Theory and Phenomenology of the Neutrino and Gamma-Ray Universe

Theory and Phenomenology of the Neutrino and Gamma-ray Universe

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Bei Zhou, M.S. Graduate Program in Physics

The Ohio State University 2020

Dissertation Committee: Professor John F. Beacom, Advisor

Professor James J. Beatty

Professor Eric A Braaten

Professor Annika H. G. Peter c Copyright by

Bei Zhou

2020 Abstract

Neutrinos, an essential piece of the standard model of particle physics, provide critical information about the universe. In reverse, astrophysical neutrinos provide indispensable tools to study particle physics. Tremendous progress, including breakthroughs, has been made in the past decades and will likely be made in the next decade. To get there, many aspects of work are needed, including the properties of neutrino sources, neutrino interactions, neutrino detectors, and more. In general, there are two major directions, one to increase the data from current sources of neutrinos, and the other is to develop new sources.

In this dissertation, I discuss a series of papers pertinent to both directions. First is studying TeV solar gamma-ray emission, as part of a larger program of work to develop the Sun as a new high-energy laboratory, including as a neutrino source. Next is studying subdominant interactions for high-energy neutrinos, including W -boson production and trident production, as demanded by the increasing precision of TeV–PeV neutrino astrophysics. The last is understanding atmospheric neutrino foregrounds for searches of the diﬀuse supernova neutrino background in Super-Kamiokande. This provides a solid theoretical foundation for further reducing the foregrounds, which will signiﬁcantly help the

first detection and subsequent precision measurements of the diffuse supernova neutrino background. Together, this series of work advances neutrino astrophysics as a broad and vibrant field.

ii To those who love me.

iii Acknowledgments

I have had an incredible time as a Ph.D. student in CCAPP and the Department of Physics at The Ohio State University, thanks to the help from the following people.

First of all, I would like to thank my advisor, Professor John F. Beacom. He is an incredible advisor, physicist, and leader. He cares about students’ success and is willing to spend plenty of time on students. I can ask for help on research, career, and life even though it was during late nights or weekends. He is very helpful in all these aspects, which is very important for international students like me. He has cultivated excellent students and postdocs who are also extremely helpful to me. His creativity, hard work, responsibility, and rigorousness set a great role model for me as a physicist, as a person, and as an advisor in the future.

I am also thankful to the current and previous graduate students of CCAPP and the physics department. I have got tremendous help and advice from Ranjan Laha, Shirley Li, and Kenny Ng before and after their graduation. I have also learned a lot from frequent chat and discussions with Benjamin Buckman, Chris Cappiello, Yang Cheng, Bin Guo, Liping

He, Paulo Montero, Bowen Shi, Wenjuan Zhang, and Guanying Zhu, and I enjoyed the time with them.

Next, I would like to thank the current and previous postdocs at OSU. I appreciate help from Tim Linden with various things especially my postdoc applications. I have enjoyed discussing particle physics with Juri Smirnov, from whom I have learned a lot. I have had a lot of fun talking about neutrinos, exploring pizza places in Columbus, and other things with Francesco Capozzi. I have also learned a lot from frequent chat and discussions with

Mauricio Bustamante, Junichiro Kawamura, Hong Zhang, Xilin Zhang, and Yinu Zhang.

iv I am also thankful to the professors. I am grateful to Annika Peter for working with us on solar gamma rays, serving on my committee, writing a letter for me, and giving me advice.

I am also grateful to Yuri Kovchegov who has taught me electrodynamics, particle physics, and quantum ﬁeld theory in his courses, plus discussing questions related to my research.

I thank Eric Braaten who taught me quantum ﬁeld theory, serving on my committee, and talking about physics with me. I also thank Stuart Raby who talked about physics and other things with me. I appreciated Jim Beatty for serving on my committee and Mary

Hall Reno of the University of Iowa who wrote a letter of reference for me.

The life in graduate school would also be harder without help from administrative people, who have given me all kinds of help in the department. So I would like to thank Jon Pelz,

Kris Dunlap, Phil Davids, Bryan Dunlap, Andrew Canale, and especially CCAPP Program

Coordinator Lisa Colarosa.

Last but not least, I would like to thank my parents for their unconditional care, love, support, patience, and everything. I also thank my friends outside physics. It is a lot of fun to be with them when I am not thinking about physics.

v Vita

2016 – 2020 ...... Graduate Research Associate, The Ohio State University 2015 – 2016 ...... University and Fowler Graduate Fellow, The Ohio State University 2015 ...... M.S. Astrophysics, Purple Mountain Ob- servatory, Chinese Academy of Sciences 2012 ...... B.S. Physics and B.E. Computer science, Guangxi University

Publications

“W -boson and Trident Production in TeV–PeV Neutrino Observatories” Bei Zhou, John F. Beacom Phys. Rev. D 101, 036010 (2020) [arXiv:1910.10720] (Ref. [1] in the bibliography)

“Neutrino-nucleus Cross Sections for W -boson and Trident Production” Bei Zhou, John F. Beacom Phys. Rev. D 101, 036011 (2020) [arXiv:1910.08090] (Ref. [2] in the bibliography)

“Constraints on Spin-Dependent Dark Matter Scattering with Long-Lived Mediators from TeV Observations of the Sun with HAWC” HAWC collaboration, plus J. F. Beacom, R. K. Leane, T. Linden, K. C. Y. Ng, A. H. G. Peter, B. Zhou (alphabetical) Phys. Rev. D 98, 123012 (2018) [arXiv:1808.05624] (Ref. [3] in the bibliography)

“First HAWC Observations of the Sun Constrain Steady TeV Gamma-Ray Emission” HAWC collaboration, plus J. F. Beacom, R. K. Leane, T. Linden, K. C. Y. Ng, A. H. G. Peter, B. Zhou (alphabetical) Phys. Rev. D 98, 123011 (2018) [arXiv:1808.05620] (Ref. [4] in the bibliography)

“Unexpected Dip in the Solar Gamma-Ray Spectrum” Qing-Wen Tang, Kenny C.Y. Ng, Tim Linden, Bei Zhou, John F. Beacom, Annika H.G. Peter Phys. Rev. D 98, 063019 (2018) [arXiv:1804.06846] (Ref. [5] in the bibliography)

vi “Evidence for a New Component of High-Energy Solar Gamma-Ray Production” Tim Linden, Bei Zhou, John F. Beacom, Annika H.G. Peter, Kenny C.Y. Ng, Qing-Wen Tang Phys. Rev. Lett. 121, 131103 (2018) (Editors’ Suggestion; Featured in Physics) [arXiv:1803.05436] (Ref. [6] in the bibliography)

“TeV Solar Gamma Rays From Cosmic-Ray Interactions” Bei Zhou, Kenny C. Y. Ng, John F. Beacom, Annika H. G. Peter Phys. Rev. D 96, 023015 (2017) [arXiv:1612.02420] (Ref. [7] in the bibliography)

“GeV excess in the Milky Way: Depending on Diﬀuse Galactic Gamma Ray Emission Template?” Bei Zhou, Yunfeng Liang, Xiaoyuan Huang, Xiang Li, Yizhong Fan, Lei Feng, Jin Chang Phys. Rev. D 91, 123010 (2015) [arXiv:1406.6948] (Ref. [8] in the bibliography)

“GRB 131231A: Implications of the GeV Emission” Bin Liu, Wei Chen, Yunfeng Liang, Bei Zhou, Haoning He, Pak-Hin Thomas Tam, Lang Shao, Zhiping Jin, Yizhong Fan, Daming Wei Astrophys. J. Lett. 787, L6 (2014) [arXiv:1401.7283] (Ref. [9] in the bibliography)

“Fast Radio Bursts as a Cosmic Probe?” Bei Zhou, Xiang Li, Tao Wang, Yizhong Fan, Daming Wei Phys, Rev. D, 89, 107303 (2014) [arXiv:1401.2927] (Ref. [10] in the bibliography)

“Model-dependent Estimate in the Connection Between Fast Radio Bursts and Ultra-High Energy Cosmic Rays” Xiang Li, Bei Zhou, Haoning He, Yizhong Fan, Daming Wei Astrophys. J. 797, 33 (2014) [arXiv:1312.5637] (Ref. [11] in the bibliography)

“High Energy Emission of GRB 130821A: Constraining the Density Proﬁle of the Circum- burst Medium as well as the Initial Lorentz Factor of the Outﬂow” Yunfeng Liang, Bei Zhou, Haoning He, Pak-Hin Thomas Tam, Yizhong Fan, Daming Wei Astrophys. J. 781, 74 (2014) [arXiv:1312.2662] (Ref. [12] in the bibliography)

“High Energy Emission of GRB 130427A: Evidence for Inverse Compton Radiation” Yizhong Fan, Pak-Hin Thomas Tam, Fuwen Zhang, Yunfeng Liang, Haoning He, Bei Zhou, Ruizhi Yang, Zhiping Jin, Daming Wei Astrophys. J. 776, 95 (2013) [arXiv:1305.1261] (Ref. [13] in the bibliography)

Fields of Study

Major Field: Physics Studies in theoretical particle and astroparticle physics: neutrinos, dark matter, astroparticle physics, gamma rays

vii Table of Contents

Page Abstract...... ii Dedication...... iii Acknowledgments...... iv Vita...... vi List of Figures ...... xi List of Tables ...... xviii

Chapters

1 Introduction1 1.1 What is exciting about neutrinos from the universe...... 1 1.2 Developing the Sun as a new high-energy neutrino source...... 3 1.2.1 Motivations...... 4 1.2.2 Review of prior work...... 5 1.2.3 My work ...... 8 1.3 Diﬀuse High-Energy Astrophysical Neutrinos ...... 10 1.3.1 Motivations...... 11 1.3.2 Review of prior work...... 12 1.3.3 My work ...... 18 1.4 Diﬀuse Supernova Neutrino Background...... 19 1.4.1 Motivations...... 20 1.4.2 Review of prior work...... 22 1.4.3 My work ...... 25

2 TeV Solar Gamma Rays From Cosmic-Ray Interactions 27 2.1 Introduction...... 27 2.2 Interplanetary and Solar Magnetic Fields ...... 31 2.3 Hadronic Gamma Rays ...... 33 2.3.1 Calculational framework...... 33 2.3.2 Calculation for the simpliﬁed case ...... 35 2.3.3 Calculation for the realistic case ...... 37 2.4 Leptonic Gamma Rays...... 41 2.5 Cosmic-Ray Electrons ...... 44 2.6 Conclusions and Outlook ...... 47

viii 3 Neutrino-nucleus cross sections for W -boson and trident production 50 3.1 Introduction...... 50 3.2 Review of W -boson and trident production ...... 52 3.2.1 Hadronic part...... 53 3.2.2 W -boson production...... 55 3.2.3 Trident production...... 58 3.3 Cross sections between neutrinos and real photons ...... 59 3.3.1 W -boson production...... 59 3.3.2 Trident production...... 62 3.4 Neutrino-nucleus cross sections: coherent and diﬀractive regimes ...... 64 3.4.1 Framework ...... 64 3.4.2 W -boson production...... 68 3.4.3 Trident production...... 71 3.5 Neutrino-nucleus cross sections: inelastic regime ...... 72 3.5.1 Framework ...... 72 3.5.2 W -boson production...... 73 3.5.3 Trident production...... 77 3.6 Total cross sections of W -boson and trident production, and ratio to CCDIS 79 3.7 Conclusions...... 81 3.8 Supplemental Material...... 84 3.8.1 W -boson production: amplitudes in the Standard Model ...... 84 3.8.2 Trident production: amplitudes in the Standard Model ...... 85 T/L 2 3.8.3 Trident production: kinematics and phase space for σνγ (ˆs, Q ) . . . 86 3.8.4 Trident production: coherent and diﬀractive cross sections for all channels...... 90

4 W -boson and trident production in TeV–PeV neutrino observatories 92 4.1 Introduction...... 92 4.2 W -boson production cross sections and implications ...... 95 4.2.1 Review of the total cross sections...... 95 4.2.2 New results for the diﬀerential cross sections ...... 96 4.2.3 Implication: Cross-section uncertainty...... 98 4.2.4 Implication: Attenuation in Earth ...... 101 4.3 Detectability ...... 102 4.3.1 Larger W -boson yields than Glashow resonance ...... 104 4.3.2 Review of detection in IceCube...... 104 4.3.3 Total shower detection spectrum...... 108 4.3.4 Unique signatures ...... 111 4.4 Conclusions...... 113 4.5 Supplemental Material...... 115 4.5.1 dσνA/dE` and dσνA/dEW ...... 115

5 First Detailed Calculation of Atmospheric-Neutrino Foregrounds for Super-Kamiokande Searches for the Diﬀuse Supernova Neutrino Back- ground 119 5.1 Introduction...... 120

ix 5.2 Framing the Problem...... 123 5.2.1 Predicted DSNB signals...... 123 5.2.2 Observed atmospheric neutrino backgrounds ...... 124 5.2.3 Atmospheric neutrinos...... 125 5.2.4 Neutrino interactions...... 126 5.2.5 Physics of detection in Super-K...... 128 5.3 Calculation Validation by Matching Super-K High-Energy Atmospheric Neu- trino Data...... 129 5.4 New Results on Super-K Low-Energy Atmospheric data: Invisible-Muon Component...... 134 5.4.1 Theoretical calculations ...... 135 5.4.2 Summary of predictions and uncertainties of the invisible-muon component...... 140 5.4.3 Comparison with data and implication for nuclear gamma rays . . . 143 5.5 New results on Super-K Low-Energy Atmospheric data: (νe +ν ¯e) CC component...... 143 5.6 Parent Neutrino Distributions...... 146 5.7 Conclusions...... 147

Bibliography 149

x List of Figures

Figure Page

1.1 Spectrum of neutrinos from astrophysical and other sources. The “Back- ground from old supernovae” corresponds to the diffuse supernova neutrino background in this dissertation. Figure taken from [14]...... 2 1.2 Dark matter annihilation at the center of the Sun. Left: standard scenario, in which standard model particles are produced directly and only neutrinos can get out. Right: long-lived dark mediators are produced and decay outside the Sun to standard model particles, including gamma rays and others that produce gamma rays. Figure taken from [15]...... 5 1.3 Integral intensity profiles above 500 MeV for the two components of solar gamma rays. Red points are the observed counts by F ermi-LAT. Dotted magenta line is the disk emission. It extends to larger than 0.26◦ due to the angular resolution of F ermi-LAT. Dashed green line is the halo emission (labeled as IC emission on the figure). Dash-dotted horizontal line is the background. Solid blue line is the sum of the background and the two components of the emission. The shaded areas around the lines show total error estimates. Figure taken from [16]...... 6 1.4 Status of the solar gamma rays after our series of work [3–7]. The dashed green line is the disk flux without magnetic effects [7], which also serves as a theoretical lower bound. The blue and red points show the Fermi- LAT measurements during and outside the solar minimum respectively [5,6]. There is a clear dip feature in each dataset. The nominal prediction from Ref. [17] is shown as the green band. The dashed blue line is a theoretical upper bound of disk flux [6]. The black upper limit is from HAWC 3 years of observation [4]. The dotted cyan line and purple solid line are the sensitivity of future LHAASO [18, 19] and 100000 m2 water Cherenkov detector (WCD) array in the Southern Hemisphere [20] respectively. Figure taken from [21]. 10 1.5 Three possible kinds of messengers to identify the sources of high-energy cosmic rays, including cosmic rays on their own, gamma rays, and neutrinos. Figure taken from Ref. [22]...... 12 1.6 Diagrams for deep-inelastic scattering. Left: CCDIS for neutrino, which is mediated by a W boson. Right: NCDIS, which is mediated by a Z boson. Figure taken from Ref. [23]...... 13

xi 1.7 Cross sections between neutrinos and 16O for CCDIS [24], NCDIS [24], and the Glashow resonance (¯ν e− W −, taking into account eight e → electrons) [25]. Figure is original to the thesis...... 15 1.8 Typical event signatures observed by IceCube. Left: An elongated track, mostly from νµ CCDIS and a small fraction from ντ CCDIS. Middle:A shower event, from NCDIS, νe CCDIS, or ντ CCDIS if the final-state τ lepton does not decay to a muon and no double-bang signature formed. Right: Double-bang, from ντ CCDIS when the τ lepton is above hundreds TeV. Each small colored spheres marks a photomultiplier tube that is triggered by Cherenkov photons, and the size of a sphere indicates the amount of photons registered. The colors represent the relative triggered time of the photomultiplier tubes, and the earlier, the redder, while the later, the bluer. The left and middle are from IceCube’s observation while the third is from simulation. Figure taken from Ref. [26]...... 16 1.9 IceCube’s 6 years of data for astrophysical neutrinos per neutrino flavor (black points with σ errorbar). The atmospheric neutrino components have been subtracted from a combined likelihood fit. The dashed blue line shows the best-fit conventional atmospheric neutrinos flux, and the dashed green line shows the best-fit upper limit on “prompt” neutrinos. The solid blue line with 1σ error band is the best fit using a single power-law spectrum. The pink line with 1σ error band shows the best fit using muon neutrinos only [27]. Figure taken from Ref. [28]...... 17 1.10 DSNBν ¯e spectrum for different neutrino emission spectra, as labeled. For each, two curves are plotted to include the uncertainties mainly due to the cosmic rate of core collapses. The shadings represent backgrounds, and their origins are labeled. Figure taken from Ref. [29]...... 21 1.11 Current status of DSNB searches in Super-K. Solid black lines are the DSNB event rate calculated by Ref. [29] based on an effectiveν ¯e temperature of 6 MeV. Two curves are plotted to include the uncertainties mainly due to the cosmic rate of core collapses. Shaded areas represent the backgrounds as labeled, details below. Colored lines represent atmospheric backgrounds in the current Super-K search window (> 16 MeV). Green lines are for the νµ andν ¯µ components, while red lines are for the νe andν ¯e components. Dashed lines are for current Super-K, while solid lines are for the Super-K-Gd phase. Figure is original to the thesis...... 24

