ACTA UNIVERSITATIS UPSALIENSIS Uppsala Dissertations from the Faculty of Science and Technology

143

Alexander Burgman

Bright Needles in a Haystack A Search for Magnetic Monopoles Using

the IceCube Neutrino Observatory

Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, 10134,Uppsala, Wednesday, 3 February 2021 at 13:15 for the degree of Doctor of Philosophy. Theexamination will be conducted in English. Faculty examiner: Professor David Milstead(Stockholm University, Stockholm, Sweden).

AbstractBurgman, A. 2020. Bright Needles in a Haystack. A Search for Magnetic MonopolesUsing the IceCube Neutrino Observatory. (Ljusstarka Nålar i en Höstack. En Sökning efterMagnetiska Monopoler med Neutrino-Observatoriet IceCube). Uppsala Dissertations fromthe Faculty of Science and Technology 143. 166 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-513-1083-1.

The IceCube Neutrino Observatory at the geographic South Pole is designed to detect the lightproduced by the daughter-particles of in-ice neutrino-nucleon interactions, using one cubickilometer of ice instrumented with more than 5000 optical sensors.

Magnetic monopoles are hypothetical particles with non-zero magnetic charge, predictedto exist in many extensions of the Standard Model of particle physics. The monopole massis allowed within a wide range, depending on the production mechanism. A cosmic flux ofmagnetic monopoles would be accelerated by extraterrestrial magnetic fields to a broad finalvelocity distribution that depends on the monopole mass.

The analysis presented in this thesis constitutes a search for magnetic monopoles with aspeed in the range [0.750;0.995] in units of the speed of light. A monopole within this speedrange would produce Cherenkov light when traversing the IceCube detector, with a smooth andelongated light signature, and a high brightness.

This analysis is divided into two main steps. Step I is based on a previous IceCube analysis,developed for a cosmogenic neutrino search, with similar signal event characteristics as in thisanalysis. The Step I event selection reduces the acceptance of atmospheric events to lowerthan 0.1 events per analysis livetime. Step II is developed to reject the neutrino events thatStep I inherently accepts, and employs a boosted decision tree for event classification. The(astrophysical) neutrino rate is reduced to 0.265 events per analysis livetime, corresponding toa 97.4 % rejection efficiency for events with a primary energy above 1E+5 GeV.

No events were observed at final analysis level over eight years of experimental data. Theresulting upper limit on the magnetic monopole flux was determined to 2.54E–19 per squarecentimeter per second per steradian, averaged over the covered speed region. This constitutesan improvement of around one order of magnitude over previous results.

Keywords: magnetic monopole, IceCube, astroparticle physics, neutrino telescope

Alexander Burgman, Department of Physics and Astronomy, High Energy Physics, Box 516,Uppsala University, SE-751 20 Uppsala, Sweden.

© Alexander Burgman 2020

ISSN 1104-2516ISBN 978-91-513-1083-1urn:nbn:se:uu:diva-425610 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-425610)

This thesis is dedicated tomy children Olivia and Victor,who are all that is best in me.

Contents

1 Units and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1 High Energy Physics Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Speed and Relativistic Lorentz Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Electric-Magnetic Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 The Dirac Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 The ’t Hooft-Polyakov Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 The Cosmic Monopole Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Magnetic Monopole Search Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Monopole Search Results from Neutrino Telescopes . . . . . . . . . . . . . . . . . . 23

3 The IceCube Neutrino Observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 The Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.1 The Detector Constituent Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 The DOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 The Detector Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.4 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Data Acquisition and Triggering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Data Filter Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Typical Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 High Energy Neutrinos in IceCube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.1 Typical Event Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Interpreting an Event View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Magnetic Monopoles in IceCube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Energy Loss in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Light Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Indirect Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Magnetic Monopole Signatures in IceCube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Magnetic Monopole Event Simulation in IceCube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1 Magnetic Monopole Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1.1 Generation Disk Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1.2 Generation Disk Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Magnetic Monopole Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Light Production and Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Simulation Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4.1 Validation with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.4.2 Magnetic Monopole Light Yield Validation . . . . . . . . . . . . . . . . . . 58

6 Event Cleaning and Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.1 Event Cleaning Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.1 The SeededRadiusTime Cleaning Method . . . . . . . . . . . . . . . . . . . . 606.1.2 The TimeWindow Cleaning Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2 Event Reconstruction Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.1 A Particle Track Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.2 The LineFit Track Reconstruction Method . . . . . . . . . . . . . . . . . . . . 626.2.3 The EHE Reconstruction Suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2.4 The CommonVariables Event Characterization Suite . . . 666.2.5 The Millipede Track Reconstruction Method . . . . . . . . . . . . . . . . 676.2.6 The BrightestMedian Track Reconstruction Method . . . . 68

7 Data Analysis and Statistical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.1 Analysis Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.1.1 Cut-and-Count Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.1.2 Multi-Variate Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.1.3 Analysis Blindness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2 Determining an Upper Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.2.1 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.2.2 Upper Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.2.3 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2.4 Including Uncertainties in the Upper Limit . . . . . . . . . . . . . . . . . . . 75

7.3 Model Rejection and Discovery Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.3.1 Model Rejection Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.3.2 Model Discovery Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.4 Boosted Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8 Analysis Structure, Exposure and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.1 Analysis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.1.1 Step I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.1.2 Step II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.2 Analysis Exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2.1 Livetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2.2 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3 Signal and Background Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.3.1 Magnetic Monopole Flux Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 828.3.2 Astrophysical Neutrino Flux Assumptions . . . . . . . . . . . . . . . . . . . . 84

9 Simulated Event Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.1 Signal Monte Carlo Event Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.2 Background Monte Carlo Event Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

10 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.1 Event Triggers and Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2 Step I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.2.1 The Offline EHE Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8810.2.2 The Track Quality Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8810.2.3 The Muon Bundle Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9210.2.4 The Surface Veto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

10.3 Step II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9310.3.1 Additional Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9610.3.2 Step II Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9610.3.3 BDT Implementation and Performance . . . . . . . . . . . . . . . . . . . . . . . 10410.3.4 Placing the Cut Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

10.4 Expected Numbers of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

11 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.1 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.2 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

12 Uncertainty on the Magnetic Monopole Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12212.1 Systematic Variation of Monte Carlo Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.2 Total Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

13 Uncertainty on the Astrophysical Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12813.1 Statistical Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12813.2 Uncertainties in the Astrophysical Flux Measurement . . . . . . . . . . . . . . 12813.3 Alternative Neutrino Flux Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12913.4 Expected Flux outside of the Simulated Energy Range . . . . . . . . . . . . 13113.5 Expected Background at Final Analysis Level . . . . . . . . . . . . . . . . . . . . . . . . . . 133

14 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13414.1 Experimental Event Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

14.1.1 Step I Accepted Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13414.1.2 Step II Accepted Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

14.2 Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

15 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

16 Swedish Summary — Svensk Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14516.1 Vad är en Magnetisk Monopol? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14516.2 Vad är IceCube? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14616.3 Att Söka Magnetiska Monopoler med IceCube . . . . . . . . . . . . . . . . . . . . . . . . . 14816.4 Resultat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

17 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A The IceCube EHE Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B Step I Observed Events over BDT Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Preface

This ThesisThe topic of this thesis is an analysis that was conducted by the author with thegoal of discovering a cosmic flux of magnetic monopoles with a speed abovethe Cherenkov threshold in ice, here restricted to the speed region between0.750 c and 0.995 c. The analysis was conducted on data collected with theIceCube detector over the course of 8 yr, and the author has been associatedwith Uppsala University and the IceCube collaboration for the entire durationof the project.

The contents of this thesis can roughly be divided into three main groupsof chapters — background, tools and methods, and the work by the author. Inaddition to this, there are a few supplementary chapters, such as Preface (thischapter), Acknowledgements and the Appendices.

The background chapters, Chapters 1–4, contain the background know-ledge needed to fully grasp the remainder of the thesis. These cover mag-netic monopoles, the IceCube detector, and the expected interactions betweena magnetic monopole and the IceCube detector medium, as well as a few use-ful units and conventions.

The next set of chapters (5–7), cover different tools and methods that havebeen used in this thesis. The IceCube magnetic monopole event simulationmethods are described in Chapter 5, and the relevant event reconstruction andcleaning methods are described in Chapter 6. Chapter 7 covers the tools thathave been used for data analysis, statistical analysis and machine learning.

Finally, the work by the author is covered in Chapters 8–14. Chapter 8marks the beginning by covering the overall strategy of this analysis, the anal-ysis scope, and additional parameter space constraints that may exist. Next,the Monte Carlo samples that were used to develop this thesis are detailed inChapter 9. Chapter 10 covers the event selection developed for this thesis,from beginning to end, and the results that can be expected from the applica-tion of this event selection is covered in Chapter 11. After this, Chapters 12and 13 cover the uncertainty on the analysis efficiency for magnetic mono-poles and the uncertainty on the neutrino astrophysical flux, respectively. Thefinal chapter, Chapter 14 covers the final results produced by this analysis.

Author’s ContributionsThe majority of the time I have spent as a Ph.D. student with Uppsala Univer-sity has been devoted to the development of the analysis that is described in

11

this thesis. During this time, I have been an active collaborator in the IceCubecollaboration, and have attended six IceCube collaboration meetings whereI have presented and discussed my work, discussed the work of my collab-orators, and formed strong professional bonds. I also attended the ParticlePhysics with Neutrino Telescopes conference in October 2019, PPNT-2019,where I presented my own work along with other active and finished magneticmonopole analyses in IceCube. Additionally, my work has been presentedtwice at the International Conference for New Frontiers in Physics, in Au-gust 2017 and July 2018, ICNFP-2017 and -2018 respectively, for the latter ofwhich I co-authored the presentation material.

I have attended four Partikeldagarna meetings, the Particle Days, which isthe annual meeting of the Swedish particle physicist community. I have hadthe opportunity of presenting my work during several of these.

During my Ph.D. student time, I have also completed a number of courses,both in and outside of Uppsala University. Within Uppsala University I havecompleted courses for a total of 42.5 HP, covering a wide scope of topics rele-vant for my Ph.D. education. I have also attended the Neutrinos Underground& in the Heavens summer school, given by the Niels Bohr Institute (Copen-hagen University) and the Detector Technologies for Particle Physics course,jointly given by the Niels Bohr Institute and Helsinki University. Addition-ally, I have attended two internal IceCube courses — the IceCube bootcampon collaboration and analysis, and the IceCube advanced course on C++.

Before commencing my IceCube data analysis work, I conduced an antennacharacterization study for the ARIANNA collaboration over ∼ 6 months. Theprimary measurements were conducted by myself and my Ph.D. student col-league Lisa Unger, and secondary measurements were conducted later by aproject worker. I performed the data analysis in this project, which resulted inan internal ARIANNA report documenting the measurements and concludedantenna characteristics. The results of the study were also used as benchmarksin an antenna Monte Carlo study conducted at the University of California,Irvine.

In addition to my main tasks with IceCube and ARIANNA, I have alsotaken on various smaller engagements:

• I was a part of the IceCube Software Strike Team for two years, withmonthly code sprints to maintain and develop the IceCube software

• I have conducted the Uppsala University monitoring shifts for the Ice-Cube detector over three years

• I have performed teaching duties in laboratory exercises for two Upp-sala University courses, on the topics of shearing in classical mechan-ics and the photoelectric effect in quantum mechanics

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1. Units and Conventions

1.1 High Energy Physics Natural UnitsIn high energy physics it is common to adopt a simplified unit convention,usually denoted by high energy physics natural units, where:

c = ε0 = h = kB = 1 (1.1)

Here, c is the speed of light in vacuum, ε0 is vacuum permittivity, h is thereduced Planck constant and kB is Boltzmann’s constant. Additionally, thevacuum permeability is one, µ0 = 1, due to:

1c2 = ε0µ0 (1.2)

These constants are commonly omitted in equations, such that all units canbe given as different powers α of energy, usually on the form GeVα . Belowfollows a non-exhaustive list of the units used for different physical quantitiesin high energy physics natural units and how they relate to the InternationalSystem of Units, SI, we are familiar with.

Energy α = 1 , 1GeV ⇔ 1.60×10−10 JMass α = 1 , 1GeV ⇔ 1.78×10−27 kgTime α =−1 , 1GeV−1 ⇔ 6.58×10−25 s

Distance α =−1 , 1GeV−1 ⇔ 1.97×10−16 mTemperature α = 1 , 1GeV ⇔ 1.16×1013 K

High energy physics natural units will be used for energy and mass through-out this thesis, while time and length will be given in SI units.

1.2 Speed and Relativistic Lorentz FactorA central quantity in the work described in this thesis is speed. Speed willhere be denoted by β , which is the speed given in units of the speed of light invacuum, c:

β =vc

(1.3)

The Lorentz factor γ is used to quantify how relativistic a particle is, and isdirectly related to the speed of the particle:

γ =1√

1−β 2(1.4)

13

The Lorentz factor also relates to the mass and the relativistic total energy ofthe particle through:

γ =Etot

m0=

(Ekin +m0)

m0(1.5)

where m0 is the rest mass of the particle, Etot is its relativistic total energy andEkin is its relativistic kinetic energy.

It is common to describe the speed of a high energy particle in terms of howrelativistic it is, e.g. nonrelativistic, relativistic or ultrarelativistic. These termsare vague, and can be adapted to the use-case at hand. In IceCube, magneticmonopoles below a speed β ∼ 0.1 are usually referred to as nonrelativistic,while a speed of β ∼ 0.5 or above is commonly said to be relativistic. Theultrarelativistic regime usually denotes Lorentz factors above γ ∼ 102 to∼ 104,i.e. speeds above β ∼ 0.99995 to ∼ 0.999999995.

14

2. Magnetic Monopoles

The existence of magnetic monopoles seems like one of the safest bets that onecan make about physics not yet seen. It is very hard to predict when and ifmonopoles will be discovered. [...] But we must continue to hope that we willbe lucky, or unexpectedly clever, some day.

J. Polchinski[1]

The quote above is an apt illustration of the current state of the pursuit ofmagnetic monopoles. On the one hand, magnetic monopoles are explicitly al-lowed in most current theories describing the fundamental laws of physics. Onthe other hand, several properties, such as the monopole mass and its currentabundance, are unknown and vary over many orders of magnitude betweentheories.

This chapter is an introduction to magnetic monopoles and current experi-mental efforts to find them. Magnetic monopole matter-interactions and theirdetectable signatures in the IceCube detector are described in Chapter 4.

2.1 Electric-Magnetic DualityThe electromagnetic interaction is classically described by J. C. Maxwell’sequations of electromagnetism:

∇ · E = ρe (2.1)

∇ · B = 0

∇× E =− ∂

∂ tB

∇× B =∂

∂ tE + je

Where E and B are the electric and magnetic fields, and je and ρe are theelectric current and charge density, respectively.

Isolated magnetic charges are not included in these equations due to theirabsence in observation. They can, however, be trivially included in the form

15

of a magnetic current jm and charge density ρm:

∇ · E = ρe (2.2)

∇ · B = ρm

∇× E =− ∂

∂ tB− jm

∇× B =∂

∂ tE + je

This would introduce a symmetry to Maxwell’s equations known as electric-magnetic duality, which entails that the equations remain unchanged whenintroducing the substitutions [2]:

E, je, ρe → B, jm, ρm (2.3)B, jm, ρm → −E, − je, −ρe

2.2 The Dirac MonopoleThe electromagnetic interaction may also be expressed using the scalar andvector potentials φ and A, replacing the electric and magnetic fields of Maxwell’sequations via [2]:

E =− ∂

∂ tA− ∇φ (2.4)

B = ∇× A

This disallows a magnetic charge density, ρm = ∇ · B, as the divergence of thecurl of a vector field is zero, ∇ · B = ∇ ·

(∇× A

)= 0. The physical interpreta-

tion of this is that magnetic field lines may not have a beginning or end.However, in 1931 P. A. M. Dirac included magnetic monopoles in quan-

tum mechanics by modeling each pole as the end of a semi-infinitely long andinfinitesimally thin idealized solenoid, a Dirac string [2; 3]. This allows themagnetic field around the end of the solenoid to be identical to that of a sin-gle magnetic pole, without explicitly including isolated magnetic charges inthe theory. The apparent charge of the pole, g, corresponds to the magneticflux inside of the Dirac string, ΦB = g. The Dirac monopole is illustrated inFigure 2.1.

In order for the Dirac monopole to truly act as a free pole, the Dirac stringmust be experimentally undetectable. Classically, an infinitesimally thin ob-ject is undetectable. However, in quantum mechanics, a solenoid with mag-netic flux ΦB can be observed by the phase shift of qΦB that is introducedto the complex phase of a particle with electric charge q that is transportedaround the solenoid. In order for the solenoid to be undetectable, this phase

16

Figure 2.1. An illustration of a Dirac magnetic monopole, with the magnetic fieldlines represented in blue and the Dirac string extending to the left. The shaded arearepresents the region where the magnetic field does not look like the field of a pointcharge, but like the end of a solenoid. This region is infinitesimally small for a Diracmonopole. Figure by the author.

shift must be a multiple of 2π [2], i.e. satisfy:

qΦB

2π=

qg2π

∈ Z (2.5)

This requires that both the electric and magnetic charges are quantized if mag-netic monopoles exist, and the statement is known as the Dirac charge quanti-zation condition [2].

Inserting the elementary electric charge as the charge of the transportedparticle, q = e, and the fine structure constant as α = e2

4π , into Equation 2.5yields that magnetic charges, g, are allowed as:

g = n1

2αe (2.6)

where n is an integer. Setting n = 1 yields the smallest allowed magneticcharge, known as the Dirac charge, gD, as:

gD =1

2αe ≈ 68.5e (2.7)

where the fine structure constant is approximated with α ≈ 1137 .

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2.3 The ’t Hooft-Polyakov MonopoleThe three fundamental forces that are included in the Standard Model of par-ticle physics are each described by a symmetry gauge group — U (1) for elec-tromagnetism, SU (2) for the weak interaction, and SU (3) for the strong in-teraction [4]. So-called grand unified theories, GUTs, attempt to unify thesethree symmetries into a single symmetry group at high energy scales, a sym-metry which is broken at lower energy scales into the three gauge groups thatare observed today. The energy scale of the grand unification, ΛGUT , dependson the details of the theory, with a current lower bound of ∼ 1016 GeV.

With this is mind, G. ’t Hooft and A. M. Polyakov independently describeda different type of magnetic monopole than Dirac [2; 5; 6; 7]. ’t Hooft andPolyakov considered the Standard Model Higgs field, which contains severalcomponents φi in internal space that may be considered as individual compo-nents of a vector. At the high energy scales of a GUT symmetry, the valuesof these may vary arbitrarily, but in the vacuum state their vector magnitude,v, is fixed. The magnitude v corresponds to the radius of the minimum ofthe Mexican hat potential. A higher symmetry, such as a GUT symmetry, mayindependently break into the electromagnetic U (1) symmetry in different spa-tial domains simultaneously, potentially resulting in vastly different directionof the Higgs field in adjacent domains. These domains may be arranged suchthat the field is always pointing away from (or always towards) a particularpoint in space, in the so-called hedgehog configuration. In this point the fielddirection is undefined, and the magnitude must be zero. This is not the vacuumconfiguration of the Higgs field, and it cannot be continuously transformed tothe vacuum state. Thus, the point constitutes a (stable) topological soliton, andmust contain a localization of energy to uphold the unbroken symmetry [2].This results in an effective massive particle with mass mMM:

mMM &ΛSB

αSB(2.8)

where ΛSB is the energy scale of the symmetry breaking and αSB is the cou-pling constant at this scale (αSB ∼ 10−2 for the simplest allowed GUT gaugegroups [8]).

Topologically, the ’t Hooft and Polyakov soliton (in the hedgehog con-figuration) can arise when any gauge symmetry of a grand unifying theorybreaks into the electromagnetic U (1) symmetry. It is also shown that thistopological soliton carries magnetic charge, and thus constitutes a magneticmonopole [2; 6]. Thus, all grand unified theories that unify the three forces(electromagnetism, weak and strong) predict ’t Hooft-Polyakov magnetic mo-nopoles [2].

In the case that a GUT symmetry breaks directly into the U (1) symmetry,the resulting magnetic monopole mass would be given by the energy scale ofthe GUT and the coupling constant at this scale. These are known as GUT

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10−15 m

10−18 m

10−31 m

[Point charge magnetic field]

[Confinement]

[EW unification]

[Grand unification]X

Y

ZZ

W+

W−

µ+µ−

τ+τ−ss

cc

bb

tt

uuds

γuude+

gνµ νµ

ddγ

νeνe

uude−

uug

e+e−

ντ ντ

Figure 2.2. An illustration of the inner structure of a GUT scale magnetic monopole.Figure by the author, inspiration from [9].

scale monopoles, with a mass between mMM ∼ 1014 GeV and 1017 GeV, de-pending on the details of the theory [2; 9]. If, on the other hand, the GUTsymmetry does not break immediately into the U (1) symmetry, but does sovia an intermediate symmetry at a lower energy scale, monopoles would arisewith a correspondingly lower mass [2]. These monopoles are known as inter-mediate mass monopoles, and would have a mass between mMM ∼ 105 GeVand 1013 GeV, depending on theoretical details [2; 9].

The mass of the magnetic monopole subsequently determines the innerstructure of the monopole, illustrated in Figure 2.2 for a GUT scale monopole.A GUT scale monopole would have a core (radius r 10−31 m) where theGUT symmetry is upheld, containing virtual X and Y GUT gauge bosons [2;9]. Here, otherwise forbidden baryon number violating processes, such asproton decay, are allowed, and a proton that passes this region may decay viap → π0 + e+. Outside of this region, there would be the electroweak unifi-cation region (r 10−18 m) containing virtual Z and W bosons, and yet out-side of this would be the confinement region (r 10−15 m), containing virtualphotons, gluons, fermion-antifermion pairs and four-fermion structures, e.g.uude+. Further out from the core, only the “classical” monopole magneticfield is observed.

An intermediate mass monopole would have a similar structure, but withthe inner region extending to the scale of its corresponding symmetry break-ing. This scale does not reach the grand unification scale, i.e. the scale wherebaryon number violation is allowed, implying that intermediate mass mono-poles do not induce baryon number violating processes.

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2.4 The Cosmic Monopole PopulationIf the grand unification hypothesis is correct, GUT scale magnetic monopoleswould have formed as the ambient temperature of the early Universe decreasedbelow the energy scale of the GUT symmetry breaking. As described above,the monopole formation takes place where several domains of simultaneoussymmetry breaking meet, resulting in a monopole number density on the sameorder as that of the domains. This mechanism is known as the Kibble mecha-nism [2; 10].

Due to magnetic charge conservation, the only magnetic monopole destruc-tion channel is annihilation with an oppositely charged monopole. This pro-cess is made exceedingly rare because of the expansion of the Universe, imply-ing that a significant fraction of the monopoles that were formed in the earlyUniverse should remain today. Calculations based on the Kibble mechanismsuggest a present GUT scale magnetic monopole number density comparableto that of baryons [8], which clearly opposes everyday observations. This isknown as the monopole problem in cosmology, and is resolved by postulat-ing a period of rapid cosmic inflation in the early Universe, thus diluting themonopole density to the currently unobserved abundance [2; 8; 9].

The speed distribution of the cosmic monopole population would dependon the magnetic monopole mass, as well as the magnitude of the ambientaccelerating magnetic fields. A magnetically charged particle is acceleratedlinearly along a magnetic field, in analogy to an electrical charge acceleratedby an electric field. The Lorentz force, FL, acting on a particle with magneticcharge ngD and velocity β is given by:

FL = ngD(B+ β × E

)(2.9)

The gained kinetic energy, Ekin, over a distance L along a path ds is calcu-lated as the integral of the Lorentz force over the path. Assuming a vanishingelectric field, E = 0, this is given by [9]:

Ekin =∫

LFLds = ngD |B|L (2.10)

Here, the traversed distance L represents the coherence length of the ambi-ent magnetic field, i.e. the size of the domain where the magnetic field direc-tion remains constant.

The typical domain length of the Milky Way galaxy is ∼ 300pc with amagnetic field strength∼ 3µG. Given these values, a Dirac charged monopolewould gain a kinetic energy of Ekin ∼ 6×1010 GeV [11].

Other astrophysical environments, with higher magnetic fields or largersizes, would yield even higher kinetic energy. For example, active galacticnuclei jets, with ∼ 100µG over . 10kpc, yield Ekin . 2×1013 GeV and largescale extragalactic magnetic field structures, called sheets, with . 1µG over. 30Mpc, yield . 5×1014 GeV [11].

20

The total kinetic energy, E totkin, of a magnetic monopole is then the sum of

the kinetic energy gained in each domain passing:

E totkin =

√N×Ekin (2.11)

where the factor of√

N approximates the effect of a random walk throughan average of N domains.

A relic magnetic monopole is expected to pass N ∼ 102 extragalactic mag-netic sheets, acquiring a total kinetic energy of ∼ 1016 GeV. This allows mag-netic monopoles within a broad range of masses to be accelerated to relativis-tic, or even ultrarelativistic, speeds.

The heaviest magnetic monopoles, on the other hand, are expected to begravitationally bound to various astronomical systems. A magnetic monopoleorbiting our solar system, the Milky Way galaxy or the local galaxy clusterwould gain an orbital speed of β ∼ 10−4, 10−3 or 10−2, respectively.

2.5 Magnetic Monopole Search MethodsMagnetic monopoles may be detectable through several different channelshere at Earth — trapped in terrestrial or extraterrestrial material, producedin high energy colliders or in the form of a cosmic flux [9]. Each of thesechannels can also be probed using several different experimental techniques.The available magnetic monopole search channels, and different experimentalsearch techniques, are summarized in this section, and several current resultson the cosmic flux of magnetic monopoles can be found in Figure 2.3.

The hypothetical cosmic flux of magnetic monopoles can be searched forusing, for example, neutrino telescopes (e.g. IceCube [12; 13], ANTARES [14],BAIKAL [15]), where a large volume of water or ice is instrumented with alarge number of optical modules that detect visible light. The typical scale of aneutrino telescope is . 1km3, with isotropic acceptance. For a wide portion ofthe allowed spectrum of magnetic monopole speeds, a monopole that passesthrough ice or water should interact with the medium in such a way that itproduces optical light that is readily registered with the detector. The relevantmatter-interactions of a magnetic monopole in ice are described in Chapter 4.The work described in this thesis is an example of a search for magnetic mo-nopoles with a neutrino telescope, and previous results of similar analyses aregiven in Section 2.6.

An alternative technique to search for a cosmic flux of magnetic monopolesis to use large area cosmic ray air shower detectors (e.g. the Pierre Auger Ob-servatory [16]). These are designed to register the particle showers producedby cosmic rays as they enter the atmosphere. Magnetic monopoles that enterthe atmosphere with an ultrarelativistic speed produce similar particle showers

21

Figure 2.3. Current upper limits on the cosmic magnetic monopole flux as a functionof speed, v/c, and of the base-10 logarithm of the Lorentz factor, log10 γ), over abroad interval. Credit: A. Pollmann [19]. Included are results from IceCube [12; 13],ANTARES [14], BAIKAL [15], MACRO [20], RICE [17], ANITA [18] and the PierreAuger Observatory [16], along with the Parker bound [21; 22; 5].

continually along their entire track, and can thus be readily detected by suchdetector arrays. Cosmic ray detector arrays are usually deployed over largeareas of land in order to observe a large volume of the atmosphere, which alsoyields a very large effective volume for magnetic monopole detection. How-ever, in order to produce enough atmospheric particle showers to be detected,the magnetic monopoles must be in the regime of γ 108, thus limiting cos-mic ray detectors to search for monopoles with mass mMM 108 GeV.

Ultrarelativistic magnetic monopoles may also be detected by the Cheren-kov radiation that they produce in the radio frequency range. This is producedas a magnetic monopole traverses a dielectric medium, and is detectable if themedium is transparent to radio waves. Upper limits on the cosmic monopoleflux have been determined through radio observations of the Antarctic ice (e.g.RICE [17], ANITA [18]).

A dedicated magnetic monopole experiment, MACRO [20], used a com-bination of different detector techniques (liquid scintillation counters, lim-ited streamer tubes and nuclear track detectors) to search for a cosmic mag-netic monopole flux. This allowed them to analyze a wide range of allowedmagnetic monopole speeds, from β = 10−5 to 1, with an acceptance for anisotropic flux of monopoles of ∼ 104 m2 sr (compare to the ∼ km2 scale and4πsr acceptance of neutrino telescopes).

Another approach to search for a cosmic monopole flux is by using a mag-netometer, where the passage of a magnetic monopole through a supercon-ducting coil would be registered by the induced current [23]. This type ofdevice has an ideal detection efficiency for magnetic monopoles, independent

22

of the monopole speed. The limiting factor of this technique is, however, theeffective area, as the magnetic monopole must pass through the loop of the su-perconducting coil in order to be detected, with a typical scale of . 1m2 [24].

A magnetometer can also be used to detect terrestrially bound magneticmonopoles, i.e. monopoles that have been trapped in a piece of material. Thepiece of material is thus passed through the superconducting coil of the mag-netometer, and a magnetic monopole present in the material would be readilydetected. Searches of this type have been conducted on several classes of ma-terial, e.g. volcanic rocks [25] and meteoritic material [26].

Additionally, instrumentation components that have been close to the colli-sion point at high energy particle colliders have also been examined for trappedmagnetic monopoles (e.g. HERA [27]) that were produced in the collision.Due to magnetic charge conservation, collider produced magnetic monopolesmust be produced in pairs with opposite magnetic charge, and are thus limitedto a mass that is lower than half of the center-of-mass energy of the collision.In addition to searching for magnetic monopoles that are trapped in the detec-tor material, collider produced magnetic monopoles can be sought by lookingfor highly ionizing tracks in general purpose detectors surrounding the colli-sion point (e.g. ATLAS [28]) or in dedicated detectors (e.g. MoEDAL [29]).

2.6 Monopole Search Results from Neutrino TelescopesSeveral efforts have been made to experimentally examine the hypotheticalflux of magnetic monopoles with a speed in the range that is relevant to theanalysis that is described in this thesis. As previously mentioned, no mag-netic monopole has been detected, but in this section four different works arehighlighted, each producing an upper limit on the cosmic monopole flux in thespeed range from β = 0.5 to 0.995. The resulting upper limits are shown inFigure 2.4 along with the so-called Parker bound [21]. The Parker bound lim-its the galactic flux of magnetic monopoles to less than 10−15 cm−2 s−1 sr−1,by arguing that a higher flux would disallow the presently observed galac-tic magnetic fields. This limit is valid for magnetic monopole masses below1017 GeV, above which the monopole is only slightly deflected by galacticfields [5; 22].

Each of the highlighted analyses is produced with neutrino telescopes whereoptical modules have been immersed in large volumes of (liquid or solid) wa-ter. Each analysis is performed by first identifying a number of characteristicsignatures of a magnetic monopole event that distinguishes them from back-ground events. From the characteristic event signatures, a number of analysis-specific variables were constructed, and used to reduce the background rate.All of the analyses are focused on events with a track-like event signature, asmagnetic monopoles should have large penetrative power in matter (see Chap-ter 4.1), and an upwards directed light distribution, in order to reject events

23

Figure 2.4. The (current) lowest upper limits on the magnetic monopole flux as afunction of speed, in a speed interval relevant for the analysis presented in this the-sis. These results (barring the Parker bound) were produced by the analysis of datacollected with neutrino telescopes.

induced by atmospheric muons. After additional event selection the remainingevents are used to obtain an upper limit on the flux of magnetic monopoles.

The IceCube-40 1 yr result was produced by the analysis of one year ofdata from the IceCube detector in the 40 string configuration [13]. For thisanalysis, four benchmark magnetic monopole speeds β were selected (β =0.760, 0.800, 0.900 and 0.995) and a sample of simulated magnetic monopoleswas produced for each. The event selection criteria were the same for allmagnetic monopole speeds, but differed for events that exhibited a higher orlower average number of detected photons per optical module. The selectioncriteria accepted a total of three events over the 1 yr of data collection with theIceCube-40 detector configuration, which is not compatible with the includedatmospheric background. However, the light distributions of the three eventsdo not correspond well to the expected distribution from a magnetic monopoleevent. The events may have been the product of the not-included backgroundflux of astrophysical neutrinos. An upper limit was set for each of the fourselected monopole speeds (see Figure 2.4), where the three observed eventswere treated as upwards fluctuations of the background flux.

