高能微中子偵測與味物理學

國 立 交 通 大 學

物

理

研

究

所

博 士 論 文

高能微中子偵測與味物理學

High Energy Neutrino Detection and Neutrino Flavor

Physics

研 究 生： 劉宗哲

指導教授： 林貴林 教授

黃明輝 教授

高能微中子偵測與味物理學

High Energy Neutrino Detection and Neutrino Flavor

Physics

研 究 生 ： 劉宗哲

Student: Tsung-Che Liu

指 導 教 授 ： 林貴林

Advisors: Guey-Lin Lin

黃明輝

M. A. Huang

國 立 交 通 大 學

物

理

研

究

所

博 士 論 文

A Thesis

Submitted to Institute of Physics

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in

Physics

July, 2010

Hsinchu, Taiwan

中華民國九十九年七月

高能微中子偵測與味物理學

學生： 劉宗哲

指 導 教 授 ： 林貴林

黃明輝

國立交通大學物理研究所

摘

要

本篇論文討論微中子由微中子源傳播至地球間的各種現象. 在本文的第一部 份,我們利用對微中子震盪角度的了解,重建微中子在不同量測條件下時於源頭的 比例. 我們發現區分不同的微中子源需要大量的觀測數據. 另外本文也考慮當微 中子衰變與震盪同時作用時, 地球上的微中子比率的對應影響. 由此我們給定了 在99.7%信心水準下, 不同衰變機制作用下, 微中子比率在地球上的允許與禁制 範圍. 在本文的第二部份, 我們介紹如何運用蒙地卡羅方法模擬微中子與地球的 各種交互作用. 依照模擬我們可得到在偵測區域(DSR)內的輕子流量. 由於濤輕 子(Tau lepton)簇射後會產生大量正負電子對並受地磁偏轉而產生同步輻射. 藉 此我們計算不同能量下濤輕子簇射所對應的同步輻射訊號強度與行為. i

High Energy Neutrino Detection and Neutrino Flavor Physics

Student: Tsung-Che Liu

Advisors: Guey-Lin Lin

M. A. Huang

Institute of Physics

National Chiao Tung University

ABSTRACT

This thesis discusses the neutrino flavor physics from the source to the Earth. In first part, we analyze the measured flavor ratios on the Earth to reconstruct the flavor ratio at the source. We also estimate the critical event numbers for distinguishing two most common astrophysical sources. We not only introduce the standard oscillation mechanism but also neutrino decay mechanisms into the propagations of neutrinos. Applying the currently understood of neutrino mixing angle ranges, we obtain the corresponding flavor ratios on the Earth for different propagation mechanisms. In the second part, we discuss the neutrino detections. We simulate neutrinos interacting with the Earth and record the lepton fluxes inside the detector sensitive region. The leptons generate the electromagnetic and hadronic showers. We use CORSIKA simulate these showers and estimate the resulting synchrotron radiations.

致

致

致

謝

謝

謝

僅以本書獻給感謝我的指導教授, 林貴林與黃明輝兩位老師. 與協力完成本論文 的每個人. 在物理所七年, 聽到稱讚林老師的話多不勝數, 但我完全沒有聽過有人抱怨過 林老師一句. 林老師的為人已達到”背後無人語”的境界. 因此每個畢業的同學在 他們的致謝中, 都會稱讚老師高尚的品格和外圓內方的研究態度. 這些好話用在 很多人身上只會讓人覺得溢美與虛假, 但用在林老師身上則是完全的貼和. 林老 師不只是我在學問上的恩師, 也是我為人處事的典範. 而黃明輝老師則是在我的 博士生涯樹立了另外一盞明燈, 在學術上,老師常常以機敏百出的思維來處理物理 問題, 使我每每都只能在心裡暗地驚嘆與佩服,難以望其項背. 而在生活及其他方 面, 黃老師待我視同己出, 常常原諒我過度的怠惰與偏執, 並妥善的指出我應當行 的路. 在此我只能說: ”選擇兩位恩師當我的指導教授是一生我的榮幸, 是我進交 大後最正確也是最重要的決定.”. 這篇論文的內容, 並不是由我一人獨立完成. 交大高能團隊的每個人都在不同 的部份有所參與與貢獻：首先感謝王正祥老師在每個章節都給予我不同的意見, 讓我可以在學術上適度地修正研究方向. 而在生活上, 則是指正我過度奢靡的生 活態度. 讓我稍稍地控管自己的財政支出. 最後我也必須感謝王老師在聯合大學 給我的各項照顧, 讓我減輕經濟上得負擔.而永順, 振軒, 光昶, 志清四位是則除了 老師們以外, 讓本論文得以完成的最主要功臣. 每當我在程式, 在物理, 甚至是硬 體上遇到難關無法突破時, 他們四位常常伸出援手, 讓我得以繼續下去. 他們是我 在博士生涯的四位貴人. 最後感謝鳳吟, 瑜隆, 峰旭, 貝禎, 禹廷 , 志榮, 念潔與高 iii

能團隊的各位的幫忙與指導. 以下是特別感謝名單: FLASH-TW 實驗: (2003-2005). 感謝同步輻射許國棟組長與許森元先生幫忙建 立整個實驗平台. 許組長以他豐富的經驗給與我許多寶貴的意見. 在他的身上我 看到一個一流的實驗學家如何由零開始一步一步建立整個實驗. 另外許森元先生 謹慎的設計.則是讓我們的實驗能被正確運作的最重要關鍵. 沒有他們兩位的關 注,本實驗絕對無法進行下去.

Neutrino Interaction Simulation in IHEP (2007). 感謝曹臻,何會海與查敏三 位老師給我在北京高能所學習的機會. 在北京我見識到微中子觀測實驗如何與電 腦模擬相結合. 他們三位常常在會議中進行廢寢忘食的激烈論戰,有系統地呈現 各方的意見,只為求得科學上的真相. 這樣的研究精神深深地讓我折服. 另外也要 感謝劉加麗, 白云翔, 肖剛, 張丙開, 張壽山等好朋友在生活與研究上給我各種協 助. 沒有他們, 別說是作研究了, 我大概連怎麼在北京生活都會成問題. Geant4 Simulation (2008). 感謝王正祥老師與陳鎰鋒老師讓我有機會參加 Geant4 團隊的研討會並實際了解KEK如何的運作. 在那次經驗, KEK的Sasaki教 授讓我見識到國際一流團隊如何攥寫程式, 如何反覆推敲每一個環節以達到真實 世界所需要的精度. 這使我在Geant4模擬的工作上有著長足的進步. 雖然在這篇 論文內, 我沒有加入質子治療的各項模擬結果. 但我必須特別感謝以上團隊的協 助.

Geosynchrotron Raiation Detector (2008-2009). 這裡必須感謝韓國梨花大學 南智悟(Nam JiWoo)教授在我們團隊期間給予關於實驗的各種實務經驗. 有他的 引領我們才漸漸的建立起如何地球同步輻射光與高能微中子的關聯. 交大物理成員(2003-2008). 感謝博士班內一同為資格考廝殺的92級兄弟們, 也 感謝半夜一同吃宵夜的胖子團隊. 無論你們現在在哪裡, 感謝你們這麼多年來的 陪伴. iv

Contents

1 Introduction 1

1.1 The Discovery of Electron Neutrino . . . 1 1.2 Other Types of Neutrinos . . . 2

I Neutrino Flavor Physics 4

2 Neutrino Flavor Ratio at the Astrophysical Source 5

2.1 Sources of Ultra-high Energetic Neutrinos . . . 5 2.1.1 AGN (Active Galactic Nucleus) Neutrinos . . . 5 2.1.2 GZK ( Greisen-Zetsepin-Kuzmin ) Neutrinos . . . 6 2.2 Pontecorvo-Maki-Nakagawa-Sakata Mixing Matrix and Flavor Transition

Proba-bility Matrix . . . 6 2.3 Observed Neutrino Events in Detector . . . 13 2.4 Reconstructing Source Flavor Ratios at Low Energies . . . 16

2.4.1 The Reconstruction of Initial Neutrino Flavor Ratio by Measuring RIAlone 18

2.4.2 The Flavor Reconstruction with Measurements on Both R and S . . . . . 18 2.4.2.1 (sin2_θ

13)best fit= 0 . . . 19

2.4.2.2 (sin2θ

13)best fit> 0 . . . 21

2.4.3 Critical Accuracies Needed for Distinguishing Astrophysical Sources. . . 21 2.5 Reconstructing Flavor Ratio of Source at High Energy. . . 23 2.5.1 The Reconstruction of Muon-damped Source . . . 25