2.1 Prospects for TeV solar gamma-ray observations, illustrated with the disk emission (details in Fig. 2.5). Points: observations with Fermi [16, 30], where the flux difference is due to time variation. Green band: the only theoretical prediction that includes magnetic effects [17]. Dashed lines: the estimated differential point-source sensitivity of HAWC [31] (scaled to one year) and LHAASO [18, 19]...... 29 2.2 Solar mass density as function of height above the photosphere (left axis, blue dashed), as well as the same for optical depth for inelastic proton-proton collisions (right axis, red solid)...... 36

xii 2.3 Solar-limb gamma-ray spectrum produced by hadronic cosmic rays. Red dotted line: semianalytic result for proton-proton interactions with 0 < τ < 0.3. Green dash-dotted line: GEANT4 results for the full range of τ; the gradual cutoff is because it cannot simulate proton interactions above 100 TeV. Blue dashed line: our empirical fit to the GEANT4 results, extrapolated to higher energies. Black solid line: our full prediction, including a correction factor for nuclei. The light grey shading approximately indicates the energies at which magnetic effects, neglected here, should be included...... 38 2.4 Normalized relative contributions of different τ values to the predicted gamma-ray flux, based on our GEANT4 simulation. We show the example of Eγ = 1 TeV; other energies give similar results...... 39 2.5 Gamma-ray spectrum of the Sun. Points: disk observations with Fermi [16, 30], where the flux difference is due to time variation. Green band: the predicted disk flux [17]. Dotted lines: the estimated differential point-source sensitivity of HAWC [31] (scaled to one year) and LHAASO [18, 19]. Our new prediction of the solar-disk signal due to cosmic-ray hadrons (from the limb) is shown by the green solid line. Our new prediction of the solar-halo signal due to inverse-Compton scattering of cosmic-ray electrons is shown by the black solid line for the nominal case and by the dashed lines for enhanced cases from Fig. 2.6...... 43 3 2.6 Diffuse flux (weighted with Ee ) of cosmic-ray electrons. Below about 5 TeV, there are measurements (points, as labeled [32–35]). Above about 70 TeV, there are limits (gray region, which combines many experiments [36, 37]). In between, the spectrum could be as large as the blue solid line, allowing enhanced contributions (pulsar or dark matter; details are in the text). HAWC should be able to immediately improve sensitivity down to 10−3 (hadronic ∼ rejection) of the proton spectrum (red dashed line)...... 45

3.1 Diagrams for (on-shell) W -boson production via photon exchange. A and A0 are the initial- and ﬁnal-state nuclei. (See Fig. 3.5 and Sec. 3.3.2 for the connection with trident production.) For antineutrinos, take the CP transformation of the elementary particles...... 52 3.2 Diagrams for trident production via photon exchange in the four-Fermi theory (see Fig. 3.5 for the full Standard Model). For antineutrinos, take the CP transformation of the elementary particles...... 54 3.3 Summary of cross sections for W -boson and trident production from previous work, with the two processes separated as labeled. To simplify the ﬁgure, for W -boson production, we show only ν e−W + on 16O (by Seckel [38] and e → Alikhanov [39, 40]), and for trident production, only the coherent regime (the dominant part) of ν ν e−µ+ on 40Ar (by Magill & Plestid [41], Ballett et µ → e al. [42], and Altmannshofer et al. [43]). Also shown, for comparison, is the cross section of charged-current deep inelastic scattering (CCDIS) [24]. . . 56

xiii 3.4 Our cross sections (actually σνγ(sνγ)/sνγ) for W -boson and trident production, between a neutrino and a real photon as a function of their CM energy. Red, green, and blue lines are νe-, νµ-, and ντ -induced channels, respectively. Solid lines are trident CC channels, and dashed lines are trident CC+NC channels (we label only the final states for both). Magenta dotted lines are trident NC channels, which depend on only the final-state charged leptons (we label both the initial and final states). The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5) of Fig. 3.5. The corresponding antineutrino cross sections (i.e., obtained by CP-transforming the processes shown) are the same. See text for details. . 60 3.5 Diagrams for trident production via photon exchange in the Standard Model, with the order, Tri– Tri, labeled in parentheses, and with the momenta M1 M5 labeled on the fourth diagram. The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5). For antineutrinos, take the CP transformation of the elementary particles. The first and second diagrams are connected to W -boson production (Fig. 3.1; also see Sec. 3.3.2 for details of the connection)...... 61 3.6 Our coherent and diffractive components of W -boson production cross sections, ν `− + W +, on 16O. Red, green, and blue lines are ν -, ν -, ` → e µ and ντ -induced channels, respectively. Solid: coherent (right bump) and diffractive (left bump) components. Dashed: longitudinal contribution to the coherent regime, which is small, even for the largest case (νe). The ντ line is not shown due to being below the bound of the y axis. Dotted: contribution from neutrons to the diffractive regime, which is small. The corresponding antineutrino cross sections are the same...... 66 3.7 Our coherent and diffractive components of W -boson production cross sections (red solid, from Fig. 3.6 but thicker), for the example of ν `−W + e → on 16O (the flavor with the largest cross section), comparing with our “EPA + no Pauli blocking” results (dashed) and previous calculations (dotted) by Seckel [38] and Alikhanov [39, 40]. Left and right bumps are coherent and diffractive components, respectively. Note our results are substantially smaller, which is important...... 67 3.8 Our coherent (solid lines) and diffractive (dashed lines) components of trident production cross sections on 16O. We show one typical channel for each category, i.e., CC, NC and CC+NC, to make the figure simple. For all the channels, see Appendix 3.8.4. Gray lines are for νe-induced W -boson production from Fig. 3.6, shown as a comparison. The corresponding antineutrino cross sections (i.e., obtained by CP-transforming the processes shown) are the same...... 70 3.9 Different components of our inelastic neutrino-nucleus cross sections for W - boson production. Only νe is shown to keep the figure simple. For νµ and ντ , the photon-initiated cross sections are smaller (Fig. 3.10), while the quark- initiated cross sections are basically the same. See text for details...... 74

xiv 3.10 Our inelastic neutrino-nucleus cross sections for W -boson production on 16O (solid lines), for all three flavors. Also shown are previous results from Alikhanov [40] and, for comparison, coherent and diffractive cross sections of νe from Fig. 3.6. The corresponding antineutrino cross sections are the same. 75 3.11 Our cross sections for trident production in the inelastic regime. Left: CC channels. Right: CC+NC and NC channels. Solid: the photon-initiated subprocess. Dashed: quark-initiated subprocess. The corresponding antineutrino cross sections are the same...... 76 3.12 Our total cross sections (actually σνA/Eν) for W -boson and trident production on 16O. The colors and line styles are same as in Fig. 3.4( red, green, and blue lines are νe-, νµ-, and ντ -induced channels, respectively; solid lines are CC channels, and dashed lines are CC+NC channels; magenta dotted lines are NC channels, which depend on only the final-state charged leptons). The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5) of Fig. 3.5. The corresponding antineutrino cross sections (i.e., obtained by CP-transforming the processes shown) are the same. See text for details...... 78 3.13 Ratios of the W -boson production cross sections to those of CCDIS ((ν + ν¯)/2) [24]. Solid lines are for water/ice targets, dotted line for iron targets, and dashed lines are for the Earth’s averaged composition. Color assignment is in the legend. Also shown is the νe (iron) case of Seckel [38], much larger than ours...... 79 3.14 Our elastic cross sections for all trident channels. We add the coherent and diffractive components together to simplify the figure. The colors and line styles are same as in Fig. 3.4( red, green, and blue lines are νe-, νµ-, and ντ -induced channels, respectively; solid lines are CC channels, and dashed lines are CC+NC channels; magenta dotted lines are NC channels, which depend on only the final-state charged leptons). The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5) of Fig. 3.5. Gray dashed lines are the coherent and diffractive cross sections for W -boson production from Fig. 3.6, shown as a comparison. For antineutrinos, which have the same corresponding cross sections, take the CP transformation of the channel labels. See text for details...... 90

4.1 Cross sections between neutrinos and 16O for W -boson production [2], compared to those for CCDIS [24], NCDIS [24], and the Glashow resonance (¯ν e− W −, taking into account eight electrons) [25]...... 94 e → 4.2 Left: Differential cross sections for W -boson production in terms of the energy of the charged lepton, shown for each neutrino flavor and two typ- 5 6 ical energies (Eν = 10 GeV and 10 GeV). The y axis is Edσ/dE = −1 (2.3) dσ/d log10 E, matching the log scale on the x axis, so that relative heights of the curves at different energies faithfully show relative contributions to the total cross section. Right: Same, in terms of the energy of the W boson...... 96 4.3 Average energy of the charged lepton (`) and W boson, divided by Eν, for each neutrino flavor...... 99

xv 4.4 Ratios of the W -boson production cross sections [2] to those of CCDIS ((ν + ν¯)/2) [24]. Solid lines are for water/ice targets, dotted for iron targets, and dashed for Earth’s average composition. Color assignments are noted in the legend. For comparison, we also show the νe (iron) result of Seckel [38], which is much larger than ours...... 100 −σC 4.5 Upper: Neutrino attenuation factor, e , for νe in Earth. Dashed lines (ADIS) are for CCDIS and NCDIS without W -boson production. Solid lines (ADIS+WBP) include W -boson production. For attenuation factors below 0.1, the event rate is too low to use, which we denote by using thin lines. Lower: The relative change in the attenuation factor due to W -boson production. 103 4.6 Relative W -boson yields due to W -boson production (νl + A l + W + 0 − − → A ) and the Glashow resonance (¯νe + e W ). We use dΦ/dEν = −2.9 −2 −1 −1 → (Eν/1 GeV) cm s GeV with unit normalization. The yield from W - boson production is 20 times that from the Glashow resonance, which can ' be seen by logarithmic integration of the peaks. The CCDIS (cyan, dashed) and NCDIS (magenta, dashed) cases are shown for comparison, though they do not produce on-shell W bosons...... 105 4.7 Left: Shower spectrum (Upper) and detection significance of W -boson production (Lower) for the conservative case as regards identifying W -boson production events. Right: Same, but for the optimistic case. The main difference between the two cases is the change with the CCDIS channel. The shaded region below 60 TeV is below the IceCube threshold for cleanly identifying astrophysical neutrinos. We use 0.5 km3 as the approximate fiducial volume of IceCube [44], and assume 10 years of IceCube data. For E > 60 TeV, where W -boson production contributes 6 shower events, dep ' the cumulative detection significance should be 1.0σ for the conservative ' case and 3.2σ for the optimistic case. See text for details...... 106

5.1 Super-K’s present DSNB searches, which start at Ee = 16 MeV [45]. Note the green and red lines are in units of per MeV instead of per 4 MeV, to match the DSNB lines. The DSNB signal band is based on a Fermi-Dirac spectrum with a 6-MeV effective (after neutrino mixing) temperature [29]. The backgrounds due to atmospheric neutrinos are based on our calculations (details in Fig. 5.5, and the text). Soon, added Gd will reduce these and other detector backgrounds...... 121 5.2 Our calculated fluxes of atmospheric neutrinos, without and with neutrino mixing, following Refs. [46–48]. Left: Results for νµ andν ¯µ. Right: Results for νe andν ¯e (note that the figure goes to lower energies; the notch is because we use different calculations above and below 0.1 GeV). Forν ¯ in both panels, we multiply by 0.5 for clarity...... 130 5.3 The GENIE cross sections for CC interactions with various interactions and targets as labeled. We only show the most important channels. Left: Results for νµ andν ¯µ. Right: Results for νe andν ¯e (note that the figure goes to lower energies). Below 1 GeV, CCQE dominates, as shown for one specific ∼ channel in the left panel...... 131

xvi 5.4 Our calculated results (lines, as labeled) for charged-lepton spectra induced in Super-K by atmospheric neutrinos, compared their measured data (points with statistical uncertainties only) [49]. Left: Muons, combining FC single- ring, multi-ring and PC data. Right: Electrons, single-ring FC data only, which dominates at lower energies, so at high energies (shaded region), some disagreement is expected. Overall, the agreement is well within the estimated theoretical uncertainties, which are dominated by the systematics (shown as the green arrow in each panel), at the level of a few tens of percent. . . . . 132 5.5 Our final calculation (last column of Table 5.1) of atmospheric neutrino backgrounds compared to Super-K-I data [45] (note the different axis scales compared to Fig. 5.1). Even though the data and curves are shown with 4-MeV steps, we have converted them to units of per 1 MeV to match Fig. 5.1. The two components (dashed) and their sum (solid) are shown. Overall, our final calculation — an absolute prediction, not a fit — agrees well with Super-K data...... 141 5.6 Distribution of parent atmospheric neutrinos relevant to the invisible muons in our final calculation (Sec. 5.4.1). Left: (ν +ν ¯ ) CC component ( 80% µ µ ' of total). Right: NC π+ component ( 20% of total)...... 144 ' 5.7 Distribution of parent atmospheric neutrinos relevant to the (νe +ν ¯e) CC component in our final calculation (Sec. 5.5). Left: Results for 16 MeV < Ee < 55 MeV. Right: Results for 55 MeV < Ee < 90 MeV. These comprise about 45% and 55%, respectively, of the events in 16 MeV < Ee < 90 MeV. The boundary of 55 MeV is roughly where the invisible-muon and electron components of the background cross in Fig. 5.5...... 145

xvii List of Tables

Table Page

1.1 List of current and future high-energy neutrino detectors...... 18

3.1 Summary of the features of previous calculations and of this work. “+” and “ ” means “considered” and “not considered” in the calculation respectively. − “Full SM” means using full Standard Model, instead of four-Fermi theory. 57

4.1 Diﬀerent ﬁnal state particles, signatures, corresponding fractions, and counts in IceCube. The counts are for greater than 60 TeV deposited energy and 10 years of IceCube observations (or 1 year for IceCube-Gen2). The numbers in the “Channel” column are the maximal ratios to the CCDIS cross section with water/ice. The numbers in “W decay” and “τ decay” columns are the branching ratios. For the “Final state” and “τ decay” columns, we omit the neutrinos; “h” means hadrons. The unique signatures are in boldface. The “/” divides the cases in which the charged lepton from the initial interaction is undetectable or detectable, which, to a good approximation, is half-half. The “Fractions” column shows the fraction of that row relative to the whole channel, which is the multiplication between the branching ratios of W and τ decay...... 110

5.1 Our predicted numbers of decay electrons (16 MeV < Ee < 90 MeV) from invisible muons, using GENIE’s default model set. The “Naive” calculation is defined in Sec. 5.4.1, and its improvement to the “Standard” calculation is defined in Sec. 5.4.1. We build on the latter by including the “Coulomb” corrections (Sec. 5.4.1) and also the “Threshold” corrections (Sec. 5.4.1). We show results for two sets of assumptions about nuclear gamma rays, favoring the second set. All calculations include signal efficiency and other detector effects. Numbers in boldface are bottom-line numbers...... 138 5.2 Same as Table 5.1, but using the GENIE’s EffSFTEM model set...... 138

xviii Chapter 1 Introduction

1.1 What is exciting about neutrinos from the universe

Neutrinos, the most mysterious particle in the standard model of particle physics, provide an indispensable tool to learn about the universe. Tremendous progress, including breakthroughs, has been made in the past decades and will likely be made in the next decade.

For example, the detection of solar neutrinos (Fig. 1.1) established the picture that nuclear fusion powers the Sun and other stars. Soon, the solar neutrinos from CNO and hep channels could be measured, and the CNO neutrinos would be the best probe of solar metallicity [23]. The detection of supernova neutrinos from SN1987A (Fig. 1.1), though only about 20 events, veriﬁed the basic picture of supernova explosion [50–52]. Now we are much more prepared for the next supernova burst in the Milky Way or its satellite galaxies [53]. For example, for a supernova burst with a distance of 10 kpc, we can detect

104 neutrinos in different flavors. ∼ Recently, the first detection of TeV–PeV astrophysical neutrinos by IceCube has opened a new window for astrophysics [44, 54]. Moreover, detecting those neutrinos is probably the only way to identify the sources of cosmic rays with energies above a few PeV, a hundred- year question of cosmic rays. The number of detected events keeps increasing rapidly so the sources of those neutrinos, and hence the sources of their above-PeV parent cosmic rays, may soon be identified. In fact, there are already candidate source detections in association with a blazar flare [55] and a tidal disruption event [56].

There is also great potential to detect new astrophysical sources of neutrinos. The

1 Figure 1.1: Spectrum of neutrinos from astrophysical and other sources. The “Background from old supernovae” corresponds to the diﬀuse supernova neutrino background in this dissertation. Figure taken from [14].

diﬀuse supernova neutrino background (DSNB, Fig. 1.1) will soon be detected by Super

Kamiokande (Super-K) in the new Super-K-Gd phase, thanks to adding Gadolinium into

Super-K water to reduce lots of backgrounds for DSNB searches [57–59]. The solar TeV–

PeV neutrinos could also be detected by IceCube, and theoretical work is still needed to understand the emission [60–64].

In reverse, astrophysical neutrinos provide an indispensable tool to study particle physics.

Same as above, tremendous progress, including breakthroughs, has been made in the past decades and can be made in the next decade. For example, the solar neutrinos and atmospheric neutrinos (Fig. 1.1) oﬀered the ﬁrst hint and evidence for neutrino mixing [65, 66].

Subsequent observations provide the most precise measurements of relevant parameters in the standard model [49, 67–69]. Recently, the cross sections of TeV–PeV neutrinos were

2 ﬁrst measured using data from IceCube [70, 71], and the precision will increase signiﬁcantly.

Neutrinos are the only particles in the standard model without a known mass origin, inspiring many physicists to study and to connect this with other big questions of particle physics, e.g., dark matter, matter-anti-matter asymmetry, and more [72–77]. Lots of such new physics have been tested using astrophysics neutrinos, thanks to the fact that they have a wide energy spectrum, directional information, and travel cosmic distances and through extremely high column densities (e.g., through Earth) [78–88].