A similar analysis was performed on the first year of data collected withthe completed IceCube detector array, denoted by IceCube-86 1 yr [13]. Thisanalysis targets magnetic monopoles in the speed range from β = 0.5 to 0.75,where the dominant light production process is indirect Cherenkov light (seeChapter 4.2.2). A continuous spectrum of magnetic monopole events was sim-ulated in the speed range from β = 0.4 to 0.99, and a boosted decision tree was

24

used to distinguish between possible magnetic monopole event candidates andbackground events at the final level of the analysis. The event selection criteriaof this analysis accepted a total of three events in the experimental data of thefirst year of data taking with the IceCube-86 detector configuration, which iscompatible with the expected background. Conservatively, the resulting upperlimit was calculated using an expected background of 0 events, and was locallyworsened around the reconstructed speeds of the observed events. The upperlimit is world-leading between magnetic monopole speeds of about β = 0.5and 0.8 (see Figure 2.4).

In addition to the two IceCube analyses, an analysis of five years of datacollected with the ANTARES detector is included, denoted by ANTARES5 yr [14]. This analysis targets magnetic monopoles over a wide speed range,from β = 0.5945 to 0.9950, and selects magnetic monopole candidate eventsbased on their detected brightness and track-likeness. The speed range wasdivided into nine equal width bins, and an individual upper limit was set foreach bin. Two events were observed in the full 5 yr of collected data, whichis compatible with the expected atmospheric muon background. The resultingupper limit is world-leading above a speed of β = 0.8615 (see Figure 2.4).

Finally, I highlight the upper limit produced by the analysis of five yearsof data collected by the BAIKAL collaboration, here denoted by BAIKAL5 yr [15]. In this analysis, a magnetic monopole event candidate was mainlyidentified by its track-likeness and brightness, and quality cuts consideringthe event apparent direction and position relative the detector array were ap-plied. A total of 3.9± 2.2 background events were expected to be acceptedby this event selection over the five years of collected data, and zero eventswere observed. This resulted in then world-leading upper limits for each ofthe three speeds β = 0.8, 0.9 and 1.0. These have now been superseded bymore than one order of magnitude, but are included here for completeness (seeFigure 2.4).

25

3. The IceCube Neutrino Observatory

The IceCube Neutrino Observatory is a multi-purpose facility that can be usedfor a wide range of physics topics. One of the primary objectives of IceCubehas been to discover the diffuse cosmic flux of high energy neutrinos, whichwas achieved in 2013, through the observation of an excess of high energyneutrino events over the expected background [30].

In addition to this, IceCube has contributed in many areas, including:Neutrino astronomy Where IceCube has several world leading searches

for point-like neutrino emitters, both steady [31] and transient [32],and an excellent sensitivity for discovering a nearby supernova [33].

Neutrino oscillations Using atmospheric neutrinos, where IceCube issensitive to baseline lengths from ∼ 10km (directly above the detec-tor) to ∼ 104 km (≈ 2REarth, directly below the detector) [34], severalneutrino oscillation parameters have been measured.

Dark matter Searches for neutrinos as dark matter decay or annihila-tion products from the galactic center [35], the Sun [36] or the centerof the Earth [37].

New physics A wide category, including searches for non-standard in-teractions [38], sterile neutrinos [39], and magnetic monopoles [13],the latter of which the work described in this thesis is an example of.

3.1 The DetectorThe IceCube detector is designed to detect the Cherenkov light produced bythe neutrino interaction products in the deep Antarctic ice. In addition to neu-trinos (astrophysical and atmospheric) the IceCube detector measures atmo-spheric muons, muon bundles and potentially also exotic particles. See Fig-ure 3.1 for a schematic illustration of the IceCube detector and its constituentcomponents.

The first IceCube string was deployed in 2005, and the detector was com-pleted in 2010 [40]. During the construction years IceCube operated and col-lected data in partial configuration modes, where the detector grew larger witheach year. Since 2011, the completed detector has operated in its full configu-ration.

Each year of operations, denoted by detector season, is identified with adesignation such as “IceCube-XX” where XX is given by the number of op-erating strings, e.g. 40, 79, 86. The seasons of full configuration are also

26

Figure 3.1. A schematic illustration of the IceCube detector. Credit: IceCube col-laboration. An illustration of the Eiffel Tower in Paris (height 324 m) is inserted todemonstrate the scale.

appended with a roman numeral, representing the year since completion. Thefirst full configuration season (2011) is thus denoted by IceCube-86 I, and theroman numeral is incremented by 1 for each subsequent year. The season des-ignation may also be abbreviated as ICXX-I, where the XX numeral and finalroman numeral behave correspondingly.

3.1.1 The Detector Constituent ArraysThe IceCube detector array consists of three sub-arrays, each one with a differ-ent purpose. Each sub-array is instrumented with a number of digital opticalmodules (DOMs), and they all feed their data to the IceCube Laboratory (ICL)at the surface for readout [40].

The main in-ice array makes up the bulk of the IceCube detector, andis also the main tool for the majority of the measurements, e.g. astrophys-ical neutrinos, atmospheric neutrinos and muons, and exotic particles. Themain in-ice array consists of 78 cables extending deep into the ice, commonlycalled strings. These are placed in a hexagonal grid (see Figure 3.2) with anearest-neighbor average spacing of 125 m. Each string is instrumented with60 DOMs with a nearest-neighbor spacing of 17 m, deployed between depthsof 1450 m and 2450 m. The DOMs on each string are labeled with a numeralfrom 1 to 60, increasing with depth. As such, the main in-ice array can be

27

Globen Arena

Figure 3.2. A schematic view of the surface footprint of the IceCube detector. Credit:IceCube collaboration. An illustration of the Globen Arena in Stockholm (diameter110 m) is inserted to demonstrate the scale.

divided into 60 layers of DOMs at similar depths, by selecting DOMs withidentical identification numbers.

An additional eight strings, distributed around the center of the main in-icearray, allow the definition of the DeepCore sub-detector [41] (see Figure 3.2).The purpose of DeepCore is to lower the energy detection threshold for inci-dent neutrinos in a region of the detector, for use in analyses on e.g. neutrinooscillations, astrophysical neutrino sources or various exotic topics. Similar tothe main in-ice array, the DeepCore strings also hold 60 DOMs each, but witha denser spacing. Of these, 50 are placed below the main dust layer with a 7 minter-DOM spacing, and the remaining 10 DOMs are placed above the dustlayer with a 10 m spacing. The DOMs on the eight DeepCore strings cannotbe trivially included in the DOM layers of the main array, due to the differinginstrumentation depths. The definition of the DeepCore sub-detector variesby use-case, but always includes the bottom 50 DOMs of the eight DeepCorestrings, often along with the DOMs on the adjacent main array strings that arevertically close to the DeepCore instrumented volume. The DeepCore stringsare instrumented with DOMs with a higher quantum detection efficiency, apartfrom strings 79 and 80, that carry both standard and high quantum efficiencyDOMs.

28

The main in-ice array and DeepCore together instrument the deep Antarcticice with 5160 DOMs.

In addition to the deep detector arrays, IceCube includes a surface array,the IceTop array, that instruments the surface footprint of the deep detectorarray. The IceTop array consists of a total of 162 frozen ice containers, tanks,(approximately 1 m3), each instrumented with two DOMs facing downwards.The tanks are placed in pairs at the surface coordinates of each of the 78 mainarray strings, and the DeepCore strings numbered 79, 80 and 82. IceTop isused as a detector for cosmic ray air showers, with a primary cosmic ray energyfrom E = 300TeV to 1EeV. It can be used to measure the arrival direction,the flux and the mass composition of the southern hemisphere cosmic rays.Additionally, IceTop enables a veto for the in-ice constituents against eventsoriginating in a cosmic ray atmospheric interaction by measuring the resultingparticle air-shower, and correlating the arrival times of the air shower and thein-ice detection.

3.1.2 The DOMThe basic building-block of the IceCube detector is the digital optical module(DOM). A total of 5484 DOMs constitute the IceCube detector, 5160 frozeninto the deep Antarctic ice (the main in-ice array and DeepCore) and 324frozen into ice-tanks at the surface level (IceTop). Each DOM operates asan independent photon detector, and communicates continuously with the sur-rounding DOMs and the control unit in the IceCube Laboratory (ICL) at thesurface [40].

The main constituents of a DOM are a photo-multiplier tube (PMT) severalcalibration LEDs and a processing main-board. These are encased in a spher-ical pressure resistant 13 mm thick glass housing, made up by two equallydimensioned hemispheres. The glass housing is pierced by a cable for dataexchange and power supply. See Figure 3.3 for a schematic illustration of aDOM.

The photo-detector in each DOM is a 10 inch (25.4 cm) R7081-02 HAMA-MATSU PMT. Each PMT is operated with an individually calibrated voltagein order to achieve a gain (amplification factor) of ∼ 107, thereby allowingsingle-photon detection. The overall detection efficiency of a DOM also de-pends on the quantum efficiency of the PMT, i.e. the probability that an elec-tron is ejected by a photon incident on the photo-cathode. By default, theIceCube DOMs have a quantum efficiency of ∼ 25% (for λ ∼ 390nm), whilesome (the majority of DOMs deployed on DeepCore strings) are instrumentedwith a higher efficiency PMT (model HAMAMATSU R7081-02 MOD) andtherefore have a higher quantum efficiency of ∼ 34%.

One source of noise in an IceCube event is the noise originating in thePMT itself. For example, a∼ 300Hz noise rate is caused by thermal electrons

29

Figure 3.3. A schematic illustration of a digital optical module, DOM. Credit: Fig-ure 2 from reference [42].

that evaporate from the PMT dynodes, and may cause a detection unrelatedto an incident photon. Additionally, photo-production on the first dynode (asopposed to the photo-cathode), or electrons that scatter past one or more dyn-odes, produce so-called pre-pulses. So-called after-pulses are produced byionization of residual gas in the PMT. Finally, radioactive decays in the DOMglass housing contribute an additional ∼ 350Hz noise rate.

The glass housing has a 93 % transmission efficiency for photons with awavelength of λ = 400nm, and 50 % and 10 % for 340 nm and 315 nm respec-tively. The PMT is coupled to the glass housing with an optical gel, which istransparent to light with wavelengths from λ ≈ 350nm to 650nm, and encasedin a mu-metal grid for partial shielding against the Earth magnetic field. Theair is evacuated from the spherical glass housing and replaced with nitrogengas at a pressure of ∼ 0.5atm. The lack of oxygen prevents corrosion of theelectronics and the low pressure ensures the tightness of the seal between thetwo hemispheres prior to deployment, even at the minimum recorded SouthPole air pressure.

Mounted in the top hemisphere of the DOM is also the processing main-board. This is where the digitization of the registered PMT signal takes placebefore transmission from the DOM to the ICL. The digitization is done withtwo different systems, the fast Analog to Digital Converter (fADC) and theAnalog Transient Waveform Digitizer (ATWD). Each DOM contains one fADCand two ATWDs, where the fADC allows a longer and coarser detection (256samples at 40 MHz) and the ATWDs yield a more detailed and short readout(128 samples at 300 MHz) The two ATWDs alternate operations in order tominimize dead-time. Additionally, the ATWDs digitization can be done with

30

one of three different multiplication factors (16, 2 or 0.25), which is deter-mined by the amplitude of the incoming signal.

In order for a detected signal to be designated as a DOM launch (alsoknown as a hit or pulse), the total registered charge by the PMT must sur-pass a discriminator threshold of 0.25 PE, where PE denotes photo-electron.A photo-electron is a unit of electric charge that represents the average chargedetected in a PMT per single detected photon. Thus, the total charge measuredin an event is often given in units of PE, or as the number of photo-electrons,nPE = [total registered charge]

1PE .Each DOM also holds a flasher-board instrumented with 12 calibration

LEDs. These are placed in pairs at equal intervals around the horizontal planeof the DOM, with each pair producing light directed horizontally through theice as well as 48° upwards. The majority of the DOMs are instrumented withLEDs that produce light with a wavelength of λ = 399nm, while sixteen areinstrumented with LEDs that yield light with λ = 340nm, 370nm, 450nm and505nm for wavelength dependence calibration.

3.1.3 The Detector MediumThe Antarctic glacial ice, where the IceCube detector is placed, was formedover many millenia as consecutive layers of snow were compressed to ice bythe pressure of new snow above. Therefore, the top layer of the glacier consti-tutes a cover of snow. Below the snow comes a layer of firn which transitionsinto ice with trapped air bubbles. These air bubbles scatter light substantially,rendering the shallow ice opaque to optical light. As the depth increases fur-ther, the high pressure forces the air to diffuse into the ice and form air-hydratecrystals, so-called clathrates, with optical properties very similar to ice [43].The majority of the air bubbles have dissipated into the ice around a depth of∼ 1400m, which is why the in-ice constituents of the IceCube detector areplaced below 1450 m of depth.

Figure 3.4 shows the absorptivity a and the effective scattering coefficientbe as functions of in-ice depth and photon wavelength. The absorptivity is thereciprocal of the characteristic absorption length, λa, a = λ−1

a , where the char-acteristic absorption length is the distance traversed by a photon as it reachesa(1− 1

e

)probability of having been absorbed. Correspondingly, the effective

scattering coefficient is the reciprocal of the effective scattering length, λe,be = λ−1

e . The effective scattering length relates to the scattering mean freepath, λs, through Equation 3.1, where avg(cos(θ)) is the average cosine ofthe scattering angle [43].

λe =λs

1− avg(cos(θ))(3.1)

31

Figure 3.4. The effective scattering coefficient be (left) and the absorptivity a (right)as functions of in-ice depth and photon wavelength. Credit: Figure 22 from refer-ence [43].

Below the depth of 1400 m the main scattering and absorbing agent consistsof micron-sized dust particles frozen in the ice. These mainly include mineralgrains, sea salt, soot and drops of acid, and are concentrated in several roughlyhorizontal layers. The dust layers correspond to ancient geological eventsresulting in higher concentration of dust particles in the air, and, thus, in theAntarctic snow. A photon incident on a dust particle has an average cosine ofthe scattering angle of avg(cos(θ))≈ 0.94 [43].

The region in the instrumented ice with the highest concentration of dust,known as the main dust layer, is found around a depth of 2000 m. The icein this region was formed ∼ 65000yr ago [44], and the effective scatteringand absorption lengths here are ∼ 5m and ∼ 20m respectively for light with awavelength λ = 400nm [43]. This can be compared to the typical scatteringand absorption lengths in the instrumented volume, ranging within [20;50]mand [80;200]m, respectively. The ice is clearer below the main dust layer thanabove it, with the longest effective scattering and absorption lengths, ∼ 100mand ∼ 250m, respectively, for λ = 400nm [43].

The (group velocity) index of refraction, nλ , decreases monotonically fromnλ = 1.38 to 1.33 for wavelengths from λ = 337nm to 532nm [43; 45]. SomeIceCube first-guess algorithms assume a constant index of refraction over thewhole detector, with a value of nλ = 1.34.

3.1.4 Coordinate SystemAn IceCube-local coordinate system is defined. The (x,y,z) coordinates ofIceCube follow a right-handed configuration where the x-y plane is horizon-tally directed, with the y axis directed along the Global prime meridian, andthe z axis is vertically directed. The origin of the IceCube coordinate sys-

32

tem, (0,0,0)IC, is located close to the center of the instrumented volume, ata vertical depth of 1946 m. The origin of the IceCube coordinate system issometimes referred to as the center of the IceCube detector.

The direction of a particle in IceCube can thus be given in terms of an(x,y,z) unit vector, or in spherical (azimuth, zenith) coordinates, (φ ,θ), where:

tan(φ) =yx, cos(θ) =

z√x2 + y2 + z2

(3.2)

The instrumented volume of IceCube can be defined in many ways, depend-ing on the purpose of the definition. For the purpose of the analysis presentedin this thesis, the IceCube instrumented volume has been defined as a hexag-onal prism extending 62.5 m outside of the outermost DOMs. This is usedwhen determining if a set of coordinates represents a point inside or outside ofthe detector, e.g. when calculating the geometric length of a particle trajectorythrough the detector (see Chapter 6.2.1). The 62.5 m margin was chosen ashalf of the average horizontal nearest-neighbor distance between DOMs, thuscontaining the volume within which the detector is evenly instrumented (asidefrom the denser DeepCore volume).

3.2 Data Acquisition and TriggeringThe majority of IceCube data readout triggers are based on so-called hardlocal coincidence hits (HLC hits). An HLC hit is designated as any hit thatis registered within a 1 µs time window around another hit in an adjacent orsecond-adjacent DOM on the same string. If a hit does not fulfill this conditionit is labeled as a soft local coincidence hit (SLC hit) [40]. HLC hits are oftenrequired by the IceCube trigger conditions as SLC hits are more likely to arisefrom noise.

Several trigger conditions are in place for triggering the data readout ofthe detector, where some are general purpose and others are tuned to specificuse-cases. The most general trigger is the simple multiplicity trigger (SMT),which requires eight or more registered HLC hits within a 5 µs time windowto trigger data readout. Other triggers may require fewer hits or longer timewindows by setting additional spatial requirements (e.g. a certain number ofhits occurring on the same string), or be specially designed for selecting slowlymoving particles. In the analysis presented in this thesis, data collected withany active trigger is considered.

The data readout triggering system continuously monitors the detector forany satisfied trigger conditions. As a trigger condition is met a time window of[−4µs;+6µs] is spanned around the trigger time window (of length 5 µs in thecase of the SMT) to define the period of data readout. Overlapping periods ofdata readout are merged, excepting the longer running triggers (e.g. the slow

33

particle trigger, for which the time window is 102–103 times longer than aregular event). Next, the (merged) readout time window is sent to the eventbuilder software. The event builder collects all hits (HLC as well as SLC)within the time window into a so-called pulse-map for the event, which formsthe basis for further data analysis.

The average total trigger rate is 2.7 kHz, with a <±10% annual modulationthat depends on atmospheric conditions, which yields an average daily datareadout of ∼ 1TB d−1.

Both the trigger conditions and filter algorithms are implemented solely insoftware, and may therefore technically be changed at any time. However, itis agreed that these procedures should be changed no more than once per year,in the transition between data collection seasons.

3.2.1 Data Filter StreamA number of higher level filters exist that monitor the event stream. Theseare designed to define specialized data substreams with different character-istics, e.g. cascade-like events, events that start inside the detector or eventscontained in the DeepCore sub-detector. An event that passes one or severaldata filter(-s) is transferred via satellite to the central IceCube data storage forfurther analysis. This constitutes a transfer rate of ∼ 100GB d−1 [40].

The majority of the current data filters base their analysis on a commonpulse map, called InIcePulses. As a part of the filter procedure, the filter algo-rithms construct a custom set of variables for use in further data analysis.

In the analysis presented in this thesis, events that pass the EHE filter areselected. The EHE filter was designed to be used as an initial step of the EHEanalysis, searching for extremely high-energy neutrinos with energies in andabove the PeV range (see Appendix A). The filter is set to reject events wherethe number of registered photo-electrons, nPE, is less than 103, and the averagedata rate is 0.8 Hz [46].

3.3 Typical EventsThe majority of events registered with the IceCube detector are caused bymuons, and bundles of muons, that were produced by cosmic ray interactionsin the atmosphere. Atmospheric muons are detected at a rate between 2.5 kHzand 2.9 kHz, depending on atmospheric conditions.

The second most common class of events is induced by atmospheric neu-trinos. These are detected with a typical rate a factor of ∼ 10−6 lower thanthe atmospheric muon rate. Like atmospheric muons, these neutrinos are pro-duced when cosmic rays interact with nuclei in the atmosphere.

In addition to the atmospheric neutrinos and muons, certain IceCube eventsare induced by extra-terrestrially produced neutrinos, commonly called astro-

34

physical neutrinos. The diffuse flux of astrophysical neutrinos can be experi-mentally measured by large neutrino telescopes, such as IceCube [30], but atvery high energies the neutrino flux measurement suffers from low statisticsand therefore exhibits a large uncertainty. At the highest energies the localflux is further reduced by neutrino absorption in the Earth.

In addition to the event classes described above, very low energy neutrinosmay arrive at the IceCube detector, originating in e.g. the atmosphere, the Sunor a nearby supernova. Such low energy neutrinos may give rise to only asingle photo-electron, and thereby mainly contribute to the ambient stochasticnoise background in IceCube. However, a close enough supernova might stillproduce enough low energy neutrinos to be detected as a significant increasein the collective rate of the ambient background [33].

3.4 High Energy Neutrinos in IceCubeA neutrino interacts rarely with matter, and is only detectable in IceCube aftera collision with the detector medium. The possible interaction channels areneutral current, NC, and charged current, CC, interactions with both the ambi-ent nuclei (dominantly) and electrons. Neutral current interactions take placethrough the exchange of a Z boson, via:

ν/ν +X Z−−→ ν/ν +X∗ (3.3)

where a momentum exchange between the neutrino ν (antineutrino ν) andnucleus X has taken place. If the momentum transfer is large enough, thenucleus X will break up (indicated by the final state asterisk) and cause aparticle shower through the medium. Charged current interactions take placethrough the exchange of a W boson, via:

ν/ν +X W±−−→ l−/l++Y (3.4)

where the W exchange implies the production of an (electrically) charged lep-ton l±, as well as the conversion of the nucleus X into Y . Similar to the NCcase, a momentum transfer takes place between the neutrino and the nucleus,which may break up in the impact.

In the analysis that is described in this thesis, neutrino events that deposit ahigh amount of light in the detector are interesting as the background channelthat is most difficult to reject. This implies neutrino events induced by veryhigh energy neutrinos. For neutrinos with Eν & 10TeV the neutrino-nuclearinteraction cross sections for NC and CC, σNC and σCC respectively, are verysimilar for neutrinos and antineutrinos. Both increase with the neutrino energy,

35

and are approximately given by the power laws [47]:

σNCνX =

(Eν

1GeV

)α

×2.31×10−36 cm2 (3.5)

σCCνX =

(Eν

1GeV

)α

×5.53×10−36 cm2 (3.6)

with the exponent α = 0.363 (for comparison, 10−36 cm2 = 1pb). This isillustrated by the fact that a 30TeV neutrino has a∼ 50% absorption probabil-ity when traversing the whole Earth, while this probability increases to above95% for 300TeV neutrinos.

In addition to the mainly dominant neutrino-nucleon interaction, electronantineutrinos incident on ambient electrons can produce resonant W− bosonswhen the collision center-of-mass energy coincides with the W boson mass,i.e. for an incident neutrino energy of Eν ≈ 6.3PeV. This is known as theGlashow resonance [48], and locally increases the electron antineutrino matter-interaction cross section by several orders of magnitude.

A neutral current interaction induced by a neutrino always manifests as ahadronic cascade in the IceCube detector, originating from the break-up prod-ucts of the target nucleus. A charged current interaction may also yield ahadronic cascade, along with the final state charged lepton. This enables vastlydifferent signatures, determined by the lepton flavor:

• A final state electron (or positron) produces a local electromagneticshower in the detector.

• A final state (anti-)muon propagates long distances through the ice(energy dependent), producing light along its path both through Cher-enkov and radiative loss processes.

• A final state (anti-)tauon propagates shorter than a muon, as its pathis promptly ended by a tauon decay (producing yet another particleshower). A tauon may thus produce a so-called double bang event,where two consecutive particle showers are produced and connectedby the track of the tauon.

3.4.1 Typical Event SignaturesThe great majority of all events detected by IceCube can be categorized intoone of two main event categories, track-like and cascade-like events.

A track-like event is an event that displays a clearly elongated light signa-ture in the detector. The elongation arises when a particle produces light whilepropagating through the detector, usually more than several hundred meters,which requires a particle with high penetrative power in ice. Outside of therealm of exotics, such as magnetic monopoles, this limits the particle optionsto muons or tauons, where the tauon is required to be highly energetic in order

36

Figure 3.5. Event viewof a track-like event, hererepresented by a simulatedmuon antineutrino event withenergy Eν = 3.1×106 GeV.

Figure 3.6. Event view ofa cascade-like event, hererepresented by a simulatedelectron neutrino event withenergy Eν = 5.2×106 GeV.

37

to propagate sufficiently before decaying. A track-like event may have a parti-cle shower at the beginning, or at the end in the case of tauons. See Figure 3.5for an event view of a simulated track-like event.

The track may start inside or outside of the detector. A track starting insidethe detector, a starting track, must be induced by a neutrino (again, outside ofthe realm of exotics such as boosted dark matter), which enables the outer lay-ers of strings of the detector to be used as a veto against non-neutrino events.A track that starts outside of the detector, a non-starting track, may still beinduced by a neutrino, but may also be an atmospheric muon or muon bundle.It is impossible to distinguish a non-starting track induced by a muon neutrinofrom an atmospheric muon, so this must be done on a statistical level based onthe directions and energies of the incoming tracks.

The light of a muon or tauon track is mainly produced as Cherenkov ra-diation from secondary interaction products that are produced by stochasticcollisions along the trajectory of the particle. The radiative cross section in-creases with energy, with the result that more energetic particles produce moresecondary light than less energetic particles do.

A cascade-like event displays a shorter and broader light signature in thedetector. A cascade-like event is induced by a neutrino that interacts withthe ice to produce a large particle shower, and may be either hadronic orelectromagnetic. These are characteristically produced by electron and tauonneutrino charged current interactions, or neutral current interactions involv-ing any neutrino flavor. The light is produced as direct Cherenkov light fromthe secondary charged particles in the particle shower, which spread out overa short distance, typically less than ∼ few meters. Therefore, the light in acascade-like event appears to have a point-like production vertex compared tothe characteristic instrumentation scale of the detector, and has a broad angu-lar distribution. See Figure 3.6 for an event view of a simulated cascade-likeevent.

3.5 Interpreting an Event ViewThis thesis, as well as other IceCube literature, contains visual representationsof registered events in the form of so-called event views. An example eventview of a magnetic monopole event is shown in Figure 3.7.

In Figure 3.7, the colored and gray spheres represent DOMs with and with-out registered charge in the selected pulse-map, respectively. The apparentsize of a sphere represents the registered charge magnitude, with a customiz-able size-to-magnitude ratio. The color scale of the colored DOMs, from redto blue, represents the detection time of the first registered pulse in the DOM,from early to late, respectively. The time interval represented by the colorscale is also customizable, in order to allow meaningful viewing of eventswith varying time widths.

38

Figure 3.7. Event view of a simulated magnetic monopole event, displaying the In-IcePulses pulse-map along with the Monte Carlo true trajectory of the particle. Thehorizontal deficiency of registered pulses close to the center of the track is caused bythe main dust layer present in the ice.

An event view may also contain one or several straight lines. These repre-sent (reconstructed or Monte Carlo) particle trajectories through the detector(see Chapter 6.2.1).

Note the horizontal deficiency of registered pulses close to the center of thetrack in the example event view, Figure 3.7. This is caused by the increasedabsorption in main dust layer present in the detector volume.

39

4. Magnetic Monopoles in IceCube

As a magnetic monopole propagates through the IceCube detector mediumit interacts with the surrounding matter via a number of different processes.Several of these processes produce optical light around the trajectory of themonopole, which is readily detected by the IceCube optical modules.

This chapter is dedicated to the interactions between a magnetic monopoleand the IceCube ice, the light that is produced, as well as the characteristicsignatures of a magnetic monopole event in IceCube.

4.1 Energy Loss in MatterAs a magnetic monopole traverses a medium, it will inevitably lose energythrough interactions with the surrounding matter. These energy losses occurthrough several different processes, depending on the speed of the monopole.

Over a large portion of the speed range, from β ∼ 0.1 to 0.99995 (γ ∼ 100),the interactions between a magnetic monopole and the surrounding mediumcan be modeled as the interactions of a heavy electrically charged particle witha charge corresponding to the effective charge of the magnetic monopole [49;50]. Formally, the substitution ze→ gβ is made, where ze is the charge of theelectrically charged particle, g is the monopole charge and β is the speed ofthe monopole in units of the speed of light.

In this speed region, the monopole mainly loses energy by ionizing andexciting the electrons in the surrounding medium, so the average energy lossdE per unit length dx is given by Equation 4.1 below [51; 52], similar to theBethe-Bloch formula.

−(

dEdx

)=

4πg2e2ne

mec2

(ln(

2β 2γ2mec2

I

)+

K2− 1+δ

2−B)

(4.1)

Here, ne is the number density of electrons in the medium, me is the electronmass, and β and γ are the monopole speed and Lorentz factor respectively.Additionally, I is the mean excitation energy, δ is a density effect correction,K is the correction given by the δ -electron ionization cross section [53] (givenby the Kazama-Yang-Goldhaber (KYG) cross section, see Section 4.2.2) and Bis the Bloch correction. The Bloch correction accounts for interactions wherethe wavefunction of the incident monopole does not fully cover the scatteringcenter of the ambient target electron.

40

The mean excitation energy and the density effect correction are givenin [54], and the KYG and Bloch corrections in [51] as 0.406 and 0.248, re-spectively, for Dirac charged magnetic monopoles in ice.

Equation 4.1 yields a slow and steady rise of the average monopole energyloss, −dE

dx , over the speed range from β = 0.1 to 0.99995 (γ = 100). At thelower boundary, β = 0.1, the energy loss is ∼ 350GeV m−1, and at the upperboundary, β = 0.99995, it is ∼ 1300GeV m−1 [52].

4.2 Light ProductionAs a magnetic monopole passes through a medium it would produce light overa wide range of frequencies through several processes, with different processesdominating in different monopole speed intervals. If the medium is transparentto visible light, such as ice or water, the visible part of the produced spectrumcould be readily detected by optical modules immersed in the medium. Thissection will treat the production of visible light by magnetic monopoles prop-agating through ice.

As described in Chapter 2.3, a GUT scale magnetic monopole is hypoth-esized to induce proton decay in the surrounding medium. The final decayproducts of the proton decay would produce visible light in the form of Cher-enkov radiation. This is the dominant source of visible light for magneticmonopoles passing through ice with a speed below β ∼ 0.1.

Faster magnetic monopoles, with a speed above β ∼ 0.1, induce lumi-nescence light as they excite the surrounding atoms that subsequently deex-cite [55]. The cross section for luminescence light production is currentlyuncertain, but this is the dominant light production process for magnetic mo-nopoles with a speed β between ∼ 0.1 and ∼ 0.5. For monopoles in the speedrange that is relevant for this analysis, β ∈ [0.750;0.995], luminescence lightconstitutes between ∼ 0.1% and ∼ 1% of the total light yield, which is domi-nated by direct Cherenkov light.

Magnetic monopoles with a speed above β ∼ 0.5 not only excite the sur-rounding medium, but also ionize it. The resulting unbound electrons mayhave a speed above the Cherenkov threshold, and produce Cherenkov light.In this manner, magnetic monopoles above β ∼ 0.5 produce indirect Cher-enkov light [53]. This is the dominant light source for magnetic monopoleswith a speed below the Cherenkov threshold in ice, and is described further inChapter 4.2.2.

Magnetic monopoles that themselves have a speed above the Cherenkovthreshold in ice induce Cherenkov light directly [56]. Due to the high ef-fective charge of a magnetic monopole it will produce almost four orders ofmagnitude more Cherenkov light than a muon with the same speed. DirectCherenkov light is the dominant light source for magnetic monopoles with a

41

speed above the Cherenkov threshold in ice, and is described further in Sec-tion 4.2.1.

In the final regime of magnetic monopole light production the light is pro-duced by radiative energy losses (pair production, bremsstrahlung and photo-nuclear interactions) along the path of a monopole with ultrarelativistic speed(Lorentz factor γ & 103) [11]. This is the dominant source of visible light forultrarelativistic magnetic monopoles passing through ice.

The amount of Cherenkov light produced by a magnetic monopole, directlyand indirectly, is shown in Figure 4.1 expressed in number of photons permeter, along with the direct Cherenkov light produced by a muon.

