with cutoffs applied. 88 Figure 4.3.5 Exit points of those µ+ and µ that are accepted in the so far built detector. . . 88

Figure 4.3.6 Invariant mass for the detected µ+µ−state. . . 90

Figure 4.4.1 Muon spectrum at IP, separating the charm production from the rest. . . 91

Figure 4.4.2 Evaluation of a vector sum, resulting in a certain angle between the components. . . 92

Figure 4.4.3 Muon spectrum with a cutoff on ηµto enter the dipole. . . 93

Figure 4.4.4 Exit angles of µ± from charms. . . 94

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List of Figures v

Figure 4.4.6 Muon spectrum at a detector designed with cutoffs applied on

exit angles. . . 95

Figure 4.4.7 Muon spectrum at a detector designed with cutoffs applied on exit angles and having identified µ+µstates. . . 96

Figure 4.4.8 Exit point distribution for muons: the region with maximum signal/background ratio is highlighted. . . 97

Figure 4.4.9 Muon spectrum at the complete detector: all requirements applied. 98 Figure 4.5.1 Kinematical region explored with our detecting strategy. . . 101

Figure 4.5.2 Kinematical region explored with our detecting strategy in a IP5-based region. . . 102

Figure 1 Sketch of the complete detecting area around the IP. . . 105

Figure 1 Geometrical representation of the parallel transport. . . 109

Figure 2 Parallel transport in a closed loop. . . 112

Figure 3 QED interaction vertex. . . 112

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L I S T O F T A B L E S

Table 1 Table of the LHC detectors covering η&5. . . 47 Table 2 Total cross-section contributions from a pp scattering at√s =

13TeV. . . 75 Table 3 Normalized production rates for simulated D mesons in the

LHCb region. . . 82 Table 4 Comparison between experimental data from LHCb and PYTHIA

predictions. . . 83 Table 5 Table showing different values for signal/background ratio. . . 97

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I N T R O D U C T I O N

The investigation of astrophysical phenomena has always been of primary interest within the scientific community. There is a consolidated belief that the study of such phenomena could allow a significant step forward in understanding the origin and evolution of our Universe.

This general assumption is mainly due to two peculiar aspects of most astrophysical observations:

1. they are signals traveling from the past, coming from regions very far away from the Earth,

2. they often come witnessing events that, if occurring in the vicinity of the Earth, would certainly lead to an unlivable environment for the humankind.

For these reasons, the study of the matter (cosmic rays and neutrinos) and the radiation (γ-rays) ejected during these extraordinary events seems to represent our unique opportunity to have an open window into the rest of the Universe.

As a prominent example, Ultra-High-Energy-Cosmic-Rays (UHECRs) are cosmic rays of the highest energies (& 1020eV) and their detection is of great interest to answer cosmological questions but also to give information on possible new physics, exploring a center-of-mass (CM) energy of√s&400TeV, impossible to reach at colliders.

However, the very low measured flux of these incoming objects in high-energy ranges let us conclude that only km3 detectors can collect an amount of data statistically sufficient to give significant information in a relatively-short period of time.

Therefore, the impossibility of exploiting detectors outside the Earth’s atmosphere implies that telescopes face a big challenge in identifying the background produced by the impact of these objects with air nuclei.

Neutrinos and γ-rays produce a very few particles in this impact; for the same weakly-interacting behavior, neutrinos in particular are suited to be direct messengers of the astrophysical events of interest. On the other hand there are cosmic rays which are primarily composed by protons and heavier nuclei: since protons are part of a family of composite objects called hadrons, bound together by the strong force, in high-energy impacts they disintegrate and generate big hadronic showers (mostly π/K and charmed hadrons, both of which later decay into neutrinos and charged leptons), that need to be accurately reconstructed.

It is then obvious that a poor knowledge of the set of processes occurring in such collisions can seriously limit the relevant astroparticle-physics information that can actually be extracted on the cosmic primaries, for instance for galactic propagation parameters or indirect dark matter searches.

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x introduction

In the search for astrophysical neutrinos, the IceCube detector at the South Pole observes an excess, in the high-energy region, that can be accommodated by a new astrophysical source with some difficulty, therefore remarking the fact that evaluation of the (background) atmospheric production could be somehow misunderstood. Specifically, there are reasons to believe that the π/K component (commonly referred to as conventional) of this neutrino production is well-understood: it is experimentally observed and has identifying marks. On the other hand, the charmed component (called prompt) has never been measured so far and its characteristics well emulate those of neutrinos from astrophysical origins.

To estimate hadronic processes, several methods making use of the parton model of hadrons and the theory of strong interactions (QCD) have been developed, but their reliability lies in the direct measurement of observables (such as cross-sections) at particle accelerators.

However, the region currently covered by detectors at the Large Hadron Collider (LHC) is mostly limited, in terms of the production angles θproduction that secondaries form with

the beam axis, to the central production, characterized by large θproduction around the interaction point.

In the forward region at small angles (as opposite to the central), where data are not available, extrapolations based on theory are possible but very often affected by large uncertainties, since, when applied in a non-perturbative regime of QCD, they suffer from the intrinsic property of the strong interactions called infrared slavery.

Within this framework, with the goal of curing this lack of experimental information, this thesis presents a study meant to show how the production of charm quark could be measured at extremely small angles around the LHC beams.

In general, the development of a small angle spectrometer is of paramount importance for a full experimental knowledge of hadron collisions. With reference to our specific problem, the interest in such a detector is strengthened by the observation that the CM energy of the LHC (√s=13TeV) is now in the range where anomalies are appearing in the IceCube data, that is10161017eV, measured in the Earth’s frame.

The work has been divided into four chapters, as described below.

Chapter 1gives an overview of the astrophysical studies showing the importance of building a device aiming at fully exploring hadronic interactions. In particular, we follow a two-fold approach:

discuss the very general outlines of cosmic ray physics with a particular focus on an open question (the problem of UHECRs);

show the most recent experimental observations made by IceCube and analyze the excess they find in their high-energy data, illustrating why a question is raised, specifically, on the charm (prompt) component of the atmospheric background.

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introduction xi

Chapter 2recalls briefly the main features of the parton model of hadronic inter-actions, as it was motivated by the historical results of Deep Inelastic Scattering experiments at SLAC and, also, its weakness deriving from the lack of experimen-tal data in the forward region (large η) at colliders, such as the LHC.

This weakness introduces large uncertainties in extrapolating information where data are not available and results in what we call poor knowledge of the atmospheric processes.

Chapter 3makes use of a simulating tool for particle transport in matter (MARS) to extract information about the trajectory of the secondaries, guided by the set of magnets already present around an interaction point of the LHC.

Chapter 4is the core of the thesis. Based on the results of Chapter 3, we use a standard event generator such as PYTHIA 8.2 to design a muon detector able to see the leptonic and semi-leptonic decay products of the charmed hadrons in the very forward region at LHC, with the best possible acceptance. As anticipated, the purpose of this experiment would be to measure the production cross-section of the charm quark, σppc(¯c).