2.5.2 The Reconstruction of Pion Source . . . 25 2.5.3 RIIand Tri-bimaximal Limit . . . 26 2.6 Reconstructing Flavor Ratio of Source by Monte Carlo Method with N Events . . 29 2.6.1 The Flavor Ratios on the Earth from Pion Source with 200 Events . . . . 29

2.6.1.1 The Reconstructed Neutrino Flavor Ratios at Low Energy, Eν≤

33.3 PeV . . . 29 2.6.1.2 The Reconstructed Neutrino Flavor Ratios at High Energy, Eν≥

33.3 PeV . . . 32 2.6.2 The flavor ratios on the Earth from the Muon-damped source with 200 events 33

2.6.2.1 The Reconstructed Neutrino Flavor Ratios at Low Energy, E_ν≤ 33.3 PeV . . . 33 2.6.2.2 The Reconstructed Neutrino Flavor Ratios at Low Energy, E_ν≥

33.3 PeV . . . 34 2.6.2.3 Summary of the Reconstructed Neutrino Flavor Ratios with 200

Events . . . 37 2.6.3 The Flavor Ratios on the Earth from Pion Source with 400 Events . . . . 37

2.6.3.1 The Reconstructed Neutrino Flavor Ratios at Low Energy, Eν≤

33.3 PeV . . . 38 2.6.3.2 The Reconstructed Neutrino Flavor Ratios at High Energy, E_ν≥

33.3 PeV . . . 40 2.6.4 The Flavor Ratios on the Earth from the Muon-damped Source with 400

Events . . . 41 2.6.4.1 The Reconstructed Neutrino Flavor Ratios at Low Energy, Eν≤

33.3 PeV . . . 41 2.6.4.2 The Reconstructed Neutrino Flavor Ratios at High Energy, Eν≥

33.3 PeV . . . 42 2.6.4.3 Summary of the Reconstructed Neutrino Flavor Ratios with 400

Events . . . 43

3.1 Q Matrix Representation . . . . 47

3.1.1 Q Matrix Representation for Oscillations of Astrophysical Neutrinos . . . 47

3.1.2 Q Matrix Representation for General Flavor Transitions of Astrophysical Neutrinos . . . 50

3.2 Neutrino Decay Mechanisms . . . 51

3.3 The Range of Neutrino Flavor Ratios on the Earth . . . 52

3.3.1 The Heaviest and Middle Mass Eigenstates decay to the Lightest and In-visible States. . . 52

3.3.1.1 Normal Mass Hierarchy . . . 52

3.3.1.2 Inverted Mass Hierarchy . . . 53

3.3.2 The Heaviest Mass Eigenstate Decays to the Middle, Lightest and Invisi-ble States . . . 54

3.3.2.1 Normal Mass Hierarchy . . . 54

3.3.2.2 Inverted Mass Hierarchy . . . 54

3.3.3 The Middle Mass Eigenstate Decays to the Lightest and Invisible States . 56 3.3.3.1 Normal Mass Hierarchy . . . 56

3.3.3.2 Inverted Mass Hierarchy . . . 56

3.3.4 The Lightest Mass Eigenstate decays to Invisible States Only . . . 58

3.3.4.1 Normal Mass Hierarchy . . . 58

3.3.4.2 Inverted Mass Hierarchy . . . 59

3.3.5 The Middle and Lightest Mass Eigenstates decay to Invisible States Only 60 3.3.5.1 Normal Mass Hierarchy . . . 60

3.3.5.2 Inverted Mass Hierarchy . . . 60

3.3.6 The heaviest and lightest mass eigenstates decay to the middle mass eigen-state and invisible eigen-states only . . . 61

3.3.6.1 Normal Mass Hierarchy . . . 61

3.3.6.2 Inverted Mass Hierarchy . . . 62

3.4 The Most Probable Neutrino Flavor Ratios on the Earth Induced by Standard Neu-trino Oscillation and NeuNeu-trino Decays . . . 63

II High Energy Neutrino Detection 65

4 Simulation of High Energy Neutrino Interacting with the Earth 66

4.1 Introduction . . . 66

4.2 Interactions . . . 67

4.2.1 Energy Loss of Leptons . . . 67

4.2.1.1 Ionization . . . 68

4.2.1.2 Bremsstrahlung . . . 69

4.2.1.3 Pair production . . . 70

4.2.1.4 Photonuclear . . . 70

4.2.2 Lepton decay . . . 70

4.2.3 Neutrino Charged Current and Neutral Current Interaction . . . 71

4.3 Environments . . . 72

4.3.1 Earth’s Density Model in SHINIE . . . 73

4.3.2 Detector Sensitive Region (DSR) . . . 74

4.4 Controls of Simulation . . . 77

4.5 Running Options . . . 77

5 Geosynchrotron Raiation Detector 79 5.1 Extensive Air Shower (EAS) . . . 80

5.1.1 Longitudinal Profile . . . 80

5.1.2 Lateral Distribution Function . . . 80

5.2 CORSIKA Simulation . . . 82

5.2.0.1 Electron Shower . . . 82

5.2.0.2 Pion Shower . . . 83

5.2.0.3 Kaon Shower . . . 83

5.3 Radiation Emission from Air Shower . . . 93

5.3.1 Radiation by a Moving Charge . . . 93

5.3.2 The Radiation form the Electron-positron Pairs . . . 94

6 Shower Experiment with electron beam 99

6.1 Instrument . . . 99

6.2 The Measurement of Shower Lateral Profile with Scintillator . . . 101

6.3 The Measurement of Cherenkov Radiations from Showers . . . 104

6.4 Geant4 Simulation . . . 105

7 Conclusions 109 A Differential cross section of lepton energy loss 111 A.1 Pair production . . . 111

A.2 Photonuclear . . . 112

List of Figures

2.1 The spectrum of AGN [1] and GZK [6] muon neutrino flux. The blue solid line shows the muon neutrino spectrum from AGN source. The red dashed line shows the muon neutrino spectrum from GZK source. . . 7 2.2 The ranges for the neutrino flavor ratios on the Earth resulting from standard

neutrino oscillation. The numbers on each side of the triangle denote the flux fraction of a specific flavor of neutrino. The blue point marks the pion source φ0(νe) : (φ0(νµ) :φ0(ντ) = 1/3 : 2/3 : 0 and the red point marks the muon-damped

sourceφ0(νe) : (φ0(νµ) :φ0(ντ) = 0 : 1 : 0. In panel (a), the red and blue squares

mark the corresponding neutrino flavor ratios observed on the Earth. In panel (b), we introduce uncertainties of neutrino mixing angles in the probability matrix. The red and dashed line denote neutrino flavor ratios observed on the Earth from muon-damped source in 1σand 3σranges of neutrino mixing angles respectively. The blue and solid line denote neutrino flavor ratios observed on the Earth from pion source in 1σ and 3σ ranges of neutrino mixing angles respectively . . . 10 2.3 Different types of neutrino-induced events. Dashed lines and solid lines

corre-spond to paths of neutrinos and leptons respectively. The ellipsoids are showers. The detectable energy range for each type of event is listed in Table 2.1. . . 16

2.4 The reconstructed ranges for the neutrino flavor ratios at the source with∆RI_/RI₌ 10% only. The left and right panels are results with the muon-damped source and the pion source as the input true source respectively. The numbers on each side of the triangle denote the flux percentage of a specific flavor of neutrino. The red point marks the muon-damped sourceΦ0,µ= (0, 1, 0) and the blue point marks the

pion sourceΦ0,π= (1/3, 2/3, 0). Gray and light gray areas respectively denote the