The work presented in this dissertation focuses on three aspects of those mentioned above. First is calculating the TeV solar gamma-ray fluxes, as a step to understanding how cosmic rays interact with the Sun. This will eventually develop the Sun as a new high-energy laboratory, including as a neutrino source. Second is studying the high-energy neutrino interactions that are subdominant to the deep-inelastic scattering. It is high time to do this, due to our increasing data statistics of the TeV–PeV neutrinos. Third is studying the physics of the backgrounds for DSNB searches. This gives hints for reducing those background, which will significantly help the first detection and subsequent precision measurement of DSNB. The rest of this chapter is organized as follows. In Sec. 1.2, I present the motivation and prior work of developing the Sun as a new high-energy laboratory, including as a neutrino source. This motivates my work in Chapter2. In Sec. 1.3, I present the motivation and prior work of detecting high-energy neutrinos, motivating my work in

Chapters3 and4. In Sec. 1.4, I present the motivation and prior work of detecting the diﬀuse supernova neutrino background, motivating my work in Chapter5

1.2 Developing the Sun as a new high-energy neutrino source

One way to make progress in neutrino astrophysics is ﬁnding new astrophysical sources of neutrinos. There is great potential for the Sun being a source of high-energy neutrinos [60–

64]. These neutrinos are predicted from cosmic rays interacting with the solar atmosphere.

However, the predicted flux has large uncertainties, especially at lower energies, due to our insufficient knowledge of solar magnetic fields and interplanetary magnetic fields, which is

3 hard to study directly.

Solar gamma rays, as also products from cosmic-ray interactions, provide an excellent tool to study how those magnetic ﬁelds aﬀect cosmic rays. Moreover, the gamma-ray spectrum provides a benchmark for the neutrino spectrum because both are from decay of pions from cosmic ray interactions. This is similar to how people predict the neutrino output of other astrophysical objects, like supernovae and active galactic nuclei, using measurements of their electromagnetic output.

Throughout this dissertation, we consider gamma rays only from the quiescent, steady- state Sun. We do not consider gamma rays from solar ﬂares [89–91], as their origin is diﬀerent in nature than those from the quiet Sun.

1.2.1 Motivations

There are more motivations to study solar gamma rays than developing the Sun as source of high-energy neutrinos.

The most obvious motivation, as mentioned above, is studying the solar magnetic fields and interplanetary magnetic fields on their own and how they affect the propagation of cosmic rays. Energy and location-dependent studies can be done with solar gamma rays, which are hard with directly measuring cosmic rays. There are small cosmic-ray detectors launched into the inner solar system but they have limited ability compared to those on or near the Earth [92, 93]. For example, those detectors have limited reach and only measure the total number of cosmic rays without energy information.

Moreover, studying solar gamma rays is essential to developing the Sun as a laboratory to test new physics [3, 15, 21, 60–64, 94–96]. Searching for new physics usually suﬀers from backgrounds from astrophysical processes, which should be understood well. For example, it is well-motivated that dark matter particles could be trapped by the Sun’s gravitational potential and accumulate at the center of the Sun. Those dark matter particles could annihilate to long-lived metastable mediators, travel outside the Sun, and eventually decay to gamma rays or standard model particles that produce gamma rays [3, 15]. This popular scenario and, as a comparison, the standard scenario are shown in Fig. 1.2.

4 γ (...... ) γ (extinguished) ν (less attenuated) γ (extinguished) γ ν γ γ › ν ν γ, ν ν (attenuated) (unattenuated)

Short-lived mediators Long-lived mediators

Figure 1.2: Dark matter annihilation at the center of the Sun. Left: standard scenario, in which standard model particles are produced directly and only neutrinos can get out. Right: long-lived dark mediators are produced and decay outside the Sun to standard model particles, including gamma rays and others that produce gamma rays. Figure taken from [15].

The last motivation is more from curiosity. The Sun has been studied in depth in multi- wavelength electromagnetic signals but not in gamma rays. We would like to complete this multi-wavelength in-depth study. Moreover, the Sun is a typical G-type main-sequence star is the universe. Therefore, the above beneﬁts also apply to studying other stars in the universe.

1.2.2 Review of prior work

There are two components of solar gamma rays, with diﬀerent emission regions, spectra, morphologies, and physical origins. They are solar disk gamma rays and solar halo gamma rays, shown in Fig. 1.3.

The solar disk gamma rays are emitted from the disk of the Sun, with an angular radius of about 0.26◦. It is point-like in terms of the resolution of current gamma-ray telescopes except at above 10 GeV energies by F ermi-LAT. They are likely produced by hadronic cosmic rays interacting with the matter in the Sun’s photosphere and convection zone. Because the physics of gamma-ray production and solar matter density are well

5 Figure 1.3: Integral intensity proﬁles above 500 MeV for the two components of solar gamma rays. Red points are the observed counts by F ermi-LAT. Dotted magenta line is the disk emission. It extends to larger than 0.26◦ due to the angular resolution of F ermi-LAT. Dashed green line is the halo emission (labeled as IC emission on the ﬁgure). Dash-dotted horizontal line is the background. Solid blue line is the sum of the background and the two components of the emission. The shaded areas around the lines show total error estimates. Figure taken from [16].

understood, observing the disk emission provides us with the information of interplanetary

and solar magnetic ﬁelds.

Those gamma rays were first mentioned in Ref. [97] and the flux was first estimated

(based on measurements of terrestrial emission) in Ref. [98]. Ref. [99] suggested them to be

detectable by the EGRET experiment on board the Compton Gamma-Ray Observatory. In

1991, Seckel, Stanev, and Gaisser made the ﬁrst, and so far the only, detailed theoretical

study of this emission [17]. In that work, they modeled the eﬀects of interplanetary magnetic

ﬁelds on cosmic rays as a purely spatial diﬀusion and time-independent problem. The solar

6 magnetic fields are modeled as vertical and homogeneously distributed magnetic flux tubes that root deeply in the convection zone of the Sun and stretch out into corona. Cosmic rays, after being modulated by the interplanetary magnetic fields, which reduces the fluxes, reach the magnetic flux tubes homogeneously and isotropically, get trapped, and propagate along a helical line into the Sun. Due to the mirror effect [100], most of those charged particles are turned around and leave the Sun at some point. When cosmic rays are close to and in the Sun, they interact with the matter of the Sun and develop hadronic showers.

The gamma rays that are produced when the parent particles are leaving the Sun could reach us as the disk emission. Therefore, Ref. [17] predicts a homogenous, isotropic, and steady-state disk emission.

For the observation of disk gamma rays, in 1995, EGRET set an upper limit of 2 × 10−7 cm−2 s−1 on the ﬂux using four years of observation data [101]. The ﬁrst detection of the disk emission was in 2018, after a reanalysis of the EGRET data [102].

The launch of F ermi-LAT gamma-ray telescope heralded a new era of studying solar gamma rays. In 2011, the F ermi-LAT collaboration analyzed their ﬁrst 1.5 years of data

(0.1–10 GeV) from the direction of the Sun [16]. Their results show that the disk ﬂux is about 7 times higher, and the spectrum is harder, than the prediction by Ref. [17]. In 2015,

Ref. [30] analyzed 6 years of public Fermi-LAT data and detected the disk emission up to

30 GeV. It also confirmed the high flux and hard spectrum found by Ref. [16]. Surprisingly, the work found an anticorrelation between solar activity and gamma-ray flux of 1–10 GeV, with unknown reasons.

Solar halo gamma rays were ﬁrst studied by Refs. [102–104]. They are produced by cosmic-ray electrons (electrons means both electrons and positrons in this section) inverse-

Compton scattering the photons ( 1 eV) emitted by the Sun. The solar photons are emit- ∼ ted from the surface of the Sun (in the photosphere) and the density follows 1/r2, where ∼ r is the distance from the Sun’s center. Therefore, the halo emission can extend to tens of degree, and the further from the Sun, the lower the intensity. Because inverse-Compton scattering and solar photon density are well understood, observing the halo emission provides us with the information of interplanetary magnetic ﬁelds.

7 On the observation side, the halo emission was also ﬁrst detected by Ref. [102] using

EGRET data and then by F ermi-LAT [16]. Their results are consistent with theoretical predictions of Refs. [102–104].

1.2.3 My work

In this section, I give an overview of the motivations and key ideas for my work on solar gamma rays.

As mentioned before, the long-term goal is to develop the Sun as a high-energy laboratory, including to comprehensively understand solar disk and halo gamma-ray emission, to understand how the solar and interplanetary magnetic ﬁelds aﬀect the parent cosmic rays, to develop the Sun as a high-energy neutrino source, and to develop the Sun as a lab to test new physics. However, this journey is far from easy, as the solar gamma rays on their own have many problems unsolved, i.e., unexplained observational features including the high

ﬂux and hard spectrum of disk gamma rays mentioned above. (In fact, our further work found more unexplained features, details below.)

A key step is that, because the problems above are highly likely due to insufficient knowledge of solar and interplanetary magnetic fields, we need to know what the emission is like if there were no magnetic effects. At GeV energies where the magnetic effects are strong, comparing the predicted gamma-ray spectrum between without and with magnetic effects and with observation gives us precious information on how magnetic fields play a role. The other key step is understanding the emission at TeV energies. As HAWC [31] and

LHAASO [18, 19] will have data soon, connecting theory and observation will provide us with very valuable information. The TeV flux is not much affected by neglecting magnetic effects, because the parent cosmic rays are too energetic. Therefore, the above two steps can be combined, i.e., calculating solar gamma-ray spectrum at both GeV and TeV energies without magnetic effects, though we need to deal with the disk and halo components separately.

My work in Ref. [7], presented in Chapter2, aims to do the above calculations, which have never been done before. The disk gamma rays without magnetic eﬀects (hence no

8 mirror eﬀects) are only from the solar limb, where cosmic rays graze the Sun and propagate through a column density that is large enough for interactions while small enough for gamma rays to escape. We calculate the ﬂux in two ways, one is semi-analytic and the other is using full simulation by Geant4 [105, 106]. This assures the robustness of the results and helps understand relevant physics. We use the most updated hadronic cosmic-ray data [107–109].

Moreover, our calculation includes the effects from multiple scattering, absorption, cascade processes, and heavier nuclei than proton. The results serve as an important theoretical lower bound on the disk emission at both GeV and TeV energies. Comparing to the case with magnetic effects and observation, it shows that magnetic effects enhance the disk flux by a factor of 10 at GeV energies. HAWC and LHAASO will be able to see if the ∼ enhancement continues at TeV energies and if there are contributions due to new physics.

For the halo gamma rays, we use the StellarICs code [110, 111]. The ﬂux is lower than the sensitivity of HAWC and LHAASO, but they can at least set constraints at these energies, where there are no measurements. Moreover, we show that the halo gamma-ray emission can be used to probe the density of TeV cosmic-ray electrons in the inner solar system, which is not well measured.

The above work [7] led by me is part of a larger program of work to develop the Sun as a new high-energy laboratory. It also sets a theoretical foundation for our further studies on the disk emission which I participated in and are published in Refs. [3–6] (not included in this dissertation). In Refs. [5,6], we analyzed 9 years of F ermi-LAT data and ' 1. Detected signiﬁcant disk gamma-ray emission from 1 GeV up to 200 GeV (Fig. 1.4);

−2.2 2. Further conﬁrmed the hard spectrum (Eγ , Fig. 1.4) compared to the prediction −2.7 (Eγ ) and the anticorrelation between gamma-ray ﬂux and solar activity. 3. Observed a spectral dip between about 30 and 50 GeV in an otherwise power-law

spectrum (Fig. 1.4), which is surprising and still unexplained;

4. Found that the gamma-ray emission is dominated by the equatorial region of the

disk during solar minimum and by the polar region outside solar minimum, which

strongly suggests that the disk emission is produced by two separate mechanisms.

In Ref. [4], we worked with HAWC collaboration and set strong constraints on the ﬂux

9 10 10− HAWC 95% C.L. (2014–2017) Fermi-LAT (2014–2017) Fermi-LAT (Solar Min.) ] 1 − s 2

− 11 10− CR Upper Bound SSG1991 LHAASO 1yr

12 10− 100,000 m

CR Lower Bound 2 WCD Array 1 yr Energy Flux [ TeV cm

13 10− 2 1 1 10− 10− 1 10 Energy [ TeV ]

Figure 1.4: Status of the solar gamma rays after our series of work [3–7]. The dashed green line is the disk flux without magnetic effects [7], which also serves as a theoretical lower bound. The blue and red points show the Fermi-LAT measurements during and outside the solar minimum respectively [5,6]. There is a clear dip feature in each dataset. The nominal prediction from Ref. [17] is shown as the green band. The dashed blue line is a theoretical upper bound of disk flux [6]. The black upper limit is from HAWC 3 years of observation [4]. The dotted cyan line and purple solid line are the sensitivity of future LHAASO [18, 19] and 100000 m2 water Cherenkov detector (WCD) array in the Southern Hemisphere [20] respectively. Figure taken from [21].

of TeV solar gamma rays (Fig. 1.4). In Ref. [3], also with HAWC collaboration, we set strong constraints on the scenario of a long-lived mediator from dark matter annihilation at the center of the Sun.

1.3 Diﬀuse High-Energy Astrophysical Neutrinos

The recent detection of extraterrestrial high-energy neutrinos by IceCube is one of the breakthroughs in neutrino physics and astrophysics [44, 54]. It also opened a new era for neutrino astronomy. Since then, we have made tremendous progress on both the theoretical and experimental sides. These include studying high-energy neutrino scatterings, poten- 10 tial production of those neutrinos by various astrophysical objects, detection techniques, source identiﬁcation, and more. Those studies are still underway and will lead to huge accomplishments.

1.3.1 Motivations

High-energy astrophysical neutrinos provide the best tool to identify the sources of high- energy cosmic rays. Although we have studied cosmic rays for more than 100 years, with tens of experiments, we are still not sure of their origins. This is probably the biggest open question for cosmic rays. For cosmic rays below a few PeV, there is more and more evidence that they are from supernova remnants in the Milky Way. But for higher energy ones, we know very little about their sources except that they should be extragalactic.

Fig. 1.5 illustrates the reasons. We could just use cosmic rays on their own. However, since they are charged, the magnetic ﬁelds in the host galaxies, intergalactic medium, and the Milky Way will deﬂect them. Therefore, they lose their directional information. Another possible messenger is their interaction products like gamma rays and neutrinos which have directional information because they are neutral. For gamma rays, however, since they are high energy and extragalactic, they are absorbed by cosmic microwave background

(CMB) photons or extragalactic background light, via γ +γ e+ +e−. Therefore, CMB/EBL → neutrinos are the best tool to identify the sources of high-energy cosmic rays, as they have

directional information and are not absorbed by matter or radiation during propagation.

Moreover, as a result of above, neutrinos are also the best messenger to study particle

acceleration mechanisms of those sources, which is also a big and long-term question.

Lastly, high-energy neutrinos have a unique power to be used to study neutrino proper-

ties and test new physics [78–88, 112–114]. This is thanks to them having very high energy

and directional information, and traveling cosmic distances and through extremely high

column densities.

11 Figure 1.5: Three possible kinds of messengers to identify the sources of high-energy cosmic rays, including cosmic rays on their own, gamma rays, and neutrinos. Figure taken from Ref. [22].

1.3.2 Review of prior work

To study high-energy neutrinos, we need ﬁrst detect them, which is a big topic. Neutrinos can not be detected directly, but instead through the secondary particles from neutrino interactions with detector material (for IceCube, it is ice).

The cross section of a neutrino with the nucleus is much larger than that with the elec-

12 Figure 1.6: Diagrams for deep-inelastic scattering. Left: CCDIS for neutrino, which is mediated by a W boson. Right: NCDIS, which is mediated by a Z boson. Figure taken from Ref. [23].

trons, which is suppressed by m /m , where m and m are masses of electron and proton, ∼ e p e p respectively (an exception is the Glashow resonance,ν ¯ + e− W − any ﬁnal states, e → → details below).

Deep inelastic scattering (DIS) dominates the neutrino-nucleus scattering, and has been a focus of study. It includes charged-current DIS (CCDIS), which can be written as

ν + A `− + X for neutrinos and (1.1) ` → ν¯ + A `+ + X for antineutrinos , (1.2) ` → and neutral current DIS (NCDIS), which can be written as

(−) (−) ν +A ν +X (1.3) ` → ` where A is the nucleus, ` the charged lepton, and X represents any hadronic ﬁnal states.

Fig. 1.6 shows the diagrams for the CCDIS, mediated by a W boson, and NCDIS, mediated by a Z boson. What the weak boson couples to is actually a quark in the nucleon in a nucleus, because of the large momentum transfer.

Fig. 1.7 shows the cross sections for CCDIS and NCDIS. The cross sections of NCDIS are about 40% of CCDIS. Neutrinos have larger cross sections than antineutrinos. This is

13 because neutrinos couple to d andu ¯ quarks while antineutrinos couple to u and d¯ quarks,

and these quarks have diﬀerent densities in a nucleon. At higher energies, neutrinos and

antineutrinos have the same cross sections because they dominantly couple to sea quarks.

The cross sections increase following E1 below 103 GeV (not shown) then the increasing ∼ ν ∼ slows down and finally saturates at E0.3 above 106 GeV. Also shown on the figure is the ν ∼ Glashow resonance,ν ¯ + e− W − any final states. It peaks at E = m2 /2m e → → ν W e ' 6.3 106 GeV. The width of the peak is due to the decay width of the W boson. The × Glashow resonance peaks at a very large cross section, but this is only for one flavor and

happens at a very narrow range of neutrino energy.

