Indirect Search for Dark Matter with the Super-Kamiokande Detector

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NATIONAL CENTREFOR NUCLEAR RESEARCH

DOCTORAL THESIS

Author: Supervisors: Katarzyna FRANKIEWICZ Prof. dr hab. Ewa RONDIO Dr Piotr MIJAKOWSKI

A thesis submitted in fulﬁllment of the requirements for the degree of Doctor of Philosophy

April 2018

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NATIONAL CENTRE FOR NUCLEAR RESEARCH Abstract

Doctor of Philosophy

Indirect Search for Dark Matter with the Super-Kamiokande Detector

by Katarzyna FRANKIEWICZ

One of the strategies for dark matter detection is to search for the products of its self-annihilation or decay, such as antimatter, photons, or neutrinos. The later are particularly interesting since they can travel unaffected through the galactic scales. Therefore, neutrinos can provide valuable information on their source location and initial energy spectra related to dark matter annihilation/decay processes.

In this dissertation, an indirect search for dark matter induced neutrinos is described. Two types of analyses are performed: the search for neutrinos originating from dark matter annihilation or decay in the Milky Way, and a search for neutrinos from dark matter annihilation inside the Earth’s core. In both presented analyses, atmospheric neutrino data set collected with the Super-Kamiokande detector from 1996 to 2016 is used. The expected dark matter induced signal would manifest itself as an excess in the number of neutrinos originating from the direction corresponding to position of the Galactic Center or the Earth’s core, when compared to the background level of atmospheric neutrinos. The selection criteria which allows us to effectively distinguish the signal from background are developed in order to estimate the potential contribution of dark matter induced neutrinos to all detected neutrino interactions.

No statistically significant excess of neutrinos coming from the direction of the considered sources has been found in the performed analyses. The allowed number of dark matter induced neutrinos which can be contained in the Super-Kamiokande data so far is estimated. In case of the Galactic Center analysis, obtained limits on dark matter induced neutrino flux are related to the upper limits on the velocity- averaged dark matter self-annihilation cross-section, < σAv >, and lower limits on dark matter particle lifetime, τ. Limits are set for wide range of masses from 1 GeV up to 10 TeV, and different possible annihilation/decay channels including: νν¯, bb¯, µ+µ− and W+W−. The influence of the dark matter halo model choice on the obtained results is also estimated for the annihilation scenario.

For the Earth’s core analysis, the spin-independent WIMP-nucleon scattering cross- section is constrained for dark matter particle masses ranging from 10 to 1000 GeV, and for bb¯, τ+τ− and W+W− annihilation channels. The limits from performed analysis are compared against the results of other direct and indirect detection experiments. High sensitivity of the Super-Kamiokande detector, allows to set the strongest limits among all neutrino experiments up to date. For the dark matter particle mass ∼ 50 GeV, limit for τ+τ− annihilation channel reaches 10−44 cm2.

NATIONAL CENTRE FOR NUCLEAR RESEARCH Streszczenie

Jednym ze sposobów na poszukiwanie ciemnej materii jest próba detekcji produk- tów jej anihilacji lub rozpadów, takich jak antymateria, fotony lub neutrina. Te os- tatnie s ˛aszczególnie interesuj ˛ace,poniewaz˙ podczas propagacji w przestrzeni kos- micznej ich kierunek oraz energia pozostaj ˛aniezmienione i mog ˛aone dostarczy´c niezwykle istotnych informacji na temat ´zródłaz którego pochodz ˛a.

W ponizszej˙ rozprawie opisane s ˛aposzukiwania neutrin pochodz ˛acychz anihilacji lub rozpadów ciemnej materii. Przeprowadzone zostały dwie analizy, poszukiwanie neutrin pochodz ˛acychz centrum Drogi Mlecznej oraz neutrin pochodz ˛acych z wn˛etrzaZiemi. Do obydwu analiz uzyto˙ przypadków oddziaływa´nneutrin zarejestrowanych w japo´nskimdetektorze Super-Kamiokande, w latach 1996-2016. W´sród wszystkich zarejestrowanych przypadków poszukiwane s ˛ate posiadaj ˛acecharak- terystyki zgodne z potencjalnym wkładem od neutrin produkowanych w wyniku anihilacji lub rozpadów ciemnej materii. Poszukiwanym sygnałem jest nadwyzka˙ neutrin pochodz ˛acaz centrum Galaktyki lub wn˛etrzaZiemi, w stosunku do tła, które stanowi ˛aneutrina produkowane w atmosferze ziemskiej przez promienio- wanie kosmiczne.

W przeprowadzonych analizach nie stwierdzono nadmiaru przypadków zwi ˛azane- go z rozpatrywanymi ´zródłami,co pozwoliło na górne ograniczenie liczby przypad- ków mog ˛acychpochodzi´cod ciemnej materii, rejestrowanych w detektorze Super- Kamiokande. Na tej podstawie dla analizy z centrum Galaktyki obliczono górne limity na wazony˙ pr˛edko´sci˛aprzekrój czynny na samo-anihilacj˛eciemnej materii, < σAv >, i dolne limity na czas zycia˙ cz ˛astekciemnej materii, τ. Limity te wyznaczono w szerokim zakresie mas od 1 GeV do 10 TeV i dla czterech rozwazanych˙ kanałów anihilacji/rozpadów ciemnej materii: νν¯, bb¯, µ+µ− oraz W+W−. W przypadku anihilacji ciemnej materii dodatkowo rozwazono˙ wpływ wybranego modelu galaktycznego halo na otrzymywane wyniki.

W przypadku analizy z centrum Ziemi, wyznaczono górne limity na niezalezny˙ od spinu przekrój czynny na rozpraszanie WIMP-nukleon, σχ−N, dla mas cz ˛astek ciemnej materii w zakresie od 10 GeV do 1 TeV,dla trzech rozwazanych˙ kanałów anihilacji: bb¯, τ+τ− oraz W+W−. Dzi˛ekiduzej˙ czuło´scidetektora Super-Kamiokande uzyskano najsilniejsze limity pochodz ˛acez eksperymentów neutrinowych na ´swiecie, si˛egaj˛ace10−44 cm2 dla cz ˛astekciemnej materii o masie ∼ 50 GeV, przy załozeniu˙ anihilacji w pary leptonów τ+τ−.

vii Acknowledgements

First of all, I would like to express my deep gratitude to Prof. Ewa Rondio. She guided me through my PhD and was an excellent supervisor. I gained a lot from her vast experience in neutrino physics and I am very happy that I decided to join her group. She would always ﬁnd time to discuss physics problems and answer any of my questions. Thank you for your support, patience, and encouragement.

I am very thankful to Dr Piotr Mijakowski, who helped supervise my work, especially through the last and most difﬁcult part – preparing this dissertation. He also introduced me to the Super-Kamiokande experiment, during the last year of my Masters, what convinced me to pursue a PhD in neutrino physics. In addition to all his help, he always was a good friend.

Super-Kamiokande is an exceptional experiment, which provides the opportunity to study a wide range of physics topics. The experiment was built and has been operated with funding from the Japanese Ministry of Education, Culture, Sports, Science and Technology and the U.S. National Science Foundation. Special thanks goes to Kamioka Mining and Smelting Company for helping to host the experiment.

I would like to thank all my collaborators for their dedication and engagement in the Super-K experiment. In particular, I want to mention the ATMPD working group conveners: Prof. Ed Kearns and Prof. Masato Shiozawa. They provided excellent leadership and guidance in my analysis group.

Moreover, a special thanks to Ed for giving me the opportunity to visit Boston and Fermilab. I had a great time working with his neutrino group at Boston University. Thanks to his help I had a chance to broaden my outlook. I really appreciate his time, support and advice.

My greatest debt is to Roger Wendell, who prepared the framework which we all use in the ATMPD working group. He was a tremendous help when dealing with nearly any kind of problems: software, statistics, physics, and much more. My analysis would not have been possible without his support. Thank you for the time you spent with us during the WIMP meetings, and for the hundreds of emails with my questions that you have answered.

Lluis Marti is a great friend and colleague, and I really appreciated his company during the long days spent in Mozumi. I enjoyed our conversations very much and I am in debt to you for the delicious coffee and cookies.

I would like to thank my family and friends. Thanks to my parents support I was able to take up my studies in Warsaw. My sister and brother were always ready to help me with every problem I have encountered.

Krzysztof was not only the best ﬂatmate, but above all, the best friend. I don’t know what I would do without his support and great advice. I would like to thank you for all this good times we had during our PhDs.

A special thanks goes to Elzbieta˙ Gaca, my ﬁrst physics teacher, who aroused my interest in science. I am very grateful for her time and commitment, and having the viii faith in me. If it was not for her I would probably never have thought that I could be a physicist.

Last but not the least, I would like to say thank you to Spencer. For everything. For his love and understanding, and always being there for me.

My work has been supported by the funds from the National Science Centre, Poland (2015/17/N/ST2/04064) and the European Union (H2020 RISE-GA641540-SKPLUS). ix

Contents

Abstract iii

Streszczeniev

Acknowledgements vii

1 Introduction1

2 Dark Matter5 2.1 Observational Evidence...... 5 2.1.1 Galaxy Clusters...... 5 2.1.2 Galaxy Rotation Curves...... 5 2.1.3 Gravitational Lensing...... 6 2.1.4 Cosmic Microwave Background...... 7 2.1.5 Big Bang Nucleosynthesis...... 7 2.1.6 Structure Formation...... 8 2.2 Dark Matter Candidates...... 8 2.2.1 Neutralino as a Dark Matter Candidate...... 9 2.3 Dark Matter Detection...... 10 2.3.1 Direct Detection...... 10 2.3.2 Indirect Detection...... 15 Antimatter...... 16 Gamma Rays...... 16 Neutrinos...... 17 2.3.3 Production in Accelerators...... 19

3 Neutrinos 21 3.1 History...... 21 3.2 Neutrino Oscillations...... 22 3.2.1 In Vacuum...... 22 3.2.2 In Matter...... 24 3.3 Neutrino Interactions...... 25 3.4 Sources of Neutrinos...... 26

4 Super-Kamiokande 31 4.1 Cherenkov Radiation...... 31 4.2 Detector...... 32 4.2.1 Photomultiplier Tubes...... 34 4.3 History...... 35 4.4 Water Puriﬁcation System...... 36 4.5 Air Puriﬁcation System...... 37 4.6 Calibration...... 38 x

5 Atmospheric Neutrinos at Super-Kamiokande 41 5.1 Data Samples...... 41 5.1.1 Data Reduction and Reconstruction...... 43 5.2 Monte Carlo...... 44 5.3 Atmospheric Neutrino Oscillation Analysis...... 46 5.3.1 Systematic Uncertainties...... 48 5.3.2 Results of the Oscillation Analysis...... 50

6 WIMPs in the Center of the Milky Way 51 6.1 Dark Matter Halo Models...... 51 6.2 Dark Matter Annihilation...... 52 6.3 Dark Matter Decay...... 54 6.4 Energy Spectrum of Dark Matter Induced Neutrinos...... 54

7 Search for the Galactic WIMPs at Super-Kamiokande 57 7.1 Equatorial Coordinate System...... 57 7.2 Simulation of WIMP Induced Neutrinos from the Galactic Center... 58 7.3 ON-OFF Source Approach...... 63 7.4 Optimization of the ON-Source Region Size...... 63

8 Results of the Search for Galactic WIMPs 67 8.1 Asymmetry Between ON- and OFF-Source Regions...... 67 8.2 Constraints on Dark Matter Self-Annihilation Cross-Section...... 69 8.3 Constraints on the Dark Matter Particles Lifetime...... 72

9 WIMPs in the Center of the Earth 75 9.1 Capture Rate of WIMPs in the Earth...... 75 9.2 WIMP Annihilation in the Earth...... 76 9.3 Neutrino Flux from WIMP Annihilation in the Earth...... 77 9.4 WimpSim Results for the Earth WIMPs...... 78

10 Search for the Earth WIMPs at Super-Kamiokande 81 10.1 Global Fit Approach...... 81 10.2 Signal and Background Predictions...... 83 10.3 Sensitivity Study for WIMP Induced Neutrinos from the Earth’s Core. 86 10.3.1 Null WIMP Contribution (β = 0)...... 87 10.3.2 Injected WIMP Contribution (β = 0.01)...... 92 10.4 Binning Optimization...... 95 10.4.1 Toy Monte Carlo Method...... 98

11 Results of the Search for the Earth WIMPs 103 11.1 Fit Results with Standard Binning...... 103 11.1.1 Constraints on the WIMP-Nucleon Spin Independent Scatter- ing Cross-Section for Mχ < 100 GeV...... 112 11.2 Fit Results with Optimized Binning...... 113 11.2.1 Constraints on the WIMP-Nucleon Spin Independent Scatter- ing Cross-Section for Mχ up to 1 TeV...... 120 11.2.2 Comparison with Other Experiments...... 122

12 Summary and Outlook 125

A Galactic WIMP Weights 129 xi

B WimpSim Results 133 B.1 bb¯ annihilation channel...... 133 B.2 τ+τ− annihilation channel...... 136 B.3 W+W− annihilation channel...... 139

C Earth WIMP Signal Illustrations 141 C.1 Standard Binning...... 141 C.2 Optimized Binning...... 146

D Bayesian Approach 149

Bibliography 151

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List of Figures

2.1 An example of a galaxy rotation curve for NGC 6503 dwarf spiral galaxy studied by Begeman, Broels, and Sanders. The solid line is the theoretical rotation curve and the data points with error bars are the measured object speed. The segmented lines show the contributions from gas, luminous matter, and dark matter. Figure is taken from (Begeman, Broeils, and Sanders, 1991)...... 6 2.2 Left: Visible light image of the Bullet Cluster from the Magellan Tele- scope. Right: The Chandra X-ray Observatory image of the Bullet Cluster showing the location of the x-ray emission of the hot gas. In both panels the mass density contours are shown with green lines. Figure taken from (Clowe et al., 2006)...... 7 2.3 Simpliﬁed diagram showing the three different dark matter detection channels. The direct detection experiments (→) look for the recoil introduced by a WIMP (χ) on a SM particle in the detector. Indirect detection experiments (↓) look for the SM annihilation products of WIMPs. Collider experiments (↑) look for the creation of WIMPs in Standard Model particle collisions...... 11 2.4 Feynman diagrams contributing to the spin-dependent elastic scattering of neutralinos from quarks...... 11 2.5 Feynman diagrams contributing to the scalar elastic-scattering amplitude of a neutralino from quarks...... 12 2.6 A sketch idea of dark matter direct detection...... 12 2.7 Latest limits on SI WIMP-nucleon scattering cross-section as a function of WIMP mass: results from combined PandaX-II Run 9 and Run 10 data (Cui et al., 2017) (red) (the green band represents the ±1σ sensitivity band), overlaid with results from PandaX-II 2016 (Tan et al., 2016) (blue), LUX 2017 (Akerib et al., 2017) (magenta), and XENON1T 2017 (Aprile et al., 2017) (black)...... 13 2.8 Latest limits on the SD WIMP-proton scattering cross-section as a function of WIMP mass: results from PICO-60 C3F8 (Amole et al., 2017) (thick blue), along with limits from PICO-60 CF3I (Amole et al., 2016a) (thick red), PICO-2L (Amole et al., 2016b) (thick purple), PICASSO (Behnke et al., 2017) (green band), SIMPLE (Felizardo et al., 2014) (orange), PandaX-II (Fu et al., 2017) (cyan), IceCube (Aartsen et al., 2017b) (dashed and dotted pink), and SuperK (Choi et al., 2015) (dashed and dotted black). Indirect limits from IceCube and SuperK assume annihilation to τ leptons (dashed) and b quarks (dotted). The purple region represents parameter space of the constrained minimal supersymmetric model of (Roszkowski, Austri, and Trotta, 2007)...... 13 xiv

2.9 Experimental 90 % C.L. exclusion limit on the SI WIMP-nucleon scattering cross section as a function of WIMP mass for the combined fit from EDELWEISS-II detectors (Hehn et al., 2016) (solid red). The green and yellow band represent the 1 and 2 σ confidence band of the expected median sensitivity (dashed black). Result of the EDEL- WEISS BDT based analysis (dashed red) is shown for comparison. Colored regions show possible signals from CDMS-II (Si) (Agnese et al., 2013) (blue), DAMA (Bernabei et al., 2013) (brown), CRESST-II (Angloher et al., 2012) (pink) and CoGeNT (Aalseth et al., 2014) (orange). Other existing exclusion limits are from EDELWEISS-II (Ar- mengaud et al., 2012) (small red dashes), CoGeNT (Aalseth et al., 2013) (orange), CRESST (Angloher et al., 2016) (pink), SuperCDMS (Agnese et al., 2014) (purple), XENON100 (Aprile et al., 2016) (black), CDMSlite (Agnese et al., 2015) (dashed violet) and LUX (Akerib et al., 2016) (green)...... 14 2.10 An illustration of a possible products of supersymmetric dark matter self-annihilation...... 15 2.11 The electron flux and the positron flux measured by AMS-02 experiment (Aguilar et al., 2016)...... 16 2.12 The upper 90% C.L. limits from neutrino experiments on dark matter self-annihilation cross section for the bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) annihilation channels. The limits from Super-Kamionkande (Frankiewicz, 2017a) are plotted with solid lines, IceCube (Aartsen et al., 2017c) with dotted lines and ANTARES (Al- bert et al., 2017a) with dashed lines...... 18 2.13 The upper 90% C.L. limits on the WIMP-proton SD cross section (left) and WIMP-nucleon SI cross section (right) for the bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channels. The limits from Super-Kamionkande (Choi et al., 2015) are plotted with solid line, Ice- Cube (Aartsen et al., 2017b) with dotted lines. For the comparison the strongest limits from direct detection experiments: LUX (Akerib et al., 2017) (orange) and PICO-60 (Amole et al., 2017) (purple) are plotted, and possible signal from DAMA (Bernabei et al., 2013)...... 18 2.14 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section from the CMS experiment (Lowette, 2016), for the mono-jet (magenta), mono-photon (red and yellow), and mono-lepton (grey) searches, as a function of the dark matter mass...... 19 2.15 90% C.L. upper limits on SD WIMP-nucleon scattering cross-section from the CMS experiment (Lowette, 2016), for the mono-jet (magenta), mono-photon (red and yellow), and mono-lepton (gray) searches, as a function of the dark matter mass. The limits from Super-Kamionkande come from Solar WIMP search analysis (Choi et al., 2015)...... 19

3.1 Charged current neutrino interactions via a W boson exchange..... 25 3.2 Summary of the current knowledge of νµ charged-current cross sections. Figure taken from (Formaggio and Zeller, 2012)...... 27 3.3 Measured and expected ﬂuxes of natural and reactor neutrinos. Fig- ure taken from (Katz and Spiering, 2012)...... 28 3.4 Schematic view of cosmic rays interactions in the atmosphere. Figure taken from (Louis et al., 1997)...... 29 xv

4.1 A schematic view of the detection of Cherenkov light by the photomultipliers...... 32 4.2 The schematic view of Super-Kamiokande detector and the experimental hall. Figure taken from (Fukuda et al., 2003)...... 33 4.3 A diagram of the module structure of the PMT support frame. Figure taken from (Fukuda et al., 2003)...... 33 4.4 Diagram of 20-inch (50 cm) photomultiplier tube used in Super-K detector (taken from (Fukuda et al., 2003))...... 34 4.5 Quantum efficiency of the photocathode as a function of the light wavelength (taken from (Fukuda et al., 2003))...... 34 4.6 Super-Kamiokande water purification system (taken from (Fukuda et al., 2003))...... 37 4.7 Super-Kamiokande air purification system (taken from (Fukuda et al., 2003))...... 37 4.8 Energy scale stability measured as a function of date since the start of Super-Kamiokande detector operations. The energy scale is taken as the average of the reconstructed momentum divided by range of stopping cosmic ray muon data in each bin. The vertical axis shows the deviation of this parameter from the mean value for each SK period separately. Error bars are statistical. Figure taken from (Abe et al., 2017a)...... 38

5.1 The schematic illustration of event categories used in Super-K: fully- contained (FC), partially-contained (PC) and upward - going muons (UPMU) samples...... 41 5.2 The expected parent neutrino energy distributions for the FC e-like (left), FC µ-like and PC (middle), and UP-µ (right) samples based on MC simulation...... 43 5.3 An example event display of single-ring µ-like (left) and e-like (right) event. The colored points indicate the quantity of the detected light by each PMT...... 43 5.4 Atmospheric neutrino fluxes for Kamioka averaged over all directions (left panel), and the flux ratios (right panel), calculated by Honda (Honda et al., 2011) (solid red), Fluka (Battistoni et al., 2003) (dotted green) and Bartol (Barr et al., 2004) (dashed blue) groups, (dash-dotted line is for the previous Honda calculation (HKKM06))...... 45 5.5 Data and MC comparisons for the entire Super-K data divided into 19 analysis samples. Samples with more than one zenith angle bin are shown as zenith angle distributions (second through fifth column) and other samples are shown as reconstructed momentum distributions (first column). Cyan (orange) lines denote the best f it MC assuming the normal (inverted) hierarchy. Narrow panels below each distribution show the ratio relative to the normal hierarchy MC. In all panels, the error bars represent the statistical uncertainty...... 47 5.6 Constraints on neutrino oscillation parameters from the Super-K atmospheric neutrino data fit. Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in cyan, by the difference in their minimum χ2 values...... 50 xvi

6.1 The dark matter density as a function of the distance from the GC for Moore (Moore et al., 1999) (dotted green), NFW (Navarro, Frenk, and White, 1997) (solid red) and Kravtsov (Kravtsov et al., 1998) (dashed blue) halo profiles. The vertical gray line indicates the Solar System position (Rsc = 8.5 kpc). The plot illustrate Eq. 6.1 with parameters from Tab. 6.1...... 52 6.2 Illustration of the line of sight l and the angle Ψ in the coordinate system related to the Galactic Center...... 53 6.3 Intensity of dark matter annihilation products versus angular distance from the Galactic Center for Moore (green), NFW (red) and Kravtsov (blue) profiles...... 53 6.4 Intensity of dark decay products versus angular distance from the Galactic Center for Moore (green), NFW (red) and Kravtsov (blue) profiles...... 54 6.5 Differential muon neutrino energy spectra for WIMP particle mass of 100 GeV annihilating into bb¯ (blue), W+W− (maroon) and µ+µ− (purple), after taking into account neutrino oscillations through the Galaxy...... 55

7.1 Definition of the coordinates (right ascension, declination) in the equatorial coordinate system...... 58 7.2 Atmospheric neutrino events from SK I-IV data sets in the equatorial coordinate system...... 58 7.3 Left: Standard atmospheric MC in the equatorial coordinate system (upper left plot for the true and bottom left for the reconstructed directions). Middle: Distribution in the cosine of the angular distance from the GC (upper middle plot) and the angular distance from the GC (bottom middle plot). Right: Distribution in right ascension (upper right plot) and declination (bottom right plot). Blue curves in middle and right plots correspond to true neutrino directions and black curves to the reconstructed directions of leptons produced in neutrino interactions...... 59 7.4 Spherical projections of the expected signal shape in the equatorial coordinate system for Moore (left), NFW (middle) and Kravtsov (right) halo profiles, assuming the dark matter annihilation scenario. Upper plots show the true neutrino directions and bottom plots show the reconstructed lepton directions...... 60 7.5 Expected signal shape in RA (upper plots) and DEC (bottom plots) for Moore (left), NFW (middle) and Kravtsov (right) halo profiles, assuming the dark matter annihilation scenario. Blue curves correspond to the true direction of neutrino and black curves to the reconstructed one. 61 7.6 Expected signal shape in the cosine of the angular distance from the GC, cos (Ψ), (upper plots) and the angular distance, Ψ, from the GC (bottom plots) for Moore (left), NFW (middle) and Kravtsov (right) halo profiles, assuming the dark matter annihilation scenario. Blue curves correspond to the true direction of neutrino and black curves to the reconstructed one...... 61 xvii

7.7 Signal expectations for the NFW halo proﬁle, assuming dark matter decay scenario. Left: Spherical projections of the expected signal shape in the equatorial coordinate system (upper left plot for the true and bottom left for the reconstructed directions). Middle: Distribu- tion in the cosine of the angular distance from the GC (upper middle plot) and the angular distance from the GC (bottom left plot). Right: Distribution in right ascension (upper right plot) and declination (bottom right plot). Blue curves on middle and right plots correspond to true neutrino direction and black curves to the reconstructed direction of lepton produced in neutrino interactions...... 62 7.8 The illustration of the ON-OFF source approach. The ON-source region is deﬁned around the GC where the signal concentration is the highest. The OFF-source region is shifted in RA by 180◦. Because ON- and OFF-source regions are equally sized, the expected number of background events is the same in both of them...... 63 7.9 Spherical projections of the expected signal shape in the equatorial coordinate system for NFW halo model, assuming the dark matter annihilation scenario for FC SubGeV, FC MultiGeV, PC and UP-µ event categories...... √ 64 7.10 The signal to square root of the background (S/ B) ratio as a function of the angular size of the ON-source region for FC SubGeV (blue), FC MultiGeV (magenta), PC (black), UP-µ (green) and all events together (red). The NFW halo model is assumed and dark matter annihilation scenario is considered...... 65

8.1 The asymmetry in the number of events between the ON- and OFF- source regions...... 69 8.2 The upper 90% C.L. limit on dark matter self-annihilation cross-section as a function of the dark matter particle mass for bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) annihilation channels. The inﬂuence of the halo model choice is shown as a band around the result for the benchmark NFW proﬁle...... 71 8.3 The upper 90% C.L. limits on dark matter self-annihilation cross-section as a function of dark matter particle mass for bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) annihilation channels (solid lines), compared with the results of independent Global Fit analysis (dotted lines) of Super-K data (Mijakowski, 2018)...... 72 8.4 The lower 90% C.L. limits on the dark matter particle lifetime as a function of dark matter particle mass for bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) decay channels, compared with limits from IceCube experiment (dotted lines) (Abbasi et al., 2012). NFW halo model is assumed...... 73

9.1 The capture rate of WIMPs in the Earth for a scattering cross section of σ = 10−44cm2, calculated in (Sivertsson and Edsjo, 2012). The normalization of the original calculation performed for σ = 10−42cm2 was adjusted due to recent exclusions from direct detection (Akerib et al., 2017). Figure taken from (Aartsen et al., 2017a)...... 76 xviii

9.2 The differential ﬂuxes of the neutrinos (left panels) and anti-neutrinos (right panels) produced by 6 GeV WIMPs annihilating into bb¯ quarks, as a function of the neutrino energy Eν, divided by the WIMP mass Mχ. The different neutrino species are indicated with a different color, black for νe, red for νµ, and green for ντ...... 79 9.3 Energy spectra of the dark matter induced muon neutrinos at the Earth’s surface generated with WimpSim, for 100 GeV WIMPs annihilating into bb¯ (blue line), τ+τ− (green line), and W+W− (maroon line) channel...... 79 9.4 Angular distribution of the dark matter induced muon neutrinos observed at the Earth’s surface, generated with WimpSim for 10 GeV (solid line), 100 GeV (dashed line), and 1 TeV (dotted line) WIMPs annihilating into b¯ quarks...... 79

10.1 Atmospheric neutrino background expectation (orange histogram) and the SK-I-IV data (black points). The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. In all panels, the error bars represent the statistical uncertainty...... 83 10.2 The signal originating from the annihilation of 25 GeV WIMPs in the Earth’s core into bb¯ quarks. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. The plots contain the expected signal from WIMP annihilation (filled blue histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty...... 84 10.3 The signal originating from the annihilation of 100 GeV WIMPs in the Earth’s core into τ+τ− leptons. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. Plots contain expected signal from WIMP annihilation (filled green histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty...... 85 10.4 The signal originating from the annihilation of 500 GeV WIMPs in the Earth’s core into W+W− bosons. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. Plots contain expected signal from WIMP annihilation (filled maroon histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty...... 86 10.5 Heat map showing the tested parameter space of WIMP masses (X - axis) and WIMP contribution (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). Null WIMP contribution (β = 0) is assumed and the color represents χ2 value of the fit...... 88 10.6 Sensitivity contours at 68% (blue), 90% (yellow) and 99% (maroon) C.L for null WIMP contribution (β = 0), for tested parameter space of WIMP masses (X - axis) and WIMP contribution (Y - axis), plotted for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom)...... 89 xix

2 2 2 10.7 The ∆χ = χ − χmin distributions minimized over the corresponding WIMP masses. The horizontal lines correspond to the 68% (blue), 90% (yellow) and 99% (maroon) C.L. for no WIMP hypothesis (β=0), plotted for τ+τ− annihilation channel...... 90 10.8 The 90% upper limits on the dark matter induced muon neutrino flux from the sensitivity study for bb¯ (dotted blue), τ+τ− (dotted green) and W+W− (maroon line) annihilation channel...... 91 10.9 The 90% upper limits on SI WIMP-nucleon scattering cross-section from the sensitivity study for bb¯ (dotted blue), τ+τ− (dotted green) and W+W− (maroon line) annihilation channel. Positive result claimed by DAMA/LIBRA (Bernabei et al., 2013) is shown for comparison... 91 10.10Heat map plot showing the tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). Positive WIMP contribution (β = 0.01) is assumed and the color rep- 2 + − resents χ value of the fit. In bb¯ channel: Mχ = 25 GeV, in τ τ : + − Mχ = 100 GeV, and in W W : Mχ = 500 GeV...... 93 10.11Sensitivity contours at 68% (blue), 90% (yellow) and 99% (maroon) C.L for positive WIMP contribution (β = 0.01), for tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) + − + − and W W (bottom). In bb¯ channel: Mχ = 25 GeV, in τ τ : Mχ = + − 100 GeV, and in W W : Mχ = 500 GeV...... 94 10.12Atmospheric neutrino background expectation (orange histogram) and the SK-I-IV data (black points) with the optimized binning. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. In all panels, the error bars represent the statistical uncertainty...... 96 10.13The signal originating from the annihilation of 1000 GeV WIMPs in the Earth’s core into bb¯ quarks with the optimized binning. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. Plots contain expected signal from WIMP annihilation (filled histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels the error bars represent the statistical uncertainty...... 97 10.14Signal expectation (filled histogram) from Fig. 10.13 and the SK-I-IV atmospheric neutrino data (black points) for UP-µ samples...... 98 10.15The distribution of ∆χ2 = χ2 − χ2 for M = 6 GeV, bb¯ annihila- toy β0=0 min χ 2 tion channel. Vertical line shows the critical value ∆χc , such that 90% 2 2 of the ToyMC experiments have ∆χtoy < ∆χc ...... 99 10.16The distribution of β90 for Mχ = 6 GeV, bb¯ annihilation channel. Ver- tical lines shows the median (black), ±1σ (green) and ±2σ (yellow)... 100 10.17The sensitivity expectation for a null WIMP hypothesis (dotted lines) and its ±1σ (green band) and ±2σ (yellow band) uncertainty is presented for the bb¯ (top), τ+τ− (middle) and W+W− (bottom) annihilation channel...... 101 10.18Comparison between 90% upper limits on SI WIMP-nucleon scattering cross-section for bb¯ annihilation channel, calculated with standard (black line) and optimized (blue line) binning. The significant improvement for Mχ > 100 GeV is visible...... 102 xx