4.2.1 Cherenkov RadiationCherenkov radiation is the electromagnetic (EM) radiation that is producedas an electromagnetically charged particle traverses a dielectric medium at aspeed higher than the speed of light in the medium, cm. It was first observed byP. Cherenkov in 1934 [57], for which he received the Nobel prize in 1958. The1958 Nobel prize was shared with I. Frank and I. Tamm, as they developed thetheory describing the production of Cherenkov radiation, deriving the Frank-Tamm formula (Equation 4.3, adapted for monopoles) [58].

When a charged particle propagates through a dielectric medium it contin-uously polarizes the medium around it. As the particle passes, the polarizationis relaxed and the medium returns to its ground state by emitting a wave ofelectromagnetic radiation. The EM waves propagate concentrically outwardsfrom the point of origin with the speed of light in the medium. As the par-ticle moves with a speed vp, the polarization origin also moves, so the inter-val between subsequent photons will be smaller in the forward direction thanthe backward. Thus, when the particle speed equals the speed of light in themedium, cm, it will propagate along with its own EM wave front, and con-structive interference will induce a shock wave front of photons in the forwarddirection. When the speed of the particle is increased further, now above cm,the wave fronts will only be produced behind the particle, and will align withearlier fronts such that a conical shock wave front with the apex at the particleposition is formed. The light in this wave front is called Cherenkov light.

The direction of the wave front relative to the particle direction, θC, dependson the ratio between the speed of light in the medium and the particle speedaccording to:

cos(θC) =cm

vp=

1nλ β

(4.2)

Here nλ = ccm

is the refractive index of the medium and β =vpc is the speed

of the particle given as a fraction of the speed of light in vacuum. The produc-tion angle of the Cherenkov photons will thus increase with the particle speed,

42

Figure 4.1. The number of photons per meter produced by a magnetic monopole inice through the Cherenkov (red line) and indirect Cherenkov (yellow line) processes.Also included is the number of direct Cherenkov photons produced by a 1e electricallycharged particle, exemplified with a muon (blue line).

and as the speed of the particle approaches the vacuum speed of light, vp → c,the characteristic Cherenkov angle of the medium is reached. The refractiveindex for visible light in the deep Antarctic ice varies between nλ = 1.38 and1.33 (see Chapter 3.1.3), and approximating it with nλ = 1.34 yields a char-acteristic Cherenkov angle of θC = 41.7°.

Equation 4.2 additionally reflects the fact that Cherenkov light is only pro-duced for combinations of particle speed β and index of refraction nλ wherethe cosine function is defined, which implies that there is a medium-dependentminimal particle speed required to produce Cherenkov light. This speed is la-beled as the Cherenkov threshold, βCT , and is βCT = 0.746 for deep Antarcticice.

The number of photons produced through the Cherenkov process, Nγ , perunit length dx and unit wavelength dλ is given by the Frank-Tamm formulafor magnetic monopoles [56; 58; 59]:

d2Nγ

dλdx=

2παλ 2

(gnλe

)2(

1− 1

(βnλ )2

)(4.3)

where α is the fine structure constant and g is the magnetic charge of themonopole.

Equation 4.3 differs from the Frank-Tamm formula for electric charges bythe factor

(gnλe

)2. This means that a magnetic monopole carrying the Dirac

43

charge, g = gD, that traverses deep Antarctic ice will produce(gDnλ

e

)2 ≈ 8430times as much Cherenkov light as a muon propagating with the same speed.

Integrating Equation 4.3 over the wavelength band detectable by an IceCubeoptical module, 350nm – 650nm, yields that a magnetic monopole travelingwith a speed β = 0.995 through the deep Antarctic ice (assuming nλ = 1.34)will produce 2.23×108 detectable photons per meter (see Figure 4.1). Thecorresponding number for a monopole with a speed close to the Cherenkovthreshold, β = 0.8, is 6.61×107 photons per meter.

4.2.2 Indirect Cherenkov RadiationIn addition to the processes described above, a high energy particle with elec-tric or magnetic charge also ionizes the medium as it traverses it, thereby re-leasing free electrons (also known as δ -rays or δ -electrons) [53]. If a highenough energy is delivered to the electron, it will exceed the Cherenkov thresh-old of the medium and produce Cherenkov light. This Cherenkov light isknown as indirect Cherenkov light, as it is not produced by the passage of theprimary particle itself, but of secondary particles in the event.

The total amount of indirect Cherenkov light produced by a magnetic mo-nopole depends on two factors — the number of produced δ -electrons, Nδ ,and the Cherenkov light yield of each δ -electron. The latter is given bythe Frank-Tamm formula, and the cross section for δ -electron production iscalculated using one of several popular models. The cross section calcula-tion used in IceCube is known as the Kazama-Yang-Goldhaber (KYG) crosssection [53], which is calculated with full consideration of quantum electro-dynamical and special relativistic effects. The KYG cross section is chosenover the Mott cross section [49], which neither accounts for quantum mechan-ical effects, nor allows the magnetic monopole to have a non-zero spin. Thedifference in light production between between the KYG and Mott cross sec-tions is negligible in the monopole speed range that is relevant for this analysis,β ∈ [0.750;0.995] [59].

The number of δ -electrons produced by a relativistic magnetic monopolepropagating through ice, Nδ , is given per unit length dx and unit of electronkinetic energy dTe by [49; 59]:

d2Nδ

dTedx=

2πneβ 2g2e2

mec2T 2e

FKYG (Te,β ) (4.4)

where ne is the electron number-density in the medium, me is the electronmass and Te is the electron kinetic energy. The form factor FKYG (Te,β ) is thesum of the contributions corresponding to the helicity flipped and non-flippedfinal states.

The magnetic monopole does not itself need to exceed the Cherenkov thresh-old to produce δ -electrons, but only exceed a speed of β ∼ 0.5. At speeds of

44

β = 0.5, 0.8 and 0.995 a total of 2.08×104, 5.16×106 and 3.14×107 pho-tons are produced per meter (see Figure 4.1). This means that indirect Cheren-kov light yields a detection channel for magnetic monopoles with speeds downto β ∼ 0.5, and also gives a significant contribution to the total light yield ofmonopoles with speeds above the Cherenkov threshold.

4.3 Magnetic Monopole Signatures in IceCubeThe analysis that is described in this thesis is designed to detect magneticmonopoles with a speed above the Cherenkov threshold. These monopolesdominantly produce light via the Cherenkov process, and subdominantly asindirect Cherenkov light via δ -electrons. Direct Cherenkov light is producedwith a characteristic shape around the magnetic monopole trajectory, whichforces monopole events to have several characteristic signatures if detectedwith the IceCube detector array. In order to be identified as a monopole event,it needs to satisfy each of these signatures. Several illustrations of possibleevent signatures in IceCube are included in Figures 4.2 and 4.3.

The major distinguishing signature of a magnetic monopole event in Ice-Cube is its brightness, i.e. the large amount of produced, and detected, light.A magnetic monopole would produce almost four orders of magnitude moredirect Cherenkov light than a muon. Similar to an ultrarelativistic magneticmonopole, a muon passing through the IceCube detector volume may also pro-duce light through stochastic collisions with the medium, resulting in a non-uniform light pattern in the detector. However, the majority of muon eventswill exhibit less light than the average magnetic monopole event, due to thelarge effective charge of the monopole. Compare Figure 4.2a (monopole) withFigures 4.2b and 4.2c (dim track and cascade).

Additionally, the magnetic monopole event should be non-starting/-stopp-ing, i.e. the event should neither appear to originate from inside the detec-tor volume nor finish before exiting the detector. The monopole should passthrough the detector with negligible changes in direction and speed due to itshigh penetrative power. Compare Figure 4.2a (monopole) with Figures 4.3aand 4.3b (cascade and starting track).

Cherenkov radiation is also produced with a highly consistent number ofphotons per unit length, yielding a very smooth, non-stochastic, light outputover the length of the detector. Compare Figure 4.2a (monopole) with Fig-ure 4.3c (stochastic track).

Finally, as this work aims to discover magnetic monopoles in the speedrange β ∈ [0.750,0.995], the final event signature is the subluminal speed ofthe primary particle in the event.

The large fiducial volume, the dense and uniform instrumentation, and theexcellent timing resolution of IceCube makes it ideally suited to detect a cos-mic flux of magnetic monopoles above the Cherenkov threshold.

45

(a) A magnetic monopoleevent.

(b) A dim cascade-likeevent.

(c) A dim track-like event,with minor stochasticlosses.

Figure 4.2. Illustration the IceCube detector volume (blue) along with a registeredevent. The color scale corresponds to the color scale of an actual event view, wherered→blue represents early→late detected pulses.

46

(a) A bright cascade-likeevent.

(b) A bright track-likeevent with stochasticenergy losses, startinginside the detector volume.

(c) A bright track-likeevent with stochasticenergy losses, startingoutside of the detectorvolume.

Figure 4.3. Illustration the IceCube detector volume (blue) along with a registeredevent. The color scale corresponds to the color scale of an actual event view, wherered→blue represents early→late detected pulses.

47

5. Magnetic Monopole Event Simulation inIceCube

This chapter covers the methods that are used to produce Monte Carlo (MC)simulated magnetic monopole events in IceCube, as well as the validation ofthe simulation process against theoretical prediction and experimental data.

The event simulation process can be divided into several steps — particlegeneration, particle propagation, light generation, light propagation, and lightdetection.

In the particle generation step of the simulation process the initial parame-ters of the simulated particles are determined. These include the particle type,initial energy (or speed), and initial position and direction. The generation stepcan either involve an isotropic incident flux of particles within a large energyor speed range, or be conducted with more specific initial parameters.

In the particle propagation step, the generated particles are propagatedthrough the detector until they either propagate too far away or lose enoughenergy to no longer produce a significant amount of light. The propagationstep includes simulating any stochastic scattering that the particle might expe-rience, along with any decay and resulting daughter particles.

The propagation step yields a list of particles that produce light in the de-tector. This list is evaluated in the light generation step of the simulationprocess, where the appropriate light production processes are set up for eachparticle, and the Monte Carlo photons are generated. The photons are thenpropagated through the detector in the light propagation step. This ties intothe light detection step where the detector acceptance to incident photons isaccounted for.

5.1 Magnetic Monopole GenerationThe IceCube magnetic monopole generation software allows both for a generalpurpose generation of magnetic monopoles, and a targeted generation usingspecific generation parameters. This is achieved by allowing tuning of eachparameter of the software to the specific use-case. Several of the simulationparameters can be set to hold a specific value or to be randomly sampled froma user-specified probability function.

Each magnetic monopole event is generated independently and assigned thefollowing parameters:

48

(0,0,0)ICd

R

Figure 5.1. Illustration of the placement of the generation disk (red) around the Ice-Cube detector volume (blue) and three examples of generated magnetic monopoleparticles (green). The center of the IceCube detector is marked by (0,0,0)IC, and thegeneration disk radius and the distance between the generation disk and the center ofIceCube are labeled R and d, respectively. Figure by the author.

• Speed, β = vc

• Mass, mMM• Initial position, x = (x,y,z)• Direction, (θzen,φazi)

The speed of the magnetic monopole is given by the user, either as onespecific value, or sampled from a given distribution (uniform or a power law)with specified boundaries.

The mass of the magnetic monopole is given by the user as a single value. Itis common to choose a mass that allows the monopole to pass “un-scattered”over the length of the detector, i.e. a mass for which the velocity vector of themonopole is not expected to change significantly. This is then generalized torepresent a wide variety of masses where this assumption is valid. The defaultvalue for the magnetic monopole mass in the IceCube simulation software is1011 GeV, which is also used in this analysis.

The initial position of the magnetic monopole is generated at random coor-dinates on a generation disk placed close to the detector volume. The gener-ation disk is placed with its normal vector directed towards the center of the

49

IceCube detector volume, (0,0,0)IC, and the magnetic monopole is directedalong the normal vector of the generation disk. The monopole generation pro-cedure is illustrated in Figure 5.1. The direction of the disk normal vector issampled separately for each monopole event from a user specified interval,where the default settings yield an isotropic flux of magnetic monopoles. Theradius of the generation disk and distance between the disk and the center ofIceCube are set by the user, and both have standard values of 1000 m. Twoseparate studies were performed in connection with this analysis to determineappropriate values for these parameters.

5.1.1 Generation Disk RadiusIn order to ensure that a sample of simulated magnetic monopole events ap-propriately represents the assumed natural magnetic monopole flux, the radiusof the generation disk must be adequately large. If a too small value is cho-sen, the simulated sample will be lacking events that in reality would triggera detector response. If, on the other hand, a too large value is chosen, a largenumber of magnetic monopole events would be generated with a trajectory faroutside of the instrumented volume (that therefore do not trigger the detector),which would be computationally expensive.

Therefore, a study was conducted to determine the minimal sufficientlylarge generation disk radius, and whether or not the 1000 m default value isappropriate to use in this analysis.

A total of 2.3×104 magnetic monopole events were simulated in the speedinterval β ∈ [0.800;0.995] with an isotropic flux generator configuration anda generator disk radius of R = 1200m. Lower speed monopoles, down tothe Cherenkov threshold, were excluded as these events would exhibit lesslight than higher speed monopoles, which is counterproductive in a studyregarding the farthest radial distance a monopole can have and still triggera detector response. The light yield for magnetic monopole events withinβ ∈ [0.800;0.995] is relatively similar.

The closest approach distance between each monopole trajectory and thecenter of the IceCube detector volume was recorded, as this is geometricallyidentical to the radial coordinate of the monopole generation position on thegeneration disk. The closest approach between a particle track and the centerof IceCube is called the centrality of the particle track, dC.

Just over half of the generated magnetic monopole events, 52.6 %, induceda trigger in IceCube. The remaining monopoles passed too far from the de-tector volume to yield enough light in the detector to induce a trigger. Thedistribution of the triggered monopoles over the centrality variable can befound in Figure 5.2. Included here is also the average distribution of the sim-ulated flux of magnetic monopoles. It was found that no magnetic monopolewith a centrality larger than ∼ 1060m induced a trigger in IceCube. There-

50

Figure 5.2. The distribution of the detector triggering subsample of a simulated setof magnetic monopoles over their radial generation coordinate (red), generated witha disk radius of 1200 m. The radial generation coordinate is equal to the centrality ofthe monopole track, dC, which is the closest approach distance between the monopoletrack and the center of the IceCube detector. Included is also the average distributionof the simulated flux of magnetic monopoles (blue), with statistical uncertainty.

fore, the default disk radius value of 1000 m is disregarded, and a value of1100 m is adopted for further simulations. Note that the present monopoletrigger acceptance is ideal for smaller generation radial distances, with cen-trality dC 700m.

5.1.2 Generation Disk DistanceThe distance between the generation disk and the center of IceCube is by de-fault set to 1000 m. This section covers a confirmation study that the defaultvalue of 1000 m is large enough. A too small value for this distance couldresult in the magnetic monopole appearing to have been generated inside (orclose to) the detector volume, as the amount of detected light in the outerDOM layers of the detector will be too small. An estimation of the minimumdisk distance that appropriately emulates an infinite track that enters the de-tector can be obtained from first principles based on the absorptivity of thedeep glacial ice. Typically, light that is produced further away from the instru-mented volume than the characteristic absorption length will not be registeredwith the detector, which therefore indicates the minimum distance betweenthe generation disk and the edge of the detector.

The described effect is most easily identified when the magnetic monopoletrajectory is parallel to the IceCube strings, as the monopole then passes a

51

500 m600 m700 m800 m900 m

1 000 m

Figure 5.3. Illustration of the IceCube detector volume (blue) along with the genera-tion disks (red) that were simulated to determine the appropriate distance between thegenerator disk and the center of IceCube. Figure by the author.

series of consecutive DOM layers when traversing the detector, where the de-tected light can be measured as a function of depth. This is achieved whenthe magnetic monopole enters from e.g. the top of the instrumented volume,where the characteristic absorption length is ∼ 100m. A dedicated MonteCarlo study was performed to validate this prediction, thus considering thefull optical properties of the ice and the detector triggering logic.

A total of 6 different distances between the generation disk and the centerof IceCube were tested, ranging from 500 m to 1000 m with an increment of100 m. The generation disk was positioned directly above the detector volumewith the normal directed parallel to the strings. The instrumented volumestarts around 500 m above the center of the detector, which means that a diskdistance of 500 m corresponds to the generation disk being placed directly atthe edge of the detector volume. Correspondingly, a 1000 m disk distanceplaces the disk 500 m outside of the instrumented volume. The radius of thedisk was set to 250 m in this study to avoid simulating partially containedevents along the edges of the detector. An illustration of the generation diskconfigurations above the IceCube detector volume is included in Figure 5.3.

For each tested disk distance a total of 1000 events were simulated in thespeed interval β ∈ [0.800;0.995]. Lower speed monopoles, down to the Cher-

52

Figure 5.4. The average registered charge per DOM layer for each tested disk distance.The variation over DOM layers is an effect of the depth-dependent optical propertiesof the Antarctic ice (see Chapter 3.1.3, compare to Figure 3.4).

enkov threshold, were excluded as their light yield decreases rapidly with de-creasing speed, while it remains relatively stable within β ∈ [0.800;0.995].The small generation disk radius forced all magnetic monopoles to pass closeto the center of IceCube, so all simulated events induced a trigger.

For each simulated event the detected light was summed per horizontalDOM layer. For each tested disk distance, the average registered charge perlayer of DOMs was then calculated. The average registered charge per DOMlayer is shown in Figure 5.4 for each tested disk distance. It was found thata disk placed 500 m from the center of IceCube, i.e. directly above of theinstrumented volume, yielded a lower average registered charge in the mostshallow DOM layers in comparison with the larger tested disk distances. Asno deviation was found for larger disk distances (600 m and upwards), it wasconcluded that it is sufficient to place the disk at least 100 m above the instru-mented detector volume, corresponding to the characteristic absorption lengthat this depth.

The absorption length of the deep glacial ice is typically less than 200 m(maximum ∼ 250m). This is combined with the largest distance betweena DOM and the center of the detector, 764 m, to conclude that the defaultvalue of the disk distance parameter, 1000 m, can be accepted for use in thisanalysis.

53

5.2 Magnetic Monopole PropagationThe propagation of a magnetic monopole in the IceCube software is based onthe assumption that the magnetic monopole has negligible interactions overits propagation in the detector volume. This implies that the monopole ex-periences negligible changes to its velocity vector (direction and speed) as itpropagates through the full length of the detector.

The magnetic monopole propagation is realized as a series of consecutivetrack segments with a set length, s. The first segment is initiated at the co-ordinates given by the generation procedure, and is extended by s along thedirection vector from the generation. The starting point of the track is giventhe time coordinate 0, and the end point time coordinate s

βc , representing thetime it would have taken a magnetic monopole to propagate along the seg-ment. Each following track segment is initiated at the end (t, x) coordinatesof the previous segment, and given a length s and duration s

βc . The series isfinished when the sequence has propagated far outside of the detector volume.

The length of a track segment, s, can be specified by the user, and has adefault value of 10 m.

5.3 Light Production and DetectionThe IceCube light production simulation is based on the series of particle seg-ments that are created by the particle propagation procedure. This procedureis identical for any type of light-producing particle, e.g. a magnetic monopoleor a muon. For each particle segment in turn, the relevant light production pro-cesses are evaluated, and the appropriate amount of light with the appropriateangular distribution is simulated.

A magnetic monopole in the speed range β ∈ [0.750;0.995] mainly pro-duces light through the Cherenkov process. The light yield is thus calculatedwith the Frank-Tamm formula for magnetic monopoles (Equation 4.3), usingthe effective charge of the monopole, its speed and the refractive index of theice. The sub-dominant light production process in this speed range is indirectCherenkov light, where the simulated light yield is based on Equation 4.4 inconjunction with the Frank-Tamm formula.

The light propagation and detection are governed by the ice-model of theIceCube simulation software. Here, “ice-model” is a slightly misleading term,as it not only describes how light propagates through the detector medium (theice), but also how it propagates through the DOM glass housing, optical geland PMT until its conversion to photo-electrons.

An IceCube ice-model is defined by several sets of parameters, each guidinga different aspect of the light propagation and detection. Three of these param-eter categories are particularly important for the uncertainty studies conductedin this analysis, and will therefore be highlighted below.

54

The scattering and absorption parameters in the ice-model represent thecharacteristic scattering and absorption lengths of photons propagating in ice.They are parametrized as functions of depth and photon wavelength.

The DOM efficiency parameter in the ice-model represents the compoundedprobability that a photon that reaches the surface of a DOM will be registeredas a detector hit. It comprises several different effects that together affect theprobability that an incident photon gives rise to a photo-electron in the PMT,e.g. the transmittance of the DOM glass housing and optical gel, the PMT effi-ciency, and the shadowing effect of the cable running alongside the DOM. Themagnitude of each of these effects has either been measured in the laboratory,in situ or as a combination of the two.

Despite the outer spherical symmetry of a DOM, it does not have isotropicphoto-sensitivity. The inner geometry of the DOM is (close to) rotationallysymmetric around the vertical axis, with the active detection unit (the PMT)directed downwards. Thus, the photo-sensitivity of the DOM is assumed tohave the corresponding symmetry. Therefore, the DOM angular sensitivityparameters in the ice-model, p0 and p1, describe the detection probability for aphoton that reaches the DOM from a particular incident zenith direction. Thedefault settings yield zero efficiency for photons incident from directly abovethe DOM, and maximum efficiency for photons from almost directly below.

In addition to the ice-model, the detector response is governed by the de-tector configuration file, that describes the status of each part of the detectorat a given time. This is where information about individual DOMs is stored,e.g. their spatial coordinates and whether they are active or not. At the startof each detector season, the current detector configuration is extracted to rep-resent the season in Monte Carlo events production. Notice that the detectorconfiguration file must be chosen by the user, and that there is some differencebetween the configuration files of each year.

5.4 Simulation ValidationIn order to produce reliable results, it must be validated that the simulationprocedure that is used in an analysis represents nature well. This is commonlydone by comparing the event distributions that are produced with experimentaldata and simulated event samples over several variables relevant to the anal-ysis. It may also be done by comparing features of the simulated events withthe results of analytical calculation.

The modular design of the IceCube simulation chain allows all IceCubesimulation to be produced with common software for the photon propagationand detection. This, in turn, allows that validation of the simulation chainperformed within one analysis may transfer to another analysis with similarevent characteristics.

55

Figure 5.5. Experimental data and simulated event samples (atmospheric muons, at-mospheric neutrinos, GZK neutrinos) at the EHE filter level over the base-10 log-arithm of the number of registered photo-electrons, log10 NPE, and the cosine of thereconstructed zenith direction of the event, cosθ . Credit: Figure 4 from reference [60].

5.4.1 Validation with Experimental DataTwo validation procedures are described below, both performed within Ice-Cube analyses that are relevant for the analysis that is presented in this the-sis. The first, the Extremely High Energy (EHE) analysis, is described in Ap-pendix A, and is developed to search for a population of high energy neutrinos(typically Eν & 108 GeV) that were produced in cosmic ray interactions withthe cosmic microwave background (via the Greisen-Zatsepin-Kusmin (GZK)mechanism). The EHE analysis event selection is used in the first step ofthe present analysis. The second analysis, measuring the flux of high energyastrophysical muon neutrinos from the northern hemisphere, is described inChapter 8.3.2. The resulting flux measurement of this analysis is assumed asthe background astrophysical neutrino flux in the present analysis.

The variables that were used in the EHE analysis event selection havebeen extensively validated through comparison between simulated event sam-ples and experimental data [60]. An example of this is shown in Figure 5.5,where experimental data collected during the first complete detector season,IceCube-86 I, is compared with simulated event samples representing the sig-nal and background of the EHE analysis over two variables. The data is shownat the EHE filter level, and over the base-10 logarithm of the number of reg-istered photo-electrons, log10 NPE, and the cosine of the reconstructed zenithdirection of the event, cosθ . The mass composition of the ultra-high energycosmic ray (UHECR) flux is currently unknown, but is commonly bracketed

56

(a) The data distributions over the cosine of the reconstructed zenith angle of theevent.

(b) The data distributions over a muon energy proxy variable.

Figure 5.6. The experimental data and simulated neutrino event samples (astrophysi-cal neutrinos, prompt atmospheric neutrinos, conventional atmospheric neutrinos) thatare included in the analysis over the cosine of the reconstructed zenith angle and amuon energy proxy variable. Notice that the sum of the simulated neutrino distribu-tions largely falls directly on top of the conventional atmospheric distribution. Credit:Figures 1 and 2 from reference [61], respectively.

57

by two extreme composition cases — pure protons and pure iron nuclei [60].The experimental data lies between the atmospheric muon flux expectationsgiven by the two extreme UHECR cases in both of the tested variables, thusimplying an agreement between the simulated samples and experimental data.

A similar comparison was made in the second analysis. An example of thisis displayed in Figure 5.6, where data collected during the detector seasonsIceCube-79 and -86 I is shown along with simulated samples of astrophys-ical neutrinos, prompt atmospheric neutrinos and conventional atmosphericneutrinos [61]. Atmospheric muons do not enter this analysis, as they cannottraverse the Earth and therefore only arrive at the detector from above (thesouthern hemisphere). The data is shown over the cosine of the reconstructedzenith angle and a muon energy proxy variable, and good agreement is seenbetween the experimental data and the simulated event samples. Note that thehigh event count in a bin around the muon energy proxy value 1.4×105 isattributed to statistical fluctuation, which would arise with this size in 9 % ofan ensemble of identical experiments. This is supported by the fact that thepresent resolution of the energy proxy variable would smear any feature basedin physics into surrounding bins [61].

5.4.2 Magnetic Monopole Light Yield ValidationWithin the context of an IceCube analysis searching for magnetic monopoleswithin the speed range β ∈ [0.1;0.5], a study was conducted to confirm that themagnetic monopole light yield corresponds well to theoretical prediction [62].This study was conducted by simulating a sample of magnetic monopoles withisotropic angular distribution around the detector for 17 discrete values of β ,evenly spaced from β = 0.1 to 0.9. For speeds β ≤ 0.75 a total of 105 eventswere simulated per speed, while 103 events were simulated for each of thehigher speeds.

The result of this study is shown in Figure 5.7. Here, the left y axis, alongwith the analytical curves, indicates the theoretical light yield prediction ofa magnetic monopole, given by luminescence light, indirect Cherenkov lightand direct Cherenkov light. The right y axis, along with the included datapoints, indicate the average number of photo-electrons detected per event, NPE.The two y axes are normalized by aligning their values for the highest testedspeed, β = 0.9. This point was separately validated through independent cal-culation and comparison.

Note the two separate sets of data points. The first, denoted by arriving atDOM, accounts for all photons that produce a pulse in an optical module. Thesecond, denoted by recorded by in-ice arrays, represents all photons that arepart of a data acquisition event in IceCube. The difference between the twosets of data points in the low speed region is attributed to the low brightness

58

0 0.2 0.4 0.6 0.8 1

β

2

3

4

5

6

7lo

g( d

Nγ

dx/c

m−

1) (t

heo

ry)

direct Cherenkov light

indirect Cherenkov light

luminescence light

total light, arriving at DOM

total light, recorded by in-ice arrays

ref. point, left and right axis

1

2

3

4

5

log( N

PE

) (sim

ula

tion

)

Figure 5.7. The theoretical light yield prediction of magnetic monopoles (blue) alongwith the average number of detected photo-electrons in simulated magnetic monopoleevents (red), as functions of the monopole speed. Credit: F. Lauber [62].

and low speed of these monopoles, which results in a decreased trigger rateand an increased event truncation probability.

The conclusion of this study was that the magnetic monopole light yield insimulated events corresponds well to the analytically predicted light yield.

59

6. Event Cleaning and ReconstructionMethods

In order to properly understand a recorded event in a detector, the low levelrecorded data must be combined into a coherent picture. This can be doneby combining the low level data into more complex features of the detection.This process is called reconstruction, and the derived features are called recon-structed features. Reconstruction can be divided into two main categories —low level reconstructions and high level reconstructions. Low level reconstruc-tions concern transforming the raw electrical signal that is output from eachdetector module into a pulse that represents the incident signal. High levelreconstructions concern combining the individual incident signals to deducemore physically relevant features of the event, such as the flavor or energy ofthe primary particle. This type of reconstruction is the topic of this chapter.

High level reconstruction can be done to a desired complexity — wherethe rule of thumb is that simple reconstructions are computationally fast, e.g.fitting a simple function to the collected time series of pulses, while morecomplex reconstructions are slower. Therefore, a common strategy entails per-forming fast and basic reconstructions early in the data selection scheme, whenthe data volume typically is large, and only performing the more advanced andcomputationally heavy reconstructions later in the selection scheme, when thedata volume is smaller.

In order to implement an accurate reconstruction scheme, the features ofthe event that likely originate from the targeted physical phenomena must beselected, while disregarding other features. This process is known as eventcleaning. In IceCube this typically entails removing DOM hits that are deter-mined to originate from stochastic noise or coincident atmospheric muons.

In this chapter, the event cleaning and reconstruction methods that are usedin this analysis are listed. The effects of several of the cleaning and recon-struction algorithms are exemplified with event views in Figures 6.1 and 6.2.

6.1 Event Cleaning Algorithms6.1.1 The SeededRadiusTime Cleaning MethodThe SeededRadiusTime (SRT) event cleaning algorithm is designed to acceptdetected light pulses that are close to each other in space and time. It is oftenused in conjunction with the TimeWindow cleaning algorithm.

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The SRT algorithm accepts a time duration, TSRT , and a spatial radius, RSRT ,as input, along with the name of the pulse-map that will be cleaned. First, aset of seed pulses is identified as the set of all HLC:s with at least two otherHLC:s within the chosen radius-time interval. The seed pulses are registeredon a list of pulses accepted by the algorithm. Next, all pulses (SLC and HLC)within the input radius-time range of any seed pulse are added to the acceptlist. A new set of seed pulses is now defined including any HLC present onthe accept list, and the procedure is repeated. This is repeated a maximum of3 times, or until no more pulses can be added to the accept list. Finally, thealgorithm produces an output in the form of a pulse-map containing the pulseson the accept list.

The values of TSRT and RSRT must be input by the analyzer, and appropriatevalues differ depending on the type of event that is cleaned. A common choicefor very bright events is a time duration of TSRT = 1000ns and radius of RSRT =150m.

The effect of the SRT cleaning method in combination with the TimeWin-dow algorithm is exemplified in Figure 6.1, where the two methods have beenapplied in the order SeededRadiusTime→ TimeWindow.

6.1.2 The TimeWindow Cleaning MethodThe TimeWindow (TW) event cleaning algorithm is a basic event cleaningalgorithm commonly used in IceCube, and often applied in conjunction withthe SRT algorithm.

The TW method accepts a pulse-map as input, along with a fixed time win-dow width, TTW . First, the pulses in the pulse-map are ordered in time. Next,a time window with duration TTW is set up, and its starting time is shifted untilthe largest number of registered pulses within the time window is found. Thepulses inside this time window are accepted, while the remaining pulses arediscarded. The TW algorithm produces a cleaned pulse-map as output.

Similar to the SRT algorithm, the appropriate width of the time windowdepends on the type of the event that is cleaned. A common choice for verybright events is a time window of TTW = 6000ns. The TW time window istypically much longer than the SRT time window, as the TW algorithm appliesits time window once per event, considering the full duration, while the SRTalgorithm applies the time window many times per event, once per registeredpulse.

The effect of the SRT and TW cleaning methods (applied in the orderSRT→TW) is exemplified in Figure 6.1.

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6.2 Event Reconstruction AlgorithmsIn this analysis, both simple event reconstruction methods and more advancedalgorithms are employed. A common practice is that an advanced recon-struction algorithm employs a simple reconstruction algorithm, or a variationthereof, as a sub-procedure in its larger reconstruction scheme. Examples ofthis can be found among the methods used in this analysis.

6.2.1 A Particle Track RepresentationThe term “track” is commonly used in IceCube to refer to the event class thatfeatures an elongated light signature, as opposed to the spheroid signature ofthe cascade event class.