Several requirements will be applied, in order to reject background muons from π/K decay. Also, an auxiliary measurement of the µ+µ−channel of light unfla-vored resonances is proposed, to improve the signal/background ratio.

Finally, a last part drives some conclusions and mentions proposals for additional future studies.

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Part I

Astroparticle Physics & Hadron

Collisions

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1

A S T R O P H Y S I C S O F C O S M I C R A Y S &

N E U T R I N O S

As pointed out in the introduction, astrophysical studies by means of cosmic rays (CR) are of large interest for cosmological reasons.

However, since it is known they are mostly proton and heavier atomic nuclei, they are electrically charged and so deflected by cosmic magnetic fields on their way to Earth, making almost impossible pointing to the sources.

Therefore, it appears natural to wonder whether or not there is the possibility to extend the searches for CRs including the investigation of electrically neutral stable particles. Theory suggests that, at their birthplace, cosmic rays interact with radiation fields and matter to produce neutrinos and high-energy electromagnetic waves (γ-rays). So, although cosmic rays themselves cannot lead us to their origins, the products they presumably generate may do so.

Actually, only neutrinos provide incontrovertible evidence for acceleration of hadrons, since γ-rays may also evolve from inverse Compton scattering of accelerated electrons and other electromagnetic processes, then, from now on, we choose not to pay attention to γ-ray astronomy.

The possibility of observing the so-called Greisen - Zatsepin - Kuz’min (GZK) neutrinos produced in the interactions of cosmic rays with microwave background photons [BZ69] was recognized since 1969 and then largely discussed [Spi12]. Decades of theoretical modeling led to the conclusion that kilometre-long detectors would be required to collect neutrinos from the cosmos in the statistically significant numbers necessary to finally identify the sources of the cosmic rays [Hal17].

The huge effort of the whole cosmic ray community payed off only recently, when the IceCube Collaboration detected the first astrophysical neutrinos [Sul13].

Nevertheless, the nature and origin of the whole cosmic radiation is still not well-understood and thus source for open questions.

1.1

cosmic radiation

First evidences of a sort of ionizing radiation were already known at the beginning of the 1900, when the general belief was that it was originating from radioactive elements present at the ground level.

Soon after, research on the nature of this radiation led to a discovery worth a Nobel Prize to Victor Hess:

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4 astrophysics of cosmic rays & neutrinos

“[...] The mystery of the origin of this radiation remained unsolved until Prof. Hess made it his problem. Hess, who was from the start of the opinion that the radiation was due to very powerful γ-rays, first investigated in detail the manner in which such rays are weakened on passing through dense layers of air. [...] Hess made a number of balloons ascent to heights up to 5300m, in 1911 and 1912. His systematic measurements showed that a decrease in ionization did occur up to 1000m, but that it increased considerably thereafter, so that at 5000m radiation was twice as intensive as on the Earth’s surface. [...] From these investigations, Hess drew the conclusion that there exists an extremely penetrating radiation coming from space which enters the Earth’s atmosphere. This radiation, which has been found to come from all sides in space, has been called cosmic radiation1

“.

In 1938, Pierre Auger and his colleagues first reported the existence of extensive air showers (EAS), showers of secondary particles caused by the collision of primary high-energy particles with air molecules. On the basis of his measurements, Auger concluded that he had observed showers with energies of1015eV [Aug+39].

1.1.1 Composition & origin

Cosmic rays are relativistic protons and heavier nuclei constantly impacting the Earth. Their spectrum follows a power law, to a few percent, of the type:

dN dE = kE

γ, (1.1.1)

where k is a constant and γ is a slowly varying spectral index, and spans an energy-range that covers from∼GeV to extremely high energy,∼1020eV.

Multiplied by E2to clearly identify its changes in the slope, CR all-particle-spectrum is reported in detail in Figure1.1.1.

Different regions and, correspondingly, different spectral indices can be recognized [Bla14]:

• E30GeV, all the spectral shapes bend down: this is understood as being due to the presence of strong magnetic fields originated from the Sun, which inhibit very low-energy particles from reaching the inner solar system;

• 30GeV. E.1015eV←→ γ∼ −2.7;

• E3·1015eV, the area around this value is called knee: there is evidence that the chemical composition of CRs changes with a trend to become increasingly richer of heavy nuclei at high energy;

• 3·1015eV. E.1018eV←→ γ∼ −3.1;

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1.1 cosmic radiation 5

• E1018eV ←→γ∼ −2.6, the area here around is called ankle: spectrum has a flattening, then it seems that a light component (protons and light nuclei) becomes more important. It is suggested a change from galactic to extragalactic sources [Ape+13];

• E1020eV, these are Ultra-High-Energy-Cosmic-Rays (UHECR). Their origin is still not clear and they undergo a severe cutoff, called GZK cutoff, which will be extensively discussed in Section1.1.2.

Figure 1.1.1: All-particle-spectrum of cosmic rays, as measured by several collaborations. In particular, the contributions from electrons, positrons and antiprotons, measured by PAMELA [Adr+11] - [Adr+13] - [Adr+10], are isolated.

The physical reasons for the changes in slope of the radiation can be several and are not well-understood, although they are thought to be changes in the composition. In a few words, heavy atomic nuclei are expected to disintegrate interacting with interstellar material and with photons from Cosmic Microwave Background (CMB), which permeates the whole Universe, as it will be seen in Section1.1.2. On the other hand, light particles like protons travel without interacting as much with CMB, at least until

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6 astrophysics of cosmic rays & neutrinos

a threshold energy, even though scattering with the interstellar material is still possible. Therefore, to a hardening of the CR spectrum corresponds a light-particle component more abundant than before the slope changed.

Correspondingly, whenever CR composition changes, it is likely that the source from which we are receiving those particles does too.

Regarding their origin, the question is still open, in particular about UHECR, and strictly connected to the study of the mechanisms able to accelerate particles to such high energies.

In his work of 1939, Pierre Auger concludes: ”[...] It is actually impossible to imagine a single process able to give a particle such an energy. It seems much more likely that the charged particles which constitute the primary cosmic radiation acquire their energy along electric fields of a very great extension” [Aug+39].

Soon after, though, they were proposed and generally accepted several models of accelerating particles involving fast-varying magnetic fields present in the interstellar medium at a very large scale,10pc [Fer49], near specific astrophysical objects, and the question turned to how these extended strong magnetic fields originate and where they can be found in the Universe.

Nowadays, there are evidences that high-energy cosmic rays are originated by SuperNovae Remnants [Ack+13], Gamma Ray Bursts [Wax10] or also Active Galactic Nuclei [Ber08] - [DC15], and accelerated by Fermi shock mechanism [Fer49].

1.1.2 UHECR: the GZK limit

It is worth to illustrate some issues related to the region of CR spectrum corresponding to UHECRs.

