On the Measurement of Atmospheric Muon-Neutrino Oscillations with IceCube-DeepCore

Transcription

1 On the Measurement of Atmospheric Muon-Neutrino Oscillations with IceCube-DeepCore von Matthias Geisler Diplomarbeit in P H Y S I K vorgelegt der Fakultät für Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen im Juni 2010 angefertigt am III. Physikalischen Institut B Prof. Dr. Christopher Wiebusch

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5 Abstract The IceCube neutrino observatory is located at the geographic South Pole and uses the deep ice as a detector medium. The DeepCore enhancement, which is centrally embedded in IceCube, reduces the energy threshold for muon neutrinos from about 100 GeV for IceCube down to 10 GeV. In this energy range atmospheric neutrino oscillations become relevant. One experimental signature is the disappearence of atmospheric muon neutrinos which depends on the energy as well as on the distance the neutrino covered through the Earth and so on the zenith angle as measured by IceCube. For verticall upwards going neutrinos maximal disappearence is expected at 25 GeV. Hence, the event rate of atmospheric muon neutrinos for this energy and angular region in IceCube-DeepCore should be reduced significantly. By application of current reconstruction methods for energy and direction the sensitivity of IceCube for this signature is analyzed using a Monte-Carlo study.

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7 Contents Abstract Contents List of Figures List of Tables i iii v vii 1 Introduction 1 2 Atmospheric Neutrinos Air Showers Measured Energy Spectrum Atmospheric Muons Atmospheric Neutrinos Neutrino Flux Ratio between Electron- and Muon-Neutrinos Ratio between Muon- and Antimuon-Neutrinos Angular distribution Uncertainties Neutrino Oscillations Motivation Quantum Mechanical Derivation Approximations Enhancement by Matter Sterile neutrinos Recent Atmospheric Neutrino Oscillation Analyses The IceCube/DeepCore Detector Neutrino Detection Neutrino Interactions with Matter The Cherenkov Effect Lepton Energy Loss Optical Neutrino Detection IceCube Detection Method The IceCube Configuration iii

8 CONTENTS 5 Atmospheric Neutrino Oscillations in IceCube Conversion to Detector Observables Spectrum without Oscillations Influence of Oscillations Spectrum with Oscillations Reconstruction Data selection Initial seed: LineFit Full Reconstruction pandelfit FiniteReco Quality Selection Optimization of the Angular Reconstruction Optimization of the Energy Reconstruction Results for Angular and Energy Reconstruction Results for Neutrino Oscillations The Log-Likelihood Method Simulation of an Experiment Generating the Reference Histograms Comparison between Experiment and Reference Histograms Results Oscillation Parameters Detector Observables Evaluation of the Systematic Error Uncertainties in the Angular Reconstruction Uncertainties in the Reconstructed Length Threshold on N Chan Consequences of the Systematic Error Summary & Outlook 83 Acknowledgements Erklärung / Declaration References I III V iv

9 List of Figures 2.1 Artistic picture of a cosmic shower The all-particle spectrum from air shower measurements Composition and components of cosmic rays Plot of the proton cross-section Muon and neutrino flux from pions and kaons Calculated energy spectra for atmospheric neutrinos Flux ration of ν µ + ν µ to ν e + ν e Flux ratio of ν µ to ν µ and ν e to ν e Zenith angle dependence for atmospheric neutrinos Observed flux of cosmic ray protons and helium Phase-space distribution for sub-gev atmospheric neutrinos Probability for ν µ oscillations for three generations Probability for ν µ oscillations for two depending generations Survival probability P ( ( ν ) µ ( ν ) µ) with the effect of matter χ 2 χ 2 min distribution for the θ 23 analyses from Super-K Results from the Super-K analyses Comparison between measured flux of the Super-Kamiokande experiment and calculated flux Schematic visualization of the Cherenkov effect Sketch illustrating the signatures for different neutrino flavor Scattering and absorption profiles for the Antarctic ice Sketch of an IceCube Digital Optical Module (DOM) Schematic view of the IceCube detector Artistic illustration of the IceCube detecter placed in the Earth Simulated neutrino spectrum without oscillations Zenith distribution of atmospheric muons The influence of neutrino oscillations in the region Simulated neutrino spectrum with oscillations Spectrum with oscillations converted into detector observables Spectrum without oscillations converted into detector observables Geometry of the signal generation process Distribution of the LLHR versus track length Quality of the zenith reconstruction for different direct length cuts Quality of the length reconstruction vs track shape probability Comparison of the zenith reconstruction before and after the cuts Correlation between reconstructed muon track length and neutrino energy Comparison of the energy distributions before and after the cuts Spectrum after reconstruction and with neutrino oscillations Spectrum after reconstruction and without neutrino oscillations Simulated neutrino spectrum with oscillations with the used cuts Simulated experimental histogram in the oscillation parameters v

10 LIST OF FIGURES 7.3 Simulated experimental histogram in the detector observables Generated reference histograms for different m Typical distribution of the log-likelihood for one experiment Typical distribution of the log-likelihood for one experiment with oscillation parameters Result of the m 2 reco reconstruction for 100 simulated experiments with oscillation parameters Result of the relative error σ m 2 reco / m 2 init for 100 simulated experiments with oscillation parameters Result for 100 simulated experiments with oscillation parameters that the case of no oscillations can be excluded Result of the m 2 reco reconstruction for 100 simulated experiments with detector observables Result of the relative error σ m 2 reco / m 2 init for 100 simulated experiments with detector observables Result for 100 simulated experiments with detector observables that the case of no oscillations can be excluded Result for 100 simulated experiments with a smeared zenith reconstruction Effects of a biased zenith on the m 2 reconstruction Result for 100 simulated experiments with a biased track length reconstruction Effects of a biased track length on the m 2 reconstruction Relative amount of events with #hitdoms > N Chan Effects of an additional N Chan threshold on the m 2 reconstruction vi

11 List of Tables 5.1 Parameters of the dataset Results of the analyses with a biased zenith value Results of the analyses with a biased reconstructed track length Results of the analyses with an additional N Chan threshold vii

12 LIST OF TABLES viii

13 CHAPTER I Introduction

14 1 INTRODUCTION Postulated in 1930 by Wolfgang Pauli, the neutrino is still one of the elementary particles with many unknown features. Therefore, the neutrino belongs to the most interesting fields in today s research. Due to the fact that the neutrino interacts only weakly it is a proper messenger for astronomical observations, too. While charged particles like protons are affected by electromagnetic fields, the neutrinos initial direction is hardly changed during their propagation through the univers. In contrast to photons, neutrinos have the advantage that their scattering and absorbtion probabability is very low and that they are not absorbed by the Earth s atmosphere. This advantage of a small interaction crosssection for neutrinos is a big challenge, too. A huge amount of detector medium is needed to detect and to study them. The mass of neutrinos is still unknown, it is not even clear if all three generations have a mass. Besides the direct estimation of the mass by the measurement of the missing energy at the beta decay, the observation of neutrino oscillations allows conclusions on the squared mass difference between the three neutrino masses. These neutrino oscillations are possible due to the fact that a discrepancy exists between the neutrino mass eigenstates and neutrino flavor eigenstates. The size of this mixing factor is a goal for current experiments. Various experiments look for different neutrinos depending on their origin. While other experiments use neutrinos generated at weak interactions in the sun or in different colliders this analysis studies atmospheric neutrinos. These neutrinos are generated in the atmosphere in the developement of air showers. Chapter 2 presents a more precise analysis of such neutrinos. The processes which generate neutrinos and features like the expected neutrino energy distribution are shown. Chapter 3 presents a quantum mechanical derivation of neutrino oscillations. Furthermore, some special cases like neutrino oscillations in matter or the assumption of a possible fourth neutrino mass eigenstates are shown. Additionally, recent results from the Super Kamiokande experiment for the neutrino mass difference and for the mixing angle are shown. In chapter 4 the IceCube detector is introduced. This detector is located at the geographic South Pole with its huge amount of deep and clear ice as the detector medium. It uses the Cherenkov effect to detect secondaries. Afterwards in chapter 5 neutrinos oscillations is applied to a dataset. The conversion from oscillation parameters to detector observables is shown, too. The main part of this analysis is shown in chapter 6. The reconstrunction of the zenith angle and the energy of an event is explained. Therefore, a few algorithms utilized in the data processing and for the event reconstruction are shortly presented. To optimize the quality of the reconstruction a few cuts are used. These cuts and their result on the 2

15 reconstruction are presented. Finally, the resulting spectra with the effects of the reconstruction are shown. Afterwards the used log-likelihood method is shown in chapter 7. The results of this analysis are presented in chapter 8. First, results are presented assuming a perfect angular and energy resolution. Afterwards the reconstruction errors are included. Furthermore, an evaluation of the systematic error is done. Therefore, a worse reconstruction is simulated as well as a defect at the signal detection at the detector itself. At the end, a short summary and outlook on possible realizations is done in chapter 9. 3