A track, when used in the context of event reconstruction features, refersto a geometrical representation of a particle trajectory with constant speedthrough the detector (or outside of it). This track carries both spatial and timeinformation about the position of the particle, and is usually defined as aninfinite straight line along which the particle propagates.

A track is specified by the following parameters:x0 Spatial coordinates of an arbitrary point along the trackt0 Time coordinate of the same arbitrary point along the trackx A directional unit vectorv The particle speed, v = βc

These parameters together define all (t, x) points along the track.In this analysis, a track is also represented by two derived properties.

GeometricLength The geometrical distance between the entry andexit points of the track in the detector volume. If the track does notenter the detector, but only propagates outside of it, the Geometric-Length is 0 m.

Centrality The closest approach distance between the track and thecenter of IceCube. This defines the centrality point, i.e. the pointalong the track that is closest to the detector central point.

6.2.2 The LineFit Track Reconstruction MethodThe LineFit track reconstruction method (LF) is a first-guess algorithm that isused to obtain a quick estimation of the primary particle trajectory. It ignoresthe propagation of the photons in the detector medium, and instead assumesthat each detected light pulse is an independent measurement of the primaryparticle position, as it moves along a straight line. The LF method accepts apulse-map as input, and produces an output in the form of a track. The trackis found by minimizing the following function:

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χ2 =

N

∑i=1

(xi− x0− vLF (ti− t0))2 (6.1)

Here, N is the total number of pulses, where (ti, xi) represent the pulse timeand DOM position coordinates of pulse number i. Additionally, (t0, x0) is anarbitrary coordinate along the track and vLF = vx is the velocity vector. Theχ2 minimization is done with respect to the track parameters x0, t0, v and x.

Before applying the χ2 minimization procedure, the pulse-map is cleanedof outlier pulses and pulses determined to likely originate from noise. Thiscleaning is designed with the focus of enhancing the LF reconstruction perfor-mance, and is thus applied regardless of if other cleaning methods have beenapplied beforehand. The cleaning is done by first rejecting pulses that aredetected late in relation to pulses in neighboring DOMs, as well as rejectingpulses that are too far from a first approximation of the track (obtained by aHuber regression on the pulse-map, that penalizes outlier pulses).

The main component of the LineFit algorithm, the χ2 minimization, is an-alytically solvable, which makes it a very fast reconstruction algorithm. How-ever, it entirely ignores that each light pulse is generated by a photon that haspropagated a non-zero distance away from the primary particle, a propaga-tion that depends on the scattering and absorption parameters of the detectormedium, as well as the characteristic shape of the Cherenkov cone. Theseapproximations reduce the accuracy of the algorithm.

On the other hand, several of the advanced IceCube track reconstructionmethods are unsuitable for use in the analysis that is presented in this thesis,as they assume that the primary particle is a muon. This includes assumptionson stochastic energy losses along the track as well as an assumption that theparticle is propagating with the speed of light, both of which are incompatiblewith the physics of a magnetic monopole event in the speed interval relevantto this analysis. The LineFit algorithm makes no assumption on the shape ofthe detected light and, importantly, leaves the speed of the fitted track as a freeparameter.

6.2.3 The EHE Reconstruction SuiteThe EHE reconstruction suite is a set of reconstruction algorithms that pro-duces a number of variables that are used to evaluate if an event passes theEHE filter (see Chapter 3.2.1).

It starts by producing a custom pulse-map directly from the recorded DOMlaunch data, selecting only HLC pulses. Based on this pulse-map severalcollective statistics are produced, such as the number of registered photo-electrons, nPE, and the number of triggered detector channels (DOMs), nCH .

Next, any pulses registered with the dedicated DeepCore strings are dis-carded, as the high DOM density of DeepCore otherwise biases further clean-

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(a) Event view with theInIcePulses pulse-map andthe true particle track.

(b) Event view with theInIcePulsesSRTTWpulse-map, the trueparticle track, and theEHE and BMreconstructed tracks.

Figure 6.1. Event views of the same simulated magnetic monopole event before (a)and after (b) event cleaning using the SRT and TW methods. The combined effect ofthe SRT and TW methods is most distinct when comparing (a) and (b) in the top-leftand bottom-right outlier regions. The red, green and blue lines represent the true trackof the magnetic monopole and the EHE and BM reconstructed tracks, respectively.See Chapter 3.5 for a description of how to interpret an event view.

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(a) Event view with theInIcePulsesSRTTWpulse-map, the trueparticle track, the BMreconstructed track and theMillipede fitted series ofenergy losses.

(b) Event view with theBrightestMedianpulse-map, the trueparticle track, and the BMreconstructed track.

Figure 6.2. Additional event views of the same simulated magnetic monopole eventas in Figure 6.1. The red and blue lines represent the true track of the magnetic mono-pole and the BM reconstructed track, respectively. The larger spheres along the BMtrack in (a) represent the Millipede fitted series of energy losses, with the size of thesphere representing the magnitude of the energy loss and the color scale correspondsto the color scale of the displayed DOMs. See Chapter 3.5 for a description of how tointerpret an event view.

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ing and reconstruction algorithms. The custom pulse-map is now cleaned us-ing the SRT and TW methods. The SRT cleaning is applied using a defaulttime of 1000 ns and a radius of 150 m. The TW cleaning is applied with adefault time window of 6000 ns.

The LineFit track reconstruction is applied on the the cleaned pulse-map,which yields a best fit track for the registered pulses. The track fitting qualityis preserved in the form of a reduced χ2 value, where the number of degreesof freedom is taken as the number of track parameters subtracted from thenumber of DOMs with registered charge.

An example of an EHE reconstructed track can be found in Figure 6.1b,along with the InIcePulsesSRTTW pulse-map.

6.2.4 The CommonVariables Event Characterization SuiteThe CommonVariables (CV) event characterization suite is a collection of in-tuitively simple event characterization packages. Each of the CommonVari-ables packages focuses on a different type of event signature, e.g. the pulsetiming information, and uses this to produce a number of statistics about theevent. The CommonVariables suite was developed within IceCube with thepurpose of providing a set of variables that could be useful in a wide varietyof analyses.

In this analysis three different characterization packages from the Common-Variables suite are used:

TimeCharacteristics Produces statistics based on the timing informa-tion of the registered pulses of an input pulse-map.

TrackCharacteristics Produces pulse distribution statistics based onan input pulse-map and an input reconstructed track.

HitStatistics Produces general statistics related to the registered pulsesof an input pulse-map.

The variables that are used from each of the event characterization packagesare listed below.

TimeLengthFWHM (TimeCharacteristics) This variable quantifiesthe duration of the event. It is given as the full width at half maximumof the time distribution of the first registered pulse in each DOM withregistered charge in the input pulse-map.

AvgDomDistQTotDom (TrackCharacteristics) This variable con-siders all DOMs with registered charge in the input pulse-map. Thedistance between each hit DOM and the input reconstructed track isobtained, and is assigned a weight equal to the total registered chargein that DOM. The variable finally constitutes the weighted average ofthe distance between all hit DOMs and the reconstructed track.

TrackHitsSeparationLength (TrackCharacteristics) This variablerepresents the geometrical span of detected light in the detector. First,

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the center of gravity coordinates of the first and fourth quartiles of thetime distribution of the registered pulses in the input pulse-map arefound. Then, the two center of gravity points are projected onto theinput track. And finally, the variable is taken as the distance betweenthe on-track projections of the two center of gravity points.

COG (HitStatistics) This variable is the center of gravity of all reg-istered pulses. It is calculated as the average position of all DOMSwith registered charge in the input pulse-map, where the weight ofeach DOM is given by its total registered charge.

6.2.5 The Millipede Track Reconstruction MethodThe Millipede reconstruction algorithm is an advanced IceCube track recon-struction package that enables full reconstruction of a track, including vertex,direction and intermediate energy losses [63]. For computational reasons, onlythe reconstruction of energy losses along the track is applied in this analysis.This requires a pulse-map and a track as input.

The energy loss reconstruction algorithm begins by modeling the track as aseries of consecutive independent cascade energy losses along a series of con-secutive minimally ionizing track segments. The cascades and track segmentsare all placed at (t, x) points along the input track, directed in the forward di-rection of the track, and placed with a separation that is given as an input tothe algorithm. The energies of the cascades and track segments are obtainedby fitting them to the light pulses registered in the input pulse-map.

The output of the Millipede energy loss reconstruction algorithm consists ofthe fitted series of consecutive cascades and track segments, for each cascadeand segment including vertex, direction and energy information.

In this analysis, the Millipede energy loss reconstruction algorithm is ap-plied on the InIcePulsesSRTTW pulse-map, i.e. the standard InIcePulses pulse-map cleaned with the SRT and TW algorithms, using the BrightestMedianreconstructed track, and setting the energy loss separation to 10 m. An inves-tigation was conducted to find the optimal track segment length and energyloss separation, based on obtaining the best separative power between eventsinduced by magnetic monopoles and astrophysical muons in the energy lossRSD variable. This variable represents the relative standard deviation of theestimated energy losses of cascades and tracks inside of the detector volume,(see Chapter 10.3.2).

An example of a Millipede fitted series of energy losses can be found inFigure 6.2a, along with the InIcePulsesSRTTW pulse-map.

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6.2.6 The BrightestMedian Track Reconstruction MethodThe BrightestMedian (BM) reconstruction method is a method that was specif-ically developed for this analysis, and is specifically tailored to reconstructevents that exhibit the unique event signatures of a magnetic monopole event(see Chapter 4.3). The algorithm is divided into two consecutive stages —the pulse cleaning stage and the track reconstruction stage. In the track recon-struction stage, the LineFit algorithm is applied to the specialized pulse-mapthat is constructed in the pulse cleaning stage. The pulse cleaning stage is thusspecifically designed to account for the characteristic monopole event features,as well as the machinery of the LineFit algorithm.

Before applying the pulse selection, the SRT and TW cleaning methods areapplied to the standard InIcePulses pulse-map with the settings used in theEHE reconstruction suite. The cleaned pulse-map is denoted by InIcePulsesS-RTTW.

Next, a custom pulse-map is constructed to form the basis of the followingtrack reconstruction. Initially, 90 % of the DOMs with registered charge in theInIcePulsesSRTTW pulse-map are rejected, accepting only the 10 % DOMswith the highest registered charge. For each remaining DOM, the distribu-tion of pulses is sorted in time. The pulse at the median time position is thenkept (favoring the earlier pulse in case of an even number of pulses), while theremaining pulses are discarded. The resulting pulse-map, i.e. the map contain-ing the median time-positioned pulses of each of the 10 % brightest DOMs inthe InIcePulsesSRTTW pulse-map, is labeled the BrightestMedianMap pulse-map.

The brightest DOMs of the event are selected as these are likely to be lo-cated close to the true trajectory of the primary particle. Since the light outputof a magnetic monopole is homogeneous over the full track length, the bright-est DOMs should be distributed evenly along the full length of the track.

Additionally, only one pulse is chosen per accepted DOM as the LineFitmethod assumes each detected pulse to be an independent measurement of the(t, x) coordinates of the particle, which is incompatible with detecting multiplepulses in the same DOM. The pulse at the median time position is chosen inorder to safeguard against outliers at the start and the end of the pulse series.

In the next stage of the BM reconstruction method, the track reconstructionstage, the LineFit method is applied to the BrightestMedianMap pulse-map.The LineFit method is chosen as it leaves the particle speed as a free parameter,which is important as the monopoles that are pursued in this work propagatewith a speed that may be significantly lower than c.

An example of a BrightestMedianMap pulse-map along with the resultingBM reconstructed track can be found in Figure 6.2b. The same BM recon-structed track can also be found in Figure 6.1b along with the correspondingInIcePulsesSRTTW pulse-map.

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In order to enable an evaluation of the BrightestMedian reconstruction algo-rithm, the reconstructed speed and direction are compared to the true particlespeed and direction below.

The magnetic monopole simulated event rate before Step II of the eventselection scheme (see Chapter 10.3) is displayed in Figures 6.3, 6.4 and 6.5,as a function of the BM reconstructed and the true particle speed, cosine of thezenith direction and azimuthal direction, respectively. In each figure the 1:1diagonal is marked with a black line. Note that there is a systematic tendencyfor overestimation of the particle speed. This variable will only be used inevent selection, so this effect will not reduce the performance of the variable.If its purpose would have been to estimate the real particle speed, a correctionwould be applied.

Additionally, the difference between the BM reconstructed speed and thetrue particle speed is shown in Figure 6.6 along with the corresponding quan-tity for the EHE reconstruction for the same sample of magnetic monopoleevents. The root-mean-squares of the distributions are 0.0373 and 0.0803 forBM and EHE, respectively.

Correspondingly, the angular difference between the BM reconstructed di-rection and the true particle track direction is shown in Figure 6.7 along withthe corresponding quantity for the EHE reconstruction. The root-mean-squaresare 5.69° and 7.90° for BM and EHE, respectively.

It is concluded that the BM track reconstruction method reconstructs thetrack more accurately than the EHE track reconstruction method, both consid-ering the speed and the direction of the magnetic monopole.

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Figure 6.3. The magneticmonopole simulated event ratebefore Step II of the event se-lection scheme as a function ofthe BM reconstructed speed,βBM, and the true particlespeed, βMC. The black linerepresents the 1:1 diagonal.

Figure 6.4. The magneticmonopole simulated eventrate before Step II of theevent selection scheme asa function of the cosine ofthe BM reconstructed zenithdirection, cos(θzen,BM), andthe cosine of the true particlezenith direction, cos(θzen,MC).The black line represents the1:1 diagonal.

Figure 6.5. The magneticmonopole simulated eventrate before Step II of theevent selection scheme as afunction of the BM recon-structed azimuthal direction,φazi,BM, and the true particleazimuthal direction, φazi,MC.The black line represents the1:1 diagonal.

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Figure 6.6. The difference between the BM reconstructed speed and the true par-ticle speed (red) along with the corresponding quantity for the EHE reconstruction(blue) for the simulated magnetic monopole events before Step II of the event selec-tion scheme.

Figure 6.7. The angular difference between the BM reconstructed direction and thetrue particle track direction (red) along with the corresponding quantity for the EHEreconstruction (blue) for the simulated magnetic monopole events before Step II ofthe event selection scheme.

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7. Data Analysis and Statistical Tools

In order to conduct an analysis in contemporary physics research a number oftools and methods may be employed to guide the procedure. These may forexample guide the overall strategy of the analysis, the statistical data treatmentor the event classification. The most important tools and methods that wereused in the present analysis are described in this chapter.

7.1 Analysis StrategiesA data analysis can be conducted using several overall strategies, concerninge.g. the data selection, the statistical data treatment, and what data may be usedfor analysis development.

A brief introduction is given below for three common strategies.

7.1.1 Cut-and-Count AnalysesAs the name suggests, a cut-and-count analysis is divided into two main stage— the cutting, where a series of acceptance criteria are employed to select asubset of data — and the counting, where the number of events in the selectedsubset are counted and used to draw conclusions on the initial hypothesis.

The cutting stage of a cut-and-count analysis is done with the purpose ofenhancing the relative contribution of the signal (the phenomenon you aresearching for) in the total data volume. Therefore, the cut criteria must bedesigned in such a way that they favor signal events over background events.To do this, a number of cut variables should be identified as quantifiable eventfeatures where signal and background events appear dissimilar. With each cutvariable, a cut value should also be defined, which represents the boundarybetween the signal- and background-like region.

After having designed a series of selection criteria, i.e. a multi-level data se-lection scheme, that increases the relative signal contribution to a satisfactorydegree, the number of both signal and background events that survive to thefinal level (after all cuts have been applied) is estimated. The final result willemerge from the statistical comparison of the expected number of final-levelbackground events, and the observed number of final-level events in real data.

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7.1.2 Multi-Variate AnalysesA multivariate analysis is, as the name suggests, an analysis that considersmultiple variables simultaneously. This differs from the cut-and-count ap-proach, where each variable is evaluated separately, and yields increased sep-arative power (between signal and background events) by examining the datasample shape in multi-dimensional space [64].

A multivariate analysis quickly grows in complexity with an increasingnumber of variables. Therefore, it is common to conduct such an analysis bythe use of multivariate analysis tools, such as boosted decision trees or neu-ral networks. These are trained on known samples of signal and backgroundevents with the purpose of creating an algorithm that is able to extract one ormore features of each event. In the present analysis, a boosted decision tree isused to classify an event on a floating scale between background- and signal-like. Additionally, a neural network has recently been used within IceCube toclassify events into event type categories such as cascade, track, double bangand starting track [65].

Due to the complexity of a multivariate analysis, a common approach en-tails including this as a final step in a cut-and-count analysis and to define acut criterion based on the final classification result of the multivariate analysistool.

7.1.3 Analysis BlindnessA blind analysis is an analysis where the data selection method is devel-oped without investigating the full sample of experimental data. This avoidsbias that can arise based on the preconceptions and expectations of the ana-lyzer [66]. An analysis can be either fully or partially blind:

Fully blind The analysis strategy cannot be based on any specificaspect of the experimental data set.

Partially blind Experimental data may be used in the design of theanalysis strategy if one or several key features are kept hidden, e.g.through randomization.

In order to construct a well-grounded analysis strategy, it is common to basethe analysis on simulated samples of events for both signal and background,or to use a subsample of experimental data (a burn sample, which is laterdiscarded) as background. Relying solely on simulated event samples requiresan accurate knowledge of the detector response, as well as detailed models ofthe signal and background, in order to reliably represent nature.

When the full analysis strategy is finalized it is time to apply the analysison experimental data, after which the analysis strategy should not be altered.This process is called analysis unblinding.

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7.2 Determining an Upper LimitAn analysis targeting an exotic phenomenon (e.g. the existence of a hypotheti-cal particle) will have one of two possible outcomes — either the discovery ofthe phenomenon — or the setting of an upper limit (UL) on the abundance ofthe phenomenon. A discovery may be claimed when the collected data cannotreasonably be described without invoking the phenomenon (usually requiringa deviation of 5σ or more from the background-only expectation value).

The upper limit represents the upper bound of the confidence interval, i.e.the interval over the interesting observable that includes the true value of theobservable in a fraction α of identical experiments. Here, α is called confi-dence level (CL), and a larger CL implies a wider confidence interval over theinteresting observable (which, in the present analysis, is the magnetic mono-pole flux). In this analysis, a confidence level of 90 % is adopted.

7.2.1 Effective AreaThe signal efficiency of an analysis scheme can be quantified in terms of theeffective area, which represents the cross-sectional area of an ideal detectorrecording with 100 % efficiency. Thus, the effective area is a general measureof the efficiency of the analysis scheme that allows comparison with otherexperimental efforts.

The effective area at event selection level LV of an analysis, ALVeff , is given

by:

ALVeff = Agen×

NLV

Ngen(7.1)

Here, Agen is the area of the generation disk that is used in the magneticmonopole simulation scheme, Ngen is the number of generated events, andNLV is the number of registered events at analysis level LV . The ratio NLV

Ngenthus

represents the signal detection efficiency at LV .

7.2.2 Upper LimitIn the analysis that is described in this thesis, the 90 % CL upper limit on themagnetic monopole flux, ΦMM

90 , is calculated with the method developed byG. Feldman and R. Cousins (F&C) [67], which is given by:

ΦMM90 = Φ

MM0 × µ90 (nOB,nBG)

nSG=

µ90 (nOB,nBG)

Aeff × t×Ω(7.2)

Here, ΦMM0 is the assumed magnetic monopole flux, nOB is the observed

number of events, and nSG and nBG are the number of expected signal andbackground events respectively. The F&C upper limit, µ90, is the upper limit

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on the number of signal events that can be set given an observed number ofevents, nOB, and an expected number of background events, nBG. The secondequality above is given by the calculation of the expected number of signalevents,

nSG = ΦMM0 ×Aeff × t×Ω (7.3)

where Aeff represents the effective area, t the analysis livetime, and Ω thecovered solid angle.

In the analysis described in this thesis, the monopole flux assumption (andthe expected number of signal events) is taken at the level of the previous bestupper limit in the relevant β range, for illustrative purposes. (see Chapters 2.6and 8.3.1). The absolute level of this assumption is irrelevant for the calcula-tion of the upper limit, which is clear from the final equality of Equation 7.2,where the factor ΦMM

0 has canceled with a factor ΦMM0 in nSG. The expected

number of background events, on the other hand, is relevant, and enters thecalculation of µ90. In this analysis, this number is evaluated using simulatedevent samples.

7.2.3 SensitivityThe sensitivity of an experiment is given by the average of the possible upperlimits that can be expected to be set by an analysis before knowing the numberof observed events, nOB. It is calculated as a Poisson-weighted average of thepossible upper limits that can be set for all possible values of nOB, and giventhe expected number of background events, nBG.

The sensitivity, ΦMM90 , is thus calculated through:

ΦMM90 =

∞

∑nOB=0

(Φ

MM90 ×

e−nBG (nBG)nOB

nOB!

)

=ΦMM

0nSG

∞

∑nOB=0

(µ90 (nOB,nBG)×

e−nBG (nBG)nOB

nOB!

)

= ΦMM0 × µ90 (nBG)

nSG=

µ90 (nBG)

Aeff × t×Ω(7.4)

Where µ90 (nBG) represents the F&C sensitivity for a given number of ex-pected background events, nBG, and is given by:

µ90 (nBG) =∞

∑nOB=0

(µ90 (nOB,nBG)×

e−nBG (nBG)nOB

nOB!

)(7.5)

7.2.4 Including Uncertainties in the Upper LimitIn the upper limit calculations above the expected number of final level signaland background events have been treated as absolute numbers with negligible

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uncertainty. This, however, is rarely true in real experiments, so the uncertain-ties on these quantities must be considered.

The uncertainties are included by transforming nSG and nBG in Equation 7.2to their corresponding average expected equivalents with a modified Poisson-weighting, given by Equation 7.6 [13]. The transformed nBG enters the calcu-lation of µ90 as before.

The nSG and nBG transformations are fully equivalent to each other, andbased on their central nSG and nBG values and absolute uncertainties σSG andσBG, given by Monte Carlo estimation. The transformations are done as below,where A represents SG or BG:

nA → nA =∞

∑n′A=0

n′A×P(n′A|nA,σA

)(7.6)

Here, n′A is the summation variable, and P(n′A|nA,σA) is the modified Poisson-weight with an included factor w that represents a Gaussian uncertainty withwidth σA:

P(n′A|nA,σA

)=∫

∞

−nA

e−(nA+x) (nA + x)n′A

n′A!w(x|σA)dx (7.7)

The Gaussian uncertainty, w(x|σA), is a weight factor with a mean value of0 and a variance of σ2

A, given by Equation 7.8. The uncertainty on the expectednumber of signal or background events, σA, is thus included as the Gaussianweight, w, along with the assumed number of events, (nA + x). The lowerboundary of the integral is set to −nA in order to avoid unphysical negativevalues of the expression (nA + x).

w(x|σA) =1

σA√

2πe−

12

(x

σA

)2

(7.8)

The second equality of the upper limit calculation (Equation 7.2) removesthe explicit dependence of the value of nSG. Therefore, the inclusion of thenSG uncertainty in the upper limit has an additional step where a multiplicativefactor nSG

nSGis included as such:

ΦMM90 → nSG

nSGΦ

MM90 (7.9)

7.3 Model Rejection and Discovery PotentialsThere are several methods for evaluating the performance of an event selectionscheme before it is applied to data, each highlighting a different quality of theselection scheme. Two methods are used in this analysis: the model rejectionpotential and the model discovery potential.

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7.3.1 Model Rejection PotentialThe model rejection potential (MRP) constitutes the ratio between the F&Csensitivity, µ90, and the number of signal events that can be expected assuminga model flux, nSG [68]:

MRP =µ90

nSG(7.10)

This expression for the MRP is recognized as the coefficient that yields theflux sensitivity, Φ

MM90 , from the assumed flux in Equation 7.4.

The MRP thus represents the sensitivity that is achieved per expected sig-nal event. Therefore, finding the minimal MRP over a range of possible cutvalues yields the optimal cut value with regards to the optimization of both thesensitivity and signal acceptance.

7.3.2 Model Discovery PotentialThe model discovery potential (MDP) represents the relation between the ex-pected number of signal events and the number of events that need to be ob-served in order to claim a discovery [69]. The MDP is calculated as the ratiobetween the least detectable signal, µLDS, and the expected number of signalevents, nSG:

MDP =µLDS

nSG(7.11)

Finding the minimal MDP over a range of cut values yields the optimalcut value with regards to optimizing both the signal acceptance and the leastdetectable signal.

In order to calculate the least detectable signal, the critical number of ob-served events, ncrit, must be calculated. The critical number of observed eventsis defined as the lowest number of observed events that is required to reject thebackground-only hypothesis with a confidence level (1−α) and is found bysolving:

P(nOB ≥ ncrit|nBG)< α (7.12)

Where P(nOB ≥ ncrit|nBG) represents the probability of observing a num-ber of events, nOB, larger than or equal to ncrit given the expectation of nBGbackground events.

The least detectable signal, µLDS, is the lowest number of expected signalevents that yield a (1−β ) probability of observing ncrit or more events. Thisis found by solving:

P(nOB ≥ ncrit|nBG +µLDS) = 1−β (7.13)

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A significance level of α = 5.73×10−7 is commonly required to claim adiscovery in high energy physics research (corresponding to the area under the5σ tails of a Gaussian distribution). This significance level is used to calculatethe MDP in this analysis, along with a confidence level of (1−β ) = 50%(commonly used for a discovery).

7.4 Boosted Decision TreesA boosted decision tree (BDT) is a machine learning based multivariate analy-sis tool that can be trained to classify events into different categories based ontheir appearance in multiple variables [70; 71]. When a BDT is implementedin an event selection scheme it will award each event with a score that signi-fies its likeness to a signal event, e.g. in the range [−1,+1] corresponding to[background-like,signal-like].

In order to employ a BDT for event classification purposes, it must betrained to distinguish between signal and background events. In the trainingstage, the BDT is exposed to two multivariate samples of events, one contain-ing signal events and the other containing background. These are known asthe training samples, and serve as the basis of the classification criteria thatare developed for the BDT event classification scheme. The training samplescan either consist of simulated events, experimental data or a mix of both.

The training of a BDT is done by sequentially training a series of decisiontrees (DTs) to distinguish signal events from background. Each DT is itself amultivariate based binary classifier tool, where the classification is defined asa series of divisions, or branchings, of the data, until the resulting subsamplescontain a pure enough sample of either signal or background events. Branch-ings that yield a low gain in separative power may be reversed, or pruned. TheDTs are trained sequentially in order to allow a boosting of wrongly classi-fied events between the training of two trees to enhance the separating perfor-mance. This is done by awarding each wrongly classified event in the trainingdata with a higher importance weighting than the correctly classified eventsbefore using the data to train the next tree in the sequence. Finally, the overallBDT score is formed as the average score awarded by the constituent DTs.

The branching depth, the pruning strength, the boost factor and the numberof constituent decision trees are all tunable parameters of the BDT trainingstage. Additionally, it is common to only include a (randomly selected) sub-sample of the training data for the training of each DT, the size of which canalso be tuned by the analyzer.

In the subsequent validation stage, the BDT is exposed to two additionaldata samples, one containing known signal and the other known backgroundevents. These are called the validation samples, and their events are classifiedusing the newly trained BDT classification scheme. The resulting BDT scoredistributions are compared to the distributions of the training samples using a

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Kolmogorov-Smirnov test (KS-test) [72]. If the BDT score distributions of thevalidation samples are determined to not originate from the same underlyingdistribution as the training samples (using the KS test resulting p-value), theBDT is labeled as overtrained, and unfit for use. This implies that the BDThas learned to recognize the individual event features in the training samples,as opposed to the common broad features of the full samples. The result isan event selection scheme that is specialized in either selecting the individualevents that make up the training signal sample (and thus fail to select othersignal events), or in rejecting the individual background training events (andthereby fail to reject other background events). An overtraining problem isusually solved by training a new BDT with tuned training parameters.

After the training and the validation are finished the BDT algorithm shouldhave produced a functioning and well performing event scoring scheme. TheBDT can now be implemented as an event classification algorithm directly inthe event characterization and selection scheme.

However, before applying the BDT event scoring algorithm to the chosendata samples, it is important to remove the training sample from the full dataset. To use the training samples in the further event selection development andevaluation would result in an overestimation of the separative power of theevent selection scheme, as the BDT performance is inherently optimal whenapplied to these samples.

Additionally, it is important to remember that it is up to the analyzer todetermine the BDT score range that corresponds to signal-like events, and therange that does not.

In this work, the BDT is implemented through the pyBDT software pack-age, a standard BDT package in IceCube [71].

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8. Analysis Structure, Exposure andAssumptions

The aim of the analysis that is described in this thesis is to examine the hypo-thetical cosmic flux of magnetic monopoles with speeds above the Cherenkovthreshold in ice. This work constitutes a fully blind cut-and-count analysiswith a multivariate final stage.

8.1 Analysis StructureThe analysis described in this thesis is divided into two main steps, Step I andStep II.

8.1.1 Step IStep I constitutes a one-fell-swoop procedure to drastically reduce the con-tribution of atmospheric neutrino and muon events, as well as dim neutrinoevents with astrophysical origin. The event selection was initially developedfor an earlier IceCube analysis, the Extremely High Energy (EHE) analy-sis [46], searching for a so-far unobserved population of astrophysical neu-trinos with extremely high energy (Eν & 106 GeV). These neutrinos wouldinduce very bright events in IceCube, typically with a registered charge higherthan ∼ 105 PE.

The EHE event selection consists of a small number of simple cuts, thusenabling a highly general selection of bright events that imposes minimal con-straints on the shape of the light distribution.

Additionally, the event selection is developed to efficiently reject the back-ground flux of atmospheric neutrinos and muons. Less than 0.085 atmo-spheric events are expected after the application of the event selection overthe full EHE analysis livetime (9 yr, IceCube-40, IceCube-59, IceCube-79,and IceCube-86 I–VI).

The full procedure of the Step I event selection is described in Chapter 10.2,and the EHE analysis is detailed in Appendix A.

8.1.2 Step IIStep II was developed to remove any neutrino events that are accepted by theEHE analysis, both the hypothetical GZK neutrinos and additional astrophys-

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ical neutrinos. This step is designed using custom reconstruction methods andemploys a boosted decision tree for particle type classification.

Using Monte Carlo simulated events for both the signal and the backgroundsamples, several distinguishing monopole event signatures were examined todetermine the degree to which they show distinction between events inducedby magnetic monopoles and astrophysical neutrinos. This involved, but wasnot limited to, designing a dedicated reconstruction algorithm to reconstructthe signal events more accurately (the BrightestMedian method), and employ-ing an advanced IceCube reconstruction algorithm in an unconventional man-ner (the Millipede method). Finally, nine cut variables were chosen to train aBDT to construct an event scoring scheme that ranks each event according tohow signal-like it is, and a final cut is made on the BDT score variable. Thevalue for the final BDT score cut criterion is set where the best model rejectionpotential is obtained.

The procedure and performance of Step II is described in Chapter 10.3.

8.2 Analysis Exposure8.2.1 LivetimeThis analysis was applied on 8 yr of data collected with the IceCube detectorarray, IceCube-86 I through IceCube-86 VIII. This constitutes all of the datathat is collected with the completed detector array at the date of the unblindingof this analysis. The livetime per year is listed in Table 8.1.

The event selection scheme of Step I is based on the event selection of theEHE analysis. The most recent EHE analysis iteration covers 6 yr of datacollected with the full detector configuration (IC86-I through IC86-VI), alongwith 3 yr of data collected in partial configuration (IC40, IC59 and IC79).When developing the event selection scheme of Step II, the total amount ofdata that would be available at completion was not known. Therefore, it waschosen that the selection would be developed assuming a six year livetime(IC86-I through IC86-VI), corresponding to the IC86 portion of the EHE anal-ysis livetime. The analysis is thus developed assuming a livetime of 1935 d,while the final results include data from a total of 2715 d.

For the purpose of this analysis, the experimental data can be divided intotwo main categories: the physics sample and the burn sample. The burn sam-ple constitutes the portion of data that is reserved for developing the analysisevent selection scheme. Therefore, in order to avoid bias, the data that belongsto the burn sample cannot be included in the evaluation of the final results. Theremaining experimental data is denoted by the physics sample.