Let us consider a proton p traveling in the deep space: we know from 1960 [PW65] that the Universe is filled with a quasi-isotropic electromagnetic radiation known as Cosmic Microwave Background (CMB), then, even in the deep space, protons will scatter against photons of this radiation, γCMB.

Since its discovery, and later confirmed by accurate measurements [Pat+16], CMB is found to be in thermal equilibrium with its environment and isotropic, except for very small mapped anisotropies, thus it is possible to define a temperature for it and treat it as blackbody radiation.

With a defined temperature of about T ≈2.7K [Pat+16], Planck’s law states [RL07]: Bλ(T) = 2hc2 λ5 1 eλkB Thc −1 (1.1.2)

where Bλ is the energy per unit time, emitting area surface and wavelength λ, emitted

in the normal direction per unit solid angle by a blackbody of temperature T, h is Planck’s constant, kB the Boltzmann’s constant and c the speed of light.

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1.1 cosmic radiation 7

Nullifying its first derivative, it is found that CMB presents a peak at λCMB =1.96mm

and we can consider its photons carrying an energy ECMB =

hc λCMB

=6.33·10−4eV.

Since protons are a bound state of uud quarks, then, at first approximation, the Standard Model QED predicts an interaction of the type shown in the Feynman diagram in Figure 1.1.2(see Appendix1) [Mag05].

γ

q

γ

f qf

Figure 1.1.2: Feynman diagram describing the scattering of one quark of a proton off a CMB photon.

The final state has the same quark composition, then there is the possibility, at the proper CM energy, to create a hadronic resonance having the same quarks: this resonance exists and is known as ∆+, m

∆+ =1232MeV/c2. This will eventually decay through its

own decay channels, the most likely being [Pat+16]: ∆+

→ N+π 99.4%,

where N is a nucleon and π can be either a charge pion π±or a neutral one π0, with the only constraint of preserving electric charge.

As an instance, let us consider the process

p+γCMB →∆+→ p+π0 (1.1.3)

and find the minimum energy Ethrp the proton needs to let this process happen. With respect to Earth’s reference frame, where CMB spectrum is measured, 4-momenta of the involved initial particles are:

PpEarth= (Ep,~pp)

PγEarthCMB = (ECMB,~pCMB),

then their invariant mass will be

sp+γCMB = (P Earth

p +PγEarthCMB) 2.

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8 astrophysics of cosmic rays & neutrinos

To find the threshold energy of the proton, we require the final products to be created at rest in their CM frame, then, in CM and set c=1, their 4-momenta are:

Pp0∗ = (mp,~0) Pπ0∗0 = (mπ0,~0), with sp+π0 = (Pp0∗+Pπ0∗0) 2 = (m p+mπ0) 2,

where the prime symbol is meant for final state particles.

Since the conservation of 4-momentum holds, in the Earth’s frame we have: PpEarth+PγEarthCMB =Pp0Earth+Pπ0Earth0 →

→ (PpEarth+PγEarthCMB)2 = (Pp0Earth+Pπ0Earth0 )2 ⇒

⇒ √sp+γCMB =

sp+π0, (1.1.4)

where we implicitly used that√s is a kinematic invariant, so√s|Earth= √s|CM.

From the last result, it follows:

sp+γCMB = (Ep+ECMB,~pp+~pCMB) 2 = (E p+ECMB)2− (~pp+~pCMB)2= =E2p− |~p|2+E2CMB− |~pCMB|2 | {z } =0 +2EpECMB−2~pp· ~pCMB,

where we used the convention ηµν = (1,−1,−1,−1) to multiply two 4-vectors in

Minkowski space.

For highly relativistic protons, Ep≈ pp, so:

sp+γCMB ≈m 2

p+2EpECMB(1−cos θ),

where θ is the angle between the directions of motion of~pp and~pCMB in the Earth’s

reference frame.

If we are to find the minimum energy the proton needs for the process (1.1.3), then we set the best condition for the scattering, which means head-on, θ=π.

This way, holding the condition (1.1.4), we have: sp+γCMB ≈m 2 p+4Ethrp ECMB sp+π0 = (mp+mπ0) 2 ⇒ Ethrp = (mp+mπ0) 2m2 p 4ECMB . (1.1.5) With: • ECMB =6.33·10−4eV, • mp≈938MeV, • mπ0 ≈135MeV,

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1.1 cosmic radiation 9

from Equation (1.1.5) we find:

Ethrp '1.07·1020eV. (1.1.6)

It has been said that, to find Ethrp , the final products are created at rest in CM reference frame. Even though this result has to be boosted to find the energy of the final proton in Earth’s frame, that is what our telescopes detect, roughly speaking we understand that part of the total energy of the final system has been used to create the pion mass, mπ0.

Since the total energy is conserved in the process, the proton in the final state will have total energy smaller than the initial proton’s:

E0p< Ethrp .

Still, it could be argued that more energetic primary protons will produce final state protons exceeding Ethrp . It is true, but then these protons will still have enough energy to interact again with CMB; this process will then occur several times until the proton energy becomes smaller than the threshold value.

For these reasons, the value reported in Equation (1.1.6) represents a limit, beyond which we expect to find no protons in our detectors; it is called GZK limit after the physicists who first discussed it, Greisen [Gre66], Zatsepin & Kuz’min [ZK66].

1.1.3 GZK paradox

Figure1.1.3shows the energy spectrum observed by AGASA Collaboration [Tak+03] in the hypothesis of uniformly distributed cosmological sources. Since nine candidates are found to exceed the GZK cutoff, represented in the trend of the dashed blue line, the spectral shape suggests its absence.

Given the experimental evidence that the cutoff may be violated, it is instructive and indispensable to get detailed into the problem; in fact, the limit found in Equation (1.1.6) is just a kinematic illustration of the process, which does not take into account the dynamics, represented by its cross-section.

Explicitly, in a specific reference frame where two particle beams have velocities~v1 and

~v2, number densities n1 and n2, energies E1and E2, the differential Lorentz-invariant

cross-section for the scattering process of the two beams is defined as [Mag05]:

= 1 4I|Mf i| 2(n), (1.1.7) where: • I = E1E2 p (~v1− ~v2)2− (~v1∧ ~v2)2, • dΦ(n) ≡ ()4 δ(4)(Pi−Pf)∏ni=1 d 3p i ()32E

i is the n-body phase space, with Pi and Pf total 4-momenta of the initial and final particles, respectively,

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10 astrophysics of cosmic rays & neutrinos 6 10 3 2 3 10 1019 1020 10 10 10 23 24 25 26 J(E) E [m sec sr eV ] 3 − 2 − 1 − 12 Energy [eV] AGASA C Uniform sources

Figure 1.1.3: Energy spectrum measured by AGASA and the exposure with zenith angles smaller than 45°up until July 2002. The vertical axis reports the differential flux multiplied by E3. Error bars represent the Poisson upper and lower limits at 68% confidence limit and arrows are 90% C.L. upper limits. Numbers attached to the points show the number of events in each energy bin.