16 1 INTRODUCTION 4

17 CHAPTER II Atmospheric Neutrinos

18 2 ATMOSPHERIC NEUTRINOS 2.1 Air Showers More than one thousand cosmic particles hit the Earth s atmosphere per second. When a single high-energy primary cosmic ray particle enters the Earth s atmosphere and interacts with its molecules, it generates a large amount of secondary particles in a cascade-like process: the hadrons and leptons produced in the interactions can further interact or decay and generate new particles (see figure 2.1). At first, the number of secondary particles multiply, then it reaches a maximum before it drops down because the energy of the particles is not sufficient enough to produce further particles. The energy spectrum of these secondaries peaks in the GeV range but extends to high energies. Figure 2.1: Artistic picture of a cosmic shower [1] Measured Energy Spectrum The particular rate of the cosmic particles depends on the particle s energy. From an energy of about 10 TeV on the spectrum can be interpolated by a power law df (E) de E γ (2.1) with different spectral indices γ for different energy regions. Up to the knee at about ev the spectral index is given by γ = 2.68 ± 0.02 [2]. Above this energy up to the supposed second knee at about ev the spectrum steepens with γ = 3.02 ± 0.03 [3], from there up to the ankle at about ev it is γ = 3.16 ± 0.08 [3]. At even higher energies the spectrum flates again and γ is estimated to drop down to 2.81 ± 0.03 [4]. Above this energy threshold at ev the GZK-cutoff might occur: Cosmic ray protons could interact with Cosmic Microwave Background photons via -resonances p + γ + p + π 0 and p + γ + n + π + In this interaction the proton looses about 20% of its initial energy so that the range of cosmic protons with such high energies is limited to less than 50 Mpc. While the 6

19 2.1 Air Showers AGASA experiment did not see a cutoff in the energy spectrum, lately submitted results from the HiRes and Auger experiments favor this cutoff, but it is still not fully confirmed, yet [4]. Since the origin of the cosmic particles is not fully understood yet, this decrease in the energy spectrum can have other reasons. Figure 2.2: All-particle spectrum at high energies from air shower measurements, multiplied by E 2.7 to display the features of the steep spectrum [5]. In general, cosmic radiation can be divided into two categories: primary cosmic rays, which are those particles accelerated at astrophysical sources like supernovae or Gamma-Ray-Bursts, and secondary cosmic rays, which were produced in interactions of primaries with the interstellar medium. The composition of the cosmic rays in comparison to the relative abundance in the solar system is shown in figure 2.3(a). As one can see lithium, beryllium and boron are typical secondary particles since their relative abundance in the solar system is less than it is in the cosmic rays. On the other hand carbon and nitrogen are apparently primary particles. This disagreement between the relative abundances in cosmic rays and in the solar system is an important tool to understand the propagation of the particles in and between the galaxies. Figure 2.3(b) shows the fluxes from figure 2.2 for different primary nuclei. The fluxes of all major components of the primary cosmic radiation follow nearly the same distribution. Also the comic ray composition is definable; for lower energies about 90% of the 7

20 2 ATMOSPHERIC NEUTRINOS (a) Cosmic ray elemental composition [5]. (b) Major components of the primary cosmic radiation [6]. Figure 2.3: Cosmic ray composition: (a) Comparison of the relative abundance in cosmic rays and in the solar system and (b) Flux for different nuclei. particles are protons, about 9% are He-nuclei and the rest are heavier nuclei. It is important to note that, except for geomagnetic effects, the primary cosmic ray flux is nearly isotropic. However, there are indications that the motion of the Earth due to galactic rotation at 300 km/s would lead to an asymmetry of the order of 1% in the distribution of arrival directions of the relativistic cosmic rays if their sources were external to the galaxy. Furthermore, the peculiar motion of the solar system at 20 km/s could lead to a smaller asymmetry of the order of 0.1% Atmospheric Muons The most dominant interactions for the production of pions (and kaons) by proton primaries are : p + p + + p p + n + π + p + p π + + X n + p 0 + p p + p + π p + γ + n + π +. 8

21 2.1 Air Showers Figure 2.4: Plot of the total, inelastic and production proton-air cross sections. Also shown is a comparison of the proton-carbon total cross section with measured accelerator data [7]. The proton-air cross sections are shwon in figure 2.4. To illustrate how well the formalism that is used for these predictions works, figure 2.4 also shows the total cross section for carbon calculated in the same way compared with measurements [8]. The pion and kaon decays are the sources of muons and neutrinos (see figure 2.5): π + µ + + ν µ e + + ν e + ν µ + ν µ π µ + ν µ e + ν e + ν µ + ν µ Below 100 GeV primary energies neutrinos are mainly generated by pion decays while for higher energies kaons become the dominant source (see figure 2.5). At higher energies generation via kaons through the following processes are also possible. K + µ + + ν µ e + + ν e + ν α + ν µ K + π 0 + µ + + ν µ π 0 + e + + ν e + ν α + ν µ K + π + + π0 K + K 0 S K 0 L π + + π + + π π + + π π ± + e + ( ν ) e KL 0 π ± + µ + ( ν ) µ KL 0 π + + π + π 0 9

22 2 ATMOSPHERIC NEUTRINOS Figure 2.5: Fractional contribution of pions and kaons to the flux of muons and neutrinos. Solid lines indicate vertical; dashed lines indicate 60 [7]. The energy spectrum of these muons and neutrinos is determined by the competition between decay and interaction of parent mesons. Above a critical energy ɛ interaction processes are dominant while below ɛ the generation by decay is dominant. This critical energy for pions in the atmosphere is ɛ π 115 GeV and ɛ K 850 GeV for kaons. For E π,k ɛ/ cos θ, (2.2) with the zenith angle θ, the muon and neutrino spectra have approximately the same spectral index as the parent meson production spectrum and hence the same as the primary energy spectrum of cosmic ray nucleons (see figure 2.3(b)). For E π,k ɛ/ cos θ the muon and neutrino spectra is one power steeper due to the interactions listed above. That is why the spectral index of primary particles γ 2.7 (see section 2.1.1) steepens to γ atm 3.7. Because ɛ K > ɛ π the energy at which the spectrum of neutrinos from pions steepens is lower than that for neutrinos from kaon decay, so that kaons become an increasingly important neutrino source at high energies. Despite of their short lifetime of τ µ =2.2 µs, atmospheric muons can be detected by underground experiments. Produced in a height of typically 20 km, they are able to cover such a distance due to their relativistic velocity. These muons are the major component of the background of Cherenkov neutrino detectors since they produce the same signatures as neutrinos. A method to discriminate between neutrinos and muons is explained in section Atmospheric Neutrinos Except for their energy spectrum which differs from the expected signal spectrum (compare to section and section 2.1.2), atmospheric neutrinos are fundamentally indistinguishable from neutrinos generated at extragalactic sources. Because of that, atmospheric neutrinos are considered as background for many astrophysical neutrino analyses. In this 10

23 2.2 Atmospheric Neutrinos study atmospheric neutrinos are signal. The kinematics of the π and µ decays listed above are such that on average roughly equal energy is carried by each neutrino in the chain Neutrino Flux For a study of neutrino oscillations with atmospheric neutrinos, it is important to know the predicted flux without oscillations. The atmospheric neutrino flux is a convolution of the primary spectrum at the top of the atmosphere (section 2.1.1) with the yield of neutrinos per primary particle. The energy spectra of atmospheric neutrinos calculated by different authors and their comparison are shown in figure 2.6. The agreement at the models around 10 GeV is within 10%. This is understood since the accuracy of recent cosmic ray flux measurements [9][10] below 100 GeV is limited to about 5% and the hadronic interaction models used in the calculations are different. First, the spectral index γ is discussed. For low-energy protons (<100 GeV) as primary particles, a power law with γ = 2.74 ± 0.01 was fitted to measured neutrino data. However, this fit is not valid for high-energy data. Therefore, the spectral index is modified to 2.71 for energies above 100 GeV. The spectrum for He-nuclei as primary particles is described by either 2.64 (low energy) or 2.74 (high energy) [7]. For heavier nuclei the uncertainty for the spectral index is larger than 2 %. Taking the flux-weighted average of these spectrum index uncertainties, an uncertainty of 1.1 % and 1.9 %, for low and high energies can be calculated, respectively. Furthermore, the Earth s geomagnetic field deflects the cosmic rays both inside and outside the atmosphere. Outside, it acts as a filter that allows particles of sufficiently high energy to enter the atmosphere but excludes those of lower energy. Inside, it bends charged secondaries. Whether a particle is allowed or forbidden is determined by its position, direction and radius of curvature. Only particles that interact in the atmosphere before curvating back into space can contribute to the flux of atmospheric neutrinos. Basis for this study is the so-called Bartol flux as presented in [12] Ratio between Electron- and Muon-Neutrinos The sum of partial branching ratios for the decays listed in section is almost 100%. Therefore, the ratios between the expected neutrinos flavors (ν e : ν µ : ν τ ) for lower energies is about 1:2:0. For higher energies this ratio rises in favor of ν µ since not every muon decays before it reaches the surface (For vertical muons this occurs at E µ 2 GeV, and at somewhat higher energy for inclined muons). Figure 2.7 illustrates the flux ratios of ν µ + ν µ to ν e + ν e as a function of neutrino energy. This ratio is essentially independent of the primary cosmic ray spectrum. 11

24 2 ATMOSPHERIC NEUTRINOS Figure 2.6: (a) The directionally averaged atmospheric neutrino energy spectrum calculated by several authors. (b) The ratio of the calculated neutrino fluxes related to the Honda model [11]. The fraction of neutrinos produced in kaon decays rises for higher energies (see figure 2.5). At 10 GeV about 10% of ν e + ν e and 20% of ν µ + ν µ, respectively, were generated in kaon decays. The fractions increase to more than 30% at about 100 GeV for both ν e + ν e and ν µ + ν µ. At these energies the ratio depends more on the K production cross section and the uncertainty of the ratio is expected to be larger. A 20% uncertainty in the K/π production ratio [12][13] causes at least a few percent uncertainty in the ν µ + ν µ to ν e + ν e ratio. However, as shown in figure 2.7 the relative disagreement between the different models ν µ + ν µ to ν e + ν e is as large as 10% at 100 GeV. Therefore, the uncertainty of the ratio increases from about 3% at 5 GeV to about 10% at 100 GeV. 12