IceCube is an ice-Cherenkov detector and it detects Cherenkov photons with its pho-

tomultiplier tubes. When a charged particle travels at a speed faster than the light speed

in the medium, it produces Cherenkov radiation. Among the interaction products of DIS,

IceCube and other high-energy neutrino detectors can detect the ﬁnal-state charged leptons

and hadrons. The volume of IceCube is about 1 km 1 km 1 km. × × Fig. 1.8 shows three possible event signatures that can be formed from the topology of

triggered photomultiplier tubes in the IceCube detector. The ﬁrst is an elongated track

from a high-energy muon. The second is a shower from a high-energy electron that ini-

tiates electromagnetic cascade or from high-energy hadron(s) that initiate hadronic cas-

cade. Currently, IceCube could not distinguish electromagnetic and hadronic showers, but

this is possible in IceCube-Gen2 [115] (next generation of IceCube) with the echo tech-

nique [116]. For the τ lepton, before decaying to muon, electron, or hadrons, it could

leave a track. However, due to its short lifetime (t 2.9 10−13 s), it only travels ' × γc t 8.7 10−5 (E /1.78 GeV) m. Taking into account the minimal spacing between ' × × τ IceCube’s photomultiplier tubes, 10 m, this means that if the τ lepton is above hundreds ∼ TeV, the ντ CCDIS could form a double-bang signature, the third in the ﬁgure, with the ﬁrst bang from the primary hadrons and the second from the displaced decay of the τ lep-

ton. IceCube has already identiﬁed two double-bang candidates [117]. (See Chapter4 for

more about IceCube and its detection technique.)

From those signatures, including their Cherenkov-light yields, spatial and temporal pro-

14 30 10 Glashow resonance ( e) 31 10 ]

2 m c

[

10 CCDIS NCDIS 33 10

34 10 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 1.7: Cross sections between neutrinos and 16O for CCDIS [24], NCDIS [24], and the Glashow resonance (¯ν e− W −, taking into account eight electrons) [25]. Figure is e → original to the thesis.

files, we can reconstruct a neutrino’s energy, arrival direction, and flavor information, because the track and double-bang signatures are mainly from νµ and ντ CCDIS respectively (τ lepton also decays to a muon, with a branching ratio of only 17%). The angular resolution of track events is less than 1 degree while 5–20 degrees for shower events. The difference is due to the topologies of the two signatures described above. The energy resolution of track events is only about a factor of 2, because tracks could leave the detector before depositing all the energies, and for shower events, it is about 15% as most of the showers deposit all of

15 Figure 1.8: Typical event signatures observed by IceCube. Left: An elongated track, mostly from νµ CCDIS and a small fraction from ντ CCDIS. Middle: A shower event, from NCDIS, νe CCDIS, or ντ CCDIS if the ﬁnal-state τ lepton does not decay to a muon and no double-bang signature formed. Right: Double-bang, from ντ CCDIS when the τ lepton is above hundreds TeV. Each small colored spheres marks a photomultiplier tube that is triggered by Cherenkov photons, and the size of a sphere indicates the amount of photons registered. The colors represent the relative triggered time of the photomultiplier tubes, and the earlier, the redder, while the later, the bluer. The left and middle are from IceCube’s observation while the third is from simulation. Figure taken from Ref. [26].

their energies in the detector. IceCube could not distinguish neutrinos from antineutrinos, as it could not distinguish the charge signs. With the above abilities, IceCube can measure astrophysical neutrinos in terms of ﬂuxes, time variations, directions, spectra, ﬂavor compositions, and more.

Fig. 1.9 shows IceCube’s data for astrophysical neutrinos per ﬂavor. The data have already had the atmospheric neutrino components subtracted from a combined likelihood

ﬁt, which is consistent with the theoretical prediction. The astrophysical neutrino spectrum can be ﬁtted using a single power law,

+0.2 −(2.89−0.19) dΦν +1.46 Ev −18 −1 −2 −1 −1 = 6.45−0.46 10 GeV cm s sr , (1.4) dEν 100 TeV ·

16 Figure 1.9: IceCube’s 6 years of data for astrophysical neutrinos per neutrino flavor (black points with σ errorbar). The atmospheric neutrino components have been subtracted from a combined likelihood fit. The dashed blue line shows the best-fit conventional atmospheric neutrinos flux, and the dashed green line shows the best-fit upper limit on “prompt” neutrinos. The solid blue line with 1σ error band is the best fit using a single power-law spectrum. The pink line with 1σ error band shows the best fit using muon neutrinos only [27]. Figure taken from Ref. [28].

for all three ﬂavors.

As shown in the ﬁgure, the spectrum is not well measured, due to a lack of data. The

flavor compositions are also poorly constrained. Moreover, the origin of those astrophysical neutrinos are not known yet, except that they should mostly be extragalactic. There are candidate source detections in association with a blazar flare [55], but the significance is less than 4σ and we are not sure if they could contribute all those astrophysical neutrinos.

Therefore, we need more data, hence more and larger detectors, for high-energy neutrinos.

17 1.3.3 My work

In this section, I give an overview of the motivations and key ideas for my work on high- energy neutrinos.

Table 1.1: List of current and future high-energy neutrino detectors.

Detector Size Detector material Lcoation Status IceCube 1 km3 Ice South Antarctica Running for 8 years ' KM3NeT 1 km3 Water Mediterranean Sea Upcoming Baikai-GVD 1 km3 Water Lake Baikal Upcoming IceCube-Gen2 10 km3 Ice South Antarctica Proposed

Table 1.1 shows current and future high-energy neutrino detectors, which shows that the future of high-energy neutrinos is promising. We are about to have more and much larger detectors, hence much more data. The rapidly increasing amount of data demand a deeper understanding of neutrino interactions, which plays a key role in neutrino detections, in order to have more precise measurements. The deep inelastic scattering has been a focus for theorists. After tremendous eﬀort, the claimed theoretical precision (from the parton- distribution functions) has reached 2% for TeV–PeV neutrinos. ' However, there are subdominant interactions that are larger than the claimed uncertainties of DIS, and even large enough that they should be included when analyzing data.

The largest subdominant processes are actually through photon coupling to the nucleus, as a neutrino can split into a charged lepton and a charged boson that can couple to a photon. Speciﬁcally, they are W -boson production and trident production. Since CCDIS is the dominant process for detecting high-energy neutrinos, we take it for comparison. Previous calculations of trident production show that the cross sections are only 10−5 of CCDIS ∼ at GeV energies, but they increase with energy much faster than CCDIS [41–43, 118, 119].

So it is very important to know how large they are at TeV–PeV energies. On the other hand, previous calculations of W -boson production shows that the cross sections are as large as 10% of CCDIS [38–40]. However, those calculations have shortcomings so a new

18 and complete calculation is eagerly needed. Then, if our calculations show the cross sections are large, we should study how they aﬀect the detection of high-energy neutrinos.

In my first work on this topic [2], presented in Chapter3, I did the first calculation of trident production at TeV–PeV neutrino energies and the first complete calculation of

W -boson production. We divided every process into three regimes, in which the whole nucleus, a nucleon, and a quark is involved, respectively. The cross section of a process is the sum of the cross sections in all three regimes. For the W -boson production, we found that the previously used equivalent photon approximation is not valid, so we use a complete formalism. More improvements in the calculations compared to earlier work are in

Chapter3. As a result, our cross sections of W -boson production are about half of before, which is very important in terms of the future statistics of high-energy neutrinos.

For the trident production, previous calculations only used four-Fermi theory, which signiﬁcantly simpliﬁes the calculation, because they were interested in lower energies. How- ever, for TeV–PeV energies, the four-Fermi theory does not work so we need to use the full standard model. Our calculation shows the cross sections are as large as 0.5% of CCDIS.

More interestingly, we found that trident and W -boson production are related processes for

Eν & 7 TeV. In my second work on this topic [1], presented in Chapter4, we studied how W -boson production aﬀects the detection of TeV–PeV neutrinos, because in Ref. [2] we found their cross sections are large enough for detecting. We found that W -boson production should be taken into account by IceCube, KM3NeT, and Baikal-GVD, and must be taken into account by IceCube-Gen2.

1.4 Diﬀuse Supernova Neutrino Background

Another new astrophysical source of neutrinos that we can detect is the diﬀuse supernova neutrino background (DSNB) [120].

A supernova burst can produce an enormous amount of neutrinos. For a galactic supernova, we expect to detect thousands of neutrinos, but the burst rate is only a few per

19 century. We definitely do not want to wait that long. Instead, we integrate all the supernova in the whole universe. Though the detectable neutrinos per burst is tiny, the burst rate is huge. This gives us a guaranteed steady neutrino flux, the DSNB. Detecting DSNB is vital to neutrino physics and astrophysics. In this section, I first give a more detailed introduction of DSNB and, as a result, the importance of detecting it (Sec. 1.4.1). Then,

I review the prior work on searching for DSNB by Super-Kamiokande (Sec. 1.4.2). Last, I brieﬂy introduce my work (Sec. 1.4.3).

1.4.1 Motivations

DSNB is a predicted isotropic ﬂux of neutrinos and antineutrinos in all ﬂavors produced from all the massive-star core collapses throughout the universe. Here the core collapses include both supernovae and those that directly produce back holes without detectable electromagnetic emission, i.e., dark collapses [121]. Such dark collapses can not be studied by electromagnetic signals at all.

The DSNB ﬂux can be calculated by a cosmological line-of-sight integral [29, 120],

Z ∞ dφ c dt (Eν) = [(1 + z) ϕ (Ev(1 + z))] R(z) dz , (1.5) dEν 0 dz where z is redshift and c is the light speed. dt/dz is the diﬀerential distance and dt/dz −1 = | | | | 31/2 H0(1+z) ΩΛ + Ωm(1 + z) , where H0,Ωm, and ΩΛ are cosmological parameters, which are relatively very well known.

There are two major ingredients in Eq. (1.5). First is ϕ (Ev(1 + z)), the neutrino emission per core collapse, which is the primary observable, because it can only be determined by neutrino experiments. It is also the most uncertain part, as we only observed about 20 neutrinos from SN1987A [50–52] and the explosion mechanism and neutrino mixings are poorly understood theoretically.

The other ingredient is R(z), the cosmic core-collapse rate, which is relatively well known, as it can be measured by electromagnetic observations. The dominant contribution to DSNB is from z 1, as the corresponding core-collapse rate is 10 times the present ∼ ∼ rate.

20 10

8 MeV 6 MeV ] 4 MeV -1 SN1987A 1 MeV

-1 Invisible µ s -2 [cm ν 0.1 / dE φ

d Invisible µ + Reactor νe Spallation 0.01 0 10 20 30 40 Eν [MeV]

Figure 1.10: DSNBν ¯e spectrum for diﬀerent neutrino emission spectra, as labeled. For each, two curves are plotted to include the uncertainties mainly due to the cosmic rate of core collapses. The shadings represent backgrounds, and their origins are labeled. Figure taken from Ref. [29].

Figure 1.10 shows the spectrum of DSNBν ¯e, which is the most promising ﬂavor to be detected, calculated by Ref. [29]. The calculation was based on a widely-used Fermi-Dirac

spectrum with zero chemical potential [122, 123], which characterizes the time-integrated

ν¯e spectrum per supernova very well. Different styles of lines are for different effective

ν¯e temperature, which is a key parameter of Fermi-Dirac spectrum. Also shown is the calculation based on the measured SN1987A spectrum. Overall, the DSNB spans from

E 0 to 40 MeV, and the energy-integrated flux per flavor is 10 cm−2 s−1. Moreover, the ν ' ∼ uncertainty in the DSNB spectrum is large. The cosmic core-collapse rate mainly affects the

21 ﬂux normalization, which has less than a factor of 2 uncertainty, while the neutrino emission per core collapse mainly aﬀects the spectral shape, which has a much larger uncertainty.

Therefore, measuring DSNB will provide us with precious information about the neutrino emission spectrum per collapse, which can not be learned from electromagnetic signals.

The data will be extremely valuable to learn about explosion mechanisms of supernovae and dark collapses, neutrino mixings (in extremely dense environment), and more. On the other hand, we could also learn this from detecting neutrinos from a galactic supernova burst.

However, the burst rate is only a few per century and we can only measure one burst.

On the contrary, DSNB is a steady and guaranteed ﬂux, so we can measure it any time and it probes the average neutrino emission per core collapse, especially including the dark collapses.

Moreover, DSNB also provides precious information that a galactic supernova burst cannot, for example, the cosmic rate of core collapses, dark collapses, and star formation [124].

We could also study particle physics using DSNB, such as electric dipole, magnetic moment, new light dark-sector particles, and many kinds of BSM physics in the neutrino sector [125–127]. Moreover, if the guaranteed DSNB ﬂux is not found, there must be surprising new physics or astrophysics [120].

1.4.2 Review of prior work

To detect DSNB, the most promising detector is Super-Kamiokande (Super-K) [128]. Super-

K is a water-Cherenkov detector. It contains 50 ktons of ultrapure water in a cylinder.

The inner detector has 32 ktons water and the ﬁducial volume has 22.5 kton. Super-K has 104 photomultiplier tubes which can detect Cherenkov radiation from relativistic ' charged particles. Super-K cannot distinguish charge signs. (See Chapter5 for more about

Super-K.)

Super-K can detect DSNB mainly through the inverse-beta decay process,

ν¯ + p n + e+ , (1.6) e → due to its large cross section [129, 130]. Interactions with nuclei are suppressed by binding

22 eﬀects and interactions with electrons are suppressed by their small mass. The total cross section is σ(E ) 10−43 cm2 (E 1.3 MeV)2, with the outgoing electron (we use electron ν ' ν¯e − to mean both electrons and positrons in this section, unless we specify otherwise) carrying

E E 1.3 MeV and emitted near-isotropically. Super-K has a trigger threshold of e ' ν¯e − 5 MeV, so the relativistic positron is easy to trigger the detector. The neutron is hard to ∼ detect [131], because the neutrons are neutral and mostly captured by hydrogen, producing an only 2.2 MeV gamma ray [132]. As predicted, Super-K has collected 100 DSNB events ∼ after more than 20 years of running. However, these events are hidden in huge backgrounds, so that Super-K has not identiﬁed them yet. (In principle, anything that produces electrons or positrons in DSNB energy range could be a background for DSNB.)

Fig. 1.11 summarizes the current status of searching for DSNB in Super-K [45, 131,

133]. The search window is for above 16 MeV electron energy, Ee, due to the backgrounds from reactor neutrinos, spallation events, and NC elastic events (from atmospheric neutrino interactions) which are hard to reduce. Above 16 MeV, the situation is better but there are still large backgrounds caused by atmospheric neutrinos. The dominant component

(dashed green line) is due to atmospheric νµ andν ¯µ, which interact with Super-K water mainly through,

ν (¯ν ) + H/O X + µ− µ+ , (1.7) µ µ → where X represents any ﬁnal state nuclei or hadrons. If the muon is below 55 MeV kinetic energy, it is invisible in Super-K because it will not produce Cherenkov radiation. Then the muon decays to a visible relativistic electron that mimics a DSNB event. (This corresponds to the “invisible µ” background in Fig. 1.10.) Another component comes from atmospheric

νe andν ¯e (dashed red line) interacting through

ν (¯ν ) + H/O X + e− e+ , (1.8) e e → and those electrons are backgrounds for DSNB events. Therefore, in the ﬁgure, we can see that those backgrounds leave only a small window for DSNB searches.

The Super-K collaboration has been working hard on searching for DSNB, by separat-

23 NC elastic

] 1.2

1 & V

e Spallation 1.0 M

1 Reactor e r y

1 0.8 ) n o t

k 0.6 ( ) 5 . 2 2 (

[ 0.4 E d / DSNB flux N 0.2 d e( e)

0.0 0 10 16 20 30 40 Ee [ MeV ]

Figure 1.11: Current status of DSNB searches in Super-K. Solid black lines are the DSNB event rate calculated by Ref. [29] based on an eﬀectiveν ¯e temperature of 6 MeV. Two curves are plotted to include the uncertainties mainly due to the cosmic rate of core collapses. Shaded areas represent the backgrounds as labeled, details below. Colored lines represent atmospheric backgrounds in the current Super-K search window (> 16 MeV). Green lines are for the νµ andν ¯µ components, while red lines are for the νe andν ¯e components. Dashed lines are for current Super-K, while solid lines are for the Super-K-Gd phase. Figure is original to the thesis.

ing them from backgrounds [45, 131, 133]. Though DSNB events are not identiﬁed yet,

tremendous progress has been made. The ﬁrst search was published in 2003 [133], in which

they used 1496 days of Super-K-I data with a search window of Ee > 18 MeV. Their upper

−2 −1 limit on DSNBν ¯e ﬂux, 1.2 cm s for Eν > 19.3 MeV, was already in tension with DSNB model with an 8 MeV eﬀectiveν ¯e spectrum [29]. The second and much more sophisticated search was published in 2012 [45], in which 2853 days of data from Super-K-I to Super-K-III

24 were used. Moreover, the searching window was pushed down to 16 MeV, signal eﬃciency improved especially at lower energies, and much more sophisticated cuts to the backgrounds

−2 −1 were performed. Those gave an upper limit on DSNBν ¯e ﬂux to be 2.8–3.1 cm s for

Eν > 17.3 MeV. In 2008, Super-K was upgraded from Super-K-III to Super-K-IV, which could detect the coincident neutrons from the inverse-beta decay (Eq. (1.6)). In 2015, a search using only Super-K-IV data (960 days) was published [131]. In the search window of 12 MeV < Ee < 31.3 MeV, the neutron detection eﬃciency is only about 17.7%. The upper limit was higher than that of Ref. [45], due to less data, high backgrounds at lower energies, low neutron detection eﬃciency, etc.

Right now, Super-K is being upgraded to Super-K-Gd with the addition of dissolved gadolinium as 0.2% (by mass) gadolinium sulfate, Gd2(SO4)3 [57–59]. Adding gadolinium signiﬁcantly improves Super-K’s ability to detect neutrons. Due to the huge neutron- capture cross section of gadolinium, in the Super-K-Gd phase, more than 90% of the neutrons are captured on gadolinium, and the energy release is about 8 MeV in a few gamma rays [132], which are much easier to detect. This neutron-tagging technique signiﬁcantly improves DSNB detectability [57]. This is because each DSNB event has exactly one neutron (Eq. (1.6)), but for atmospheric neutrino backgrounds, more than half of them have zero or multiple neutrons, and those can easily be reduced. This is shown as the solid green and dashed lines in Fig. 1.11. This enlarges the DSNB detection window, though still small.