The livetime of the physics and burn samples, tphysics and tburn respectively,are listed on a per-season basis in Table 8.1, where the burn sample roughlyconstitutes 10 % of the full per-season data sample. The total livetime, ttotal, isalso listed in Table 8.1.

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Table 8.1. The per-season livetime — listing the physics, burn and total livetimes.

Detector season tphysics [d] tburn [d] ttotal [d]

IC86-I 309 31.5 341IC86-II 295 32.5 327IC86-III 322 34.1 356IC86-IV 325 35.0 360IC86-V 328 37.5 365IC86-VI 357 0.0 357IC86-VII 411 0.0 411IC86-VIII 369 0.0 369Total 2715 170.6 2886

The experimental data is only divided into physics and burn subsamplesfor the first five seasons (IC86-I through IC86-V), while the full experimen-tal data set is used as the physics sample for the remaining seasons (IC86-VIthrough IC86-VIII). This is because the Step I event selection (from the EHEanalysis) was developed only using the first five seasons. The sixth season ofIC86 that was used in the EHE analysis was included after the event selectionstrategy was finalized, and therefore did not need to be subdivided. Addition-ally, no burn sample is used for the design of the Step II event selection, as theexperimental data rate is too low after the application of Step I.

For the reminder of this thesis the full livetime refers to the total livetimeavailable for physics analysis (tphysics over the 8 yr in Table 8.1), i.e. excludingthe burn sample livetime.

8.2.2 Solid AngleThis analysis is designed to find magnetic monopoles isotropically over thefull sky. The event selection scheme may reduce the signal efficiency to almostzero over certain portions of the sky. However, the results are translated tocorrespond to a solid angle, Ω, of 4π .

8.3 Signal and Background Parameter Space8.3.1 Magnetic Monopole Flux AssumptionsHere I describe the conditions on the magnetic monopole parameter spacethat constrain the fiducial region. Magnetic monopoles that lie outside of thefiducial region are not targeted by this analysis, but may still be selected bythe analysis event selection.

The first and most basic constraint is that the analysis is designed to searchfor magnetic monopoles carrying the Dirac magnetic charge, gMM = 1gD. Theresults obtained in this analysis will not be trivially generalizable to other

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charges, as the number of photons produced through the Cherenkov effect,Nγ , depends strongly on the charge of the monopole, Nγ ∝ g2

MM (see Chap-ter 4.2.1).

The second constraint is one that has been touched upon previously in thisthesis, namely that the magnetic monopole should have a speed that is abovethe Cherenkov threshold in ice. The Cherenkov threshold for light withinthe sensitive range of IceCube in deep Antarctic ice is around a speed of β =0.746. In addition to this, the speed is bounded from above to the region wheredirect Cherenkov light is the dominant light production mechanism i.e. up toa Lorentz factor of γ ∼ 100, which corresponds to a speed of β ≈ 0.99995. Inorder to draw conclusions about monopoles in a higher speed range, radiativeenergy loss processes would need to be included in the simulation software,which is beyond the scope of this work. The speed region between β = 0.995and 0.99995 is omitted in this analysis, which is developed using simulatedmagnetic monopole events in the speed range β ∈ [0.750;0.995].

An upper bound of the magnetic monopole mass mMM can be deduced fromthese speed constraints in combination with the allowed kinetic energy range(see Chapter 2.4 along with Equation 1.5). A magnetic monopole must belighter than ∼ 1015 GeV in order to fall within this speed range. A highermonopole mass is expected to display the same event shape in the detector,and thus be accepted by the analysis, but is not expected to fully populate ofthe presently studied monopole speed range. Therefore, this upper mass boundrepresents a conservative order-of-magnitude estimation.

Additionally, the monopole velocity vector is required to remain unchangedover the monopole path through the detector, which in turn requires that themonopole has negligible energy losses along its trajectory. The energy loss perunit length is independent of the monopole mass in the interval from β = 0.1 toγ = 100 (see Chapter 4.1), and ranges from 350 GeV m−1 to 1300 GeV m−1.

The scale of IceCube, LIC, is ∼ 1km, yielding a maximal energy loss of∼ 106 GeV when crossing the IceCube volume. The total energy loss overIceCube must be negligible compared with the monopole kinetic energy Ekin

MM,i.e.:

EkinMM LIC×

dEdx

(8.1)

Equations 1.4 and 1.5 can be used to determine the ratio between the par-ticle kinetic energy and its mass, depending on its speed, β . A particle speedβ = 0.75 yields the kinetic energy as Ekin ∼ 1

2 m0 (and β = 0.995 yields Ekin ∼9m0). This, in combination with Equation 8.1, yields the lower mass boundaryof this analysis as

12

mMM & LIC×dEdx

(8.2)

i.e. mMM & 108 GeV.

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Formally, the analysis is developed using Monte Carlo simulated magneticmonopoles with a mass of mMM = 1011 GeV, which is assumed to be repre-sentative of the entire allowed mass range.

The final assumption considers the overall level of the monopole flux, ΦMM0 ,

which is taken as the approximate average of the most recent flux upper limitsin the relevant parameter range [13]:

ΦMM0 = 3.46×10−18 cm−2 s−1 sr−1 (8.3)

The actual value of the assumed flux has no effect on the calculation of thefinal result. This flux assumption is thus included here for illustrative pur-poses only, and will be used to calculate concrete numbers for the expectedmagnetic monopole rate at each level of the analysis. Thus, for the remainderof this thesis, the expected number of magnetic monopole events refers to theexpected number of events assuming the flux given above.

8.3.2 Astrophysical Neutrino Flux AssumptionsAs described above, the Step I event selection efficiently rejects any significantatmospheric contribution. Therefore, the remaining background consists ofastrophysical neutrinos. For this analysis the astrophysical neutrino flux Φν isassumed to be well described by a single power law

Φν = φν ×(

Eν

100TeV

)−γν

×10−18 GeV−1 cm−2 s−1 sr−1 (8.4)

with the astrophysical flux normalization φν = 1.01+0.26−0.23 and spectral index

γν = 2.19±0.10. These values are the result of an IceCube analysis targetingthe diffuse flux of upwards directed muon neutrino events with astrophysicalorigin, which was presented at the International Cosmic Ray Conference in2017 (ICRC-2017) [73]. The analysis made use of 8 yr of recorded high en-ergy events, with 90 % of the likelihood contribution originating in the energyrange Eν ∈ [199TeV;4.8PeV]. The validation of an earlier iteration of thisanalysis, through the comparison of simulated event samples with experimen-tal data, is presented in Chapter 5.4.1. For the remainder of this thesis, this as-trophysical neutrino flux will be denoted by Φ2017

DIF-νµ. Two alternative IceCube

measurements of the astrophysical neutrino flux are discussed in Chapter 13.3.In addition to this, the astrophysical neutrino flux is assumed to arrive at

Earth with a 1:1:1 flavor ratio (for [νe:νµ :ντ ]) and consisting of equal amountsof neutrinos and antineutrinos. The 1:1:1 ratio at Earth is expected after flavor-oscillation of astrophysical neutrinos produced via pion decay considering theIceCube energy resolution for high energy events [74; 75; 76], and is consis-tent with recent measurements [74].

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9. Simulated Event Samples

The event selection that is used in this analysis is developed by only consid-ering Monte Carlo simulated events. It is crucial that these MC event samplesare as similar as possible to the natural flux that they represent. Therefore,the parameters that are used to guide the simulation procedure must be chosenwith care, and the software packages that are used must be thoroughly vali-dated. The validation of two IceCube analyses that are relevant for the presentanalysis is described in Chapter 5.4.1. Additionally, the magnetic monopolesimulated light yield has been validated against theoretical prediction, whichis described in Chapter 5.4.2.

Several samples of simulated magnetic monopole events have been pro-duced specifically to represent the signal in this analysis. Additionally, severalsimulated neutrino event samples that are available for use within IceCubehave been selected to represent the background. This chapter is dedicated tothe description of these simulated event samples.

9.1 Signal Monte Carlo Event SamplesThe magnetic monopole simulation procedure is detailed in Chapter 5.

A set of 4×105 magnetic monopole events was simulated for the develop-ment of the event selection for this analysis. The simulation was performed as-suming a uniform speed distribution over the speed interval β ∈ [0.750;0.995],and with an isotropic directional distribution. The magnetic monopole masswas set to 1011 GeV, and the IC86-VI detector configuration was used. Thisset of simulated magnetic monopole events is denoted by the baseline set.

In addition to the baseline set an additional 10 event samples were sim-ulated with varied simulation parameters, each one containing 105 magneticmonopole events. The purpose of the additional samples is to study the ef-fect of systematic uncertainties in the modeling of the detector medium andits performance on the analysis signal efficiency. One type of parameters werevaried per simulated event sample, while the others were kept at the standardvalues. The varied parameter settings and the full procedure of the systematicuncertainty study are described in Chapter 12.

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Table 9.1. The Monte Carlo simulated event samples that were used to represent thebackground flux of astrophysical neutrinos. The spectral indices given here were usedto promote the simulation of high energy events. Before being used in the developmentof the analysis, the event samples were weighted to represent the Φ2017

DIF-νµspectrum.

Blank table entries indicate the value(-s) given above.

Energy Number of Spectral DetectorIdentification Neutrino range simulated index, configurationnumber flavor (GeV) events γ season

20364 νe[5×103;107

]1.2×108 1.5 2016

20493 νµ 2.4×108

20622 ντ 1.2×108

20407 νe[106;108

]4.9×104 1.0 2016

20536 νµ 1.0×105

20665 ντ 5.0×104

11070 νµ

[107;109

]2.0×106 1.0 2012

11297 ντ 2.0×106

9.2 Background Monte Carlo Event SamplesThe simulated event samples that were used to emulate the background fluxof astrophysical neutrinos were combined to represent the background to thebest possible degree.

A total of eight samples of neutrino events were used, each representing adifferent portion of parameter space. For each sample, the primary neutrino in-teraction vertices were simulated using the NuGen neutrino interaction pack-age, and the particle propagation and detection was done with the standardIceCube software. The samples are listed with their respective internal Ice-Cube identification numbers in Table 9.1. Here, the primary neutrino flavorsand covered energy ranges are also listed, along with the numbers of simulatedevents, the power law spectral indices, and the detector configuration seasons.

The selected MC event samples cover the energy ranges[5×103;108

]GeV

for electron neutrinos and[5×103;109

]GeV for muon and tauon neutrinos.

No available sample of νe events extends to a primary energy of 109 GeV. Theimpact of the high energy deficiency is evaluated in Chapter 13.4. Addition-ally, the samples were truncated from below at a primary energy of 105 GeV,which is allowed as the EHE analysis event selection has negligible acceptancefor neutrinos with an energy below 105.5 GeV.

Spectral indices of 1.5 and 1.0 were used in MC sample production insteadof more realistic values (such as ∼ 2.2) in order to promote event generationwith higher primary energy. The samples were subsequently weighted to rep-resent the Φ2017

DIF-νµspectrum.

The choice of using simulated events produced with two different detectorconfiguration seasons, 2012 and 2016, has no effect on the final result.

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10. Event Selection

As was described in Chapter 8, this analysis follows a cut-and-count scheme intwo steps, with the goal of rejecting the majority of background events whilemaintaining a high acceptance of signal events.

Before Steps I and II are implemented, an initial event selection is per-formed on the detector event stream, in the form of data acquisition triggersand filters. Next, the Step I event selection is applied, based on a small numberof selection variables that originate from a set of simple reconstruction algo-rithms, and Step II employs a set of additional reconstruction algorithms inorder to promote the selection of magnetic monopole events over the remain-ing astrophysical neutrino flux. A more sophisticated selection is required inStep II as many neutrino and monopole events share similar characteristicsafter the Step I selection scheme.

This chapter covers the full event selection scheme for this analysis, fromthe initial data acquisition triggers to the final classification of an event as acandidate magnetic monopole or a background event.

10.1 Event Triggers and FiltersThe trigger and filter algorithms that are centrally applied to the IceCube datastream are described in Chapter 3.2.

In this analysis no trigger information is used to discriminate the eventstreams, but events that trigger any of the active trigger algorithms are ac-cepted. Next, events that pass the EHE filter (see Chapter 3.2.1) are selected.This filter was designed for use with the EHE analysis and rejects all eventsthat exhibit too low registered brightness. All events that pass the EHE fil-ter are also subject to the EHE reconstruction scheme, which is described inChapter 6.2.3.

Assuming the magnetic monopole flux given in Chapter 8.3.1, a total of244 magnetic monopole events with a speed above the Cherenkov thresholdare expected to pass the initial trigger conditions during the 8 yr livetime of thisanalysis. Of these, a total of 178 events are expected to pass the EHE filter cri-teria. The corresponding numbers for astrophysical neutrino events with anenergy above 105 GeV are 838 and 371 respectively. Atmospheric muons trig-ger data acquisition with a rate between 2.5 kHz and 2.9 kHz, and atmosphericneutrinos with an event rate a factor of∼ 10−6 lower. The corresponding ratesat the EHE filter level are 0.8 Hz and 7.6×10−6 Hz (Table A.1).

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10.2 Step IThe Step I event selection, replicated from the EHE analysis (Appendix A),employs a series of simple selection criteria that efficiently reject the majorityof atmospheric background events while selecting high energy astrophysicalevents with a high detected brightness. The cut variables originate from theEHE reconstruction scheme, and include:

• The number of registered photo-electrons, nPE, and its base-10 loga-rithm, log10 (nPE).

• The number of detector channels (DOMs) with registered charge, nCH .• The fit quality (the reduced χ2 parameter) of the EHE track recon-

struction, χ2red,EHE.

• The zenith direction of the EHE reconstructed track, cos(θzen,EHE).The EHE selection scheme is efficient for any search that targets events

with a bright signature in IceCube, including the present search for magneticmonopoles with a speed above the Cherenkov threshold. The scheme alsopresents an efficient rejection of events with atmospheric origin, with a finalexpected number of atmospheric events lower than 0.085 over the detectorseasons IC40 through IC86-VI.

Below, the details of the Step I event selection scheme are described alongwith its performance on the simulated samples of signal and background eventsthat are used in this analysis. The corresponding performance on the signal andbackground samples of the EHE analysis can be found in Table A.1.

10.2.1 The Offline EHE CutAt this cut level, three cut criteria are defined to reject events with too littleregistered light. A cut criterion is applied to the three variables nPE, nCH andχ2

red,EHE separately, such that an accepted event must satisfy the following:

nPE ≥ 25000 (10.1)

nCH ≥ 100

χ2red,EHE ≥ 30

The distributions of magnetic monopole and astrophysical neutrino eventsover these variables are shown in Figure 10.1. These cuts accept 50.6 % and15.4 % of the assumed magnetic monopole and astrophysical neutrino eventsrespectively.

10.2.2 The Track Quality CutThe next cut level is specifically designed to reject a population of atmosphericelectron neutrino events. This is done by setting a brightness requirement that

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(a) Event distributionsover the registered numberof photo-electrons, nPE.

(b) Event distributionsover the number ofchannels with registeredcharge, nCH .

(c) Event distributionsover the EHEreconstructed trackfit-quality, χ2

red,EHE.

Figure 10.1. Event distributions at the Step I offline EHE cut level over the three cutvariables, nPE, nCH and χ2

red,EHE. The selection criteria are set to accept events thatsatisfy the conditions given in Equation 10.1, and represented by vertical black linesin the figures (with the acceptance region to the right).

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(a) Magnetic monopole events.

(b) Astrophysical neutrino events.

Figure 10.2. Event distributions at the Step I track quality cut level over the registerednumber of photo-electrons, nPE, and the track fit quality of the EHE track reconstruc-tion, χ2

red,EHE. The selection criterion is set to accept events that satisfy the conditionsgiven in Equation 10.2, represented by a black line (with the acceptance region above).

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(a) Astrophysical electronneutrino events.

(b) Astrophysical muonneutrino events.

(c) Astrophysical tauonneutrino events.

Figure 10.3. Event distributions at the Step I track quality cut level over the registerednumber of photo-electrons, nPE, and the track fit quality of the EHE track reconstruc-tion, χ2

red,EHE. The selection criterion is set to accept events that satisfy the conditionsgiven in Equation 10.2, represented by a black line (with the acceptance region above).

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depends on the track fit quality of the EHE track reconstruction. An eventmust satisfy the following relation in order to pass this selection criterion:

log10 (nPE)≥

4.6 if χ2red,EHE < 80

4.6+0.015×(χ2

red,EHE−80)

if 80≤ χ2red,EHE < 120

5.2 if 120≤ χ2red,EHE

(10.2)

The distributions of magnetic monopole and astrophysical neutrino eventsover these variables are shown in Figure 10.2. The corresponding distribu-tions for each neutrino flavor in turn are shown in Figure 10.3. The selectioncriterion is represented by a black line. An event that falls below this line isrejected, while events falling above the line are accepted to the next level ofthe analysis. Figure A.1 contains the corresponding distributions for the signaland background samples of the EHE analysis.

The total acceptance rate for this cut is 71.3 % for magnetic monopoleevents and 35.7 % for astrophysical neutrino events. A large fraction of themagnetic monopole events is accepted as these are typically well fitted by theEHE track reconstruction.

10.2.3 The Muon Bundle CutThis cut level is designed to reject bright events induced by atmospheric muons,muon bundles or neutrinos. This is achieved by setting a brightness require-ment that depends on the reconstructed incoming direction of the event, wherea significantly higher brightness is required for a downwards directed eventthan an event with an upwards directed trajectory. An event is accepted at thislevel if it satisfies the following relation:

log10 (nPE)≥

4.6 if cos(θzen,EHE)< 0.06

4.6+1.85×√

1−(

cos(θzen,EHE)−10.94

)2

if 0.06≤ cos(θzen,EHE)

(10.3)

The boundary value of cos(θzen,EHE)= 0.06 corresponds to θzen,EHE = 86.6°,i.e. events with a slightly downwards directed trajectory.

The distributions of magnetic monopole and astrophysical neutrino eventsover these variables are shown in Figure 10.4. The corresponding distribu-tions for each neutrino flavor in turn are shown in Figure 10.5. The selectioncriterion is represented by a black line. An event that falls below this line isrejected, while events falling above the line are accepted to the next level ofthe analysis. Figure A.2 contains the corresponding distributions for the signaland background samples of the EHE analysis.

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The total acceptance rate for this cut is 55.4 % for magnetic monopoleevents and 49.6 % for astrophysical neutrino events.

10.2.4 The Surface VetoThe final event selection level of the EHE analysis consists of a veto againstany event that shows a correlation with an atmospheric particle shower in-duced by a cosmic ray interaction. This is achieved by examining pulses thatwere registered in the IceTop array, and correlating them to the reconstructedparticle trajectory of the event seen in the main IceCube detector.

The surface veto cut is only applied to downwards directed events, here de-fined by θzen,EHE < 85°. An event is rejected at this level if two or more pulsesare registered in IceTop within a time window of [−1000ns;+1500ns] aroundthe time coordinate of the closest approach between the reconstructed particletrajectory and IceTop. Several alternative time windows were examined withinthe context of the EHE analysis, and the selected time window was chosen asthe best compromise between background rejection and coincidental veto rate.

A study was also performed within the EHE analysis to estimate the coinci-dental veto rate where two or more IceTop pulses fall in the given time windowof an unrelated EHE event in the main IceCube detector. A coincidental vetorate of 10.6 % was found. Since the simulated Monte Carlo samples that wereused to develop the selection scheme of the present analysis do not include Ice-Top simulation, the surface veto cut was applied as a 10.6 % down-weightingof any event with θzen,EHE < 85° at this level.

As the subsample of events with a downwards pointing direction was heav-ily reduced at the previous cut level, only 0.222 % of the remaining mag-netic monopole events and 12.6 % of the remaining astrophysical neutrinoevents has θzen,EHE < 85°. This results in a fractional event rate reductionof 2.35×10−4 for magnetic monopole events and 1.34 % for astrophysicalneutrino events.

10.3 Step IIThe event selection of Step I was designed to search for an astrophysical fluxof EHE neutrinos, and thus has an inherent acceptance for such neutrinos.Therefore, a second analysis step is required in the present analysis, to rejectthese neutrino events while maintaining a high efficiency for magnetic mono-pole events.

To achieve this, several high level reconstructions are applied, targeted atdifferent characteristic magnetic monopole event signatures. From these a setof variables are concretized and subsequently used to train a boosted decisiontree for the final classification of events.

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(a) Magnetic monopole events.

(b) Astrophysical neutrino events.

Figure 10.4. Event distributions at the Step I muon bundle cut level over the registerednumber of photo-electrons, nPE, and the zenith angle of the EHE reconstructed track,cos(θzen,EHE). The selection criterion is set to accept events that satisfy the conditionsgiven in Equation 10.3, represented by a black line (with the acceptance region above).

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(a) Astrophysical electronneutrino events.

(b) Astrophysical muonneutrino events.

(c) Astrophysical tauonneutrino events.

Figure 10.5. Event distributions at the Step I muon bundle cut level over the registerednumber of photo-electrons, nPE, and the zenith angle of the EHE reconstructed track,cos(θzen,EHE). The selection criterion is set to accept events that satisfy the conditionsgiven in Equation 10.3, represented by a black line (with the acceptance region above).

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10.3.1 Additional ReconstructionThe additional reconstructions applied in Step II are performed on a noise-cleaned pulse-map. This is the same pulse-map as was used for the EHEreconstructions, labeled InIcePulsesSRTTW, and was subject to the SRT andTW cleaning methods, as well as the purging of any pulses registered withDOMs on the DeepCore detector strings. These pulses are discarded as thehigh DOM density of DeepCore otherwise may bias further cleaning and re-construction algorithms. Due to the high brightness of the selected events atthis level, no reconstruction accuracy is lost by this rejection.

The EHE reconstruction of Step I was designed for high speed and compar-atively low accuracy. For the application of further advanced reconstructionsa more accurate reconstruction of the particle trajectory must be performed.

This was the motivation for developing the BrightestMedian (BM) recon-struction algorithm specifically for the present analysis, and tailoring it to thespecific event characteristics of magnetic monopole events with a speed abovethe Cherenkov threshold. The BM reconstruction yields a parameterized trackrepresenting the trajectory of the magnetic monopole, which serves as the de-fault track in the subsequent reconstruction algorithms.

In addition to the BrightestMedian reconstruction, three of the CommonVa-riables (CV) event characterization packages were applied.

TimeCharacteristics applied on the InIcePulsesSRTTW pulse-map.TrackCharacteristics applied on the InIcePulsesSRTTW pulse-map

and seeded with the BrightestMedian track reconstruction.HitStatistics also applied on the InIcePulsesSRTTW pulse-map.

In addition to this, the Millipede track reconstruction package was usedto reconstruct stochastic energy losses along the BM reconstructed track. TheBM reconstructed track along with the InIcePulsesSRTTW pulse-map are usedas input track and pulse-map, and the energy loss along the track is discretizedwith a 10 m interval. The algorithm returns an array of reconstructed consec-utive energy losses along the BM track, with space and time coordinates aswell as the fitted magnitude of the energy loss. Note that these energy lossesare reconstructed under the assumption that they are produced as particle cas-cades along a muon track, originating from muon-nucleus interactions in theice. Magnetic monopoles in the speed range of this analysis do not collide within-ice nuclei, and mainly produce light directly via the Cherenkov process, sothe fitted series of energy losses should be homogeneous along the track.

10.3.2 Step II VariablesNine variables are input into the BDT for the characterization of an event assignal- or background-like. The majority of the variables are chosen to repre-sent a typical event signature of a magnetic monopole event that differentiatesit from a neutrino event. These variables are labeled as signature variables.

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The remaining variables are chosen to represent other features of the event thatare not directly related to characteristic event signatures of a magnetic mono-pole event, but that could affect the appearance of the event in the signaturevariables. These are labeled helper variables.

There are four major magnetic monopole event signatures in the parameterspace of this analysis. These are described in Chapter 4.3, and summarizedbelow:

Brightness The high effective charge of a magnetic monopole yieldsa high production of direct Cherenkov light.

Non-starting/-stopping The magnetic monopole enters from outsideof the detector and passes through it with completely negligible changesin direction and speed.

Non-stochastic The magnetic monopole is assumed to have negli-gible stochastic energy losses along the track, leading to negligiblestochastic cascades.

Subluminal speed The speed is allowed in the range β ∈ [0.750;0.995],i.e. distinctly separate from the speed of light in vacuum, which isnot the case for more commonly detected particles such as electrons,muons and tauons

The first of the monopole signatures given above, the high brightness of theevent, was aggressively selected for in Step I. There, only the brightest mo-nopole and neutrino events were selected, with the effect that the remainingmonopole events no longer are significantly brighter than the other remain-ing events. Nonetheless, the event brightness is included among the Step IIvariables, as some discriminating power still remains in the differently shapedevent distributions of monopoles and neutrinos.

The nine Step II variables are listed below, along with a brief descriptionof each. A discussion of the shape of the signal and background distributionsover each variable is also included.

A common feature among several of these variables is that the populationof background events is divided into two sub-populations, one that is dom-inated by muon neutrinos and one by electron neutrinos, with tauon neutri-nos populating both. The former of these populations generally takes simi-lar values as the magnetic monopole distribution, whereas the latter is gener-ally more dissimilar to monopoles. As muon neutrinos give rise to track-likeevents, electron neutrinos give cascade-like events, and tauon neutrinos giveboth (see Chapter 3.4.1), the two populations will be labeled the the track-likeand cascade-like populations, respectively, throughout the remainder of thischapter. Note that no formal event-type classification has been carried out,and the track- and cascade-like labels are only used for illustrative purposes.

As was described in Chapter 5.4, the selection variables of Step I have beenthoroughly validated through comparison between simulated event samplesand experimental data. Additionally, the common IceCube software, as well asthe monopole light yield, has been validated, which is exemplified in the same

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chapter. As the Step II variables are derived from the same underlying pulsepopulations as the previously validated variables, they are also considered asvalidated. A direct comparison between MC events and experimental datais not meaningful after the application of Step I due to the low experimentalevent count (see Chapter 14.1.1).

Step II Variable — Speed• Signature variable — subluminal speed

Taken as the reconstructed speed of the BrightestMedian track reconstruc-tion, βBM. This variable is used to distinguish primary particles that propagatewith the speed of light from particles that propagate slower.

See Figure 10.6a for the distributions of magnetic monopole and neutrinoevents over βBM. Overall, events are registered with reconstructed speeds fromβ = 0.4 to 1.3. Of course, true speeds above β = 1.0 are disallowed by spe-cial relativity, but this does not influence the allowed parameter space of theLineFit reconstruction.

The magnetic monopole events are mainly distributed between a recon-structed speed of βBM = 0.8 and 1.1, while the two neutrino event popula-tions — track-like and cascade-like — are distributed around βBM = 1.0 andbetween βBM = 0.5 and 0.9 respectively. The speed variable is a powerful sep-arator not only between magnetic monopole events and light-speed track-likeevents, but also between track-like events and cascade-like events.

Step II Variable — Energy Loss RSD• Signature variable — non-stochastic

The energy loss RSD (relative standard deviation) variable, rsd(EMIL), isbased on the track reconstruction of the Millipede package. The Millipedetrack reconstruction is used to reconstruct the stochastic energy losses alongthe BM reconstructed track. The energy loss RSD is calculated as the relativestandard deviation of all reconstructed stochastic energy losses with origincoordinates inside of the IceCube detector volume.

This variable is included to distinguish events that show a high degree ofstochastic energy losses along the track from the uniform light production thatis expected from a magnetic monopole.

See Figure 10.6b for the distributions of magnetic monopole and neutrinoevents over rsd(EMIL). The distributions cover roughly the same range, fromrsd(EMIL) = 1 to 12, and peak at approximately the same value, rsd(EMIL) =3. Both the magnetic monopole and neutrino population show a monotonicdecrease towards higher values, but the monopoles have a steep decrease, i.e.a sharp peak at 3, while neutrino events have a flatter decrease and therefore ahigher ratio of events in the high value tail.

The relative standard deviation of a variable is unitless, as it is calculatedas the ratio of the standard deviation and the mean value of the variable, thatboth carry the same unit.

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(a) The Step II speedvariable, βBM.

(b) The Step II energy lossRSD variable, rsd(EMIL).

(c) The Step II averagepulse distance variable,avg(dDOM,Q)CV-TrackChar.

Figure 10.6. Simulated magnetic monopole and neutrino event distributions over theStep II BDT variables.

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Step II Variable — Average Pulse Distance• Signature variable — non-stochastic

The average pulse distance variable, avg(dDOM,Q)CV-TrackChar, is identicalto the AvgDomDistQTotDom parameter of the CommonVariables TrackChar-acteristics reconstruction on the InIcePulsesSRTTW pulsemap and using theBM track reconstruction.

The purpose of this variable is, similar to the energy loss RSD variable, toseparate events that have a smooth light production along the track from eventsthat have stochastic losses.

See Figure 10.6c for the distributions of magnetic monopole and neutrinoevents over avg(dDOM,Q)CV-TrackChar. The two neutrino event populations (track-like and cascade-like) are not well separated over this variable. However, thevariable does show discrimination power between neutrino and magnetic mo-nopole events. Neutrino events cluster around a value of avg(dDOM,Q)CV-TrackChar =60m, whereas the magnetic monopole events cluster around a lower value of40m. Additionally, the shapes of the monopole and neutrino distributions aredistinctly different, which can be beneficial for the BDT classification.

Step II Variable — Pulse-Time FWHM• Signature variable — non-starting/-stopping

The pulse-time FWHM variable, tFWHM,CV-TimeChar, is identical to the Time-LengthFWHM parameter of the CommonVariables TimeCharacteristics re-construction on the InIcePulsesSRTTW pulse map. This variable is includedto distinguish between through going track events and events that are (fullyor partially) contained in the detector, such as cascade and starting/stoppingtrack events.

See Figure 10.7a for the distributions of magnetic monopole and neutrinoevents over tFWHM,CV-TimeChar. The neutrino population shows two peaks overthis variable, one at shorter times, tFWHM,CV-TimeChar = 2µs, populated mainlyby electron and tauon neutrinos, and one at longer times, slightly below 3 µs,populated by all three neutrino flavors. The magnetic monopole population isclustered around slightly longer times, centered around 3 µs.

Therefore, the cascade-like population is well distinguishable from the mag-netic monopole population, while the track-like population is not.

Step II Variable — Length Fill Ratio• Signature variable — non-starting/-stopping

The length fill ratio variable, LFRCV-TrackChar, is constructed to representto what extent the geometric length of the track, i.e. the distance betweenthe track entry and exit points in the IceCube volume, corresponds to thelength of the light pattern in the detector, represented by the TrackHitsSepara-tionLength parameter from the CommonVariables TrackCharacteristics pack-

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(a) The Step II pulse-timeFWHM variable,tFWHM,CV-TimeChar.

(b) The Step II length fillratio variable,LFRCV-TrackChar.

(c) The Step II relativeCoG offset variable,RCOCV-HitStats.

Figure 10.7. Simulated magnetic monopole and neutrino event distributions over theStep II BDT variables.

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age. The LFRCV-TrackChar is thus calculated through:

LFRCV-TrackChar =[TrackHitsSeparationLength]

[GeometricLength](10.4)

The purpose of this variable is to distinguish fully through going track eventsfrom cascade events and starting/stopping track events.

See Figure 10.7b for the distributions of magnetic monopole and neutrinoevents over LFRCV-TrackChar. The magnetic monopole and track-like neutrinoevent populations both peak around a value of LFRCV-TrackChar = 0.7, whilethe cascade-like neutrino event population peaks around a value of 0.3.

The length fill ratio is unitless, as it is calculated as the ratio of two variableswith the same dimension (length).

Step II Variable — Relative CoG Offset• Signature variable — non-starting/-stopping

The relative CoG offset variable, RCOCV-HitStats, is constructed to identifyevents where the center of gravity, CoG, of pulses is separated from the mid-point of the reconstructed track. This allows the separation of tracks that ex-hibit a uniform light production along the full track, i.e. with a small distancebetween the CoG and the track mid-point, and events that have a significantclustering of pulses separate from the track mid-point. The latter is a commonfeature among cascade events, and is also seen among muon tracks with highstochastic losses.