25 2.2 Atmospheric Neutrinos Figure 2.7: The flux ration of ν µ +ν µ to ν e +ν e averaged over all angles versus neutrino energy for three different models [11] Ratio between Muon- and Antimuon-Neutrinos As a result from the decays listed in section the ration of ν µ to ν µ should be of the order of 1 for low energies. As shown in figure 2.8, all three calculations agree with an uncertainty of 5% below 10 GeV. However, the disagreement increases with the neutrino energy. The reason for this are the uncertainties in K cross section and decay length and the fact that not every high-energy µ decays before it reaches the ground. Therefore, the systematic errors in the ν to ν ratios are about 5% at 10 GeV and rise to about 10% and 25% at 100 GeV for the ν e to ν e and ν µ to ν µ ratios, respectively Angular distribution Although the primary cosmic ray spectrum is nearly isotropic the angular distribution of atmospheric neutrinos is not. This is due to the characteristics of their production from their primary mesons (compare with equation 2.2). Figure 2.9 shows the zenith angle dependence of the atmospheric neutrino flux for several energy ranges. At low energies the fluxes of neutrinos with θ 0 are lower than those of neutrinos with θ 180. This is due to the deflection of primary cosmic rays by the geomagnetic field, roughly 13

26 2 ATMOSPHERIC NEUTRINOS Figure 2.8: Flux ration of ν µ to ν µ and ν e to ν e versus neutrino energy for three different models [11]. characterized by a minimum rigidity cutoff. For neutrinos with energies higher than a few GeV this effect is negligible. Hence, in the zenith angle dependence in figure 2.9(b) the influence of the geomagnetic field has been neglected so that the distribution is essentially up-down symmetric. The increase at the horizon is a result of the geometry of atmospheric neutrino production [7]. The differential rate E ν dn/de ν from atmospheric patches ds below and above the horizon is inversly proportional to the zenith angle cos θ and proportional to a yield factor Y νi (E ν, Ω). This yield factor is proportional to cos θ cr, which is the projection of the isotropic cosmic ray flux onto the surface element ds. As energy increases, the neutrino production rate peakes in the forward direction (cos θ cr = cos θ) (see section 2.2.5). Therefore, Y νi (E ν, Ω) cos θ, and the horizontal increase in the differential rate cancels. For low-energy neutrinos, the deviation from the direction of the primary parent particle cannot be neglected and the cancellation is not complete. The mean transverse momentum of pions produced in cosmic-ray interactions sets the energy scale for horizontal enhancement through equation

27 2.2 Atmospheric Neutrinos (a) Zenith angle dependence for low energetic neutrinos [11]. (b) Zenith angle dependence for high energetic neutrinos [7]. Figure 2.9: Zenith angle dependence for atmospheric neutrinos (a) for low energetic neutrinos for the three models: Honda (continuous line), Bartol (dashed line) and Fluka (dotted line) (b) for high energetic neutrinos Uncertainties Some uncertainties on the neutrino energy spectrum and on the production rate of neutrinos have explained above. The remaining uncertainties on the total flux of atmospheric neutrinos originate from the primary spectrum and from the treatment of hadronic interactions. Additional uncertainties of technical nature are caused by approximations in present calculations. As shown above, the proton primary spectrum measured by BESS [10] and AMS [9] has an estimated uncertainty of ±5% below 100 GeV/nucleon increasing to ±10% at 10 TeV/nucleon. If all valid measurements are included (see figure 2.10) for protons and helium nuclei (those with even larger uncertainties) to estimate the uncertainty in the primary spectrum, a more complete estimation is received. There is still a lack of recent data in the TeV range which is important for neutrino-induced muons with θ 180. Even if one assumes that the high-energy data should be normalized down to better connect with the low-energy data, the measurement covers a range of ±20% below 100 GeV and ±30% above [7]. The uncertainties due to hadronic interactions are more difficult to quantify. As shown in figure 2.11 large regions in the phase-space for proton-air interactions 15

28 2 ATMOSPHERIC NEUTRINOS (a) Observed flux of cosmic ray protons. (b) Observed flux of cosmic ray helium nuclei. Figure 2.10: Observed flux of cosmic ray protons (a) and helium (b). The dashed lines show the fits described in section The data point are measured by: Webber [14], crosses; LEAP [15], upward solid triangles; MASS1 [16], open circles; CAPRICE [17], vertical solid diamonds; IMAX [18], downward solid triangles; BESS98 [10], solid circles; AMS [9], solid squares; Ryan [19], horizontal solid diamonds; JACEE [20], downward open triangles; Ivanenko [21], upward open triangles; Runjob [22], open diamonds; Kawamura [23], open squares; [7]. are not measured, yet, and models need to interpolate and extrapolate into this region. Combining the systematic uncertainties in the primary spectrum and in pion production as if they were statistically independent, one can therefore assign a theoretical uncertainty of about ±20% to ±25%. 16

29 2.2 Atmospheric Neutrinos Figure 2.11: Phase-space distribution for sub GeV atmospheric neutrinos. squares indicates the density in the phase-space [7]. The size of the The deviations between the direction of particles originating from the cascade and the direction of the primary have two major sources. One is the deflection of the muon due to the geomagnetic field which can be estimated as follows: A muon with a Lorentz factor γ = E µ /m µ typically bends by an angle Θ γcτ µ /r L, (2.3) where r L is its gyroradius in the geomagnetic field. Therefore, the dependence of E µ cancels out in equation 2.3 and for a field of the order of 0.3 Gauss the typical bending is 3. As a consequence, muon bending is noticeable even in the multi- GeV range. The effect of transverse momentum on the other hand is most important at low energies. Since the distribution of the transverse momenta of secondaries in hadronic interactions is nearly independent of the energy, with p T 300 MeV for pions, the corresponding angular deviation is inversly proportional to energy. Finally, the decay processes such as π + ν µ + µ + followed by µ + e + + ν e + ν µ in which the 17

30 2 ATMOSPHERIC NEUTRINOS neutrinos are produces lead to a deviation of an order of p T E π 300MeV 0.1 rad. (2.4) E π E ν /GeV Characteristic 3D-effects are therefore most important for neutrinos with sub-gev energies. Other sources for uncertainties are that calculations usually assume a spherical Earth with the flat surface at sea level. When parent mesons enter the ground before decaying, they rapidly loose energy by ionizations or nuclear interaction and produce only very-low-energy neutrinos. Therefore, a high mountain above a neutrino detector reduces the neutrino flux to some extent. Except for downward going particles the effect is negligible and then it is significant only for decays of muons. For the Super-Kamiokande detector the effect of the Ikenoyama mountain as a sink for ν e has been estimated to be 2% 3% in the range of GeV [24]. Calculations are also made for standard atmosphere conditions without noticable effect from pressure and seasonal variations on the surviving muon and neutrino fluxes at ground level. Since this effect is small for neutrinos, variations at 1% for muons have been estimated [24]. 18

31 CHAPTER III Neutrino Oscillations

32 3 NEUTRINO OSCILLATIONS 3.1 Motivation In 1957 the Italian theoretical physicist Bruno Pontecorvo predicted a quantum mechanical process which could change a neutrino s flavor α = e, µ, τ from one to another. This implies that neutrinos have mass and that leptons mix[25]. Both facts are not predicted by the Standard Model. In recent years, experimental evidences for such flavor oscillations have been accumulated in solar, reactor, accelerator and in particular in atmospheric neutrino data[25]. These oscillations are a result of the discrepancy between the flavor eigenstates and the mass eigenstates of the neutrinos. 3.2 Quantum Mechanical Derivation If neutrinos have mass, then there is a spectrum of three or more possible neutrino mass eigenstates ν 1, ν 2, ν 3,.... These are the counterparts of the three lepton mass eigenstates e, µ, τ. The weak interaction, coupling a W ± boson to a charged lepton and a neutrino, can also couple any charged-lepton mass eigenstate l α, α = e, µ, τ to any neutrino mass eigenstate ν i. The amplitude for the decay of a W boson to generate a specific combination l α + + ν i is given by the unitary coupling matrix Uαi [25]. This matrix is the so-called Pontecorvo-Maki-Nakagawa-Sakata matrix U. It can be written as a product of three rotation matrixes U 23 U 13 U 12 : U = c 23 s 23 0 s 23 c 23 c 13 0 s 13 e iδ s 13 e iδ 0 c 13 c 12 s 12 0 s 12 c (3.1) with c ij = cos θ ij and s ij = sin θ ij. Each mixing angle θ ij parametrizes a rotation U ij, between the flavor eigenstates and the mass eigenstates. In general the coupling matrix U contains a phase factor δ which would cause CP-violation. This is indeed anticipated but has not been observed by any experiment, yet[25]. In the following δ is neglected. The neutrino flavor state ν α generated by a W ± decay is a quantum superposition of the mass eigenstates ν i : 3 ν α = Uαi ν i. (3.2) And vice versa using the unitarity of U i ν i = α U αi ν α. (3.3) As the evolution in time occurs in the free neutrino mass eigenstates, Schrödinger s equa- 20