1.4.3 My work

In this section, I give an overview of the motivations and key ideas for my work on DSNB detection.

According to above, the key next step for DSNB is to further enlarge the detection window of DSNB, which is helpful to both the ﬁrst detection and the subsequent precision measurement of DSNB. Therefore, we need to further reduce the backgrounds. One way forward is to further reduce the spallation backgrounds between about 10 and 16 MeV [134–

136]. Another way forward is to further reduce the atmospheric neutrino backgrounds at higher energies (Fig. 1.11), which is my work. On the other hand, the atmospheric neutrino

25 events are interesting as a signal, which provides important tests of neutrino-interaction modeling and neutrino mixings [137].

Chapter5 presents my thorough study of the underlying physics of atmospheric neutrino backgrounds for DSNB detection. This is crucial to both seeking ideas to reduce those backgrounds and using them as a signal (e.g., Ref. [137]). In this study, I devel- oped a detailed framework to calculate those backgrounds (both the νµ/ν¯µ component and the νe/ν¯e component) in Super-K for the ﬁrst time. The framework combined all the underlying physics pieces, including atmospheric neutrino ﬂuxes, neutrino mixings, neutrino interactions (mainly using GENIE [138–140], plus our own corrections for Coulomb distortion), propagation of secondaries in water (using FLUKA [141, 142]), interpretations of

Super-K analysis cuts, and detector physics. Using the framework, I successfully reproduced those atmospheric neutrino backgrounds. From this process, I have found several ideas that will signiﬁcantly help reduce the backgrounds further. The most promising one stems from the neutrons from atmospheric neutrinos backgrounds (tens MeV) being usually much more energetic than those from DSNB interactions (. 2 MeV). Neutrons with higher energies propagate further in water than those with lower energies, and the distances can be measured in the Super-K-Gd phase, so that the atmospheric neutrino backgrounds can be reduced further. Our study in Chapter5 also sets a solid theoretical foundation for reducing atmospheric neutrino backgrounds for other searches in Super-K like proton decay [143–145], neutron-antineutron oscillation [146], dark matter [147–149], and more.

26 Chapter 2 TeV Solar Gamma Rays From Cosmic-Ray Interactions

The Sun is a bright source of GeV gamma rays, due to cosmic rays interacting with solar matter and photons. Key aspects of the underlying processes remain mysterious. The emission in the TeV range, for which there are neither observational nor theoretical studies, could provide crucial clues. The new experiments HAWC (running) and LHAASO (planned) can look at the Sun with unprecedented sensitivity. In this paper, we predict the very high-energy (up to 1000 TeV) gamma-ray flux from the solar disk and halo, due to cosmic-ray hadrons and electrons (e+ + e−), respectively. We neglect solar magnetic effects, which is valid at TeV energies; at lower energies, this gives a theoretical lower bound on the disk flux and a theoretical upper bound on the halo flux. We show that the solar-halo gamma-ray flux allows the first test of the 5–70 TeV cosmic- ∼ ray electron spectrum. Further, we show that HAWC can immediately make an even stronger test with nondirectional observations of cosmic-ray electrons. Together, these gamma-ray and electron studies will provide new insights about the local density of cosmic rays and their interactions with the Sun and its magnetic environment. These studies will also be an important input to tests of new physics, including dark matter.

The contents of this chapter were published in [7].

2.1 Introduction

The Sun is a passive detector for cosmic rays in the inner Solar System, where direct measurements are limited. It shines in gamma rays from its disk and from a diﬀuse halo [16, 17, 30, 102–104]. Disk emission is expected due to cosmic-ray hadrons interacting with solar matter, which produces pions and other secondaries of which the decays

27 and interactions lead to gamma rays. Halo emission is expected due to cosmic-ray electrons

(e+ + e−) interacting with solar photons via inverse-Compton scattering. There are no other important astrophysical mechanisms for steady solar gamma-ray production; solar-

ﬂare gamma rays are episodic, and are observed up to only a few GeV [89, 150–152].

Gamma-ray observations thus open the possibility of detailed cosmic-ray measurements near the Sun. The hadronic and leptonic components can be distinguished because the disk and halo emission can be separated by direction. Further, the energy spectra of the cosmic rays can be inferred from the gamma-ray spectra, which can be measured over a wide energy range. This would give a signiﬁcant advance compared to typical satellite detectors in the inner Solar System, which only measure the energy-integrated all-particle ﬂux (e.g.,

Refs [153, 154]), and are thus dominated by low-energy particles. Further, gamma-ray data can trace the full solar cycle, testing how solar modulation of cosmic rays depends on energy and position [155, 156].

Figure 2.1 shows that the prospects for measuring TeV solar gamma rays are promising.

The solar-disk ﬂuxes measured in the GeV range with Fermi data [16, 30] are high, and naive extrapolation suggests that HAWC and LHAASO may detect gamma rays in the

TeV range. Further, the GeV observations are significantly higher than the theoretical prediction of Seckel et al. [17], who proposed a compelling mechanism by which the solar- disk gamma-ray flux could be enhanced by magnetic effects. Evidently, even this expected enhancement is not enough, which increases the need for new observations to reveal the underlying physical processes. Even if HAWC and LHAASO only set limits on the TeV gamma-ray flux, that would be important.

Our goal here is to provide a theoretical foundation to quantitatively assess the TeV detection prospects. At low energies, cosmic rays are affected by magnetic modulation in the inner Solar System, as well as by magnetic fields in the solar atmosphere, all of which are complicated [17]. At high energies, where magnetic effects can be neglected, the calculations are relatively straightforward but have not been done before this paper.

The energy separating the two regimes is not known. We estimate that neglecting magnetic eﬀects is appropriate for TeV–PeV gamma rays and show that it leads to useful benchmarks

28 10-7 Fermi2011 ]

1 Ng2016 − s

2 -8 − 10 m c V e G

[ SSG1991 prediction γ HAWC 1 yr

E -9 d 10 / F d 2 γ E LHAASO 1 yr

10-10 -1 1 4 5 10 100 10 102 103 10 10

Eγ [ GeV ]

Figure 2.1: Prospects for TeV solar gamma-ray observations, illustrated with the disk emission (details in Fig. 2.5). Points: observations with Fermi [16, 30], where the flux difference is due to time variation. Green band: the only theoretical prediction that includes magnetic effects [17]. Dashed lines: the estimated differential point-source sensitivity of HAWC [31] (scaled to one year) and LHAASO [18, 19].

for GeV–TeV gamma rays. In future work, we will treat magnetic eﬀects in detail. For a broader context on our program of work on understanding the gamma-ray emission of the

Sun — aimed toward eventual new measurements of cosmic rays, among other goals — see

Ref. [30].

We now provide more information about gamma-ray observations and prospects. Over

29 the 0.1 GeV–TeV range, the Sun has been well observed. Following the upper limits given by EGRET [157] and the ﬁrst detection using EGRET archival data [102], more detailed measurements were reported in Ref. [16] by the Fermi Collaboration, based on 1.5 years of data. Over the range 0.1–10 GeV, they separately measured the disk and halo ﬂuxes,

ﬁnding spectra E−2, plus a hint of time variation in the disk ﬂux. In Ref. [30], where ∼ we used six years of Fermi data and a newer version of the data processing (Pass 7 vs Pass

6), we detected the disk flux up to 100 GeV, finding that its spectrum falls more steeply than E−2. We also made the first robust detection of time variation, showing that the disk flux decreased by a factor of 2.5 from solar minimum to maximum. While the solar- halo gamma-ray flux is reasonably well understood, our results deepen the mysteries of the solar-disk gamma-ray flux. New observations are needed, especially at higher energies, which will critically test emissions models. However, this is difficult with Fermi due to the low gamma-ray flux.

In the TeV–PeV range, the only ground-based gamma-ray experiments that can observe the Sun are those that directly detect shower particles. (For air-Cherenkov detectors, based on detecting optical photons, the Sun is too bright.) The HAWC experiment began full operations in 2015, and is now reporting ﬁrst results. The LHAASO experiment, under construction, is expected to begin operations in 2020. These experiments will greatly improve upon the energy range and ﬂux sensitivity of their predecessors, e.g., Milagro [158],

ARGO-YBJ [159], and Tibet AS-gamma [160]. Those and other experiments have observed the “Sun shadow,” a deﬁcit of shower particles caused by the solar disk blocking cosmic-ray hadrons [161, 162], but none have detected an excess gamma-ray emission from the Sun.

The shadow is displaced by 1◦ from the Sun’s position due to magnetic deﬂections of ∼ cosmic rays en route to Earth, but the gamma-ray excess will be centered on the Sun.

HAWC and LHAASO observations in the TeV range, combined with Fermi observations in the GeV range, will provide a long lever arm to test models of solar gamma-ray emission.

This paper makes steps toward a comprehensive understanding of solar gamma rays. In

Sec. 2.2, we discuss the eﬀects of magnetic ﬁelds and justify why we can neglect them here.

The next three sections are ordered by the directionality of the signals. In Sec. 2.3, we detail

30 our calculation of the hadronic gamma-ray emission from the limb of the solar disk. This calculation has not been done before. We also estimate the ﬂux of other secondary products

(electrons, positrons, and neutrons), discussing if they are signiﬁcant background for the gamma rays. In Sec. 2.4, we detail our calculation of the leptonic gamma-ray emission from the solar halo. We extend earlier calculations to higher energies and are the ﬁrst to include allowed new contributions to the electron spectrum. In Sec. 2.5, we discuss the all-sky signal of directly detected cosmic-ray electrons. Our points about these prospects are new and exciting. In Sec. 2.6, we present our conclusions and the outlook for further work.

2.2 Interplanetary and Solar Magnetic Fields

The flux of cosmic rays near the Sun is altered by magnetic effects. Throughout the So- lar System, there are magnetic disturbances sourced by the Sun and carried by the solar wind [163, 164]. These form an interplanetary magnetic field (IMF) that repels galactic cosmic rays (“solar modulation”) [155, 156]; the effects and their uncertainties increase at low energies and at small distances from the Sun. In addition, near the Sun, within approximately 0.1 AU, there are solar magnetic fields (SMF) that are quite strong, especially ∼ in the photosphere and corona [165]. Because the SMF are complex and not completely measured, their effects may be varied and are quite uncertain.

In this paper, we focus on gamma-ray signals in the energy regime where magnetic effects can be neglected. When this is appropriate for the solar-disk signal, it will be even more so for the solar-halo signal, for which cosmic rays interact farther from the Sun. We begin by discussing SMF effects on the solar-disk signal, as these turn out to be dominant over IMF effects.

SMF effects enhance gamma-ray production from the solar disk. A likely physical mechanism was proposed in Ref. [17], although the authors’ predictions still fall far below observations [16, 30]. The enhancement is due to the mirror effect of solar magnetic flux tubes on charged hadronic cosmic rays, which can reverse the directions of cosmic rays before they interact, thus producing outgoing gamma rays that are not absorbed by the Sun. At

31 high enough energies, this mirroring becomes ineffective, and the enhancement ends. To estimate the critical energy Ec for this transition, where magnetic-field effects on cosmic rays can be neglected, we compute the Larmor radius, L, using the typical SMF strength

10 near the Sun, B 1 G, and the solar radius, R 7 10 cm [163, 164], ﬁnding ∼ ' × 4 L B Ec 10 GeV . (2.1) ∼ R 1 G

A similar value is obtained for a single ﬂux tube, for which the magnetic ﬁeld strength can be

103 times larger and the distance scale 103 times smaller [163, 164]. (Ref. [17] estimated ∼ ∼ E to be between 3 102 GeV and 2 104 GeV, so our choice is conservative.) Because c ' × ' × E 0.1E for typical hadronic interactions, SMF eﬀects should therefore be negligible γ ∼ p for gamma-ray energies above about 1 TeV. However, SMF models are uncertain, and it is important to test them with new data.

IMF effects reduce gamma-ray fluxes. Near Earth, IMF effects on the cosmic-ray spectrum are well described by the widely used force-field approximation [16, 166, 167] and detailed simulations [168, 169], which are informed by extensive measurements. For cosmic rays in the inner Solar System, both modeling and data are sparse. A key clue is that the

MESSENGER probe to Mercury found only . 10% modulation of the cosmic-ray spectrum above 0.125 GeV near solar distances around 0.4 AU [153, 154]. Using a force-field model with appropriate parameters to be consistent with these data (potentials . 400 MV), we find that IMF effects can be neglected for cosmic rays with energies above 100 GeV (and thus gamma rays above 10 GeV), even near the solar surface. However, IMF models are also uncertain, heightening the need for new data.

At energies where magnetic effects can be neglected, the solar-disk signal should thus be wholly due to the limb contribution. This emission is caused by cosmic rays that graze the Sun, encountering a column density that is large enough for them to interact but small enough for their gamma rays to escape. Because this signal can be calculated with minimal uncertainty, a gamma-ray measurement consistent with its flux prediction would confirm that magnetic effects are negligible. In principle, this could also be checked by the angular distribution of the signal, where the Sun would appear as a bright ring with

32 a dark center, although planned TeV–PeV experiments may not have adequate angular resolution [18, 19, 31, 170]. Finally, tests could also be made by the time variation, as there should be none.

At lower energies, where magnetic effects are important, several distinctive signatures of the solar-disk signal should emerge. The flux should be larger, as SMF effects that enhance the gamma-ray flux dominate over IMF effects that decrease it [17]. That is, our solar-disk prediction neglecting magnetic effects provides a theoretical lower bound on

the disk ﬂux, which is especially interesting at GeV–TeV energies. The angular variation

of the signal should tend toward illumination of the full disk. And there should be time

variations that reveal the nature of the dominant magnetic eﬀects. IMF eﬀects decrease

gamma-ray production near solar maximum, due to cosmic-ray modulation [16, 30]. Perhaps

surprisingly, SMF eﬀects must act in the same sense, as the IMF eﬀects are too small to

explain the observed time variation [30].

For the solar-halo signal, IMF eﬀects dominate over SMF eﬀects [16], except perhaps

very near the Sun. The comparison of disk and halo signals will thus help disentangle IMF

and SMF eﬀects. It also means that neglecting magnetic eﬀects provides a theoretical upper

bound on the halo ﬂux.

2.3 Hadronic Gamma Rays

2.3.1 Calculational framework

In the direction of the solar disk, the dominant source of gamma rays is the interactions of

hadronic cosmic rays with matter in the solar atmosphere [16, 17, 30]. Of these interactions,

the most important are inelastic proton-proton collisions that produce neutral pions, which

promptly decay to gamma rays. (In Sec. 2.4, we calculate gamma-ray production by leptonic

cosmic rays, including near the direction of the solar disk, although the interactions occur

well away from the solar surface.)

Here we calculate the gamma-ray emission from the solar limb — the small fraction of

the Sun encountered by cosmic rays that just graze its surface on trajectories toward Earth.

33 We use the straight-line approximation, where gamma rays maintain the direction of their parent hadrons, appropriate because the particle energies are so high. We ignore emission from the disk because we neglect magnetic eﬀects that can reverse the directions of cosmic rays before they interact [17] and because the contributions of back-scattered pions are tiny [171]. As the ingredients of the calculation are reasonably well known, the predicted limb emission is robust and, as noted, sets a theoretical lower bound on the solar-disk ﬂux.

We calculate the total ﬂux from the limb, integrating over its solid angle. Here we assume that it cannot be resolved, as single-shower angular resolution of HAWC and LHAASO near

1 TeV is comparable to the 0.5◦-diameter of the Sun [18, 19, 31, 170]. The solid angle of the limb is tiny, 10−3 of that of the solar disk. If the limb could be resolved, it would appear ∼ as a thin, bright ring, with the intensity (ﬂux per solid angle) enhanced by 103 over the ∼ intensity averaged over the solar disk. The angular resolutions of HAWC and LHAASO

improve at higher energies, which may allow partial resolution of the limb, especially with

stricter cuts to select events with the best angular resolution. In the long term, hardware

upgrades to improve this should be considered.

We begin in Sec. 2.3.2 by discussing gamma-ray production in a simpliﬁed case —

proton-proton production of neutral pions in the thin-target limit — which can be handled

semianalytically, following Ref. [172]. Then, in Sec. 2.3.3, we include the eﬀects of multiple

scattering and absorption, cascade processes, and nuclear composition through a simulation

using GEANT4 [105, 106]. In ths simpliﬁed case, the ﬂux is

dF Z Z Z dI dN (Eγ) = dΩ ds np(~s) dEp (Ep) σinel(Ep) (Ep,Eγ), (2.2) dEγ dEp dEγ where np is the number density of target protons at the line-of-sight coordinate ~s, dI/dEp is

the cosmic-ray proton intensity, σinel is the inelastic proton-proton scattering cross section, and dN/dEγ is the spectrum of gamma rays per interaction. The length of the chord

1/2 4 through the solar atmosphere is ∆s (8R h ) 2.6 10 km, where the 8 comes from ∼ 0 ∼ × 5 geometry, R 7 10 km is the radius of the Sun, and h 120 km is the scale height ' × 0 ' of the solar matter density in the photosphere. In the realistic case, the most important interactions occur at proton optical depths τ 2, so this simpliﬁed case is not adequate for ∼ 34 our full results, although it does introduce the framework well.

2.3.2 Calculation for the simpliﬁed case

Figure 2.2 shows the solar mass density ρ from Refs. [173, 174]. Above the photosphere, the density declines exponentially, following the Boltzmann distribution of gravitational potential energy in the nearly isothermal atmosphere. Figure 2.2 also shows the proton optical depth τ as a function of height above the photosphere. The cross section for inelastic proton-proton collisions changes only modestly with energy and is 30 70 mb for proton ' − energies 1 107 GeV [108]. In the optically thin limit, gamma-ray production is dominated − by the decay of neutral pions, which, at the low densities considered here, always decay in

ﬂight before interacting. Kelner, Aharonian, and Bugayov [172] have extensively studied the yields of secondaries in proton-proton collisions, in which their results are based on a

ﬁt to data and to particle-interaction simulations. The yield of gamma rays has a broad energy spectrum, but the most important gamma ray typically has E 0.1E . The shape γ ∼ p of τ(h) closely follows that of ρ(h), due to the exponential dependence, with the conversion

factor σ s/m 5 107 cm3 g−1. ' inel p ' × The cosmic-ray ﬂux can be taken to be that at Earth, as we neglect magnetic eﬀects.