• |Mf i|is the matrix element between the initial and final states of the interaction

Lagrangian, computed by means of Feynman diagrams. The cross-section has the dimensions of an area and its unit is:

[σ] =barn=10−24cm2.

|Mf i| is the most important point to discuss: in fact, Figure 1.1.2 reports only one

example of possible interaction between p and γCMB.

In order to correctly compute the cross-section of the process, one has to take into account that there is a term of type|Mf i|2for each quark of the initial proton, and each

|Mf i| itself contains the QED vertex plus corrections coming from Standard Model

QCD vertices.

Figure1.1.4reports the total cross-section for the process γp plotted against the photon energy in the rest frame of the proton, with a fit of the data [Muc+00].

The photon threshold energy for the process (1.1.3) is reported in the caption, corre-sponding to the minimum for the total cross-section; this is reasonable, as only one process is activated, so that the final phase-space is smaller.

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1.1 cosmic radiation 11

Figure 1.1.4: Total cross-section for the inclusive process γp: ENRFis the energy of the photon in the rest frame of the nucleon (NRF). Notice that, in this frame, following the same steps that led to Equation (1.1.6), we find the threshold energy of the photon to be ECMBNRF =mπ0  1+mπ0 2mp  ≈145MeV.

The minimum cross-section corresponds to the best probability for p to travel without interacting, thus it will give a hint on the maximum value of the distance it can walk freely.

In a semiclassical but realistic picture, this hint defines the concept of mean free path for the proton, as:

λp= 1

tot, (1.1.8)

where N is the number density of targets in the crossed medium, CMB, and σtot the

total cross-section. With:

σtot'80µb, as seen from the plot, for the smallest probability of interaction,

• N'411cm−3 [Pat+16],

we get the upper bound for the mean free path that a proton above the threshold can walk without interacting:

λp '3.04·1025cm'10Mpc.

As mentioned in Section1.1.1, the mechanisms for accelerating cosmic particles have been extensively studied and led to the conclusion that they involve magnetic fields

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12 astrophysics of cosmic rays & neutrinos

generated by charged-particle plasmas moving inside astrophysical objects.

Without going into details, a very quoted step forward has been made by Hillas [Hil84] in 1984. Figure1.1.5is generally known as Hillas plot: magnetic field is plotted against the size of the objects producing it.

Figure 1.1.5: Hillas plot: magnetic field produced by a source against its radius. The dashed line is where, based on Hillas acceleration mechanism, protons can be accelerated to Ep ∼1020eV. It is to notice that, for heavier nuclei, the corresponding line is underneath.

The core of Hillas argument is that, to be accelerated from a certain source, a particle has to travel within an effective radius, that is as long as it feels the magnetic field. On the other hand, a magnetic field is able to contain particles up to a certain energy, that thus will be their final energy.

Since it shows from the figure that galaxy halos are below the dashed line, in which objects are found to accelerate protons to Ep ∼100EeV=1020eV, then we can conclude

that protons beyond the GZK cutoff cannot origin in the vicinity of our Milky Way. Among the objects able to produce those energies, the closest to the Earth is the Virgo cluster, which is∼20Mpc from the Earth and, by the way, very well-studied.

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1.2 neutrino 13

In conclusion, no evident sources of acceleration for protons above the threshold are found within the distance they are able to walk across CMB. That is why this problem is often referred to as GZK paradox.

In treating the UHECR problem, a set of assumptions have been made:

• they are only protons. However, it is not clear why heavier nuclei would not disintegrate by photon interactions;

• CMB presents a blackbody spectrum, thus ECMB = 6.33·10−4eV is the most

populated energy. However, there is an amount of∼10/15% photons below that value, increasing the energy required for the proton to activate the process (1.1.3);

• full reliability of Hillas criterion;

semiclassical approach by defining the mean free path λpfor the protons.

Besides, there are also a number of attempts of treating the paradox involving dark matter, non-detectable astrophysical sources and theories beyond the Standard Model.

Nevertheless, the question still holds and remains very puzzling [Dov15]. Several experiments are studying this extremely high range of cosmic ray energy, one example being the Auger Observatory in Argentina [Aab+15].

1.2

neutrino

Neutrinos are neutral particles, first theorized by Pauli in order to preserve conserva-tion of energy and angular momentum within a long debate about the experimental outcomes of β-decay, the decay of a neutron into a proton and an electron.

Pauli himself writes:

“[...] Because of the "wrong" statistics of the N and6Li nuclei and the continuous β-spectrum, I have hit upon a desperate remedy to save the exchange theorem of statistics and the law of conservation of energy. Namely, the possibility that there could exist in the nuclei electrically neutral particles, that I wish to call neutrons, which have spin

1

2 and obey the exclusion principle and which further differ from light quanta in that

they do not travel with the velocity of light. The mass of the neutrons should be of the same order of magnitude as the electron mass and in any event not larger than 0.01 proton masses. The continuous β-spectrum would then become understandable by the assumption that in β-decay a neutron is emitted in addition to the electron such that the sum of the energies of the neutron and the electron is constant“.

Although slightly wrong in few details, his idea was basically correct. The name neutrino was later assigned to that new particle by Enrico Fermi.

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14 astrophysics of cosmic rays & neutrinos

For what concerns astrophysical interests, as pointed out in the introduction, the chargeless, weakly interacting neutrinos are ideal astronomical messengers. As they can escape much denser celestial environments than light, they can be tracers of processes which stay hidden to traditional optical and γ-ray astronomy.

From the point of view of their Standard Model description (see also Appendix 1.3), neutrinos are leptons (spin-1

2 elementary particles) well described within the

electroweak sector [Mag05], which presents them as being weakly interacting and massless, coming in three different lepton flavors:

νewith the same lepton quantum-number as the electron,

νµwith the same lepton quantum-number as the muon,

ντ with the same lepton quantum-number as the tau lepton.

Of course, this weak interaction is what makes them notoriously difficult to detect. Today, there are evidences that a neutrino created with a specific lepton-flavor can later be measured to have a different flavor. This quantum mechanical phenomenon, observed in 1998 by SuperKamiokande Collaboration [Fuk+98] and called neutrino oscillation, is not allowed in the Standard Model and implies that neutrinos have a mass (Pauli was not that wrong, after all) which is extremely tiny, even though non-zero. In fact, since a neutrino is always produced in a flavor eigenstate, its eigenstate wave function will be a mixture of the three mass eigenstates such that at the time of production it is a pure flavor eigenstate.

However, as the neutrino wave function propagates, the three mass eigenstates will effectively move at different speeds so that, at the point in space where the propagating neutrino interacts with the measuring apparatus, it will be a different mixture of flavor eigenstates.