11.1 The momentum (1st column) and zenith angle (2nd-5th columns) distributions before and after the fit are shown for 19 samples used in the analysis. See detailed description in text...... 104 11.2 Heat map plot showing the tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). The color represents χ2 value of the fit to data...... 105 11.3 The 68%, 90% and 99% C.L contours of allowed regions in tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom)...... 106 11.4 The fitted number of dark matter induced neutrinos together with 90% and 99% C.L. sensitivity contours as a function of WIMP mass for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom)...... 107 11.5 The distribution of fitted values of the systematic errors ei, given in units of σ, at the global best fit point from the fit with standard binning (bb¯ annihilation channel, Mχ = 150 GeV, β = 0.0007)...... 108 11.6 Global best fit values of the systematic error parameters, ei, given in units of σ, from the fit with standard binning (part I)...... 109 11.7 Global best fit values of the systematic error parameters, ei, given in units of σ, from the fit with standard binning (part II)...... 110 11.8 Global best fit values of the systematic error parameters, ei, given in units of σ, from the fit with standard binning (part III)...... 111 11.9 The 90% upper limits on dark matter induced muon neutrino flux for bb¯ (blue), τ+τ− (green) and W+W− annihilation channel, compared with expectations from the sensitivity studies (dashed lines)...... 112 11.10The 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section + − for bb¯ (blue) and τ τ (green) annihilation channels, for Mχ < 100 GeV. The visible peaks correspond to resonant capture on the most abundant elements 16O, 24Mg, 28Si, 56Fe, and their isotopes. Positive result claimed by DAMA/LIBRA (Bernabei et al., 2013) is shown for comparison...... 113 11.11The momentum (1st column) and zenith angle (2nd-5th columns) distributions before and after the fit are shown for 19 samples used in the analysis. See detailed description in text...... 114 11.12The fitted number of dark matter induced neutrinos together with 90% and 99% C.L. sensitivity contours calculated with optimized binning, plotted as a function of WIMP mass for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom)..... 115 11.13The 90% C.L. upper limit on the fitted number of dark matter induced neutrinos of all flavors from WIMP annihilation into bb¯ (top), τ+τ− (middle) and W+W− (bottom) channel as a function of the mass of relic particles. The sensitivity expectations for a null WIMP hypothesis (dotted lines) and their ±1σ (green bands) and ±2σ (yellow bands) uncertainty calculated in Sec. 10.4 are shown...... 116 11.14The distribution of fitted values of the systematic errors ei, given in units of σ, at the global best fit point from the fit with optimized bin- + − ning (W W annihilation channel, Mχ = 100 GeV and β = 0.00035)...... 117 xxi

11.15Global best fit values of the systematic error parameters, ei, given in units of σ, from the fit with optimized binning (part I)...... 118 11.16Global best fit values of the systematic error parameters, ei, given in units of σ, from the fit with optimized binning (part II)...... 119 11.17Global best fit values of the systematic error parameters, ei, given in units of σ, from the fit with optimized binning (part III)...... 120 11.18The 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section for bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channels. The visible peaks correspond to resonant capture on the most abundant elements 16O, 24Mg, 28Si, 56Fe, and their isotopes. The results are compared with previous SK result (Desai et al., 2004) (dotted cyan line), as well as positive result claimed by DAMA/LIBRA (Bern- abei et al., 2013)...... 121 11.19The 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section for bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channels. For comparison the results from IceCube (Aartsen et al., 2017a) (dotted line) and ANTARES (Albert et al., 2017b) (dashed line) are shown, as well as positive result claimed by DAMA/LIBRA (Bern- abei et al., 2013)...... 122

12.1 Favored regions obtained by interpreting the observed positron and electron excesses as due to dark matter annihilation in µ+µ−. Green region is favored by PAMELA (at 3σ), red region is favored by the global ﬁt of FERMI, HESS and PAMELA data (at 3σ) (Meade et al., 2010), and blue region is favored by AMS-02 data (at 2σ) (Di Mauro et al., 2016). The 90% C.L. limits from SK data are plotted with solid purple line for "ON-OFF souce" analysis (this thesis), and with solid line for "Global ﬁt" approach (Mijakowski, 2018). NFW halo model is assumed in all cases...... 126 12.2 The upper 90% C.L. limits on the WIMP-nucleon SI scattering cross section for the bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channel from the SK data, compared with the results from direct detection experiments: LUX (Akerib et al., 2017) (orange) and PICO- 60 (Amole et al., 2017) (purple), and possible signal from DAMA (Bern- abei et al., 2013)...... 127

D.1 An example of a measurement αˆ which can give results in unphysical region. If one assumes that α, the quantity one tries to measure, cannot lie in the unphysical region, but can lie anywhere in the physical region (no prior knowledge), then Bayes’ theorem says the new knowledge of distribution of α (given our measurement αˆ ) is given by the shaded function after appropriate normalization...... 150

xxiii

List of Tables

2.1 Table of the SM particles and their SUSY partners. Particles with spin j in the SM have supersymmetric partners with spin |j − 1/2|...... 10

3.1 The best-fit values and 3σ allowed ranges of the neutrino oscillation parameters, derived from a global fit of the current neutrino oscillation data (based on (Capozzi et al., 2016)). The values (values in brackets) correspond to normal (inverted) hierarchy m1 < m2 < m3 2 2 2 2 (m3 < m1 < m2). The definition of ∆m used is ∆m = m3 − (m2 + 2 2 2 2 m1)/2. Thus, ∆m = ∆m31 − ∆m21/2 > 0, if m1 < m2 < m3, and 2 2 2 ∆m = ∆m32 + ∆m21/2 < 0 for m3 < m1 < m2 ...... 23 4.1 The values of threshold energy for e±, µ± and π±...... 32 4.2 Summary of the differences between each of the SK periods...... 36

5.1 Livetime of SK-I, SK-II, SK-III and SK-IV data sets...... 41 5.2 Summary of event subsamples for atmospheric neutrino events at Super-Kamiokande. For each subsample, the standard bins in cosine of the zenith angle and momentum (or visible energy) are specified... 44 5.3 The global best fit values of neutrino oscillation parameters from (Olive, 2014) fixed during the fit. The uncertainties on the parameters values are treated as systematic errors in the fit...... 46

6.1 The parameters of Eq. 6.1 for Moore (Moore et al., 1999), NFW (Navarro, Frenk, and White, 1997) and Kravtsov (Kravtsov et al., 1998) halo pro- ﬁles...... 52

7.1 The results of the optimization of an ON-source region size assuming dark matter annihilation for NFW, Moore and Kravtsov halo proﬁle. The results for decay scenario are the same for all models, and the size of 60◦ is the result of the requirement that ON- and OFF-source regions cannot overlap...... 65

8.1 The number of events observed in ON-source (NON) and OFF-source (NOFF) regions for each considered event classes together with resulting asymmetry (A) and the corresponding statistical errors (σ∆N and σA). The optimal size of the ON-source region is determined separately for each group of event categories...... 68 8.2 The number of neutrino events observed in ON-source and corresponding OFF-source regions for each considered µ-like event class. The optimal size of the ON-source region was determined separately for each class. The last column shows the upper 90% C.L limit on the allowed difference in number of signal events between two considered regions. Results for dark matter annihilation scenario...... 70 xxiv

8.3 The number of neutrino events observed in ON-source and corresponding OFF-source regions for each considered µ-like event class. The optimal size of the ON-source region is constrained to be 60◦ in order to no not have overlapping between ON- and OFF-source regions. The last column shows the upper 90% C.L limit on the allowed difference in number of signal events between two considered regions. Results for dark matter decay scenario...... 73

9.1 The composition of the Earth’s core and mantle (McDonough, 2003).. 76

10.1 Optimized binning in cosine of the zenith angle for FC SubGeV, FC MultiGeV, PC and UP-µ samples...... 95 10.2 The calculated atmospheric neutrino background expectation for ﬁrst four bins in UP-µ samples...... 98

11.1 The best f it points from the ﬁt with standard binning, for the three considered annihilation channels (bb¯, τ+τ− and W+W−). The corresponding χ2 values are compared with atmospheric neutrino background only hypothesis...... 108 11.2 The best f it points from the ﬁt with optimized binning, for the three considered annihilation channels (bb¯, τ+τ− and W+W−). The corresponding χ2 values are compared with atmospheric neutrino background only hypothesis...... 117 xxv

List of Abbreviations

ATIC Advanced Thin Ionization Calorimeter BR Branching Ratio CC Charged Current CDM Cold Dark Matter CDMS Cryogenic Dark Matter Search CL Confidence Level CMB Cosmic Microwave Background CMS Compact Muon Solenoid CoGeNT Coherent Germanium Neutrino Technology CRESST Cryogenic Rare Event Search with Superconducting Thermometers DEC Declination DONUT Direct Observation of the Nu Tau EROS Expérience pour la Recherche d’Objets Sombres FC Fully Contained FERMI-LAT Fermi Large Area Telescope GC Galactic Center GPS Global Positioning System GUT Grand Unification Theory HST Hubble Space Telescope ID Inner Detector LHC Large Hadron Collider LSP Lightest Supersymmetric Particle LUX Large Underground Xenon experiment MACHO Massive Compact Halo Objects MC Monte Carlo MOND Modified Newtonian Dynamics MSSM Minimal Supersymmetric Model MSW effect Mikheyev Smirnov Wolfenstein effect NC Neutral Current NOνANuMI Off-Axis νe Appearance experiment OD Outer Detector OGLE Optical Gravitational Lensing Experiment PAMELA Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics PANDAX Particle and Astrophysical Xenon Detector PC Partially Contained PDF Probability Density Function PMNS matrix Pontecorvo Maki Nakagawa Sakata matrix PMT Photomultiplier Tube PQSM Peccei-Quinn extended Standard Model RA Right Ascension RENO Reactor Experiment for Neutrino Oscillations SD Spin Dependent SI Spin Independent xxvi

SK Super-Kamiokande SM Standard Model SNO Sudbury Neutrino Observatory SUSY Supersymmetry T2K Tokai to Kamioka TPC Time Projection Chamber UPMU Upward-going Muon WIMP Weakly Interacting Massive Particle WMAP Wilkinson Microwave Anisotropy Probe 1

Chapter 1

Introduction

Detecting and elucidating the nature of a dark matter is one of the main goals of the astrophysics and particle physics nowadays. Dark matter neither emits or absorbs light, nor does it interact electromagnetically and cannot be observed directly with telescopes. However, its existence and properties can be inferred from its gravitational effects on visible matter, radiation, and on the large-scale structure of the Universe. According to studies of the cosmic microwave background and on predictions made by the standard cosmological model (ΛCDM), dark matter is expected to account for a large part of the total mass in the Universe (Ade et al., 2016).

Various hypotheses regarding the nature of the dark matter have been proposed. Most of them suggest composition of new type, non-baryonic particles, and their production mechanisms going beyond the Standard Model of particle physics. In this thesis we will focus on widely accepted proposal that dark matter consist of non relativistic, neutral, long-lived Weakly Interacting Massive Particles (WIMPs). It is expected that they are present in the whole Universe forming large invisible structures which encompass visible matter formations (Einasto, 2009). One of the commonly considered example of WIMP is the lightest supersymmetric particle, neutralino χ (Jungman, Kamionkowski, and Griest, 1996)

There are three complementary strategies for dark matter detection consisting of direct, indirect, and collider-based searches. The direct methods are based on the assumption that WIMPs can be elastically scattered on baryonic nuclei. Many experiments attempt to measure the kinetic energy of the resulting nuclear recoils, but current results are inconsistent and controversial (see Chapter2). The other, indirect method, focuses on a search for products of dark matter self-annihilation or decay, such as antimatter, photons or neutrinos. Alternatively, to those methods, there are attempts to produce dark matter particles in the high energy particle accelerators like Large Hadron Collider (LHC).

This thesis aims at the indirect search for dark matter, concentrating on a search for WIMP-induced neutrinos. Neutrinos are expected to be produced in the annihilation of WIMPs taking place in the entire cosmic space - both in the center parts of galaxies as well as in their outer regions. The decay of the dark matter is also discussed in the literature for non-standard WIMP candidates, which may lead to production of neutrinos as well (Feldstein and Fitzpatrick, 2010).

Neutrinos can provide very precise information on their source position while travers- ing unaffected through galactic scales. Moreover, their energy remains unchanged during propagation providing valuable information about energy spectra generated 2 Chapter 1. Introduction in dark matter annihilation/decay processes. Neutrinos originating from the annihilation/decay of the relic particles could be created directly or in subsequent decays of mesons and heavy leptons. The energy spectrum of these neutrinos and branching ratio for their production is model dependent. In the following thesis, various WIMP annihilation/decay scenarios and various models of the anticipated dark matter distribution in the galactic halo are studied.

In the analyses of data from neutrino telescopes, the reconstructed direction of in- coming neutrino is the main selection criterion. The searches focus on the regions where dark matter density is expected to be larger. For instance, it is believed that massive celestial objects like the Sun, the Earth, or the region of the Galactic Center could gravitationally bound dark matter particles. It is expected that non-relativistic WIMPs present in the halo can interact elastically with baryonic matter inside such objects with a cross section of the electroweak scale (< 10−38cm2) and occasionally loose enough energy so they cannot escape the strong gravitational potential. Due to the consecutive interactions, WIMPs would naturally accumulate in the core of heavy celestial objects and effectively annihilate there. Among many annihilation products, only neutrinos could escape dense annihilation region. Their investigation could constrain the WIMP-nucleon scattering cross section based on the relationship between capture and annihilation rate.

This thesis presents the search for dark matter induced neutrinos based on the data collected with the Super-Kamiokande (SK) detector in years 1996-2016. It stars with an overview of the dark matter in Chapter2, including the observational evidence and possible candidates, methodology of dark matter searches and current experimental results. Neutrinos which are one of the possible products of dark matter annihilation or decay are described in Chapter3, which starts with a short historical introduction, then discuss neutrino oscillation formalism and their interactions, and at the end, the natural and artiﬁcial neutrino sources. The atmospheric neutrinos are discussed in more details since they cover wide energy range (from hundreds of MeV up to tens of TeV) where dark matter induced neutrinos are expected. The description of the large water Cherenkov neutrino detector and principle of its operation is the subject of Chapter4.

The author of this thesis joined the Super-K collaboration in November 2013. The main author’s activity in the experiment is related to the search for dark matter induced neutrinos using atmospheric neutrino data. The data and Monte Carlo samples, as well as results of standard atmospheric neutrino oscillation analysis are presented in Chapter5. The two separated analyses were performed during the time of the PhD by the author. The first one covers the search for dark matter induced neutrinos from the Milky Way, and it is described in Chapters6-8. Chapter6 discuss the three different Milky Way halo models investigated in the analysis, dark matter annihilation and decay scenario and resulting expectation for neutrino signal. The simulation of WIMP induced neutrinos, which was developed for the purpose of this analysis is discussed in Chapter7. This chapter contains also the concept of ON- OFF source search. The results including upper limits on the self-annihilation cross- section and lower limits on dark matter lifetime, obtained for dark matter particle masses ranging from 1 GeV to 10 TeV are presented in Chapter8. The second analysis performed by the author covers search for dark matter induced neutrinos from the Earth’s core and is described in Chapters9- 11. Chapter9 discusses WIMP annihilation in the center of the Earth, including the calculations of annihilation rate and Chapter 1. Introduction 3 expected neutrino flux. The analysis technique, signal simulation and performed sensitivity studies for the Super-Kamiokande detector are discussed in Chapter 10. Finally Chapter 11 contains the fit results and constraints on WIMP-nucleon spin independent scattering cross-section. Both of the analysis were performed indepen- dently by the author. The results were officially approved by the Super-Kamiokande Collaboration, presented during international conferences and will be publish. The last Chapter 12 contains summary and outlook.

Chapter 2

Dark Matter

2.1 Observational Evidence

There are many compelling evidence indicating the presence of dark matter in the Universe (Bertone, Hooper, and Silk, 2005). They originate at very different scales: from cosmological ones (through the analysis of angular anisotropies in the cosmic microwave background radiation), down to galactic scales (considering gravitational lensing and the dynamics of galaxies).

2.1.1 Galaxy Clusters The ﬁrst observational clue for the existence of dark matter was by Fritz Zwicky in 1933 (Zwicky, 1933). He noticed that galaxies in the Coma Cluster were rotating faster than expected when only visible matter was considered. Based on these observations, he suggested that the major part of matter is invisible, hence the name dark matter. At ﬁrst, this claim was not fully approved by most scientists. In time, however, more evidence have emerged. Similar observations in the Virgo cluster were made later by Sinclair Smith at the Mount Wilson Observatory in 1936 (Smith, 1936). He had came to the same conclusion as Zwicky.

2.1.2 Galaxy Rotation Curves The distributions of rotational velocities of stars in galaxies are another form of evidence which support the dark matter hypothesis. This observation was ﬁrst made by Vera Rubin’s group in the 70s an 80s of the 20th century (Rubin and Ford Jr, 1970). The rotational velocities of stars were measured at different distances from the galactic center, and did not agree with the theoretical predictions based solely on visible matter contribution. Most notably, the stars in outer region of the galaxy move faster than expected. This lead to the conclusion that there must be additional invisible mass, which creates large structures that extend far beyond the visible edges of galaxies.

In 1991, Begeman, Broels, and Sanders presented an analysis of the rotation curves of ten carefully selected galaxies (Begeman, Broeils, and Sanders, 1991). Using accurate and high resolution 21 cm emission lines from neutral hydrogen, they modeled the mass distribution of the galaxies (see example in Fig. 2.1). The study determined the contribution to the rotation curve from three types of matter (gas, luminous and dark matter), and showed that the dark matter contribution becomes dominant with increasing distance from the galactic centre. 6 Chapter 2. Dark Matter

FIGURE 2.1: An example of a galaxy rotation curve for NGC 6503 dwarf spiral galaxy studied by Begeman, Broels, and Sanders. The solid line is the theoretical rotation curve and the data points with error bars are the measured object speed. The segmented lines show the contributions from gas, luminous matter, and dark matter. Figure is taken from (Begeman, Broeils, and Sanders, 1991)

Dark matter is the widely accepted explanation for this observations, however, numerous alternatives have also been proposed. Most of them postulate a modification to the law of gravity, replacing the laws established by Sir Isaac Newton and Albert Einstein. These theories can be referred to as Modified Newtonian Dynamics (MOND) (Milgrom, 1983). However, even when MOND is taken into consideration there is still not enough luminous matter. Further, the MOND theory is incompatible with general relativity. Later modifications of the theory based on general relativity have been more successful (and it is still an active area of research), but it is un- likely to be successful at resolving the dark matter problem. Detailed studies of the cosmic microwave background, gravitational lensing and Big Bang primodial nucleosynthesis strongly support the dark matter hypothesis and cannot be explain by MOND theories.

2.1.3 Gravitational Lensing Dark mater existence can also be inferred from observations of gravitational lensing. When light emitted from a distant source passes near a massive object, the trajectory of the photons is bent due to the gravitational field, and can cause the distant source image to be magnified or distorted. Based on this effect, the total mass in the gravitational field can be inferred. As one example among many, Fig. 2.2 shows the mass profile of the Bullet Cluster (as measured in August 2006 (Clowe et al., 2006)). Here, the collision between two galaxy clusters caused a separation of dark and baryonic matter. The electromagnetic interactions between passing gas particles caused the baryonic matter to slow down near the impact point, as indicated by the emission of X-rays. Careful measurements show that the hot gas in the Bullet Cluster contains 2.1. Observational Evidence 7 about 7 times more baryonic matter than the stars in the cluster’s galaxies combined. On the other hand, gravitational lensing observations show that the majority of the mass composition of these galaxies simply passed through each other without interacting. Unlike the galactic rotation curves, this evidence for dark matter existence cannot be explained by the modification of laws of gravity.

FIGURE 2.2: Left: Visible light image of the Bullet Cluster from the Magellan Telescope. Right: The Chandra X-ray Observatory image of the Bullet Cluster showing the location of the x-ray emission of the hot gas. In both panels the mass density contours are shown with green lines. Figure taken from (Clowe et al., 2006) .

2.1.4 Cosmic Microwave Background The Cosmic Microwave Background (CMB) is the oldest electromagnetic radiation in the Universe, dating back to the epoch of recombination (approximately 300,000 years after the Big Bang). Measurements of the CMB allow for an independent measurement of the total mass and energy contained in the Universe. Dark matter does not interact directly with electromagnetic radiation, but it does affect the CMB by its gravitational potential (mainly on large scales) and by altering the density and velocity of ordinary matter. Ordinary and dark matter perturbations therefore evolve differently with time, and leave different imprints on the CMB.

The Wilkinson Microwave Anisotropy Probe (WMAP) was the first experiment to develop a full sky map of temperature fluctuations in the CMB, and its measurements played the key role in establishing of the current Standard Model of Cos- mology, namely the ΛCDM model. ΛCDM implies a flat Universe dominated by dark energy, supplemented by dark and baryonic matter (Larson et al., 2011). The latest results obtained by Planck mission team implies that total mass - energy of the Universe contains lest then 5% ordinary matter, 26% dark matter and 69% dark energy (Ade et al., 2016).

2.1.5 Big Bang Nucleosynthesis Big Bang nucleosynthesis (also known as primordial nucleosynthesis) makes predictions on the amount of helium, hydrogen, and lithium contained in the Universe, and further, the required baryonic matter densities needed to produce these abundances. Big Bang nucleosynthesis predictions match the observed abundances only when the baryonic matter accounts for 4 - 5% of the critical density of the Uni- verse (Schramm and Turner, 1998). This is strong indication for non - baryonic dark matter. 8 Chapter 2. Dark Matter

2.1.6 Structure Formation Dark matter seems to be essential in evolution of the Universe. Its presence is required in cosmological models of the Big Bang. Signiﬁcant amount of non-baryonic, cold dark matter (CDM) is necessary to explain the large-scale structure formation of the Universe (Tegmark, 2004). If the majority of the non-baryonic dark matter consists of the hot dark matter (HDM) - particles that were relativistic at the time when structure formed, the small sub-structures in the large scale galaxy are expected to be ﬂattened, which is inconsistent with observations. If there were only ordinary matter in the Universe, there would not have been enough time for density perturbations to grow into the galaxies and clusters that we see today.

2.2 Dark Matter Candidates

Despite the evidence for the existence of dark matter, its nature is still unknown. At ﬁrst, non-luminous heavy baryonic objects, such as black holes, neutron stars and brown dwarfs (collectively named as Massive Compact Halo Objects, or MA- CHOs), were considered a possible candidate for dark matter. However, astronomical surveys for gravitational microlensing, including the MACHO (Alcock et al., 2001), EROS (Tisserand et al., 2007) and OGLE (Udalski et al., 1992) projects, along with the Hubble Space Telescope (HST) searches for ultra-faint stars, have not found enough of MACHOs, and indicated that they could constitute only a very small portion of dark matter (Freese, Fields, and Graff, 2000). Since it has been proved that neutrinos have mass and are abundant in the Universe, they were also considered a good candidate. However, neutrinos can only form a small fraction of the dark matter, due to limits from a large-scale structure formation models and high-redshift galaxies observations (Hannestad et al., 2010).

One can distinguish two possible creation mechanisms among the dark matter candidates, thermal creation in the early Universe and non-thermal creation in a phase transition. Thermal and non-thermal relics have a different relic abundance and properties such as mass and couplings, so the distinction is especially important for dark matter detection efforts. Here we focus on thermally created dark matter. In the early Universe, thermal equilibrium was obtained. In case of particles (which are created thermally), their number density was roughly equal to the number density of photons. Then, as the Universe cooled, the number of dark matter particles would decrease until the temperature of Universe ﬁnally dropped below the dark matter particle mass. At this point, their creation would require being on the tail of the thermal distribution, the number density would drop exponentially, and the probability of pair annihilation would become small. The number density would "freeze-out" and the Universe would be left with a substantial number of dark matter −26 3 −1 particles today, approximately ΩWIMP ≈ 10 cm s / < σAv >, where < σAv > is the thermally averaged cross section for dark matter annihilation into Standard Model (SM) particles (Steigman, Dasgupta, and Beacom, 2012).

The dark matter must be stable or long-lived, since it was produced after the Big Bang and is still present today. Dark matter is also required to be dynamically cold (not relativistic) so that it can clump and form gravitationally bound structures. The SM does not provide a viable non-baryonic candidates for dark matter particles and the existence of dark matter is one of the most compelling evidence for physics beyond the Standard Model. The best motivated candidates which would be able to 2.2. Dark Matter Candidates 9 fulﬁll the above criteria are those which arise in high-energy completions of the SM, which solve also other fundamental problems of particle physics and cosmology. Examples of the considered particles are: • Neutralinos or Gravitinos in supersymmetric extensions of the SM, which solve also the hierarchy problem. • The lightest sterile neutrino in the νMSM which explains also neutrino masses and mixing as well as the baryon asymmetry of the Universe via the introduction of right - handed sterile neutrinos featuring a Majorana mass. • Axion in the Peccei - Quinn extended SM (PQSM) which solves also the strong CP problem. In this thesis, we will focus on neutral particles, which would be able to reproduce the required dark matter density in the Universe derived from CMB measurements, additionally having a mass of the order of the electroweak scale and belonging to a group referred to as Weakly Interacting Massive Particles (WIMPs). Presently WIMPs, with mass in the range from several GeV to few TeV, are considered the best motivated dark matter candidate supported by supersymmetry theories (SUSY), which provide one of the most popular WIMP candidate: the lightest supersymmetric particle (LSP), namely the neutralino χ (Jungman, Kamionkowski, and Griest, 1996). In this thesis, we will consider neutralino as a dark matter candidate (or any other with similar phenomenology). More details about neutralinos can be found in the following Sec. 2.2.1.

2.2.1 Neutralino as a Dark Matter Candidate Supersymmetry (SUSY) was originally introduced to solve some of the outstanding problems of the Standard Model (SM), such as the hierarchy problem, gauge coupling uniﬁcation, and radiative electroweak symmetry breaking. SUSY also predicts new ideas for matter-antimatter asymmetry of the Universe and prospects for the GUT/Planck scale where all forces uniﬁed.

In SUSY, each SM particle has a supersymmetric partner with spin |j − 1/2|. The minimal extension of the SM of particle physics is described by the Minimal Super- symmetric Standard Model (MSSM). The MSSM contains the smallest possible ﬁeld content necessary to give rise to all the ﬁelds of the SM. In Tab. 2.1, the SM particles and corresponding SUSY partners are shown.

A new symmetry, R-parity is introduced for particles in the SUSY theory, which is deﬁned as: R = (−1)3B+L+2j, (2.1) where B is the baryon number, L is the lepton number, and j is particle spin. This implies that R = +1 for SM particles and R = −1 for their supersymmetric partners. Therefore, if R-parity is conserved, supersymmetric particles can only be created or annihilated in pairs, in reactions of ordinary particles. It also means that a single supersymmetric particle can only decay into ﬁnal states containing an odd number of supersymmetric particles. In particular, this makes the lightest supersymmetric particle stable (LSP), since there is no kinematically allowed state with negative R- parity which it can decay to. 10 Chapter 2. Dark Matter

In a supersymmetric extensions of the SM, the superpartners of the Z boson (Zino), the photon (photino) and two neutral Higgs bosons (higgsino), have the same quantum numbers, so they can mix to form four eigenstates of the mass operator called 0 0 0 0 0 neutralinos (χ˜1, χ˜2, χ˜3, χ˜4). In many models, the lightest of the four neutralinos χ˜1 (from now on simply noted as χ) turns out to be the LSP. As a heavy, stable particle, which is electrically neutral and thus neither absorbs nor emits light, the LSP is an excellent candidate for WIMP.

TABLE 2.1: Table of the SM particles and their SUSY partners. Par- ticles with spin j in the SM have supersymmetric partners with spin |j − 1/2|.

Particle Spin Name Superpartner Spin Name d, c, b, u, s, t 1/2 quarks d˜, c˜, b˜, u˜, s˜, t˜ 0 squarks e, µ, τ 1/2 leptons e˜, µ˜, τ˜ 1/2 sleptons νe, νµ, ντ 1/2 neutrinos ν¯ 0 sneutrinos G 2 graviton G˜ 3/2 gravitino W± 1 weak boson W˜ ± 1/2 Wino Z 1 weak boson Z˜ 1/2 Zino γ 1 photon γ˜ 1/2 photino g 1 gluon g˜ 1/2 gluino h 0 higgs boson h˜ 1/2 higgsino

2.3 Dark Matter Detection

Weakly Interacting Massive Particle as a dark matter candidate is a well posed scien- tiﬁc hypothesis and a subject of experimental veriﬁcation. Numerous experiments attempt to detect dark matter using different methods and a wide variety of techniques. The experiments can be divided into three categories: direct detection experiments, which search for the scattering of dark matter particles off atomic nuclei; indirect detection, which search for the products of WIMPs annihilations; and an alternative approach to the detection of WIMPs is their production in the laboratory in very high energy collisions (see Fig. 2.3).

2.3.1 Direct Detection It is expected that dark matter local density in the Galaxy at the Solar system position is 0.3 GeV/cm3 (Yüksel et al., 2007). That implies that many of dark matter particles must pass through every square centimeter of the Earth in each second. In direct detection method, one tries to detect the nuclear recoils originated from the rare interactions of the dark matter particles with the target nuclei (see Fig 2.6).

Two kind of interactions can be expected from WIMP-nucleon scattering. The ﬁrst interaction is a spin dependent (SD) axial-vector interaction, where WIMPs couple to the spin content of a nucleon. The cross-section for SD scattering ∝ J(J + 1), where J is spin of the nucleus. The SD scattering can be efﬁcient when a nucleus has large number of unpaired protons or neutrons. The second interaction is a spin independent (SI) scalar interaction, where the WIMP couples to the nucleus as a whole. The 2.3. Dark Matter Detection 11

FIGURE 2.3: Simpliﬁed diagram showing the three different dark matter detection channels. The direct detection experiments (→) look for the recoil introduced by a WIMP (χ) on a SM particle in the detector. Indirect detection experiments (↓) look for the SM annihilation products of WIMPs. Collider experiments (↑) look for the creation of WIMPs in Standard Model particle collisions. cross section for SI scattering ∝ A2, where A is an atomic mass number. Therefore, the cross section increases dramatically with the mass of the target nuclei, and typically dominates over spin-dependent scattering in current experiments which use heavy atoms as targets. The possible tree level Feynman diagrams for SD and SI scattering are shown in Fig. 2.4 and 2.5 respectively.

FIGURE 2.4: Feynman diagrams contributing to the spin-dependent elastic scattering of neutralinos from quarks.

The expected energy of nuclear recoils is in the range from several to hundreds of keV depending on the mass of dark matter particle and type of the detector. The most effective scattering can be achieved if target nucleus mass is similar to WIMP mass. In this energy range the main contribution to the experimental background comes from interactions of α-particles, neutrons, electrons, and photons on the detector target material. These backgrounds originate from decays of radioactive isotopes present in the materials surrounding the detectors, in airborne contaminants, or from within the detectors themselves. Therefore, appropriate shielding with passive and active materials as well as veto detectors around the experiment are very important. 12 Chapter 2. Dark Matter

FIGURE 2.5: Feynman diagrams contributing to the scalar elastic- scattering amplitude of a neutralino from quarks.

FIGURE 2.6: A sketch idea of dark matter direct detection.