33 3.2 Quantum Mechanical Derivation tion has to be applied to the ν i component of ν α from equation 3.2 with i t ν i (t, L) = 1 2m ν i (t, L) (3.4) ν i (t, L) = ν i (0, 0) exp [ i (E i t p i L)] (3.5) as a solution, where E i and p i are the energy and momentum and t and L are the time and covered distance of the ν i in the laboratory frame. As the neutrinos will be highly relativistic v c t L (with c = 1) in the laboratory frame the phase factor becomes exp [ i (E i p i ) L]. Picturing that the ν α was generated with a fixed energy E and assuming that every component ν i received the same energy E i = E then every component ν i has the momentum p i = E 2 m 2 i E + m 2 i /2E. This leads to a phase factor of [ ] exp [ i ( E E + m 2 i /2 ) L ] = exp Substituting equation 3.6 into equation 3.2 leads to i m2 i 2E L. (3.6) ν α (L) i U αie i(m2 i /2E)L ν i. (3.7) Applying equation 3.3 for flavor β in equation 3.7 leads to ν α (L) [ ] Uαie i(m2 i /2E)L U βi ν β. (3.8) β i The initial ν α has turned into a superposition of all flavor eigenstates e, µ, τ after travelling a distance L. The probability to measure flavor β, P (ν α ν β ), is ν β ν α (L) 2. This leads to ν β ν α (L) 2 2 = U αiu βi e i(m2 i /2E)L i [ ] = UαiU βi e i(m2 i /2E)L U αj Uβje i(m2 j /2E)L i j = i,j U αiu βi U αj U βje i ( m 2 i m2 j 2E ) L = i UαiU βi UαiU βi U αj Uβje i>j m 2 ij i 2E L (3.9) where m 2 ij denotes the squared mass difference between neutrino mass eigenstate i and j. Using the unitarity of the coupling matrix U leads to UαiU βi 2 2 = U αiu βi 2 UαiU βi U αj Uβj i i i>j }{{} δ αβ 21

34 3 NEUTRINO OSCILLATIONS with the Kronecker-delta δ αβ. Therefore, equation 3.9 can be written as [ ν β ν α (L) 2 = δ αβ 2 i>j U αiu βi U αj U βj 1 e m 2 ij i 2E L ]. (3.10) With e iϕ = cos ϕ i sin ϕ and 2 sin 2 (x/2) = 1 cos (x) the exponent in equation 3.10 becomes m 2 ij/ev 2 L/eV 4E/eV. With evs and c m/s this leads to m 2 ij ev 2 L ev Finally, equation 3.10 becomes P ( ( ν ) α ( ν ) β) = ν β ν α (L) 2 ( ) 4E 1 = m2 ij ev ev 2 L m 1 c 1.27 m2 ij ev 2 = δ αβ 4 R ( [ ) UαiU βi U αj Uβj sin 2 i>j + 2 I ( [ ) UαiU βi U αj Uβj sin i>j ( 4E ev L km ) 1 ( E GeV 1.27 m2 ij ev m2 ij ev 2 ) 1. ( L E km GeV L km ( E GeV ) 1 ] ) 1 ] (3.11) with R and I as the real and imaginary part, respectively. Since the phase factor δ in equation 3.1 is neglected, the imaginary part in equation 3.11 is neglected, too. Figure 3.1 shows the probabilty for the oscillations of a ν µ with recent results of the paramaters[25]: m 2 12 = (7.59 ± 0.20) 10 5 ev 2, m 2 13 m 2 23 = (2.43 ± 0.13) 10 3 ev 2, sin 2 (2θ 12 ) = (0.87 ± 0.03), sin 2 (2θ 23 ) > 0.92, sin 2 (2θ 13 ) < Approximations In the case of atmospheric neutrino oscillations, experiments led to the hypothesis that the oscillations driven by m 2 12 can be neglected (if a non-zero θ 13 in equation 3.1 is assumed) and hence, atmospheric neutrino oscillations are purely ν µ ν τ [25]. Therefore, these are two-neutrino oscillations with a mass difference m 2 atm := m 2 23 and θ atm := θ 23. Consequently the coupling matrix equation 3.1 simplifies to the relevant part of U 23 : U = ν µ ν τ [ ν 2 ν 3 cos θ atm sin θ atm sin θ atm cos θ atm ] (3.12) 22

35 3.3 Approximations Figure 3.1: Probability for ν µ oscillations in case of a three neutrino mass spectrum with the parameters as given below, sin 2 (2θ 23 ) = 1 and sin 2 (2θ 13 ) = 0.19 ; Blue: ν µ ν µ, Black: ν µ ν e, Red: ν µ ν τ. where θ atm is the mixing angle. Thus, equation 3.11 simplifies considerably to P ( ( ν ) α ( ν ) β) S αβ sin 2 [ ( ) 1.27 m2 atm L E 1 ] ev 2 km GeV (3.13) with S αβ 4 2 UαiU βi iup (3.14) where iup denotes a sum over only those neutrino mass eigenstates which lie above m 2 atm. For β = α one obtains P ( ( ν ) α ( ν ) α) 1 4 T α [ ( ) 1.27 m2 atm L E 1 ] ev 2 km GeV (3.15) with T α Uαi 2 (3.16) iup 23

36 3 NEUTRINO OSCILLATIONS Figure 3.2: Probability for ν µ oscillations in case of two depending neutrino masses without the influence of matter; Blue: ν µ ν µ, Black: ν µ ν τ. Substituting equation 3.12 into equation 3.14 and equation 3.16 leads to S αβ = sin 2 2θ atm and 4 T α (1 T α ) = sin 2 2θ atm, so that finally equation 3.13 and equation 3.15 become [ ( ) P ( ( ν ) α ( ν ) β) = sin 2 2θ atm sin m2 atm L E 1 ] ev 2 km (3.17) GeV and P ( ( ν ) α ( ν ) α) = 1 sin 2 2θ atm sin 2 [ ( ) 1.27 m2 atm L E 1 ] ev 2 km. (3.18) GeV The probability for the oscillations of a ν µ for only two neutrino mass eigenstates is shown in figure Enhancement by Matter Atmospheric neutrinos are typically generated in a height of about 20 km. Therefore, the oscillations on their way down to the Earth s surface can be well approximated by oscillations in vacuum[26]. For neutrinos propagating in matter like the Earth, the Hamiltonian H has to be modified to H = H 0 + H M where H 0 is the undisturbed Hamiltonian leading to the results discussed in section 3.2. H M is the effective weak interaction Hamiltonian due to the coherent interaction with the electrons, protons and neutrons in a medium. The neutrino flavors in equation 3.2 are eigenstates of H M H M ν α = V α ν α 24

37 3.4 Enhancement by Matter where V α is the effective potential depending on the neutrino flavor: and V e = V NC + V CC (3.19) V µ = V τ = V NC. (3.20) Here V NC and V CC are the charged- and neutral-current effective potential of neutrinos, given by V CC = 2G F N e, (3.21) 2 V NC = 2 G F N n (3.22) where G F is the Fermi constant and N e and N n are the electron and neutron number density in the medium[26]. Due to the fact that the matter is considered to be free of muons and tauons ν µ and ν τ interact only by neutral current (see equation 3.20). Considering a homogeneous, eletrically neutral medium with an equal number density of protons and electrons the neutral-current effective potential of protons and electrons cancels out. Hence, only the neutron number density has to be considered for equation In the Earth this typically leads to V CC V NC ev. Because of the flavor conservation of the neutral-current it affects all neutrino flavors equally (compare equation 3.19 and equation 3.20). The additional charged-current term in equation 3.19 for ν e leads to effective masses and mixing angles which are different than in vacuum (for a one-mass splitting see equation 3.23 and equation 3.24). Figure 3.3 shows the effect of matter for a three mass spectrum (left panel) on the survival probability P ( ( ν ) µ ( ν ) µ). While the y-axis represents the energy of the neutrino the x- axis represents the measured zenith angle which is connected to the distance the neutrino travelled through the Earth. Matter which does not contain anti-particles has no effects on the oscillations of anti-muon neutrinos (lower left panel). In the upper plot the regular oscillations pattern dominated by the vacuum oscillations ν µ ν τ is seen with certain distortions in the region from 3 GeV to 10 GeV: In the region from cos Θ ν = 1 to cos Θ ν 0.8 the neutrino crosses the Earth s core as well as the Earth s mantle. Due to the higher density of the core, the core s matter is responsible for the resonances in this area. In the region from cos Θ ν 0.8 to cos Θ ν = 0 the neutrino crosses only the Earth s mantle. The resonance in this part is a result from the mantle matter. For the case of only two depending neutrino flavors, one can determine the effective squared mass difference in matter to [27] µ 2 atm = m 2 atm (cos 2θ atm 2E V CC / m 2 atm) 2 + sin 2 2θ atm (3.23) 25

38 3 NEUTRINO OSCILLATIONS and the mixing angles in matter to [27] sin 2 2θ atm,m = sin 2 2θ atm sin 2 2θ atm + (cos 2θ atm 2E V CC / m 2 atm) 2. (3.24) Therefore, equation 3.18 has to be modified to P ( ( ν ) α ( ν ) α) = 1 sin 2 2θ atm,m sin 2 [ ( ) 1.27 µ2 atm L E 1 ] ev 2 km GeV (3.25) The right panels in figure 3.3 show the difference between the survival probability for a three neutrino mass spectrum and a two neutrino mass spectrum with the effect of matter. As seen the difference is comparably small and follows to a large extend the structure of the distribution for the three mass spectrum. 3.5 Sterile neutrinos While there are only three charged lepton mass eigenstates (e, µ, τ), it is suggested that there are three or even more neutrino mass eigenstates (i = 1, 2, 3...). Supposing a possible fourth eigenstate, one of the linear combinations s ν s = i U si ν i, does not have any charged lepton partner. Hence, it does not couple to the Standard Model W ± -boson. Furthermore, since the decays Z ν α ν α of the Standard Model Z- boson have been found to yield only three distinct neutrinos ν α [29], ν s does not couple to the Z either. Such a neutrino, which does not have any Standard Model couplings, is referred to as a sterile neutrino. In this thesis a sterile neutrino is neglected. 3.6 Recent Atmospheric Neutrino Oscillation Analyses The Super-Kamiokande detector which is placed at Kamioka in Japan, is water-filled tank equipped with optical modules. Thereby, atmospheric neutrinos of all three flavors can be detected. Additionally, a comparison can be made between downward going neutrino events and upward going neutrino events. The latter cover a distance so that neutrino oscillations might occur and create a disappearence of neutrinos at certain energy and zenith ranges. The data which was used for this analysis was taken in three different detector configurations from 1996 till The analysis is divided into two parts, the first concentrates on the analysis of θ 23 while the second concentrates on θ