Thus, the possibility of flavor oscillation requires that the masses of the mass eigen-states are not equal.

1.2.1 Expected flux & origin

Figure 1.2.1 reports the expected flux of neutrinos generated in different processes [Spi12]. Based on the process which created them, and consequently on their energy, different spectra can be isolated.

In the range E < 1eV only cosmological neutrinos are found, part of a background radiation, like CMB. Currently, there are no feasible ideas on how and if they could be detected, but their existence is theoretically predicted [LP14].

The energy range 103eV. E.107eV is dominated by neutrinos from the Sun, the SuperNovae and from artificial origins such as nuclear reactors.

About reactor origin, even though of artificial creation and thus not important for astro-physics purposes, they are worth to be mentioned, as this marked the first experimental

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1.2 neutrino 15

Figure 1.2.1: Measured and expected fluxes of natural (astrophysical and terrestrial) and artificially-generated (reactor) neutrinos. As discussed in [Spi12], the experimentally-observed neutrinos are those in the central region, while cos-mological ν, ν from AGN and cosmogenic ν are only predicted.

observation of neutrinos back in 1956 [Cow+56].

For what concerns the solar source, of major historical importance is the measurement by Davis [DHH68] in an underground experiment at Homestake Gold Mine in US. Davis’ purpose was to detect neutrinos coming from one of the branches of the proton-proton chain, a set of processes occurring in the Sun to transform hydrogen nuclei into helium nuclei.

The disagreement of his result with the theoretical predictive model was later recog-nized as the first evidence of neutrino oscillations and Davis shared a Nobel Prize for this discovery.

In the energy range under examination, the flux of cosmic rays is very high, therefore neutrino detectors are placed underground to prevent direct or indirect background.

The range 107eV.E.1017eV is dominated by atmospheric neutrinos. Primary cosmic

rays, interacting in the upper atmosphere, generate a number of secondary particles which in turn generate neutrinos. In fact, as it is well-known from studies at particle accelerators, among the most likely, the following processes may occur [Pat+16]:

π+/K+→µ++νµ

D+/D0→µ++νµ+anything

µ+→e++νe+ ¯νµ

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16 astrophysics of cosmic rays & neutrinos

and charge conjugated.

The problem of detecting atmospheric neutrinos is of great interest, as it also directly involves the search for cosmic rays. In fact, a poor knowledge of the atmospheric processes leads immediately to large uncertainties in cosmic ray flux, measured by reconstructing their shower.

A discussion on the consequences here coming will be given in Section1.3.2.

Finally, the highest energies are the domain of neutrinos from sources like Super-Novae Remnants, Gamma Ray Bursts or Active Galactic Nuclei, or from the interactions of UHECR with the CMB.

In fact, as described in Section1.1.2, the interaction between γCMB and protons can generate pions, which will eventually decay via the process (1.2.1), thus giving birth to both electron and muon neutrinos. These neutrinos are called cosmogenic and their occurrence was properly considered soon after the discovery of the GZK cutoff [BZ69]. Following the clues from astronomical telescopes, IceCube has been searching for neutrinos arriving from the direction and at the time of Gamma Ray Bursts. After more than 1000 follow-up observations, none was found, resulting in a limit on the neutrino flux from GRB of less than∼1%. This has shifted the focus to an alternative explanation for the sources of extragalactic cosmic rays to AGN [Hal17].

1.3

search for astrophysical neutrinos: icecube

IceCube Neutrino Observatory is a particle detector located at the geographic South Pole, built with the aim to search for very high energy neutrinos created in the most extreme cosmic environments. The improved performance at EeV (=1018eV) energies has opened a window to search for cosmogenic neutrino interactions.

1.3.1 Detector & working principle

A sketch of the IceCube detector layout is reported in Figure1.3.1.

There are 5160 Digital Optical Modules (DOM) divided into 86 strings, 60 per string, vertically spaced with∆y'17m, while the horizontal spacing is∆x'125m. Out of these 86 strings, an exception to this configuration is made by the DeepCore, drawn as a cylinder in the Figure, which consists of 8 closer strings with vertical and horizontal spacing of∆yDeepCore'7m and∆xDeepCore '70m, respectively.

The DOMs are glass spheres containing a 25cm-photomultiplier (PMT) and an on-board computer to send the signal up to the laboratory.

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1.3 search for astrophysical neutrinos: icecube 17

Figure 1.3.1: IceCube detector within the environment of the South Pole ice.

contain the signal that a particle, traveling at the speed of light c'3·108m/s, would

leave in two modules separated by a 7m-distance. In fact: ∆t= ∆yDeepCore

c '

7m

3·108m/s '2.3·10−

8s=23ns>2ns.

A detailed review on the features of the Collaboration, including the experimental equipment, is given in [AO].

IceCube detection principle makes use of neutrino weak interaction, discussed in Appendix1.3, with the environment nuclei, as it is shown in Figure1.3.2.

First, let us take into account the interaction via the charged W± boson, Figure1.3.5a and treat the question in a very naive way.

Once a neutrino νl interacts with a f -flavored quark qf ≡q inside the nucleus N, their

4-momenta are:

Pν= (Eν,~pν)

Pq,N= (mq,~0),

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18 astrophysics of cosmic rays & neutrinos q f qf q f'

(

(

W± Z⁰ ( (

l

l

+

-l

± ( ( ν

-( -(( ( q f qf g qf qf q f q f'

l

± ν(

-

( l N X q f q f' ν(

-

( l N X Z⁰ ν(

-

( l (a) q f qf q f'

(

(

W± Z⁰ ( (

l

l

+

-l

± ( ( ν

-( -(( ( qf qf g qf q f q f q f'

l

± ν(

-

( l N X q f q f' ν(

-

( l N X Z⁰ ν(

-

( l (b)

Figure 1.3.2: Feynman diagrams showing the interaction of neutrinos with environment nuclei: (a) represents the interaction via the charged W± boson, giving birth to a cherged lepton, while (b) represents the interaction via the neutral Z0boson, which gives another neutrino in the final state.

supposing the nucleus initially at rest and, with a bit of abuse, also the quarks inside it. After the reaction, the final quark will have its flavor changed, qf0≡ q0:

Pl0 = (El,~pl)

Pq00,X = (mq0,~0),

(1.3.2) where the prime symbols are for the final quantities and we supposed the final quark at rest.

The 4-momentum conservation law holds:

Pν+Pq= Pl0+Pq00,X → (Pν+Pq)2 = (Pl0+Pq00,X)2 ⇒

⇒ m2ν+m2q,N+2Pν·Pq,N =ml2+m2q0,X+2Pl0·Pq00,X

where mν can be set to zero.