1σ and 3σ ranges for the reconstructed neutrino flavor ratios at the source. We choose parameter set 1 in Table 2.4 for this analysis. . . 19 2.5 The reconstructed ranges for the neutrino flavor ratios for an input muon-damped

source with∆RI/RI_{= 10% and∆S}I_/SI_{related to the former by the Poisson} statis-tics, E.q. (2.17). Gray and light gray areas in the left (right) panel denote the reconstructed 1σ and 3σranges with the parameter set 1 (2) in Table 2.4. . . 20 2.6 The reconstructed ranges for the neutrino flavor ratios for an input pion source

with ∆RI/RI _{= 10% and ∆S}I_/SI _{related to the former by the Poisson statistics,} E.q. (2.17). Gray and light gray areas in the left (right) panel denote the recon-structed 1σand 3σ ranges with the parameter set 1 (2) in Table 2.4. . . 20 2.7 The reconstructed ranges for the neutrino flavor ratio at the source for an input

muon-damped source with∆RI/RI_{= 10% and}∆_SI_/SI_{related to the former by the} Poisson statistics, E.q.(2.17). The left panel is obtained with θ13 and θ23 taken

from the parameter set 1 in Table 2.4 and the input CP phase taken to be 0,π/2

andπrespectively. The right panel is obtained with the parameter sets 3a, 3b and 3c in Table 2.4. Light gray area, dashed blue and dashed red lines correspond to the 3σranges for the reconstructed neutrino flavor ratio at the source for cosδ= 1,

cosδ = 0 and cosδ_{= −1 respectively. Gray area, blue and red lines correspond to}

the 1σranges for the reconstructed neutrino flavor ratio at the source for cosδ= 1,

cosδ = 0 and cosδ _{= −1 respectively. The effect from the CP phase} δ only appears in the right panel. . . 22

2.8 The reconstructed 1σ and 3σ ranges for the neutrino flavor ratio at the source for an input pion source with∆RI/RI _{= 10% and∆S}I_/SI_{related to the former by the} Poisson statistics. The choices of parameter sets are identical to those of Fig. 2.7. Once more, the effect from the CP phaseδ only appears in the right panel. . . . 22 2.9 Critical accuracies needed to distinguish between the pion source and the

muon-damped source. In the left panel where the muon-muon-damped source is the true source, the reconstructed 3σ range for the neutrino flavor ratio just touches the pion source at ∆RI/RI _{= 11% and} ∆_SI_/SI _{related to the former by the Poisson} statistics, E.q. (2.17). In the right panel where the pion source is the true source, the reconstructed 3σ range for the neutrino flavor ratio just touches the muon-damped source at∆RI/RI_{= 4% and∆S}I_/SI _{related to the former by the Poisson} statistics, E.q. 2.17.. We choose parameter set 1 in Table 2.4 for this analysis. . . 23 2.10 Reconstructed ranges for muon-damped source with∆Ra/Ra_{= 10% and∆S}a_/Sa₌

12%. The dark and light shaded areas denote the range of reconstructed neutrino flavor ratios under 1σ and 3σ limits. The left and right panels correspond to the condition I and II. The pion source can be ruled out at the 3σ level for both conditions. . . 25 2.11 Reconstructed ranges for muon-damped source with∆Ra/Ra _{= 10% only. The}

dark and light shaded areas denote the range of reconstructed neutrino flavor ratios under 1σ and 3σ limits. The left and right panels correspond to the condition I and II. The pion source can be ruled out at 3σ level for the condition II but not for the condition I even at 1σlevel. . . 26 2.12 Reconstructed ranges for pion source with ∆Ra/Ra_{= 10% and∆S}a_/Sa_{= 12%.}

The dark and light shaded areas denote the range of reconstructed neutrino flavor ratios under 1σ and 3σ limits. The left and right panels correspond to the condi-tion I and II. The muon-damped source can be ruled out at the 1σ level for both conditions. . . 27

2.13 Reconstructed ranges for pion source with ∆Ra_/Ra _{= 10% only. The dark and} light shaded areas denote the range of reconstructed neutrino flavor ratios under 1σ and 3σ limits. The left and right panels correspond to the condition I and II. For condition I, 3σ limit covers all flavor ratio of source. But the muon-damped source can be ruled out at 1σlevel for condition II. . . 27 2.14 Measured flux ratios for input source ratios,φ0(νe) :φ0(νµ) :φ0(ντ) =α: 1−α:

0 with 0_≤α _{≤ 1. The thick and long dashed lines correspond to R}II and SII respectively. The thin and short-dashed lines denote RI and SIrespectively. It is seen that SI and RIIare more sensitive toα; while RI and SII are less sensitive to this parameter. . . 28 2.15 The distribution of neutrino flavor ratio on the Earth from pion source for 200

events. Each flavor ratio on the ternary plot is generated by 200 events. There are 10 million flavor ratios generated on the ternary plot, which form the distribution of flavor ratios. The blue point denotes the original flavor ratio of pion source. In panel (a), the distribution of flavor ratios is cut off at 99.7%. In panel (b), the

distribution of flavor ratios is cut off at 68.2%. . . 30 2.16 The reconstructed ranges for pion source with 200 events and neutrino energy

below 33.3 PeV. The range for possibly measured flavor ratio is cut off at 99.7%.

The dark and light shades area denote 1σ and 3σ reconstructed ranges. . . 31 2.17 The reconstructed ranges for pion source with 200 events and neutrino energy

below 33.3 PeV. The range for possibly measured flavor ratio is cut off at 68.2%.

The dark and light shades area denote 1σ and 3σ reconstructed ranges. . . 32 2.18 The reconstructed ranges for pion source with 200 events and neutrino energy

above 33.3 PeV. The range for possibly measured flavor ratio is cut off at 99.7%.

The dark and light shades area denote 1σ and 3σ reconstructed ranges. . . 33 2.19 The reconstructed ranges for pion source with 200 events and neutrino energy

above 33.3 PeV. The range for possibly measured flavor ratio is cut off at 68.2%.

2.20 The distribution of neutrino flavor ratio on the Earth from muon-damped source for 200 events. Each flavor ratio on the ternary plot is generated by 200 events. There are 10 million flavor ratios generated on the ternary plot, which form the distribution of flavor ratios. The red point denotes the original flavor ratio of muon-damped source. In panel (a), the distribution of flavor ratios is cut off at 99.7%. In

panel (b), the distribution of flavor ratios is cut off at 68.2%. . . 35 2.21 The reconstructed ranges for muon-damped source with 200 events and neutrino

energy below 33.3 PeV. The range for possibly measured flavor ratio is cut off at

99.7%. The dark and light shades area denote 1σ and 3σreconstructed ranges. . 36 2.22 The reconstructed ranges for muon-damped source with 200 events and neutrino

energy below 33.3 PeV. The range for possibly measured flavor ratio is cut off at

68.2%. The dark and light shades area denote 1σ and 3σreconstructed ranges. . 36 2.23 The reconstructed ranges for muon-damped source with 200 events and neutrino

energy above 33.3 PeV. The range for possibly measured flavor ratio is cut off at

99.7%. The dark and light shades area denote 1σ and 3σreconstructed ranges. . 37 2.24 The reconstructed ranges for muon-damped source with 200 events and neutrino

energy above 33.3 PeV. The range for possibly measured flavor ratio is cut off at

68.2%. The dark and light shades area denote 1σ and 3σreconstructed ranges. . 38 2.25 The distribution of neutrino flavor ratio on the Earth from pion source for 400

events. Each flavor ratio on the ternary plot is generated by 400 events. There are 10 million flavor ratios generated on the ternary plot, which form the distribution of flavor ratios. The blue point denotes the original flavor ratio of pion source. In panel (a), the distribution of flavor ratios is cut off at 99.7%. In panel (b), the

distribution of flavor ratios is cut off at 68.2%. . . 39 2.26 The reconstructed ranges for pion source with 400 events below 33.3 PeV. The

range for possibly measured flavor ratio is cut off at 99.7%. The dark and light

2.27 The reconstructed ranges for pion source with 400 events below 33.3 PeV. The

range for possibly measured flavor ratio is cut off at 68.2%. The dark and light

shades area denote 1σand 3σ reconstructed ranges. . . 41 2.28 The reconstructed ranges for pion source with 400 events and neutrino energy

above 33.3 PeV. The range for possibly measured flavor ratio is cut off at 99.7%.

The dark and light shades area denote 1σ and 3σ reconstructed ranges. . . 42 2.29 The reconstructed ranges for pion source with 400 events and neutrino energy

above 33.3 PeV. The range for possibly measured flavor ratio is cut off at 68.2%.