(Technically, the ﬂux at Earth includes some modulation eﬀects, but these are negligible

at such high energies.) Up to 1 TeV, we use the precisely measured proton spectrum from

the Alpha Magnetic Spectrometer (AMS-02) [107]. At higher energies, it is suﬃcient to

extrapolate this using dI/dE 1 (E/GeV)−2.7 cm−2 s−1 sr−1 GeV−1 [108]. ' Figure 2.3 shows the resulting gamma-ray spectrum for the case where we integrated over 0 < τ < 0.3, up to roughly the largest value for which an optically thin calculation is appropriate (the probability for a proton to interact twice is then . 10%, so the gamma-ray spectrum scales linearly with τ). We checked the results of our semianalytic calculation by a Monte Carlo simulation with the particle-interaction code GEANT4 [105, 106], for which the results matched to within . 10%. This shows that eﬀects beyond those in Ref. [172], e.g., particle cascades in the medium, are unimportant.

Lastly, compared to the Sun, the gamma-ray ﬂux from the limb of the Earth’s atmo-

35 Figure 2.2: Solar mass density as function of height above the photosphere (left axis, blue dashed), as well as the same for optical depth for inelastic proton-proton collisions (right axis, red solid).

sphere has been measured by Fermi up to nearly 1 TeV and compared to simulations,

finding good agreement with predictions, which demonstrates the robustness of theoretical calculations [175, 176]. In principle, in the thin-target limit, the limb flux from the Sun could simply be expressed in terms of the limb flux from Earth, nullifying several potential uncertainties, such as the energy spectrum, composition, and cross section.

36 2.3.3 Calculation for the realistic case

To include proton-proton interactions in the optically thick case, we use GEANT4 [105, 106].

This allows protons to interact several times, and takes into account their particle and energy

losses from all processes. It also includes gamma-ray production by cascade processes, such

as bremsstrahlung by electrons. The number density of target photons is 104 times ∼ smaller than that of solar matter so energy losses and gamma-ray production by inverse-

Compton processes can be neglected [177]. The density is low enough that charged pions

below 1 PeV will typically decay in ﬂight before interacting. Neutrons and muons may

escape, and the neutrons may survive to Earth without decay.

Figure 2.4 shows the range of τ values that contribute most to gamma-ray production,

based on our GEANT4 simulation. The y axis is weighted to properly compare diﬀerent

logarithmic ranges of τ. The peak is near τ 2, where 90% of cosmic-ray protons will ∼ ∼ interact at least once. To the left of the peak, the linear decline is due to reduced optical

depth. To the right, the exponential decline is due to proton cooling and especially gamma-

ray absorption, which happen to have similar interaction lengths (for pion production and

electron-positron pair production, respectively). In combination, about 90% of the total

ﬂux arises in the range τ 0.1 10. ∼ − Using this range of τ values, we use Fig. 2.2 to determine the corresponding range of

heights above the photosphere and corresponding mass densities, ﬁnding h 60 600 km ∼ − and ρ 10−7 10−9 g cm−3. This leads to important insights about the physical conditions ∼ − in which interactions occur. In this range, the solar properties are reasonably well known

and are stable in time. The conditions for the production of solar atmospheric gamma rays

are quite diﬀerent from those for Earth atmospheric neutrinos [178]; for the latter, ρ 10−4 ∼ g cm−3 at an altitude of 10 km, and the distance scales are short but the proton optical

depth is high (τ 20). Lastly, this information will be useful for assessing interactions in ∼ the presence of magnetic eﬀects, which we will consider in future work.

With GEANT4, we can simulate proton interactions only up to a laboratory energy of

100 TeV, which leads to a gradual cutoﬀ of the gamma-ray spectrum near 10 TeV. To

37 Figure 2.3: Solar-limb gamma-ray spectrum produced by hadronic cosmic rays. Red dotted line: semianalytic result for proton-proton interactions with 0 < τ < 0.3. Green dash-dotted line: GEANT4 results for the full range of τ; the gradual cutoff is because it cannot simulate proton interactions above 100 TeV. Blue dashed line: our empirical fit to the GEANT4 results, extrapolated to higher energies. Black solid line: our full prediction, including a correction factor for nuclei. The light grey shading approximately indicates the energies at which magnetic effects, neglected here, should be included.

extend our results to higher energies, we develop an empirical ﬁt to the GEANT4 results at

lower energies. We modify our semianalytic approach, Eq. (2.2), by including a correction

factor, e−ατ(Ep,~s), that only becomes important in the optically thick regime. For the free

parameter α, we ﬁnd that 0.65 gives a good match to the GEANT4 results. This is shown in

38 Fig. 2.3.

Figure 2.4: Normalized relative contributions of diﬀerent τ values to the predicted gamma- ray ﬂux, based on our GEANT4 simulation. We show the example of Eγ = 1 TeV; other energies give similar results.

Finally, we consider the eﬀect of nuclei in the cosmic rays and in the solar atmosphere.

Besides protons, the only important constituent is helium, which has a 10% relative ' number abundance in both the beam and target [108, 173, 174]. We use the cosmic-ray helium data from Ref. [109] up to 1 TeV. Above that, we use a power law and extrapolate 39 up to 10 PeV with spectral index 2.7, which roughly describes the data compilation in ∼ Ref. [108]. Following Ref. [179], we calculate the gamma-ray flux enhancement factor due to cosmic-ray helium. We find that the gamma-ray flux is increased by an overall factor

1.8, with a small energy dependence due to a slightly diﬀerent spectral shape between the ' proton and helium. We also consider the case in which the helium spectrum may be harder than 2.7 at high energies [180–182]. If we use a spectral index of 2.58 [181] for extrapolation, our result changes by less than 20% near 10 TeV. Thus, we can safely ignore the spectral hardening.

Figure 2.3 shows our full prediction for the gamma-ray spectrum from the solar limb.

The gamma-ray spectrum closely follows the cosmic-ray proton spectrum [172]. This is because the pions and gamma rays typically carry ﬁxed fractions of the parent proton energy, the cross sections and multiplicities for (high-energy) pion production and gamma- ray absorption have only mild energy dependence, and the pions decay before interacting.

(For the same reasons, Earth atmospheric neutrinos at sub-TeV energies also follow the cosmic-ray spectrum [178].) In Fig. 2.3, the gamma-ray ﬂux has normalization 2 10−9 ' × sr times the proton intensity (ﬂux per solid angle). This factor can be roughly reproduced

using ∆Ω τ (0.1)1.7, where ∆Ω 10−7 sr is the relevant solid angle of the limb, τ 1 × × ∼ ∼ is a typical value, and the last factor comes from assuming that each proton produces 1 ∼ gamma-ray at E 0.1E . γ ∼ p The hadronic-interaction processes discussed here also produce neutrinos, electrons (in-

cluding positrons), and neutrons [17]. The neutrino ﬂux [183–186] is an important back-

ground for dark matter searches with neutrino telescopes [187–189], and constitutes a sen-

sitivity ﬂoor [60–62]. The other species could be useful messengers to study cosmic-ray

interactions with the Sun, using detectors such as Fermi [190], AMS-02 [191], CALET [192],

and DAMPE [193]. A dedicated study of their detectability, is beyond the scope of this

paper, and will be considered elsewhere. Here we brieﬂy comment on their relevance to

gamma-ray observations.

Electrons can be eﬀectively separated from gamma rays in space-borne detectors. How-

ever, this separation is diﬃcult for ground-based experiments, as both particles induce

40 electromagnetic showers in the atmosphere. In principle, the inclusion of electrons en- hances the detectability of the Sun for ground-based experiments. The flux of the electrons can be estimated similarly to that of gamma rays, described above, also by first ignoring magnetic-field effects. The electron flux is found to 2 times lower than that of the gamma ∼ rays, due to receiving a smaller fraction of the pion energy. Further, the detection of these secondary electrons with ground-based experiments is more complicated than gamma rays, as the effects of solar, interplanetary, and Earth magnetic fields all need to be taken into account, demonstrated by cosmic-ray shadow studies [161, 162]. The deflections and diffusion they cause will reduce the electron flux per solid angle. Therefore, for the current study, we neglect the addition of the electron flux to the total electromagnetic signal observable by ground-based experiments.

Neutrons, the most important secondary hadrons, travel without being affected by the magnetic fields. The Sun is therefore a point source of neutrons, and could in principle be detectable by ground-based experiments. Compared to gamma-ray production in pionic processes, secondary neutrons carry a smaller fraction of the primary energy. However, spallation of helium is efficient at producing secondary neutrons. Combining these two factors, the limb neutron flux is comparable to that of the gamma rays (also the disk flux [17]). In practice, it is difficult for these neutrons to be confused with gamma rays by ground-based experiments, due to the excellent hadron rejection factor, 10−3. The detection in the ∼ hadron channel is also likely to be difficult due to the much higher background, compared

to that of gamma rays and electrons. A more careful treatment of hadrons, in particular at

lower energies, is the subject of a separate paper (Zhou et al., in prep.).

2.4 Leptonic Gamma Rays

In directions away from the solar disk, there is a solar halo of gamma-ray emission, of

which the dominant source is the interactions of cosmic-ray electrons (e+ + e−) with solar

photons [16, 102–104]. Of these interactions, the most important is inverse-Compton scat-

tering. There is also a contribution in the direction of the solar disk. We estimate that

41 other interactions with solar photons are irrelevant; these include Bethe-Heitler [194, 195] and photo-pion interactions of protons [196] and deexcitation interactions of nuclei following photodisintegration [197–200].

Here we calculate this leptonic gamma-ray emission, mostly following prior work [102–

104, 201]. For the ﬁrst time, we calculate results up to 1 PeV and show that uncertainties in the electron spectrum at very high energies allow larger signals than in the nominal case

(a broken power-law spectrum for cosmic-ray electrons). As above, we neglect magnetic effects and assume straight-line propagation for the parent-daughter kinematics. Although the solar halo flux is present in all directions, its intensity (flux per solid angle) is greatest near the Sun, falling approximately as θ−1 [102–104] , where θ is the angle away from the center of the Sun. The flux within a given angle thus grows as θ, but the backgrounds — especially significant for ground-based detectors — grow as θ2. Therefore, the solar-halo signal is most interesting at relatively small angles. We calculate the leptonic signal within

1.5◦ degrees of the solar center; this value matches what we used for our Fermi analysis [30] and will allow HAWC and LHAASO to treat it as a near-point source.

In the optically thin regime, the gamma-ray ﬂux from the inverse-Compton interactions of cosmic-ray electrons is

Z Z Z Z dF dnph dσ dI = dΩ ds dEph (Eph, ~s) dEe (Ee,Eph,Eγ) (Ee, ~s) (2.3) dEγ dEph dEγ dEe

where dnph/dEph(~s) is the number-density spectrum of target photons at the line-of-sight

coordinate ~s, dI/dEe is the cosmic-ray intensity, and dσ/dEγ is the electron-photon diﬀer- ential cross section including Klein-Nishina eﬀects.

The column density of the solar photon ﬁeld is n R2 /Dθ for small angles θ [102], ∼ ph where nph is the number density of photons at the solar surface and D = 1 AU. For electron energies below about 0.25 TeV, the inverse-Compton cross section is in the Thompson

regime, where the total cross section is constant with energy. At higher energies, it is

in the Klein-Nishina regime, where the total cross section falls with increasing energy.

An electron passing close to the Sun has an optical depth of 10−2 (in the Thompson ∼ regime; less at higher energies), so the optically thin assumption of Eq. (2.3) is appropriate.

42 10-7 Fermi2011 Ng2016 S ] -8 SG1 991 IC Max

1 10 I − − nv s er se 2 C − -9 h om 10 a p C 1 yr m d HAW r to c o n ni 1 c, .5 V lim ◦ e b LHAASO 1 yr -10 o G n 10 ly [ γ E

d -11

/ 10 F d 2 γ -12 E 10 IC Pulsar − 10-13 1 4 5 100 10 102 103 10 10 106

Eγ [ GeV ]

Figure 2.5: Gamma-ray spectrum of the Sun. Points: disk observations with Fermi [16, 30], where the flux difference is due to time variation. Green band: the predicted disk flux [17]. Dotted lines: the estimated differential point-source sensitivity of HAWC [31] (scaled to one year) and LHAASO [18, 19]. Our new prediction of the solar-disk signal due to cosmic-ray hadrons (from the limb) is shown by the green solid line. Our new prediction of the solar- halo signal due to inverse-Compton scattering of cosmic-ray electrons is shown by the black solid line for the nominal case and by the dashed lines for enhanced cases from Fig. 2.6.

To calculate the gamma-ray spectrum, we use the StellarICs code [110, 111], slightly modiﬁed to include a parametrization of the electron spectrum at the highest energies.

The solar photons are taken to have a blackbody spectrum with temperature 5780 K and corresponding typical energy of 1 eV. The photon density falls as distance squared far ∼ 43 from the Sun but less quickly near its surface, where it varies as with radial distance r as

[1 (1 R2 /r2)] [102, 103]. The cosmic-ray electron ﬂux has been precisely measured by − − AMS-02 up to almost 1 TeV [32], and measured moderately well by H.E.S.S. [33, 34] and

VERITAS [35] up to 5 TeV. We use a broken power-law fit to these data. As discussed in detail in Sec. 2.5, the electron spectrum at very high energies might be much larger than expected from this nominal case, in which the flux above 5 TeV is assumed to fall off quickly. Our calculation is the first to show how allowed contributions to the electron spectrum above 5 TeV would enhance the solar-halo gamma-ray signal.

2.5 Cosmic-Ray Electrons

Figure 2.5 shows our results for the leptonic gamma-ray emission in the nominal case plus some enhanced cases. (Below 10 GeV, where there are measurements from Fermi [16], not shown here, our prediction is consistent.) In the Thomson regime, the gamma-ray spectrum is less steep than the electron spectrum due to the nature of the differential cross section. In the Klein-Nishina regime, the gamma-ray spectrum steepens sharply due to the suppression of the total cross section (in addition to the steepening electron spectrum). The nominal predictions are not detectable with HAWC and LHAASO. In fact, only the most extreme enhanced scenarios — with the cosmic-ray electron flux as large as the proton flux — are

(lines labeled “Max” in Figs. 2.5 and 2.6). If no solar-halo signals are detected, as is likely, that will make it easier to isolate hadronic gamma-ray ﬂux in the direction of the solar disk.

Section 2.5 introduces a better way to probe cosmic-ray electrons.

Figure 2.5 also recaps our result for the hadronic gamma-ray emission from the solar limb. This is well below the leptonic gamma-ray emission from the solar halo near the disk

(below about 1 TeV), as well as the sensitivity of HAWC and LHAASO. However, this prediction leads to several important points. The gamma rays observed from the solar disk must be hadronic, with their flux enhanced by magnetic effects, and the ratio of the data to our limb prediction provides a first direct measure of the strength of that enhancement.

The hadronic gamma-ray spectrum must eventually bend toward and join with our limb

44 3 Figure 2.6: Diffuse flux (weighted with Ee ) of cosmic-ray electrons. Below about 5 TeV, there are measurements (points, as labeled [32–35]). Above about 70 TeV, there are limits (gray region, which combines many experiments [36, 37]). In between, the spectrum could be as large as the blue solid line, allowing enhanced contributions (pulsar or dark matter; details are in the text). HAWC should be able to immediately improve sensitivity down to 10−3 (hadronic rejection) of the proton spectrum (red dashed line). ∼ prediction. Until the energy at which that occurs, there is positive evidence for interesting processes (magnetic effects) beyond the limb emission. It may be that the leptonic gamma- ray emission is never dominant in the data, despite its apparent dominance in Fig. 2.5.

Here we show that HAWC and LHAASO can directly measure the cosmic-ray electron

45 (e+ + e−) spectrum, which is of great interest [33–37, 190–193, 202–206]. Compared to the method of Sec. 2.4, this is simpler and more powerful. The ﬂux is expected to be isotropic. If a nearby pulsar or dark matter halo contributes signiﬁcantly, the resulting anisotropy would enhance the detection prospects, but we neglect this possibility. Because cosmic-ray electrons lose energy quickly, by synchrotron and inverse-Compton processes, the highest-energy electrons must come from quite nearby, e.g., a few hundred pc at 10

TeV.

Figure 2.6 summarizes present knowledge of the cosmic-ray electron spectrum. Below 5

TeV, there are measurements from various detectors, including AMS-02 [32], H.E.S.S. [33,

34], and VERITAS [35]. Above 70 TeV, there are strong limits from ground-based arrays

(summarized in Refs. [36, 37]). Importantly, at 5–70 TeV, there have been no experimental probes, as emphasized in Ref. [36]. At those energies, the only limit, which is quite weak, comes from requiring that the electron ﬂux not exceed the all-particle ﬂux. New sensitivity is needed to probe the electron spectrum in this energy range, where new components could appear. Intriguingly, there are hints of a new component starting to emerge at 5 TeV, seen by both the southern-sky H.E.S.S. [34] and the northern-sky VERITAS [35].

HAWC and LHAASO detect electrons and gamma rays with comparable eﬃciency [207].

However, the flux sensitivity for electrons is worse because, like the background protons, they are isotropic. The sensitivity depends on just the hadronic rejection factor. (Gamma rays are not a background, except in the direction of point sources; the diffuse flux of TeV electrons, even in the nominal case, exceeds that of gamma rays, even in the direction of the Milky Way plane [208].) We assume a hadronic rejection factor of 103, which should ∼ be reachable (Segev BenZvi, private communication). Performance close to this has been demonstrated by some analyses with a partially complete HAWC detector [209, 210]. More importantly, HAWC has already shown preliminary limits that approach our estimated sensitivity [211].