If we agree that the detector measures the energy of the final lepton, regardless how it does that, we have:

Eν =

m2l +m2q0,X−m2q,N+2Elmq0

2mq

that, at the high-energy range at which IceCube works, can be approximated considering the masses as very low and the two quarks as having the same masses:

Eν 'El. (1.3.3)

Although this toy model considers the final nucleus at rest, which is very unlikely, Equation (1.3.3) teaches how it is possible to use the energy of the final state charge lepton as a proxy variable for the primary neutrino energy, where as proxy variable we mean a quantity that is used in place of another, which is unobservable.

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1.3 search for astrophysical neutrinos: icecube 19

the neutrino energy, but it deviates from Equation (1.3.3), besides the experimental uncertainties, for two reasons:

1. especially for the muon case, the relatively short section of the muon’s total track which is observed allows to measure a deposited energy which is not the total energy of the muon,

2. the target nucleus N will start scattering thereafter; the energy missing from the final lepton, in respect to the parent neutrino, is taken into account by measuring the hadronic shower initiated by the final state nucleus X.

The same approximated configuration can be adopted in the neutral current interac-tion via Z0 boson.

The detection procedure exploits the radiation produced by the passage of the charged lepton in the ice: in fact, as long as the lepton is traveling in ice with speed faster than the speed light has in this medium, a cone of equally-phased electromagnetic radiation is generated. This effect has been discovered in 1937 and is well-known as

ˇ

Cerenkov effect [Cer37].

The intensity of the radiation thus generated, that is the number of photons N produced per unit path length x of a particle with charge ze traveling at β and per unit wavelength λof the photons, is given by the Frank-Tamm formula [FT37] - [Pat+16]:

d2N dx dλ = 2παz2 λ2  1− 1 β2n2(λ)  , (1.3.4)

where α is the fine-structure constant [Pat+16], α= e¯hc2, n(λ)is the index of refraction of the crossed medium.

It is straightforward to see that, to have ˇCerenkov light, the speed of the charged particle has to be β> 1

n.

ˇ

Cerenkov light will then be collected and amplified inside the DOMs, thus the number of photons emitted by each DOM acts as a proxy variable for the energy that the lepton deposited in it.

This method has different outcomes for each lepton-flavor. In fact, the main responsi-ble for particle energy-loss in matter is ionization [Pat+16], that is when the particle takes away one or more electrons from matter’s nuclei:

µ’s have large inertia, because mµ  me and tend to travel a rather long path

through the detector and to keep going outside. The result is a straight line which even allows to point to the source of the neutrino, with angular resolution<1◦, Figure1.3.3a;

• e’s, scattering off other electrons, rapidly loose all their energy, then producing an electromagnetic shower event, likely to be fully contained in the detector, Figure 1.3.3b;

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20 astrophysics of cosmic rays & neutrinos

τ’s have very short lifetime, ττ = 2.9·10−13s, then are only seen when created

inside the detector. Besides, they will also decay inside it, thus, as a result, energy deposition occurs twice and the event signature is a double bang, Figure1.3.3c. For these reasons, the three charged-current interactions leave information on the flavor and source of the incoming neutrino: in the µ case, the last information is rather accurate, while for e and τ the angular resolution is∼15◦above 100TeV.

On the other hand, for neutral current interactions, the only information we can obtain comes from the hadronic shower and no information is left on lepton flavor.

(a) (b) (c)

Figure 1.3.3: Neutrino signatures from three different lepton-flavors: each dot is from a single struck DOM. The size of the circles indicates the number of detected photons, while the color gives the time, from red (earliest) to blue (latest). From left to right, (a) an observed muon in the 40 string configuration (νµN → µX), (b) a

simulated cascade (νeN→cascade) and (c) a simulated tau double-bang event (ντN→τcascade1→cascade1cascade2).

1.3.2 IceCube latest results

By now, IceCube, in its completed 86-string configuration, has collected several years of data from 2010. Figure1.3.4and the following discussion here carried out are based on the results of the latest full analysis conducted by the Collaboration [Aar+15a].

The astrophysical neutrinos IceCube is interested in detecting are thought to be produced together with cosmic rays, therefore we expect them to have a flux scaling with a related power law of the formΦ ∝ E−γ, where γ should be2 [Wax13].

Based on the processes in Equation (1.2.1), atmospheric neutrinos are usually sepa-rated into two groups:

1. those produced by the decays of pions and kaons, referred to as conventional, 2. those produced by the decays of heavier mesons, particularly those containing

charm quarks, referred to as prompt, because of the very short lifetimes of charmed hadrons.

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1.3 search for astrophysical neutrinos: icecube 21

Figure 1.3.4: The distribution of the reconstructed muon-energy proxy for up-going muon neutrino events selected in [Aar+15a], compared to the expected distributions for the backgrounds and an E−2 astrophysical neutrino spectrum (green line). Only statistical errors are shown. The energy proxy does not have a linear relationship to actual muon energy, but values of 3·103are roughly equivalent to the same quantity in GeV. Larger proxy values increasingly tend to underestimate muon energies, while smaller values tend to overestimate.

Regarding the conventional atmospheric neutrinos, their parent mesons are relatively long-lived [Pat+16], with γcτπ/K∼ γ·5m, where γ= √1

1−β2 depends on energy.

This means that, in walking inside a medium like air or ground, they can interact before decaying, suppressing the production of background neutrinos, especially at high energies, large γ. The thicker the volume of crossed medium is, the smaller the amount of neutrinos found in the corresponding direction will be, thus conventional atmospheric ν’s have a characteristic distribution in direction.

For the same reason, their spectrum is steeper (∝ E−3.7) than the one of the cosmic rays

from which they are produced (∝ E−2.7) and significantly softer than the hypothesized

spectrum of astrophysical ν’s, approximated to be∝ E−2. Plus, the lepton-flavor they have is almost completely muonic.

For what concerns prompt neutrinos, their parent hadrons have much shorter lifetime, with γcτD ∼γ·10−4m, hence no time to interact with the crossed medium. Also, their lepton-flavor distribution is nearly flat, because of their decay channels [Pat+16], like the flavor-distribution of astrophysical neutrinos, due to flavor oscillation with time.

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22 astrophysics of cosmic rays & neutrinos

For these reasons, we can conclude that prompt neutrinos require to be better understood both because the theoretical predictions depend on understanding heavy-quark production in cosmic ray-air collisions at high energies and, also, because they have not been experimentally observed yet.

In fact, as it can be seen in Figure1.3.5, the production of prompt neutrinos is based on several models among which is the Enberg-Reno-Sarcevic (ERS) model [ERS08] and is likely to be corrected in light of experimental results at particle accelerators, as it already happened [Bha+15].

(a) (b)

Figure 1.3.5: The atmospheric νe flux and νµ flux are shown, as measured by several

Col-laborations, presented online in the panels. (a) comes from [Sul13], (b) from

[Aar+15b]. The magenta bands show the Enberg-Reno-Sarcevic predictions in light

of heavy-flavor measurements made in only one run at LHC in 2008 (a) and, slightly modified after the second run, in 2015 (b). References in the text.