The dark and light shades area denote 1σ and 3σ reconstructed ranges. . . 43 2.30 The distribution of neutrino flavor ratio on the Earth from muon-damped source

for 400 events. Each flavor ratio on the ternary plot is generated by 400 events. There are 10 million flavor ratios generated on the ternary plot, which form the distribution of flavor ratios. The red point denotes the original flavor ratio of muon-damped source. In panel (a), the distribution of flavor ratios is cut off at 99.7%. In

panel (b), the distribution of flavor ratios is cut off at 68.2%. . . 44 2.31 The reconstructed ranges for muon-damped source with 400 events below 33.3

PeV. The range for possibly measured flavor ratio is cut off at 99.7%. The dark

and light shades area denote 1σ and 3σ reconstructed ranges. . . 45 2.32 The reconstructed ranges for muon-damped source with 400 events below 33.3

PeV. The range for possibly measured flavor ratio is cut off at 68.2%. The dark

and light shades area denote 1σ and 3σ reconstructed ranges. . . 45 2.33 The reconstructed ranges for muon-damped source with 400 events above 33.3

PeV. The range for possibly measured flavor ratio is cut off at 99.7%. The dark

and light shades area denote 1σ and 3σ reconstructed ranges. . . 46 2.34 The reconstructed ranges for muon-damped source with 400 events above 33.3

PeV. The range for possibly measured flavor ratio is cut off at 68.2%. The dark

3.1 The ranges for the neutrino flavor ratios on the Earth resulting from standard neu-trino oscillation. The left panel shows the range of neuneu-trino flavor ratios from the source flavor ratios,φ0(νe) :φ0(νµ) :φ0(ντ) =α : 1−α : 0, with 0≤α≤ 1. The

violet and orange colors correspond the 1σ and 3σ level of neutrino mixing an-gles respectively. The left panel shows the range of neutrino flavor ratios from the source flavor ratios,φ0(νe) :φ0(νµ) :φ0(ντ) =α:β: 1−α−β, withα> 0,β> 0

andα+β _{≤ 1. The violet and orange colors correspond the 1}σ and 3σ level of neutrino mixing angles respectively. The values of mixing angles are same as the set 1 in Table 2.5. . . 48 3.2 The range of flavor ratios on the Earth from the heaviest and middle mass

eigen-states decay to the lightest mass eigenstate and invisible eigen-states. The final branching ratios of the heaviest and lightest mass eigenstates are BrH→L+ BrH→invisible = 1

and BrM→L+ BrM→invisible = 1. Panel (a) shows the range of flavor ratios on the

Earth in the normal mass hierarchy. Panel (b) shows the range of flavor ratios on the Earth in the inverted mass hierarchy. . . 53 3.3 The range of flavor ratios on the Earth from the heaviest mass eigenstate decays

to the middle, lightest and invisible state in the normal mass hierarchy. Panel (a) shows the range of flavor ratios on the Earth from the heaviest eigenstate decays to invisible eigenstates only. Panel (b) shows the range of flavor ratios on the Earth from the heaviest eigenstate decays to the middle eigenstate only. Panel (c) shows the range of flavor ratios on the Earth from the heaviest eigenstate decays to the lightest eigenstate only. Panel (d) shows the all possible flavor ratios on the Earth which satisfy BrH_→M+ BrH_→L+ BrH_→invisible= 1. . . 55

3.4 The range of flavor ratios on the Earth from the heaviest mass eigenstate decays to the middle, lightest and invisible state in the inverted mass hierarchy. Panel (a) shows the range of flavor ratios on the Earth from the heaviest eigenstate decays to invisible eigenstates only. Panel (b) shows the range of flavor ratios on the Earth from the heaviest eigenstate decays to the middle eigenstate only. Panel (c) shows the range of flavor ratios on the Earth from the heaviest eigenstate decays to the lightest eigenstate only. Panel (d) shows the all possible flavor ratios on the Earth which satisfy BrH_→M+ BrH→L+ BrH→invisible= 1. . . 57

3.5 The range of flavor ratios on the Earth from the middle mass eigenstate decays to the lightest and invisible states. The branching ratio, BrM→L+ BrM→invisible= 1,

for both hierarchy. Panel (a) shows the flavor ratios on the Earth from the mid-dle mass eigenstate decays to the lightest and invisible states in the normal mass hierarchy. In this case, the range of flavor ratios on the Earth from the heaviest mass eigenstate decay in the inverted mass hierarchy is same as the middle mass eigenstate decay in the normal mass hierarchy. Panel (b) shows the flavor ratios on the Earth from the middle mass eigenstate decays to the lightest and invisible states in the inverted mass hierarchy. . . 58 3.6 The range of flavor ratios on the Earth from the lightest mass eigenstate decays to

invisible states only. The lightest neutrino mass eigenstate decays to invisible state only. The branching ratio, BrL→invisible = 1, for both hierarchy. Panel (a) shows

the lightest eigenstate decays to invisible states in the normal mass hierarchy. The possible flavor ratios are same as the case of the middle mass eigenstate decay. Panel (b) shows the lightest eigenstate decays to invisible state only in the inverted mass hierarchy. The possible flavor ratios are same as the case of the heaviest mass eigenstate decay in the normal mass hierarchy. . . 59

3.7 The range of flavor ratios on the Earth from the middle and the lightest mass eigenstates decay to invisible states. The middle mass eigenstate can decay to the lightest and invisible eigenstates, but the lightest neutrino mass eigenstate only can decay to invisible state. Hence the final branching ratios of the middle and lightest mass eigenstates are BrM→invisible= 1 and BrL→invisible= 1 for both

hier-archy. Panel (a) shows the middle and lightest mass eigenstates decay to invisible states in the normal mass hierarchy. Panel (b) shows the middle and lightest mass eigenstates decay to invisible states only in the inverted mass hierarchy. . . 61 3.8 The range of flavor ratios on the Earth from the heaviest and lightest mass

eigen-states decay to the middle mass eigenstate and invisible eigen-states. The heaviest mass eigenstate can decay to the middle mass eigenstate and invisible eigenstates, but the lightest neutrino mass eigenstate only can decay to invisible eigenstates. Hence the final branching ratios of the heaviest and lightest mass eigenstates are

BrH_→M+ BrH_→invisible= 1 and BrL_→invisible= 1 for both hierarchy. Panel (a) shows

the possible flavor ratios on the Earth in the normal mass hierarchy. Panel (b) shows the possible flavor ratios on the Earth in the inverted mass hierarchy. . . . 62 3.9 The neutrino flavor ratios on the Earth induced by different flavor transition

mech-anisms. The left panel shows the range of flavor ratios on the Earth from decay and oscillation in 1σ level. The right panel shows the ranges at 3σ level. Both panels show the most probable flavor ratios on the Earth is aroundφ0(νe) :φ0(νµ) :

φ0(ντ) = 1/3 : 1/3 : 1/3 . . . 64

4.1 The range of tau leptons in materials. Blue, red and orange lines correspond to the range of tau leptons in water, standard rock, and iron respectively. The range of tau in iron is much lower than standard rock and water at 1020 eV. . . 68 4.2 Charged current (CC) and neutral current (NC) interaction cross section in SHINIE,

according the the CTEQ6-DIS parton distributions. The red and blue curves are CC and NC interaction cross section, respectively, for 105_{≤ Eν ≤ 10}11 GeV. . . 73 4.3 density model of earth . . . 74 4.4 The grid map data with altitudes. . . 75

4.5 Detector sensitive region (DSR). The size, material and position of DSR can be

defined by user. SHINIE record all information of particles within this area. . . . 76

4.6 Graphic interface system . . . 78

5.1 The relationship between slant depth and propagation distance in the air with con-stant density. Blue, red, yellow and green lines correspond to the transition rate for horizontal shower develop in sea level, 2 km, 4 km and 6 km altitudes. The densities of those altitudes are 1.23 kg/m3_{, 1.01 kg/m}3_{, 0.82 kg/m}3_{and 0.66 kg/m}3_{. 81} 5.2 Mean longitudinal profile of 30 electron showers at five input energies. . . 84

5.3 Lateral profile of electron showers at five input energies. . . 85

5.4 Lorentz factor distribution of electron showers at five input energies. . . 86

5.5 longitudinal profile of pion showers at five input energies. . . 87

5.6 longitudinal profile of pion showers at five input energies. . . 88

5.7 The longitudinal profile of pion showers at five input energies. . . 89

5.8 longitudinal profile of kaon showers at five input energies. . . 90

5.9 The lateral profile of kaon showers at five input energies. . . 91

5.10 The Lorentz factor distribution of kaon showers at five input energies. . . 92

5.11 Geometry of synchrotron radiation. Blue point denotes the particle moving with velocity v. Red line is the trajectory of particle lies in x_{− y plane with radius}ρ. ˆ e_|| is the unit vector corresponding to the polarization in the plane of orbit and ˆ e_⊥ is the unit vector of the other polarization, which perpendicular to the ˆe_||and emission direction n. . . . 94

5.12 Off-set dependence of _|E(R,2πν_{)| of a 10}17 eV shower at the observation dis-tance of 10 km. Curves in blue, red and orange represent signals in observing frequencies of 50 MHz, 75 MHz and 100 MHz respectively. . . 95