The variable is calculated using the centrality point and geometrical lengthof the BrightestMedian reconstructed track, along with the COG variable ofthe CommonVariables HitStatistics package. The relative CoG offset is cal-culated as the distance between the centrality point of the BrightestMedianreconstructed track and the HitStatistics COG position divided by the geomet-ric length of the track.

See Figure 10.7c for the distributions of magnetic monopole and neutrinoevents over RCOCV-HitStats. The magnetic monopole event population peaksclose to 0 and has a steep decrease towards higher values of the variable. Theneutrino event populations are more flatly distributed and also extend to highervalues of relative CoG offset.

The relative CoG offset is unitless, as it is calculated as the ratio of twovariables with the same dimension (length).

Step II Variable — Log-Brightness• Helper variable

The log-brightness, log10 (nPE), is the base-10 logarithm of the number ofphoto-electrons, nPE, which is given by the initial EHE reconstructions. Thisvariable is included as a helper variable. It shows significant correlation withother variables, e.g. the average pulse distance variable, where brighter eventsgenerally have higher values.

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(a) The Step IIlog-brightness variable,log10 (nPE).

(b) The Step II cos-zenithvariable, cos(θzen,BM).

(c) The Step II centralityvariable, dC,BM.

Figure 10.8. Simulated magnetic monopole and neutrino event distributions over theStep II BDT variables.

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See Figure 10.8a for the distributions of magnetic monopole and neutrinoevents over log10 (nPE). The magnetic monopole distribution is peaked arounda value slightly below log10 (nPE) = 5.0, and decreases sharply above thisvalue. The neutrino event distributions, on the other hand, extend to valuesabove 6.0 — significantly higher than the magnetic monopole population.

Step II Variable — Cos-Zenith• Helper variable

This variable, cos(θzen,BM), is the cosine of the zenith direction of theBrightestMedian reconstructed track, and is included as a helper variable. Itshows significant correlations with other variables, e.g. the pulse-time FWHMvariable, where diagonally directed events generally have higher values astheir in-detector path is longer than that of vertically or horizontally directedevents.

See Figure 10.8b for the distributions of magnetic monopole and neutrinoevents over cos(θzen,BM). Magnetic monopole events are only expected inthe up-going region of parameter space, while neutrino events are expected inthe down-going region as well. This is an effect of the muon bundle cut inStep I, where only the brightest events are allowed to be downwards directed.Additionally, high energy neutrino events interact substantially in the Earth,and are therefore expected to have a lower upwards directed flux.

Step II Variable — Centrality• Helper variable

The centrality of a track represents its closest-approach distance to the cen-ter of the IceCube detector volume. The centrality of the event, dC,BM, is cal-culated on the BrightestMedian reconstructed track, and is included as a helpervariable. It shows significant correlations with other variables, e.g. the pulse-time FWHM variable, where more central events have a longer in-detectorpath than less central events.

See Figure 10.8c for the distributions of magnetic monopole and neutrinoevents over dC,BM. All included distributions are highly similar, but separativepower is achieved in combination with other variables.

10.3.3 BDT Implementation and PerformanceBefore training the boosted decision tree (BDT), the simulated signal andbackground event samples were randomly divided into training and valida-tion samples with a 1:3 ratio. This ratio was chosen in order to preserve themajority of the simulated events, as the training sample must be discardedfrom further event selection development to avoid bias.

The BDT was trained with the designated training samples, and possi-ble overtraining was evaluated by applying a Kolmogorov-Smirnov test (KS-test) [72] to the training and validation samples. The null hypothesis that is

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Figure 10.9. The distributions of astrophysical neutrino and magnetic monopoleevents over the score of the Step II BDT.

evaluated with the KS-test is that the training and validation samples are drawnfrom the same underlying distribution, for which the test yielded p-values of38 % and 84 % for signal and background respectively. This indicated that thenull hypothesis can be rejected with (< 1)σ for both signal and background.Therefore, the BDT is labeled as not overtrained.

The event scoring scheme of the trained and validated BDT was now ap-plied to the non-training event samples, and the resulting event distributionsover the BDT score are shown in Figure 10.9 for each particle flavor. Anal-ogous to several of the nine BDT classification variables, the neutrino eventsform two clusters over the BDT score variable — one dominated by electronneutrino events, labeled as cascade-like, and centered around a BDT scorevalue of −0.6 — and dominated by muon neutrino events, labeled track-like,and centered around a score of −0.05. The magnetic monopole event distri-bution is centered around a BDT score of +0.2.

Figure 10.10 contains the correlation matrices for the nine BDT variablesand the BDT score, for the events of the signal and background samples sep-arately as they enter the BDT. Several variables show some degree of corre-lation (or anticorrelation) with each other. This is expected, as they may beconstructed to represent the same monopole event signature. The expectedcorrelation is also demonstrated by the common separation of the backgroundinto two distributions among several of the variables — the track- and cascade-like populations.

Figure 10.10 implies that the signal and background samples mainly arecharacterized by different variables. The signal sample shows a strong cor-

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(a) Signal event sample.

(b) Background event sample.

Figure 10.10. The correlation matrices of the signal and background event sampleswith the nine BDT variables and the BDT score. The diagonal values have beenremoved for viewing purposes. Note that the top row and leftmost column display thecorrelation between each variable and the BDT score, of which the latter is not a BDTvariable.

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Figure 10.11. The correlation matrix for the nine BDT variables and the BDT score forthe full Step I MC sample (signal combined with background). The diagonal valueshave been removed for viewing purposes. Note that the top row and leftmost columndisplay the correlation between each variable and the BDT score, of which the latteris not a BDT variable.

relation between the speed and log-brightness variables, which is expectedfrom the Frank-Tamm formula, Equation 4.3. Additionally, a strong anti-correlation is found between the centrality and pulse-time FWHM variables,which is expected from the geometry of a through-going track-like event. Thebackground sample shows a low internal correlation, with the most signifi-cant correlation between the pulse-time FWHM and length-fill-ratio variables.This corresponds to the division of the background sample into its two sub-populations.

All of the variables, apart from the energy loss RSD variable, show notice-able correlation with the BDT score for either signal or background or both.For both the signal and background samples, the BDT score shows a rela-tively high correlation with the pulse-time FWHM. Additionally, consideringthe signal sample, the BDT score is mainly correlated with the reconstructedspeed, while for the background sample the BDT score is mainly correlated tothe length fill ratio, and the log-brightness.

It is also interesting to study the correlation between the variables and theBDT score for the full sample of events, i.e. the combination of signal andbackground. This is shown in Figure 10.11. The correlation of a variable withthe BDT score indicates the importance of that variable in the event classifi-

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cation scheme. It appears that all variables, including the energy loss RSDvariable, have some correlation with the BDT score except for the centralityvariable. This implies that the energy loss RSD has some importance for theclassification scheme between signal and background despite showing weakcorrelation with the BDT score for the signal and background samples sep-arately. The centrality variable shows weak correlation with almost all vari-ables, both for the signal and background samples, with the exception of thepulse-time FWHM for the signal sample, with which it is highly anticorre-lated. The fact that it has such a weak correlation with the BDT score indicatesthat it would be safe to remove from the scoring scheme without reducing se-lection efficiency significantly. However, allowing it to remain does not implyany reduction in efficiency.

Generally, there is weak correlation between the variables, with the strongestcorrelation found between the pulse-time FWHM and the length-fill-ratio aswell as the pulse-time FWHM and the energy loss RSD variable. If two vari-ables show a strong correlation (or anticorrelation) with each other they wouldprovide equivalent information to the BDT, which implies that it would be re-dundant to include both in the classification scheme. This redundancy doesnot imply a reduced algorithm efficiency or increased algorithm bias.

Additionally, a strong (anti-)correlation between a variable and the BDTscore indicates that this variable has high importance in the BDT scoringscheme, but the opposite — that a weak correlation indicates a low importanceof the variable — is not strictly true. This is exemplified with the reconstructedspeed variable, which was shown to have good separative power between themonopole population and the two neutrino populations (Figure 10.6a). It does,however, only show weak correlation with the BDT score. The explanationlies in the bimodal shape of the background sample, and the fact that the mo-nopole population lies between the two peaks. The BDT algorithm makesfull use of this separation, but it also results in a weak correlation to the BDTscore.

10.3.4 Placing the Cut CriterionThe BDT score exhibits a powerful discrimination between signal and back-ground events. But in order to use this in the event selection scheme, a cutcriterion on the BDT score spectrum must be defined.

The model rejection and discovery potentials, MRP and MDP, respectively,are two measures of the performance of an event selection scheme (see Chap-ter 7.3). The optimal cut criteria according to the MRP or MDP can thus befound by scanning all possible cut values, along with the resulting nSG andnBG, to find the cut value that yields the minimal MRP or MDP, respectively.The expected nSG and nBG that result from each possible cut value can be seenin Figure 10.12, assuming the model fluxes, and a livetime of 6 yr.

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Figure 10.12. The total expected numbers of signal and background events assumingthe model fluxes, nSG and nBG respectively, as functions of the cut criterion on theBDT score.

The model rejection potential represents the sensitivity that is achieved perexpected signal event, and is calculated with Equation 7.10 as the ratio be-tween the F&C sensitivity of the analysis, µ90, and the number of expectedsignal events, nSG. Figure 10.13 contains µ90 and nSG as functions of the BDTscore cut, and the resulting MRP is shown in Figure 10.14. For lower val-ued BDT score cuts, the MRP is mainly affected by the decreasing sensitivity,showing the bimodal structure of the background shape (one dominated byelectron neutrinos, on by muon neutrinos), and for higher BDT score cuts bythe decreasing signal rate. It is minimized at a BDT score cut value of 0.047,with a rejection potential value of 0.11. The signal and background accep-tances at this cut value are 93.5 % and 2.65 % respectively.

The model discovery potential is calculated with Equation 7.11 as the ratiobetween the least detectable signal, µLDS, i.e, the lowest number of observedsignal events that is required to claim a discovery, and the expected number ofsignal events, nSG. Thus, the minimum of the MDP as a function of possiblecut values represents the cut value that yields the highest expected number ofsignal events compared to the number of signal events required to claim a dis-covery. Determining the value of µLDS (Equation 7.13) requires knowledge ofthe number of expected background events, nBG, as well as the total number ofobserved events that is required to claim that the observation is not compatibleexclusively with the background expectation, ncrit, the latter calculated usingEquation 7.12.

The least detectable signal is shown in Figure 10.15 as a function of theBDT score cut, along with the expected number of signal events. The saw-

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Figure 10.13. The F&C sensitivity, µ90, and the expected number of signal events,nSG, assuming the model fluxes over 6 yr of data as functions of the Step II BDT scorecut value.

Figure 10.14. The model rejection potential for 6 yr of data as a function of the Step IIBDT score cut value. The preferred BDT score cut value of 0.047 is marked with adark red line.

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tooth shape of µLDS originates in the discrete nature of ncrit, so in order tomimic a continuous curve, a cubic spline interpolation has been computedusing the saw-tooth tips as spline knots. The spline and spline knots are alsoshown in Figure 10.15.

The MDP is shown as a function of the BDT score cut in Figure 10.16,where the MDP as calculated with the µLDS spline and the accompanyingspline knots are also included.

The model discovery potential is minimized at a plateau between BDT scorecut values of 0.101 and 0.309 (marked in Figures 10.16 and 10.15), wherethe discovery potential takes a value around 0.17 with less than a 3 % differ-ence over the plateau. The signal and background acceptances range between82.7 % and 16.4 % (signal) and 0.857 % and 7.13×10−8 (background) overthe plateau, monotonously decreasing with increasing cut value. The rejec-tion factor takes values between 0.12 and 0.59 over the width of the plateau,increasing with higher cut value.

The MDP plateau arises as the slopes of nSG and the µLDS spline are propor-tional in this region, as can be obtained from Figure 10.15. Several differentmethods can be employed to search for a preferred cut value within the plateaurange, e.g. to fit a function (such as a parabola) to a suitable set of MDP valuesaround the minimum range. This method would yield a minimum value at onedefinitive BDT score cut value, but this minimum would be without physicalmeaning. The reason for this is that the MDP does not come from a proba-bility distribution with a particular shape that should be mimicked, but it isthe combination of two separate distributions that both originate directly fromsimulated events.

As no cut value is preferred solely based on the MDP, an additional criterionis imposed to select a cut value over the MDP plateau. This is taken as the cutvalue with the best model rejection potential. The lowest MRP is found at thelow- BDT score edge of the plateau, with a cut value of 0.101, and is wherethe signal and background acceptances are maximized. This cut value is takenas the cut value that is preferred by the MDP.

The signal and background acceptance of the cut values preferred by theMRP and MDP minimization procedures are shown in Table 10.1 along withthe expected numbers of signal and background events for 8 yr analysis live-time, as well as the corresponding model rejection and discovery potentialvalues. The MRP preferred cut has a discovery potential of 0.23 (∼ 34%higher than the minimum), and a signal and background acceptance of 94 %and 2.6 %, respectively. The MDP preferred cut has a rejection potential of0.12 (∼ 8.1% higher than the minimum), and a signal and background accep-tance of 83 % and 0.86 %, respectively. Both cut options predict a number ofbackground events significantly lower than one per full analysis livetime.

The two cut options offer similar rejection potentials as well as similar dis-covery potentials, and the signal and background acceptances are also similar,with the MDP preferred cut being slightly more aggressive.

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Figure 10.15. The least detectable signal, µLDS, and the expected number of signalevents, nSG, assuming the model fluxes over 6 yr of data as functions of the Step IIBDT score cut value. Included are also the µLDS spline and accompanying splineknots, as well as the range of the MDP plateau (blue shaded region).

Figure 10.16. The model discovery potential for 6 yr of data as a function of the Step IIBDT score cut value. The MDP minimum plateau, between scores of 0.101 and 0.309,is marked with a blue shaded region, and the MDP spline and spline knots are alsoincluded.

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Table 10.1. The minima of the model rejection and discovery potentials over the BDTscore cut range, and their accompanying preferred BDT cut values. Listed are also thecorresponding expected numbers of signal and background events assuming the modelfluxes over 8 yr analysis livetime, as well as the signal and background acceptancesrelative the Step I expectations.

BDT Signal Backgroundscore nSG, nBG, acceptance acceptance

cut MRP MDP 8 yr 8 yr [%] [%]

MRP 0.047 0.110 0.231 33.2 0.265 93.5 2.65MDP 0.101 0.119 0.172 29.4 0.0856 82.7 0.857

The final result of this analysis — either a flux upper limit or a magneticmonopole discovery — will be shown as a function of the speed of the particle,β . Therefore, it is important to understand the signal efficiency as a functionof the particle speed for each of the suggested cut values before making a finaldecision. The expected number of signal events is shown as a function of thetrue particle speed in Figure 10.17 for the preferred MRP and MDP cut values.

Both cut options mainly reject signal in the high-β region, and higher BDTscore cut values yield higher reduction over the full speed range. The MRPpreferred cut shows negligible signal rejection for speeds below 0.95, and therejection in the highest speed region is . 20%. Correspondingly, the MDPpreferred cut shows negligible rejection below 0.90, and the rejection in thehighest speed region is & 40%.

The reason for why the signal rejection is concentrated in the high speedregion is that there is a strong anticorrelation between the BDT score andthe true monopole speed, βMC. This is indicated in Figure 10.18, where theexpected number of signal events is shown as a function of the BDT score aswell as the particle speed. A clear anticorrelation can be seen, indicating thathigher speed monopoles are awarded lower BDT scores, presumably sincehigh speed monopoles appear more muon-like in the eyes of the BDT thanthose with low speed. Similarly, a strong anticorrelation is found between theBDT score and the reconstructed particle speed, βBM, for magnetic monopoleevents (see Figure 10.10a).

In the context of this analysis, it is important to produce a competitive resultover the full included β range. Therefore, the MDP preferred cut value isrejected in favor of the MRP preferred cut.

Thus, the Step II signal and background acceptances become 93.5 % and2.65 % respectively. The per-flavor neutrino acceptance is 6.97×10−4, 6.37 %and 1.11 % for νe, νµ and ντ respectively. The final level expected magneticmonopole flux (assuming the model flux, Chapter 8.3.1) and astrophysicalneutrino flux are thus 33.2 and 0.265 events per analysis livetime, respec-tively. The expected neutrino flavor ratio at final level is 1.1% : 91% : 7.9%for [νe : νµ : ντ ].

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Figure 10.17. The signal event rate as a function of true particle speed for the alterna-tive BDT score cut values.

Figure 10.18. The magnetic monopole expected event rate as a function of the trueparticle speed and the Step II BDT score.

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Figure 10.19. The model rejection potential for 6 yr of collected data and for the full8 yr data sample as functions of the Step II BDT score cut value. The preferred BDTscore cut value of 0.047 is marked with a dark red line.

Finally, as was described in Chapter 8.2.1, the event selection was devel-oped assuming a livetime of 6 yr, while the results are produced considering atotal of 8 yr collected data. The MRP calculated using the full data sample isshown together with the 6 yr MRP in Figure 10.19. The full data MRP is min-imized at the same BDT score cut value as the 6 yr MRP, at the value 0.047,where it has a rejection potential value of 0.080.

10.4 Expected Numbers of EventsThis section summarizes the outcome of the event selection scheme describedabove. The expected signal and background event rates over the full analysislivetime after each cut level are listed in Table 10.2. The signal and back-ground acceptance relative to the trigger rate after each cut level is listed inTable 10.3. For the corresponding event rates of atmospheric muons and neu-trinos, see Table A.1. Note that the 2.7 kHz trigger rate of atmospheric muonevents corresponds to 6.3×1011 triggered events over the 8 yr livetime of thepresent analysis. Similarly, the EHE filter acceptance rate of atmosphericmuons, 0.8 Hz, corresponds to 1.9×108 events over 8 yr.

The Step I and Step II acceptances of triggered magnetic monopole eventsare 14.6 % and 13.6 % respectively. The corresponding acceptances for astro-physical neutrino events are 1.19 % and 0.0316 %. Assuming the model flux,this indicates a final expected number of signal events of 35.5 and 33.2 afterthe Step I and Step II event selections, respectively. Correspondingly, astro-

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Table 10.2. The expected number of events per analysis livetime (8 yr) at each levelof the analysis, assuming the model signal and background fluxes.

Analysis level nSG nBG nνe nνµnντ

Trigger 244 838 146 548 144EHE filter 178 371 90.1 202 78.2

Step I Offline EHE cut 89.9 57.2 23.4 20.0 13.8Track quality cut 64.1 20.4 6.77 9.91 3.72Muon bundle cut 35.5 10.1 4.39 3.83 1.90Surface veto 35.5 9.99 4.33 3.78 1.88

Step II 33.2 0.265 0.00302 0.241 0.0209

Table 10.3. The expected number of events, n, per analysis livetime after each cutlevel, relative to the number of expected triggering events, nTrigger, assuming the modelsignal and background fluxes.

nSG/nTriggerSG nBG/nTrigger

BGAnalysis level [%] [%]

Trigger 100.0 100.0EHE filter 72.9 44.2

Step I Offline EHE cut 36.9 6.83Track quality cut 26.3 2.43Muon bundle cut 14.6 1.21Surface veto 14.6 1.19

Step II 13.6 0.0316

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Figure 10.20. The (bin-wise) expected signal event rate per analysis livetime, assum-ing the model monopole flux. This is given as a function of the true particle speed,βMC, and for the generation, trigger, EHE filter, Step I and Step II analysis levels.

physical neutrino events are expected to levels of 9.99 and 0.265 after Step Iand Step II, respectively. This can be compared with the 0.085 expected atmo-spheric events over the 9 yr livetime of the EHE analysis (after the EHE eventselection, i.e. Step I). This is more than one order of magnitude lower than theStep I expected astrophysical background, and a factor of ∼ 3 lower than afterStep II.

As already mentioned, it is important to understand the acceptance of mag-netic monopoles over the relevant range of the true particle speed. The signalacceptance as a function of the particle speed is shown in Figure 10.20. Thetrigger and EHE filter has an approximate uniform acceptance of magneticmonopole events over this β range. The Step I selection scheme preservesthis uniform acceptance over most of the range, barring the low β region. Areduced acceptance is expected in this region, as the main selection variable ofStep I, the event brightness, is highly correlated with the speed of the particle.The opposite trend is seen in the Step II selection, where the acceptance is onlyreduced for high speed monopoles. This stems from the strong correlation be-tween β and BDT score, where a high speed yields a low BDT score (seeFigure 10.18). This can be attributed to the fact that higher speed monopolesappear more muon-like in the BDT classification scheme.

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11. Sensitivity

Chapter 10 covered the event selection scheme that was used to search formagnetic monopoles above the Cherenkov threshold within the context of thisanalysis. It is concluded with a listing of the expected numbers of signal andbackground events over the full analysis livetime, under the assumptions ofthe model magnetic monopole and astrophysical neutrino fluxes (Chapter 8.3).This, however, does not constitute the final result of the analysis, which willeither be the discovery of a magnetic monopole or an upper limit on the cosmicmagnetic monopole flux.

This chapter covers the upper limit that is expect to be set given the expectedfinal level event rates, barring the extraordinary discovery of a magnetic mo-nopole. The average expected upper limit is known as the sensitivity of theexperiment.

11.1 Effective AreaThe signal efficiency of an analysis scheme can be quantified in terms of an ef-fective area, representing the cross sectional area of an ideal detector recordingwith 100 % signal efficiency. This is calculated as the size of the generationarea times the detected fraction of generated events (Equation 7.1).

The effective area at the trigger, EHE filter, Step I and Step II levels of thisanalysis can be found in Table 11.1, averaged over the β range of the analysis.The effective area is comparable in order of magnitude to the geometrical crosssectional area of the detector, commonly quoted as 1 km2, which is due to thehigh detectability of magnetic monopoles above the Cherenkov threshold. Thetrigger level effective area of 2.39 km2 is significantly larger than this, whichmainly is attributed to the fact that bright magnetic monopoles can cause a

Table 11.1. The average effective area at the trigger, EHE filter, Step I and Step IIlevels of the analysis.

Analysis Effective arealevel [km2]

Trigger 2.39EHE Filter 1.74Step I 0.348Step II 0.326

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Figure 11.1. The effective area as a function of true particle speed at the trigger, EHEfilter, Step I and Step II levels of the analysis.

trigger in IceCube even if they pass far outside of the instrumented detectorvolume.

Since the final results of the analysis will be presented as a function of thetrue particle speed, it is important to understand the effective area over the fullrange of considered speeds. The effective area at the trigger, EHE filter, Step Iand Step II levels of the analysis are shown as a function of the true particlespeed in Figure 11.1.

Here, the effective area is calculated using a kernel density estimator (KDE)[64; 77; 78] with the events that remain at each level of the analysis. In a KDE,each event is described by a kernel with a specific shape in the chosen variable,and the KDE curve is given as the sum of all individual single-event kernels.Thus, a KDE is used to smoothen statistical fluctuations between adjacenthistogram bins.

For the present effective area calculation, a Gaussian kernel shape is usedover the true particle speed, with the kernel width chosen such that the statis-tical fluctuations observed in the total event count of Figure 10.20 are smooth-ened, but the overall shape of the curve is maintained.

The effective area is truncated to the speed range [0.780;0.995], which cor-responds to the range containing final level MC monopole events.

As is shown in Figure 11.1, the final level effective area is increasing withthe true particle speed from βMC = 0.78 to approximately 0.87, where it plateausbetween 0.45 km2 and 0.50 km2.

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11.2 SensitivityThe sensitivity is calculated using Equation 7.4. At final analysis level thenumber of expected background events is nBG = 0.265 over the full 8 yr live-time, 2715 d. This yields a F&C sensitivity of µ90 (nBG) = 2.67. The solidangle covered by the analysis is 4π and the average effective area Aeff =0.326km2. The 90 % CL sensitivity to the magnetic monopole flux thus be-comes 2.78×10−19 cm−2 s−1 sr−1 over the full monopole speed range.

The sensitivity is shown as a function of the true particle speed in Fig-ure 11.2. The shape of the sensitivity is the inverse of the effective area shape,as the effective area is the only parameter in Equation 7.4 that varies withthe particle speed. The sensitivity can thus be described as rapidly decreas-ing to a speed of approximately βMC = 0.87, whereafter it plateaus around2×10−19 cm−2 s−1 sr−1.

Additionally, the sensitivity of this analysis is shown in Figure 11.3 alongwith the current best limits on the magnetic monopole flux. The results of thepresent analysis are thus expected to be competitive for speeds above β = 0.79,and yield an improvement of around an order of magnitude over the previousresults for speeds above 0.82.

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Figure 11.2. The 90 % CL sensitivity of this analysis to the cosmic flux of magneticmonopoles as a function of true particle speed.

Figure 11.3. The 90 % CL sensitivity of this analysis to the cosmic flux of mag-netic monopoles as a function of true particle speed (green dashed curve, denotedby IceCube-86 8 yr, sens.). Included are also the current best upper limits on themagnetic monopole flux (see Chapter 2.6 for further description and references).

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12. Uncertainty on the Magnetic MonopoleEfficiency

Any scientific result carries some degree of uncertainty, and the size and natureof this uncertainty is determined by several aspects of the underlying analy-sis. This chapter covers the statistical and systematical uncertainties that con-tribute to the final uncertainty on the acceptance of signal events. In additionto this, there will be an uncertainty on the flux of background events. This istreated in the next chapter.

The statistical uncertainty in an experiment is related to the fact that we donot have infinite data. This inherently yields the result as an approximation ofthe true, unknown, value.

In addition to the statistical uncertainty, there is an uncertainty introducedby systematic effects that are due to limited intrinsic accuracies in the descrip-tion of the detector performance or in the physical models used to describe theunderlying physical processes. A physics or detector model with limited accu-racy can introduce a systematic shift of the final result such that it is no longercentered around the true value of the measured parameter. The magnitude ofthis effect can be estimated by simulating an event sample with shifted modelparameters and applying the analysis selection criteria to the new simulatedevents. The appropriate parameter shift is determined by the estimated un-certainties of the parameters themselves, and is commonly taken as the ±1σ

interval.Such a study has been conducted within the scope of this work, where three

properties of the IceCube detector and detector medium have been varied. Pa-rameters of the signal modeling, such as the magnetic monopole mass andcharge, are not varied here. This is because they are considered as modelchoices that define the fiducial region of the analysis, as opposed to measuredparameters with testable uncertainty regions. The parameters that define thefiducial region are described in Chapter 8.3.1.

In the present chapter, the final signal efficiency uncertainty is presented inthe form of the relative uncertainty corresponding to the one standard deviationrange. Due to the definitions of the effective area and the number of expectedsignal events (Equations 7.1 and 7.3, respectively), the relative uncertaintyon the signal efficiency, the effective area and the number of expected signalevents will be identical.

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12.1 Systematic Variation of Monte Carlo SettingsThe main sources of systematic uncertainty in the present analysis are relatedto the propagation and detection of light in the detector. The optical propertiesof the ice, and the in-situ detection efficiency of the optical modules, are areaswith significant uncertainties, due to the difficulty of measuring them.

These properties are all described by the ice-model of the IceCube sim-ulation software (Chapter 5.3). The ice-model determines each aspect of aphoton’s life in IceCube, from the production by a charged particle, via itspropagation through the bulk ice, to the triggering of a photo-electron in aPMT or to being absorbed by the surrounding material.

Therefore, in order to estimate the systematic uncertainties introduced bythe ice-model in the event simulation, three ice-model parameter categorieshave been studied, one of which considers the light propagation through thebulk ice, and two that consider different aspects of the DOM photon detectionefficiency:

Scattering and absorption The characteristic scattering and absorp-tion lengths of optical light in the ice.

DOM efficiency The probability that a photon hitting the DOM surfaceis detected by the DOM.

DOM angular sensitivity The relative sensitivity of a DOM depend-ing on the incident zenith angle of the photon. Based on the internalgeometry of a DOM, its sensitivity is assumed to be rotationally sym-metric around the vertical axis.

A total of 10 specific variations, listed in Table 12.1, were selected to coverthe true parameter values of the given parameter categories. For each variation105 magnetic monopole events were simulated at generation level and broughtthrough the full event selection scheme of the analysis. The effective area foreach case was calculated, and compared to the effective area of the baselinecase.

Table 12.1. The systematic variation parameter sets to be tested for each parametercategory.

Parameter category Systematic variation

Scattering and absorption Scat. +5%, abs. +5%Scat. +5%, abs. −5%Scat. −5%, abs. +5%Scat. −5%, abs. −5%

DOM efficiency DOM eff. +10%DOM eff. −10%

Angular sensitivity Ang. sens. set 5Ang. sens. set 9

Ang. sens. set 10Ang. sens. set 14

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As they are both affected by the same impurities in the detector ice, the scat-tering and absorption characteristic lengths are highly correlated (see Chap-ter 3.1.3). These are therefore varied together in the four combinations givenby changes of±5% on each variable. This range has been determined throughsystematic studies of the detector response to the calibration LEDs installed ineach DOM. The variations are relative as both the scattering and absorptionlengths fluctuate strongly with the in-ice depth and the photon wavelength(see Figure 3.4). The resulting shifts in effective area from baseline are listedin Table 12.2 and Figure 12.1, and range from −4.6% to +4.5%.

The DOM efficiency setting was shifted by −10% and +10%, resultingin shifts in effective area by −6.9% and +5.0% with respect to the baselinevalue, respectively (see Table 12.2 and Figure 12.1). The range of ±10%has been determined through a combination of several internal IceCube com-parative studies of the detector response to atmospheric muon events. Thevariations are applied as overall shifts in the absolute DOM efficiency, whichvaries over the photon wavelength spectrum.

The DOM angular sensitivity is parametrized by two parameters, p0 andp1. Studies of the effect of the systematic uncertainties on the DOM angularsensitivity on the event selection are conducted by examining the effect of sev-eral sets of (p0, p1) from the region likely to contain the true (p0, p1) values.A total of 50 ice-models with varied (p0, p1) values are available for this pur-pose. The corresponding parameter values are shown in Figure 12.2 as greencrosses, where the baseline (p0, p1) set is marked with a red rhombus.

A very bright event is assumed to be less sensitive to variations in the angu-lar sensitivity of each DOM, and more sensitive to variations in the absoluteDOM efficiency. Therefore, a simplified approach is adopted in this analysisfor estimating the systematic uncertainty introduced by the uncertainty on theDOM angular sensitivity. Here, four of the available (p0, p1) sets that approxi-mately frame the recommended distribution are selected for evaluation. Thesesets, numbered 5, 9, 10 and 14, are marked with yellow circles in Figure 12.2.

The difference in effective area between the selected (p0, p1) sets and thebaseline are listed in Table 12.2 and Figure 12.1, and they range between−1.5% and lower than +1%. These variations are thereby found to be sig-nificantly smaller than the differences in effective area found for the otherparameter categories. Therefore, an extended study using the full sample of50 (p0, p1) sets is omitted, and the DOM angular sensitivity systematic uncer-tainty is assumed to be on this level.

The final-level effective areas of the systematic variations are asymmetri-cally distributed around the baseline effective area, with a tendency towardslower values. At first glance, this may be unexpected, as the shifted param-eter settings are symmetrically distributed around the baseline settings. Theasymmetry is not present at the previous step of the event selection, Step I,(see Table 12.3) so it must arise in the Step II BDT classification. The signalefficiency of the BDT cut is listed for each varied event sample in Table 12.3

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Figure 12.1. The relative difference between the baseline effective area and the effec-tive area for each tested systematic variation. The points representing the ang. sens.sets 10 and 14 lie at similar values, and may be difficult to distinguish. Note thatthere is a 1.1 % statistical uncertainty on the effective area of the varied sets. See alsoTable 12.2.

Figure 12.2. The (p0, p1) values of the 50 ice-models that may be used to evaluate theeffect of the systematic uncertainties on the DOM angular sensitivity of an analysis(green crosses). Also shown are the (p0, p1) baseline values (red rhombus), and thefour numbered sets (5, 9, 10, 14) that were used in this analysis (yellow circles).