Depending on the type of detector, energy of nuclear recoils can be transferred to scintillation, ionization, phonon signals, or a combination thereof. Some experiments (e.g. XENON (Aprile et al., 2005), LUX (Akerib et al., 2013), PANDA-X (Cao et al., 2014)) are using liquid xenon (LXe), which has high radiopurity and high density, and has proven to be very good target material. In dual phase xenon time projection chambers (TPC), scintillation light from nuclear recoil and the ionization from electrons accelerated in the electric field can be detected. The ratio of ionization to scintillation signal allows for a very effective separation of electron recoil, from nuclear recoil events. This is very important sice the dominant background for dark matter searches consists of electrons. The flux of neutrons and α-particles is approximately six orders of magnitude lower than flux of electrons. However, interactions of neutrons and α-particles are expected to yield similar signatures as a WIMP-induced signal. The only difference is that any WIMPs that interact in the detector will have negligible chance of repeated interaction. Neutrons and α-particles, on the other hand, have a reasonably large chance of multiple collisions within the target volume, with the frequency which can be accurately predicted. Therefore, only the multiple scattering of neutrons and α-particles in the detector allows to distinguish them from interactions of WIMPs, which are not expected to scatter more than once. 2.3. Dark Matter Detection 13

The technology used in ton-scale detectors yields very strong limits on the SI WIMP- nucleon scattering cross-section, reaching below 10−46cm2 for WIMP masses of 30- 40 GeV (see Fig. 2.7). Next generation of these experiments plan to increase target volume to multi-ton scale and is expected to reach solar neutrino ﬂoor (the region of WIMP parameter space where coherent neutrino-nucleus scattering constitute a dominant background, see (Billard, Figueroa-Feliciano, and Strigari, 2014) for more details). 4 ] -1 0 1 2 3 4 2 10-37 60 Neutron 40 WIMP search 10-38 20 Counts

0 -39 1 10

0.5 10-40

NN score PICO-60 C F 0 3 8 -1 0 1 2 3 4 10-41 log(AP) SD WIMP-proton cross section [cm 101 102 103 WIMP mass [GeV/c2] FIG. 2. Top: AP distributions for AmBe and 252Cf neutron calibration data (black) and WIMP search data (red) at FIG. 3. The 90% C.L. limit on the SD WIMP-proton cross 3.3 keV threshold. Bottom: AP and NN score for the same FsectionIGURE from2.8: PICO-60 Latest C3 limitsF8 plotted on inthe thick SD blue, WIMP- along dataset. The acceptance region for nuclear recoil candidates, with limits from PICO-60 CF3I (thick red) [10], PICO-2L defined before WIMP search acoustic data unmasking using proton scattering cross-section as a func- (thick purple) [9], PICASSO (green band) [14], SIMPLE (or- neutron calibration data, are displayed with dashed lines and FIGURE 2.7: Latest limits on SI WIMP- tionange) [15], of WIMPPandaX-II mass: (cyan) [35], results IceCube from (dashed PICO- and dot- reveal no candidate events in the WIMP search data. Alphas 222 60ted pink) C F [36],(Amole and SuperK et (dashed al., 2017 and dotted) (thick black) blue), [37, 38]. fromnucleon the Rn scattering decay chain can cross-section be identified by their as atime func- sig- 3 8 The indirect limits from IceCube and SuperK assume anni- nature and populate the two peaks in the WIMP search data along with limits from PICO-60 CF I (Amole tion of WIMP mass: results from combined hilation toτ leptons (dashed) andb quarks (dotted).3 The at high AP. Higher energy alphas from 214Po are producing etpurple al., region2016a represents) (thick parameter red), PICO-2L space of the (Amole constrained et largerPandaX-II acoustic signals. Run 9 and Run 10 data (Cui et al.,minimal 2016b supersymmetric) (thick purple), model of[39]. PICASSO Additional (Behnke limits, not al., 2017) (red) (the green band represents shown for clarity, are set by LUX [40] and XENON100 [41] the ±1σ sensitivity band), overlaid with re- et(comparable al., 2017 to PandaX-II)) (green and band), by ANTARES SIMPLE [42, 43] (Fe- (com- tosults be 0.25 from± 0.09 PandaX-II (0.96± 0.34) 2016 single(multiple)-bubble (Tan et al., 2016) lizardoparable to et IceCube). al., 2014) (orange), PandaX-II (Fu 133 events.(blue), PICO-60 LUX was 2017 exposed (Akerib to a et 1 mCi al., 2017Ba) source (ma- et al., 2017) (cyan), IceCube (Aartsen et al., bothgenta), before and and after XENON1T the WIMP 2017 search (Aprile data, which, et al., 2017b) (dashed and dotted pink), and Su- compared against a Geant4 [21] Monte Carlo simulation, perKWIMP (Choi mass, are et shown al., 2015 in Fig.) (dashed 3 and 4. These and dot-limits, gives a measured nucleation2017) e(black).fficiency for electron recoil corresponding to an upper limit on the spin-dependent events above 3.3 keV of (1.80± 0.38)×10 −10. Combin- tedWIMP-proton black). cross Indirect section limits of 3.4× from10 −41 IceCubecm2 for a ing this with a Monte Carlo simulation of the external and30 GeV SuperK c−2 WIMP, assume are currently annihilation the world-leading to τ leptons con- gammaflux from [16, 22], we predict 0.026± 0.007 events (dashed)straints in the and WIMP-protonb quarks spin-dependent (dotted). The sector pur- and due to electron recoils in the WIMP search exposure. The pleindicate region an improved represents sensitivity parameter to the dark space matter of signal the background from coherent scattering of 8B solar neutri- of a factor of 17, compared to previously reported PICO nos is calculated to be 0.055± 0.007 events. constrainedresults. minimal supersymmetric model We use the same shapes of the nucleation efficiency ofA (Roszkowski, comparison of ourAustri, proton-only and Trotta, SD limits 2007). with curves forfluorine and carbon nuclear recoils as found in neutron-only SD limits set by other dark matter search Ref. [8], rescaled upwards in recoil energy to account for experiments is achieved by setting constraints on the the 2% difference in thermodynamic threshold. We adopt effective spin-dependent WIMP-neutron and WIMP- theBubble standard chamber halo parametrization experiments [23], with (e.g. the PICO follow- (Amoleproton couplings et al., 2017a n and))a provep that are to calculated be extremely according −2 −3 ingcompetitive parameters:ρ D for=0.3 SD GeV searches. c cm ,v esc For= 544 example, km/s, to PICO-60 the method uses proposed a bubble in Ref. [29]. chamber The expectation filled 19 vEarth = 232 km/s, andv o = 220 km/s. We use the effec- values for the proton and neutron spins for the F nu- with 52 kg of C3F8 located in the SNOLAB underground laboratory, sets the most tivefield theory treatment and nuclear form factors de- cleus are taken from Ref. [24]. The allowed region in − −2 scribedstringent in Refs. direct-detection [24–27] to determine constraint sensitivity to to both datethe ona then a WIMP-protonp plane is shown for SD a scattering 50 GeV c WIMP cross- in spin-dependentsection reaching and spin-independent 10−41cm2 for dark WIMP matter inter- massesFig. of 5. 30 We GeVfind that (see PICO-60 Fig. 2.8 C).3F8 improves the con- actions. For the SI case, we use theM response of Table straints ona n anda p, in complementarity with other 1 in Ref. [24], and for SD interactions, we use the sum dark matter search experiments that are more sensitive ofMany theΣ ′ experimentsandΣ ′′ terms from use the cryogenic same table. detectors To im- operatedto the WIMP-neutron at very low coupling. temperatures, which plementdetect these heat interactions produced and inform electron factors, we and use the nuclearThe recoils. LHC has Cryogenic significant sensitivity detectors to dark (such matter, as publicly available dmdd code package [27, 28]. The calcu- but to interpret LHC searches, one must assume a spe- latedCRESST Poisson (Angloher upper limits at et the al., 90% 2016 C.L.), for CDMS the spin- (Agnesecific model et al., to generate 2015), the EDELWEISS signal that is then (Hehn looked et for dependental., 2016 WIMP-proton)) have proven and spin-independent to be very effective WIMP- inin the exploring data. Despite the light this subtlety, dark matter the convention region has nucleonwith elasticWIMP scattering mass

FIGURE 2.9: Experimental 90 % C.L. exclusion limit on the SI WIMP- nucleon scattering cross section as a function of WIMP mass for the combined ﬁt from EDELWEISS-II detectors (Hehn et al., 2016) (solid red). The green and yellow band represent the 1 and 2 σ conﬁdence band of the expected median sensitivity (dashed black). Result of the EDELWEISS BDT based analysis (dashed red) is shown for comparison. Colored regions show possible signals from CDMS-II (Si) (Agnese et al., 2013) (blue), DAMA (Bernabei et al., 2013) (brown), CRESST-II (Angloher et al., 2012) (pink) and CoGeNT (Aalseth et al., 2014) (orange). Other existing exclusion limits are from EDELWEISS- II (Armengaud et al., 2012) (small red dashes), CoGeNT (Aalseth et al., 2013) (orange), CRESST (Angloher et al., 2016) (pink), Super- CDMS (Agnese et al., 2014) (purple), XENON100 (Aprile et al., 2016) (black), CDMSlite (Agnese et al., 2015) (dashed violet) and LUX (Ak- erib et al., 2016) (green).

The event rate of a dark matter signal is expected to modulate annually due to relative motion of the Earth around the Sun. The expected event rate R is proportional to the density of dark matter particles ρ, cross section for WIMP-nucleus elastic scattering σ and the speed of the Earth with respect to the dark matter halo V:

R ∝ ρ · σ · V. (2.2)

The speed of the Earth with respect to the halo changes throughout a year due to combination of the Earth’s motion around the Sun and the Sun’s motion around the Galactic Center. Therefore, with a high statistics of collected dark matter particle interactions one would expect to see annual modulation of an observed event rate. Such an effect was observed by DAMA/LIBRA experiment (Bernabei et al., 2013), which has been in operation since 1996. DAMA/LIBRA claims the observation of modulation at 9.3 σ obtained with a total exposure of 1.33 ton×year collected over 14 cycles. The observed annual modulation is inconsistent with indirect and other direct detection bounds, but they were obtained using different target materials. The long standing DAMA anomaly is expected to be conﬁrmed or resolved in the near future with multiple new experiments (e.g. SaBRE (Tomei, 2017), PICO- LON (Fushimi et al., 2017), ANAIS (Amaré et al., 2016), COSINE (Adhikari et al., 2.3. Dark Matter Detection 15

2017)). They are coming on-line now and rely on the identical detector technology as DAMA/LIBRA.

2.3.2 Indirect Detection Indirect detection experiments search for WIMP annihilation or decay products in the Universe by investigating cosmic rays. The annihilation rate is proportional to the velocity-averaged dark matter self-annihilation cross-section (< σAv >) and nu- 2 merical density squared (nχ):

ρ2 < > × 2 =< > × χ ΓA ∝ σAv nχ σAv 2 , (2.3) Mχ where ρχ corresponds to dark matter particles density in the local halo. Most of the SUSY models assume that dark matter particles would be their own antiparticles. The particle-antiparticle pairs of WIMPs could annihilate, and depending on their mass, produce leptons, quarks, gauge or Higgs bosons (see Fig. 2.10).

FIGURE 2.10: An illustration of a possible products of supersymmetric dark matter self-annihilation.

After subsequent decays of the primal annihilation products, various Standard Model particles can be created. The indirect detection searches focus on channels and energy ranges where it is possible to reasonably disentangle dark matter induced signals from the ordinary astrophysical background. There are few viable possibilities such as gamma rays, neutrinos, and antimatter. Additionally, if the WIMPs are un- stable, they could also decay into SM particles. In this case, the decay rate would be proportional to dark matter density (see Chapter6). 16 Chapter 2. Dark Matter

Antimatter Since antimatter is much less abundant in the Universe than the ordinary matter, its excess in the cosmic rays may suggest that it originated in dark mater annihilation or decay. The searches are conducted using detectors placed on the Earth’s orbit on satellites (FERMI-LAT (Atwood et al., 2009), PAMELA (Picozza et al., 2007) and AMS-02 (Aguilar et al., 2016)), or on balloons (HEAT (Barwick et al., 1997), ATIC (Chang et al., 2008) and GAPS (Ong et al., 2017)). High altitudes are required to reduce background induced by primary cosmic ray interactions in the atmosphere. Data from the PAMELA satellite showed an excess of positrons from 10 GeV up to 100 GeV as compared to standard prediction of positron production in cosmic rays ﬂuxes. The result was compatible with previous results from HEAT and AMS-01 experiments, and later was conﬁrmed in independent measurements by FERMI-LAT and AMS-02 experiments. The most recent results from AMS-02 experiment are shown in Fig. 2.11.

FIGURE 2.11: The electron ﬂux and the positron ﬂux measured by AMS-02 experiment (Aguilar et al., 2016).

The excess could originate from self-annihilating or decaying dark matter and has been extensively discussed in the literature, e.g. (Di Mauro et al., 2016). However, this interpretation is in tension with indirect bounds based on gamma-rays and neutrinos. On the other hand, astrophysical sources, such as pulsars could explain the observed spectra. Recent measurement of nerby pulsars by HAWC and Milagro indicate that these objects generate signiﬁcant ﬂux of high-energy positrons that is similar in spectrum and magnitude to the positron fraction measured by PAMELA and AMS-02 (Hooper et al., 2017).

Gamma Rays Since some hints of the dark matter annihilations in the positron spectrum have been observed, it is very important to consider the associated signals in the gamma ray fluxes. Photons can be produced in many WIMP annihilation or decay channels. Moreover, photons propagate in space without significant deflection, so they can 2.3. Dark Matter Detection 17 provide good angular information of the source position, and their observed energy spectra also could be a good estimate of primary WIMP annihilation/decay spectra.

FERMI-LAT (Albert et al., 2017c) experiment attempts to search for gamma-ray lines from the sky, looking at satellite galaxies, at the isotropic diffuse background, at clusters of galaxies, and at diffuse gamma rays from wide regions of the galactic halo. In addition to satellite born experiments, such as FERMI-LAT, the gamma rays can be measured in Cherenkov telescopes such as MAGIC (Doro, 2017), HESS (Oakes et al., 2017) or VERITAS (Zitzer, 2017) and HAWC (Yapici, 2017). HESS performed a large number of studies which are relevant for dark matter searches, such as observations towards the Galactic Center, the Galactic Ridge, the Galactic Center Halo, a couple of globular clusters, a number of dwarf galaxies (Sagittarius, Carina and Sculptor and Canis Major), and the Fornax galaxy cluster. HESS has also looked at possible signals from spikes of dark matter accumulated around Intermediate Mass Black Holes. The VERITAS telescope has investigated a few dwarf spheroidal galaxies and the Coma galaxy cluster. The MAGIC telescope has investigated a few dwarf galaxies and the Perseus galaxy cluster. HAWC performed a search for GeV-TeV photons resulting from dark matter annihilation or decay considering dwarf spheroidal galaxies (dSphs), the M31 galaxy and the Virgo cluster as sources, with 507 days of data collected in the full detector conﬁguration. No statistically signiﬁcant excess from these sources has been observed.

Neutrinos Neutrinos could be one of the direct products of dark matter annihilation/decay or be produced in subsequent decays of baryons, bosons and leptons. Neutrinos provide very good information about the source position and they can provide in- sight into areas where the dark matter induced signal to astrophysical background ratio can be maximized. Neutrino telescopes such as IceCube (Aartsen et al., 2017d), ANTARES (Ageron et al., 2011), and Super-Kamiokande (Fukuda et al., 2003) are searching for signals originating from the center of the Milky Way, the Sun or the Earth. The latest results of searches for dark matter induced neutrinos from the center of our Galaxy are presented in Fig. 2.12.

Some fraction of WIMPs passing through the Sun or the Earth may scatter off proton or nuclei and loose energy. Due to these processes a large population of WIMPs may accumulate at the center of these celestial bodies, increasing the chance that two of them will collide and annihilate. As the Sun is a large hydrogen target, the high sensitivity to the spin-dependent dark matter scattering can be achieved. Searches for dark matter induced neutrinos from the Sun have been performed by various neutrino detectors. The latest ones are presented in Fig. 2.13. No positive signal in neutrinos has been found so far.

This thesis presents the search for dark matter induced neutrinos from the Galac- tic Center and the Earth’s core using the data from the Super-Kamiokande detector. The results of two analysis performed by the author are describes in Chapters6- 11. 18 Chapter 2. Dark Matter

FIGURE 2.12: The upper 90% C.L. limits from neutrino experiments on dark matter self-annihilation cross section for the bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) annihilation channels. The limits from Super-Kamionkande (Frankiewicz, 2017a) are plotted with solid lines, IceCube (Aartsen et al., 2017c) with dotted lines and ANTARES (Albert et al., 2017a) with dashed lines.

FIGURE 2.13: The upper 90% C.L. limits on the WIMP-proton SD cross section (left) and WIMP-nucleon SI cross section (right) for the bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channels. The limits from Super-Kamionkande (Choi et al., 2015) are plotted with solid line, IceCube (Aartsen et al., 2017b) with dotted lines. For the comparison the strongest limits from direct detection experiments: LUX (Akerib et al., 2017) (orange) and PICO-60 (Amole et al., 2017) (purple) are plotted, and possible signal from DAMA (Bernabei et al., 2013). 2.3. Dark Matter Detection 19

2.3.3 Production in Accelerators An alternative approach for the detection of dark matter particles is to produce them in a laboratory. The searches for the supersymmertic WIMP candidates at the accelerators aim at recreating the conditions in the early Universe and production of the relic particles. The evidence of dark matter might present itself in observations of processes that deviate from the SM predictions.

The strategy for WIMP searches in hadron colliders is to use mono-jet, mono-photon and mono-lepton signatures with energy or momentum imbalance in the ﬁnal state. The missing reconstructed energy or momentum could be an indication of a production of heavy neutral stable particles in a collision. The WIMP-nucleon scattering cross section can be deduced by these searches. However, it is a model dependent approach relying on effective ﬁeld theories and assumptions on masses of mediating particles.

The experiments at Large Hadron Collider (LHC) has not found any signs of the production of dark matter particles so far. The results of several analysis of the CMS experiment data are shown in Fig 2.14 for SI and in Fig. 2.15 for SD interactions, and compared with some of the previously discussed results from direct and indirect dark matter searches.

FIGURE 2.14: 90% C.L. upper limits on SI FIGURE 2.15: 90% C.L. upper limits on SD WIMP-nucleon scattering cross-section from WIMP-nucleon scattering cross-section from the CMS experiment (Lowette, 2016), for the the CMS experiment (Lowette, 2016), for the mono-jet (magenta), mono-photon (red and mono-jet (magenta), mono-photon (red and yellow), and mono-lepton (grey) searches, as yellow), and mono-lepton (gray) searches, as a function of the dark matter mass. a function of the dark matter mass. The limits from Super-Kamionkande come from So- lar WIMP search analysis (Choi et al., 2015).

Chapter 3

Neutrinos

3.1 History

The existence of neutrinos was postulated by Wolfgang Pauli in 1930, in an attempt to explain the observed continuous energy spectrum of β particles emitted in nuclear decays (Brown, 1978). He considered that the new particle was emitted from the nucleus together with an electron in the process of beta decay in order to conserve energy, momentum, and angular momentum. The predicted particle was initially named "neutron", but in 1932, a much more massive nuclear particle discovered by James Chadwick claimed the name. The name "neutrino" (little neutral one in Italian) was introduced by Enrico Fermi, who used it during the July 1932 Solvay Conference, hosted in Paris.

Experimental conﬁrmation of the neutrino existence was made by Frederick Reines and Clyde Cowan in 1956 (Cowan Jr et al., 1991). In their experiment, antineutrinos created in a nuclear reactor via beta decays, reacted with protons to produce neutrons and positrons: + 0 + ν¯e + p → n + e . The positron quickly annihilates with an encountered electron, resulting in two back - to - back 511 keV gamma rays. The neutron can be detected by its capture on an appropriate nucleus, releasing another gamma ray. The coincidence of both events: positron annihilation and neutron capture, gives a unique signature of an antineutrino interaction.

In 1962, Leon M. Lederman, Melvin Schwartz, and Jack Steinberger showed that more than one type of neutrino exists by making the ﬁrst measurement of a muon neutrino (Danby et al., 1962). After the discovery of a third type of lepton, the tau, in 1975 at the Stanford Linear Accelerator Center, the existence of associated neutrino (the tau neutrino) was expected. The ﬁrst detection of tau neutrino interactions was announced in the summer of 2000 by the DONUT collaboration at Fermilab (Ko- dama et al., 2001).

The first measurement of the electron neutrino flux from the core of the Sun was made in the 1960s by the Homestake Experiment. The measured value of the flux was between one third and one half the number predicted by the Standard Solar Model (Cleveland et al., 1998). Many subsequent radiochemical and water Cherenkov detectors confirmed the deficit, including the Kamioka Observatory and Sudbury Neutrino Observatory (SNO). This discrepancy, known as the solar neutrino problem, remained unresolved for thirty years. The problem was resolved with an improved understanding of the properties of neutrinos. It was realized that neutrinos 22 Chapter 3. Neutrinos have a non-zero mass and that the masses of the individual states are different. Al- lowing for the interacting flavor states to be different than the mass states would manifest itself as an oscillation, where a certain flavor of neutrino produced at some location will have a quantifiable probability of being measured as a different flavor some distance away. This means that some portion of the electron neutrinos created in the Sun would then oscillate into tau and muon flavor neutrinos by the time they arrive at the Earth. This hypothesis has been investigated by a new series of experiments (Shirai, 2005), thereby opening a new major field of research that still continues.

3.2 Neutrino Oscillations

A practical method for investigating neutrino oscillations was first suggested by Bruno Pontecorvo in 1957, who proposed that neutrino - antineutrino transitions may occur in analogy with neutral kaon mixing. Although such matter - antimatter oscillation has not been observed, this idea formed the conceptual foundation for the quantitative theory of neutrino flavor oscillation, which was first developed by Maki, Nakagawa, and Sakata in 1962 (Maki, Nakagawa, and Sakata, 1962) and further expanded on by Pontecorvo in 1967 (Pontecorvo, 1968). Experimental detection and confirmation of the phenomenon of neutrino oscillation led to two Nobel prizes. First, in 2002, Nobel Prize was divided, and one half was awarded jointly to Raymond Davis Jr. and Masatoshi Koshiba for pioneering contributions to astrophysics, in particular for the detection of cosmic neutrinos. Later, in 2015, Nobel Prize has been awarded to Takaaki Kajita (SK) and Arthur B. McDonald (SNO), for discovering that neutrinos can change from one type to another, evidence that they must have mass.

3.2.1 In Vacuum Neutrino oscillation is the quantum mechanical effect which arises from the non- equivalence of neutrino mass states and the eigenstates of the weak interaction (the ﬂavor states). The ﬂavor state |ναi (α = e, µ, τ) can be described as a linear combination of different mass eigenstates |νii with masses (i = 1, 2, 3):

3 ∗ |ναi = ∑ Uαi|νii. (3.1) i=1

U is 3x3 rotation matrix between these states, called Pontecorvo-Maki-Nakagawa- Sakata (PMNS) matrix:

   −iδCP    1 0 0 c13 0 s13e c12 s12 0 U = 0 c23 s23  0 1 0  −s12 c12 0 , (3.2) −iδCP 0 −s23 c23 −s13e 0 c13 0 0 1 where cij = cosθij, sij = sinθij. The propagation of these states according to their vacuum Hamiltonians leads to the standard oscillation formula for relativistic neutrinos in vacuum:

∗ ∗ 2 ∗ ∗ P(να → νβ) = δαβ − 4 ∑ <(UαiUβiUαjUβj)sin ∆ij ± 2 ∑ =(UαiUβiUαjUβj)sin2∆ij, i j i j (3.3) 3.2. Neutrino Oscillations 23 where 2 2 1.27∆mij[eV ]L[km] ∆ = . ij E[GeV] The sign before second summation is positive for neutrinos and negative for anti - neutrinos.

Neutrino oscillations in vacuum are fully characterized by 6 parameters: three mix- 2 2 ing angles θ13, θ12, θ23, two mass splittings ∆m21, ∆m31, and one CP - violating phase δCP. Majority of neutrino mixing parameters have been experimentally determined by reactor, atmospheric, solar, and long-baseline neutrino experiments. This includes parameters such as magnitude of the two mass splittings, the ordering of the mass states with the smallest splitting, and the values of the mixing angles. In particular, recent measurements by reactor experiments (Daya Bay, RENO, and Dou- ble Chooz) (Olive, 2014) and T2K (Abe et al., 2014) have established that the mixing angle θ13 is small but non - zero. The values of these parameters were derived from a global ﬁt of the current neutrino oscillation data (Capozzi et al., 2016), and are listed in Tab. 3.1.

2 The two remain unknown parameters in the PMNS formalism are the sign of ∆m23, which relates to the ordering of the mass states with the largest splitting (commonly referred to as the neutrino mass hierarchy) and CP-violating phase δCP. If δCP is neither 0 nor π then the neutrinos and antineutrinos have a different oscillation probabilities. The precise value of δCP is still unknown, but the T2K (Abe et al., 2017b) and NOvA (Adamson et al., 2017) long-baseline experiments are beginning to constrain it. It is known that the muon and tau neutrino mixing is nearly maximal, i.e. θ23 is near π/4, however, it is not known if θ23 takes exactly that value, or is slightly larger or slightly smaller.

TABLE 3.1: The best-fit values and 3σ allowed ranges of the neutrino oscillation parameters, derived from a global fit of the current neutrino oscillation data (based on (Capozzi et al., 2016)). The values (values in brackets) correspond to normal (inverted) hierarchy 2 m1 < m2 < m3 (m3 < m1 < m2). The definition of ∆m used is 2 2 2 2 2 2 2 ∆m = m3 − (m2 + m1)/2. Thus, ∆m = ∆m31 − ∆m21/2 > 0, if 2 2 2 m1 < m2 < m3, and ∆m = ∆m32 + ∆m21/2 < 0 for m3 < m1 < m2 Parameter Best fit 3σ 2 −5 2 ∆m12 [10 eV ] 7.37 6.93 – 7.97 |∆m2| [10−3 eV2] 2.50 (2.46) 2.37 – 2.63 (2.33 – 2.60) 2 sin θ12 0.297 0.250 – 0.354 2 2 sin θ23, ∆m > 0 0.437 0.379 – 0.616 2 2 sin θ23, ∆m < 0 0.569 0.383 – 0.637 2 2 sin θ13, ∆m > 0 0.0214 0.0185 – 0.0246 2 2 sin θ13, ∆m < 0 0.0218 0.0186 – 0.0248 ∗ δCP/π 1.35 (1.32) 0.92 – 1.99 (0.83 – 1.99) * For the Dirac phase δCP we give the best fit value and the 2 σ allowed ranges (at 3σ no physical values of δCP are disfavored). 24 Chapter 3. Neutrinos

3.2.2 In Matter If neutrinos travel long distances through matter, they undergo weak interactions. Due to the presence of electrons in matter, electron neutrinos may additionally interact with them via the W± boson, while muon and tau neutrinos do not. This asymmetry induces an effective potential which is proportional to the density of electrons, Ne, in the surrounding matter: √ V = ± 2GF Ne, (3.4) where GF is the Fermi constant, the plus sign is for νe, and the negative sign is for ν¯e. The inﬂuence of matter on neutrino oscillations was ﬁrst pointed out by Wolfenstein, Mikheyev and Smirnov and is often referred to as the MSW effect (Smirnov, 2005).

Introducing this potential changes the energy levels of the propagation eigenstates (mass eigenstates) of neutrinos. When neutrinos travel through matter, the effective Hamiltonian is modiﬁed from its vacuum form due to the difference in the forward scattering amplitudes of νe and νµ,τ. In the mass eigenstate basis, the effective Hamil- tonian can be written as:

 m2  1   2E 0 0 V 0 0  m2  † H = 2 + U 0 0 0 U, (3.5) matter  0 2E 0     2  m3 0 0 0 0 0 2E where U is the PMNS matrix. Since the Hamiltonian is now position-dependent, it is not possible to write a simple closed form for the oscillation probability. Following the approach presented in (Barger et al., 1980), we can deﬁne the matrix X, whose row vectors are the pure mass eigenstates and therefore contain the evolution of each of the mass states:

" 2 # ! 2EHmatter − M I M2L = j − k X ∑ ∏ 2 2 exp i , (3.6) k j6=k Mk − Mj 2E

2 where Mi /2E are the eigenvalues of the constant-density matter Hamiltonian Hmatter, and I is the identity matrix. The oscillation probability can then be written as:

† 2 P(να → νβ) = |UXUαβ| . (3.7)

In the two ﬂavor approximation, the solution of the corresponding Schroedinger equation is simple in case where the matter density is constant. Then, Hmatter can be re-diagonalised to obtain the mixing matrix and mass eigenstates in matter via a rotation matrix, similar to the one for vacuum. If we note the effective mixing angle in matter as θM: 2 2 sin 2θ sin 2θM = , (3.8) sin2 2θ + (Γ − cos 2θ)2 and the effective difference of squared masses as ∆M2: q ∆M2 = ∆m2 sin2 2θ + (Γ − cos 2θ)2, (3.9) 3.3. Neutrino Interactions 25

√ 2 where Γ = ±2 2GF NeE/∆m , we can write the equation in matter as:

2 ∆M cos 2θM sin 2θM i∂xΨ(x) = Ψ(x). (3.10) 4E − sin 2θM cos 2θM

For constant density, the matter evolution leads to an oscillation probability:

1.27∆M2L P(ν → ν ) = sin2 2θ sin2 . (3.11) e α M E

Therefore, to observe signiﬁcant matter effects, long baselines or high matter densities are required. In the limit ∆M2L/4E << 1 the vacuum probabilities can be retrieved. Since the "matter" mixing angle now depends on the local matter density, maximal mixing occurs when the vacuum mixing angle θ is non-maximal. That is, a resonance condition can be achieved for any set of vacuum mixing parameters if

f ρE g/cm3 · GeV cos 2θ = Γ = ± , (3.12) 6.5 × 103∆m2 eV2 where ρ is the matter density in g/cm3 and f is the proton to nucleon ratio in the matter. In the resonance region, oscillations can be significantly enhanced. The resonant condition depends on the sign of ∆m2, which can be used to determine the neutrino mass hierarchy. For example, long baseline experiments, which look for 2 2 νµ → νe oscillations in the region of ∆m ∼ ∆m23, are sensitive to the mass hierarchy through matter effects in the Earth if the baseline is sufficiently long and the energy is sufficiently high, so that matter effects are significant. Oscillation probabilities for neutrinos and antineutrinos can be different due to matter effects (because of the ± sign), even if neutrino interactions with matter do not violate CP.

3.3 Neutrino Interactions

Neutrinos do not have electric charge, which means that they are not affected by the electromagnetic forces and interact only via weak interactions (and gravity). There- fore, neutrinos pass through normal matter unimpeded and are very difﬁcult to detect. Occasionally neutrinos can interact with the electrons or nuclei and produce a detectable charged particles (see Fig. 3.1). Neutrino couplings to atomic electrons

FIGURE 3.1: Charged current neutrino interactions via a W boson exchange. are typically 3 orders of magnitude smaller then to nucleons, and can be neglected in most experimental cases. The value of cross section is proportional to the energy of neutrino. The ratio of neutrino to anti-neutrino cross section is approximately 2. 26 Chapter 3. Neutrinos

One can distinguish two types of neutrino interactions based on exchanged boson: • Charged current (CC): ν + N → l + N0 + .. - with exchange of W± boson,

• Neutral current (NC): ν + N → ν + N0 + .. - with exchange of Z0 boson. Here, N, N0 are nucleons and l is a charged lepton of the same flavor as interacting neutrino. Neutral current processes are usually difficult to reconstruct as part of the energy available in the final state is carried away by the unobservable neutrino. Therefore, neutrino detection is based mostly on the detection of a lepton produced in charged current interactions. There is a correlation between the direction of a parent neutrino and direction of a charged lepton produced in the neutrino interaction. The more energetic the neutrino is, the closer the lepton follows the original direction of its parent neutrino.

There are a few different types of CC neutrino interactions which are shown in Fig. 3.2. • Charged current quasi elastic interactions (CCQE): ν + N → ν + N0 In these processes, there are no additional particles in the ﬁnal state apart from the charged lepton l and recoiling nucleon N0. CCQE processes (red line) are dominant interaction mode for neutrino energies below 1 GeV. They signiﬁ- cantly contribute to the total cross section up to around 10 GeV.