5.13 Reconstructed pulses from emission of 1017_{and 10}17.5 _{eV electron shower at the}

observation distance of 10 km, using an idealized rectangular filter spanning 30₋ 80 MHz. The strength of signal is proportional to the energy of shower. The curves in blue, red, orange and green denote pulses measured at center, at lateral distances of 500 m 1000 m and 1500 m, respectively. The pink and gray curves are measured at lateral distances of 2000 m and 2500 m and close to the x axis, and the signals are too weak to be distinguished. . . 97 5.14 Reconstructed pulses from emission of a 1017.5eV pion shower at the observation

distance of 10 km, using an idealized rectangular filter spanning 30_{− 80 MHz.} The curves in blue, red, orange and green denote pulses measured at center, at lateral distances of 500 m 1000 m and 1500 m, respectively. The pink and gray curves are measured at lateral distances of 2000 m and 2500 m and close to the x axis, and the signals are too weak to be distinguished. . . 98 5.15 Reconstructed pulses from emission of a 1017.5eV kaon shower at the observation

distance of 10 km, using an idealized rectangular filter spanning 30_{− 80 MHz.} The curves in blue, red, orange and green denote pulses measured at center, at lateral distances of 500 m 1000 m and 1500 m, respectively. The pink and gray curves are measured at lateral distances of 2000 m and 2500 m and close to the x axis, and the signals are too weak to be distinguished. . . 98

6.1 Top view of the experimental platform. The 1.5 GeV electron beams enter the

chambers from the left hand side. The secondary charge particles are generated while incident electrons are passing through the aluminum targets. The secondary charge particles hit the scintillation screen placed in the second chamber and gen-erate scintillation light, which is recorded by the CCD camera. . . 100 6.2 Results from the decay-time measurement. This decay-time of the scintillation

light from scintillator is measured by photo-diode and digital multi-meter. The first decay pattern has a short decay time about 3.4 ms, the second one has a longer decay time about 6.7 ms and the third one has a decay time longer than 15

6.3 The fluorescence spectrum of AF995r scintillator (Al2O3:Cr3+). Two close peaks

located at wavelengths 692.8 nm and 694.0 nm are clearly seen. . . 102

6.4 The shower image recorded by the CCD camera at 2.5 radiation length. The x and

y axises labels the pixels number of CCD chip. The z axis is the count of photos record by CCD camera. . . 103 6.5 The shower longitudinal profile record by the CCD camera. The CCD camera

integrates the fluorescence photons spread over the X_{−Y plane for each radiation} length and obtain the shower longitudinal profile. . . 103 6.6 The number of photons emitted from an electron with in the wavelength between

380 nm to 780 nm. The threshold energy is about 21 MeV with refractive index of air is 1.00029. The photons numbers saturate close to 35 photons per meter above

100 MeV. . . 105 6.7 The platform structure setting of Geant4 simulation. The blue lines indicate the

track of electrons, pink lines are emitted photons by charge. The incident electron beam hit the 2 mm thick aluminium window first, then lose the energy within aluminium radiators. The electrons emit the Cherenkov photons while electron pass the air, but the CCD record the photons only after electrons pass the third aluminium window. . . 107 6.8 The longitudinal profiles of experimental data and Geant4 simulation result. The

blue squares are experiment data, which record the Cherenkov photons by CCD. The red line is the result of Geant4 simulation. Both data and simulation result show the shower maximum are close to 2.3 r.l.. . . 108

List of Tables

1.1 The properties of lepton generations. Since the neutrino flavor eigenstates are combinations of neutrino mass eigenstates, we do not list the neutrino mass in this table. The relations beetween those three neutrino flavor eigenstates and mass eigenstates are discussed in chapter 2. . . 3

2.1 Different types of neutrino induced events. . . 13 2.2 The definitions of R and S at different energy ranges. . . . 15 2.3 Parameter sets chosen for our analysis . . . 18 2.4 True values of neutrino flavor ratios on the Earth . . . 18 2.5 Neutrino mixing parameters . . . 24

3.1 The neutrino decay and oscillation scenarios. The suffix “H”, “M”, and “L” la-bel the heaviest, middle and lightest mass eigenstates. The number “1”, “2” and “3” label ν1, ν2 and ν3 mass eigenstates. The red numbers and black numbers

correspond to the unstable and stable eigenstates respectively. . . 52

4.1 The parameters in Bethe-Bloch formula . . . 69 4.2 The Density of Earth, Rf is fraction of earth radius. The density of the Earth. The

symbol “R” is the distance from the center of the Earth to an observable location. 75 4.3 Status identifications as used in SHINIE. . . 76 4.4 Particle identifications as used in SHINIE. . . 77 4.5 Running command . . . 78

5.2 The major branching ratios ofτ. There are 31 basic decay mode ofτ. The total branching ratio of first 6 decay channels is more than 90%. . . 82

Chapter 1

Introduction

1.1

The Discovery of Electron Neutrino

The history of neutrino began from theβ-decay experiment. In 1914, J. Chadwick discovered that energy spectrum of theβ-decay electrons is a continuous distribution. This result cannot be explained by the nuclear model ofβ-decay at that time and implies that theβ-decay may not be a two-body decay process. W. Pauli tried to interpretβ-decay as a three-body decay process and introduced a new neutral particle carrying spin 1/2 to ensure the a angular momentum

conserva-tion. In 1933 E. Fermi named that particle as neutrino,ν, and extended Pauli’s idea to his weak interaction theory. The reaction ofβ-decay is

(Z, A) → (Z + 1, A) + e−+ ¯νe,

where Z and A denote the atomic number and mass number respectively and the subscript “e” indicts the flavor of neutrino. Based on Pauli’s theory, H. Bethe and R. Peierls predicted the cross section of neutrinos interact with matter should be much smaller than the one of electrons. The first discovery of neutrinos is made by F. Reines and C. Cowen from inverseβ-decay reaction,

νe+ p → n + e+.

F. Reines and C. Cowen detected the radiatedγfrom positron annihilation and neutron interacting with cadmium to establish the neutrino events.

1.2

Other Types of Neutrinos

Muon is a particle with a negative electric charge and carries spin 1/2. Most properties of muon are

the same as those of the electron, but the rest mass of muon, 105.7 MeV/c2_{, is 200 times heavier}

than the rest mass of the electron. The first discovery of muon was made by C. D. Anderson and S. Neddermeyer in 1936 from cosmic ray experiment. Those muons are generated by cosmic ray colliding with nuclei in the atmosphere. Since the muon is not the lightest lepton, it decays to electron. The lifetime of muon is 2_{.2 × 10}−6 s and the process of muon decay is

µ−_{→ e}−_+ν

e+νµ.

This process conserves the muon lepton number, L_µ, and char\ge. Table 1.1 lists properties of lepton generations. In order to conserve the lepton number, muon neutrino and anti-electron neu-trino are generated in muon decay. The other lepton is tau, which was discovered in linear collider experiment in 1975. Tau is more massive than other leptons and its rest mass is 1.777 MeV/c2_.

Since the tau is the heaviest lepton, it could decay to electron or muon. The lifetime of tau is about 2_{.9 × 10}−13s. The tau has several decay channels. We list the major decay channels in Table 5.2. The purely leptonic decay channels are

τ−_{→ e}−_{+ ¯ν}

e+ντ (17.85 ± 0.06%) and

τ−_→µ−_{+ ¯νµ+ντ (17.36 ± 0.05%).}

Unlike other particles, neutrinos interact with matter very weakly and may not decay to other particles. Hence they keep a lot of information about their origin. One can probe the astrophysical object by measuring neutrino. Based on the understanding of neutrino oscillation and mixing

Generations Name Electric charge Lepton number Mass First electron ₋₁ Le= 1, Lµ= 0, Lτ= 0 0.511 MeV/c2

electron neutrino 0 Le= 1, Lµ= 0, Lτ= 0

Second muon ₋₁ Le= 0, Lµ= 1, Lτ= 0 105.658 MeV/c2

muon neutrino 0 Le= 0, Lµ= 1, Lτ= 0

Third tau ₋₁ Le= 0, Lµ= 0, Lτ= 1 1.777 GeV/c2

tau neutrino 0 Le= 0, Lµ= 0, Lτ= 1

Table 1.1: The properties of lepton generations. Since the neutrino flavor eigenstates are combi-nations of neutrino mass eigenstates, we do not list the neutrino mass in this table. The relations beetween those three neutrino flavor eigenstates and mass eigenstates are discussed in chapter 2.

angles, we can establish the relation of neutrino flavor ratio at the source and that on the Earth. In chapter 2, we reconstruct the possible flavor ratio at the astrophysical source by measuring the flavor ratio on the Earth. If neutrino can decay, the neutrino flavor ratio on the Earth should be different from that predicted by neutrino oscillation. In chapter 3, we demonstrate the possibly observed flavor ratio if heavier neutrino mass eigenstates can decay to lighter one. Since the cross section of high energy neutrinos interacting with matter is very small, the detection of neutrinos are very difficult. Most of experiments use the Earth as the target for neutrino to interact with it. In chapter 4, we build a Monte-Carlo program to simulate the neutrino interacting with the Earth. The charged current interaction of a neutrino produces the lepton and hadrons. If the lepton is muon or tau and generated inside the Earth, it may penetrate the Earth and decay to showers. The experiment can observe those showers to estimate the energy of the original neutrino. In chapter 5, we use CORSIKA to simulate the tau shower with different energy and calculate the electric field of synchrotron radiation generated by this shower. We estimate the energy of original neutrino by measuring the signal of synchrotron radiations. In chapter 6, we design a experiment in laboratory to measure the development of shower and compare the data with GEANT4 simulation.