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Table 12.2. The effective area and relative effective area difference to the baselinecase for each tested systematic variation at final level of the analysis. Note that thereis a 1.1 % statistical uncertainty on the effective area of the varied sets. See alsoFigure 12.1.

Effective area Relative difference toSystematic variation [km2] baseline [%]

Baseline 0.326Scat. +5%, abs. +5% 0.312 −4.3Scat. +5%, abs. −5% 0.340 +4.5Scat. −5%, abs. +5% 0.311 −4.6Scat. −5%, abs. −5% 0.334 +2.5DOM eff. +10% 0.342 +5.0DOM eff. −10% 0.303 −6.9Ang. sens. set 5 0.327 +(< 1)Ang. sens. set 9 0.321 −1.5Ang. sens. set 10 0.323 −(< 1)Ang. sens. set 14 0.323 −(< 1)

Table 12.3. The efficiency of the BDT cut for each tested systematic variation afterStep I of the analysis.

Step I Step I relative BDT cuteffective area difference to baseline signal efficiency

Systematic variation [km2] [%] [%]

Baseline 0.348 93.5Scat. +5%, abs. +5% 0.336 −3.4 92.7Scat. +5%, abs. −5% 0.365 +4.8 93.3Scat. −5%, abs. +5% 0.337 −3.2 92.2Scat. −5%, abs. −5% 0.358 +2.8 93.3DOM eff. +10% 0.372 +6.8 91.9DOM eff. −10% 0.324 −6.9 93.5Ang. sens. set 5 0.351 +(< 1) 93.1Ang. sens. set 9 0.348 −(< 1) 92.3Ang. sens. set 10 0.345 −(< 1) 93.5Ang. sens. set 14 0.347 −(< 1) 93.1

Table 12.4. The largest relative effective area differences to baseline for each testedsystematic settings category.

Systematic Specific Magnitude of relativevariation category systematic variation difference to baseline [%]

Scattering and absorption Scat. −5%, abs. +5% 4.6DOM efficiency DOM eff. −10% 6.9Angular sensitivity Ang. sens. set 9 1.5Total 8.4

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(final column), where the highest efficiency is achieved for the baseline set.This confirms that the BDT classification favors monopole events that wereproduced with the baseline settings over events produced with alternative set-tings, which is expected as the BDT was trained specifically on baseline mag-netic monopole events.

12.2 Total UncertaintyThe total uncertainty, σtot, on the analysis acceptance of final level magneticmonopole events is calculated as the quadratic sum of the systematic and sta-tistical uncertainties, σsyst and σstat, respectively:

σ2tot = σ

2syst +σ

2stat (12.1)

The total systematic uncertainty on the final level effective area, σsyst, is con-servatively taken as a symmetric interval around the baseline value, given bythe quadratic sum of the largest values of each parameter category, σScat. abs.,σDOM eff. and σAng. sens. (see Table 12.4):

σ2syst = σ

2Scat. abs. +σ

2DOM eff. +σ

2Ang. sens. (12.2)

This results in a total systematic uncertainty of 8.4 %.The three contributions in Equation 12.2 are uncorrelated by design, which

allows this calculation. The scattering and absorption characteristic lengths,being properties of the bulk ice, are naturally uncorrelated with the two DOM-related parameter categories. The two DOM-related categories are also or-thogonal, as only the direction-dependent relative sensitivity is varied in theDOM angular sensitivity category (preserving the overall efficiency betweenvariations), while only the absolute efficiency is varied in the DOM efficiencycategory (with no directional dependence).

The statistical uncertainty on the final level effective area, σstat, is directlygiven by the total number of baseline Monte Carlo signal events that remain atfinal level, NFinal level

MM , through:

σstat =

√NFinal level

MM

NFinal levelMM

(12.3)

The present NFinal levelMM = 34306 yields a total statistical uncertainty of σstat =

0.5%, negligible in comparison to the systematic uncertainty.The total uncertainty on the acceptance of magnetic monopoles remains at

σtot = 8.4%.

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13. Uncertainty on the Astrophysical NeutrinoFlux

The main background to the present search for a cosmic flux of magnetic mo-nopoles are astrophysical neutrinos. The topic of this chapter is the uncertaintythat exists in the assumed background flux measurement, as well as the uncer-tainties that are introduced by the assumption of this particular flux and by thelimitations of the available Monte Carlo event samples.

The uncertainties presented in this chapter will be presented both in theform of an absolute and a relative uncertainty on the expected number of back-ground events, and the distinction between the two will be made clear by thecontext.

13.1 Statistical UncertaintySimilarly to the statistical uncertainty on the signal event efficiency, the num-ber of expected background events will have an uncertainty originating in thefinite size of the employed Monte Carlo samples. The background statisticaluncertainty, σstat,BG, is calculated with:

σstat,BG =

√NFinal level

ν

NFinal levelν

(13.1)

A total of 4191 neutrino events remain at final level, giving a statistical un-certainty of σstat,BG = 1.5%. This is negligible with respect to the uncertaintiespresented below.

13.2 Uncertainties in the Astrophysical FluxMeasurement

In the present analysis, the background flux has been assumed to be well de-scribed by the Φ2017

DIF-νµflux, detailed in Chapter 8.3.2. The total uncertainty

(the sum of the statistical and systematic uncertainties) on the assumed flux isspecified in terms of the uncertainty on the astrophysical flux normalization,φν , and on the spectral index, γν , that enter the formula for a single power law(Equation 8.4):

Φν = φν ×(

Eν

100TeV

)−γν

×10−18 GeV−1 cm−2 s−1 sr−1 (13.2)

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The flux normalization and spectral index of Φ2017DIF-νµ

are:

φν = 1.01+0.26−0.23 , γ = 2.19±0.10 (13.3)

The quoted errors propagate to the final level of the monopole analysis andgive an expected number of background events of 0.265+0.265

−0.135 over the in-cluded 8 yr of data. The relatively small uncertainties of the flux normaliza-tion and spectral index are amplified to the present factor of ∼ 2 uncertaintyon the number of expected background events. This is attributed to the spec-tral index uncertainty that enters exponentially in the calculation, resulting ina large uncertainty in energy ranges away from the energy normalization (here100 TeV).

This effect is demonstrated in Figure 13.3, where the number of expectedbackground events after Step I, along with its uncertainty, is shown as a func-tion of the neutrino energy. Note the increasing relative uncertainty betweenthe low and high energy regions of the distribution.

13.3 Alternative Neutrino Flux AssumptionsThis section is dedicated to investigating what the average final backgroundflux would be if nature is better represented by a different flux than the as-sumed Φ2017

DIF-νµ. Two additional flux measurements, both performed by Ice-

Cube and presented at the International Cosmic Ray Conference in 2019 (ICRC-2019), are examined here:

Φ2019DIF-νµ

An update to Φ2017DIF-νµ

, presented at the ICRC-2019 [79]

Φ2019HESE Result of the high energy starting events (HESE) analysis, pre-sented at the ICRC-2019 [75]

The Φ2019DIF-νµ

measurement is an update to the Φ2017DIF-νµ

measurement, wherethe same analysis has been applied to an additional 2 yr of collected data. Theupdated analysis obtains 90 % of the likelihood contribution originating in theenergy range Eν ∈ [40TeV;3.5PeV]. The latter flux measurement, Φ2019

HESE, isthe result of the high energy starting events (HESE) analysis, targeting eventsof all flavors with a total deposited energy in the detector of at least 60 TeV.As indicated by its name, this analysis only includes starting events, whichimplies events that are caused by a neutrino.

Both the Φ2019DIF-νµ

and the Φ2019HESE flux measurements are given in the form of

single power laws, which is the same parametrization as the Φ2017DIF-νµ

flux. Theastrophysical flux normalization and spectral index of each tested astrophysi-cal flux are given in Table 13.1.

The simulated samples of astrophysical neutrinos were weighted to eachof these fluxes in turn, and the number of remaining neutrino events wererecorded at each level of the analysis. The expected number of events, nν ,given each astrophysical neutrino flux assumption can be found in Table 13.1.

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Table 13.1. The astrophysical flux normalization and spectral index of each testedastrophysical flux measurement, along with the expected numbers of events at Steps Iand II, nStep I and nStep II, respectively.

Astrophysical SpectralFlux normalization, φν index, γν nStep I nStep II

Φ2017DIF-νµ

1.01+0.26−0.23 2.19+0.10

−0.10 9.99+10.3−5.13 0.265+0.265

−0.135

Φ2019DIF-νµ

1.44+0.25−0.24 2.28+0.09

−0.08 9.41+7.32−3.93 0.251+0.192

−0.105

Φ2019HESE 2.15+0.49

−0.15 2.89+0.20−0.19 1.14+1.74

−0.618 0.029+0.046−0.016

Table 13.2. The Step II rejection efficiency of the highest energy decade of the avail-able simulated event samples for each neutrino flavor along with the correspondingenergy range.

Energy range of highest MC Step II rejectionNeutrino available energy decade efficiencyflavor [GeV] [%]

νe[107;108

]100.0

νµ

[108;109

]98.3

ντ

[108;109

]97.0

Table 13.3. The expected number of background events at Steps I and II that areoutside of the energy ranges of the simulated event samples.

Analysis level νtot νe νµ ντ

Step I 1.59+3.14−1.06 0.24+0.44

−0.16 0.71+1.38−0.47 0.65+1.31

−0.43Step II 0.032+0.063

−0.021 0 0.012+0.024−0.008 0.019+0.039

−0.013

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The Φ2017DIF-νµ

flux (assumed in this analysis) yields the highest number ofexpected events at Steps I and II of the analysis, indicating that this flux repre-sents an upper limit on the astrophysical neutrino flux. Note, though, that theflux may be as low as indicated by the Φ2019

HESE flux.

13.4 Expected Flux outside of the Simulated EnergyRange

The Monte Carlo simulated samples that were used to estimate the backgroundregion for this analysis cover the energy range Eν ∈

[105;108

]GeV for elec-

tron neutrinos and Eν ∈[105;109

]GeV for muon and tauon neutrinos. The

single power law that is assumed to represent the natural flux, on the otherhand, has an exponential falloff that continues to infinity. This section is ded-icated to estimating the final level background contribution of the high energytail that lies outside of the available Monte Carlo samples.

In order to achieve this, two main ingredients are needed — an assumptionon the astrophysical neutrino flux and the effective area of the analysis above108–109 GeV. The former is naturally the same background flux assumptionthat is made throughout this thesis, the Φ2017

DIF-νµflux, shown as a function of

energy in Figure 13.1. The latter was calculated for Step I within the EHEanalysis, using other samples of Monte Carlo events than in the present mono-pole analysis, extending the upper energy bound to 1011 GeV. This effectivearea is shown as a function of energy in Figure 13.2 for each neutrino flavor.Major changes have been made both in the trigger and filter systems of Ice-Cube and in the IceCube reconstruction software since these event sampleswere produced. These changes have negligible effect on the simple variablesused in the Step I event selection, but render the MC samples incompatiblewith data when considering the advanced reconstructions in Step II.

In order to estimate the number of neutrinos after Step I, i.e. after EHE anal-ysis event selection, the EHE analysis effective area to neutrinos was foldedwith the Φ2017

DIF-νµneutrino flux, along with the total livetime of the analysis, the

solid angle covered by the analysis, and the energy bin width.Using this method, the total number of astrophysical neutrino events ex-

pected after Step I, nν , is 11.1+12.2−5.79. See Figure 13.3 for the number of ex-

pected events as a function of energy. Truncating this to the energy rangecovered by the available simulated samples yields an expected number of9.50+9.05

−4.72. This leaves a total of 1.59+3.14−1.06 expected events that lie outside

of the Monte Carlo sample energy ranges after Step I (see Table 13.3, Row 1for a per-flavor listing).

This estimate of the number of Step I expected events, 9.50+9.05−4.72, allows a

validity check of the procedure through a comparison with the number of ex-pected events that was obtained through direct evaluation of the Monte Carlo

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Figure 13.1. The Φ2017DIF-νµ

flux as a function of neutrinoenergy, including the ±1σuncertainty region (blueband).

Figure 13.2. The Step I ef-fective area for astrophysicalneutrino events as a functionof neutrino energy, shown foreach neutrino flavor in turn(yellow, turquoise, purple) aswell as the sum of all flavors(blue).

Figure 13.3. The numberof expected astrophysicalneutrino events over the8 yr livetime of the presentmonopole analysis, as afunction of neutrino energy.This is calculated by foldingthe incident neutrino flux withthe neutrino effective area,the energy bin-width, and thetotal livetime and solid angleof the analysis. Included isalso the ±1σ uncertaintyregion (blue band).

132

samples, 9.99+10.3−5.13. These numbers are expected to agree significantly bet-

ter than their uncertainty region indicates, as they should represent the sameastrophysical flux measured with the same detector. The two numbers agreewithin 5 %, a small difference that may be attributed to the major differencesin the IceCube data treatment between the MC samples that were used in thetwo estimates.

In order to estimate the expected number of Monte Carlo events outside thesimulated energy range after Step II, the Step II rejection efficiency for theseevents must be evaluated per flavor. This is approximated as the rejectionefficiency of the events in the highest energy decade available in the simulatedevent samples of each corresponding flavor. The Step II rejection efficiency inthe highest energy decade can be found in Table 13.2 for each neutrino flavor,along with the corresponding energy range.

The result of this estimation is shown in Table 13.3. The final level to-tal expected number of background events outside of the simulated energyrange, 0.032+0.063

−0.021, is small with respect to the number of expected backgroundevents within the simulated range, 0.265+0.265

−0.135, and its uncertainty region.

13.5 Expected Background at Final Analysis LevelThe chosen astrophysical neutrino flux model, Φ2017

DIF-νµ, results in an expected

number of events at final level of nBG = 0.265. This number of events carriesan uncertainty of a factor of∼ 2 originating in the uncertainties on the flux nor-malization, φν , and spectral index, γν , yielding a background of 0.265+0.265

−0.135.The statistical uncertainty originating in the the finite MC sample size is

∼ 1.5%, and therefore negligible.Additionally, two alternative astrophysical neutrino flux models were eval-

uated (Φ2019DIF-νµ

and Φ2019HESE). The Φ2019

DIF-νµflux yielded an expectation on the

number of events at final level consistent with the Φ2017DIF-νµ

flux, with slightlysmaller uncertainty. The Φ2019

HESE flux, on the other hand, yielded an nBG expec-tation ∼ 1 order of magnitude lower than Φ2017

DIF-νµ, 0.029+0.046

−0.016.Finally, the contribution of the high energy tail of the Φ2017

DIF-νµflux, outside

of the energy ranges of the MC event samples, was evaluated. This region maycontribute a total of 0.032+0.063

−0.021 events at the final level of the analysis.This results in an expected number of astrophysical neutrino events at final

level over the full analysis livetime ranging from ∼ 0.01 (assuming the lower1σ bound of the Φ2019

HESE flux) to ∼ 0.6 (assuming the upper 1σ bound of theΦ2017

DIF-νµflux, including the estimated high energy tail). This is a wide range,

but its upper bound is still significantly below one event per livetime of theanalysis. Considering this, in the case of a non-detection the upper limit willbe evaluated using a background-free assumption, i.e. with nBG = 0.

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14. Result

This chapter covers the results of the analysis that is described in this thesis,which are produced in the counting stage of a cut-and-count analysis.

14.1 Experimental Event RateThis section covers the number of events that, after unblinding the analysis,are observed in experimental data at each level of the analysis. These arelisted in Table 14.1 along with the corresponding expected numbers of signaland background events, assuming the model fluxes (compare to Table 10.2).At the lower analysis levels, a significant flux contribution is expected fromneutrinos and muons produced in the atmosphere (see Table A.1) as well asfrom astrophysical neutrinos with a primary neutrino energy below 105 GeV.These event classes are not covered by the used Monte Carlo samples, sincethey contribute negligibly after Step I.

After Step I of the analysis, a total of 9.99+10.3−5.13 background events are ex-

pected over the full analysis livetime, where 3 events are observed in experi-mental data.

None of the three experimental data events are accepted by the Step II eventselection, with a total number of expected background events of 0.265+0.265

−0.135events.

14.1.1 Step I Accepted EventsThe three events that were accepted by the Step I event selection scheme wereobserved in the IC86-IV, -VI and -VIII seasons, and are listed with their obser-vation date in Table 14.2. Throughout this section the events will be denotedby Events A, B and C.

Event views of each of the three observed events A, B and C are displayedFigures 14.2, 14.3 and 14.4 respectively. The values taken by the observedevents over the variables of the event selection are also listed in Table 14.3.The awarded BDT scores are also displayed in Figure 14.1 along with thedistributions of signal and background events over the BDT score. The eventsare displayed over the variables of the Step II BDT in Appendix B.

None of the Step I observed events were accepted by the selection criteria ofStep II. Event A was awarded the highest BDT score of the three,−0.089, and

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Table 14.1. The number of observed events, nOB, at each level of the analysis, alongwith the corresponding numbers of expected signal and background (astrophysicalneutrino) events, nSG and nBG, assuming the model fluxes.

Analysis level nOB nSG nBG

EHE filter 1.63×108 178 371

Step I Initial cuts 3.16×104 89.9 57.2Cascade cut 8.46×103 64.1 20.4Down-going cut 3 35.5 10.1Surface veto 3 35.5 9.99+10.3

−5.13

Step II 0 33.2 0.265+0.265−0.135

Table 14.2. The observation date and season of the three Step I observed events.

Season Date

Event A IC86–IV 2014–06–11Event B IC86–VI 2016–12–08Event C IC86–VIII 2019–03–31

Table 14.3. The values taken by the three observed events at Step I over the eventselection variables of this analysis.

Event A Event B Event C

log10 (nPE) 5.10 5.30 5.32nCH 219 227 633χ2

red,EHE 52.3 102 124cos(θzen,EHE) −0.209 −0.0489 −0.172

βBM 1.127 0.628 0.942rsd(EMIL) 3.20 4.97 7.25avg(dDOM,Q)CV-TrackChar 42.8 m 67.6 m 54.1 mtFWHM,CV-TimeChar 2.76 µs 2.78 µs 2.56 µsLFRCV-TrackChar 0.615 0.362 0.344RCOCV-HitStats 0.0458 0.434 0.0886log10 (nPE) 5.10 5.30 5.32cos(θzen,BM) −0.203 0.0205 0.391dC,BM 314 m 418 m 137 m

BDT score −0.089 −0.742 −0.626

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Figure 14.1. The BDT scores of the Step I observed events A, B and C, along with themagnetic monopole and neutrino event distributions over the Step II BDT score. Thecut criterion at a BDT score of 0.047 is also included (black).

Figure 14.2. Event view of theStep I observed Event A. SeeChapter 3.5 for a descriptionof how to interpret an eventview. The blue line representsthe BM track reconstruction.

136

Figure 14.3. Event view of theStep I observed Event B. SeeChapter 3.5 for a descriptionof how to interpret an eventview. The blue line representsthe BM track reconstruction.

Figure 14.4. Event view of theStep I observed Event C. SeeChapter 3.5 for a descriptionof how to interpret an eventview. The blue line representsthe BM track reconstruction.

137

was thus closest to be accepted as a magnetic monopole candidate. The frac-tion of magnetic monopole events at this level of the analysis that are awardeda lower BDT score than −0.089 is 0.22 %. Correspondingly, 32 % of the ex-pected muon neutrino events are scored higher than −0.089.

Note that the Step II event classification and selection procedures were de-veloped fully blind to these events, i.e. without using any information aboutthe experimental acceptance of the Step I selection.

14.1.2 Step II Accepted EventsThe Step II event selection did not accept any experimental data events overthe full 8 yr livetime of the analysis.

14.2 Final ResultAs no events were observed at the final level of this analysis, the analysiscannot result in a discovery of magnetic monopoles. The result is thus anupper limit on the magnetic monopole flux.

The number of final level expected background events over the livetimeof the analysis is estimated to 0.265+0.265

−0.135, assuming the Φ2017DIF-νµ

flux. Analternative flux measurement, Φ2019

HESE, yields an expectation on the numberof final level background events of 0.029+0.046

−0.016, and it is currently unknownwhich of these represents reality best. The conservative choice is to assumean entirely background-free analysis, i.e. to calculate the final result using afinal level background expectation of zero. This is motivated by the largeuncertainty on the number of background events, and is expected to differfrom the true number of background events by less than < 0.6 events over thefull livetime of the analysis (discussed in Chapter 13.5). Therefore, the finalresult will be reported with the background-free assumption.

The upper limit is calculated through Equation 7.2 with the effect of the nSGuncertainty σSG included according to Equation 7.9.

Here, the number of observed events, nOB, and the expected number ofbackground events, nBG, are both zero, resulting in a F&C upper limit ofµ90 (nOB,nBG) = 2.44. The (relative) signal efficiency uncertainty is σSG =8.4%. The 90 % CL upper limit on the magnetic monopole flux thus becomes2.54×10−19 cm−2 s−1 sr−1 averaged over the full monopole speed range.

If the expected number of background events was taken as 0.265, i.e. thenumber that arises from Monte Carlo estimation, with a 100 % uncertainty,the F&C upper limit would be 2.15 and the final flux upper limit would be2.24×10−19 cm−2 s−1 sr−1 over the full monopole speed range. This is ∼13% lower (better) than the present result.

The upper limit is also shown as a function of the true particle speed inFigure 14.5 along with the sensitivity. The numerical values of the upper limit

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and sensitivity are listed over the true particle speed in Table 14.4. Note thateach upper limit point in Figure 14.5 represents an individual result on theflux, as if having been produced with an analysis only considering that speed.

The shape of the upper limit over the particle speed is identical to the shapeof the sensitivity, as they both are shaped by the inverse of the effective areaover the true particle speed. The upper limit and the sensitivity differ by:

ΦMM90

ΦMM90−1 =

µ90

µ90−1 = 9.5%

Analogous to the sensitivity, the upper limit can be described as sharply de-creasing to a speed of approximately βMC = 0.87, whereafter it plateaus slightlybelow 2×10−19 cm−2 s−1 sr−1.

Additionally, the upper limit is shown in Figure 14.6 in the context of thecurrent upper limits on the cosmic flux of magnetic monopoles (compare toFigure 2.4). It is concluded that the results of the current analysis are compet-itive for speeds above β = 0.79, and yield an improvement of around an orderof magnitude over the previous results for speeds above 0.82.

In several of the previous analyses that have been conducted in this β rangethe upper limit has been extended from β = 0.995 to β = 1. This is allowed ifthere is no reason to believe that the event selection scheme will select againstevents in the range β ∈ [0.995;1]. In the present analysis, however, this is notallowed, as the particle speed is strongly anticorrelated with the BDT score ofthe Step II BDT, i.e. that high speed events are selected against in Step II ofthe analysis.

Additionally, a magnetic monopole in the ultrarelativistic regime (Lorentzfactor γ & 100) will produce particle showers along its track, which is a char-acteristic signature of an ultrarelativistic muon event. Events induced by an ul-trarelativistic magnetic monopole would therefore be heavily selected againstby the Step II BDT, and a search for a cosmic flux of these would require adedicated analysis strategy separate from the one described in this thesis. Con-sidering this, it is decided that the final upper limit on the magnetic monopoleflux will only cover the simulated range, and extend to β = 0.995.

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Figure 14.5. The 90 % CL upper limit on the cosmic flux of magnetic monopoles thatcan be set through this analysis, as a function of the true particle speed. Included isalso the sensitivity.

Figure 14.6. The 90 % CL upper limit on the cosmic flux of magnetic monopoles thatcan be set by employing this analysis (yellow curves, denoted by IceCube-86 8 yr), asa function of the true particle speed. Included are also the current best upper limits onthe magnetic monopole flux (see Chapter 2.6 for further description and references).

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Table 14.4. The 90 % CL upper limit and sensitivity on the cosmic magnetic monopoleflux that can be set with this analysis, as functions of the true particle speed.

Speed [c] Sensitivity [cm−2 s−1 sr−1] Upper limit [cm−2 s−1 sr−1]

0.780 1.26×10−17 1.15×10−17

0.785 5.98×10−18 5.46×10−18

0.790 3.18×10−18 2.90×10−18

0.795 1.88×10−18 1.71×10−18

0.800 1.23×10−18 1.12×10−18

0.805 8.72×10−19 7.96×10−19

0.810 6.65×10−19 6.07×10−19

0.815 5.34×10−19 4.88×10−19

0.820 4.46×10−19 4.08×10−19

0.825 3.84×10−19 3.51×10−19

0.830 3.39×10−19 3.10×10−19

0.835 3.07×10−19 2.80×10−19

0.840 2.83×10−19 2.58×10−19

0.845 2.65×10−19 2.42×10−19

0.850 2.50×10−19 2.29×10−19

0.855 2.38×10−19 2.17×10−19

0.860 2.26×10−19 2.06×10−19

0.865 2.16×10−19 1.97×10−19

0.870 2.08×10−19 1.90×10−19

0.875 2.02×10−19 1.84×10−19

0.880 1.98×10−19 1.81×10−19

0.885 1.96×10−19 1.79×10−19

0.890 1.94×10−19 1.77×10−19

0.895 1.92×10−19 1.75×10−19

0.900 1.89×10−19 1.73×10−19

0.905 1.87×10−19 1.70×10−19

0.910 1.85×10−19 1.69×10−19

0.915 1.84×10−19 1.68×10−19

0.920 1.84×10−19 1.68×10−19

0.925 1.84×10−19 1.68×10−19

0.930 1.84×10−19 1.68×10−19

0.935 1.84×10−19 1.68×10−19

0.940 1.84×10−19 1.68×10−19

0.945 1.84×10−19 1.68×10−19

0.950 1.85×10−19 1.69×10−19

0.955 1.86×10−19 1.70×10−19

0.960 1.89×10−19 1.72×10−19

0.965 1.90×10−19 1.74×10−19

0.970 1.92×10−19 1.75×10−19

0.975 1.92×10−19 1.76×10−19

0.980 1.93×10−19 1.76×10−19

0.985 1.94×10−19 1.77×10−19

0.990 1.95×10−19 1.79×10−19

0.995 1.97×10−19 1.80×10−19

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15. Summary and Outlook

The purpose of the analysis described in this thesis was to discover a cosmicflux of magnetic monopoles with a speed within the range β ∈ [0.750;0.995],where β = v

c is the magnetic monopole speed. In the case of a non-detection,the aim was to determine a competitive upper limit on the cosmic monopoleflux.

The cosmic flux search is conducted using data collected with the IceCubedetector (Chapter 3), that instruments ∼ 1km3 of the deep Antarctic ice with5160 optical modules, during 8 yr of operation. For the development of theanalysis strategy, the (background) flux of astrophysical neutrinos is assumedto be described by a power law as given by a previous IceCube measurementof the diffuse flux of upwards directed muon neutrino events (Chapter 8.3.2).

The analysis strategy of the present search is divided into two steps (Chap-ter 8.1) — Step I and Step II.

In Step I the event selection scheme of a previous IceCube analysis, theEHE analysis (Appendix A), is employed (Chapter 10.2). This analysis wasselected as it was designed to search for neutrino events that deposit a largeamount of light in the detector, similar to magnetic monopoles with speedabove the Cherenkov threshold in ice. Additionally, the EHE event selectionconsisted of a small number of simple cuts, thus enabling a highly general se-lection of bright events that imposes minimal constraints on the shape of thelight distribution. The EHE analysis is conducted with the aim of discoveringa flux of astrophysical neutrinos, and is therefore also made to reject atmo-spheric events with a high efficiency. At the final level of the EHE analysis,the atmospheric contribution is expected to be lower than 0.085 events overthe full 9yr livetime.

Since the aim of the Step I event selection is to select high energy neutrinoevents, and not limited to magnetic monopole events, an additional step of theanalysis — Step II — was developed (Chapter 10.3). The aim of Step II is toefficiently reject astrophysical neutrino events, while maintaining a high ac-ceptance for magnetic monopole events. To achieve this, a dedicated sampleof magnetic monopole Monte Carlo events was produced (Chapter 9.1), andexamined together with several simulated samples representing the astrophys-ical neutrino events (Chapter 9.2), in order to determine event signatures thatdistinguish magnetic monopole events from neutrinos. The astrophysical neu-trino Monte Carlo samples were truncated at Eν = 105 GeV, as events with alower energy than this would not contribute to the final distributions. A se-lection of nine variables were used to train a boosted decision tree (BDT) to

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identify the primary particle of an event (Chapters 10.3.2 and 10.3.3). TheBDT awards a score to each event that represents its likeness with a monopoleevent, and a cut criterion over the BDT score was determined by finding theoptimal model rejection potential (Chapter 10.3.4).

With the flux assumptions given above, the basic IceCube trigger condi-tions would accept a total of 244 magnetic monopole events over the full 8 yranalysis livetime, as well as 838 astrophysical neutrino events with an energyabove 105 GeV (Chapter 10.4). Over the full analysis livetime, the Step I eventselection would accept 35.5 magnetic monopole events and 9.99 astrophysicalneutrino events. The Step II selection would accept 33.2 monopole events,corresponding to 93.5 % of the Step I monopoles, and accept only 0.265 as-trophysical neutrino events, i.e. 2.65 % from Step I.

The sensitivity to the cosmic magnetic monopole flux of this analysis thusbecomes 2.78×10−19 cm−2 s−1 sr−1 averaged over the considered speed range(Chapter 11.2). See Table 14.4 and Figure 11.2 for the sensitivity as a functionof speed.

A study of the systematic uncertainty on the signal detection efficiency ofthis analysis was also conducted (Chapter 12). The total uncertainty was foundto be 8.4%, with negligible statistical contribution.

After the analysis procedure had been approved by the IceCube collabora-tion, it was applied to experimental data. The Step I event selection accepted atotal of three events in experimental data, all of which were rejected in Step II(Chapter 14.1). An upper limit of 2.54×10−19 cm−2 s−1 sr−1, average overthe full β range, on the magnetic monopole flux was obtained (Chapter 14.2).See Table 14.4 and Figure 14.5 for the upper limit as a function of speed.This result constitutes an improvement of about one order of magnitude overthe previously published results in the relevant β range, most recently by theANTARES collaboration [14] (Figure 14.6).

This result is valid if the magnetic monopole mass, mMM, lies in the rangemMM ∈

[108,1015

]GeV (Chapter 8.3.1). Here, the upper constraint arises

from the upper bound on the magnetic monopole kinetic energy (Chapter 2.4),and the lower constraint comes from the requirement that the monopole losesnegligible energy while traversing the length of the IceCube detector (Chap-ter 4.1).

Future magnetic monopole analyses with IceCube may enjoy improved re-construction and particle identification techniques, such as an event type clas-sifying neural network [65], and a perpetually increasing data volume. How-ever, when the magnetic monopole efficiency approaches perfect, and if thebackground is kept low, the IceCube monopole sensitivity will only improvewith the inverse of the collected livetime (Equation 7.4). This means thatadding one additional year of data to an analysis covering 8 yr would at bestproduce a 12.5 % improvement of the sensitivity. Similarly, an improvementof a factor of 10 would require a total of 80 yr of observation.

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Therefore, in order to continue to make significant progress in the search fora cosmic flux of magnetic monopoles, a larger detector is required, such as theIceCube Gen2 detector [80]. The IceCube Gen2 detector is proposed to havean effective volume V 1/3

eff one order of magnitude larger than that of IceCube,corresponding to an effective area a factor of ∼ 4.6 larger than IceCube, asA1/2

eff ≈V 1/3eff . The sensitivity to a magnetic monopole flux would thus improve

with the collected livetime ∼ 4.6 times faster than for IceCube, but the samelimitation would be reached, where each added year of data yields a smallerrelative sensitivity improvement.

This may seem discouraging, as no larger future detectors are planned, butthe search for a cosmic flux of magnetic monopoles must continue. The dis-covery of magnetic monopoles would constitute a monumental feat in physics,and the flux may lie just beneath the upper limit presented in this thesis.