39 3.6 Recent Atmospheric Neutrino Oscillation Analyses Figure 3.3: Shown are the contours of constant probability P ( ( ν ) µ ( ν ) µ) (left) as well as the difference in P ( ( ν ) µ ( ν ) µ) between a three neutrino mass spectrum and a two neutrino mass spectrum (right), both for muon neutrinos (upper panels) and antimuon neutrinos (lower panels). The oscillation parameters are sin 2 (2θ 13 ) = 0.05, m 2 12 = ev 2 and tan 2 (θ 12 ) = For the two neutrino mass spectrum m 2 = m 2 13 is used [28]. For the analysis of θ 23 two different fits were perfomed on the data. The first neglects the m 2 12 mass splitting (see section 3.3). Hence, the fit was done on a twodimensional space of m 2 23 and sin 2 (2θ 23 ). Additionally, this space was extended to four dimensions including also m 2 12 and sin 2 (2θ 12 ). To identify the best fit a χ 2 -method was used. Figure 3.4 shows the χ 2 χ 2 min distribution as a function of 27

40 3 NEUTRINO OSCILLATIONS sin 2 (2θ 23 ) for the two-dimensional as well as for the four-dimensional space. The best fit values are sin 2 (2θ 23 ) = 0.50, m 2 23 = ev 2, sin 2 (2θ 12 ) = 0.30 and m 2 12 = ev 2 for the four-dimensional space and sin 2 (2θ 23 ) = 0.50 and m 2 23 = ev 2 for the two-dimensional space, respectively [27]. For the analysis of θ 13 a grid was scanned of oscillation points in three variables: log 10 m 2 23, sin 2 (2θ 23 ) and sin 2 (2θ 13 ). The best fit was placed at m 2 23 = ev 2, sin 2 (2θ 13 ) = 0.0 and sin 2 (2θ 23 ) = 0.50 [27]. Figure 3.5 summarizes the results of these two analyses. Figure 3.4: χ 2 χ 2 min distribution as a function of sin2 (2θ 23 ) for the two-dimensional space (dotted line) and the four-dimensional space (solid line). The horizontal line corresponds to the 68 %(90 %) confidence level [27]. Figure 3.6 illustrates the comparison between the neutrino fluxes measured at the Super-Kamiokande detector with calculations based on the two different models introduced in chapter 2 not including oscillations. Shown is the number of electron and muon neutrino events depending on the neutrino energy. The difference between calculations and experimental results is consistent with what is expected for atmospheric neutrino oscillations in this energy range that involve ν µ and ν τ but not ν e. Therefore, there is a deficit of ν µ (right panel) whereas the measured flux of ν e (left panel) is consistent with the models. 28

41 3.6 Recent Atmospheric Neutrino Oscillation Analyses Figure 3.5: Results of the Super-K analyses for sin 2 (2θ 13 ), sin 2 (2θ 23 ) and m 2 23 with the allowed regions at 68 % (thin line), 90 % (medium) and 99 % (thick) confidence level. The shaded region in the first panel shows the Chooz 90 % exclusion region [27]. Figure 3.6: Comparison between calculated flux (see chapter 2) without neutrino oscillations and the flux measured with the Super-Kamiokande detector [7]. 29

42 3 NEUTRINO OSCILLATIONS 30

43 CHAPTER IV The IceCube/DeepCore Detector

44 4 THE ICECUBE/DEEPCORE DETECTOR 4.1 Neutrino Detection Due to the fact that neutrinos interact only the via weak interaction and gravitation it is a big challenge to detect them. Most detectors are sensitive to the reaction products from charged current interactions (see section 4.1.1). One possible detection technique is the radiochemical method, which is based on the counting of neutrino-induced reaction products. Unfortunately, this type of experiments doesn t allow a reconstruction of the neutrino s original direction. For the measurement of the neutrino s direction and its energy the observation of the produced lepton or the cascade is necessary. An adequate way for this purpose is the Cherenkov detection method which is explained in section Neutrino Interactions with Matter There are two channels of weak interaction for neutrinos. One is the neutral current by a Z 0 -boson ( ν ) l + N ( ν ) l + X (4.1) where the neutrino interacts with a nucleus N and produces a hadronic cascade X. The neutrino survives with a change of its energy. The cross-section for interactions with the matter s electrons is comparatively small and can be neglected. The other channel is the charged current by a W ± -boson ( ν ) l + N l + X (4.2) where the neutrino produces a lepton with the same flavor that receives a dominant part of its energy. The hadronic rest X also triggers a hadronic jet. The initial direction of the lepton is well connected with the one of the neutrino; the average angle mismatch is approximately ( ) 0.7 TeV ψ = 0.7 [30]. (4.3) E ν 4.2 The Cherenkov Effect A common detection principle for high-energy neutrinos is the optical detection by Cherenkov radiation. This phenomenom occurs when a charged particle moves through a dielectric medium with a velocity v higher than the speed of light in this medium c = c/n, where n denotes the refraction index of the medium. For ice, n ice 1.33 and therefore the condition for Cherenkov light is v c ice = c = 0.75c (4.4) n ice 32

45 4.3 Lepton Energy Loss and β = v c c ice/c = 1/n ice = (4.5) The effect can be explained as follows: The charged particle polarises the molecules of the surrounding dieleectric medium. When these molecules return to their equilibrium they emit light comparable to dipole radiation. If the speed of the charged particle is too slow, the emitted photons interfere destructively. If the speed of the particle exceeds v given in equation 4.4, the interference becomes constructive. Thereby a wavefront of light is built which forms a cone. The opening angle of this cone the so-called Cherenkov angle θ C depends on the refraction index and on the speed of the particle cos θ C = 1 nβ (4.6) For ice and v c the Cherenkov angle becomes θ C 41. This effect is similar to that effect that occurs if an aircraft moves with a velocity greater than the speed of sound and generates a Mach cone. Figure 4.1: A lepton - here a muon - travelling through a lattice of photomultipliers emitting light under the Cherenkov angle [31]. Figure 4.1 illustrates this effect of a muon which travels through a lattice of optical sensors and emits cherenkov photons. 4.3 Lepton Energy Loss The charged lepton generated in equation 4.2 produces the most important signature for neutrino detectors. Common to all three flavors is the hadronic jet at the interaction point, which contains roughly 40% of the neutrino energy. The propagation of the lepton and therefore the detected signal structure varies depending on the flavor. 33

46 4 THE ICECUBE/DEEPCORE DETECTOR Electrons quickly lose energy in dense matter due to bremsstrahlung as long as their energy is above the threshold at which ionizations losses become dominating, called critical energy. For ice, the critical energy is E e crit =81 MeV[32]. The photons produced by bremsstrahlung generate new electrons and positrons via pair production until the energy of the photon falls below 2m e. The relevant processes for the energy loss of muons are ionization of the detector medium, bremsstrahlung, pair production and nuclear reactions with ice molecules. In contrast to the other processes ionization yields a continuous energy loss, while the others occur stochastically. In these interactions charged secondary particles are generated. These secondaries produce Cherenkov photons, too. The energy loss due to ionization of the medium is described by the Bethe-Bloch formula [33] de µ dx Z 1 Ionization A β 2 [ 1 2 ln 2m ec 2 β 2 γ 2 T Max β 2 δ ] I 2 2 (4.7) with the atomic number Z and atomic mass A of the medium, the electron rest energy m e c 2, velocity β = v and c γ 1 = 1 β 2 of the muon, the maximum transfer of kinetic energy in a muon-electron collision T Max, the mean excitation energy I and the density effect correction term δ. The energy required to ionize an ice molecule is of the order of some ev. Up to an energy of E µ crit 500 GeV in ice ionization is the dominant process. At higher energies stochastic processes like bremsstrahlung, pair-production and nuclear reactions become important. If the muon is deflected in the electromagnetic field of a nucleus, it emits bremsstrahlung. The average energy loss per distance is given by [33]: de ( ) µ Z2 183 dx Bremsstrahlung A ln E Z 1/3 µ. (4.8) The total energy loss of a muon in ice is appoximately given by de µ dx = a + b E µ[34], (4.9) with a 2.4 MeV and b 4 cm 10 6 cm 1, where a (E µ ) describes the contribution by ionization and b E µ is a parametrization for high-energy processes (b = b pair + b brems + b nucl ). The energy loss for tau particles is described by equation 4.9 as well with a very similar value of a and b m 1 [34]. Up to high energies the generated track of taus is comparably short because of their short lifetime of τ τ = 0.29ps. Usually a second cascade is triggered at the decay of a tauon. The branching ratio for a tauon to decay into τ µ + ν µ + ν τ is % and the remaining decays generate at least one electron or charged meson [35]. For energies greater than a few 34