Part I

Chapter 2

Neutrino Flavor Ratio at the

Astrophysical Source

Due to neutrino oscillations, the neutrino flavor ratio at the astrophysical source could be quite different from that observed on the Earth. This chapter discusses the reconstruction of neutrino flavor ratios at the astrophysical source.

2.1

Sources of Ultra-high Energetic Neutrinos

2.1.1 AGN (Active Galactic Nucleus) Neutrinos

Active galactic nuclei (AGN) are the most luminous astrophysical objects in the sky [1]. AGN have two jets in opposite directions and perpendicular to the accretion disc of AGN. The jets accelerate the particles to extremely high energies by Fermi acceleration. The interacting chain between high energy protons and gamma rays for generating the pions via∆+resonance is

p+γ_→∆+_{→ p +}πo p+γ_→∆+_{→ n +}π+

π+_→µ+_+ν

µ→ e++νe+νµ+ ¯νµ

(2.1)

If the muon decays without losing too much of its energy, the energies of neutrinos produced by the muon decay are close to the energy of neutrinos produced by from π+ decay . In this

chapter, we do not distinguish between neutrino and anti-neutrino. Hence this chain leads to the neutrino flavor ratioφ0(νe) :φ0(νµ) :φ0(ντ) = 1 : 2 : 0, where φ0(να) is the sum ofνα and ¯να

fluxes. Since these neutrinos are produced by the decay of charged pion, such a source referred to as the pion source. In the case that the muon loses a huge part of its energy by interacting with the strong field [2] or matter [3], the energies of neutrinos produced by the muon decay are much lower than the energy of neutrino produced by pion decay. Hence this chain leads to the neutrino flavor ratioφ0(νe) :φ0(νµ) :φ0(ντ) = 0 : 1 : 0, which is referred to as muon-damped source. Fig.

2.1.2 shows the neutrino spectrum from AGN.

2.1.2 GZK ( Greisen-Zetsepin-Kuzmin ) Neutrinos

The distribution of ultra high energy cosmic ray is isotropic and homogeneous. The major com-position of cosmic ray are protons and nuclei. The ultra high energy proton, Ep> 1019eV, has a possibility of interacting with the 2.7 K microwave background radiation during its propagation

[4, 5]. In this interaction, proton and microwave background photon collide into the resonance state∆+, which decays to neutron and pion. Finally, the pion decays to neutrinos. The chain of interactions and decays is the same as Eq. (2.1) in Sec.2.1.1. The GZK effect leads to a cutoff of the cosmic ray spectrum around 1020eV. The energy of neutrinos produced in this decay chain is around 1017to 1018eV. Fig. 2.1.2 shows the GZK tau neutrino spectrum [6].

2.2

Pontecorvo-Maki-Nakagawa-Sakata Mixing Matrix and Flavor

Transition Probability Matrix

In neutrino oscillation theory, the relationship between neutrino mass eigenstates_|νi> and neu-trino flavor eigenstates_|ν_α> are described by Pontecorvo-Maki-Nakagawa-Sakata mixing matrix

U (PMNS matrix)[7, 8]: |να>= 3

∑

i=1 Uαi|νi>, (2.2)

Figure 2.1: The spectrum of AGN [1] and GZK [6] muon neutrino flux. The blue solid line shows the muon neutrino spectrum from AGN source. The red dashed line shows the muon neutrino spectrum from GZK source.

U=        1 0 0 0 cosθ23 sinθ23 0 _−sinθ23 cosθ23              

cosθ13 0 sinθ13e−iδ

0 1 0 −sinθ13eiδ 0 cosθ13               cosθ12 sinθ12 0 −sinθ12 cosθ12 0 0 0 1        =       

cosθ12cosθ13 sinθ12cosθ13 sinθ13e−iδ

−sinθ12cosθ23− cosθ12sinθ23sinθ13eiδ cosθ12cosθ23− sinθ12sinθ23sinθ13eiδ sinθ23cosθ13

sinθ12cosθ23− cosθ12cosθ23sinθ13eiδ −cosθ12sinθ23− sinθ12cosθ23sinθ13eiδ cosθ23cosθ13

       . (2.3)

The time-dependent mass eigenstates_|

ν

i>can be written as

|

ν

i(x,t) >= e−iEit|

ν

i(x, 0) >= e−iEiteipx|

ν

i> .

The transition amplitude for flavor eigenstates|

ν

_α >to flavor eigenstates|

ν

_β >is

<

ν

_β|

ν

α >=<

ν

j|U_β∗jUαie−iEiteipx|

ν

i>=

∑

i U_β∗_iU_αie−iEiteipx=

∑

i U_β∗_iU_αie−i miL 2E,

whereL is the propagation distance of neutrino, E and mi correspond the energy and mass of

as the neutrino. Then the time-dependent transition probabilities for flavor

β

to flavor

α

can be written as | <

ν

α|

ν

β > |2=

δ

αβ− 4

∑

Re(UαiU_β∗jUα∗iUβj) sin2 ∆m2_{i j}L 4E ! + 4

∑

Im(U_αiU_β∗_jUα∗iUβj) sin ∆m2_{i j}L 4E ! cos ∆m 2 i jL 4E ! , (2.4)

where∆m2_{i j≡ m}2_{i − m}2_j. In the limit of large neutrino propagation distance, the neutrino oscilla-tion probability only depend on the mixing angles

θ

i j andCPphase. In this case, the transition

probability for flavor

β

to flavor

α

can be expressed in terms of unitary matrixU purely: P_αβ =

3

∑

i=1|U

αi|2|Uβi|2. (2.5)

Hence he neutrino flux at the astrophysical site and that detected on the Earth is related by

    

φ

(

ν

e)

φ

(

ν

µ)

φ

(

ν

τ)      =      Pee Peµ Peτ P_µe Pµµ Pµτ P_τe Pτµ Pττ          

φ

0(

ν

e)

φ

0(

ν

µ)

φ

0(

ν

τ)      ≡ P     

φ

0(

ν

e)

φ

0(

ν

µ)

φ

0(

ν

τ)      , (2.6)

where

φ

(

ν

_α)is the neutrino flux measured on the Earth while

φ

0(

ν

α)is the neutrino flux at the

source, and the matrix elementP_αβ is the probability for the oscillation

ν

_{β →}

ν

_α. The exact analytic form of the probability matrixPis given by

Pee =  1₋1 2

ω

 (1 − D2)2+ D4, Peµ = 1 4(1 − D 2₎ 

ω

(1 +∆_{) + (4 −}

ω

_{)(1 −}∆)D2+ 2 q

ω

(1 −

ω

)(1 −∆2_{)D cos}

δ

 , Peτ = 1 4(1 − D 2₎ 

ω

(1 −∆) + (4 −

ω

)(1 +∆)D2_{− 2} q

ω

(1 −

ω

)(1 −∆2_{)D cos}

δ

 , P_µµ = 1 2(1 +∆ 2 ) − (1 −∆)2D2_{(1 − D}2) − 1 8

ω

(1 +∆) 2 + (1 −∆)2D4_{− (1 −}∆2)D2(2 + 4 cos2

δ

) − 1₂ q

ω

(1 −

ω

)(1 −∆2_{)(1 +}∆_{) − (1 −}∆_)D2_{ D cos}

_δ

_, P_µτ = 1 2(1 −∆ 2 )(1 − D2+ D4) − 1₈

ω

(1 −∆2_{)(1 + 4D}2_cos2

_δ

_{+ D}4 ) − 2(1 +∆2)D2 + 1 2 q

ω

(1 −

ω

)(1 −∆2₎∆_{(1 + D}2_{)D cos}

δ

_, P_ττ = 1 2(1 +∆ 2 ) − (1 +∆)2D2_{(1 − D}2) − 1 8

ω

(1 −∆) 2_{+ (1 +∆)}2 D4_{− (1 −}∆2)D2(2 + 4 cos2

δ

) + 1 2 q

ω

(1 −

ω

)(1 −∆2_{)(1 −}∆_{) − (1 +}∆_)D2_{ D cos}

_δ

_{, (2.7)}

with

ω

≡ sin22

θ

12,∆≡ cos2

θ

23,D≡ sin

θ

13, and

δ

the CP phase.