Figure 2.6 shows the estimated HAWC sensitivity to the electron ﬂux (LHAASO’s will likely be similar), along with possible enhancements to the 5–70 TeV electron spectrum.

HAWC and LHAASO can reach higher energies than air-Cherenkov detectors because of

46 their huge advantages in ﬁeld of view and uptime.

Probing the 5–70 TeV cosmic-ray electron spectrum for the first time will allow interesting tests of pulsars, dark matter, and possible surprises. For pulsars, we use predictions from Refs. [204, 205], which may explain the positron excess [191, 203]. (Even larger fluxes can be found in Ref. [212].) For dark matter, we use the PPPC4DMID code [213, 214] to calculate the electron spectra from dark matter decay, in this case with a mass of 100 TeV and a lifetime of 2 1026 s, which is comparable to current constraints [215–217]. × While simple, our results are important. Although the gap in coverage of the cosmic- ray electron spectrum was known [36], as was the possibility of using HAWC to detect electrons [207], this paper is the first to combine those points and quantify the prospects.

In the near future, there will be good sensitivity to high-energy cosmic-ray electrons from the CALET [192], DAMPE [193] and CTA [218] experiments. Even so, they may only reach . 20 TeV. With more than a year of data already collected, HAWC has a unique opportunity now, and we encourage swift action to complete an analysis.

2.6 Conclusions and Outlook

The Sun’s high-energy gamma-ray emission — seemingly due to irradiation by cosmic rays

— is not well understood. Above 10 GeV, the Sun is one of the 20 brightest sources ∼ detected by Fermi, and its disk emission is nearly an order of magnitude brighter [16, 30] than predicted [17]. In the TeV range, there have been no theoretical or observational studies.

Now there is a convergence of two opportunities: the recognition that the high-energy

Sun can reveal important physics and the unprecedented sensitivity of the already running

HAWC experiment. These opportunities will be enhanced by ongoing theoretical work and the sensitivity gain due to the coming LHAASO experiment.

This paper has three main results. The ﬁrst calculation of the gamma-ray emission due to hadronic cosmic rays interacting with the solar limb. At high enough energies (& 1 TeV), magnetic eﬀects can be neglected,

47 and the complete emission from the solar disk should be from only the thin ring of the limb. This flux can be robustly calculated. Further, it serves as an important theoretical lower bound on the solar-disk emission at all energies. The enhancement of the disk flux by magnetic fields can be deduced by the ratio of the observed flux to this prediction. In the

GeV range, this is a factor 10. As illustrated in Fig. 2.1, HAWC and LHAASO will provide ∼ new sensitivity to solar gamma rays in the TeV range, and can test if this enhancement continues, plus if there are new contributions, e.g., due to dark matter. (Limits from ARGO-

YBJ [159] are already in preparation [219], and is about one order of magnitude weaker than HAWC sensitivity at TeV energies.) Finally, the limb ﬂux would be signiﬁcantly more detectable if the solar disk could be resolved, due to lower backgrounds per solid angle.

Although we have conservatively neglected this possibility, it seems attainable. New results on the gamma-ray emission due to cosmic-ray electrons interacting with solar photons. This emission forms a gamma-ray halo around the Sun, and the intensity peaks near the disk. For the ﬁrst time, we calculate the TeV–PeV gamma-ray ﬂux, including the possibility of new components in the 5–70 TeV electron spectrum. HAWC and LHAASO can at least set constraints at these energies, where there are no measurements. A new perspective on allowed enhancements to the cosmic-ray electron spectrum and direct tests of such. Lastly, we show that direct observations of electromagnetic showers by

HAWC and LHAASO can provide unprecedented sensitivity to the 5–70 TeV cosmic-ray electron spectrum. This search, based on nondirectional signals, will be a powerful probe of the high-energy electron spectrum, testing some realistic models.

This paper is part of a larger program of work to develop the Sun as a new high-energy laboratory (see Ref. [30] for further discussion). With a good theoretical understanding of magnetic eﬀects, the Sun could be used as a passive detector for cosmic rays in the inner

Solar System, allowing measurements that are differential in particle type and energy, a capability unmatched by any existing or planned detector. Currently, the major roadblock to this goal is taking into account the complicated magnetic field effects, but this problem is tractable in principle, and progress is being made (Zhou et al., in preparation). The Sun is already a calibration source for direction, and could become one for flux. Interestingly, un-

48 like any other astrophysical source, the Sun’s hadronic and leptonic emission can be clearly separated using angular information alone. Finally, a thorough understanding of cosmic-ray interactions with the Sun is crucial for testing dark matter and neutrino physics [15, 95].

Acknowledgments We thank Andrea Albert, Mauricio Bustamante, Rebecca Leane, Shirley Li, Shoko

Miyake, Carsten Rott, Qingwen Tang, and especially Segev BenZvi, Igor Moskalenko, Elena

Orlando, and Andrew Strong for helpful discussions. BZ was supported by Ohio State

University’s Fowler and University Fellowships. KCYN was supported by NASA Grant No.

NNX13AP49G, Ohio State’s Presidential Fellowship, and NSF Grant No. PHY-1404311.

JFB was supported by NSF Grant No. PHY-1404311. AHGP was supported by NASA

Grant No. NNX13AP49G.

49 Chapter 3 Neutrino-nucleus cross sections for W -boson and trident production

The physics of neutrino-nucleus cross sections is a critical probe of the Standard Model and beyond. A precise understanding is also needed to accurately deduce astrophysical neutrino spectra. At energies above 5 GeV, the cross section ∼ is dominated by deep inelastic scattering, mediated by weak bosons. In addition, there are subdominant processes where the hadronic coupling is through virtual photons, γ∗: (on-shell) W -boson production (e.g., where the underlying interaction is ν + γ∗ `− + W +) and trident production (e.g., where it is ` → ν+γ∗ ν+`− +`+). These processes become increasingly relevant at TeV–PeV → 1 2 energies. We undertake the first systematic approach to these processes (and those with hadronic couplings through virtual W and Z bosons), treating them together, avoiding common approximations, considering all neutrino flavors and final states, and covering the energy range 10 –108 GeV. In particular, we present the first complete calculation of W -boson production and the first calculation of trident production at TeV–PeV energies. When we use the same assumptions as in prior work, we recover all of their major results. In a companion paper [1], we show that these processes should be taken into account for IceCube-Gen2.

The contents of this chapter were published in [2].

3.1 Introduction

The interactions of neutrinos with quarks, nucleons, and nuclei are a cornerstone of the

Standard Model. These test neutrino couplings to hadrons and probe the internal structure of hadronic states [138, 220–225]. Increasingly precise measurements of cross sections allow increasingly precise tests of neutrino mixing and beyond the Standard Model physics [78, 88, 50 112, 226–229]. Understanding the cross section is also crucial to neutrino astrophysics [24,

230–239]. In the laboratory, neutrino scattering has been well measured up to E 102 ν ∼ GeV [240–242]. Above 5 GeV, the dominant interaction is deep inelastic scattering (DIS), ∼ where neutrinos couple via weak bosons to the quark degrees of freedom, with the nucleon

and nuclear structure being less important but still relevant.

New scientiﬁc opportunities have arisen with IceCube, as atmospheric and astrophysical

neutrinos have been detected up to E 107 GeV [54, 243]. Even though the spectra ν ∼ are not known a priori, and the statistics are low, important progress can be made. For example, the neutrino cross section can be determined by comparing the event spectra due to neutrinos that have propagated through substantial Earth matter or not [70, 71, 237, 244–

246]. And to the extent that the cross section is understood — e.g., the claimed theoretical precision (from the parton-distribution functions) at 107 GeV is 2% [24] or 1.5% [237] ' ' — the measured event spectra can be used to accurately deduce neutrino spectra and

ﬂavor ratios, allowing tests of both astrophysical emission models and neutrino properties

(e.g., Refs. [85–87, 247–257]). As IceCube accumulates statistics, and larger detectors are under consideration [115, 258], the opportunities — and the need for a better theoretical understanding of neutrino-nucleus scattering — increase.

There are neutrino-nucleus interactions in which the hadronic coupling is via a virtual photon, γ∗, and the diagrams are more complex than in ordinary DIS. Although these photon interactions are subdominant, their importance grows rapidly with energy, becoming relevant in the TeV–PeV range. In (on-shell) W -boson production (Fig. 3.1), the neutrino interacts with a virtual photon from the nucleus to produce a W boson and a charged lepton. The cross section for this process has been claimed to reach 10% of the DIS ∼ cross section at 105–107 GeV [38–40]. (To set a scale, in the past 7.5 years, IceCube ∼ has detected 60 starting events with reconstructed energies above 60 TeV [259, 260].) More careful calculations are needed. In trident production (Fig. 3.2), the neutrino interacts with a virtual photon from the nucleus to produce a neutrino, a charged lepton, and a charged lepton of opposite sign [41–43, 118, 119, 226]. The cross section for this process has never been calculated at TeV–PeV energies. A ﬁrst calculation is needed.

51 p − + l k1 1 l l W + − W p + l − 2 W l q PP’ A A’ AA’

Figure 3.1: Diagrams for (on-shell) W -boson production via photon exchange. A and A0 are the initial- and ﬁnal-state nuclei. (See Fig. 3.5 and Sec. 3.3.2 for the connection with trident production.) For antineutrinos, take the CP transformation of the elementary particles.

In this paper, we provide the ﬁrst full calculations of both processes. We treat them

in a uniﬁed way, avoiding common approximations, considering all neutrino ﬂavors and

ﬁnal states, and covering the energy range 10–108 GeV. We recover all previous major

results when we adopt their inputs. In our companion paper [1], we detail the implications

for IceCube-Gen2 measurements of neutrino spectra and ﬂavor ratios, tests of neutrino

properties, and tests of new physics.

In Sec. 3.2, we review the W -boson and trident production processes, identifying the

shortcomings of previous work. In Sec. 3.3, we calculate the neutrino-real photon cross

sections for both processes. The more complicated neutrino-nucleus cross sections in diﬀer-

ent regimes are calculated in Secs. 3.4 and 3.5, then added up (Fig. 3.12) and discussed in

Sec. 3.6. We conclude in Sec. 4.4.

3.2 Review of W -boson and trident production

The (on-shell) W -boson and trident production processes are, respectively,

ν + A `− + W + + A0 , (3.1) ` →

52 ν + A ν + `− + `+ + A0 , (3.2) → 1 2 where A and A0 are the initial and ﬁnal-state nuclei and ` is a charged lepton. For trident production, for now we simplify the ﬂavor information (for details, see Eq. (3.5)). For antineutrinos, take the CP transformation of the elementary particles.

Figures 3.1 and 3.2 show the diagrams for W -boson and trident production processes, respectively. We also calculate diagrams, not shown, with W and Z boson couplings to the hadronic side; this is discussed in Sec. 3.5. For trident production, (for Fig. 3.2 only) we use the four-Fermi theory for simplicity, the diagrams of which nicely show the “trident” feature though hiding the connection to W -boson production (see Fig. 3.5 for the full Standard-

Model diagrams, on which our calculation is based). In both processes, a neutrino splits into charged particles (leptonic part) that couple to the photon from the nucleus (hadronic part). The leptonic part is straightforward but depends on the process, while the hadronic part is complicated but independent of the process.

In the rest of this section, we review the hadronic part (Sec. 3.2.1), which also sets the foundation, then discuss the two processes respectively (Secs. 3.2.2 and 3.2.3).

3.2.1 Hadronic part

At most energies, the hadronic part is connected by a virtual photon from the nucleus.

8 Above & 10 GeV, the contributions of virtual weak bosons from the nucleus and mixing with the photon are not negligible (see Sec. 3.5).

The hadronic coupling can be in diﬀerent regimes, including coherent (σ Z2), diﬀrac- ∝ tive (σ Z), and inelastic (σ Z), in which the virtual photon couples to the whole ∝ ∝ nucleus, nucleon, and a single quark, respectively. (These three regimes are analogous to

the coherent elastic neutrino-nucleus scattering, quasi-elastic scattering, and deep-inelastic

scattering, respectively, for the usual neutrino-nucleus interaction, in which the hadronic

coupling is through a W/Z boson.) Adding the cross sections in diﬀerent regimes gives the

total cross section.

53 l l

+ − l l l l − l l+

A A’ A A’

Figure 3.2: Diagrams for trident production via photon exchange in the four-Fermi theory (see Fig. 3.5 for the full Standard Model). For antineutrinos, take the CP transformation of the elementary particles.

The coherent (A0 = A) and diﬀractive (A0 = A) regimes are both elastic, on the nucleus 6 and nucleon, respectively. The former is usually described by a nuclear form factor and the latter by a nucleon form factor.

Two different calculational frameworks have been used in previous work, i.e., using or not using the equivalent photon approximation (EPA, or Weizsäcker–Williams Approxima- tion) [261–263]. The EPA assumes the photon mediator (to the nucleus) to be on shell, i.e., q2 = 0. This is motivated by the fact that the photon is usually very soft when the beam particle is very energetic (e.g., high-energy electron scattering on nuclei). Using real photons significantly simplifies the calculation, because then one does not need to take into consideration the photon virtuality or longitudinal polarization. However, Refs. [42, 264–

266] pointed out that EPA is not valid for most cases, especially for electron ﬁnal states, leading to an overestimation of the cross section that, in some cases, is by more than 200%. The simplest reason is that, though the beam neutrino is very energetic, the charged particle that directly couples to the photon may not be.

Inelastic scattering (A0 = A) could also happen, with nucleon breakup. The hadronic 6 part is usually described by parton-distribution functions (PDFs) for photon, quarks, etc.

The inelastic regime has two subprocesses, photon-initiated (related to the photon PDF) and quark-initiated (related to the quark PDFs) [267]. See Sec. 3.5 for details.

54 3.2.2 W -boson production

The W -boson production process (Fig. 3.1) initially raised interest in the 1960s and 1970s.

The hypothetical (at that time) W boson could be directly produced by a beam of νµ scattering oﬀ the Coulomb ﬁeld of a nucleus (Fig. 3.1; e.g., Refs. [268–273]). If W bosons

were not detected, a lower bound on their mass could be set. Later, the discovery of the

W boson at a proton-antiproton collider [274], and especially its large mass, signiﬁcantly

reduced the motivation to search for this process at ﬁxed-target neutrino experiments.

The interest in this process came back due to high-energy astrophysical neutrino de-

tectors [54, 258, 275], and was studied by Seckel [38] and Alikhanov [39, 40]. In Ref. [38],

for the neutrino-nucleus cross sections, only the ratio of ν e−W + to charged-current e → (CC) DIS on 16O and 56Fe were shown, and only the coherent regime was considered (see

Table 3.1). In Refs. [39, 40], all three ﬂavors were considered and shown, and all three

scattering regimes were considered. However, all three regimes used EPA, and nuclear ef-

fects (mainly Pauli blocking) were not included. Moreover, for the inelastic regime, only

the photon-initiated subprocess was calculated (see Table 3.1).

Figure 3.3 shows their results. All three scattering regimes are important. The high

threshold is set by the W -boson mass and the hadronic structure functions. The diﬀractive

regime has a lower threshold than the coherent regime because larger Q2 ( q2; virtuality ≡ − of the photon) can be probed by the nucleon form factor than the nuclear form factor.

Above threshold, the coherent cross section ( Z2) is larger than the diﬀractive cross ∝ section ( Z). ∝ The coherent cross section of Seckel [38] is about two times that of Alikhanov [39, 40],

possibly due to their treating the nuclear form factor diﬀerently (as pointed out by Ref. [40]).

(The origin of the factor of two between them could not be traced, as the details of the

calculations are not given in Ref. [38].)

Importantly, this cross section is claimed to be 10% of the charged-current deep ∼ inelastic scattering (CCDIS) cross section [24], indicating this process is detectable by high-

energy neutrino detectors like IceCube [243], KM3NeT [258], and especially the forthcoming

55 37 10 Coherent Diffractive 38 CCDIS 10 Inelastic Alikhanov 2016

] 39

1 10 V

e 40

G 10

m 41 c

10 Magill & Plestid 2017

[ Altmanshofer+ 2019

42 Seckel 1998 E

/ 10 (coherent only) 43 10 Ballett+ 2018 + + 44 Ar e e Ar e O e W X 10 Tridents, coherent only W boson production 0.1 1 10 102 103 104 105 106 107 108 E [ GeV ]

Figure 3.3: Summary of cross sections for W -boson and trident production from previous work, with the two processes separated as labeled. To simplify the ﬁgure, for W -boson production, we show only ν e−W + on 16O (by Seckel [38] and Alikhanov [39, 40]), and e → for trident production, only the coherent regime (the dominant part) of ν ν e−µ+ on 40Ar µ → e (by Magill & Plestid [41], Ballett et al. [42], and Altmannshofer et al. [43]). Also shown, for comparison, is the cross section of charged-current deep inelastic scattering (CCDIS) [24].

IceCube-Gen2 [115]. With 60 starting events with energies above 60 TeV [259, 260], Ice-

Cube already has a nominal precision scale of 13%, and IceCube-Gen2 would be 10 times larger. However, on the theory side, due to the limitations above, more complete and careful calculations are needed (see Table 3.1).

56 Table 3.1: Summary of the features of previous calculations and of this work. “+” and “ ” means “considered” and “not considered” − in the calculation respectively. “Full SM” means using full Standard Model, instead of four-Fermi theory.

Full SM Coherent Diﬀractive Beyond EPA Pauli blocking Inel., photon Inel., quark W -boson Seckel [38] + + − − − − −

57 production Alikhanov [39, 40] + + + + − − − Altmannshofer et al. [226] + − − − − − − Magill & Plestid [41] + + + Trident − − − − Ge et al. [118] + production − − − − − − Ballett et al. [42] + + + + − − − Altmannshofer et al. [43] + + + + − − − Both, uniﬁed This work + + + + + + + 3.2.3 Trident production

The trident processes (Fig. 3.2) raised interest at a similar time to W -boson production, also as a process to probe the then-hypothetical W/Z propagators in the weak interactions.