• Resonant production (RES): ν + N → l + r∗ → l + N0 + meson(s) Here r∗ indicates a produced resonance, most commonly a ∆(1232). Mesons are usually charged or neutral pions, but contribution from heavier particles like kaons and ρ mesons is also possible. The resonant production processes (blue line) start to play role in neutrino interactions for Eν from around 1 GeV up to several GeV.

• Deep inelastic scattering (DIS): ν + N → l + other particles In these processes neutrino interact with partons (quarks and gluons) inside nucleons and as a result, many particles can be produced. Deep inelastic reactions (green line) are dominant for neutrino energies of several tens of GeV and higher. Due to the very small cross-section for neutrino interactions, their detection is very challenging. It requires high intensity neutrino sources and large neutrino detectors associated with long exposition time to gather a sufﬁcient number of events.

3.4 Sources of Neutrinos

Neutrinos are the second (after photons) most abundant particles in the Universe. They can be produce by different nautral and artiﬁcial sources. Neutrino energy spectra from various sources are shown in Fig. 3.3.

Natural sources: • Relic neutrinos: As the Universe expanded and cooled, neutrinos decoupled from matter. Just like the cosmic microwave background, these relic neutrinos are still around acting as an echo of the Big Bang. 3.4. Sources of Neutrinos 27

FIGURE 3.2: Summary of the current knowledge of νµ charged- current cross sections. Figure taken from (Formaggio and Zeller, 2012).

• Solar neutrinos: Neutrinos are produced in the nuclear fusion reactions inside the Sun and other stars. The dominant process, fusion of proton plus proton (pp) to deu- + terium (p + p → d + e + νe) is responsible for 98% of the energy production of the Sun. From this reaction, 86% of all solar neutrinos are produced, however, they have very low energies, so the detectability of solar neutrinos relies on the presence of fusion side chains giving higher-energy neutrinos from boron-8 and beryllium-7. The neutrino ﬂux from the Sun is enormous, ∼60 billion/cm2/s.

• Supernova neutrinos: 99% of the energy released by a core-collapse supernova comes in the form of neutrinos. Extreme heat and pressure force protons and electrons to com- bine together to form neutrons, releasing electron neutrinos νe. Conditions are so extreme that most of the neutrinos are produced thermally, as neutrino- antineutrino pairs.

• Atmospheric neutrinos: Cosmic rays striking the Earth’s atmosphere produce showers of pions which in subsequent decays can produce signiﬁcant ﬂux of neutrinos.

• Ultra high energy neutrinos (UHE): High-energy neutrinos produced by astrophysical objects (like active galaxies or gamma-ray bursts).

• Geoneutrinos: Neutrinos and antineutrinos emitted in decay of radionuclide naturally occurring in the Earth. 28 Chapter 3. Neutrinos

Artificial sources: • Reactor neutrinos: In fission reactors, electron-type antineutrinos ν¯e are produced in beta decays of fission fragments. Fission reactors are the best source of low energy antineutrinos.

• Accelerator neutrinos: In proton accelerators, the proton beam is directed on to a target, producing a large number of secondary pions (among other particles). Muon-neutrinos νµ and antineutrinos ν¯µ can be produced by allowing pions of the appropriate charge to decay in ﬂight, and then stopping the non-neutrino products with a dense absorber.

FIGURE 3.3: Measured and expected ﬂuxes of natural and reactor neutrinos. Figure taken from (Katz and Spiering, 2012)

Among all that sources it is worth to discuss atmospheric neutrinos in more details, as they would have the same energies as expected for dark matter induced neutrinos. Atmospheric neutrinos are produced in interactions of primary cosmic rays with atomic nuclei in the Earth’s atmosphere. Primary cosmic rays mainly consist of protons (90%) and alpha particles (9%), with small contribution of nuclei of heavier elements (Berezinskii et al., 1984). The isotropically distributed cosmic rays interact with the air molecule nuclei, producing copious amounts of pions and kaons. The neutral pion decays and produces gamma rays and leptons, but no neutrinos. ± The decay of charged pions π leads to the production of two muon neutrinos νµ and one electron neutrino νe. The whole reaction chain is shown schematically in 3.4. Sources of Neutrinos 29

Fig. 3.4. The average energy of atmospheric neutrinos is several hundreds of MeV and the energy spectra has a long high energy tail reaching TeV scale. While most of the neutrinos pass through Earth (travelling 10-10,000 km) without interaction, small fraction of them could interact and be detected. Detection of Cherenkov radiation from charged leptons produced by neutrino interactions in water or ice is a widely used technique. Super-Kamiokande is large water Cherenkov neutrino detector operating in the Kamioka Observatory, and it is described in Chapter4.

FIGURE 3.4: Schematic view of cosmic rays interactions in the atmosphere. Figure taken from (Louis et al., 1997).

If dark matter exist, it is expected that observed atmospheric neutrinos may contain some contribution of dark matter induced neutrinos. The aim of the analysis presented in this thesis is to evaluate the allowed contribution from this hypotheti- cal additional source.

Chapter 4

Super-Kamiokande

The Super-Kamiokande (Super-K, SK) is a large water Cherenkov detector located in the Mt. Ikenoyama in Japan (Fukuda et al., 2003). It is run by the Kamioka Obser- vatory of the Institute for Cosmic Ray Research, University of Tokyo. The detector is operated by the international collaboration of more than 40 institutes from Japan, United States, Korea, China, Canada, Poland, Spain, UK, Italy and France. The observatory was designed to search for a proton decay, study solar, atmospheric and man-made neutrinos, and keep a watch for Supernovae. Super-K also serves as the far detector for the Tokai-to-Kamioka (T2K) long baseline neutrino oscillation experiment (Abe et al., 2011).

4.1 Cherenkov Radiation

The detection of neutrinos in the Cherenkov detectors, is based on examining the characteristic light emitted by a charged particle produced during the interaction of a neutrino in water or ice. When the generated charged particle moves faster than the speed of light in a dielectric medium (water, in the case of Super-K), it produces electromagnetic shock wave known as Cherenkov radiation. The wavefront of the resulting radiation propagates at an angle

cos θc = 1/nβ, (4.1) where n is refractive index of the medium, and β is velocity of the particle divided by speed of light, c. In pure water where n ≈ 1.34 and β ≈ 1, this corresponds to ◦ θc ≈ 42 for 580 nm photons. The number of Cherenkov photons emitted (N) per unit wavelength (dλ) and per unit length (dx) is given by:

d2N 2πα 1 = 1 − , (4.2) dxdλ λ2 n2β2 where α is the ﬁne-structure constant. For particles traveling near the speed of light, '3400 photons per cm are emitted between 300 and 500 nm, where the Super-K photomultiplier tubes (PMTs) are most sensitive.

The threshold energy Ethr of a charged particle to emit Cherenkov radiation (if β = 1) can be written as n 2 Ethr = √ mc , (4.3) n2 − 1 2 where m is the mass of the charged particle. For pure water Ethr ∼ 1.5mc . In Table 4.1 one can see values of threshold energy for charged particles detected in Super-K. The cone of Cherenkov light when projected onto the walls of the detector forms a 32 Chapter 4. Super-Kamiokande

± ± ± TABLE 4.1: The values of threshold energy for e , µ and π .

2 Particle Rest mass (MeV/c ) Ethr (MeV) e± 0.511 0.767 µ± 105.7 157.4 π± 139.7 207.9 characteristic shape of rings (see Fig. 4.1). This light is recorded by the PMTs. The energy, direction and neutrino interaction point can be reconstructed based on the amount of charge deposited in the PTMs and the time when the light was detected.

FIGURE 4.1: A schematic view of the detection of Cherenkov light by the photomultipliers.

4.2 Detector

The Super-Kamiokande detector is located 1,000 m underground (2,700 m water equivalent) in Japan, inside the Mozumi Mine near the Kamioka section of the city of Hida in Gifu Prefecture, owned and operated by the Kamioka Mining and Smelt- ing Coperation. The geographic coordinates of SK are 36◦250N latitude and 137◦180E longitude. The mountain above the detector reduces the rate of cosmic ray muons to ∼3 Hz, which corresponds to a reduction factor of approximately 1/100,000 with respect to the ground level. The detector is a cylindrical stainless steel tank, 41.4 m tall and 39.3 m in diameter, holding 50,000 tons of ultra-pure water. The layout of the experimental site, including a schematic of the SK tank is shown in Fig. 4.2.

The tank volume is divided into an inner detector (ID) region, which is extending 2.7 m inward from the walls of the SK cylinder and 2.6 m from its top and bottom, and outer detector (OD), which consists of the remaining tank volume with total mass of 18 kton. The ID and OD are optically isolated from each other. In-between, there is a 55 cm dead-space spanned by a steel support structure that houses photomultiplier tubes viewing each of the detector regions. The ID contains 11,146 20-inch PMTs mounted on the tank wall. They are arranged in modules, each one consisting of 3x4 PMTs spaced 70 cm from each other. The space between the PMTs is lined with a reﬂective black sheet to prevent light leaks into the OD. In the OD houses 1,885 8-inch PMTs grouped in pairs and installed facing outside (see Fig. 4.3). 4.2. Detector 33

FIGURE 4.2: The schematic view of Super-Kamiokande detector and the experimental hall. Figure taken from (Fukuda et al., 2003).

FIGURE 4.3: A diagram of the module structure of the PMT support frame. Figure taken from (Fukuda et al., 2003). 34 Chapter 4. Super-Kamiokande

To enhance total light collection efficiency, the walls, top and bottom of the OD are covered with the reflective Tyvek. Tyvek’s reflectivity is 90% at 400 nm falling to about 80% at 340 nm. The OD serves as both an active and passive veto, that is, it enables us to identify muons coming from outside the tank and also to makes the inner part free from gamma rays and neutrons from the rock.

The surface of the Super-Kamiokande dome cavity above the tank is coated with radon-tight plastic material manufactured by Mineguard in order to prevent naturally occurring radon in the rock from finding its way into the detector volume. To maximize water transparency, water in the detector is ultra-purified by a special multi-step purification scheme which includes filtration, reverse osmosis, and de- gasifications (see Sec. 4.4 for more details). To neutralize the geomagnetic field that would otherwise affect photoelectron trajectories in the PMTs, 26 sets of horizontal and vertical Helmholtz coils are arranged around the inner surface of the tank. They reduce magnetic field at the site from 450 mG to 50 mG. While the detector is in operation, the performance and data acquisition is monitored 24 hours a day by physicists.

4.2.1 Photomultiplier Tubes The inner detector of Super-K uses custom built, 20-inch (50 cm) type R3600-5 PMTs developed by Hamamatsu Photonics in cooperation with the Kamiokande collaboration (Suzuki et al., 1993). A schematic view of the ID PMTs is shown in Fig. 4.4. The photocathode is approximately hemi-spherical in shape and consists of a bialkali (Sb-K-Cs) photocathode with a peak quantum efﬁciency of about 21% at 360-400 nm, and a sensitivity ranging from 280-660 nm (see Fig. 4.5). The average transit-time spread for 1 pe signal is 2.2 ns. The average dark noise rate at the 0.25 pe threshold used in Super-Kamiokande is about 3 kHz. The ID PMTs are operated with gain of 107, which corresponds to a supply voltage ranging from 1700 to 2000 V. Since SK-II phase (see Sec. 4.3), each ID PMT is encased in a ﬁber reinforced plastic shell (FRP) at the base and topped with an acrylic window over the photo-sensitive area. The transparency of the acrylic case is higher than 96% at 350 nm wavelength.

FIGURE 4.4: Diagram of 20-inch (50 cm) FIGURE 4.5: Quantum efﬁciency of the photomultiplier tube used in Super-K de- photocathode as a function of the light tector (taken from (Fukuda et al., 2003)). wavelength (taken from (Fukuda et al., 2003)). 4.3. History 35

The outer detector uses 8-inch (20 cm) Hamamatsu PMTs, which were recycled from the IMB experiment (Becker-Szendy et al., 1993), after it was decommissioned in 1991. The OD PMTs are distributed as 302 units on the top layer, 308 on the bottom, and 1275 on the barrel wall. To enhance the light collection efficiency, wavelength shifting plates are attached to each PMT, which absorbs light in a wide range of wavelengths, and re-radiates the light at a wavelength where the PMT sensitivity peaks. The plates are made from 1.3 cm thick by 60 cm square acrylic panels. These panels, increases the light collection efficiency of the OD by approximately 60%, but also lower the time resolution to approximately 15 ns. Since the OD functions pri- marily as a veto, this gain in the photon yield provides sufficient compensation for the loss in the timing resolution.

4.3 History

The Super-Kamiokande is a successor to the Kamioka Nucleon Decay Experiment (KamiokaNDE) (Koshiba, 1986), which was designed to search for a proton decay. The construction of the detector was completed in April, 1983. It was a tank 16.0 m in height and 15.6 m in width, containing 3,000 tons of pure water and about 1,000 photomultiplier tubes. After an upgrade in 1985, the detector ran as KamiokaNDE-II and was able to observe solar and atmospheric neutrinos. It detected neutrinos from the Supernova 1987A, and set what was then the world’s best limit on the lifetime of the proton.

The construction of the Super-Kamiokande detector started in 1991 and the ﬁrst data-run began on April 1st, 1996. It was continually operating until July 2001, when it was shutdown for scheduled upgrades and refurbishing dead PMTs. This period of operation is called SK-I.

In November 2001, when tank was reﬁlling with water, one of the bottom PMTs imploded. 6,779 ID PMTs and 885 OD PMTs were broken in a chain reaction due to the resulting shock wave. Since this accident, each ID tube is is covered by acrylic covers and an FRP case to protect against shock waves from imploding neighboring tubes. The detector was rebuilt using the remaining PMTs for the ID, and remaining and newly-introduced PMTs for the OD. It was working from October 2002 to Octo- ber 2005. In this period, called SK-II, 5182 ID PMTs with acrylic covers and 1885 OD PMTs were used, and the photo-coverage of ID was 20%.

In July 2006, after a break to restore lost PMTs, the experiment resumed full operation with 11,129 working ID PMTs, reaching 40% of photo-coverage. In addition, a vertical layer of Tyvek was installed to optically separate the barrel and end cap regions. The optical segmentation was done in order to better separate background corner-clipping cosmic ray muons from neutrino interactions with an exiting lepton. This period is known as SK-III and lasted until September 2008, when the detector was shutdown once more for an upgrade of electronics. Thanks to the new electronics, the dead time associated with the data acquisition system was reduced, the dynamic range was expanded, and the energy threshold lowered. The period which started in September 2008 and continues to this day is calld SK-IV. A summary of the detector conﬁguration for the SK experiment phases up to the current moment can be found in Table 4.2. 36 Chapter 4. Super-Kamiokande

TABLE 4.2: Summary of the differences between each of the SK periods.

Phase SK-I SK-II SK-III SK-IV Start of data taking 1996 Apr. 2002 Oct. 2006 Jul. 2008 Sep. End of data taking 2001 Jul. 2005 Oct. 2008 Sep. (running) Number of ID PMTs 11146 5182 11129 11129 Number of OD PMTs 1885 Photo-coverage 40% 19% 40% 40% Anti-implosion PMT cases no yes yes yes OD segmentation no no yes yes Energy threshold 5 MeV 7 MeV 4.5 Mev 4 MeV Front-end electronics ATM (ID), OD QTC (OD) QBEE

In 1998, the Super-Kamiokande Collaboration announced the evidence of atmospheric neutrino oscillation, which was the first experimental observation support- ing the theory that the neutrino has a non-zero mass (Fukuda et al., 1998). Soon after, in 2001, solar neutrino oscillations were confirmed (Fukuda et al., 2001). The precision measurement of the solar neutrino flux opened a new window that en- abled scientists to study the inner workings of the Sun. The experiment continues to operate, collecting data on various natural sources of neutrinos, as well as acting as the far detector for the Tokai-to-Kamioka (T2K) long baseline neutrino oscillation experiment (Abe et al., 2011).

On June 2015, the Super-Kamiokande Collaboration approved the SuperK-Gd project, which aim at dissolving gadolinium to the Super-K water. By adding gadolinium into water Cherenkov detector, inverse beta decay reactions will have two signals, the prompt one is positron signal and the delayed one is a ∼ 8 MeV gamma cascade from neutron capture on gadolinium. Due to this effect, background for Supernova Relic Neutrino (SRN) can be largely reduced by detecting prompt and delayed signals in coincidence. It is expected to detect ∼ 5 neutrinos from SNR after a year of detector operation. Moreover, SuperK-Gd will also have the ability to distinguish neutrino and anti-neutrino (see (Beacom and Vagins, 2004) for more details).

4.4 Water Puriﬁcation System

Water conditions affect photon propagation in the tank, so it is very important to keep the water transparency as high as possible. The water used in SK is sourced from natural water inside the Kamioka mine. It is purified by a dedicated system (see Fig. 4.6) and is continually recirculated through parts of that system during normal detector operation. Water passes throug a 1µm filter in order to remove large particles, a heat exchanger maintains the temperature at 13◦C to suppress bacteria growth, ion exchangers remove metal ions in the water, and any bacteria remaining in the water are killed by an ultra-violet sterilizer. Radon-less air (Rn-less-air) systems dissolve radon free air into the water to improve the radon removal capabilities of the vacuum degasifier. Reverse osmosis (RO) filter stops the contaminants heavier than a mass of 100 molecular weight using a high performance membrane. A vacuum degasifier is used to remove gas dissolved in the water. It is able to remove 4.5. Air Purification System 37

FIGURE 4.6: Super-Kamiokande water purification system (taken from (Fukuda et al., 2003)). about 99% of the oxygen gas and 96% of the radon gas. Cartridge Ion Exchangers se- lectively remove ions with 99% elimination efficiency. To remove small dust even of the order of nanometers Ultra filter is used. Membrane Degasifier further removes dissolved radon and oxygen gases. The water is taken from the top of the tank using a pump and returned to the bottom of the tank at a flow rate of 50 tons/hour. This systems enables the water to be transparent up to ∼90 m.

4.5 Air Puriﬁcation System

Purified (Rn-reduced) air is supplied to the gap between the water surface and the top of the Super-Kamiokande tank. A schematic view of the air purification system is shown in Fig. 4.7. It consists of three compressors, a buffer tank, dryers, filters, and an activated charcoal filter.

FIGURE 4.7: Super-Kamiokande air puriﬁcation system (taken from (Fukuda et al., 2003)). 38 Chapter 4. Super-Kamiokande

The air in the mine housing SK naturally contains 100-5,000 Bq/m3 of radon throughout the year. Varied concentrations occur due to changes in the ventilation of air in the mine. In the cool season, air ﬂows from the mine entrance closest to the experiment. In the summer months though, air ﬂows in the opposite direction and is exposed to more rock before reaching SK and picks up more radon. To limit the amount of this background reaching the detector volume, radon free air is provided to the control room and dome areas of the experiment.

Air collected from outside the mine, with intake far from the mine entrance, is pu- riﬁed in a "radon hut" located outside of the mine via a series of compressors and activated charcoal ﬁlters and pumped in at a rate of 10 m3/min. Radon concentrations at the dome are consistently below 100 Bq/m3 throughout the year. Typical radon concentration inside the top of SK tank is measured to be 20-30 Bq/m3.

4.6 Calibration

Neutrino oscillations studied with the Super-Kamiokande experiment are a function of the neutrino energy. Therefore, in the determination of particle’s energy it is important to know the relation between the number of photoelectrons recorded in PMTs and the total deposited energy. That matching is called the energy scale determination and it is crucial for precision measurements.

FIGURE 4.8: Energy scale stability measured as a function of date since the start of Super-Kamiokande detector operations. The energy scale is taken as the average of the reconstructed momentum divided by range of stopping cosmic ray muon data in each bin. The vertical axis shows the deviation of this parameter from the mean value for each SK period separately. Error bars are statistical. Figure taken from (Abe et al., 2017a).

During the detector operation, changes in the run conditions are unavoidable. They may impact the water transparency and subsequent performance of the detector. Therefore, they must be corrected through calibrations. Since the range of energies of interest to atmospheric neutrino analysis spans from tens of MeVs to tens of TeVs, 4.6. Calibration 39 the calibration through radioactive isotopes is impossible. The water transparency is continuously monitored using cosmic ray muons as a calibration source (see (Ashie, 2005) for the detailed description).

A plot showing the energy scale stability since SK-I is presented in Fig. 4.8. Note the SK-III period was subject to poor and volatile water transparency conditions, resulting in a comparatively turbulent energy scale. The stability seen in the SK-IV period is a result of improvements in the water puriﬁcation system and in correc- tions for the time variation of the PMT response.

Chapter 5

Atmospheric Neutrinos at Super-Kamiokande

The search for dark matter induced neutrinos is based on atmospheric neutrino data collected with the Super-Kamiokande detector from 1996 to 2016 (SK-I, SK-II, SK-III and SK-IV data taking periods). The livetime of each sample, which corresponds to the period of time when the detector was operational, is listed in Table 5.1.

TABLE 5.1: Livetime of SK-I, SK-II, SK-III and SK-IV data sets.

Sample SK-I SK-II SK-III SK-IV ALL FC/PC 1489.2 days 798.6 days 518.1 days 2519.9 days 5325.8 days UP-µ 1645.9 days 827.7 days 635.6 days 2519.9 days 5628.2 days

Before we discuss the search for dark matter induced neutrinos, we must ﬁrst describe the atmospheric data and Monte Carlo samples used in analysis, and the standard oscillation analysis.

5.1 Data Samples

In the Super-Kamiokande detector, the observed atmospheric neutrino events are divided into three main categories: fully-contained (FC), partially-contained (PC), and upward - going muons (UP-µ), which are schematically illustrated in Fig. 5.1. These categories are further sub-divided into ﬁnal 19 analysis samples.

FIGURE 5.1: The schematic illustration of event categories used in Super-K: fully-contained (FC), partially-contained (PC) and upward - going muons (UPMU) samples.

If the charged particle produced in neutrino interaction stops in the inner detector, 42 Chapter 5. Atmospheric Neutrinos at Super-Kamiokande the interaction is classified as fully-contained. FC events have a reconstructed vertex within the 22.5 kton fiducial volume, which is defined as the region located more than 2 m from the ID wall. In additon, we require there to be no activity in the OD. The FC data is further subdivided based on the number of observed Cherenkov rings (single- or multi-ring), the particle ID (PID) of the most energetic ring (e-like or µ-like), and visible energy or momentum (SubGeV with E < 1.33 GeV or MultiGeV with E > 1.33 GeV). There are also additional selections made based on the number of observed electrons from muon decays and the likelihood of containing a π0 (detailed description can be found in (Wendell et al., 2010)). After all the selections, there are 14 FC analysis samples.

Events are classified as partially-contained (PC) if they have a reconstructed vertex inside the fiducial volume, but the produced high energy lepton exits the ID and deposit energy also in the outer veto region. Based on the energy deposition within the OD, PC events are further classified into "stopping" and "through-going" subsamples. The energies of neutrinos which produce PC events are typically 10 times higher that those which produce FC events.

Neutrinos can also interact with the rocks surrounding the detector and produce high energy muons which intersect the tank. Downward-going muons produced in neutrino interactions cannot be distinguished from the constant ﬂux of cosmic ray muons. However, muons travelling in upward direction must be neutrino induced. The upward-going muon (UP-µ) sample, based on light deposition in both the OD and ID, is further divided into "through-going" and "stopping" subsamples for events that cross or stop within the ID, respectively. Through-going events with energy deposition consistent with radiative losses are separated into a "showering" subsample.

The expected number of neutrino events in each class as a function of the true neutrino energy for the FC, PC, and UP-µ events is presented in Fig. 5.2. The event rate of each subsample over the lifetime of the experiment have been stable at 8.3 FC events per day, 0.73 PC events per day, and 1.49 UP-µ events per day. A summary of all 19 subsamples deﬁned for atmospheric events at the Super-Kamiokande is presented in Table 5.2. Each subsample is binned in momentum or visible energy and the cosine of the zenith angle. In this way, 520 analysis bins are created for each of the SK period (2080 bins in total). Note that for FC and PC events, the cos θz bins cover the full sky:

(−1, −0.8. − 0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 1), and for UP-µ events the are deﬁned only below the horizon:

(−1, −0.9, −0.8, −0.7, −0.6, −0.4, −0.3, −0.2, −0.1, 0).

In this way, the size of the UP-µ bins is half of the size of FC and PC bins. The binning is choosen is such a way to have at least 6 events expected in each bin after scaling to data livetime of this analysis. 5.1. Data Samples 43

FIGURE 5.2: The expected parent neutrino energy distributions for the FC e-like (left), FC µ-like and PC (middle), and UP-µ (right) samples based on MC simulation.

5.1.1 Data Reduction and Reconstruction At Super-Kamiokande, roughly 10 atmospheric neutrino interaction events per day are expected, but the detector records approximately 106 events per day. The majority of them come from the low background radioactivity surrounding the detector (around 11 Hz) and cosmic ray muons (around 3 Hz). Therefore, dedicated reduction algorithms are implemented to select neutrino events from this background. Special care is taken to ensure high efficiency of the selection. The initial steps of the reduction are not computationally intense and are designed to remove apparent backgrounds quickly and efficiently, while later steps focus on finer subtleties between events. Data reduction is done separately for each category of events. The details of the reduction process can be find in (Li, 2017).

FIGURE 5.3: An example event display of single-ring µ-like (left) and e-like (right) event. The colored points indicate the quantity of the detected light by each PMT.

Following the reduction, the remaining events are put through a reconstruction algorithm in order to convert the PMT charge and timing information into physical variables. The reconstruction of the initial interaction vertex is based on PMT timing information, with a correction implemented for the photon time of ﬂight. The type of the charged particle produced in neutrino interaction can be inferred from 44 Chapter 5. Atmospheric Neutrinos at Super-Kamiokande

TABLE 5.2: Summary of event subsamples for atmospheric neutrino events at Super-Kamiokande. For each subsample, the standard bins in cosine of the zenith angle and momentum (or visible energy) are speciﬁed.

Sample Energy/momentum bins cos θz bins Fully Contained (FC) SubGeV e-like 0 decay-e 5 e± momentum 10 in [-1,1] e-like 1 decay-e 5 e± momentum 1 in [-1,1] ± Single-ring π0-like 5 e momentum 1 in [-1,1] µ-like 0 decay-e 5 µ± momentum 10 in [-1,1] µ-like 1 decay-e 5 µ± momentum 10 in [-1,1] µ-like 2 decay-e 5 µ± momentum 1 in [-1,1] 0 Multi-ring π0-like 5 π momentum 1 in [-1,1] Fully Contained (FC) MultiGeV ± νe-like 4 e momentum 10 in [-1,1] ± ν¯e-like 4 e momentum 10 in [-1,1] µ-like 2 µ± momentum 10 in [-1,1] MultiRing νe-like 3 visible energy 10 in [-1,1] MultiRing ν¯e-like 3 visible energy 10 in [-1,1] MultiRing µ-like 4 visible energy 10 in [-1,1] MultiRing Other 4 visible energy 10 in [-1,1] Partially Contained (PC) Stopping 2 visible energy 10 in [-1,1] Through-going 4 visible energy 10 in [-1,1] Upward-going Muons (UP-µ) Stopping 3 visible energy 10 in [-1,0] Through-going Non-showering 1 10 in [-1,0] Through-going Showering 1 10 in [-1,0] the sharpness of the edge of the ring and the opening angle. Muons and charged pions are classiﬁed as µ-like or non-showering, because they do not form electromagnetic showers, and as a result, they produce Cherenkov rings with sharp edges (see Fig. 5.3 left). On the other hand, particles are divided into e-like or showering categories if they produce electromagnetic showers, resulting with fuzzy edges caused by many overlapping rings (see Fig. 5.3 right). Moreover, based on the topology of the ring pattern, the particle directions can be determined. Based on the observed number of photons in the ring, the momentum is assigned to each reconstructed ring in an event. Particles with higher momentum produce more photons in Cherenkov ring. The momentum of a multi-ring event is a sum of individual ring momenta. The only exception from this rule is the momentum estimation of π0-like events which is based only on the most energetic gamma-induced ring which is e-like. In case of events with decay electrons (originated from muon), the delayed electron ring is not included in the estimation of an event direction and momentum.

5.2 Monte Carlo

Atmospheric neutrinos at Super-Kamiokande are simulated with large statistics, corresponding to 500 years of livetime for each running period (SK-I, SK-II, SK-III and SK-IV). The Monte Carlo (MC) simulation is performed in stages: ﬁrst modeling the 5.2. Monte Carlo 45 initial neutrino interactions in the water, then tracking subsequent particles inside of the detector volume, and ﬁnally checking the detector’s response to those particles.

The simulation of atmospheric neutrinos is performed following the flux calculation of Honda (Honda et al., 2011), presented in Fig. 5.4. Two other flux calculations, the Fluka flux (Battistoni et al., 2003) and Bartol flux (Barr et al., 2004), are used to compute flux - related systematic errors.

FIGURE 5.4: Atmospheric neutrino ﬂuxes for Kamioka averaged over all directions (left panel), and the ﬂux ratios (right panel), calculated by Honda (Honda et al., 2011) (solid red), Fluka (Battistoni et al., 2003) (dotted green) and Bartol (Barr et al., 2004) (dashed blue) groups, (dash-dotted line is for the previous Honda calculation (HKKM06)).

Neutrino interactions within the detector and in the surrounding rocks are simulated using the NEUT (Hayato, 2002) model (software version 5.3.6). In NEUT, the interactions of neutrinos on protons, oxygen, and sodium are considered through the following modes:

• quasi-elastic elastic scattering ν + N → l + N0

• single meson production ν + N → l + N0 + m

• meson exchange current ν + NN? → l + N0 N0?

• coherent pion production ν + 16O → l + 16O + π

• deep inelastic scattering ν + N → l + other particles where l is a lepton, m is a meson and N, N0 are nucleons. Charged-current quasi- elastic interactions are simulated using the Llewellyn-Smith model (Smith, 1972) with nucleons distributed according to the Smith-Moniz relativistic Fermi gas (Smith 2 and Moniz, 1972) assuming an axial mass MA = 1.21 GeV/c and form factors from (Bradford et al., 2006). Interactions on correlated pairs of nucleons, so-called meson exchange currents (MEC), have been included following the model of Nieves (Nieves, Amaro, and Valverde, 2004). Pion-production processes are simulated using the Rhein-Seghal model (Rein and Sehgal, 1981) with Graczyk form factors (Graczyk and Sobczyk, 2008). Since the MEC simulation includes delta absorption processes, 46 Chapter 5. Atmospheric Neutrinos at Super-Kamiokande the pionless ∆ decay process, ∆ + N0 → N00 + N0, in NEUT’s previous pion production model has been removed in the present version.

Detector simulation provide an accurate model of the Super-K tank itself as well as its electronic response to particles propagating in the detector volume. It includes tracking the particles, modeling the Cherenkov radiation and photon propagation, and simulating the PMT response. The Super-K has custom detector simulation package called SKDETSIM, based on GEANT3 (Brun et al., 1993).

The same data reduction and reconstruction procedure is applied to MC events as to real data. Therefore after scaling to the data livetime, one can obtain expected number and characteristics of the atmospheric neutrino interaction events seen in the experiment.

5.3 Atmospheric Neutrino Oscillation Analysis

The fit using atmospheric neutrino data is performed to estimate the values of the 2 2 2 oscillation parameters sin θ23 and |∆m32| (normal hierarchy)/|∆m31| (inverted hierarchy). The value of the other oscillation parameters is fixed in the fit at the global best fit values listed in Table 5.3, but their uncertainties are treated as a source of systematic error in the analysis.

TABLE 5.3: The global best fit values of neutrino oscillation parameters from (Olive, 2014) fixed during the fit. The uncertainties on the parameters values are treated as systematic errors in the fit.