47 4.4 Optical Neutrino Detection PeV the second cascade is seperated in space from the first one. These two cascades are connected by a faint Cherenkov track. Thus, tauons produce a characteristic signature called double bang. If one of the these cascades lies outside the detector volume, the resulting signature is called lollipop. Figure 4.2 illustrates the three signatures of the charged leptons in Cherenkov media. Figure 4.2: Sketch illustrating the signatures of charged current interactions of neutrinos in Cherenkov media [36]. 4.4 Optical Neutrino Detection The Frank-Tamm formula describes d 2 N dxdλ = 2πα λ 2 ( 1 1 ) n 2 β 2 the number of photons produced per length in a wavelength certain range. Here, α = 1/137 is the Sommerfeld fine-structure constant. In the wavelength range from 300 nm to 500 nm a muon in ice produces about 250 photons per cm. The energy loss for a muon in ice due to Cherenkov radiation is de µ dx = dn 500nm Cherenkov dx E γ = 300nm ( 2πα 1 1 ) dλ hc λ 2 n 2 β 2 λ 850 ev cm [33]. In comparison to the energy losses by other processes (see section 4.3) the Cherenkov effect is negligible. Due to the fact that the cross-section for neutrino interactions is small, a huge detector is needed. The medium has to be transparent for blue to ultraviolet light which is favored in Cherenkov radiation. Air is a possible detection medium but air is not very dense and has a small refraction index n air 1 which requires high velocities β for the Cherenkov light. H 2 O in liquid or solid state is a proper matter for huge detectors. Compared to 35

48 4 THE ICECUBE/DEEPCORE DETECTOR bubble free ice, water has less scattering (up to a factor of 10 [37]) but higher absorption (about a factor 2). Detectors in lakes or seas with deep and clear water have to face several challenges like a strong background of bioluminescence from various lifeforms as well as from radioactive dacays of unstable isotopes as 40 K and, furthermore, erosion of the detector components. 4.5 IceCube The IceCube Neutrino Observatory is an underground Cherenkov detector that uses ice as a detection medium. Because of the huge available amount of clear ice, it is built up at the Amundsen-Scott South Pole Station at the geographic South Pole. IceCube s layout is heavily influenced by the the properties of the Antarctic ice. As every glacier, it has grown (and still grows) in an annual cycle by accumulation and compression of the snowfall. At the South Pole the ice reaches depths of about 3000 m before a bedrock is reached. The two most important optical properties of a detector medium are the effective scattering length λ e and the absorption length λ a. Due to changes in the atmospheric conditions in the last hundreds of centuries, the ice is not fully homogeneous. Figure 4.3 shows profiles for scattering and absorption of light as a function of depth and wavelength. Above 1300 m the main reason for scattering are air bubbles. Under the higher pressure at greater depth, these bubbles decrease in size. Below 1300 m, absorption and scattering are caused by µm-sized dust grains. At depths between 2000 m and 2100 m, a particularly pronounced dust layer is present. The ice in this depth is approximately years old. Below this dust layer the ice is exceptionally clear [38]. The scattering length λ s for IceCube is defined as the mean free path between two scattering processes. The effective scattering length is given by λ e = λ s 1 cos θ, where cos θ is the mean angle in a single scattering process. This is an estimation for the length after which a beam of light has become fairly isotropic. In the dust layer the effective scattering length is less than 10 m while it is approximately 20 m outside the dust layer [40]. The absorption length is defined as the length after which the light s intensity dropped by a factor of 1/e. As one can notice in figure 4.3 the absorption length is significantly higher than the effective scattering length (about a factor 5 at 2000 m). Typical values for the absorption length are λ a = 90m 110m. Recent IceCube measurements have shown that the deep ice might be clearer than expected, with λ e 50m and λ a 200m at a blue wavelength (405 nm) [41]. 36

49 4.5 IceCube Figure 4.3: Scattering (left) and absorption (right) profiles for the ice at the South Pole. In both profiles the structure caused by dust layers is clearly visible [39] Detection Method Figure 4.4: Sketch of an IceCube Digital Optical Module (DOM) [42]. The Cherenkov light is detected with Digital Optical Modules (DOMs) (see figure 4.4). The task for the DOMs is to detect the light and to convert it into digitized waveforms. The main component of every DOM is a photomultiplier tube PMT, a flasher board and a mainboard which contains most of the electronics. The flasher board has twelve LEDs 37

50 4 THE ICECUBE/DEEPCORE DETECTOR which can be used to calibrate the positioning and timing of the DOMs. This is done by flashing the LEDs in on DOM and measuring the arrival time of the light in the neighboring DOMs. The diameter of a DOM is 32.5 cm. The noise rate of a DOM is 650Hz [43]. If a Cherenkov photon reaches the DOM it produces a lot of electron in the PMT and thereby an electronic signal. This DOM is called a hit DOM. Till 2009, DOMs were only read out if at least one of their neighbouring DOMs triggered within a microsecond, too. This is called Hard Local Coincidence (HLC). Isolated hits were cut out of the data. Since 2009, DOM hits that don t fulfill HLC are kept irrespectively if other DOMs triggered HLC. To reduce the data volume only the charge stamp and time is stored for these hits. This is called Soft Local Coincidence (SLC). Then, the hits are analyzed by triggers. The simplest trigger is the Simple Multiplicity Trigger (SMT). It occurs when a fixed number of DOMs are hit within a certain time window. If these hits meet at least one of those criteria the launches are aggregated into an event The IceCube Configuration IceCube itself consists of 80 long cables called strings, equipped with 60 DOMs each. The DOMs are deployed into the ice in a depth of 2450 m up to 1450 m with a vertical spacing of 17 m. The strings are arranged in a hexagonal structure with a horizontal spacing of 125 m (see figure 4.5). These dimensions corresponds to an energy threshold of > 100GeV [44]. IceCube has its best sensitivity at E TeV. The total volume is 1 km 3. The DeepCore detector is a low-energy enhancement of IceCube. It consists of six additional strings which are deployed around one string in the center of IceCube (see figure 4.5). Each of these strings is equipped with 50 high-efficiency DOMs with a vertical spacing of 7 m below the dust layer and with 10 high-efficiency DOMs with a vertical spacing 10 m above. These high-efficiency DOMs differ in that they have an improved photocathode which leads to a higher efficiency of up to 33 % compared to the standard IceCube DOMs. This configuration lowers the energy threshold for IceCube combined with DeepCore to 10GeV. Since the main part of the DeepCore DOMs is placed in the bottom center of IceCube, the surrounding IceCube can be used as a veto for the atmospheric muon background. By selecting signal events which have the interaction vertex inside DeepCore the search for neutrinos from the full sky becomes possible [31]. For the deployment of the DOMsholes are drilled with hot water and the strings are inserted into the water-filled holes. Then the water hole refreces. The deployment began in 2004 with one experimental string. Due to environmental conditions during Antarctic winters that make the deployment impossible, this is only possible during Antarctic summers. Therefore, the time to deploy strings is limited to less than three months which leads to a maximum number of 20 deployed strings per season. Currently 79 of the 86 38

51 4.5 IceCube Figure 4.5: 2D-illustration of the layout of the IceCube detector with the DeepCore enhancement and its precursor AMANDA. The grey stripe marks the dust layer in a depth of 2000m to 2100m. On the top there is the top view which illustrates the hexagonal structure of IceCube and on the bottom there is the side view which shows the configuration of the DOMs at the strings [45]. 39

52 4 THE ICECUBE/DEEPCORE DETECTOR strings are deployed and the detector will be completed in the next season. At this time six DeepCore strings are installed. Future plans propose that two additional strings, which couldn t be deployed at their regular position, will be installed into Deep- Core to make it even more densely instrumented. In parallel to IceCube the IceTop array is installed on the ice surface. IceTop is an air shower Cherenkov detector which consists of two tanks on top of each string filled with two DOMs each and with altogether 2.5 m 3 bubble-free ice. IceTop allows to measure cosmic ray induced air showers with a primary particle energy threshold of 150 TeV. Furthermore, it can be used as a muon veto detector for IceCube. 40

53 CHAPTER V Atmospheric Neutrino Oscillations in IceCube

54 5 ATMOSPHERIC NEUTRINO OSCILLATIONS IN ICECUBE 5.1 Conversion to Detector Observables Figure 5.1: Artistic illustration of the IceCube detecter placed in the Earth with the connection between the zenith angle ϑ and the base length. As calculated in section 3.3 the probability of a ( ν ) µ generated in a cosmic air shower in the atmosphere to be measured as a ( ν ) µ again (the so-called Survival Probability) after travelling the distance L (in the following named base length) is given by P ( ( ν ) α ( ν ) α) = 1 sin 2 2θ atm sin 2 [ ( ) 1.27 m2 atm L E 1 ] ev 2 km. (5.1) GeV Therefore, with fixed m 2 atm and θ atm, the base length L and the neutrino energy E are the only observables the survival probability depends on. For a conversion of the base length and neutrino energy into observables which can be detected by the experiment the zenith angle and the reconstructed track length of the muon are chosen. A first rough conversion between L and the zenith angle ϑ is given by (see figure 5.1) L = d Earth cos ϑ. However, this does not take into account the depth of the IceCube detector. A more accurate connection calculates the intersection of a straight line (the neutrino track) and a circle (the Earth). This yields L = r 2 Earth M 2 z 1 tan2 α 1 + tan 2 α (5.2) 42

55 5.2 Spectrum without Oscillations with M z = r Earth Z IC as the distance of IceCube from the Earth s center and α = ϑ π/2. For the estimation of the neutrino energy the reconstructed muon track length is used. By a Log-Likelihood-method the most probable muon track length l is reconstructed (see section 6.3.2). With equation 4.9 one gets ( ) E µ (l) α β + eβ l E µ,0 + α β (5.3) with a rest energy of the muon of about E µ,0 = GeV. This conversion assumes a continuous energy loss and constant values α and β. If additionally a constant relation between neutrino energy and muon energy is assumed the neutrino energy can be estimated. Howevery, the analysis is applied on the parameters used in equation 5.1 E ν and L or directly on the reconstructed observables ϑ and l. Therefore, the conversion is not used and serves only for test purposes. 5.2 Spectrum without Oscillations Figure 5.2: Simulated neutrino spectrum without oscillations: BLength vs Energy E. Shown is the base length Figure 5.2 shows the ν µ + ν µ spectrum of a Monte-Carlo dataset without the influence of oscillations (for more information about the dataset see table 5.1). The dataset was generated with an E 2 spectrum, hence, the histogram entries are reweighted to an atmospheric spectrum. To oppress the background of atmospheric muons only those events are taken which have a zenith angle of at least ϑ = 90 (from the horizon down to upwardsgoing).in contrast to neutrinos, muons are not able to pass more than a few kilometers 43