If we take the best-fit values of neutrino mixing angles from parameter set I in Tab. 2.3, the neutrino flavor ratios observed on the Earth from the pion source and muon-damped source after oscillation are shown in Fig. 2.2(a). Considering the uncertainties of mixing angle measurement, the flavors ratios observed on the Earth are shown in Fig. 2.2(b).

It is seen that Peµ = Peτ and Pµµ = Pµτ = Pττ in the limit∆= 0 = D, i.e.,

θ

23 =

π

/4

and

θ

13= 0. In this case, the probability matrixPis singular. In general, this singularity is only

slightly broken since both∆andDare expected to be small. For∆= 0 = D, the eigenvectors of the matrixPare

(a) (b)

Figure 2.2: The ranges for the neutrino flavor ratios on the Earth resulting from standard neutrino oscillation. The numbers on each side of the triangle denote the flux fraction of a specific flavor of neutrino. The blue point marks the pion sourceφ0(νe) : (φ0(νµ) :φ0(ντ) = 1/3 : 2/3 : 0 and

the red point marks the muon-damped source φ0(νe) : (φ0(νµ) :φ0(ντ) = 0 : 1 : 0. In panel (a),

the red and blue squares mark the corresponding neutrino flavor ratios observed on the Earth. In panel (b), we introduce uncertainties of neutrino mixing angles in the probability matrix. The red and dashed line denote neutrino flavor ratios observed on the Earth from muon-damped source in 1σand 3σ ranges of neutrino mixing angles respectively. The blue and solid line denote neutrino flavor ratios observed on the Earth from pion source in 1σ and 3σ ranges of neutrino mixing angles respectively

Va=_√1 3      1 1 1      , Vb= _√1 2      0 −1 1      , Vc=_√1 6      −2 1 1      , (2.8)

with the corresponding eigenvalues

λ

a= 1,

λ

b= 0,

λ

c= 1

4(4 − 3

ω

), (2.9)

where

ω

= sin22

θ

12. Therefore, those initial flavor ratios that differ from one another by a

multiple ofVbshall oscillate into the same flavor ratio on the Earth. To illustrate this explicitly, we write the initial fluxΦ0at the astrophysical source as

Φ0=      1 0 0      − √ 2 2

φ

0(

ν

µ) −

φ

0(

ν

τ)
V b₊ √ 6 2

φ

0(

ν

µ) +

φ

0(

ν

τ)
V c_, (2.10)

where we have imposed the normalization condition

φ

0(

ν

e) +

φ

0(

ν

µ) +

φ

0(

ν

τ) = 1. This

nor-malization convention will be adopted throughout this thesis. The first term on the right-hand side (RHS) of Eq. (2.10) can be expressed as(√3Va₋√6Vc)/3. Hence the neutrino flux measured by the terrestrial neutrino telescope is

Φ= P

φ

0= √ 3 3 V a − √ 6 3 (1 − 3 4

ω

)V c₊ √ 6

λ

c 2

φ

0(

ν

µ) +

φ

0(

ν

τ)
V c_{. (2.11)}

It is seen that the vectorVb, with a coefficient proportional to

φ

0(

ν

µ) −

φ

0(

ν

τ), does not appear in

the terrestrially measured fluxΦ. Hence the terrestrial measurement can not constrain

φ

0(

ν

µ) −

φ

0(

ν

τ)in this case.

The above degeneracy is lifted by either a non-vanishing

θ

13 (D6= 0) or a deviation of

θ

23

from

π

/4(∆_{6= 0}). To simplify our discussions, let us takeD= 0and ∆_{6= 0}. One can show that the flux combination(1 + 4

ω

∆_{/(4 − 3}

ω

))

φ

0(

ν

µ) −(1 − 2

ω

∆/(4 − 3

ω

))

φ

0(

ν

τ)remains

poorly constrained due to the suppression ofdet P. To demonstrate this, we observe that P= 1 8      8_{− 4}

ω

2(1 +∆)

ω

2_{(1 −}∆)

ω

2(1 +∆)

ω

_{(4 −}

ω

)(1 +∆2_{) − 2}∆

ω

_{(4 −}

ω

_{)(1 −}∆2) 2_{(1 −}∆)

ω

_{(4 −}

ω

_{)(1 −}∆2) _{(4 −}

ω

)(1 +∆2) + 2∆

ω

     (2.12)

forD= 0and∆_{6= 0}. The eigenvalues ofPexpanded to the second order in∆are given by

λ

′ a= 1,

λ

b′=  4 − 4

ω

4_{− 3}

ω

 ∆2 ,

λ

c′= 1 4(4 − 3

ω

) + 3

ω

2∆2 4_{(4 − 3}

ω

), (2.13) and the corresponding eigenvectors to the same order in∆are

V′a = Na      1 1 1      , V′b = Nb      2r∆(1 + r∆) −1 − 2r∆(1 + r∆) 1      , V′c = Nc      −2 + 6r∆ 1_{− 6r}∆_{(1 − 3r}∆) 1      , (2.14)

withr=

ω

_/(4−3

ω

)andNa,b,cthe appropriate normalization factors. It is interesting to note that the corrections to the eigenvectors ofPbegin atO₍∆_{)while the corrections to the corresponding} eigenvalues begin atO₍∆2_{). With the above eigenvectors, we write the source neutrino flux as}

Φ0 = NaV′a−(1 + 4r∆)

φ

0(

ν

µ) − (1 − 2r∆)

φ

0(

ν

τ) − 2r∆ NbV′b + 3  (1 − 4r∆)

φ

0(

ν

µ) + (1 − 2r∆)

φ

0(

ν

τ) − 2 3(1 − 3r∆)  NcV′c. (2.15)

particle major processes signal type symbol in Fig. 2.3

e EM shower shower A

µ energy loss track B

τ(E_ν< 3.3PeV) CC int. andτ-decay shower C

τ(3.3PeV< Eν< 33PeV) CC int. andτ-decay 2 separate showers D (double-bang event) τ(E_ν> 3.3PeV) energy loss and decay track and shower E (lollipop event)

τ(E_ν> 3.3PeV) CC int. and energy loss shower and track F (inverted lollipop event)

τ(E_ν> 33PeV) energy loss track G

X hadron shower shower

Table 2.1: Different types of neutrino induced events.

It is easy to show that the measured fluxPΦ0depends onV′bthrough the combination−B

λ

_b′NbV′b

with

B=(1 + 4r∆)

φ

0(

ν

µ) − (1 − 2r∆)

φ

0(

ν

τ) − 2r∆ . (2.16)

Clearly the flux combination(1 + 4r∆)

φ

0(

ν

µ) − (1 − 2r∆)

φ

0(

ν

τ)is poorly constrained due to

the smallness of

λ

_b′, of the order∆2.

2.3

Observed Neutrino Events in Detector

Neutrinos must interact with matters to produce observable signals. The major channel for such interactions is the charged-current (CC) interaction,

ν

l+ N −→ l + X, where l is the lepton

associated with

ν

l and X denotes the hadronic states. The sub-dominant channel is the

neutral-current interaction (NC),

ν

l+ N −→

ν

l+ X. The detail of CC and NC interaction mechanisms

are discussed in section 4.2.3. In Fig. 2.3 and Table 2.1, we summarize different types of neutrino induced events and their detectable energy ranges.