Parameter Value 2 −5 2 ∆m21 (7.53 ± 0.18) × 10 eV 2 sin θ12 0.304 ± 0.014 2 sin θ13 0.0219 ± 0.0012

The zenith angle distributions of data and MC for each sample are shown in Fig. 5.5. MC predictions for normal hierarchy are plotted with cyan, while the inverted hierarchy is shown in orange. Note that the predictions for normal hierarchy are plotted on top of predictions for inverted hierarchy, which makes the latter ones difﬁcult to noticed.

The likelihood is based on Poisson probabilities and can be written as:

− MC DATA e Nn (NMC)Nn L( MC DATA) = n N , N ∏ DATA , (5.1) n Nn !

MC DATA where Nn and Nn are expected and observed number of events in the n-th bin. According to Wilks’ theorem (Wilks and Daly, 1939), log likelihood ratio follows χ2 distribution with n degrees of freedom:

L(NMC, NDATA) NDATA 2 ≡ − = MC − DATA + DATA n χ 2 ln DATA DATA 2 ∑ Nn Nn Nn ln MC . (5.2) L(N , N ) n Nn 5.3. Atmospheric Neutrino Oscillation Analysis 47

FIGURE 5.5: Data and MC comparisons for the entire Super-K data divided into 19 analysis samples. Samples with more than one zenith angle bin are shown as zenith angle distributions (second through ﬁfth column) and other samples are shown as reconstructed momentum distributions (ﬁrst column). Cyan (orange) lines denote the best f it MC assuming the normal (inverted) hierarchy. Narrow panels below each distribution show the ratio relative to the normal hierarchy MC. In all panels, the error bars represent the statistical uncertainty.

In order to incorporate the effect of systematic errors into the bin expectation for independent sources of error, we use the prescription given in (Fogli et al., 2002). The expected number of events in each bin changes within an added systematic error parameter: MC MC i Nn → Nn (1 + ∑ fnei), (5.3) i i where fn represents the fractional change in the number of events in n-th bin due to a variation of the systematic error parameter ei. This is the so called "pull" method. For each systematic error, the parameter ei is estimated during the ﬁt and resulting distribution of the ei/σi is Gaussian univariate. In this way, the strength of systematic error’s is constrained through the addition of a penalty term to Eq. 5.2. The full χ2 can be written as:

NDATA e 2 2 = MC − DATA + DATA n + i χ 2 ∑ Nn Nn Nn ln MC ∑ (5.4) n Nn i σi 48 Chapter 5. Atmospheric Neutrinos at Super-Kamiokande

! NDATA e 2 2 = MC( + i ) − DATA + DATA n + i χ 2 ∑ Nn 1 ∑ fnei Nn Nn ln MC i ∑ n i Nn (1 + ∑i fnei) i σi (5.5) 2 2 In the ﬁt, χ is minimized over ei, and at the minimum ∂χ /∂ei = 0 for every i.

5.3.1 Systematic Uncertainties In the "pull" method we use, the effect of systematic errors is incorporated by the sys- i tematic error coefﬁcients fn. Calculation of these coefﬁcients is based on atmospheric MC simulations. It is assumed that they have a linear effect on the bin content B0, i with fn being the slope of the line between its contents at Bn(+σi) and Bn(−σi): B (+σ ) − B (−σ ) i = n i n i fn 0 . (5.6) 2Bn In the analysis, we consider 160 systematic error sources, which are listed below. More details about systematic errors can be found in (Abe et al., 2017a).

Neutrino ﬂux and production errors - common for all SK periods:

• Flux normalization

• Flux ratio (νµ + ν¯µ)/(νe + ν¯e)

• Flux ratio ν¯e/νe

• Flux ratio ν¯µ/νµ • Up/down ration

• Horizontal/vertical ratio

• K/π ratio in ﬂux calculation

• Neutrino production height

• Energy spectrum and sample-by-sample normalization

• Matter effects

Neutrino interaction, particle production, and PMNS oscillation parameter systematic errors that are common to all SK run periods:

• Axial mass in quasi-elastic CC interaction

• Single π production

• CCQE cross section

• CCQE ν¯/ν ratio

• CCQE µ/e ratio

• DIS cross section

• DIS model comparisons 5.3. Atmospheric Neutrino Oscillation Analysis 49

• DIS Q2 distribution

• Coherent π production

• NC/CC ratio

• ντ cross section • NC fraction from hadron simulation

• π+ decay uncertainty

• Meson exchange current

2 • ∆m21 2 • sin θ12 2 • sin θ13 Systematic errors that are independent in SK-I, SK-II, SK-III, and SK-IV: • FC reduction

• PC reduction

• FC/PC separation

• PC stopping/through-going separation

• Non-ν background

• Fiducial volume

• Ring separation

• Particle identiﬁcation

• Energy calibration

• Up/down asymmetry energy calibration

• UP-µ reduction

• UP-µ stopping/through-going separation

• Energy cut for stopping UP-µ

• Path length cut for through-going UP-µ

• Through-going UP-µ showering separation

• Background subtraction for UP-µ

• νe/ν¯e Separation • SubGeV 1-ring π0 selection

• SubGeV 2-ring π0

• Decay-e tagging

• Solar Activity 50 Chapter 5. Atmospheric Neutrinos at Super-Kamiokande

5.3.2 Results of the Oscillation Analysis The results presented in this section are based on the ofﬁcial analysis published by the Super-Kamiokande collaboration (Abe et al., 2017a). The constraints on neutrino 2 2 2 2 2 oscillation parameters |∆m32,31|, sin θ23 and δCP are based on ∆χ = χ − χmin distri- 2 2 butions, where χmin is the χ value at the best f it in the parameter space. The results for normal and inverted hierarchy are shown in Fig. 5.6 with cyan and orange lines respectively. The ﬁt results show that data prefer the normal hierarchy over the in- 2 2 2 verted hierarchy with ∆χ = ∆χNH,min − ∆χIH,min = −4.34.

The normal hierarchy ﬁt to the atmospheric mixing parameters yield:

2 +013 −3 2 • |∆m32| = (2.50−0.20) × 10 eV ,

2 +0.031 • sin θ23 = 0.588−0.067,

+1.37 • δCP = 4.19−1.59.

The full list of the best f it values of the 160 systematic error parameters ei can be found in (Abe et al., 2017a). Note that the Super-K has vey small sensitivity to δCP parameter and its value cannot be constrained within 3σ by Super-K data.

FIGURE 5.6: Constraints on neutrino oscillation parameters from the Super-K atmospheric neutrino data ﬁt. Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in cyan, by the difference in their minimum χ2 values. 51

Chapter 6

WIMPs in the Center of the Milky Way

Indirect searches aim at detecting dark matter annihilation/decay products, such as charged particles, photons, or neutrinos (e±, p¯, γ, ν) in the cosmic ray ﬂux. Since the expected signals depend on the square of the dark matter density for annihilation, or are proportional to the density in case of decay, the distribution of dark matter in the Universe is crucial for robust predictions.

6.1 Dark Matter Halo Models

The Milky Way is one of the considered sources in the searches for dark matter induced signals, due to an expected increase in the density towards the Galactic Cen- ter. The largest source of uncertainty in the dark matter density proﬁle of the un- derlying halo model, comes from the cuspiness of the inner gradient. However, the signal averaged over larger regions of the halo is much less dependent on the model as long as the density normalization is chosen such that it is consistent with the observed data, e.g., rotation curves.

It is expected that the dark matter halo envelops the galactic disk and extends far beyond the edge of the visible galaxy. In the ΛCDM model (Frieman, Turner, and Huterer, 2008), dark matter halos are assumed to form hierarchically via gravitational amplification of initial density fluctuations (bottom-up). Cosmological N- body simulations with large dynamic range (Diemand, Kuhlen, and Madau, 2007) and gravitational lensing observations (Clowe et al., 2006) seem to indicate that the dark matter density profiles can be described in the form: ρ ρ(r) = 0 , (6.1) γ α (β−γ)/α (r/rs) [1 + (r/rs) ] where the parameters α, β, γ and rs take different numerical values for different models. Parameter γ is the inner cusp index, rs is the scale radius, and the normalization ρ0 is chosen so that the mass contained within the solar circle (Rsc = 8.5 kpc) provides the appropriate dark matter contribution to the local rotational curves. In addition, to avoid numerical divergences due to very cuspy profiles (which may be an artifact of simulations), one assumes a flat core for all the profiles in the innermost 0.1◦ around GC.

The three commonly used proﬁles are NFW (Navarro, Frenk, and White, 1997), Moore (Moore et al., 1999) and Kravtsov (Kravtsov et al., 1998), and parameters corresponding to these models are listed in Table 6.1. The comparison of the dark 52 Chapter 6. WIMPs in the Center of the Milky Way

TABLE 6.1: The parameters of Eq. 6.1 for Moore (Moore et al., 1999), NFW (Navarro, Frenk, and White, 1997) and Kravtsov (Kravtsov et al., 1998) halo proﬁles.

3 Halo Model α β γ rs [kpc] ρ(Rsc) [GeV/cm ] NFW 1 3 1 20 0.3 Moore 1.5 3 1.5 28 0.27 Kravtsov 3 3 0.4 10 0.37

FIGURE 6.1: The dark matter density as a function of the distance from the GC for Moore (Moore et al., 1999) (dotted green), NFW (Navarro, Frenk, and White, 1997) (solid red) and Kravtsov (Kravtsov et al., 1998) (dashed blue) halo proﬁles. The vertical gray line indicates the Solar System position (Rsc = 8.5 kpc). The plot illustrate Eq. 6.1 with parameters from Tab. 6.1.

matter density distribution, ρ(r), as a function of distance from the Galactic Center for considered profiles is shown in Fig. 6.1. For the outer regions of the halo (several kpc away from the Galactic Center) one can observe very similar behavior and good agreement between the models. However, the inner gradient of the halo profile is the most uncertain part, due to the spatial resolution limit of the existing simulations. In the analysis presented in this thesis, the NFW profile will be used as a benchmark model, while the Moore and Kravtsov profiles will be applied as extreme cases to estimate the influence of halo model choice on the obtained results.

6.2 Dark Matter Annihilation

The intensity J (number ﬂux per solid angle) of dark matter annihilation products in the galactic halo at an angle Ψ with respect to the GC direction is proportional to the dark matter density squared integrated along the line of sight (Yüksel et al., 2007) (see Fig. 6.2 for deﬁnition of the variables):

lmax 1 Z q ( ) = 2( 2 − + 2) J Ψ 2 ρ Rsc 2lRsc cos Ψ l dl. (6.2) Rscρsc 0

1 The prefactor 2 , where ρsc = ρ(Rsc), is an arbitrary scaling which makes J(Ψ) Rscρsc dimensionless. The upper limit for integration lmax is deﬁned as: q 2 2 2 lmax = RMW − Rsc sin Ψ, (6.3) where RMW = 40 kpc is the adopted Milky Way halo size. Any enhancement of the density would result in a sharp increase in the ﬂux of annihilation products. 6.2. Dark Matter Annihilation 53

FIGURE 6.2: Illustration of the line of sight l and the angle Ψ in the coordinate system related to the Galactic Center.

Since dark matter radial halo profiles are peaked towards the Galactic Center, one can expect very large enhancements for small angles Ψ around this point. This is presented in Fig. 6.3 for the NFW, Moore, and Kravtsov profiles. One can see that the intensity strongly depends on the chosen profile and the differences can be several orders of magnitude at small angles. Presented density and intensity profiles had been calculated based on Eq. 6.1 and 6.2 using computational software program Mathematica (Wolfram, 1999).

FIGURE 6.3: Intensity of dark matter annihilation products versus angular distance from the Galactic Center for Moore (green), NFW (red) and Kravtsov (blue) proﬁles.

The relevant quantity in a measurement is related to overall intensity of dark matter annihilation products received from a given cone half-angle. The integral of J(Ψ) over the solid angle ∆Ω = 2π(1 − cos Ψ) can be deﬁned as:

1 Z J = J(Ψ)dΩ, (6.4) ∆Ω ∆Ω ∆Ω 54 Chapter 6. WIMPs in the Center of the Milky Way where ∆Ω is the chosen search region. Then, the differential ﬂux of annihilation products from this ﬁeld of view can be expressed in the form:

dΦ < σ v > R ρ2 dN ∆Ω = A sc sc J∆Ω 2 . (6.5) dE 2 4πMχ dE

The factor 1/2 accounts for dark matter being its own antiparticle, 1/4π is for isotropic emission of the annihilation products, Mχ is assumed mass of the dark matter particle and dN/dE is the spectrum of the annihilation products.

6.3 Dark Matter Decay

For decaying dark matter, the expected neutrino ﬂux can be written in form:

dΦ∆Ω 1 Rscρsc dN = Jd∆Ω , (6.6) dE τ 4πMχ dE where τ is the dark matter particle lifetime. The intensity Jd is the line of sight integral over the dark matter density,

l Zmax q 1 2 2 Jd(Ψ) = ρ( Rsc − 2lRsc cos Ψ + l )dl. (6.7) Rscρsc 0

The calculated intensity of dark matter decay products is shown in Fig. 6.4. In this case the differences between the halo models are smaller.

FIGURE 6.4: Intensity of dark decay products versus angular distance from the Galactic Cen- ter for Moore (green), NFW (red) and Kravtsov (blue) pro- ﬁles.

6.4 Energy Spectrum of Dark Matter Induced Neutrinos

In the presented analysis four generic annihilation/decay channels of the relic particles are considered: νν¯, bb¯, W+W− and µ+µ−. When considering the annihilation/decay of dark matter into pairs of neutrinos, νν¯, the spectrum of neutrinos per 2 flavor is a monochromatic line with dN/dE = 3 δ(E − Mχ), where the prefactor 2/3 arises under the assumption that 2 neutrinos are produced per one annihilation of dark matter particles pair and all neutrino flavors are equally populated. Such mono-energetic neutrinos are of a specific interest as they can be used to set a model independent limit on the total dark matter self-annihilation cross-section or decay livetime into Standard Model final states. 6.4. Energy Spectrum of Dark Matter Induced Neutrinos 55

FIGURE 6.5: Differential muon neutrino energy spectra for WIMP particle mass of 100 GeV annihilating into bb¯ (blue), W+W− (maroon) and µ+µ− (purple), after taking into account neutrino oscillations through the Galaxy.

For the calculation of the neutrino energy spectrum dN/dE, each time 100% branching ratio (BR) to a given annihilation/decay channel is assumed. The realistic annihilation/decay neutrino energy spectra, will be a mixture of different channels. Therefore, the spectra assuming 100% BR to "hard" leptonic channel µ+µ− and "soft" hadronic channel bb¯ will bracket the realistic scenario.

Differential multiplicity spectra for bb¯, W+W− and µ+µ− dark matter annihilation or decay modes are obtained using DarkSUSY (Gondolo et al., 2001) simulator . The example, differential muon neutrino energy spectra from annihilation of 100 GeV WIMPs is shown in Fig. 6.5.

Full mixing due to long-baseline oscillations is assumed for the calculation of the dark matter induced neutrino flux at the Earth. The expected number of neutrino events observed in the detector can be found by integrating the differential neu- dΦ∆Ω trino flux dE over the detector live-time and the direction- and energy-dependent effective area for annihilation. Therefore, using Eq. 6.5 one can relate the velocity- averaged self-annihilation cross-section < σAv > with dark matter induced flux and convert it into expected number of neutrinos.

DarkSUSY is a publicly-available advanced numerical package for neutralino dark matter calculations (Gondolo et al., 2001). The neutralino density in the Universe, the direct detection rate with various nuclei, and indirect detection rates for neutrinos, gamma rays, and charged particles today can be computed. In this analysis, version 5.0.6 is used in order to predict differential multiplicity spectra of neutrinos produced in dark matter annihilation into one of considered channels (bb¯, W+W−, µ+µ−), in the Milky Way. The hadronization and/or decay of the annihilation products are simulated with Pythia (Sjöstrand, Mrenna, and Skands, 2006) (v.6.154).

Chapter 7

Search for the Galactic WIMPs at Super-Kamiokande

The methodology of the indirect search for dark matter with the Super-Kamiokande detector is presented in this chapter. It is assumed that Super-K atmospheric neutrino data could have one additional origin related to dark matter induced neutrinos from the center of the Milky Way. The goal of the search is to evaluate the allowed contribution from such a source.

The presented analysis focuses on the search for angular anisotropy in the number of observed neutrino events between the region around the Galactic Center (GC) and the other part of the sky, where the expected ﬂux of dark matter annihilation/decay products is low. It is referred to as an "ON-OFF source approach". In this method, Monte Carlo simulation of WIMP induced neutrinos is used to optimize the analysis. In order to estimate the background, we use data itself, which allows to avoid dependence on the simulation and related systematic uncertainties. The impact of the dark matter halo model choice is discussed.

7.1 Equatorial Coordinate System

In the search for dark matter induced neutrinos originating in the Galactic Center, it is very convenient to use the equatorial coordinate system (see Fig. 7.1). Its origin is deﬁned at the center of the Earth and the fundamental plane is formed by the projection of the Earth’s equator onto the celestial sphere, forming the celestial equator. The primary direction is pointing towards the vernal equinox, which is one of the two points where the ecliptic intersects the celestial equator.

To determine the position of a point in this system, two coordinates are used. The ﬁrst one is the declination (DEC), which measures the angular distance of an object perpendicular to the celestial equator. It extends from 0◦ at the celestial equator to +90◦ at the north celestial pole, and to −90◦ at the south celestial pole. Declination is analogous to terrestrial latitude. The second coordinate is right ascension (RA) which measures the angular distance of an object eastward along the celestial equator from the vernal equinox and is similar to terrestrial longitude. It can be measured in degrees or hours, from 0◦/0h at the vernal equinox up to 360◦/24h.

One of the advantages of the equatorial coordinate system is the fact that it does not rotate with the Earth, but remains ﬁxed against the background stars. This is very convenient since then the stars positions do not depend on the position of the observer and the position of the GC is ﬁxed, which is crucial for this analysis. 58 Chapter 7. Search for the Galactic WIMPs at Super-Kamiokande

FIGURE 7.1: Deﬁnition of the coordinates (right ascension, declination) in the equatorial coordinate system.

The distribution of atmospheric neutrino events in equatorial coordinate system is shown in Fig. 7.2. In contrast, in the horizontal coordinate system, the stars positions differ among observers based on their positions on the Earth’s surface, and are continuously changing with the Earth’s rotation.

FIGURE 7.2: Atmospheric neutrino events from SK I-IV data sets in the equatorial coordinate system.

7.2 Simulation of WIMP Induced Neutrinos from the Galac- tic Center

The GC region is expected to contain the largest density of dark matter. For the halo models (NFW, Moore and Kravtsov) considered in Chapter6, neutrinos originating from dark matter annihilation/decay would introduce a significant modification to 7.2. Simulation of WIMP Induced Neutrinos from the Galactic Center 59 distribution of neutrino events observed at Super-K in equatorial coordinate system, where the position of the Galactic Center is fixed. The intensity of expected dark matter induced neutrino signal is proportional to the square of the dark matter density in the halo (in case of annihilation) or to the density (in case of decay). Therefore, the distribution of dark matter in the Milky Way is crucial for any robust predictions. For the assumed halo profile one can simulate the dark matter induced neutrino events based on calculations of the intensity of dark matter annihilation/decay products (see Fig. 6.3 and Fig. 6.4), in order to investigate expected signal properties.

The Monte Carlo simulation of the dark matter induced neutrino signal is based on the standard atmospheric MC described in Chapter5, which is available for the Super-K data in large statistics (500 years for each data taking period, 2000 years in total). Standard atmospheric MC plotted in the equatorial coordinates (RA, DEC and Ψ-angular distance from the GC) is shown in Fig. 7.3. For each plot, the true neutrino direction and reconstructed direction (based on direction of lepton produced in neutrino interaction) are shown. Due to Earth’s rotation, the detector gets an equal exposure for atmospheric neutrinos in RA.

FIGURE 7.3: Left: Standard atmospheric MC in the equatorial coordinate system (upper left plot for the true and bottom left for the reconstructed directions). Middle: Distribution in the cosine of the angular distance from the GC (upper middle plot) and the angular distance from the GC (bottom middle plot). Right: Distribution in right ascension (upper right plot) and declination (bottom right plot). Blue curves in middle and right plots correspond to true neutrino directions and black curves to the reconstructed directions of leptons produced in neutrino interactions.

Based on the "date" and "time" information of each event, the position of GC at a certain time can be calculated. Technically this calculation is done using Super-K astrophysical libraries. First, the ﬁxed position of GC in right ascension and declination is translated to the local frame (azimuth and zenith angle), and then, the GC position in the local frame can be compared with the position of the given event. Right ascension and declination of each event are calculated as well. The MC "time" 60 Chapter 7. Search for the Galactic WIMPs at Super-Kamiokande and "date" are assigned randomly based on the distribution of the year, month, day, hour, minute, second of the real data for each SK period. Therefore, MC reﬂects data taking periods of the real events.

In order to simulate the dark matter induced neutrinos, for every event in atmospheric MC, a WIMP weight is assigned. The WIMP weight value is based on the intensity of dark matter annihilation/decay products, that can retrieve the characteristic shape associated with dark matter density distribution for a given halo model. At a certain angular distance Ψ > 1◦ from the GC, the WIMP weight is proportional to intensity in 1◦ intervals: ( J(Ψ) for dark matter annihilation, WIMP weight(Ψ) ∼ (7.1) Jd(Ψ) for dark matter decay.

In this analysis, six sets of WIMP weights were used in order to reproduce the signal shape for three considered dark matter halo models (NFW, Moore, and Kravtsov), and for both the dark matter annihilation and decay scenarios. The numerical values of the calculated weights used in this analysis are listed in AppendixA. The intensity proﬁles for dark matter annihilation scenario have a very sharp shape for small angular distances from the GC. To take this effect into account, a ﬁner binning scheme is used for Ψ < 1◦ (see AppendixA).

FIGURE 7.4: Spherical projections of the expected signal shape in the equatorial coordinate system for Moore (left), NFW (middle) and Kravtsov (right) halo proﬁles, assuming the dark matter annihilation scenario. Upper plots show the true neutrino directions and bottom plots show the reconstructed lepton directions.

The resulting event distribution obtained in the reweighting procedure are presented in Fig 7.4- 7.7. Fig. 7.4 shows the spherical projections of the expected dark matter neutrino signal in the equatorial coordinate system, for the three halo proﬁles considered in the analysis, assuming dark matter annihilation scenario. 7.2. Simulation of WIMP Induced Neutrinos from the Galactic Center 61

FIGURE 7.5: Expected signal shape in RA (upper plots) and DEC (bottom plots) for Moore (left), NFW (middle) and Kravtsov (right) halo proﬁles, assuming the dark matter annihilation scenario. Blue curves correspond to the true direction of neutrino and black curves to the reconstructed one.

FIGURE 7.6: Expected signal shape in the cosine of the angular distance from the GC, cos (Ψ), (upper plots) and the angular distance, Ψ, from the GC (bottom plots) for Moore (left), NFW (middle) and Kravtsov (right) halo proﬁles, assuming the dark matter annihilation scenario. Blue curves correspond to the true direction of neutrino and black curves to the reconstructed one. 62 Chapter 7. Search for the Galactic WIMPs at Super-Kamiokande

The 1D projections showing RA (upper row) and DEC (lower row) distributions are presented in Fig. 7.5. The visible peak corresponds to GC direction (266◦ RA, −29◦ DEC). In Fig. 7.6 the projections showing the cosine of the angular distance from the GC, cos (Ψ) (upper plots) and the angular distance Ψ from the GC (bottom plots) are presented.

The results of the simulation are consistent with expectations. For each halo model, the annihilation signal is peaked and pointing towards to the GC direction. The differences between the three considered proﬁles and the effect of reconstruction procedure are clearly visible. One can expect differences in the obtained results due to the adopted halo model.

For the dark matter decay scenario, the expected signal intensity is linearly proportional to the dark matter density distribution in the halo. Therefore, the differences between halo models are smaller. The dark matter decay signal expectations for the NFW halo proﬁle are shown in Fig. 7.7.

FIGURE 7.7: Signal expectations for the NFW halo proﬁle, assuming dark matter decay scenario. Left: Spherical projections of the expected signal shape in the equatorial coordinate system (upper left plot for the true and bottom left for the reconstructed directions). Middle: Distribution in the cosine of the angular distance from the GC (upper middle plot) and the angular distance from the GC (bottom left plot). Right: Distribution in right ascension (upper right plot) and declination (bottom right plot). Blue curves on middle and right plots correspond to true neutrino direction and black curves to the reconstructed direction of lepton produced in neutrino interactions. 7.3. ON-OFF Source Approach 63

7.3 ON-OFF Source Approach

The dark matter induced neutrino flux would manifest itself through a large scale anisotropy, with the largest excess of neutrinos at the area of the GC, as shown in the previous section. To test this hypothesis, two regions on the sky are defined. The ON-source region is centered on the GC position (266◦ RA, -29◦ DEC) and defined by a circle around this point with a half-opening angle Ψ. The OFF-source region is equally-sized, but offset by 180◦ in RA (see Fig. 7.8).

FIGURE 7.8: The illustration of the ON-OFF source approach. The ON-source region is deﬁned around the GC where the signal concentration is the highest. The OFF-source region is shifted in RA by 180◦. Because ON- and OFF-source regions are equally sized, the expected number of background events is the same in both of them.

In both regions, signal and background events are expected. Signal events are de- ﬁned as dark matter induced neutrinos and they are concentrated in the ON-source region around the GC. Background events consist of atmospheric neutrinos and are expected to be equally distributed in the ON- and OFF-source regions. Therefore, the difference in the number of neutrino events between ON- and OFF-source regions should contain no background events. In this way, the background can be estimated directly from the data. This method is independent of the atmospheric Monte Carlo simulations and related systematic uncertainties as they should equally affect ON- and OFF-source regions. The difference in the number of neutrino events between ON- and OFF-source regions correspond to difference in the signal events:

bkg sig bkg sig sig sig sig ∆N = (NON + NON) − (NOFF + NOFF) ≈ NON − NOFF = ∆N . (7.2)

The difference in signal events is directly proportional to the velocity-averaged dark matter self-annihilation cross-section < σAv >, or inversely proportional to dark matter particle lifetime τ, if dark matter decay is considered.

7.4 Optimization of the ON-Source Region Size

The accuracy of the neutrino direction reconstruction strongly depends on its energy. Leptons produced in the interactions of low energy neutrinos (< few GeV) loosely follow the direction of the parent neutrinos. The average neutrino energy differs 64 Chapter 7. Search for the Galactic WIMPs at Super-Kamiokande between event categories used in Super-K atmospheric data analyses (see Fig. 5.2). Therefore, it is advantageous to carry out the optimization separately for FC Sub- GeV, FC MultiGeV, PC and UP-µ events. The expected neutrino signal shape from dark matter annihilation for NFW halo model is shown in Fig. 7.9 for FC SubGeV, FC MultiGeV, PC and UP-µ event categories.

FIGURE 7.9: Spherical projections of the expected signal shape in the equatorial coordinate system for NFW halo model, assuming the dark matter annihilation scenario for FC SubGeV, FC MultiGeV, PC and UP-µ event categories.

The goal of the optimization procedure is to choose the size of the ON-source region, so that the signal to background ratio is maximal. In order to do this, we calculate the ratio between the expected number of signal events, S (for arbitrary chosen annihilation cross section), and square root of the expected number of background events, B, for a range of different angular sizes of the ON-source region. The procedure is repeated for ON-source region radius varying from 5◦ to 60◦, with the step of 5◦. The maximal tested region sized is set to be 60◦ in order to avoid overlapping between ON- and OFF-source regions. The results of optimization of the ON-source region size for different event classes are listed in Tab. 7.1. The optimization results performed for our benchmark model, the NFW proﬁle, are used in further analysis. They are shown in Fig. 7.10, for the dark matter annihilation scenario.

The optimization of the size of the ON-source region was performed entirely with the simulated events. However, this is the only part of analysis where the Monte 7.4. Optimization of the ON-Source Region Size 65

TABLE 7.1: The results of the optimization of an ON-source region size assuming dark matter annihilation for NFW, Moore and Kravtsov halo proﬁle. The results for decay scenario are the same for all models, and the size of 60◦ is the result of the requirement that ON- and OFF-source regions cannot overlap.

Optimal size [◦] Event class Annihilation Decay NFW Moore Kravtsov All Models FC SubGeV 60 60 60 60 FC MultiGeV 30 25 55 60 PC 20 10 45 60 UPMU 10 5 40 60 ALL 35 20 60 60

√ FIGURE 7.10: The signal to square root of the background (S/ B) ratio as a function of the angular size of the ON-source region for FC SubGeV (blue), FC MultiGeV (magenta), PC (black), UP-µ (green) and all events together (red). The NFW halo model is assumed and dark matter annihilation scenario is considered.

Carlo events are used, later, the analysis is based solely on data in order to exclude any dependence on the MC simulations and related systematic uncertainties.

Chapter 8

Results of the Search for Galactic WIMPs

This chapter presents results of the searches for dark matter induced neutrino events with SK-I, II, III and IV data sets, using the ON-OFF source approach. The search for a large-scale neutrino anisotropy in equatorial coordinate system is performed. The analysis is focused on the area of the Galactic Center (ON-source) where the expected contribution from dark matter induced neutrinos should be higher than in the other part of the sky (OFF-source).

The results presented here were ofﬁcially approved by the Super-Kamiokande Col- laboration and presented by the author during XXIX Rencontres de Physique de la Vallee d’Aoste (La Thuile 2015) in Italy (Frankiewicz, 2016) and the Meeting of the American Physical Society, Division of Particles and Fields (DPF 2015) in Michigan, USA (Frankiewicz, 2015).

8.1 Asymmetry Between ON- and OFF-Source Regions

The comparison between the number of observed events in ON- and OFF-source regions is shown in Table 8.1. Different event categories are considered separately, and for each group of classes (FC SubGeV, FC MultiGeV, PC and UP-µ), the optimal size of the ON-source region is determined based on the simulated dark mater induced neutrino interactions (see previous Chapter7). Optimization assumes dark matter annihilation and NFW proﬁle as a benchmark halo expectation.

The resulting asymmetries between the ON- and OFF-source regions, deﬁned as A = (NON − NOFF)/(NON + NOFF) are also shown in the Table 8.1, and are plotted in Fig. 8.1. No excess of dark matter induced neutrino events from the Galactic Cen- ter has been observed. This is the main result of the performed analysis, which can be further interpreted under the assumption of various dark matter halo models and WIMP annihilation modes. 68 Chapter 8. Results of the Search for Galactic WIMPs

TABLE 8.1: The number of events observed in ON-source (NON) and OFF-source (NOFF) regions for each considered event classes together with resulting asymmetry (A) and the corresponding statistical errors (σ∆N and σA). The optimal size of the ON-source region is determined separately for each group of event categories.

sig Sample NON NOFF ∆N σ∆N A σA Fully Contained (FC) SubGeV e-like 0 decay-e 2597 2576 21 71.924 0.004 0.014 e-like 1 decay-e 283 290 -7 23.937 -0.012 0.042 Single-ring π0-like 154 141 13 17.176 0.044 0.058 µ-like 0 decay-e 677 710 -33 37.242 -0.024 0.027 µ-like 1 decay-e 1909 1924 -15 61.911 -0.004 0.016 µ-like 2 decay-e 146 167 -21 17.692 -0.067 0.056 Multi-ring π0-like 475 477 -2 30.854 -0.002 0.032 Fully Contained (FC) MultiGeV νe-like 47 58 -11 10.247 -0.105 0.097 ν¯e-like 158 193 -35 18.735 -0.100 0.053 µ-like 150 160 -10 17.607 -0.032 0.057 MultiRing νe-like 75 62 13 11.705 0.095 0.085 MultiRing ν¯e-like 45 47 -2 9.592 -0.022 0.104 MultiRing µ-like 139 137 2 16.613 0.007 0.060 MultiRing Other 100 120 -20 14.832 -0.091 0.067 Partially Contained (PC) Stopping 18 11 7 5.385 0.241 0.180 Through-going 81 91 -10 13.115 -0.058 0.076 Upward-going Muons (UP-µ)* Stopping 9.73 14.01 -4.29 4.872 -0.181 0.202 Through-going Non-showering 49.68 49.61 0.07 9.964 0.001 0.100 Through-going Showering 5.54 12.52 -6.98 4.249 -0.387 0.217 ∗ In case of UP-µ samples, estimated rate of background events due to horizontal cosmic ray muons is subtracted from the ﬁnal data set resulting in fractional entries. 8.2. Constraints on Dark Matter Self-Annihilation Cross-Section 69

FIGURE 8.1: The asymmetry in the number of events between the ON- and OFF-source regions.