56 5 ATMOSPHERIC NEUTRINO OSCILLATIONS IN ICECUBE Figure 5.3: Zenith distribution of atmospheric muons (solid line) and atmospheric neutrinos (dashed line) for IceCube s precursor AMANDA [46]. through the Earth. Figure 5.3 shows the reconstructed zenith distribution of atmospheric muons and atmospheric neutrinos measured with AMANDA. Due to uncertainties in the angular reconstruction a cut on ϑ = 90 does not sort out all atmospheric muons but reduces their amount significantly. The seperation between atmospheric muons and atmospheric neutrinos induced events which are close to the horizon can be expanded by the selection of events with a starting muon track in IceCube. Moreover, atmospheric neutrinos as signal have priority. Therefore, the selection presented in section 6.4 is optimized on the quality of the reconstruction and assumes that the muon background is cutted out in this was, too. However, the zenith range from ϑ = 90 to ϑ = 180 leads to a range of the baselength of BLength ϑ=90 157km to BLength ϑ= km. This corresponds to a range from 2.2 to 4.1 in the logarithmic scale. The lower boundary of the energy scale in figure 5.2 is given by the energy threshold of IceCube combined with DeepCore of about 10 GeV (see section 4.5.2). 5.3 Influence of Oscillations The survival probabality due to neutrino oscillations within the boundaries of figure 5.2 is shown in figure 5.4. The oscillations parameters of equation 5.1 are set to sin 2 2θ atm = 1.0 and m 2 atm = ev 2, respectively. 44

57 5.4 Spectrum with Oscillations dataset 2111 # events E min /GeV E max /GeV θ min / θ max / γ Table 5.1: Parameters of the dataset 2111: The number of generated files, the number of simulated neutrinos per file, the energy and zenith ranges and the spectral index γ. Figure 5.4: The influence of neutrino oscillations in the same parameter region as figure 5.2. Shown is the survival probability as calculated in equation 5.1. For IceCube with a maximal base length of d Earth = 12714km from the North Pole to the South Pole [47] equation 5.1 leads to an energy of E ν Minimum 25GeV log 10 (E ν /GeV) Minimum 1.4 for the first minimum. This means that IceCube combined with DeepCore should detect very few vertically upward going ν µ + ν µ events with an energy 25GeV. Remembering that the energy-threshold for IceCube only was > 100GeV it can be understood that the DeepCore enhancement is necessary to measure the disappearence of atmospheric ν µ + ν µ due to neutrino oscillations. 5.4 Spectrum with Oscillations To estimate the ν µ +ν µ spectrum with oscillations one has to multiply the spectrum shown in figure 5.2 with the survival probabilty shown in figure 5.4. Figure 5.5 shows the spectrum with the influence of neutrino oscillations. In the upper left 45

58 5 ATMOSPHERIC NEUTRINO OSCILLATIONS IN ICECUBE corner one can clearly see the expected disappearence. Compared to figure 5.2 neutrino oscillations significantly changes the shape of the distribution. Figure 5.5: Simulated neutrino spectrum with oscillations (Compare with figure 5.2 and figure 5.4). Figure 5.6 shows the spectrum of figure 5.5 converted into the detector observables with neutrino oscillations while figure 5.7 shows the spectrum without neutrino oscillations. Here, no reconstructed parameters but ideal assumptions as in equation 5.3 are used like a continuous energy loss with constant values α and β and a fixed energy transfer of 80 % from the parent neutrino to the muon. In the lower left part the influence of neutrino oscillations is clearly visible. 46

59 5.4 Spectrum with Oscillations Figure 5.6: Spectrum with neutrino oscillations as shown in figure 5.5 converted into the detector observables zenith angle ϑ and muon track length T Length. Figure 5.7: Spectrum without neutrino oscillations converted into the detector observables zenith angle ϑ and muon track length T Length. 47

60 5 ATMOSPHERIC NEUTRINO OSCILLATIONS IN ICECUBE 48

61 CHAPTER VI Reconstruction

62 6 RECONSTRUCTION 6.1 Data selection The used simulated Monte-Carlo dataset is processed by different algorithms. First, the data from all 86 strings is selected. Then only the HLC hits (see section 4.5.1) are kept. Following, two selections are set. The first selects only those events which have at least eight hits in the whole IceCube+DeepCore detector within 5000 ns. The second chooses only the six DeepCore strings and the one IceCube string that is surrounded by those six and selects only those events which have at least six hits within 5000 ns. 6.2 Initial seed: LineFit The first reconstruction process is a fast, simple algorithm called linefit which serves as a seed for the following fits. During this fit the Cherenkov cone is ignored, as are optical properties of the ice. The algorithm yields a fit only on the basis on the hit times t i. It assumes light travelling in a one-dimensional line projection of the detector (from one DOM to the next) with a velocity v. Assuming r i as the position of one DOM the propagation of the light can be described approximately by a stright line: r i r + v t i. Then a simple χ 2 -method by minimisation of χ 2 = N hit i=1 (r i r v t i ) 2 is used. This minimisation can be done analytically by calculating the fit parameters r and v: r = r i v t i and v = r i t i r i t i t 2 i r i 2. The direction of the track is given by e = v/ v and the position of the track by r. 6.3 Full Reconstruction In the following, two algorithms are presented which are used for the reconstruction of the zenith angle and the muon track length. The pandelfit module improves the result of the linefit and provides a good reconstruction for the zenith angle. This information is used by the FiniteReco module which reconstructs the length of the muon track. 50

63 6.3 Full Reconstruction pandelfit As in almost every reconstruction procedure, in the pandelfit track parameters emerge as the solution of an optimization problem. Here, these parameters are the azimuth angle ϕ, the zenith angle ϑ and the position x, y, z of the track. The pandelfit module uses only the time of the first pulse of a hit DOM for the reconstruction. By minimization of the likelihood L = p (a, t res,j ) (6.1) j the values of the parameters for the best fit can be found. Here, a denotes the parameters listed above and the probability density function (PDF) p (a, t res,j ) describes a photon arrival at the DOM. It is convenient to use the time residual t res as a PDF variable t res = t hit t geo, (6.2) where t geo is the expected photon arrival time. Latter time is calculated as follows (see figure 6.1): the particle starts at a chosen reference point X on the track at the time t X and arrives at point C when the Cherenkov front hits the DOM B at t B. The track point labeled A indicates the point at which the Cherenkov photon was emitted that hit the DOM. The time t geo is the difference between t X and t B. Therefore, with the known distance from X to B, the speed of light in ice c ice = c/n ice m/s and the known Cherenkov angle θ C 41 and assuming that the muon travells with the speed of light c the time without absorption and scattering t geo can be calculated. Figure 6.1: Geometry of the signal generation process with the DOM lattice, the muon track and the Cherenkov front. To include absorption and scattering in ice a PDF named Pandel is suggested: p (ρ, ξ, t) = ρξ t ξ1 Γ (ξ) e ρt (6.3) with ρ and ξ as phenomenological parameters. These parameters can be associated with the characteristics of the medium, such as the mean photon scattering length λ s and the absorption length λ a. Thereby, the distance between emission and detection locations of a Cherenkov photon depends on the parameters a. 51

64 6 RECONSTRUCTION Furthermore, a PDF for realistic signals should account for the finite time resolution of the detector. Therefore, the Pandel PDF is convolved with a time jitter function, which is assumed as a gaussian. Additionally a constant noise term is added to the convoluted PDF. Initially, to avoid problems with negative time residuals at the calculation of equation 6.1 the smallest t res is found and added to all hit times t hit,i and the reference time t X, respectively. The used pandelfit module initially puts the reference point X very close to the center of gravity (COG). This center of gravity is calculated with all DOMs that got hit in an event. The linefit, that is used as a seed, lays the track automatically through the COG. For the reconstruction the azimuth angle ϕ, the zenith angle ϑ and the position x, y, z of the track are varied by the Simplex algorithm and the track with the smallest L equation 6.1 is chosen. Furthermore, the direct length is calculated. A so-called direct hit is a hit with a time residual within a range from 15 ns to 75 ns. For every DOM which was hit directly in an event the closest point on the muon track is calculated (for the hitted DOM B in figure 6.1 it is point D). The distance on the track between first and last hit in space is the direct length FiniteReco The FiniteReco module uses only the hit pattern and the track information from the pandelfit as a seed. It requires for each event the direction and the position of a reconstructed track. At the beginning, the neutrino interaction vertex is reconstructed. For that for each hit DOM within a cylinder with a radius of r c = 200m around the track the position on the track is calculated at which the photon was emitted (position A in figure 6.1). The projection of the first hit DOM on the track defines the neutrino interaction vertex and so the starting point of the track. Moreover, the projection of the last hit DOM on the track defines the stopping point of the track. The following reconstructions can be divided into two processes, which do nearly the same calculation but for different purposes. The first process calculates the most probable value for the length of the muon track. Therefore, first the starting point of the track is kept fixed and the stopping point varies depending on the track length. This track length is changed from a minimal length of 1 m up to a maximal length of 3000 m with a step size of 10 m. For each stopping point all DOMs are selected which are placed in the direction of the track but behind the stopping point. Again only those DOMs are selected which are placed within a cylinder with a radius of r c = 200m around the imaginary track. All these DOMs could have been hit by an infinite track. Then, two track 52