Type-A event in Fig. 2.3 is an electron production through

ν

eCC interaction. The electron has

a large interaction cross section with the medium and produces a shower within a short distance after its production. A similar shower signature can be produced by hadrons resulting from NC interactions of all neutrino flavors. Type-B event is a muon produced by

ν

_µ CC interaction. Contrary to the electron, a muon can travel a long distance in the medium before it loses all its energy or decay. The muon range in ice is more than10km forE_ν = 1PeV (1015eV). Hence,

at this energy or higher, there is hardly any decay of muon occurring within the fiducial volume of the detector, which is about a few km3. A muon does, however, lose a small fraction of its energy and emits dim lights so that only those optical detectors which are near to the muon track can be triggered. As a result, a muon produces a track-like signal.

The

ν

_τ-induced events are listed as types C-G where the tau lepton produced by

ν

_τ CC inter-action behaves differently at different energies for a fixed detector design. For a neutrino telescope such as IceCube [9], the distance between each string of optical detectors is125m, which cor-responds to the decay length of a2.5PeV tau lepton. Such a tau lepton could be produced by the CC interaction of a

ν

_τ withE_ν = 3.3PeV. Therefore, for a

ν

_τ with an energy significantly smaller than this, the separation between the first hadronic shower produced by CC interaction and the second shower produced by the tau-lepton decay is too small to be resolved. Such an event is classified as type C. ForE_ν > 3.3PeV, one can resolve the above double-bang event (classified as type D) until the separation of two showers exceeds the effective size of the detector. Such a size is estimated to be the sum of IceCube dimension (_{≈ 1}km) and two extinction lengths of optical photons in ice (_{≈ 250}m), which corresponds to the decay length of a tau lepton withE_τ = 25 PeV. The average energy of

ν

_τ capable of producing such a tau lepton is around33.3PeV. Hence the configuration of IceCube detector determines the observable energy range for the double bang event to be3.3 PeV < E_ν < 33.3 PeV[10]. For an under-sea experiment, such as KM3NeT [11], the observable energy range for the double bang event is similar.

Type-E event is referred to as the lollipop event. In such an event, a high energy tau lepton enters the detector and decays within it, producing a track signal followed by a shower. The probability for observing a lollipop event increases with the neutrino energy, and it is about5_× 10−4 forE_ν = 1EeV [12]. Type-F event is the inverted lollipop which consists of a hadronic shower from

ν

_τ CC interaction and a subsequent tau-lepton track. Both muons and tau leptons produce inverted lollipop events and it is not easy to separate these two types of events. Finally, type-G event is a through-going tau-lepton track which is produced by

ν

_τ CC interaction with E_ν> 33.3PeV.

Flux ratio parameter for

E_ν < 3.3

PeV.—

In this energy range, type-A and type-C events

Condition I : Eν < 33 PeV Condition II : Eν> 33 PeV

RI=φ(νµ)/(φ(νe) +φ(ντ)) RII=φ(νe)/(φ(νµ) +φ(ντ)) SI=φ(νe)/φ(ντ) SII=φ(νµ)/φ(ντ) Table 2.2: The definitions of R and S at different energy ranges.

can only measure the ratioNtrack/Nshower whereNshower is the total event number for electron

showers, tau-lepton showers and the hadronic showers from NC interactions. In IceCube, this ratio can be measured in a good precision, which is useful for deducing the flux ratio RI =

φ

(

ν

_µ)/(

φ

(

ν

e) +

φ

(

ν

τ))[13].

Flux ratio parameters for

3.3 PeV < E_ν < 33.3 PeV

.—

In this energy range, one can

detect the type-D and type-E

ν

_τ events (double bang and lollipop). Hence it is also possible to measure the flux ratioSI_≡

φ

(

ν

e)/

φ

(

ν

τ)in addition toRI[14]. However, the double bang and

lollipop events are both rare so that the error associated withSI is large. We demonstrated that a large number of events is necessary for lowering down the errors ofRIandSIto the point that one can distinguish the pion source from the muon-damped source [15].

Flux ratio parameter for

E_ν > 33.3

PeV.—

In this high energy regime, the tau-lepton

range becomes long enough so that a tau lepton could pass through the detector fiducial volume without decaying. In this case, the tau-lepton loses its energy just like a muon does and the signal appears like a track event [16]. Thus, from an experimental point of view, one should classify such a signature as a track event (type G). In this energy range, there are also type-E and type-F events where tau leptons also behave like tracks. We therefore define a new flavor ratio parameter RII=

φ

(

ν

e)/(

φ

(

ν

µ) +

φ

(

ν

τ))since the electron shower can be easily separated from the muon

and tau-lepton tracks. It is however more challenging to distinguish the tau- lepton track from the muon track so that the second flux ratio parameterSII=

φ

(

ν

µ)/

φ

(

ν

τ)can not be measured as accurately asRII. Table 2.2 present the appropriate flux ratio parameters for high energy are low energy respectively.

Figure 2.3: Different types of neutrino-induced events. Dashed lines and solid lines correspond to paths of neutrinos and leptons respectively. The ellipsoids are showers. The detectable energy range for each type of event is listed in Table 2.1.

2.4

Reconstructing Source Flavor Ratios at Low Energies

Most of previous studies concerningRI andSI assume a good knowledge of neutrino flavor ratio at the source and explore possible flavor ratios to be observed on the Earth [12, 13]. We take a reversed approach to identify source flavor ratio from the observed flavor ratio on the Earth.

Given a precision on measuringRI,∆RI/RI_{, we estimate∆S}I_/SI _{with two approaches. The}

first approach assumes that both∆RI and∆SI are dominated by the statistical errors. In this case, one has ∆ SI SI  = 1+ S I √ SI s RI 1+ RI ∆ RI RI  , (2.17)

Using values ofRI andSI from Table 2.4, we obtain∆SI/SI _{= (1.1 − 1.2)(}∆_RI_/RI_)from

the pion source and∆SI/SI _{= (1.1 − 1.4)(}∆_RI_/RI_{)from the muon-damped source. The second}

approach takes into account the specific complications for identifying tau neutrinos. Since tau lepton decays before it loses a significant fraction of its energy, tau neutrino is identified by the so-called double-bang or lollipop events [10, 12, 19]. In IceCube or other detector with a comparable

size, double-bang events are observable only in a narrow energy range between2PeV and20PeV [10, 19] while the probability for observing a lollipop event, though increasing with the neutrino energy, is still less than10−3forE_ν = 1EeV [12]. In view of these, we do not correlate∆SI/SI

with∆RI/RI _{in the second approach. Rather we fix} ∆_SI_/SI _{while vary} ∆_RI_/RI _{for achieving}

the goal of distinguishing astrophysical neutrino sources. The results of both approaches will be presented. Before presenting the details of our analysis, we point out that the decays

τ

_→

ν

_τ

µ

ν

¯_µ

and

τ

→

ν

_τ

µ

ν

¯_µ, each with a18%branching ratio, produce extra muon events or secondary

ν

e

and

ν

_µ [12, 20]. Cares are needed to separate these events from those of primary

ν

e and

ν

µ or

muons produced by the charged current interaction.

The fitting to the neutrino flavor ratios at the source is facilitated through

χ

2₌ R I th− RIexp

σ

RI exp !2 + S I th− SIexp

σ

SI exp !2 +

∑

jk=12,23,13 s2 jk− (sjk)2best fit

σ

s2 jk !2 , (2.18) with

σ

_RI exp = (∆R I_/RI_)RI exp,

σ

SI exp = (∆S I_/SI_)SI exp, s2jk ≡ sin 2

_θ

jk and

σ

_s2

jk the 1

σ

range for

s2_jk. HereRI_thandSI_thare theoretical predicted values forRI andSI respectively whileRI_exp and SI_exp are experimentally measured values. The values forRI_exp and SI_exp are listed in Table 2.4, which are generated from input true values of neutrino flavor ratios at the source and input true values of neutrino mixing parameters. InRI_thandS_thI , the variabless2_jkcan vary between0and1

whilecos

δ

can vary between₋₁and 1. We note that similar

χ

2 functions have been used for fitting the CP violation phase and the mixing angle

θ

23respectively [17, 21], assuming the source

flavor ratio is known. In our analysis, we scan all possible neutrino flavor ratios at the source that give rise to a specific

χ

2 value. Since we have takenRI_exp and SI_exp as those generated by input true values of initial neutrino flavor ratios and neutrino mixing parameters, we have(

χ

2₎

min=0

occurring at these input true values of parameters. Hence the boundaries for1

σ

and 3

σ

ranges of initial neutrino flavor ratios are given by ∆

χ

2 = 2.3 and ∆

χ

2 = 11.8 respectively where ∆

χ

2_≡

χ

2_{− (}

χ

2₎