8.2 Constraints on Dark Matter Self-Annihilation Cross-Section

Based on the null results in the considered samples, one can derive the upper limits on the allowed number of dark matter induced neutrinos, which can be further translated to upper limits on the velocity-averaged dark matter self-annihilation cross-section < σAv >. In order to do this, we have to assume the dark matter halo model and the annihilation channel.

The νν¯ annihilation channel is considered separately, since the annihilating dark matter particles of a given mass are resulting with neutrinos of the same energy. In order to investigate νν¯ annihilation channel we would consider only νµν¯µ (1/3 of the expected total νν¯ ﬂux), as muon neutrinos are the most abundant in SK data and cover the whole tested energy range. We group together µ-like events from FC SubGeV, FC MultiGeV, PC and UP-µ categories, keeping the optimal size of the ON-source region (60◦, 30◦, 20◦ and 10◦ respectively). For these samples, the difference in number of events between the ON- and OFF-source regions (∆Nsig) and the sig 90% C.L. upper limit on this value (∆N90 ) are calculated. These results are listed in Table 8.2. If the difference in number of signal events happen to be negative (in unphysical region), the upper limit is calculated using the Bayesian approach described in details in AppendixD.

For the remaining annihilation channels: bb¯, W+W− and µ+µ−, the energy spectra of neutrinos produced in the dark matter annihilation processes covers wide range of the energies (see example in Fig. 6.5). In this case we expect that the dark matter induced neutrinos would be present in all event categories. Again, for the purpose 70 Chapter 8. Results of the Search for Galactic WIMPs of this calculation we use only muon neutrino ﬂavor. Therefore we group together all µ-like events, and choose the ON-source region size to be 35◦. The resulting numbers are shown in the last row of Table 8.2.

TABLE 8.2: The number of neutrino events observed in ON-source and corresponding OFF-source regions for each considered µ-like event class. The optimal size of the ON-source region was determined separately for each class. The last column shows the upper 90% C.L limit on the allowed difference in number of signal events between two considered regions. Results for dark matter annihilation scenario.

◦ sig sig Sample Size [ ] On-source Off-source ∆N ∆N90 µ-like FC SubGeV 60 2732 2801 -69± 74.38 87.51 µ-like FC MultiGeV 30 289 297 -8± 24.21 35.24 PC 20 99 102 -3± 14.18 21.56 UP-µ∗ 10 64.95 76.14 -11.19±11.88 13.91 ALL 35 2503.78 2626.69 -122.91± 71.63 65.25 ∗ In case of UP-µ samples, estimated rate of background events due to horizontal cosmic ray muons is subtracted from the ﬁnal data set resulting in fractional entries.

It is possible to translate the upper limit on a difference in number of dark mat- sig ter induced muon neutrinos (∆N90 ) to corresponding difference in average dark d Φ∆ΩON matter induced muon neutrino fluxes between ON-source ( dE ) and OFF-source d Φ∆ΩOFF ( dE ) regions: sig dΦ dΦ ∆N → ∆ΩON − ∆ΩOFF . (8.1) 90 dE dE The translation from the number of events to the flux value can be done based on MC samples of the atmospheric neutrinos. One can match the number of observed dΦatm atmospheric events (Natm) with corresponding true value of neutrino flux ( dE ) for a given neutrino energy. The same relation can be obtained for signal events. The ratio of upper limit on number of dark matter induced neutrino events to the upper limit on corresponding difference in fluxes between ON-source and OFF-source region should be the same as similar ratio for atmospheric neutrinos:

sig N ∆N atm = 90 · d d d scaling factor. (8.2) Φatm Φ∆ΩON Φ∆ΩOFF dE dE − dE Thus, from the following proportion one can derive the upper limit on a difference in ﬂuxes of signal events in the investigated region of the sky (∆Ω = ∆ΩON = ∆ΩOFF) as: sig dΦ dΦ 4π ∆N dΦ livetime(MC) ∆ΩON − ∆ΩOFF = 90 · atm · , (8.3) dE dE ∆Ω Natm dE livetime(data) where one takes into account the scaling for the adopted size of ON-source region 4π ( ∆Ω ) and the different livetime for data and atmospheric MC.

Based on the theoretical expectations on dark matter halo proﬁle (see Sec. 6.1), one can translate the limit on the dark matter induced neutrino ﬂux (obtained for different event classes with the corresponding ON-source region ∆Ω) into the upper limit for velocity-averaged self-annihilation cross-section < σAv >. The Eq. 6.5 can 8.2. Constraints on Dark Matter Self-Annihilation Cross-Section 71 be rewritten in the form: dΦ dΦ < σ v > R ρ2 dN ∆ΩON − ∆ΩOFF = A ( − ) sc sc J∆ΩON J∆ΩOFF 2 . (8.4) dE dE 2 4πmχ dE

This upper limit can be evaluated for NFW, Moore and Kravtsov proﬁles, by using the different average signal intensities J∆Ω corresponding to the assumed model. dN The spectrum of annihilation products dE for the considered annihilation channel is calculated using DarkSUSY (see Sec. 6.4).

In this way one can determine the corresponding limit on < σAv > for assumed mass of dark matter particle and annihilation channel. For the νν¯ annihilation channel, the upper limit in the given WIMP mass range is determined based on that sample of FC SubGeV, FC MultiGeV, PC or UP-µ events, which dominates in that energy range (see Fig. 5.2). The results for bb¯, W+W− and µ+µ− are based on all µ-like events together, considered within 35◦ around GC. The results for the three considered proﬁles are presented in Fig. 8.2. Each time 100% BR to a given annihilation channel is assumed.

FIGURE 8.2: The upper 90% C.L. limit on dark matter self- annihilation cross-section as a function of the dark matter particle mass for bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) annihilation channels. The inﬂuence of the halo model choice is shown as a band around the result for the benchmark NFW proﬁle.

The derived constraints on the value of < σAv > strongly depend on the adopted halo model. Obtained differences between benchmark model (NFW) and models used as extreme cases (Moore and Kravtsov) can reach the order of magnitude. If the size of an ON-source region is smaller, the difference between obtained limits is even greater due to large discrepancies in the expected intensity of dark matter annihilation products close to the Galactic Center for considered halo proﬁles (see Fig. 6.3). Therefore, focusing on the GC makes the results very sensitive to the chosen proﬁle, but on the other hand, allows to obtain the best sensitivity for dark matter 72 Chapter 8. Results of the Search for Galactic WIMPs induced neutrino signal.

The results for NFW profile are compared to limit obtained in the independent analysis of Super-Kamiokande data referred to as "Global Fit" (Mijakowski, 2018), which is shown in Fig. 8.3. This independent analysis was performed by different author. It relies on a fit of simulated dark matter induced neutrino signal and the atmospheric neutrino background to the same data sample as used in this analysis. In the Global Fit analysis, all SK data samples (FC, PC, UP-µ) including both electron and muon neutrinos, are binned in angular and momentum bins and are used in the fit providing the best search sensitivity among the two presented methods. The ON- OFF source approach was designed as an independent cross-check for the Global Fit method. The advantage of ON-OFF source approach is being data driven and independent on atmospheric MC and related systematic uncertainties.

FIGURE 8.3: The upper 90% C.L. limits on dark matter self- annihilation cross-section as a function of dark matter particle mass for bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) annihilation channels (solid lines), compared with the results of independent Global Fit analysis (dotted lines) of Super-K data (Mijakowski, 2018).

8.3 Constraints on the Dark Matter Particles Lifetime

In analogical way, one can calculate the limits on WIMP particles lifetime. The same ON-OFF source number of events comparison is repeated in the cone of 60◦ around the GC. In this case the expected signal covers significant part of the sky (see Fig. 7.7), and size of the ON-source region is limited by the requirement that ON- and OFF- source regions cannot overlap with each other. The comparison between the number sig of observed events in ON- and OFF-source regions and obtained limits for N90 are shown in Table 8.3. Since there is no statistically significant excess in the search cone, one can derive the upper limits on dark matter particle lifetime τ, using the Eq. 6.6. The results for four considered decay channels bb¯, W+W−, µ+µ− and νν¯, assuming NFW halo profile are shown in Fig. 8.4. Each time 100% BR to a given decay channel is assumed. 8.3. Constraints on the Dark Matter Particles Lifetime 73

The results are compared with the limits form IceCube experiment (Abbasi et al., 2012). Super-Kamiokande detector is located in the northern hemisphere which gives better conditions to search for dark matter induced neutrino flux from the GC, as it can be seen with UP-µ events ∼71% of the time. Given IceCube’s location at the South Pole, a search for a neutrino flux from the GC, located in the southern sky is impacted by a significant background of cosmic-ray induced atmospheric muons and atmospheric neutrinos. Part of the detector has to be used as a veto for cosmic muons.

TABLE 8.3: The number of neutrino events observed in ON-source and corresponding OFF-source regions for each considered µ-like event class. The optimal size of the ON-source region is constrained to be 60◦ in order to no not have overlapping between ON- and OFF- source regions. The last column shows the upper 90% C.L limit on the allowed difference in number of signal events between two considered regions. Results for dark matter decay scenario.

◦ sig sig Sample Size [ ] On-source Off-source ∆N ∆N90 µ-like FC SubGeV 60 2732 2801 -69±74.38 87.51 µ-like FC MultiGeV 60 1133 1087 46±47.12 111.10 PC 60 783 777 6±39.50 68.76 UP-µ∗ 60 2353.55 2453.85 -100.30±69.34 68.69 ALL 60 7001.55 7118.85 -117.30±118.83 136.99 ∗ In case of UP-µ samples, estimated rate of background events due to horizontal cosmic ray muons is subtracted from the ﬁnal data set resulting in fractional entries.

FIGURE 8.4: The lower 90% C.L. limits on the dark matter particle lifetime as a function of dark matter particle mass for bb¯ (blue), W+W− (maroon), µ+µ− (purple) and νν¯ (orange) decay channels, compared with limits from IceCube experiment (dotted lines) (Abbasi et al., 2012). NFW halo model is assumed.

Chapter 9

WIMPs in the Center of the Earth

Neutrinos can be used as a probe of dark matter present in massive baryonic bodies like the Sun or the Earth. It is expected that WIMPs from the Galactic Halo can be bound in the gravitational potential of the Solar system, as it passes through the Milky Way. WIMPs can scatter off a nucleus inside a massive celestial bodies (the Sun or the Earth), lose energy, and be gravitationally trapped. Once captured, they eventually sink to the core due to subsequent scatters and then annihilate into leptons, quarks or bosons, producing neutrinos in the subsequent decays. Among the annihilation products, only neutrinos can escape from the center of the Sun or the Earth.

The idea to search for dark matter induced neutrino ﬂux was proposed by Press and Spergel (Press and Spergel, 1985), where capture rates for dark matter particles in the Sun have been calculated. The dedicated analysis to search for neutrinos from the annihilation of the captured low-mass WIMPs in the Sun was previously performed with the Super-Kamiokande data (Choi et al., 2015). The analysis dedicated to the Earth, performed by the author, will be described in the next three chapters.

9.1 Capture Rate of WIMPs in the Earth

The history of determining the capture rates in the Earth has gone through a series of signiﬁcant changes over the years, summarized in (Lundberg and Edsjo, 2004). This thesis follows the approach presented in (Sivertsson and Edsjo, 2012).

The capture rate of WIMPs depends on their mass, velocity in the halo, and local density. When the WIMP mass (Mχ) matches the mass of an element present in the Earth, the Earth can efﬁciently capture relic particles directly from the galactic halo. Therefore, the capture rate depends on the mass and distribution of the elements in the Earth. The most important elements are listed in Table 9.1. The Earth’s density proﬁle, given in (Seyfert, 1998) is used.

The WIMP velocity in the halo cannot be measured and it is estimated through simulations. A Maxwell-Boltzmann WIMP velocity distribution is assumed, with a velocity dispersion of 270 km/s, and circular velocity at the Solar system’s location of 220 km/s. The local dark matter halo density can be estimated from observations 3 and it is taken to be ρχ = 0.3 GeV/cm (Yüksel et al., 2007). Fig. 9.1 shows the rate at which dark matter particles are captured to the interior of the Earth, for a scattering cross-section of σ = 10−44 cm2. The visible peaks correspond to resonant capture on the most abundant elements 16O, 24Mg, 28Si, 56Fe, and their isotopes. 76 Chapter 9. WIMPs in the Center of the Earth

TABLE 9.1: The composition of the Earth’s core and mantle (Mc- Donough, 2003).

Element Atomic Number Core mass fraction Mantle mass fraction Oxygen, O 16 0.0 0.440 Silicon, Si 28 0.06 0.210 Magnesium, Mg 24 0.0 0.228 Iron, Fe 56 0.855 0.0626 Calcium, Ca 40 0.0 0.0253 Phosphor, P 30 0.002 0.00009 Sodium, Na 23 0.0 0.0027 Sulphur, S 32 0.019 0.00025 Nickel, Ni 59 0.052 0.00196 Aluminum, Al 27 0.0 0.0235 Chromium, Cr 52 0.009 0.0026

FIGURE 9.1: The capture rate of WIMPs in the Earth for a scattering cross section of σ = 10−44cm2, calculated in (Sivertsson and Edsjo, 2012). The normalization of the original calculation performed for σ = 10−42cm2 was adjusted due to recent exclusions from direct detection (Akerib et al., 2017). Figure taken from (Aartsen et al., 2017a).

9.2 WIMP Annihilation in the Earth

The differential equation governing the time evolution of the number of WIMPs, N, in a celestial body can be written in the form:

dN = C − C N2 − C N, (9.1) dt C A E where the three constants describe rate of WIMP capture (CC), annihilation (CA), and evaporation (CE). WIMP evaporation, means that once captured, WIMPs could be ejected from the celestial body by hard elastic scattering from nuclei. Only WIMPS 9.3. Neutrino Flux from WIMP Annihilation in the Earth 77 with low masses < 5 GeV may undergo this process (Jungman, Kamionkowski, and Griest, 1996), so this term can be neglected in our calculations.

WIMP annihilation, depends on the WIMP annihilation cross-section (σA) and effective volumes available for WIMP capture in the Earth:

3/2 hσAvi · (Mχ/(10 GeV)) CA = √ . (9.2) 2 8 · 1025cm3

The total annihilation rate (ΓA) can be written as: 1 Γ = C N2. (9.3) A 2 A The capture of WIMPs mainly depends on the WIMP-nucleon scattering cross-section (σχ−N), the WIMP mass (Mχ), and the mass and distribution of the elements in the Earth. Therefore, for capture in the Earth, the main contribution comes from the SI component σχ−N, which is quadratically proportional to the atomic mass of the elements in the Earth. If the capture rate is constant in time and we solve Eq. 9.1 for the annihilation rate, we ﬁnd:

1 t Γ (t) = C tanh2 , (9.4) A 2 C τ where τ is the time scale for capture and annihilation equilibrium to occur, and is equal to 1 τ = √ . (9.5) CCCA 9 The present rate can be calculated using age of the Solar system, t = t ∼ 4.6 · 10 years. If t/τ >> 1, the annihilation and capture are in equilibrium and dN/dt = 0. In this case, the annihilation rate depends only on the capture rate:

1 Γ = C . (9.6) A 2 C The equilibrium time scale (τ) is determined by the WIMP-nucleon scattering cross section σχ−N, the WIMP self-annihilation cross section < σAv >, and the WIMP mass 11 Mχ. For the Earth, this equilibrium time is of the order of 10 years for the SI WIMP- SI −43 2 nucleon scattering cross-section, σχ−N ∼ 10 cm (Jungman, Kamionkowski, and Griest, 1996). As t /τ << 1, the equilibrium between capture and annihilation has typically not been reached in the Earth and the annihilation rate goes as the square of the capture rate: 2 ΓA ∝ CC. (9.7) Therefore, for the higher capture rate, we have higher annihilation rate and thus the expected dark matter induced neutrino ﬂux.

9.3 Neutrino Flux from WIMP Annihilation in the Earth

WIMPs accumulated in the Earth’s core can self-annihilate at the rate ΓA, which is proportional to the square of its density (see Eq. 9.3). The neutrino ﬂux produced at 78 Chapter 9. WIMPs in the Center of the Earth the Earth’s surface can be written as: dΦ Γ dN = A ν 2 , (9.8) dEν 4πR⊕ dEν where R⊕ is the Earth radius and dNν/dEν is neutrino spectrum for all annihilation channels.

In order to calculate the neutrino ﬂux from WIMP annihilation in the Earth, we need an explicit WIMP candidate. We consider the lightest neutralino (χ), that arises as a natural dark matter candidate in supersymmetric extensions of the standard model (see Sec. 2.2.1). In the considered model, the direct annihilation into neutrinos χχ → νν¯ is strongly suppressed. We will consider three annihilation channels: bb¯, which gives the softest (less energetic) spectrum, and τ+τ− and W+W−, which give the hardest (the most energetic) spectrum. In this case µ+µ− channel is not considered, since produced muons will be stopped before they have a chance to decay and produce neutrinos. Neutrino spectra at the detector from the considered annihilation channels are generated with WimpSim (Blennow, Edsjö, and Ohlsson, 2008) software (see following Sec. 9.4).

9.4 WimpSim Results for the Earth WIMPs

Neutrino yields from WIMP annihilation in the Earth’s core are simulated with the WimpSim simulation package (Edsjö, 2007), which is based on DarkSusy (Gondolo et al., 2001). The simulation includes WIMP captured in the Earth, annihilation into 11 channels, hadronization of the products, and their decays leading to the production of neutrinos. The propagation through the Earth to the detector is also covered. Neutrino oscillations are included in a full three-ﬂavor framework including both vacuum and matter oscillations.

For the purposes of this analysis, the simulation was performed by the author for the following WIMP masses: 3, 6, 10, 25, 50, 80.3, 91.2, 100, 150, 176, 200, 250, 350, 500, 750, and 1000 GeV and three annihilation channels: bb¯, W+W−, and τ+τ−. For each mass and annihilation channel, 2.5 · 106 annihilations were simulated, keeping track of the all produced neutrinos and antineutrinos, and their ﬂavors.

The example of dark matter induced neutrino spectra at the surface of the Earth generated with WimpSim is shown in Fig. 9.2 for 6 GeV WIMPs annihilating into the bb¯ channel. Plots for a wide range of WIMP masses and three annihilation channels considered in the analysis can be ﬁnd in AppendixB.

The output of WimpSim, used in this analysis, consists of text files containing 50 lines for each neutrino and antineutrino flavor. Each line is for one of 50 bins in z = 0.01, 0.03, . . . , 0.99, where z is the ratio between neutrino energy and assumed dark matter particle mass: z = Eν/Mχ. Each line contains 91 values corresponding to d2N/dzdθ, where θ is the angular distance from the Earth’s core, and is binned in 91 bins from 0 to 30◦. The θ bins are as follows: • The first 50 bins are for θ = 0.1◦ , 0.3◦ , . . . , 9.9◦ • The next 40 bins are for θ = 10.25◦ , 10.75◦ , . . . , 29.75◦ • The final bin in θ is for θ > 30◦. 9.4. WimpSim Results for the Earth WIMPs 79

FIGURE 9.2: The differential ﬂuxes of the neutrinos (left panels) and anti-neutrinos (right panels) produced by 6 GeV WIMPs annihilating into bb¯ quarks, as a function of the neutrino energy Eν, divided by the WIMP mass Mχ. The different neutrino species are indicated with a different color, black for νe, red for νµ, and green for ντ.

The comparison of dark matter induced muon neutrino spectra at the surface of the Earth generated for the same WIMP mass and for the three annihilation channels considered in the analysis is presented in Fig. 9.3. The fact that WIMP annihilation does not occur exactly in the center of the Earth (the size of the region depends on the WIMP mass) is also taken into account, and is shown in Fig. 9.4. The annihilation 2 region radius scales as ∼ 1/Mχ, so for higher WIMP masses, neutrinos produced in dark matter annihilation are more concentrated around the Earh’s core.

FIGURE 9.3: Energy spectra of the dark FIGURE 9.4: Angular distribution of the dark matter induced muon neutrinos at the matter induced muon neutrinos observed at Earth’s surface generated with WimpSim, the Earth’s surface, generated with Wimp- for 100 GeV WIMPs annihilating into bb¯ Sim for 10 GeV (solid line), 100 GeV (dashed (blue line), τ+τ− (green line), and W+W− line), and 1 TeV (dotted line) WIMPs annihi- (maroon line) channel. lating into b¯ quarks.

The results from WimpSim allow us to obtain the angular and energy characteristics expected for dark matter induced neutrinos from the center of the Earth. These distributions are used as a ﬁrst step in the simulation of the signal in the Super-K detector, which is described in the next chapter. WimpSim also provides a conversion script that can convert between neutrino ﬂux, annihilation rate, and corresponding SI WIMP-nucleon scattering cross-section for an assumed dark matter particle mass and annihilation channel, taking into account different thresholds and angular cuts 80 Chapter 9. WIMPs in the Center of the Earth for the detector. This conversion script is used for the interpretation of results discussed in Chapter 11. 81

Chapter 10

Search for the Earth WIMPs at Super-Kamiokande

In this chapter, the methodology of the search for dark matter induced neutrinos from the Earth’s core with the Super-Kamiokande data is presented. The approach applied here is different than the "ON-OFF source" technique used in the previously presented analysis focused on neutrinos from the Galactic Center, and the latter one would be referred to as a "Global ﬁt" approach.

In this analysis, it is assumed that the atmospheric neutrino data collected with the SK detector can be described by two components: atmospheric neutrinos (background) and neutrinos produced in dark matter annihilation in the Earth’s core (signal). The goal of the search is to evaluate the allowed signal contribution. In order to do this, a ﬁt method similar to the one used in the standard neutrino oscillation analysis (see Section 5.3) is adopted. Instead of scanning the oscillation parameters 2 2 (∆m32, sin θ23 and δCP), we will be scanning over the WIMP masses (Mχ) and dark matter contribution (β), which is normalized to the number of atmospheric neutrino data.

10.1 Global Fit Approach

Similarly to the standard oscillation analysis, data is ﬁt to the combination of signal and background MC, using a binned χ2 method that is built assuming Poisson statistics and incorporating systematic errors as a scaling factors on the MC in each bin (Fogli et al., 2002):

2 MC i DM k χ = 2 ∑ Nn 1 + ∑ fnei + βNn 1 + ∑ fn ek n i k ! NDATA − DATA n (10.1) Nn ln MC i DM k Nn (1 + ∑i fnei) + βNn (1 + ∑k fn ek) e 2 e 2 + ∑ i + ∑ k . i σi k σk

MC In this equation, Nn represents the background MC expectation (atmospheric neu- DM trinos with oscillations), Nn represents the signal MC expectation (DM induced DATA neutrinos) and Nn is the corresponding data in the n-th analysis bin. Systematic errors are incorporated into the ﬁt via the systematic error parameter ei, where i is i the systematic error index and fn is the fractional change in the MC expectation in the n-th bin for a σi change in the i-th systematic error. Systematic errors penalize 82 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

2 the χ based on their corresponding ﬁtting parameters ei. Parameter β represents the varied normalization of the simulated signal and the determination of its value is the main goal of this analysis.

2 In the fit, the system of equations defined by the requirement ∂χ /∂ei = 0, is solved for each systematic error, which brings the data and MC into the best agreement allowed by the systematic errors. This minimization in the systematic error parameters is repeated over a grid of WIMP parameters (Mχ and β), and global minimum of χ2 defines the best f it point.

The following WIMP parameters are considered in this analysis:

• β: 301 points from -0.01 to 0.02

• Mχ: 15 points for bb¯ annihilation channel (6, 10, 25, 50, 80.3, 91.2, 100, 150, 176, 200, 250, 350, 500, 750, 1000 GeV) 16 points for τ+τ− annihilation channel (3, 6, 10, 25, 50, 80.3, 91.2, 100, 150, 176, 200, 250, 350, 500, 750, 1000 GeV) 11 points for W+W− annihilation channel (80.3, 91.2, 100, 150, 176, 200, 250, 350, 500, 750, 1000 GeV)

The lower value of the tested WIMP mass range for a given annihilation channel is due to kinematical restrictions on the production of b quarks, τ leptons, and W bosons. The upper analysis threshold, Mχ = 1000 GeV, is related to limited statistics in the MC samples which are used to simulate signal and background component at such high neutrino energies.

There are 160 sources of systematic uncertainties taken into account in the analysis (see Sec. 5.3.1). All of them are applied to the background part describing atmospheric neutrinos. For the signal part describing WIMP induced contribution, solar activity and ﬂux related systematic errors are not taken into account. Uncertain- ties corresponding to neutrino interactions, particle production, PMNS oscillation parameters and detector related are applied to signal in the same way as for the background.

The simulated component describing background predictions (NMC) depends on the values of neutrino oscillation parameters: three mixing angles θ13, θ12, θ23, two 2 2 mass splittings ∆m21, ∆m32, and one CP-violating phase δCP.

2 Oscillation parameters: θ13, θ12 and ∆m21 are treated in the same way as during standard oscillation analysis described in Chapter5. They are ﬁxed at the values from the global ﬁt, listed in Tab. 5.3, and varied only through the systematic errors.

2 The values of θ23 and ∆m32 are ﬁxed during the ﬁt at their best f it point from the 2 standard oscillation analysis of SK data, assuming normal mass hierarchy: sin θ23 = 2 −3 2 0.588 and |∆m32| = 2.50 × 10 eV . Since super-Kamiokande has a very small sensitivity to δCP, this parameter is set to be 0.

The ﬁt is performed using 19 data samples, including both e-like and µ-like event categories. Each sample is binned in lepton momentum and cosine of the zenith angle, which gives in total 520 analysis bins for each of the SK running period (see Chapter5 for detailed description). 10.2. Signal and Background Predictions 83

10.2 Signal and Background Predictions

In order to simulate the detector response for the signal (WIMP induced neutrinos from the Earth’s core), the existing simulation of atmospheric neutrino interactions is used (see Sec. 5.2). This Monte Carlo assumes realistic ﬂuxes of neutrinos, and includes their interactions in the Super-K tank or in the rock surrounding the detector. The simulation also includes the detector response for the produced Cherenkov light. The same set of data reduction steps and classiﬁcation cuts are applied to simulated events and to the real data.

The background component is described by the half of the events from the standard atmospheric MC (250 years for each SK period). The plot with the background expectations is shown in Fig. 10.1.

Background illustration with the standard binning

FIGURE 10.1: Atmospheric neutrino background expectation (orange histogram) and the SK-I-IV data (black points). The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. In all panels, the error bars represent the statistical uncertainty.

Second half of the events from the standard atmospheric MC is reweighted in a way to obtain characteristics expected for dark matter induced neutrinos. The assigned weights are based on the results obtained using WimpSim simulation (see Sec. 9.4 for the details). The Earth WIMP weight is assigned to each event, based on its energy 84 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande and angular distance from the Earth’s core. After the reweighting, the signal part of MC is normalized to the background part (describing atmospheric neutrinos), therefore the β parameter used in the ﬁt correspond to the fraction of atmospheric neutrinos.

The dark matter induced neutrino signal from the Earth’s core is simulated for three annihilation channels: bb¯, τ+τ−, and W+W−. Plots of the signal contributions are shown in Fig. 10.2 for 25 GeV WIMPs annihilating into bb¯ quarks, in Fig. 10.3 for 100 GeV WIMPs annihilating into τ+τ− leptons, and in Fig. 10.4 for 500 GeV WIMPs annihilating into W+W− bosons.

Signal illustration for Mχ = 25 GeV, β = 0.02, bb¯ annihilation channel

FIGURE 10.2: The signal originating from the annihilation of 25 GeV WIMPs in the Earth’s core into bb¯ quarks. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. The plots contain the expected signal from WIMP annihilation (ﬁlled blue histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty.

All data samples which are being used in the analysis are presented there. The samples in the ﬁrst column (SubGeV e-like 1 decay electron, SubGeV π0-like and SubGeV µ-like 2 decay electron) have only one angular bin, therefore the reconstructed momentum distributions are shown. Other samples with more than one 10.2. Signal and Background Predictions 85

+ − Signal illustration for Mχ = 100 GeV, β = 0.02, τ τ annihilation channel

FIGURE 10.3: The signal originating from the annihilation of 100 GeV WIMPs in the Earth’s core into τ+τ− leptons. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. Plots contain expected signal from WIMP annihilation (filled green histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty. zenith angle bin are shown as zenith angle distributions (second through fifth column). More illustrations for the wide range of considered WIMP masses can be find in AppendixC.

The WIMP induced signal appears in various samples, depending on the assumed WIMP mass and annihilation channel. According to the expectations, the signal is always located in bins close to the Earth core (cos θ = −1) .

All neutrino ﬂavors (νeν¯e, νµν¯µ and ντν¯τ) are used in the signal and background MC simulations. Tau neutrinos are produced in the subsequent decays of WIMP annihilation products. They would be mainly categorized as MultiGeV e-like and Multi-Ring event samples due to the complex topology of secondary particles produced in tau neutrino interactions (τ decays immediately producing leptons and mesons). 86 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

+ − Signal illustration for Mχ = 500 GeV, β = 0.02, W W annihilation channel

FIGURE 10.4: The signal originating from the annihilation of 500 GeV WIMPs in the Earth’s core into W+W− bosons. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. Plots contain expected signal from WIMP annihilation (ﬁlled maroon histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty.

10.3 Sensitivity Study for WIMP Induced Neutrinos from the Earth’s Core

Dedicated studies are performed in order to estimate the SK detector sensitivity to the dark matter induced neutrino signals from the Earth’s core. Based on the prepared simulation sets, the expected sensitivity to each of the WIMP parameters (Mχ, β) can be computed. In order to do this, we choose a point in the WIMP parameter space, for which we have prepared the signal and background expectations. This point is treated as the observed data. Then, we preform "pulled" χ2 test for all 10.3. Sensitivity Study for WIMP Induced Neutrinos from the Earth’s Core 87 points from our parameter space against the template that is the assumed observation. Based on the obtained χ2 value of the ﬁt, we can draw sensitivity contours. Note that the background expectation is the same for all tested points from WIMP parameter space.

During the analysis we consider three annihilation channels separately: (bb¯, τ+τ− and W+W−). Each time, a 100% branching ratio (BR) to a given annihilation mode is assumed as a model independent approach.

10.3.1 Null WIMP Contribution (β = 0) First of all, we want to test null hypothesis when β = 0. This case corresponds to the result from the standard oscillation analysis (described in Sec. 5.3). Here we assume that all data observed by the SK detector can be explained by atmospheric neutrino oscillations. We calculate "pulled" χ2 for this hypothesis against all points in our WIMP parameter space (deﬁned in Sec. 10.1) . Results of this test are plotted in Fig. 10.5 for the three considered annihilation channels. For each tested WIMP 2 mass (Mχ - X axis) and WIMP contribution (β - Y axis), the resulting χ value of the ﬁt against no WIMP contribution hypothesis is plotted as color, creating so called "heat map". It is visible that the lowest χ2 values are located around β = 0 line (dark blue) as expected. As we move away from β = 0 line, χ2 values are getting higher, which corresponds to increasing WIMP contribution. The same results can be presented as sensitivity contours. The 68% (blue), 90% (yellow) and 99% (maroon) C.L. contours for β = 0 hypothesis are shown in Fig. 10.6.