Bearing in mind such considerations, in Figure1.3.4, different spectra are reported, one for each relevant, here mentioned, contribution.

While it is acceptable to rely on conventional atmospheric neutrinos, represented by the red line, still it seems clear that the mentioned sources cannot fully accommodate for the experimental data, as an evident excess appears at high-energies.

In this context, we stress that the prompt component could be underestimated, espe-cially in light of the very large uncertainties affecting the hypothesized astrophysical neutrinos’ spectrum, supposed to scale as∼E−2.

To summarize, whatever the search for neutrinos is focused on, it is indispensable to have a detailed knowledge of the set of reactions occurring when primary cosmic rays hit the atmosphere.

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1.4 summary of the chapter 23

1.4

summary of the chapter

In conclusion, from the considerations discussed in this chapter, it emerged that the high range of cosmic ray energies needs to be extensively explored, because too many questions are still left open, for lack of both sufficiently developed experimental equip-ment and theoretical knowledge.

Also, the difficulties in studying particles of astrophysical origins outside the atmo-sphere, because too large detectors would be required, make absolutely indispensable a more accurate and detailed knowledge of the processes occurring when cosmic rays scatter on atmosphere nuclei, to reconstruct primary particles from the shower they produce.

In this framework, the production cross-section of the charm quark, indicated as σppc(¯c), turned out to be of large interest.

As a result, it appears natural that astrophysics studies and the physics of particle interactions are intimately related.

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2

H A D R O N I C I N T E R A C T I O N S &

F O R W A R D S P H Y S I C S : A N O V E R V I E W

It was mentioned in the previous chapter that cosmic radiation is composed by high-energy protons and heavier atomic nuclei, with very small percentage of electrons, positrons and antiprotons. Besides, it is now well-known that atmosphere mainly consists of nitrogen (∼ 78%), oxygen (∼ 20%) and argon (∼ 1%), with minimum contributions (less than0.1%) coming from other molecules.

Therefore, to study the impact of cosmic rays with the terrestrial atmosphere, we can make use of particle accelerators, by performing proton-proton and A-proton collisions, where A represents a heavy nucleus.

The energy level of√s = 13TeV in the center-of-mass, recently achieved at LHC, allows for the direct study of the ∼ 1016−1017eV component of cosmic radiation (see Appendix2.A). Therefore, it should represent a significant step forward in the comprehension of the excess in neutrino data at IceCube, Figure1.3.4, either we assume that this excess has atmospheric origins or astrophysical.

Besides, as it will be clear by the end of the chapter, experimental data on the forward region, never covered so far at the LHC, should provide very useful information, with the aim of giving reliable predictions also at higher energy ranges.

To sum up, far from being a detailed description of the set of processes occurring in high-energy collisions, modeled in several ways from the sixties during the years, this chapter has rather the purpose of giving an overview on a series of reasons justifying the large use physicists do of the parton model and why theoretical predictions regarding the charm’s forward production are not to be considered very reliable.

All across the chapter, the Feynman-diagram approach to the Standard Model will be often exploited, with the conventions discussed in Appendix1.

2.1

parton model in hadron collisions

Because the electromagnetic interactions of leptons are well understood and leptons do not show any internal structure, they serve as useful probes of the structure of hadrons, which, on the other hand, already in the early years of particle collisions were thought to behave as composite bodies.

In order to give a physical interpretation of such lepton-hadron collisions, it is very useful to look at the experimental results by analyzing the differences arising from treating them as elastic or inelastic processes.

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26 hadronic interactions & forwards physics: an overview

2.1.1 Elastic electron-proton scattering

Let us consider at first an elastic scattering of an electron off a proton, as it was a point-like particle, that is the following process:

e (P1) + p (P2) → e (P3) + p (P4)

shown in Figure2.1.1aand described by the diagram in Figure2.1.1b.

e

ϑ z p E₁ E₃ b (a) q f q f q f'

(

(

W± Z⁰

(

(

l

l

+

-l

±

( (

ν

-( -(

( ( qf q f g q f q f q f q f'

l

± ν(

-

( l N X q f q f' ν(

-

( l N X Z⁰ ν(

-

( l

e

e

γ

p p P₃ P₂ P₁ P₄

e

e

γ

p P P₃ P₂ P₁

}

x Q Q (b)

Figure 2.1.1: Elastic scattering of an electron off a proton at rest: (a) is its graphical visualization, b is the impact parameter. (b) is the Feynman diagram associated to the process: Q is the momentum that the electron transfers to the proton and depends on b. The way how the electron and the proton interact is unknown and this is expressed by having inserted a shaded circle instead of a defined vertex.

In Figure2.1.1b, it is implicitly meant that the interaction between the electron and the proton is not known at this point, as shown by the presence of a circle instead of a precise Standard Model vertex. Also, it is important to remark that the final proton is identified as such, by definition of elastic process.

The details on this and the following calculations can be found in the first edition (1987) of Griffiths’ book on elementary particles [Gri87].

The matrix element described by the diagram, averaged over spins, is |Mf i| el,point = g4e Q4L µν electronLµν, proton (2.1.1) where Lµνelectron =2{P1µPν 3 +P1νP µ 3 +ηµν[(mec)2− (P1·P3)]},

Q is the momentum the initial electron transfers to the initial proton, Q = P1−P3,

(P1·P3)is the scalar product defined in Minkowski space with the metric tensor ηµν =

(1,1,1,1), me the electron mass and gethe coupling constant of electromagnetic

interactions, ge =

4πα, α= ¯hce2 the fine-structure constant, e the electric charge. Similarly for Lµν, proton, with me→mpand 1, 3→2, 4.

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2.1 parton model in hadron collisions 27

The differential cross-section, as defined in Equation (1.1.7), becomes the common Mott formula for relativistic elastic scattering of an electron off a point-like charge.

As the proton is not a point-like charge, but rather an extended object, despite its internal structure, the matrix element (2.1.1) can be modified as follows:

|Mf i| el,extended= g4e Q4L µν electronKµν, proton (2.1.2)

where Kµν, proton is now a rank-two unknown tensor.

Even though the exact form of the tensor in not known, several considerations can be made on it [Gri87] and its final expression can be reduced to:

Kµν proton = K1  −ηµν+ Q µQν Q2  + K2 (mpc)2  Pµ 2 + 1 2Q µ   Pν 2 + 1 2Q ν 

where K1, K2are two functions, to be experimentally determined, of the three available

scalar variables Q2, P22, Q·P2. However, P22 =mp ≡constant and the conservation of the

4-momentum law states:

P1+P2 =P3+P4 ⇒ P1−P3≡ Q= P4−P2 −→

−→ (Q+P2)2= P42 ⇒ Q2+2Q·P2=0 ⇒

⇒ − Q

2

2Q·P =1. (2.1.3)

Thus, only one out of the three scalars is independent: K1, K2 ≡ f(Q2).

In order to have observable quantities to compare with the experiments, we have to work in a defined reference frame and set an operational mode.