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16. Swedish Summary — SvenskSammanfattning

Avhandlingens Titel Ljusstarka Nålar i en Höstack — En Sökning efterMagnetiska Monopoler med Neutrino-Observatoriet IceCube

Målet med det forskningsprojekt, den analys, som beskrivs i den här avhand-lingen var att upptäcka en typ av exotiska partiklar som bär magnetisk ladd-ning, så kallade magnetiska monopoler. Sökningen efter monopolerna genom-fördes med data som insamlats med IceCube-detektorn under loppet av åttaår. Projektet har genomförst vid Uppsala Universitet, och inom ramarna förIceCube-kollaborationen.

16.1 Vad är en Magnetisk Monopol?En magnetisk monopol är en hypotetisk partikel, alltså en partikel som inte harobserverats (ännu). En monopol bär endast en magnetisk laddning, antingenen nord- eller sydpol. Detta kan jämföras med exempelvis en elektriskt laddadpartikel, som bär antingen positiv eller negativ elektrisk laddning.

I den här analysen har vi sökt efter magnetiska monopoler som kommer frånrymden med en hastighet nära ljusets hastighet, mellan 75 % och 99,5 % avljusets hastighet. Dessa hastigheter uppnås då monopolerna acceleras kraftigtav de starka magnetfält som finns i Universum, exempelvis kring aktiva galax-kärnor. De monopoler vi söker har även väldigt hög massa, vilket i kombina-tion med den höga hastigheten gör att de kan passera genom hela Jorden.

Inom det här hastighetsspannet skulle en magnetisk monopol sända ut Cher-enkovljus när den passerar genom vissa material, bland annat is och vatten.Cherenkovljus produceras med många olika våglängder, däribland vanligt op-tiskt ljus. Ljuset sänds ut jämnt längs med partikelns spår, med en vinkel sombestäms av partikelns hastighet. Hastigheten bestämmer även mängden ljussom produceras, där de snabbaste monopolerna ger mest ljus.

Man vet i nuläget inte om magnetiska monopoler existerar, men en storgrupp av fundamentala teorier, alltså teorier om hur världen fungerar på grund-läggande nivå, anger att magnetiska monopoler borde finnas. Magnetiska mo-nopoler beskrevs 1931 av Paul Dirac, vilket förfinades 1974 av Gerard ’t Hooftoch Aleksandr Polyakov, oberoende av varandra. Dirac beskrev att att fri mag-netisk laddning endast får existera om den är kvantiserad, vilket innebär att

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laddningen byggs upp av ett antal minsta-möjliga laddningar. I och med dettamåste även den elektriska laddningen vara kvantiserad. Dessa krav är kändasom Diracs kvantiseringsvillkor.

Man har fastställt att elektrisk laddning är kvantiserad i verkligheten, menhittills inte kunnat bekräfta varför. Om man upptäcker en magnetisk monopolskulle man därmed fastställa varför den elektriska laddningen är kvantiserad,och då lösa ett av de stora olösta mysterierna i modern fundamentalfysik.

16.2 Vad är IceCube?IceCube-detektorn är placerad vid den geografiska sydpolen och färdigställdesår 2011. Detektorn drivs och underhålls av IceCube-kollaborationen — ett in-ternationellt samarbete mellan nästan 300 forskare vid över 50 universitet itolv länder, däribland cirka 15 forskare vid Stockholms och Uppsala Univer-sitet. I den här analysen behandlas data som samlats in med IceCube-detektornmellan år 2011 och 2019.

IceCube-detektorn kan användas till forskning på flera olika områden, ochutvecklades för att registrera neutrin-partiklar som producerats i rymden. Deregistrerade neutrinerna kan analyseras för att exempelvis dra slutsatser omolika kosmiska objekt, för att söka efter mörk materia eller för att undersökaneutrinerna själva. Utöver detta kan detektorn även användas till studier avden glaciäris i vilken den är placerad.

IceCube-detektorn är uppbyggd av tre övergripande komponenter, var ochen med ett specialiserat syfte. Dessa är den huvudsakliga detektorn, DeepCoreoch IceTop, samt en kontrollenhet, IceCube-Laboratoriet, där all data samlasoch kategoriseras. DeepCore och IceTop avser del-detektorer som fokuserarpå lågenergi-neutriner respektive partiklar som bildats i atmosfären, och ärplacerade i center av den huvudsakliga detektorn respektive på glaciärens yta.Datan som behandlas i den här analysen har insamlats med den huvudsakligadetektorn. I figur 16.1 visas en illustration av IceCube-detektorn.

Huvud-detektorn består av nästan 5000 digitala optiska moduler (DOMar),fördelade i längder om 60 på totalt 78 kablar som är nedsänkta djupt ner iden Antarktiska glaciärisen. En DOM är ett instrument som mäter ljusmängdmed så pass hög noggrannhet den kan uppfatta enstaka fotoner (ljuspartiklar).Detektor-kablarna är nedsänkta i ett hexagonalt mönster med ett avstånd på125 meter mellan varje kabel. På varje kabel är de 60 DOMarna placerademed 17 meters mellanrum längs med den nedersta kilometern, mellan 1450meters och 2450 meters djup. Således har totalt en kubikkilometer av dendjupa glaciärisen på Antarktis instrumenteras med ljusdetekterande DOMar.

Data samlas kontinuerligt in med IceCube-detektorn, dag liksom natt ochsommar liksom vinter. Den data som samlas in med IceCube delas upp i så-kallade händelser. Varje händelse innehåller den data som samlats in från enoch samma ursprungspartikel, exempelvis en atmosfärisk myon eller neutrin.

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Globen Arena

Figure 16.1. En illustration över IceCube-detektorn, inklusive IceCube-laboratorietoch dess del-detektorer. Credit: IceCube-kollaborationen. En illustration av Globen iStockholm (diameter 110 meter) inkluderas för att demonstrera detektorns skala.

De vanligaste partiklarna att detekteras med IceCube, cirka 2700 gångerper sekund, är atmosfäriska myoner. Dessa bildas när partiklar i den kosmiskastrålningen krockar med atomer i atmosfären, och färdas sedan ner genomluften och isen tills de passerar genom detektorn. Myoner sänder ut Cheren-kovljus när de passerar genom isen, på samma sätt som magnetiska monopolerskulle, och det här ljuset detekteras sedemera av IceCubes DOMar. MängenCherenkovljus som sänds ut av en myon är dock mycket mindre än för en mo-nopol, men detta kompenseras för myoner med hög energi, då de framkallarljusgivande partikelskurar längs med sin bana genom isen. Då myonerna kanfärdas långväga genom isen, och sänder ut ljus längs med hela sitt spår, synsmyoner som avlånga streck i detektorn.

Näst vanligast i detektorn är atmosfäriska neutriner, vilka också bildats närkosmisk strålning krockar med atmosfären. Atmosfäriska neutriner detekterascirka en miljon gånger mer sällan än atmosfäriska myoner. Dessa färdas ocksåner genom atmosfären och isen, men växelverkar så lite med sin omgivningatt de inte själva syns i detektorn. En liten andel av dessa neutriner krockardock med en atom i isen, och ger då upphov till antingen en myon eller en par-tikelskur. Myonerna sänder ut ljus längs med sitt spår, liksom de atmosfäriskamyonerna, medan partikelskurarna endast sänder ut ljus nära sin start-punkt.Ljuset som sänds ut detekteras sedan av detekorn.

Utöver de atmosfäriska myonerna och neutrinerna detekteras också astro-fysikaliska neutriner, alltså neutriner som bildats långt ifrån Jorden. I detek-torn ser dessa likadana ut som de atmosfäriska neutrinerna, och kommer be-höva särskiljas på statistisk nivå (baserat på deras energi och ingångsvinkel).

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Magnetisk monopol Genomfarande myon Partikelskur Ljussvag myon

Figure 16.2. Illustrerade exempel på händelser i IceCube — en magnetisk monopol-händelse och tre andra händelser med distinkt särskiljda signaturer. Det blå områdetrepresenterar IceCube-detektorn i glaciärisen, och det rödgrön-färgade området repre-senterar det ljus som detekterats i händelsen. Färgskalan från rött till grönt represen-terar tiden då ljuset detekterats, där rött indikerar tidigt i händelsen och grönt indikerarsent. Pilarna representerar de ljusskapande partiklarnas väg genom detektorn.

16.3 Att Söka Magnetiska Monopoler med IceCubeDet första man gör i en sökning efter en okänd partikel är att ta reda precis vadman letar efter, alltså hur en sådan händelse borde se ut i detektorn, och hurdetta skiljer sig ifrån händelser framkallade av andra partiklar. Detta gör mangenom att konstruera en modell av sin detekor i datorn, och simulera ett antalhändelser med partikeln i modellen.

För den här analysen har vi simulerat 400 000 magnetiska monopol-händelser,och jämfört dessa med ca 500 miljoner simulerade händelser med astrofysik-aliska neutriner. Med hjälp av detta har vi tagit fram fem signaturer för mag-netiska monopoler, alltså specifika egenskaper som särskiljer magnetiska mo-nopol-händelser ifrån neutrin-händelser. Illustrerade exempel på händelsermed olika signatur kan ses i figur 16.2.

I nästa etapp av analysen analyseras alla simulerade händelser, och ett urvalutvecklas som ska sortera fram händelser som ser ut att härröra magnetiskamonopoler. Detta urval kallas för ett händelseurval, och kriterierna för händ-elseurvalet baseras på de signaturer som tidigare har tagits fram.

En karakteristisk signatur för magnetiska monopoler är att de producerarväldigt mycket ljus, vilket leder till att magnetiska monopol-händelser in-nehåller väldigt mycket detekterat ljus. I samband med en tidigare analysinom IceCube, den så kallade EHE-analysen (Extremely High Energy — Ex-tremt Hög Energi), utvecklades ett händelseurval som sorterar bort alla händ-elser som inte uppvisar väldigt mycket detekterat ljus i detektorn. Målet medEHE-analysen var att hitta en viss typ av astrofysikaliska neutriner, så urvaletsorterade även bort atmosfäriska neutriner och myoner väldigt effektivt. I våranalys återanvände vi händelseurvalet från EHE-analysen i vad vi kallar Steg Iav analysen. Detta hjälpte oss att sortera bort atmosfäriska händelser och sam-tidigt behålla en stor andel monopol-händelser.

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Då EHE-analysens händelseurval utformades för att behålla astrofysikaliskaneutriner, medan vi söker efter magnetiska monopoler, måste vi lägga till yt-terligare ett steg i analysen — Steg II. I Steg II vill vi behålla så stor an-del av de magnetiska monopol-händelserna som möjligt, och samtidigt kastaså många neutrin-händelser som möjligt. Detta åstadkommer vi genom attanvända de magnetiska monopol-signaturer som vi tagit fram, och träna ettBDT (Boosted Decision Tree — Förbättrat BeslutsTräd) till att särskilja mag-netiska monopol-händelser ifrån neutrin-händelser. Ett BDT är ett maskin-inlärningsverktyg som man tränar till att känna igen ett antal aspekter hos enhändelse, och som baserat på dessa poängsätter varje händelse på en flytandeskala mellan −1 och +1. I vår analys anger en högre poäng att en händelse ärmonopol-lik, medan lägre poäng anger att den är neutrin-lik. Det beslutadesatt kasta alla händelser med en poäng under 0,047, vilket är det värde som geranalysen bäst känslighet för lägsta möjliga monopol-flöde.

16.4 ResultatDen här analysen har utförts i blindo, vilket innebär att analysmetoden harutformas till fullo innan den har använts på experimentell data insamlad meddetektorn. På detta sätt kan vi utveckla analysen så opartiskt som möjligt,och utformar inte händelseurvalet specifikt med hänsyn till de individuellahändelser som detekterats. Därmed vet vi heller inte på förhand vilket resultatsom analysen kommer att ge, utan detta vet vi först i efterhand då analysen intelängre får redigeras. Däremot kan vi på förhand jämföra med resultaten fråntidigare sökningar efter magnetiska monopoler inom samma hastighetsspann.

Det har tidigare gjorts flera sökningar efter monopoler med hastigheterinom det intervall som den här analysen avser. Som bekant har inga magnet-iska monopoler hittats, utan alla har resulterat i en övre gräns på flödet. Detsenaste resultatet inom detta intervall satte en övre gräns på flödet till färre än3,46×10−18 monopoler per kvadratcentimeter per sekund per steradian (en-heten för rymdvinkel). Den gränsen innebär att vårt händelseurval borde sefärre än 33,2 detekterade monopoler under de åtta år som analysen behandlar.

Utöver detta beräknade vi hur effektivt vårt händelseurval är på att avvisaastrofysikaliska neutrinhändelser. Händelseurvalet borde registrera i genom-snitt 0,256 neutriner under analysens åtta år, vilket innebär att vi skulle behövaobservera nästan fyra gånger så lång tid för att detektera en enda neutrin.

När vi sedemera använde vår analysmetod på experimentell data visade detsig att ingen experimentell händelse uppfyllde urvalsvillkoren, alltså att ingenhändelse bedömdes som monopol-lik. Den övre gräns som vi då kan sätta är attvi har observerat färre än 2,44 magnetiska monopoler under hela analysens åttaår, vilket motsvarar en övre gräns på flödet på färre än 2,54×10−19 monopolerper kvadratcentimeter per sekund per steradian. Vårt resultat är alltså mer än10 gånger bättre än det senaste motsvarande resultatet.

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17. Acknowledgements

This thesis is dedicated tomy children Olivia and Victor,who are all that is best in me.

The five years that have led up to the writing of this thesis have been world-altering for me. Not only was I introduced to the role of a modern day re-searcher, and allowed to immerse myself in the international collaboration thatis IceCube, but I also got married and had two children, as well as bought andsold two apartments and bought my first house, In addition to this, the finalyear of my Ph.D. period has been afflicted with a global pandemic, that hasaffected every aspect of our lives.

In this section I would like to highlight and thank some of the people whohave contributed to my endeavor over the last five years.

To my main supervisor — Carlos Pérez de los Heros — thank you for yourguidance over the last five years. Thank you indulging my unconventionalideas and discussions about all aspects of our work, from the scientific methodto the nature of particle physics, from the state of Swedish particle physics tocareer advice, and personal topics such as vacations, children, Spain, etc. Yourengagement and (figuratively) always open door is an inspiration for me.

To my former main supervisor, now co-supervisor — Allan Hallgren —thank you for brining me into the group and for your generous support overthese years. I have met few people with such a high physics creativity asyou have, and many a problem has been solved by your unprecedented “buthave you tried this way?” or “have you checked this aspect?”. This boundlesscreativity is something that I continue to try to to internalize.

To my co-supervisor — Olga Botner — thank you for your inspiration andsupport over these years. Never have I met someone with such a piercingthought processes as yours. No matter what the issue is, you always find thecore and twist and turn it until a solution can be found. You are the epitome ofa physicist, with an unmatched physics intuition that I will always strive for.

To all of my supervisors, thank you for allowing me the freedom to trialdifferent topics within our group, before finally selecting beyond the standardmodel astroparticle physics.

Thank you to Rickard and Henric for welcoming me to the group. Andthank you to all of my previous and current colleagues at Uppsala University— Jim, Max, Mikael, Jan, Maja, Myrto, Olga S. G., Thomas, Venu, Walter,Elisabetta, Johan, Erin, Nora, Bo, and all others — for having made my timehere very enjoyable through all of the lunches, fika and discussions. And thankyou to Arnaud for your thorough feedback on this thesis. An additional thank

150

you to my colleagues in the Stockholm University IceCube group — Samuel,Jon, Martin, Marcel, Kunal, Matti, Maryon, Chad, Klas, Christian.

In the wider IceCube collaboration I would like to thank Anna P., who hasgrown into a close friend, for your mentorship and your patience with myendless questions about monopoles, about simulations, about weighting andupper limits, and for your support both in work and in life. And to Sophie —welcome to the world! Additionally, I also want to extend thanks to severalcolleagues within IceCube — Morten, Joakim, Mike, Liz, Ward, Christoph,Shivesh, Elim, Anna N, Michael, Justin, Frederik — for the good times wehad in collaboration meetings and bootcamps.

Of all my colleagues I most want to send thanks to Lisa. You welcomed mewith open arms for my very first job-interview with the group, and the timewe shared as office mates will always shine bright in my memory. When youwere there the office was large and warm and welcoming, and when you wereout it was cold and small and unforgiving. Countless hours have we spentdiscussing, trouble-shooting and laughing, and you always provided supportwhen needed. It was impossible to be unhappy when you were around. Andthank you to Jerome for your endless patience, and to Jeli just for existing.

Thank you to my father, my mother and my brother — Douglas, Anne andChristian — for having been there through the good times and the bad, and foralways believing in me. And thank you to my mother-in-law — Annette —for the numerous times you have helped us solve our life-puzzle over the lastfew years. Also thank you to my remaining family — Anneli, Pierre, Orvokki& Jari, Christian & Jenny, Linus, Hampus, Pontus, Linnea & Ted, Simon &Elin, Boel & Bosse, Janne & Lena, Anne-Lie, Kicki — for cheering me onand for providing support when needed.

A special thanks to my closest friends — Niklas, Camilla, Jonas — as wellas Edda and Mira — for always being there, and for your patience with mywork. An additional thank you to my friends — Martin, Marcus, Daniel, Filip,Douglas, Chuck, Alexandra, Veronica, Angelica & Marcus, Sofi & Simon.

I want to thank my children — Olivia and Victor — for bringing the biggestpossible joy to my life. You are the reasons that I get up in the morning, andwho I think about before falling asleep. Thank you for existing, for surprisingme with your new capabilities, for your humor and for providing me withunconditional love and kindness.

Last but not least, as the cliché goes, I want to thank my loving wife —Angelica. Thank you for having put up with all of this. Thank you for takingon all of the roles that I needed in my Ph.D. endeavor — for cheering me onwhen I was out, for pushing me when I was lazy, for being my sounding boardwhen I needed to vent and break things down, for strengthening me when Ineeded to stand up for myself and thank you for pulling me back up when Iwas down. Without your partnership and your support the last five years wouldnot have been possible, thank you for making them the best years so far. I loveyou.

151

A. The IceCube EHE Analysis

Step I of the analysis that is presented in this thesis is formed by the eventselection of another IceCube analysis — the Extremely High Energy (EHE)analysis. The purpose of the EHE analysis [46] is to discover a flux of cosmo-genic neutrinos with high energy (typically Eν & 108 GeV) that was formedthrough the Greisen-Zatsepin-Kusmin (GZK) mechanism [81]. These neutri-nos are produced through the interaction between ultra high energy cosmic rayparticles (protons or nuclei) and the cosmic microwave background, and areexpected to have a different energy distribution than the diffuse astrophysicalneutrino flux that is the background in the analysis presented in this thesis.

The EHE event selection is designed to accept as many neutrinos as possiblewith a high incident energy, while rejecting the majority of events with anatmospheric origin, and is described in Chapter 10.2. The event selectionmainly selects based on the registered brightness of an event, which is highlycorrelated with the event deposited energy, and the cut value is determinedby the reconstructed direction of the incident neutrino and its track fit quality.The selection variables in the EHE analysis are:

• The number of registered photo-electrons, nPE, and its base-10 loga-rithm, log10 (nPE).

• The number of detector channels (DOMs) with registered charge, nCH .• The fit quality (the reduced χ2 parameter) of the EHE track recon-

struction, χ2red,EHE.

• The cosine of the zenith direction of the EHE reconstructed track,cos(θzen,EHE).

The selection criteria per analysis level are summarized below:The EHE filter Data reduction by selection on nPE:

nPE ≥ 1000 (A.1)

The offline EHE cut Quality and data reduction cuts on nPE, nCH andχ2

red,EHE separately:

nPE ≥ 25000 (A.2)

nCH ≥ 100

χ2red,EHE ≥ 30

152

Figure A.1. Distributions of the signal (cosmogenic neutrinos) and background (atmo-spheric muons, conventional atmospheric neutrinos, prompt atmospheric neutrinos) ofthe EHE analysis, over event brightness, denoted by NPE, and fit quality, denoted byχ2/ndf, along with the track quality cut criterion (Equation A.3). Credit: Figure 1from reference [46].

Figure A.2. Distributions of the signal (cosmogenic neutrinos) and background (atmo-spheric muons, conventional atmospheric neutrinos, prompt atmospheric neutrinos) ofthe EHE analysis, over event brightness, denoted by NPE, and fit quality, denoted bycos(θLF), along with the muon bundle cut criterion (Equation A.4). Credit: Figure 2from reference [46].

153

The track quality cut Rejection of prompt atmospheric electron neutrinos:

log10 (nPE)≥

4.6 if χ2red,EHE < 80

4.6+0.015×(χ2

red,EHE−80)

if 80≤ χ2red,EHE < 120

5.2 if 120≤ χ2red,EHE

(A.3)

The muon bundle cut Rejection of atmospheric muon and muon neutrinoevents from above.

log10 (nPE)≥

4.6 if cos(θzen,EHE)< 0.06

4.6+1.85×√

1−(

cos(θzen,EHE)−10.94

)2

if 0.06≤ cos(θzen,EHE)

(A.4)

The surface veto Rejecting downwards directed events (θzen,EHE < 85°) incoincidence with two or more registered photons in the IceTop surface array.Simulated event distributions of the EHE analysis signal (cosmogenic neu-

trinos) and background (atmospheric muons, conventional atmospheric neu-trinos, prompt atmospheric neutrinos) are shown in Figures A.1 and A.2. Fig-ure A.1 shows the distributions over event brightness, here denoted by NPE,and fit quality, denoted by χ2/ndf, along with the applied selection criterion ofthe track quality cut. Correspondingly, Figure A.2 shows the distributions overevent brightness and reconstructed zenith direction, here denoted by cos(θLF),along with the applied selection criterion of the muon bundle cut.

This results in a total rejection of all events with a brightness of log10 (nPE)<4.6 (i.e. nPE . 4.0×104) and full acceptance of all events with log10 (nPE)≥6.45 (i.e. nPE & 2.8×106). Between these values, the acceptance dependson the direction and fit quality of the event. The atmospheric muon and neu-trino event rates over each analysis level are displayed in Table A.1 along withthe corresponding acceptance of cosmogenic neutrinos relative the EHE filterlevel.

Table A.1. The expected event rate of atmospheric muons and neutrinos, along withthe acceptance of cosmogenic neutrinos relative the EHE filter level, for the EHEanalysis levels [46].

Atmospheric Atmospheric Cosmogenicmuon event neutrino event neutrino relative

Analysis level rate [Hz] rate [Hz] acceptance

EHE filter 0.8 7.6×10−6 1.00Offline EHE cut 6.7×10−4 1.0×10−8 0.74Track quality cut 1.6×10−4 6.1×10−10 0.61Muon bundle cut 3.0×10−10 3.6×10−10 0.43

154

Before applying the selection criteria, the selection variables in this analysiswere validated against experimental data. This is described in Chapter 5.4.1.

The EHE analysis has been applied to 9 yr of experimental data, constitutedby the detector seasons IceCube-40, -59, -79 and IceCube-86 I–VI. The EHEanalysis was initially developed for the first four of these seasons, and thefollowing seasons were added incrementally. One effect of this was that thefull final season, IC86-VI, was designated as physics sample, as opposed tobeing divided into physics and burn samples.

Over the full 9 yr period, the EHE event selection was expected to accept anaverage of less than 0.085 events of atmospheric origin. The neutrino effectivearea of the event selection using the full detector configuration (IceCube-86)is shown in Figure 13.2.

The accepted neutrino events are expected to originate both from the dif-fuse astrophysical neutrino flux, measured by IceCube up to ∼ PeV ener-gies [73; 75; 79], and the GZK neutrino flux, which is not yet experimentallydiscovered. The diffuse astrophysical neutrino flux thus forms the dominantbackground for the EHE analysis. In the EHE analysis, the astrophysical fluxis distinguished from the GZK neutrinos through statistical methods, as well asan event-by-event consideration. The result is an upper limit on the abundanceof cosmogenic neutrinos.

155

B. Step I Observed Events over BDT Variables

The Step II BDT score and variable values of the Step I observed events A, Band C are displayed in Figures B.1, B.2, B.3 and B.4, and listed in Table B.1.

Figure B.1. The Step I observed events A, B and C over the Step II BDT score, alongwith the simulated magnetic monopole and astrophysical neutrino event distributions.

Table B.1. The values taken by the three Step I observed events A, B and C in theStep II BDT variables, as well as the BDT score.

Event A Event B Event C

BDT score −0.089 −0.742 −0.626

βBM 1.127 0.628 0.942rsd(EMIL) 3.20 4.97 7.25avg(dDOM,Q)CV-TrackChar 42.8 m 67.6 m 54.1 mtFWHM,CV-TimeChar 2.76 µs 2.78 µs 2.56 µsLFRCV-TrackChar 0.615 0.362 0.344RCOCV-HitStats 0.0458 0.434 0.0886log10 (nPE) 5.10 5.30 5.32cos(θzen,BM) −0.203 0.0205 0.391dC,BM 314 m 418 m 137 m

156

(a) The Step II speedvariable, βBM.

(b) The Step II energy lossRSD variable, rsd(EMIL).

(c) The Step II averagepulse distance variable,avg(dDOM,Q)CV-TrackChar.

Figure B.2. The Step I observed events A, B and C over the Step II BDT variables,along with the simulated magnetic monopole and astrophysical neutrino event distri-butions.

157

(a) The Step II pulse-timeFWHM variable,tFWHM,CV-TimeChar.

(b) The Step II length fillratio variable,LFRCV-TrackChar.

(c) The Step II relativeCoG offset variable,RCOCV-HitStats.

Figure B.3. The Step I observed events A, B and C over the Step II BDT variables,along with the simulated magnetic monopole and astrophysical neutrino event distri-butions.

158

(a) The Step IIlog-brightness variable,log10 (nPE).

(b) The Step II cos-zenithvariable, cos(θzen,BM).

(c) The Step II centralityvariable, dC,BM.

Figure B.4. The Step I observed events A, B and C over the Step II BDT variables,along with the simulated magnetic monopole and astrophysical neutrino event distri-butions.

159

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1–11: 1970–197512. Lars Thofelt: Studies on leaf temperature recorded by direct measurement and

by thermography. 1975.13. Monica Henricsson: Nutritional studies on Chara globularis Thuill., Chara zey-

lanica Willd., and Chara haitensis Turpin. 1976.14. Göran Kloow: Studies on Regenerated Cellulose by the Fluorescence Depolar-

ization Technique. 1976.15. Carl-Magnus Backman: A High Pressure Study of the Photolytic Decomposi-

tion of Azoethane and Propionyl Peroxide. 1976.16. Lennart Källströmer: The significance of biotin and certain monosaccharides

for the growth of Aspergillus niger on rhamnose medium at elevated tempera-ture. 1977.

17. Staffan Renlund: Identification of Oxytocin and Vasopressin in the Bovine Ade-nohypophysis. 1978.

18. Bengt Finnström: Effects of pH, Ionic Strength and Light Intensity on the Flash Photolysis of L-tryptophan. 1978.

19. Thomas C. Amu: Diffusion in Dilute Solutions: An Experimental Study with Special Reference to the Effect of Size and Shape of Solute and Solvent Mole-cules. 1978.

20. Lars Tegnér: A Flash Photolysis Study of the Thermal Cis-Trans Isomerization of Some Aromatic Schiff Bases in Solution. 1979.

21. Stig Tormod: A High-Speed Stopped Flow Laser Light Scattering Apparatus and its Application in a Study of Conformational Changes in Bovine Serum Albu-min. 1985.

22. Björn Varnestig: Coulomb Excitation of Rotational Nuclei. 1987.23. Frans Lettenström: A study of nuclear effects in deep inelastic muon scattering.

1988.24. Göran Ericsson: Production of Heavy Hypernuclei in Antiproton Annihilation.

Study of their decay in the fission channel. 1988.25. Fang Peng: The Geopotential: Modelling Techniques and Physical Implications

with Case Studies in the South and East China Sea and Fennoscandia. 1989.26. Md. Anowar Hossain: Seismic Refraction Studies in the Baltic Shield along the

Fennolora Profile. 1989.27. Lars Erik Svensson: Coulomb Excitation of Vibrational Nuclei. 1989.28. Bengt Carlsson: Digital differentiating filters and model based fault detection.

1989.29. Alexander Edgar Kavka: Coulomb Excitation. Analytical Methods and Experi-

mental Results on even Selenium Nuclei. 1989.30. Christopher Juhlin: Seismic Attenuation, Shear Wave Anisotropy and Some

Aspects of Fracturing in the Crystalline Rock of the Siljan Ring Area, Central Sweden. 1990.

31. Torbjörn Wigren: Recursive Identification Based on the Nonlinear Wiener Model. 1990.

32. Kjell Janson: Experimental investigations of the proton and deuteron structure functions. 1991.

33. Suzanne W. Harris: Positive Muons in Crystalline and Amorphous Solids. 1991.34. Jan Blomgren: Experimental Studies of Giant Resonances in Medium-Weight

Spherical Nuclei. 1991.35. Jonas Lindgren: Waveform Inversion of Seismic Reflection Data through Local

Optimisation Methods. 1992.36. Liqi Fang: Dynamic Light Scattering from Polymer Gels and Semidilute Solutions.

1992.37. Raymond Munier: Segmentation, Fragmentation and Jostling of the Baltic Shield

with Time. 1993.

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On Coupling-Effects of Stochastic Resonators and Spectral Optimization of Fluctu-ations in Random Network Switches. 2004.

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Importance for Future Oil Production. 2007.70. Anna Davour: Search for low mass WIMPs with the AMANDA neutrino telescope.

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bances. 2008.75. Johan Petrini: Querying RDF Schema Views of Relational Databases. 2008.76. Noomene Ben Henda: Infinite-state Stochastic and Parameterized Systems. 2008.77. Samson Keleta: Double Pion Production in dd→αππ Reaction. 2008.78. Mei Hong: Analysis of Some Methods for Identifying Dynamic Errors-invariables

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of 40 and 72 MeV. 2009.86. Jan Henry Nyström: Analysing Fault Tolerance for ERLANG Applications. 2009.87. John Håkansson: Design and Verification of Component Based Real-Time Sys-

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actions for the PANDA Experiment. 2009.90. Agnes Rensfelt. Viscoelastic Materials. Identification and Experiment Design. 2010.91. Erik Gudmundson. Signal Processing for Spectroscopic Applications. 2010.92. Björn Halvarsson. Interaction Analysis in Multivariable Control Systems. Applica-

tions to Bioreactors for Nitrogen Removal. 2010.93. Jesper Bengtson. Formalising process calculi. 2010. 94. Magnus Johansson. Psi-calculi: a Framework for Mobile Process Calculi. Cook

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100. Patrik Adlarson. Studies of the Decay η→π+π-π0 with WASA-at-COSY. 2012.101. Erik Thomé. Multi-Strange and Charmed Antihyperon-Hyperon Physics for PAN-

DA. 2012.102. Anette Löfström. Implementing a Vision. Studying Leaders’ Strategic Use of an

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2014.104. Linda Åmand. Ammonium Feedback Control in Wastewater Treatment Plants.

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Study of FeCo Thin Films and Trilayers. 2015.113. Marcus Björk. Contributions to Signal Processing for MRI. 2015. 114. Alexander Madsen. Hunting the Charged Higgs Boson with Lepton Signatures

in the ATLAS Experiment. 2015.115. Daniel Jansson. Identification Techniques for Mathematical Modeling of the

Human Smooth Pursuit System. 2015. 116. Henric Taavola. Dark Matter in the Galactic Halo. A Search Using Neutrino

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118. Li Caldeira Balkeståhl. Measurement of the Dalitz Plot Distribution for η→π+π−

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and String Variable Types. 2016.121. Andrej Andrejev. Semantic Web Queries over Scientific Data. 2016.

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verse opals. 2016.127. Per Mattsson. Modeling and identification of nonlinear and impulsive systems.

2016.128. Lars Melander. Integrating Visual Data Flow Programming with Data Stream

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and τ leptons in the ATLAS experiment. Search results using LHC Run 2 data and prospect studies at the HL-LHC. 2020.

142. Carl Kronlid. Engineered temporary networks. Effects of control and temporality on inter-organizational interaction. 2020.

143. Alexander Burgman. Bright Needles in a Haystack. A Search for Magnetic Mono-poles Using the IceCube Neutrino Observatory. 2020.