One can notice that the sensitivity contours are getting wider for lower WIMP masses (the left side of top and middle plots). Also If we compare the sensitivity contours in Fig. 10.6 between bb¯ and τ+τ− for the same WIMP mass, the contour for the soft channel (bb¯) is wider then for the hard channel (τ+τ−). In general, it is easier to distinguish a signal when it consists of neutrinos of higher energies. This effect can be explain by the fact that the WIMP annihilation does not occur exactly in the center of the Earth, and the size of the region depends on the WIMP mass (see 2 Sec. 9.4). The annihilation region radius scales as ∼ 1/Mχ. Therefore, for higher WIMP masses, neutrinos produced through dark matter annihilation are more concentrated around the Earth’s core, and it makes them easier to distinguish from the atmospheric neutrino background. Moreover, the angular resolution of the detector strongly depends on the energy of the parent neutrinos. Leptons produced in interactions of low energy neutrinos (< few GeV) loosely follow the direction of the parent neutrino. Therefore, the average angular separation between the true parent neutrino direction and the reconstructed lepton direction is greater. These effects are also visible in the signal plots for τ+τ− channel presented in AppendixC. For low WIMP masses (Mχ = 3 GeV), the signal is distributed over many angular bins and present in many samples. For high masses (Mχ ≈ 1000 GeV), almost the whole signal is concentrated in narrow UP-µ bins. 88 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

FIGURE 10.5: Heat map showing the tested parameter space of WIMP masses (X - axis) and WIMP contribution (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). Null WIMP contribution (β = 0) is assumed and the color represents χ2 value of the ﬁt. 10.3. Sensitivity Study for WIMP Induced Neutrinos from the Earth’s Core 89

FIGURE 10.6: Sensitivity contours at 68% (blue), 90% (yellow) and 99% (maroon) C.L for null WIMP contribution (β = 0), for tested parameter space of WIMP masses (X - axis) and WIMP contribution (Y - axis), plotted for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). 90 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

In order to better illustrate the sensitivity of the fit, we can plot 1D projection of ∆χ2 value as a function of β parameter (see Fig. 10.7), for each considered WIMP mass. The ∆χ2 is defined as a difference between χ2 obtained for a given value of 2 beta parameter and χmin, the value obtained for the best f it point. In case of a null 2 2 WIMP contributions, χmin = 0 for β = 0, where all obtained ∆χ parabolas have their minimum. The horizontal lines show the ∆χ2 equal to 1 (blue), 2.71 (yellow), and 6.63 (maroon), corresponding to 68%, 90% and 99% C.L. limit on β value respectively. The intersection between ∆χ2 parabola and 90% C.L. horizontal line defines 2 β90 value. One can notice that the width of ∆χ parabola is changing with mass.

2 2 2 FIGURE 10.7: The ∆χ = χ − χmin distributions minimized over the corresponding WIMP masses. The horizontal lines correspond to the 68% (blue), 90% (yellow) and 99% (maroon) C.L. for no WIMP hypothesis (β=0), plotted for τ+τ− annihilation channel.

Based on the obtained values of β90 corresponding to 90% C.L. limit on dark matter induced neutrino contribution for a given WIMP mass, the upper limits on the WIMP induced muon neutrino ﬂux are calculated using MC (in analogical way as described in Sec. 8.2) and shown in Fig. 10.8. These results are translated to limits on the SI WIMP-nucleon scattering cross-section presented in Fig. 10.9. Since the equilibrium between WIMP capture and annihilation in the Earth core has not 10.3. Sensitivity Study for WIMP Induced Neutrinos from the Earth’s Core 91 been achieved (see Sec. 9.2), this limits depend on the velocity averaged dark matter self-annihilation cross-section. For the calculation we assumed a typical value of −26 3 −1 < σAv >= 3 · 10 cm s , for which the WIMP is a thermal relic (see Sec. 2.2). For the translation, we use conversion script provided by WimpSim (see Sec. 9.4).

FIGURE 10.8: The 90% upper limits on the FIGURE 10.9: The 90% upper limits on SI dark matter induced muon neutrino ﬂux WIMP-nucleon scattering cross-section from from the sensitivity study for bb¯ (dotted the sensitivity study for bb¯ (dotted blue), τ+τ− blue), τ+τ− (dotted green) and W+W− (dotted green) and W+W− (maroon line) an- (maroon line) annihilation channel. nihilation channel. Positive result claimed by DAMA/LIBRA (Bernabei et al., 2013) is shown for comparison.

The structure of the limits corresponds to enhanced WIMP capture in the Earth’s core, when WIMP mass (Mχ) matches the mass of an element present in the Earth. The peaks visible in Fig. 10.9 correspond to most abundant elements: 16O, 24Mg, 28Si, 56Fe, and their isotopes (see Fig. 9.1 for comparison).

From the sensitivity study results, we can estimate the possible range of SI WIMP- nucleon scattering cross-section that can be probed with Super-Kamiokande data. One can see that most of the region of the positive signal claimed by DAMA/LIBRA experiment can be accessed with this analysis.

The presented results of the sensitivity study for null WIMP contribution were of- ﬁcially approved by the Super-Kamiokande Collaboration in June 2016. They were presented by the author during the 38th International Conference on High Energy Physics (ICHEP 2016) in Chicago, USA (Frankiewicz, 2017b), and XXVII Interna- tional Conference on Neutrino Physics and Astrophysics (Neutrino 2016) in London, UK (Frankiewicz, 2017a). 92 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

10.3.2 Injected WIMP Contribution (β = 0.01) After testing null hypothesis, we also test the ﬁt behavior for the positive WIMP contribution. The three example points:

• Mχ = 25 GeV, β = 0.01, bb¯ annihilation channel, + − • Mχ = 100 GeV, β = 0.01, τ τ annihilation channel, + − • Mχ = 500 GeV, β = 0.01, W W annihilation channel, are now treated as the observed data and we ﬁt all points from our parameter space against these presumed observations. The results are shown in Fig. 10.10, as the heat maps. One can clearly see that the lowest χ2 values are located around tested points with β = 0.01 and at injected WIMP masses. The same results presented as 68% (blue), 90% (yellow) and 99% (maroon) C.L. contours are shown in Fig. 10.11.

This simple test shows that the ﬁtting procedure is sensitive to the possible WIMP contribution. Therefore, if the Super-Kamiokande data contains dark matter induced neutrinos on a level greater than the limits derived from the null hypothesis, we will be able to distinguish it from atmospheric neutrino background. We have working framework which can be used to analyze the data. 10.3. Sensitivity Study for WIMP Induced Neutrinos from the Earth’s Core 93

FIGURE 10.10: Heat map plot showing the tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). Positive WIMP contribution (β = 0.01) is assumed 2 and the color represents χ value of the ﬁt. In bb¯ channel: Mχ = + − + − 25 GeV, in τ τ : Mχ = 100 GeV, and in W W : Mχ = 500 GeV. 94 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

FIGURE 10.11: Sensitivity contours at 68% (blue), 90% (yellow) and 99% (maroon) C.L for positive WIMP contribution (β = 0.01), for tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), + − + − τ τ (middle) and W W (bottom). In bb¯ channel: Mχ = 25 GeV, in + − + − τ τ : Mχ = 100 GeV, and in W W : Mχ = 500 GeV. 10.4. Binning Optimization 95

10.4 Binning Optimization

During the preformed sensitivity studies, it was noticed that for the higher WIMP masses (Mχ > 100 GeV), the signal is located very close to the direction of the Earth’s core (cos θz = −1), even for the "soft" bb¯ annihilation channel. As it is shown in Fig. 9.4, for Mχ = 1 TeV, dark matter induced neutrinos are expected to be contained in a region with radius smaller than 3◦, which is much smaller than the bin size used in the oscillation analysis 5.1.

◦ The size of the first bin in the standard oscillation analysis is 36.9 (0.2 in cos θz) ◦ for FC and PC, and 25.8 (0.1 in cos θz) for UP-µ events. This size was chosen to provide at least 6 background events expected in each bin after scaling to the data livetime of the analysis. We decided to perform the analysis with the finer binning, which is defined in the Table 10.1.

TABLE 10.1: Optimized binning in cosine of the zenith angle for FC SubGeV, FC MultiGeV, PC and UP-µ samples.

Sample cos θz bins Fully Contained (FC) SubGeV 10 in [-1,1] Fully Contained (FC) MultiGeV 12 in [-1,1] Partially Contained (PC) 12 in [-1,1] Upward-going Muons (UP-µ) 13 in [-1,0]

FC SubGeV events are binned in the cos θz in the same way as in the standard oscillation analysis (see Sec. 5.1):

(−1, −0.8. − 0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 1).

For FC MultiGeV and PC events we divide the first and the last bin (for symmetry) in half: (−1, −0.9, −0.8. − 0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 0.9, 1), and for UP-µ events, which are defined only below the horizon, we divide the first bin in four:

(−1.0, −0.9994, −0.9962, −0.94, −0.9, −0.8, −0.7, −0.6, −0.5, −0.4, −0.3, −0.2, −0.1, 0.0).

The extra UP-µ bins correspond to 2◦, 5◦, 20◦ and 25.8◦ angular distance from the Earth’s core. As a result of this procedure, 595 analysis bins are created for each of the SK periods (2380 bins in total). 96 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

The plot with the atmospheric neutrino background expectations with the optimized binning is shown in Fig. 10.12.

Background illustration with the optimized binning

FIGURE 10.12: Atmospheric neutrino background expectation (orange histogram) and the SK-I-IV data (black points) with the optimized binning. The momentum (1st column) and zenith angle (2nd- 5th columns) distributions are shown for 19 samples used in the analysis. In all panels, the error bars represent the statistical uncertainty.

Fig. 10.13 shows the SK data and the signal expectations for Mχ = 1000 GeV and bb¯ annihilation channel, with the optimized binning. Majority of the signal is concentrated in very narrow UP-µ bins, close to the Earth’s core. More illustrations for the wide range of considered WIMP masses can be ﬁnd in AppendixC. This optimized binning improves the sensitivity for dark matter induced neutrino signal from the Earth’s core. 10.4. Binning Optimization 97

Signal illustration for Mχ = 1000 GeV, β = 0.02, bb¯ annihilation channel

FIGURE 10.13: The signal originating from the annihilation of 1000 GeV WIMPs in the Earth’s core into bb¯ quarks with the optimized binning. The momentum (1st column) and zenith angle (2nd- 5th columns) distributions are shown for 19 samples used in the analysis. Plots contain expected signal from WIMP annihilation (ﬁlled histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels the error bars represent the statistical uncertainty.

To better present this effect, the first four bins from the Earth’s core are plotted separately in Fig. 10.14, for the three UP-µ samples. Note that first four new bins together cover the size of 25.8◦, which is a size of the first UP-µ bin in standard analysis. Due to the very small size of the new bins, the number of expected background events could be very low. This expectation is calculated based on the MC simulation and shown in Table 10.2. For this reason, we have to use different method to compute the sensitivity, which is based on Toy MC data sets and described in the following section. 98 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

Bin Expected bkg Up Stop µ 0◦ − 2◦ 0.65 2◦ − 5◦ 3.07 5◦ − 20◦ 47.91 20◦ − 28.5◦ 35.46 Non showering µ 0◦ − 2◦ 0.87 2◦ − 5◦ 4.62 5◦ − 20◦ 117.50 20◦ − 28.5◦ 97.83 Showering µ 0◦ − 2◦ 0.42 2◦ − 5◦ 1.99 5◦ − 20◦ 38.63 20◦ − 28.5◦ 29.04

TABLE 10.2: The calculated atmospheric neutrino background expectation for ﬁrst four bins in UP-µ samples.

FIGURE 10.14: Signal expectation (ﬁlled histogram) from Fig. 10.13 and the SK-I-IV atmospheric neutrino data (black points) for UP-µ samples.

10.4.1 Toy Monte Carlo Method For the analysis with optimized binning, we use a method to compute conﬁdence intervals numerically, developed by Feldman and Cousins (Feldman and Cousins, 1998). The method is based on toy Monte Carlo (ToyMC) experiments, which are simulated data sets drawn from the assumed probability density function (PDF). We adopt this method in order to determine 90% C.L. upper limit on dark matter induced neutrino contribution (β) for a given WIMP mass (Mχ), assuming null WIMP contribution (β = 0).

The algorithm proposed by Feldman and Cousins computes the interval [β1, β2] for each value of the true parameter (β0) in a sufﬁciently ﬁne grid. We test 401 β 10.4. Binning Optimization 99 values from 0 to 0.02 for each assumed WIMP mass and annihilation channel. In case of this analysis we do not test negative β contribution. This is motivated by a very low atmospheric background expectation in the narrow UP-µ bins around the Earth’s center (see Table 10.2). If we would add the negative contribution from the dark matter induced neutrino signal, we may end up with resulting negative number of events (signal+background) expected in these bins, which we want to avoid.

The steps of the procedure for each tested WIMP mass and assumed annihilation channel, are as follow:

• Generate 1,000 ToyMC experiments at the considered value of the true parameter β0, in our case with null WIMP contribution (β0 = 0). We generate the fake data sets representing atmospheric neutrino background by setting the number of events in each bin according to a Poisson distribution. During the ToyMC generation we ﬂuctuate data set according to the data livetime and within systematic errors (see Sec. 5.3.1).

• For each generated ToyMC experiment perform a pulled χ2 test and calculate ∆χ2 = χ2 − χ2 , where χ2 corresponds to the best f it point from a given toy β0=0 min min Toy data set, and χ2 corresponds to point β = 0 from the same data set. β0=0 0

2 • Find the critical value ∆χc , such that 90% of the ToyMC experiments have 2 2 2 ∆χtoy < ∆χc . Note that the ∆χc is different for each tested WIMP mass and assumed annihilation channel. The example for 6 GeV WIMPs annihilation into bb¯ quarks is shown in Fig. 10.15.

2 2 2 FIGURE 10.15: The distribution of ∆χ = χ − χ for Mχ = toy β0=0 min 6 GeV, bb¯ annihilation channel. Vertical line shows the critical value 2 2 2 ∆χc , such that 90% of the ToyMC experiments have ∆χtoy < ∆χc .

• For each generated ToyMC calculate β , such that χ2 − χ2 = ∆χ2. 90 β90 min c

• 90% C.L. upper limit on β (β90%CL) at given Mχ and annihilation channel, is given by the median from β90 distribution. The example for 6 GeV WIMPs 100 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

annihilation into bb¯ quarks is shown in Fig. 10.16. The ±1σ and ±2σ ﬂuctua- tion of the β90%CL value are given by 68.27% and 95.45% of the β90 distribution respectively (green and yellow vertical lines).

FIGURE 10.16: The distribution of β90 for Mχ = 6 GeV, bb¯ annihilation channel. Vertical lines shows the median (black), ±1σ (green) and ±2σ (yellow).

• The procedure is repeated for each tested WIMP mass and annihilation channel.

The summary of the sensitivity study with the ToyMC data sets is shown in Fig 10.17. The resulting β90%CL is translated to allowed number of dark matter induced neutrinos in the ﬁt, and plotted as a function of WIMP mass for three considered annihilation channel. The ±1σ and ±2σ systematic and statistical uncertainty is plotted as a green and yellow band respectively.

The 90% C.L. upper limit on allowed number of dark matter induced neutrinos from sensitivity can be translated to the limit on SI WIMP-nucleon scattering cross-section in the same way as described is Sec. 10.3.1. For bb¯ channel, the limit resulting from sensitivity studies with standard binning (see Fig. 10.9) is compared with the new result with the optimized binning. This comparison is shown in Fig. 10.18. One can notice that for low tested WIMP masses below 100 GeV, we achieve only slight improvement, but for higher WIMP masses > 100 GeV, the improvement is signiﬁcant. The optimized binning gives ∼ 3 times stronger limits. 10.4. Binning Optimization 101

FIGURE 10.17: The sensitivity expectation for a null WIMP hypothesis (dotted lines) and its ±1σ (green band) and ±2σ (yellow band) uncertainty is presented for the bb¯ (top), τ+τ− (middle) and W+W− (bottom) annihilation channel. 102 Chapter 10. Search for the Earth WIMPs at Super-Kamiokande

FIGURE 10.18: Comparison between 90% upper limits on SI WIMP- nucleon scattering cross-section for bb¯ annihilation channel, calculated with standard (black line) and optimized (blue line) binning. The signiﬁcant improvement for Mχ > 100 GeV is visible. 103

Chapter 11

Results of the Search for the Earth WIMPs

In this chapter, the results of the search for dark matter induced neutrinos from the Earth’s core are presented. The atmospheric neutrino data collected with the Super- Kamiokande detector in years 1996-2016 is used in the analysis (see Chapter5 for the detailed description). In the ﬁrst section (11.1), we present the results of the analysis with standard binning (2080 bins in cosine of zenith angle and momentum or visible energy). This part was approved by the Super-Kamiokande Collaboration in June 2017 and results were presented by the author during the Meeting of the American Physical Society, Division of Particles and Fields (DPF 2017) and 26th International Workshop on Weak Interactions and Neutrinos 2017 (WIN2017) conferences (Rott, 2017). The second part of this chapter (Sec. 11.2), presents the results with ﬁner bins, described in Section 10.4, optimized for WIMP masses > 100 GeV. They were approved by the Collaboration in November 2017 and presented by Piotr Mijakowski during the 53rd Recontres de Moriond 2018 (Mijakowski, 2018).

11.1 Fit Results with Standard Binning

In order to ﬁnd the allowed contribution of dark matter induced neutrinos in the SK data, the "pulled" χ2 ﬁt described in Sec. 10.1 is performed. For one of the tested hypothesis: 150 GeV WIMPs annihilating into bb¯ channel, Fig. 11.1 shows:

• expected signal contribution from the dark matter induced neutrinos before the ﬁt (ﬁlled blue histogram), enlarged for better visibility,

• atmospheric neutrino background before the ﬁt (orange histogram),

• the best conﬁguration of signal and background after the ﬁt (purple histogram),

• SK I-IV data (black points) with error bars represent statistical uncertainty.

The minimal value of the χ2 corresponds to β = 0.0007. Due to extremely small signal contribution, result after the ﬁt (purple) is very close to background only expectations (orange). Note that subtle differences can be visible e.g. in ﬁrst bin for stopping UP-µ events (upper right corner).

In the analysis, three annihilation channels are considered: bb¯, τ+τ− and W+W−. The resulting χ2 value of the ﬁt is shown in Fig. 11.2 with color (Z axis), for the tested parameter space of WIMP masses (Mχ - X axis) ranging from 3 GeV to 1 TeV, and added WIMP contribution (β - Y axis) ranging from -0.01 to 0.02. It is visible that the low χ2 regions (dark blue) are concentrated around β = 0. The same results 104 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.1: The momentum (1st column) and zenith angle (2nd-5th columns) distributions before and after the ﬁt are shown for 19 samples used in the analysis. See detailed description in text. are presented as 68% (blue), 90% (yellow) and 99% (maroon) C.L. contours and are shown in Fig. 11.3.

For each tested WIMP mass, Mχ, the point with dark matter contribution β resulting with the lowest χ2 value is called best f it point. The value of β parameter corresponding to best f it for a given WIMP mass, can be translated to number of dark matter induced neutrinos of all ﬂavors. The results for three considered annihilation channels are shown in Fig. 11.4 together with 90% and 99% C.L. sensitivity contours corresponding to null WIMP contribution hypothesis (described in Sec. 10.3.1). It is visible that no signiﬁcant signal contribution is allowed by the SK data in addition to the atmospheric neutrino background. All results are consistent with null hypothesis, β = 0. 11.1. Fit Results with Standard Binning 105

FIGURE 11.2: Heat map plot showing the tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). The color represents χ2 value of the ﬁt to data. 106 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.3: The 68%, 90% and 99% C.L contours of allowed regions in tested parameter space of WIMP masses (X - axis) and WIMP contribution β (Y - axis) for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). 11.1. Fit Results with Standard Binning 107

FIGURE 11.4: The ﬁtted number of dark matter induced neutrinos together with 90% and 99% C.L. sensitivity contours as a function of WIMP mass for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). 108 Chapter 11. Results of the Search for the Earth WIMPs

The best f it points for three considered annihilation channels are listed in Table 11.1, together with the corresponding χ2 values from the ﬁt. For the reference, the result of the ﬁt for atmospheric neutrino background only (β = 0) is shown in the last row 2 2 2 of the table. The obtained differences ∆χ = χatm − χbest are very small. From all of the tested hypothesis, the lowest χ2 value corresponds to bb¯ annihilation channel, 2 Mχ = 150 GeV and β = 0.0007 (global best fit point). Corresponding ∆χ is equal to 1.627.

For the global best fit point the ﬁtted values of systematic parameters ei in units of σ are plotted in Fig. 11.6- 11.8 for all considered sources described in Sec. 5.3.1. The resulting distribution of systematic errors values in units of σ is plotted in Fig. 11.5. According to expectations, ﬁtted values are distributed around 0.

TABLE 11.1: The best f it points from the ﬁt with standard binning, for the three considered annihilation channels (bb¯, τ+τ− and W+W−). The corresponding χ2 values are compared with atmospheric neutrino background only hypothesis.

2 2 Channel Mχ [GeV] β χ χ syst. χ2/bins bb¯ 150 0.0007 580.108 39.759 1.116 τ+τ− 25 0.0008 580.496 39.928 1.116 W+W− 80.3 0.0007 580.554 38.620 1.116 Bkg only − 0.0 581.735 39.3268 1.119

FIGURE 11.5: The distribution of ﬁtted values of the systematic errors ei, given in units of σ, at the global best fit point from the ﬁt with standard binning (bb¯ annihilation channel, Mχ = 150 GeV, β = 0.0007).

For other points from the ﬁt, the resulting distributions of systematic errors values have very similar behavior. This proofs that ﬁt is stable and no anomalies have been observed. 11.1. Fit Results with Standard Binning 109

FIGURE 11.6: Global best fit values of the systematic error parameters, ei, given in units of σ, from the ﬁt with standard binning (part I). 110 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.7: Global best fit values of the systematic error parameters, ei, given in units of σ, from the ﬁt with standard binning (part II). 11.1. Fit Results with Standard Binning 111

FIGURE 11.8: Global best fit values of the systematic error parameters, ei, given in units of σ, from the ﬁt with standard binning (part III).

Based on the fit results, the upper limit on allowed dark matter contribution and the corresponding neutrino flux can be calculated for each tested WIMP mass. If the resulting best f it β for a given WIMP mass happens to be negative, the upper limit on allowed dark matter contribution is calculated using the Bayesian approach, described in details in AppendixD. The 90% C.L. limits on muon neutrino flux are presented in Fig. 11.9, for the three considered annihilation channels. They are compared with the sensitivity predictions calculated in the previous chapter (see Fig 10.8).

The limits obtained from data are very close to the predictions from sensitivity. This is consistent with ﬁt results being close to zero, as it is shown in Fig. 11.4. 112 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.9: The 90% upper limits on dark matter induced muon neutrino ﬂux for bb¯ (blue), τ+τ− (green) and W+W− annihilation channel, compared with expectations from the sensitivity studies (dashed lines).

11.1.1 Constraints on the WIMP-Nucleon Spin Independent Scattering Cross-Section for Mχ < 100 GeV The upper limit on muon neutrino ﬂux can be further translated to limit on SI WIMP- nucleon scattering cross section. For this purpose, the conversion script provided with the WimpSim software (Edsjö, 2007) is used. The calculation is done for each assumed dark matter particle mass and annihilation channel, taking into account different thresholds and angular cuts of the detector. In the calculation, a typical −26 3 −1 value for the dark matter self-annihilation cross-section < σAv >= 3 · 10 cm s , for which the WIMP is a thermal relic, is assumed.

The results presented in Fig. 11.10, covering WIMP masses < 100 GeV for bb¯ and τ+τ− annihilation channels, were approved by the Super-Kamiokande Collabora- tion in June 2017. However, based on results from sensitivity studies presented in Section 10.4, the Collaboration made a decision to use optimized binning for the whole range of tested WIMP masses. These results are discussed in the following section. 11.2. Fit Results with Optimized Binning 113

FIGURE 11.10: The 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section for bb¯ (blue) and τ+τ− (green) annihilation channels, for Mχ < 100 GeV. The visible peaks correspond to resonant capture on the most abundant elements 16O, 24Mg, 28Si, 56Fe, and their isotopes. Positive result claimed by DAMA/LIBRA (Bernabei et al., 2013) is shown for comparison.

11.2 Fit Results with Optimized Binning

In this section we discuss the results of the "pulled" χ2 ﬁt performed with optimized binning described in Sec. 10.4. For one of the tested hypothesis: 100 GeV WIMPs annihilating into W+W− channel, Fig. 11.11 shows:

• expected signal contribution from the dark matter induced neutrinos before the ﬁt (ﬁlled maroon histogram), enlarged for better visibility,

• atmospheric neutrino background before the ﬁt (orange histogram),

• the best conﬁguration of signal and background after the ﬁt (purple histogram),

• SK I-IV data (black points) with error bars represent statistical uncertainty.

The minimal value of the χ2 corresponds to β = 0.00035. Due to extremely small ﬁtted signal contribution, result after the ﬁt (purple) is very close to background only expectations (orange). Therefore, the difference between purple and orange histograms is hardly visible.

The best f it points resulting with the lowest χ2 value for each tested WIMP mass Mχ, are shown in Fig. 11.12 for the three considered annihilation channels. Here, the β parameter from the ﬁt is translated to allowed number of the dark matter induced neutrinos of all ﬂavors. The expected sensitivity contours at 90% and 99% C.L. calculated for null WIMP contribution hypothesis in Sec. 10.4.1 are also shown for comparison. The obtained results are consistent with null WIMP hypothesis. 114 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.11: The momentum (1st column) and zenith angle (2nd- 5th columns) distributions before and after the ﬁt are shown for 19 samples used in the analysis. See detailed description in text.

Based on critical χ2 values determined for each WIMP mass and annihilation channel, the upper limit on ﬁtted number of WIMP-induced neutrinos of all ﬂavors is calculated and presented in Fig. 11.13. It is compared with the sensitivity expectations and their ±1, 2σ systematic and statistical uncertainties, which are plotted as a green and yellow bands respectively. It is visible that limits from data are consistent with sensitivity results for null WIMP hypothesis. All tested points are within ±2σ band (yellow).

The best f it points from the fit with optimized binning are listed in Table 11.2, for three considered annihilation channels. The corresponding χ2 values from the fit are also shown. For the reference, the result of the fit for atmospheric neutrino background only (β = 0) is shown in the last row of the table. Note that due to the optimized binning, number of bins in the analysis changed from 520 to 595, resulting with the different χ2 value for background only scenario. The obtained differences 2 2 2 ∆χ = χatm − χbest are slightly larger than for the fit with standard binning. From all of the tested hypothesis, the lowest χ2 value correspond to W+W− annihilation channel, Mχ = 100 GeV and β = 0.00035 (global best fit point). Corresponding ∆χ2 = 2.631. 11.2. Fit Results with Optimized Binning 115

FIGURE 11.12: The ﬁtted number of dark matter induced neutrinos together with 90% and 99% C.L. sensitivity contours calculated with optimized binning, plotted as a function of WIMP mass for three considered annihilation channels: bb¯ (top), τ+τ− (middle) and W+W− (bottom). 116 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.13: The 90% C.L. upper limit on the ﬁtted number of dark matter induced neutrinos of all ﬂavors from WIMP annihilation into bb¯ (top), τ+τ− (middle) and W+W− (bottom) channel as a function of the mass of relic particles. The sensitivity expectations for a null WIMP hypothesis (dotted lines) and their ±1σ (green bands) and ±2σ (yellow bands) uncertainty calculated in Sec. 10.4 are shown. 11.2. Fit Results with Optimized Binning 117

TABLE 11.2: The best f it points from the ﬁt with optimized binning, for the three considered annihilation channels (bb¯, τ+τ− and W+W−). The corresponding χ2 values are compared with atmospheric neutrino background only hypothesis.

2 2 Channel Mχ [GeV] β χ χ syst. χ2/bins bb¯ 176 0.0004 668.715 39.519 1.124 τ+τ− 100 0.0004 669.638 39.584 1.125 W+W− 100 0.00035 667.962 39.456 1.123 Bkg only − 0.0 670.593 39.373 1.127

For the global best fit point with optimized binning, (W+W− annihilation channel, Mχ = 100 GeV and β = 0.00035), the fitted values of systematic parameters ei in units of σ are plotted in Fig. 11.15- 11.17 for all considered sources (see Sec. 5.3.1). The resulting distribution of systematic errors values in units of σ is plotted in Fig. 11.14. Similar to the results for standard binning, again, fitted values are distributed around 0. This is in agreement with expectations and proofs that fit is stable and there are no anomalies. The resulting distributions of systematic errors values for other points tested in the fit have very similar behavior.

FIGURE 11.14: The distribution of ﬁtted values of the systematic errors ei, given in units of σ, at the global best fit point from the ﬁt with + − optimized binning (W W annihilation channel, Mχ = 100 GeV and β = 0.00035). 118 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.15: Global best fit values of the systematic error parameters, ei, given in units of σ, from the ﬁt with optimized binning (part I). 11.2. Fit Results with Optimized Binning 119

FIGURE 11.16: Global best fit values of the systematic error parameters, ei, given in units of σ, from the ﬁt with optimized binning (part II). 120 Chapter 11. Results of the Search for the Earth WIMPs

FIGURE 11.17: Global best fit values of the systematic error parameters, ei, given in units of σ, from the ﬁt with optimized binning (part III).

11.2.1 Constraints on the WIMP-Nucleon Spin Independent Scattering Cross-Section for Mχ up to 1 TeV Once again we can translate the upper limits on number of dark matter induced neutrinos to the limits on the SI WIMP-nucleon scattering cross section. This time the limits are set for the whole tested WIMP mass range and three annihilation channels (bb¯, τ+τ− and W+W−). The results are presented in Fig. 11.18, together with the previous result published by the Super-Kamiokande Collaboration (Desai et al., 2004). 11.2. Fit Results with Optimized Binning 121

FIGURE 11.18: The 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section for bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channels. The visible peaks correspond to resonant capture on the most abundant elements 16O, 24Mg, 28Si, 56Fe, and their isotopes. The results are compared with previous SK result (Desai et al., 2004) (dotted cyan line), as well as positive result claimed by DAMA/LIBRA (Bernabei et al., 2013).

The previous ofﬁcial result of indirect WIMP search was based only on upward through-going muons. In the analysis presented in this thesis all SK samples, including FC, PC and UP-µ, are used. This results in signiﬁcant increase of signal acceptance, especially in the resonant capture region (Mχ < 100 GeV). Moreover, the livetime of analyzed data increased from 1679.6 days to 5325.8 days for FC/PC, and 5628.2 days for UP-µ samples.

The adopted analysis methods are very different. The old analysis simply search for an excess of high energy muon neutrinos from WIMP annihilation, as compared to the number expected from the atmospheric neutrino background, in the small cone around the Earth’s core. In a way, it is similar to "ON-OFF source" approach described in Chapter7, but in this case, we cannot define the proper "OFF source" region and background cannot be estimated directly from data. The MC simulation has to be used, so we lose the advantages of the method. The new method used in this thesis is based on the global fit of simulated WIMP induced neutrino signal and expected atmospheric neutrino background to SK data. All available information (angular and energy distributions of signal and background components) are used in the fit, providing the best search sensitivity. Moreover, the 160 sources of systematic uncertainties are taken into account during the fit.