We then choose to parametrize the measurement by observing the final energy of the electron in the frame in which the proton is at rest; therefore, setting natural units c=1:

P1 = (E1, 0, 0, p1)

P2 = (mp,~0)

P3 = (E3,~p3)

P4 = (E4,~p4).

Defining θ the angle formed by the directions of the two initial and final electrons respectively, setting me ≈ 0 in the limit E1, E3  me, the differential cross-section is

given by dΩ el,extended = α¯h 4mpE1sin2(θ2) !2 E3 E1 [2K1sin2(θ/2) +K2cos2(θ/2)], (2.1.4) which is known as Rosenbluth formula [Ros50].

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28 hadronic interactions & forwards physics: an overview

Equation (2.1.3) holds, whether or not we neglect its recoil, the kinematics is fixed by measuring only one variable chosen between the angle θ and the final energy E3of the

electron, therefore, in Equation (2.1.4), E3 is a function of E1and θ.

Equation (2.1.4) is very important as it directly provides an observable to measure. In fact, by counting the number of electrons scattered at a certain angle and their final energy, which is also a measure for the transferred momentum Q, a fit of the experimental data allows for the calculation of the two functions K1(Q2), K2(Q2).

2.1.2 Deep Inelastic Scattering & Bjorken scaling

At sufficiently high energy, however, the proton is seen disintegrating, therefore, in the final state, the only measurable quantity related to the initial proton is the sum of the 4-momenta of the n hadrons detected:

PX= n

i=1

Ki. (2.1.5)

The process we are discussing is then: e (P1) + p (P2) → e (P3) + X (PX) .

In the mathematical treatment of the problem, the crucial difference from the elastic case is that the invariant mass of the final hadronic state is not the squared mass of the proton, as it is clear squaring both members of Equation (2.1.5), and thus Equation (2.1.3) does not hold anymore.

Formally speaking, Figure 2.1.2 reports the Feynman diagram associated to the inelastic electron-proton scattering, with obvious meaning of the quantities.

q f q f q f'

(

(

W± Z⁰

(

(

l

l

+

-l

±

( (

ν

-( -(

( ( q f q f g q f q f q f q f'

l

± ν(

-

( l N X q f q f' ν(

-

( l N X Z⁰ ν(

-

( l

e

e

γ

p p P₃ P₂ P₁ P₄

e

e

γ

p P P₃ P₂ P₁

}

x Q Q

Figure 2.1.2: Feynman diagram representation of the inelastic electron-proton scattering: again the photon interacts with the initial proton in an unknown way and PX is the 4-momentum of the hadronic final state, which cannot be identified anymore as a proton.

Then, for 4-momentum conservation:

P1+P2= P3+PX ⇒ P1−P3 =Q=PX−P2 −→

−→ (Q+P2)2= PX2 ⇒ Q2+2P2·Q=W2−m2p

(41)

2.1 parton model in hadron collisions 29

where W is the invariant mass of the hadronic final state and W2m2p > 0, as the electron transferred momentum to the initial proton.

If W2 =m2p, we would be back to the elastic case and recover Equation (2.1.3). Here, instead, we have:

Q2+2P2·Q>0 ⇒

Q2

2P2·Q

>−1.

As Q2 is the momentum transferred by a virtual photon γ, then Q2 <0 and we can define a positive quantity

Q2 ≡ −q2 in light of which we can write:

(

−2Pq2·2Q > −1

q2= Q2.

Therefore, there is a kinematical region allowed for the following quantity 0< q

2

2P2·Q

<1 (2.1.7)

that, for this reason, is a simple kinematical variable of common use within inelastic scattering processes, referred to as Bjorken variable:

xBj def= q2 2P2·Q −→  xBj ∈ ]0, 1[ inelastic xBj =1 elastic. (2.1.8) Physically, when an electron collides with a proton at a relatively high energy breaking it up, the final energy of the electron E3cannot be expressed in a simple way

as a function of its initial energy E1 and the angle θ, because the final hadronic state

has a spread angular distribution.

This can be explicitly seen working in the frame where the protons is at rest, the laboratory; here:

P1 = (E1, 0, 0, p1)

P2 = (mp,~0)

P3= (E3,~p3)

PX =unknown

and we have then:

P·Q=mp(E1−E3)

Q2 = (E1−E3)2− (~p1− ~p3)2' −2E2E3(1−cos θ),

where θ is, again, the angle formed by the directions of the initial and final electrons and we neglected the masses in the problem.

In this framework, the Bjorken variable reads

xBj|LAB = −E1E3(1−cos θ)

mp(E1−E3)

(42)

30 hadronic interactions & forwards physics: an overview

In case of elastic scattering xBj|LAB = 1 and thus, by knowing, for example, the final energy of the electron, the scattering angle would be fixed.

In case of inelastic process this does not happen and the kinematic would be fixed by measuring the final energy of the electron and the scattering angle, and the problem has evidently two degrees of freedom.

For what just said, the observable quantity of interest in the inelastic process is a double differential cross-section, calculated in a certain range of angular distribution and also final electron-energy.

The matrix element referred to the Feynman diagram in Figure2.1.2is:

|Mf i| inel = g4e Q4L µν electronKµν(X). (2.1.10)

The same considerations that led to the Rosenbluth formula can be applied, with the exception that, out of the three scalar variable Q2, P22, P2·Q, not one anymore but two

of them are independent, because Q2and P2·Q are now connected by a measurable

variable, not a constant, as seen.

With these assumptions, the observable quantity to compare with experiments is the following double-differential cross-section [Bjo69]:

dΩdE3 inel = α¯h 2E1sin2(θ2) !2 [2W1sin2(θ/2) +W2cos2(θ/2)] (2.1.11) where, in light of the given definitions for the common variables in deep inelastic processes,

W1, W2= f(q2, xBj).

Figure2.1.3reports the described differential cross-section measured for E1 =10GeV electrons scattered off hydrogen nuclei at an angle θ=6° [FK72]: the first elastic peak corresponding to the proton mass is clearly evident, plus three other peaks at different masses of proton’s excitations, for example∆+(1232).

In addition, from W ∼ 1.8GeV to W ∼ 2.3GeV, a continuum of peaks can be distin-guished from the data.

If we work in the frame of the nucleon at rest, based on the expressions computed above in (2.1.6), with a fixed angle θ=6°, W2is the following function of E3:

W2 =2E1E3(1−cos θ) +2mp(E1−E3) +m2p' (19.64−1.985E3)[GeV2] (2.1.12)

thus, as W increases, we are subtracting a smaller and smaller E3and this means that

the initial electron lost an increasingly amount of its energy.

This fact, together with the peaks present in the plot in the range W ∈ [1.8, 2.3]GeV, can be interpreted by picturing a certain number of proton’s constituents that can be resolved only at a large energy-loss of the lepton and an elastic scattering off these.

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