65 6.3 Full Reconstruction hypothesis are evaluated: An infinite track and a track stopping at the adjusted stopping point. For the first case, p (nohit Track) is calculated. In this case, the hit probability depends on track parameters and ice properties. The track parameters are provided by the pandelfit and determine the distance between the track and the DOM. The ice properties are provided by the Photorec tables of the Photonics project [48]. In the second case, the hit probability p (nohit notrack) of a stopping track is calculated. This is equal to the probability of a noise hit. To calculate these two probabilities the following algorithm is used: For each DOM and a given track the expected number of photons can be extracted from the Photorec tables. The probability for a hit in a certain DOM is calculated from the number of photo electrons assuming Poisson statistics. Hence, the probability for no hit is given by p λ (nohit) = p λ (0) = λ0 0! e λ = e λ with λ as the expected number of photo electrons. In the next step the product of the individual hit probabilities for all DOMs is calculated: P (nohit notrack) = p (nohit notrack) (6.4) and P (nohit Track) = selecteddoms selecteddoms p (nohit Track) (6.5) respectively. Finally, the logarithm of the ratio of these probabilities is taken: LLHR = log 10 P (nohit Track) P (nohit notrack). (6.6) The stopping point calculated with the length of the minimal LLHR is assumed as the most probable one. A typical so-called landscape of a distribution of the probabilty for different track lengths can be seen in figure 6.2. This procedure is repeated with fixed stopping point to estimate the starting point. Afterwards the most probable starting and stopping points are calculated. The distance between them is taken as track length. It is also possible to use the hit information, but the method as described here was found out to be superior to the other invented algorithms. The second process calculates the probabilities for three different track shapes: starting tracks have the neutrino interaction point inside the detector volume while the stopping point of the track lies somewhere outside, stopping tracks have the neutrino interaction point somewhere outside the detector volume while the muon track ends inside it, infinite tracks have both points, starting and stopping point of the muon track, somewhere outside the detector volume. 53

66 6 RECONSTRUCTION Therefore, the length of the muon track is changed as follows: The starting or stopping point is kept fixed and the other one is placed outside the detector volume. Additionally both points are placed outside the detector volume. The calculation for these three probabilities that the track shape fits to the given DOM hit pattern is calculated according to equation 6.4 and equation 6.5. For starting tracks, the DOMs in front of the fixed starting point as well as the DOMs behind the virtual reconstructed stopping point are selected. For stopping tracks, the DOMs in front of the virtual reconstructed starting point as well as the DOMs behind the fixed stopping point are selected. For infinite tracks the DOMs in the front of the virtual reconstructed starting point as well the the DOMs behind the virtual reconstructed stopping point are selected. Finally, the loglikelihood ratio for each track shape is calculated according to equation 6.6. For a starting or a stopping track, P (nohit Track) is small, whereas P (nohit notrack) is close to 1. The resulting LLHR for starting or stopping is therefore negative with a large absolute value. Thus, the algorithm delivers LLHR-values between 0 and any negative number. The larger the absolute value of the LLHR for a starting or a stopping track in comparison to the LLHR for an infinite track is, the higher is the track s probability to be starting or stopping. Hence, the LLHR can be used for each track as degree of believe of being starting, stopping or infinite. Figure 6.2: Distribution of the LLHR versus track length with the reconstructed track length (black). The true track length is marked in grey. 54

67 6.4 Quality Selection 6.4 Quality Selection To optimize the resolution of the reconstruction few quality cuts are made. Altogether there are four cuts which can be divided into two groups depending on the observable they are used for Optimization of the Angular Reconstruction To optimize the zenith reconstruction a cut on the direct length is made. The direct length is calculated with the pandelfit module (see section 6.3.1). Solving equation 5.3 with respect to the track length one gets TLength/cm = ln ( ) β E+α β E µ,0 +α. (6.7) β This corresponds to a track length of about 80m for a muon of 20 GeV. That muon with an energy corresponding to the energy at which the ( ν ) µ disappearence is expected will hardly generate a track length > 80m. Therefore, the cut on the direct length has to be lower than 80 m. On the other hand a longer direct length allows a better zenith reconstruction due to the longer mechanical advantage. Hence, an analysis was made to estimate for which direct length cut the quality of the zenith reconstruction reaches an adequate value. Figure 6.3: Quality of the zenith reconstruction for different direct length cuts; from top to bottom: DirectLength > 40m (black), DirectLength > 50m (red), DirectLength > 60m (green), DirectLength > 70m (blue), DirectLength > 80m (yellow). Figure 6.3 illustrates the zenith reconstruction quality for different direct length cuts. On the x-axis the absolute value of the difference between reconstructed zenith and true 55

68 6 RECONSTRUCTION zenith is plotted in logarithmic scale: log 10 ( ϑ / ) = log 10 ( ϑ true ϑ reco / ). As expected the quality of the zenith reconstruction improves for harder direct length cuts. On the other hand, the number of events that survive the cut falls rapidly. As a direct length cut of good compromise between the number of surviving events and the quality of the zenith reconstruction, 70 m is chosen. About half of the events survive the cut and the median of the reconstruction error distribution is about 5.3. Additional figure 6.7 shows the effect of the cut on the energy distribution and shows that not all low energetic events are cut out Optimization of the Energy Reconstruction The cuts which are done for the optimization of the correlation between neutrino energy and the reconstructed muon track length are mainly tests if the length reconstruction was successful. The first cut is done on the reconstructed muon track length. If the reconstruction did not converge the value is set to the minimal length of 1 m. Due to the stepsize of 10 m the smallest value the reconstruction could estimate is 11 m. To distinguish between these two lengths, all events with a track length log 10 (TLength/m) < 0.05 are cut out. Figure 6.4 shows the difference between the reconstructed muon track length and the idealized track length (see equation 6.7 with the same assumptions as for equation 5.3) versus the difference of the LLHR for an infinite track and starting or stopping track, respectively. As seen the majority of events is reconstructed with a LLHR difference close to 0. These tracks could be starting or stopping but with a track length close to the width of the detector. Therefore, only few DOMs would be selected for the estimation of the LLHR of these tracks. In a realistic analysis only those events are viewed which have a track that is starting and stopping in the detector volume. In doing so the energy reconstruction should work best due to the fact that the muon looses all its energy within the detector volume. Assuming that the algorithm presented in section really provides a value with which the shape of the track can be guessed two cuts are applied at 0 in both plots of figure 6.4. That means that the probability for an infinite track has to be smaller than the probability for a starting and a stopping track, respectively. The fraction of events that survive these three cuts are: log 10 (TLength/m) 0.05: 98.0 % LLHR(starting) > LLHR(infinite): 91.6 % LLHR(stopping) > LLHR(infinite): 94.7 % 56

69 6.5 Results for Angular and Energy Reconstruction (a) L vs LLHR(starting)-LLHR(infinite) (b) L vs LLHR(stopping)-LLHR(infinite) Figure 6.4: Quality of the length reconstruction vs difference of LLHRs for the given track shape for starting (a) and stopping (b). 6.5 Results for Angular and Energy Reconstruction Figure 6.5 illustrates the comparison between the zenith reconstruction before and after the four quality cuts. The fraction of well reconstructed events that survive the cuts is much higher than for badly reconstructed events. For a deviation of 10 less than 57

70 6 RECONSTRUCTION Figure 6.5: Comparison of the zenith reconstruction before (black) and after (red) the cuts. 30 % of the reconstructed events survive the cuts. After all cuts the median of the zenith reconstruction falls from to 5.8 while about 47 % of the events survive the cuts. Figure 6.6 illustrates the correlation between the neutrino energy on the x-axis and the reconstructed muon track length on the y-axis. Without cuts the correlation factor is 0.67 while it is 0.73 with the cuts. For this analysis the correlation factor is sufficiently good to allow satisfying results. 6.6 Results for Neutrino Oscillations Figure 6.7 illustrates the energy distributions of atmospheric neutrinos in a year before and after the quality cuts are applied. The relative amount of events that survive the cuts rises with the neutrino energy. While at energies below 20GeV only 10 % of the events survive the cuts, at energies above 150GeV more than half of the events survive the cuts. Although in the relevant region around 25 GeV only a small fraction of events survive, a good neutrino oscillations reconstruction is possible due to the fact that well reconstructed events survive the cuts. Figure 6.8 shows the same spectrum as figure 5.6 simulated with the same oscillations parameters (sin 2 2θ atm = 1.0 and m 2 atm = ev 2 ) but this time with the reconstructed detector observables and the cuts. In contrast figure 6.9 shows the histogram in detector observables without the influence of neutrino oscillations (compare with figure 5.7). Especially in the lower part of figure 6.8 the influence of neutrino oscillations is still 58

71 6.6 Results for Neutrino Oscillations (a) The logarithm of the reconstructed muon track length vs the logarithm of the initial neutrino energy prior cuts. (b) The logarithm of the reconstructed muon track length vs the logarithm of the initial neutrino energy post cuts. Figure 6.6: Correlation between reconstructed muon track length and neutrino energy prior (a) and post application of the cuts (b). visible. In comparison to figure 5.6 the change of the features of the distribution is not that significant in figure

72 6 RECONSTRUCTION Figure 6.7: Comparison of the energy distribution of the detected atmospheric neutrinos before (black) and after (red) the cuts. Figure 6.8: Spectrum in the reconstructed detector observables zenith angle and muon track length with neutrino oscillations (compare with figure 5.6). 60

73 6.6 Results for Neutrino Oscillations Figure 6.9: Spectrum in the reconstructed detector observables zenith angle and muon track length without neutrino oscillations (compare with figure 5.7). 61