The other difference is that in the previous analysis, the limit on SI WIMP-nucleon scattering cross section was based on WIMP-induced muons from both, the Sun and 122 Chapter 11. Results of the Search for the Earth WIMPs the Earth. In case of WIMP masses < 80 GeV, the strength of the limit is dominated by the constraints from the Earth, due to resonant capture on heavy elements in the core, but for WIMP masses > 80 GeV, limit is dominated by constraints from the Sun. Therefore, the previous limit is stronger for high WIMP masses. The visible discontinuities in the old limit around 80 GeV and 174 GeV are the result of using the annihilation channel, which gives the strongest constraints. For Mχ < 80 GeV, limit is based on bb¯ annihilation channel, for Mχ > 80 GeV and < 174 GeV limit is + − + − based on W W channel, and ﬁnally for Mχ > 174 GeV limit is based on τ τ .

Therefore, the two limits cannot be compared directly in the whole range of tested WIMP masses. However, in the region corresponding to the iron peak (Mχ ≈ 56 GeV), the old limit is based on bb¯ annihilation channel, and is dominated by the constraints from the Earth. Here, we can compare it with the new limit from bb¯ channel (blue line). The signiﬁcant improvement of the limit strength is visible, reaching the order of magnitude.

11.2.2 Comparison with Other Experiments Finally, the limits on SI WIMP-nucleon scattering cross-section from the Super-Kamio- kande experiment can be compared with the results of the searches performed by other experiments. In Fig. 11.19, the latest results from IceCube (Aartsen et al., 2017a) and ANTARES (Albert et al., 2017b) neutrino telescopes are plotted together with the positive result claimed by DAMA/LIBRA (Bernabei et al., 2013).

FIGURE 11.19: The 90% C.L. upper limits on SI WIMP-nucleon scattering cross-section for bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channels. For comparison the results from Ice- Cube (Aartsen et al., 2017a) (dotted line) and ANTARES (Albert et al., 2017b) (dashed line) are shown, as well as positive result claimed by DAMA/LIBRA (Bernabei et al., 2013). 11.2. Fit Results with Optimized Binning 123

The limits from the Super-Kamiokande experiment are the strongest among all neutrino experiments due to high sensitivity of the SK detector. For ∼ 50 GeV WIMPs, the limit from τ+τ− annihilation channel can reach 10−44 cm2. This high sensitivity can be achieved by including FC and PC events in the WIMP search sample, which signiﬁcantly increase signal acceptance, especially for WIMP masses in the resonant capture region. In the analysis, we take into account all neutrino ﬂavors (νe, ν¯e, νµ, ν¯µ, ντ, ν¯τ), and available momentum or visible energy information. Together with good angular resolution of the detector, they allow to effectively distinguish the signal from the atmospheric neutrino background. Moreover, the Super-K limits rule out a majority of the WIMP parameter space favored by the DAMA/LIBRA experiment, using independent technique.

This ﬁnal results of the analysis provide strong constrains on SI WIMP-nucleon scattering cross-section in wide range of tested dark matter particle masses, from 10 GeV up to 1 TeV. The obtained results were approved by the Super-Kamiokande Collab- oration in November 2017.

125

Chapter 12

Summary and Outlook

Dark matter searches are very difficult, but also extremely interesting. In spite of overwhelming observational evidence for the dark matter existence, and many detection attempts, its nature remains unknown. Various attempts to determine what the dark matter is and what are its properties, led to many discoveries, but did not give the definitive answers. The multi-messenger approach is required in order to create a consistent picture, and any positive results have to be verified before claim- ing the discovery. Only one observation method is not sufficient to fully investigate and understand the nature of dark matter.

As it is outlined in Chapter2, the existing results from various searches for dark matter particles are inconsistent with each other and require veriﬁcation in the independent measurements. In particular, the positive signal claimed by DAMA/LIBRA has been already excluded by many direct detection experiments, e.g. PANDA, XENON and LUX. Moreover, recent LHC results provide tight constraints on supersymmetry models, hence on the most popular dark matter candidate, supersymmetirc neutralino χ. However, over the last several years few unexplained astrophysical observations have been interpreted as potential dark matter signals. It is very important to verify the excess in positron fraction seen by PAMELA, FERMI and AMS-02 experiments, which may be explained by dark matter annihilation, using observations of other annihilation products, such as neutrinos. The positive results from direct detection experiment DAMA/LIBRA also require independent veriﬁcation, and investigation of the signal region with a very different indirect method is very valuable.

According to many theoretical expectations, a significant flux of dark matter induced neutrinos produced in dark matter annihilation or decay processes is anticipated. This gives a strong motivation for an indirect search for dark matter with neutrinos. The two analyses presented in this dissertation aim at searching for dark matter induced neutrinos with the Super-Kamiokande water Cherenkov detector. Super- Kamiokande is an exceptional experiment, which provides the opportunity to study a wide range of physics topics, search for proton decay, study solar and atmospheric neutrinos, and keep watch for supernovae. Almost 50,000 atmospheric neutrino events collected by the detector over 20 years of operation is analyzed in order to find characteristics expected for neutrinos produced in dark matter annihilation or decay processes.

The first analysis described in this thesis (see Chapters6-8) focuses on the Milky Way. Self-annihilating or decaying dark matter in our Galaxy might produce a large- scale anisotropy in observed neutrino flux. The excess of dark matter induced neutrinos is expected from the direction of the Galactic Center due to increased dark 126 Chapter 12. Summary and Outlook matter density in this region. The search was performed in the equatorial coordinate system, where the position of the GC is fixed, and the signal is easy to distinguish from atmospheric neutrino background. The adopted method, so called "ON-OFF source" approach, was used for the first time to analyze the Super-Kamiokande neutrino data. Its great advantages are the simplicity and possibility to estimate the background directly from data. Therefore, it was possible to be independent on Monte Carlo simulations and related systematic uncertainties.

The expected anisotropy in neutrino flux has not been observed. Main outcome of this analysis, not observing any asymmetries in data (see Sec. 8.1), is a data driven and model independent result. Introduction of additional assumptions (halo model, dark matter particle mass and annihilation/decay channel) allows to constrain the velocity-averaged dark matter self-annihilation cross section, < σAv >, in case of dark matter annihilation scenario, or dark matter particle lifetime, τ, in case of the decay. The resulting limits are calculated for various WIMP masses in range from 1 GeV to 10 TeV, and for four considered annihilation channels: νν¯, bb¯, µ+µ− and W+W−. For the dark matter annihilation scenario, the significant impact of assumed dark matter halo model on the obtained results is shown. The differences between the benchmark model (NFW) and models used as extreme cases (Moore and Kravtsov) can reach the order of magnitude. Hence, focusing on the central region of our Galaxy makes the results very sensitive to the chosen profile, but on the other hand, allows to obtain stronger constraints on dark matter induced neutrino flux.

FIGURE 12.1: Favored regions obtained by interpreting the observed positron and electron excesses as due to dark matter annihilation in µ+µ−. Green region is favored by PAMELA (at 3σ), red region is favored by the global ﬁt of FERMI, HESS and PAMELA data (at 3σ) (Meade et al., 2010), and blue region is favored by AMS-02 data (at 2σ) (Di Mauro et al., 2016). The 90% C.L. limits from SK data are plotted with solid purple line for "ON-OFF souce" analysis (this thesis), and with solid line for "Global ﬁt" approach (Mijakowski, 2018). NFW halo model is assumed in all cases. Chapter 12. Summary and Outlook 127

The preferred regions obtained by interpreting the positron excess seen by AMS- 02 (Di Mauro et al., 2016) and PAMELA, and electron data from HESS and FERMI (Meade et al., 2010) as being due to dark matter annihilation into µ+µ− are shown in Fig 12.1. They are compared to limits for µ+µ− annihilation channel from the Super- Kamiokande experiment. It is visible that interesting region of parameter space can be probed with neutrino data.

The second performed analysis (see Chapters9- 11) presents the search for the neutrinos from dark matter annihilation in the Earth’s core, using so called "Global fit" approach. This method is based on a fit of a simulated signal component (dark matter induced neutrinos) and background component (atmospheric neutrinos) to the data. The simulation is based on the existing simulation of atmospheric neutrino interactions performed for the Super-Kamiokande detector. This Monte Carlo simulation assumes the realistic fluxes of neutrinos, their interaction in the detector or in the surrounding rocks. It includes also the detector response for the produced Cherenkov light. The signal part is reweighted in such a way that retrieves characteristics expected for dark matter induced neutrinos based on the results from the WimpSim software. In this way, the angular distributions peaked from the direction of the Earth’s core and energy spectra as expected for various dark matter particle masses and annihilation channels are obtained. All available Super-K data samples binned in angular and momentum or visible energy (595 bins in total), are used in the fit providing the best search sensitivity. For each tested WIMP mass and annihilation channel, the "pulled" χ2 fit to the collected data is perform and the number of dark matter induced neutrinos that could be contained in the SK data is estimated.

FIGURE 12.2: The upper 90% C.L. limits on the WIMP-nucleon SI scattering cross section for the bb¯ (blue), τ+τ− (green) and W+W− (maroon) annihilation channel from the SK data, compared with the results from direct detection experiments: LUX (Akerib et al., 2017) (orange) and PICO-60 (Amole et al., 2017) (purple), and possible signal from DAMA (Bernabei et al., 2013). 128 Chapter 12. Summary and Outlook

The obtained results are consistent with null WIMP contribution. Therefore, the upper limits on the dark matter induced neutrino ﬂux and SI WIMP-nucleon scattering cross-section, σχ−N, are set as a function of mass of dark matter particles, for the three considered annihilation channels: bb¯, τ+τ− and W+W−. Due to high sensitivity of the Super-Kamiokande detector, the obtained limits are the strongest among all neutrino experiments. They can reach 10−44 cm2 for 50 GeV WIMPs annihilating into τ+τ− pairs.

Such a search is complementary to direct detection experiments, which look for an interaction of WIMPs with a nucleus in a low background detector. Both direct and indirect detection experiments can probe the coupling of WIMPs to nuclei which is shown in Fig 12.2. The direct detection experiments are able to set stronger limits than indirect searches for dark matter induced neutrinos. However, the constraints obtain in this analysis allow to rule out a majority of the WIMP parameter space favored by the DAMA/LIBRA experiment, with a very different, independent technique. The analysis presented in this thesis is the ﬁrst one entirely dedicated to the search for dark matter induced neutrinos from the WIMP annihilation in the Earth’s core, with the data collected by the Super-Kamiokande experiment.

The analyses presented in the dissertation are of a great interest to the scientiﬁc community. Their results were presented during the important conferences in the ﬁeld. In particular, the author had a chance to present them during six international conferences (including La Thuile 2015, Neutrino 2016, ICHEP 2016, WIN 2017 and DPF 2015 and 2017), and three domestic ones (TMEX 2014, Astroparticle Physics in Poland 2015 and Polish Physical Society Meeting 2017). The posters with presented results were awarded with "Best Astrophysics Poster" (DPF 2015) and "Dis- tinguished Poster Award" (ICHEP 2016) prizes. Moreover, the obtained results were the subject of a few seminar presentations given by the author.

The results of both presented analyses were ofﬁcially approved by the Super-Kamio- kande Collaboration. The ﬁrst analysis from the Galactic Center will be a part of the paper "Indirect Search for Dark Matter from the Galactic Center and Halo with the Super-Kamiokande Detector", together with the independent Milky Way analysis of Super-Kamiokande data referred to as "Global Fit" (see Sec. 8.2). The results of the second analysis will be published in the dedicated paper "Search for Neutrinos from Dark Matter Annihilation in the Earth’s Core with the Super-Kamiokande Detector". Both publications are currently under the internal review in Super-Kamiokande Col- laboration.

Dark matter search is a highly challenging field, encompassing very different experimental techniques and detection methods. The existing results in the field, which are not conclusive, demonstrate the importance of a multi-messenger approach to dark matter searches and validate the interest in a neutrino channel. Moreover, the null results of the searches also make a big contribution to the field. They reduce the available dark matter parameter space, inform the future search strategies and provide better guidance where to look next. In recent years, dark matter searches are expanding in their diversity. Further research in this area is extremely important, because the nature of dark matter is still a huge gap in our knowledge about the Universe. 129

Appendix A

Galactic WIMP Weights

The numerical values of Galactic WIMP weights used to reproduce the signal shape for three considered dark matter halo models (NFW, Moore, and Kravtsov). Weights are calculated for the dark matter annihilation and decay scenarios.

... Annihilation Decay Angle NFW Moore Kravtsov NFW Moore Kravtsov 0.1 7319.7600 722448.0000 41.5995 - - - 0.2 3624.8600 181116.0000 39.3704 - - - 0.4 1779.0400 45265.6000 36.8077 - - - 1 678.5680 7232.8800 32.8201 17.4933 37.9280 7.6833 2 317.1910 1802.7000 29.2611 14.6826 25.7160 7.4892 3 199.1760 798.2230 26.9183 13.0422 20.3075 7.3289 4 141.3530 447.1210 25.1186 11.8820 17.0849 7.1857 5 107.3590 284.8600 23.6331 10.9854 14.8869 7.0533 6 85.1553 196.8680 22.3547 10.2560 13.2656 6.9281 7 69.6197 143.9100 21.2240 9.6423 12.0066 6.8083 8 58.2074 109.6080 20.2047 9.1136 10.9927 6.6925 9 49.5146 86.1408 19.2730 8.6499 10.1539 6.5798 10 42.7047 69.3931 18.4124 8.2376 9.4452 6.4696 11 37.2491 57.0312 17.6111 7.8671 8.8363 6.3616 12 32.7979 47.6526 16.8604 7.5312 8.3060 6.2554 13 29.1105 40.3729 16.1535 7.2244 7.8390 6.1507 14 26.0162 34.6124 15.4852 6.9425 7.4236 6.0474 15 23.3910 29.9781 14.8516 6.6821 7.0513 5.9455 16 21.1422 26.1963 14.2492 6.4404 6.7151 5.8447 17 19.1998 23.0714 13.6755 6.2153 6.4097 5.7451 18 17.5096 20.4607 13.1281 6.0049 6.1307 5.6467 19 16.0291 18.2583 12.6053 5.8075 5.8747 5.5493 20 14.7247 16.3839 12.1054 5.6219 5.6387 5.4531 21 13.5693 14.7762 11.6271 5.4471 5.4204 5.3579 22 12.5409 13.3874 11.1691 5.2818 5.2177 5.2639 23 11.6216 12.1800 10.7305 5.1255 5.0289 5.1710 24 10.7965 11.1240 10.3103 4.9772 4.8526 5.0793 25 10.0532 10.1955 9.9076 4.8363 4.6875 4.9888 26 9.3812 9.3750 9.5217 4.7023 4.5325 4.8995 27 8.7720 8.6467 9.1518 4.5747 4.3866 4.8114 28 8.2178 7.9974 8.7973 4.4529 4.2492 4.7245 29 7.7125 7.4163 8.4575 4.3367 4.1194 4.6389 30 7.2505 6.8943 8.1319 4.2255 3.9965 4.5547 130 Appendix A. Galactic WIMP Weights

31 6.8271 6.4239 7.8199 4.1191 3.8801 4.4717 32 6.4382 5.9985 7.5209 4.0172 3.7697 4.3901 33 6.0803 5.6128 7.2345 3.9195 3.6647 4.3098 34 5.7502 5.2619 6.9601 3.8257 3.5648 4.2309 35 5.4451 4.9420 6.6973 3.7356 3.4696 4.1534 36 5.1627 4.6496 6.4457 3.6490 3.3788 4.0772 37 4.9009 4.3816 6.2047 3.5657 3.2920 4.0024 38 4.6577 4.1355 5.9740 3.4856 3.2091 3.9290 39 4.4314 3.9091 5.7531 3.4084 3.1298 3.8570 40 4.2207 3.7003 5.5416 3.3340 3.0538 3.7864 41 4.0241 3.5074 5.3392 3.2622 2.9810 3.7172 42 3.8405 3.3288 5.1454 3.1930 2.9111 3.6494 43 3.6688 3.1633 4.9600 3.1262 2.8440 3.5829 44 3.5080 3.0096 4.7825 3.0617 2.7795 3.5178 45 3.3572 2.8666 4.6127 2.9993 2.7175 3.4541 46 3.2157 2.7334 4.4501 2.9390 2.6579 3.3917 47 3.0827 2.6091 4.2946 2.8807 2.6004 3.3306 48 2.9576 2.4931 4.1457 2.8244 2.5451 3.2709 49 2.8399 2.3845 4.0032 2.7698 2.4918 3.2125 50 2.7289 2.2828 3.8668 2.7169 2.4403 3.1553 51 2.6242 2.1875 3.7362 2.6657 2.3907 3.0995 52 2.5253 2.0980 3.6113 2.6161 2.3427 3.0449 53 2.4319 2.0139 3.4916 2.5681 2.2965 2.9915 54 2.3435 1.9347 3.3771 2.5215 2.2517 2.9393 55 2.2599 1.8602 3.2674 2.4762 2.2085 2.8884 56 2.1806 1.7899 3.1624 2.4324 2.1666 2.8386 57 2.1055 1.7236 3.0618 2.3898 2.1261 2.7899 58 2.0342 1.6609 2.9654 2.3485 2.0869 2.7424 59 1.9665 1.6017 2.8731 2.3084 2.0490 2.6960 60 1.9022 1.5456 2.7847 2.2694 2.0122 2.6507 61 1.8410 1.4925 2.6999 2.2316 1.9766 2.6065 62 1.7828 1.4422 2.6186 2.1948 1.9421 2.5633 63 1.7275 1.3945 2.5407 2.1590 1.9086 2.5212 64 1.6747 1.3492 2.4660 2.1243 1.8761 2.4800 65 1.6244 1.3061 2.3944 2.0905 1.8446 2.4398 66 1.5764 1.2652 2.3257 2.0576 1.8140 2.4006 67 1.5306 1.2263 2.2598 2.0256 1.7843 2.3624 68 1.4869 1.1892 2.1965 1.9945 1.7554 2.3250 69 1.4451 1.1539 2.1358 1.9643 1.7274 2.2886 70 1.4052 1.1202 2.0775 1.9348 1.7002 2.2530 71 1.3670 1.0881 2.0215 1.9061 1.6737 2.2183 72 1.3305 1.0575 1.9678 1.8782 1.6480 2.1844 73 1.2955 1.0282 1.9161 1.8510 1.6230 2.1514 74 1.2620 1.0003 1.8665 1.8245 1.5987 2.1191 75 1.2299 0.9735 1.8188 1.7987 1.5750 2.0876 76 1.1991 0.9480 1.7729 1.7736 1.5520 2.0569 77 1.1696 0.9235 1.7289 1.7491 1.5297 2.0270 78 1.1412 0.9001 1.6865 1.7252 1.5079 1.9977 79 1.1140 0.8776 1.6457 1.7019 1.4867 1.9692 80 1.0879 0.8561 1.6064 1.6792 1.4660 1.9414 Appendix A. Galactic WIMP Weights 131

81 1.0628 0.8355 1.5686 1.6571 1.4459 1.9142 82 1.0387 0.8157 1.5323 1.6356 1.4263 1.8877 83 1.0155 0.7967 1.4973 1.6145 1.4073 1.8618 84 0.9932 0.7784 1.4635 1.5940 1.3887 1.8366 85 0.9718 0.7609 1.4311 1.5740 1.3706 1.8119 86 0.9511 0.7441 1.3998 1.5545 1.3530 1.7879 87 0.9312 0.7279 1.3696 1.5355 1.3358 1.7645 88 0.9121 0.7123 1.3405 1.5170 1.3191 1.7416 89 0.8936 0.6973 1.3125 1.4989 1.3027 1.7192 90 0.8759 0.6829 1.2855 1.4812 1.2868 1.6974 91 0.8587 0.6691 1.2594 1.4639 1.2713 1.6762 92 0.8422 0.6557 1.2342 1.4471 1.2562 1.6554 93 0.8263 0.6428 1.2100 1.4307 1.2414 1.6352 94 0.8109 0.6304 1.1865 1.4147 1.2271 1.6154 95 0.7960 0.6185 1.1639 1.3990 1.2130 1.5961 96 0.7817 0.6070 1.1421 1.3838 1.1994 1.5773 97 0.7679 0.5958 1.1210 1.3689 1.1860 1.5590 98 0.7545 0.5851 1.1006 1.3543 1.1730 1.5410 99 0.7416 0.5748 1.0809 1.3401 1.1603 1.5236 100 0.7291 0.5648 1.0619 1.3263 1.1479 1.5065 101 0.7171 0.5551 1.0435 1.3127 1.1359 1.4899 102 0.7054 0.5458 1.0258 1.2995 1.1241 1.4736 103 0.6941 0.5368 1.0086 1.2866 1.1126 1.4578 104 0.6832 0.5281 0.9920 1.2740 1.1014 1.4423 105 0.6727 0.5197 0.9759 1.2617 1.0904 1.4272 106 0.6625 0.5116 0.9604 1.2497 1.0798 1.4125 107 0.6526 0.5037 0.9454 1.2380 1.0693 1.3981 108 0.6431 0.4961 0.9309 1.2266 1.0592 1.3841 109 0.6339 0.4888 0.9168 1.2154 1.0493 1.3704 110 0.6249 0.4817 0.9032 1.2045 1.0396 1.3571 111 0.6162 0.4748 0.8900 1.1939 1.0302 1.3441 112 0.6079 0.4682 0.8773 1.1835 1.0210 1.3314 113 0.5997 0.4617 0.8650 1.1734 1.0120 1.3190 114 0.5919 0.4555 0.8530 1.1635 1.0032 1.3069 115 0.5843 0.4495 0.8415 1.1539 0.9947 1.2952 116 0.5769 0.4437 0.8303 1.1444 0.9864 1.2837 117 0.5697 0.4380 0.8194 1.1353 0.9782 1.2725 118 0.5628 0.4325 0.8089 1.1263 0.9703 1.2616 119 0.5561 0.4273 0.7988 1.1176 0.9626 1.2509 120 0.5496 0.4221 0.7890 1.1090 0.9550 1.2406 121 0.5433 0.4172 0.7794 1.1007 0.9477 1.2304 122 0.5372 0.4124 0.7702 1.0926 0.9405 1.2206 123 0.5313 0.4077 0.7613 1.0847 0.9336 1.2110 124 0.5256 0.4032 0.7526 1.0770 0.9268 1.2016 125 0.5200 0.3988 0.7442 1.0695 0.9201 1.1925 126 0.5147 0.3946 0.7361 1.0622 0.9137 1.1837 127 0.5095 0.3905 0.7283 1.0550 0.9074 1.1750 128 0.5044 0.3866 0.7207 1.0481 0.9013 1.1666 129 0.4995 0.3827 0.7133 1.0413 0.8953 1.1584 130 0.4948 0.3790 0.7062 1.0347 0.8895 1.1505 132 Appendix A. Galactic WIMP Weights

131 0.4902 0.3754 0.6993 1.0283 0.8839 1.1427 132 0.4858 0.3719 0.6926 1.0220 0.8784 1.1352 133 0.4815 0.3686 0.6862 1.0160 0.8730 1.1279 134 0.4773 0.3653 0.6799 1.0101 0.8678 1.1208 135 0.4733 0.3622 0.6739 1.0043 0.8628 1.1138 136 0.4694 0.3591 0.6680 0.9987 0.8579 1.1071 137 0.4657 0.3562 0.6624 0.9933 0.8531 1.1006 138 0.4620 0.3533 0.6569 0.9880 0.8485 1.0943 139 0.4585 0.3506 0.6516 0.9829 0.8440 1.0881 140 0.4551 0.3479 0.6465 0.9780 0.8396 1.0821 141 0.4518 0.3454 0.6416 0.9731 0.8354 1.0764 142 0.4487 0.3429 0.6369 0.9685 0.8313 1.0708 143 0.4456 0.3405 0.6323 0.9639 0.8273 1.0653 144 0.4426 0.3382 0.6279 0.9596 0.8235 1.0601 145 0.4398 0.3360 0.6236 0.9553 0.8198 1.0550 146 0.4370 0.3338 0.6195 0.9512 0.8162 1.0501 147 0.4344 0.3317 0.6155 0.9472 0.8127 1.0454 148 0.4318 0.3298 0.6117 0.9434 0.8094 1.0408 149 0.4294 0.3278 0.6081 0.9397 0.8061 1.0364 150 0.4270 0.3260 0.6045 0.9362 0.8030 1.0321 151 0.4248 0.3242 0.6012 0.9327 0.8000 1.0280 152 0.4226 0.3226 0.5979 0.9294 0.7971 1.0241 153 0.4205 0.3209 0.5948 0.9262 0.7943 1.0203 154 0.4185 0.3194 0.5918 0.9232 0.7917 1.0166 155 0.4166 0.3179 0.5890 0.9203 0.7891 1.0131 156 0.4148 0.3165 0.5863 0.9175 0.7866 1.0098 157 0.4131 0.3151 0.5837 0.9148 0.7843 1.0066 158 0.4114 0.3138 0.5812 0.9122 0.7821 1.0036 159 0.4098 0.3126 0.5789 0.9098 0.7799 1.0007 160 0.4083 0.3114 0.5767 0.9075 0.7779 0.9979 161 0.4069 0.3103 0.5746 0.9053 0.7760 0.9953 162 0.4056 0.3093 0.5726 0.9032 0.7742 0.9928 163 0.4043 0.3083 0.5707 0.9013 0.7725 0.9905 164 0.4032 0.3074 0.5689 0.8994 0.7709 0.9883 165 0.4021 0.3065 0.5673 0.8977 0.7693 0.9863 166 0.4010 0.3057 0.5658 0.8961 0.7679 0.9843 167 0.4001 0.3050 0.5643 0.8946 0.7666 0.9826 168 0.3992 0.3043 0.5630 0.8932 0.7654 0.9809 169 0.3984 0.3037 0.5618 0.8919 0.7643 0.9794 170 0.3976 0.3031 0.5607 0.8908 0.7633 0.9780 171 0.3970 0.3026 0.5597 0.8897 0.7624 0.9768 172 0.3964 0.3021 0.5588 0.8888 0.7615 0.9757 173 0.3959 0.3017 0.5581 0.8879 0.7608 0.9747 174 0.3954 0.3014 0.5574 0.8872 0.7602 0.9738 175 0.3950 0.3011 0.5568 0.8866 0.7597 0.9731 176 0.3947 0.3008 0.5564 0.8861 0.7592 0.9725 177 0.3945 0.3006 0.5560 0.8857 0.7589 0.9721 178 0.3943 0.3005 0.5557 0.8855 0.7587 0.9717 179 0.3942 0.3004 0.5556 0.8853 0.7585 0.9715 180 0.3941 0.3004 0.5555 0.8853 0.7585 0.9715 133

Appendix B

WimpSim Results

The differential ﬂuxes of the neutrinos (left panels) and anti-neutrinos (right panels) originating from dark matter annihilation in the Earth’s core, plotted as a function of the neutrino energy, Eν, divided by the WIMP mass, Mχ. The different neutrino species are indicated with a different colors. The presented spectra are calculated at the Earth’s surface, using the WimpSim software (Edsjö, 2007). Neutrino oscillations are included in a full three-ﬂavor framework including both vacuum and matter oscillations.

B.1 bb¯ annihilation channel 134 Appendix B. WimpSim Results B.1. bb¯ annihilation channel 135 136 Appendix B. WimpSim Results

B.2 τ+τ− annihilation channel B.2. τ+τ− annihilation channel 137 138 Appendix B. WimpSim Results B.3. W+W− annihilation channel 139

B.3 W+W− annihilation channel 140 Appendix B. WimpSim Results 141

Appendix C

Earth WIMP Signal Illustrations

The neutrino signal originating from dark matter annihilation in the Earth’s core, expected at the Super-Kamiokande detector. The momentum (1st column) and zenith angle (2nd-5th columns) distributions are shown for 19 samples used in the analysis. The plots contain the expected signal from WIMP annihilation (ﬁlled histogram) and the SK-I-IV atmospheric neutrino data (black points). In all panels, the error bars represent the statistical uncertainty.

C.1 Standard Binning

+ − τ τ annihilation channel, Mχ = 3 GeV, β = 0.02 142 Appendix C. Earth WIMP Signal Illustrations

+ − τ τ annihilation channel, Mχ = 6 GeV, β = 0.02

+ − τ τ annihilation channel, Mχ = 10 GeV, β = 0.02 C.1. Standard Binning 143

+ − τ τ annihilation channel, Mχ = 25 GeV, β = 0.02

+ − τ τ annihilation channel, Mχ = 50 GeV, β = 0.02 144 Appendix C. Earth WIMP Signal Illustrations

+ − τ τ annihilation channel, Mχ = 100 GeV, β = 0.02

+ − τ τ annihilation channel, Mχ = 500 GeV, β = 0.02 C.1. Standard Binning 145

+ − τ τ annihilation channel, Mχ = 1000 GeV, β = 0.02 146 Appendix C. Earth WIMP Signal Illustrations

C.2 Optimized Binning

+ − τ τ annihilation channel, Mχ = 3 GeV, β = 0.02 C.2. Optimized Binning 147

+ − τ τ annihilation channel, Mχ = 6 GeV, β = 0.02

+ − τ τ annihilation channel, Mχ = 10 GeV, β = 0.02 148 Appendix C. Earth WIMP Signal Illustrations

+ − τ τ annihilation channel, Mχ = 100 GeV, β = 0.02

+ − τ τ annihilation channel, Mχ = 1000 GeV, β = 0.02 149

Appendix D

Bayesian Approach

This is the description of the procedure which is used to derive the upper limit on the number of signal events, in case when the results of ON-OFF approach (see Chap- ter8), or ﬁtted parameter β (see Chapter 11) are negative. This approach is adapted from (Barnett et al., 1996) and referred in literature to as "Bayesian approach".

In this method, for the statistical variable α, αˆ is the estimator of α from a measurement. Therefore, the measurement provide conditional probability density function f (αˆ |α). We want to know the probability g(α|αˆ )dα, which is telling that "true" value of α lies between α and α + dα, when we have measured αˆ . Bayes’ theorem provide the connection between f and g:

f (αˆ |α)π(α) g(α|αˆ ) = . (D.1) R f (αˆ |α)π(α)dα

In this equation, π(α) represents prior knowledge of α value. We do not have any prior knowledge of α, which means that all physically reasonable values of α are equally probable. Therefore, π(α) is a constant over the region of interest and 0 in the unphysical region. In our case, for testing the possible dark matter contribution, the region of interest is α ≥ 0, and: ( f (αˆ |α)π(α)/ R f (αˆ |α)π(α)dα for α ≥ 0, g(α|αˆ ) = (D.2) 0 for α < 0.

The integral is over the physical region. If the experiments would provide the values of αˆ as shown in Fig. D.1, so with the signiﬁcant probability of obtaining results in unphysical region, with assumed step function π(α), the Eq. D.2 will replace this distribution with the function shown by the shaded region.

In order to derive the 90% C.L. on α value, one needs to ﬁnd such a value of α90 that the integral: Z α90 g(α|αˆ )dα, 0 yields 90% of the total area of g(α|αˆ ) distribution. In Fig. D.1, conﬁdence limit level is indicated as (1 − e) and equals 0.9 in our case.

In the analyses presented in this thesis, the upper limit on the ﬁtted number of dark matter induced neutrinos can be obtained for the negative (unphysical) and positive (physical) result, by using the method described above. The upper limit calculation is computed numerically. 150 Appendix D. Bayesian Approach

FIGURE D.1: An example of a measurement αˆ which can give results in unphysical region. If one assumes that α, the quantity one tries to measure, cannot lie in the unphysical region, but can lie anywhere in the physical region (no prior knowledge), then Bayes’ theorem says the new knowledge of distribution of α (given our measurement αˆ ) is given by the shaded function after appropriate normalization. 151

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