Even if the weak bosons were not produced directly due to, e.g., their large masses, their existence could make the trident production rate diﬀerent from that of the pure V-A theory

(e.g., Refs. [264–266, 276–281])

So far, only the ν ν µ−µ+ process has been observed, by the Charm-II [282] and µ → µ CCFR [283] experiments; NuTeV [284] set an upper limit. These results are consistent with

SM predictions.

The trident processes have been popular again in recent years, due to currently running and upcoming accelerator neutrino experiments (e.g., Refs. [285–288]) as well as Ref. [226] showing first trident constraints on new physics such as Z0 models. Table 3.1 summarizes the calculations of trident cross sections by Refs. [41, 118, 226] using EPA and by Refs. [42, 43] using an improved calculation. Usually, only the electron and muon flavors are considered, as the tau flavor is rare for accelerator neutrinos. The inelastic regime is very small, so not considered (except in Ref. [41]).

All previous work used the four-Fermi theory, instead of the full Standard Model (see

Table 3.1). One reason is that these papers focused on accelerator neutrinos below ∼ 100 GeV. Another reason is that the hadronic part (Sec. 3.2.1) complicates the calculation a lot, so using the four-Fermi theory for leptonic part is signiﬁcantly simpler.

Figure 3.3 summarizes previous calculations. The threshold is set by the ﬁnal-state lepton masses and hadronic structure functions. The diﬀerence between Magill & Plestid [41] and Ballett et al. [42], Altmannshofer et al. [43] is due to the former using EPA, while the latter two not.

Though at GeV energies the cross sections are 10−5 of CCDIS [24], they increase ∼ quickly. Therefore, it is interesting and important to know the cross sections at TeV–PeV energies. To this end, the full Standard Model is needed instead of the four-Fermi theory.

In addition, our calculations ﬁx several other shortcomings (see Table 3.1).

58 3.3 Cross sections between neutrinos and real photons

In this section, we calculate the cross sections of W -boson and trident production between a neutrino and a real photon. This shows the underlying physics and the basic behavior of the cross sections. The connection between the two processes is also clearly revealed.

For the cross sections between elementary particles, we calculate the matrix elements and phase space integrals ourselves, and check the results using the public tools MadGraph

(v2.6.4) [289] and CalcHEP (v3.7.1) [290]. The calculational procedures set the basis for the oﬀ-shell cross sections in Sec. 3.4.

3.3.1 W -boson production

The leptonic part of W -boson production is (Fig. 3.1),

ν + γ `− + W + . (3.3) ` →

The cross section can be calculated using

Z 1 1 X WBP 2 σνγ(sνγ) = dPS2 , (3.4) 2sνγ 2 |M | spins where 1/2s is the Lorentz-invariant ﬂux factor, s (k + q)2, 1 P WBP 2 the νγ νγ ≡ 1 2 spins |M | photon-spin averaged matrix element (Appendix 3.8.1), and dPS (= √pCM d cos θ ) is the two- 2 sνγ 8π body phase space, of which pCM is the momentum of the outgoing particle in the CM frame, with angle θ respect to the incoming particle. This process has been calculated by

Refs. [38, 39]. Our calculation gives the same results.

The diagrams for Eq. (3.3) are similar to those in Fig. 3.1, but replacing the photon from the nucleus with a free (real) photon. Both diagrams, with a relative minus sign, need to be included to assure gauge invariance. Numerically, the ﬁrst diagram dominates at small sνγ, while the second dominates at large sνγ. Neutrinos and antineutrinos have the same total and diﬀerential cross sections, as the matrix element is invariant under CP transformation.

Figure 3.4 shows σνγ(sνγ)/sνγ for W -boson production. We divide out sνγ, the dominant trend, to highlight the deviations over the wide range of the x axis. The threshold is set

59 37 10 e +

38 + W 10 -b Tridents (CCo and CC+NC) so + n p

] r

39 o d 2 10 u c ti V o

e n G

2 40 10 ee m

c +

[ e + ee e Tridents (NC)

s 41 / 10 e + + , e + + , + +

+ e e, e, 42 e 10

, e

+ + 43 10 , e

e 10 6 10 5 10 4 10 3 10 2 0.1 1 10 102 103 104 105 106 s [ GeV2 ]

Figure 3.4: Our cross sections (actually σνγ(sνγ)/sνγ) for W -boson and trident production, between a neutrino and a real photon as a function of their CM energy. Red, green, and blue lines are νe-, νµ-, and ντ -induced channels, respectively. Solid lines are trident CC channels, and dashed lines are trident CC+NC channels (we label only the final states for both). Magenta dotted lines are trident NC channels, which depend on only the final- state charged leptons (we label both the initial and final states). The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5) of Fig. 3.5. The corresponding antineutrino cross sections (i.e., obtained by CP-transforming the processes shown) are the same. See text for details.

2 by sνγ = (mW + m`) . Just above threshold, the lepton propagator in the first diagram (Fig. 3.1) gives a logarithmic term, log[(...)/m2] , which leads to σ > σ > σ [38]. ∼ ` νeγ νµγ ντ γ 6 2 For sνγ > 10 GeV , the cross sections become constant and different flavors converge, with σ 2√2αG 10−34 cm−2. νγ ' F '

60 − l1 l1 + l2

l1 W + + W l2 l2 W+ − + l1 (1) (2) l2

A A’ A A’ − l1 l1 W+ l2

(3) + l2

A A’

l1 k1 k2 l1 l1 l1 Z + Z − p1 l2 l2

(4) p − (5) + 2 l2 l2 q P P’ A A’ A A’ Figure 3.5: Diagrams for trident production via photon exchange in the Standard Model, with the order, Tri– Tri, labeled in parentheses, and with the momenta labeled on the M1 M5 fourth diagram. The trident CC, NC, and CC+NC channels correspond to diagrams (1)– (3), (4)–(5), and (1)–(5). For antineutrinos, take the CP transformation of the elementary particles. The ﬁrst and second diagrams are connected to W -boson production (Fig. 3.1; also see Sec. 3.3.2 for details of the connection).

61 3.3.2 Trident production

The leptonic part of trident production, for each incoming neutrino ﬂavor, is (Fig. 3.5)

ν + γ `− + ν + `+ (CC), (3.5a) `1 → 1 `2 2 ν + γ ν + `− + `+ (NC), (3.5b) `1 → `1 2 2 ν + γ ν + `− + `+ (CC+NC), (3.5c) ` → `

where ` , ` = e, µ or τ and ` = ` . So there are two, two and one CC, NC and CC+NC 1 2 1 6 2 channels, respectively; details below. For antineutrinos, take the CP transformation of the

elementary particles; details below.

The cross section can be calculated using [41, 226, 291]

Z 1 1 X Tri 2 σνγ(sνγ) = dPS3 , (3.6) 2sνγ 2 |M | spins where 1/2s is the Lorentz-invariant ﬂux factor, 1 P Tri 2 the photon-spin averaged νγ 2 spins |M | matrix element, and dPS3 is the three-body phase space (see below). Diﬀerent from previous calculations, here we need to use the full Standard Model, instead of the four-Fermi theory, as we are also interested in TeV–PeV energies.

Figure 3.5 shows the ﬁve possible diagrams of trident production in the Standard Model.

Note that the diagram involving a W W γ vertex is not included by the four-Fermi theory, though it is suppressed at low energies. When ` = ` , the top three diagrams (exclusively 1 6 2 mediated by W ) lead to CC channels, and the bottom two (exclusively mediated by Z) lead to NC channels. When `1 = `2 = `, all ﬁve diagrams give the same ﬁnal states, which lead to the CC+NC channels.

We work in the unitarity gauge, which is simpler for the tree level. The amplitudes for each diagram, Tri— Tri, can be found in Appendix 3.8.2, and relative signs between M1 M5 these diagrams are

Tri = ( Tri Tri + Tri) ( Tri + Tri) . (3.7) M M1 − M2 M3 − M4 M5

The matrix element is calculated using FeynCalc [292, 293].

62 For antineutrinos, the total cross sections are the same as neutrinos, due to CP invariance [118, 266]. For the diﬀerential cross sections, they are only the same for the NC channels due to interchange symmetry of two charged leptons, which is not the case for CC and CC+NC channels [118]. Therefore, for the following discussion, we take neutrinos only.

The three-body phase space in the case of real photon is [41, 226, 291]

1 1 dt dl dΩ00 dPS3 = 2 β(l) , (3.8) 2 (4π) 2sνγ 2π 4π where t 2q (k k ), l (p + p )2,Ω00 the solid angle with respect to q in the rest frame ≡ · 1 − 2 ≡ 1 2 of p + p , and 1 2 r 2(m2 + m2) (m2 m2)2 β(l) = 1 1 2 + 1 − 2 , (3.9) − l l2

2 where m1, 2 is the mass of p1, 2. The integration over l is done from (m1 + m2) to sνγ,

and t from l to sνγ. Using these variables, we ﬁnd that the numerical integration converges reasonably fast for both the four-Fermi theory case and the Standard-Model case.

Figure 3.4 shows σνγ(sνγ)/sνγ for all 15 trident channels. The thresholds are set by the

2 masses of ﬁnal states, i.e., sνγ = (m1 + m2) . The cross sections increase from threshold until 106 GeV2. For s > 106 GeV2, same as for W -boson production, the cross sections ∼ νγ become constant.

For the CC and CC+NC channels, very interestingly, just above s = m2 6.5 νγ W ' × 103 GeV2, there is a sharp increase. This is due to the s-channel like part of the ﬁrst and

2 second diagrams in Fig. 3.5, which are mediated by W bosons. For sνγ > mW , W -boson production is turned on. In the view of trident production, this is a W -boson resonance followed by decay to a neutrino and a charged lepton. Therefore the CC and CC+NC trident cross sections are enhanced by the W -boson production cross section (of same incoming neutrino flavor) times the corresponding decay branching ratio, Γ − /Γ ( 11%, with W →ν`` W ' slight deviation for specific flavors [242]). The W -resonance contribution keeps dominating

2 for sνγ > mW . This is diﬀerent from usual resonance features, like the Glashow resonance ν¯ + e− W − or e−e+ Z. The reason is that the charged lepton, `−, could take e → → 2 2 away additional 4-momentum, keeping the s-channel W propagator on shell (qW = mW ).

63 Contributions from the non-resonant part and from the other three non-resonant diagrams

2 are negligible. So for sνγ > mW , the six CC and three CC+NC channels basically form 6 2 three groups due to the three neutrino ﬂavors. For sνγ > 10 GeV , they all converge and

−35 −2 become constant, i.e., σ 2√2αG Γ − /Γ , which 10 cm . ' F × W →ν`` W ' For the NC channels, the cross sections are much smaller, as there is no resonance. The channels that have same charged lepton ﬁnal states have same cross sections, independent of incoming neutrino ﬂavor, due to the same couplings and lepton propagators.

3.4 Neutrino-nucleus cross sections: coherent and diﬀractive regimes

In this section, we calculate the photon-mediated neutrino-nucleus cross sections, σνA, for W -boson and trident production in the coherent and diﬀractive regimes, which are elastic on the nucleus and nucleon, respectively. We focus on the hadronic coupling through virtual photons, as the contribution through weak bosons is highly suppressed due to their large masses. We ﬁrst describe the framework (Sec. 3.4.1), which is independent of the leptonic part. Then we calculate the W -boson (Sec. 3.4.2) and trident (Sec. 3.4.3) production processes.

The framework we use, which is from Ballett et al. [42], is a complete treatment of the hadronic part instead of using EPA, which is known to be not a good approximation for trident production (see Fig. 3.3 and Sec. 3.2). Moreover, the major nuclear effect, Pauli blocking, is included (see Refs. [42, 266] for details). In this work, for the first time, we show that the EPA also does not work well for W -boson production. Moreover, for trident production, we calculate all 15 possible channels, including for the τ flavor, and go to

TeV–PeV energies, using the full Standard Model instead of the four-Fermi theory.

3.4.1 Framework

Both the coherent and diﬀractive cross sections can be calculated using [42]

d2σ 1 1 νX = [hT Q2, sˆ σT Q2, sˆ + hL Q2, sˆ σL Q2, sˆ] , (3.10) dQ2dsˆ 32π2 sQˆ 2 X νγ X νγ 64 where X is to distinguish coherent (X = c) and diﬀractive (X = d) regimes, Q2 q2 the ≡ − photon virtuality, ands ˆ 2(p q) = s + Q2. Note that Eq. (3.10) decomposes the σ ≡ 1 · νγ νX T/L into 2 2 parts: transverse (“T”) and longitudinal (“L”), leptonic (σνγ ) and hadronic × T/L (hX ). T/L 2 The leptonic parts, σνγ (Q , sˆ), may be viewed as the cross sections between a neutrino and an oﬀ-shell photon, and it can be calculated as

1 Z 1 X 4Q2 σ = gµν + kµkν L L∗ dPS , (3.11a) T 2ˆs 2 − sˆ2 1 1 µ ν n spins 1 Z X 4Q2 σ = kµkνL L∗ dPS , (3.11b) L sˆ sˆ2 1 1 µ ν n spins

where Lµ is the leptonic amplitudes, details in Appendices 3.8.1 and 3.8.2, and dPSn is the phase space of the leptonic part, with n = 2, 3 for the W -boson and trident production processes, respectively .

A factor of 1/2 appears in the ﬁrst equation because a virtual photon has two transverse polarizations. The Q2 dependence should also be included in both the leptonic matrix element and phase space, which are process dependent. In the limit Q2 = 0, the transverse cross section is the same as the real-photon case (Eqs. (3.4) and (3.6)), and the longitudinal cross section vanishes.

T/L 2 The hadronic parts, hX (Q , sˆ) are dimensionless factors that involve the nuclear/nucleon form factors. For the coherent regime, we use the Woods-Saxon (nuclear)

form factor. For the diﬀractive regime, we use the nucleon form factors that have a dipole

parametrization. More details are given in Ref. [42].

For the diﬀractive regime, in addition, the Pauli-blocking eﬀects are included by mul-

tiplying Eq. (3.10) by a factor derived from modeling the nucleus as ideal (global) Fermi

gas of protons and neutrons with equal density, which is (derived by Ref. [270] and used by

Refs. [42, 43, 276]),

 3 3 ~q 1 ~q  | | | | , if ~q < 2 kF , f( ~q ) = 2 2 kF − 2 2 kF | | (3.12) | |  1, if ~q 2 k , | | ≥ F

65 38 Coherent and diffractive only 10

39 10 C CoherentCD

] e

1 × 0. Diffractive 1 V 40 e 10 G

2 m c

[

41 10 Diff. (neutrons only) E

/ Coh. (longitudinal only) A

42 10

43 10 4 5 6 7 8 9 10 10 10 10 10 10 E [ GeV ]

Figure 3.6: Our coherent and diffractive components of W -boson production cross sections, ν `− + W +, on 16O. Red, green, and blue lines are ν -, ν -, and ν -induced ` → e µ τ channels, respectively. Solid: coherent (right bump) and diffractive (left bump) components. Dashed: longitudinal contribution to the coherent regime, which is small, even for the largest case (νe). The ντ line is not shown due to being below the bound of the y axis. Dotted: contribution from neutrons to the diffractive regime, which is small. The corresponding antineutrino cross sections are the same.

where kF = 235 MeV is the Fermi momentum of the gas, which sets the kinetic boundary for the ﬁnal states, and ~q is the magnitude of the transferred 3-momentum in the lab | | p 2 2 2 frame, which can be derived to be, for the virtual photon case, (Q /2MN ) + Q , where

66 Coherent and diffractive only

Seckel, coh. only Alikhanov (EPA) Our EPA 39 Our full calculation 10 Coherent ]

1 C C

V D

e IS

G × 0 2 .1

m Diffractive c

[

/ 40

A 10

4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 3.7: Our coherent and diffractive components of W -boson production cross sections (red solid, from Fig. 3.6 but thicker), for the example of ν `−W + on 16O (the flavor e → with the largest cross section), comparing with our “EPA + no Pauli blocking” results (dashed) and previous calculations (dotted) by Seckel [38] and Alikhanov [39, 40]. Left and right bumps are coherent and diffractive components, respectively. Note our results are substantially smaller, which is important.

MN is the mass of the nucleon. This reduces the diﬀractive cross section by about 50% for protons and 20% for neutrons.

2 T/L 2 The EPA formalism can be obtained by setting Q = 0 in σνγ (Q , sˆ) of Eq. (3.10). This is basically the same as that initially derived by Ref. [281] and later used by Refs. [41, 226].

67 Below, we also show the “EPA + no Pauli blocking” results for comparison.

As a validation of our understanding of the formalism, we reproduced the cross section results of Ballett et al. [42] using the four-Fermi theory and other same input. Our calculations agree with theirs to within a few percent, with the remaining diﬀerences due to numerical precision. (Note that we decompose the phase-space (Appendix 3.8.3) in a diﬀerent, but equivalent, way from them [42, 266].)

3.4.2 W -boson production

T/L 2 Oﬀ-shell cross sections, σνγ (ˆs, Q )

The process is the same as Eq. (3.3), but replacing the real photon, γ, by a virtual photon

∗ T/L 2 γ . We calculate the oﬀ-shell cross sections, σνγ (ˆs, Q ), in the CM frame (consistent with Sec. 3.3.1), using Eq. (3.11). For the leptonic matrix element, the photon virtuality can be included by writing

2 2 sνγ + Q sνγ + Q k1 = , 0, 0, , (3.13a) 2√sνγ 2√sνγ s Q2 s + Q2 q = νγ − , 0, 0, νγ . (3.13b) 2√sνγ − 2√sνγ

The 4-momenta of the outgoing particles do not have Q2 dependence, and the phase space is the same as that in Eq. (3.4). When Q2 = 0, all results return to the real-photon case

(Sec. 3.3.1).

T/L 2 T 2 2 The major features of σνγ (ˆs, Q ) are the following. First, σνγ(ˆs, Q ) decreases with Q , 2 2 especially when Q & m` , because Q enters the denominator of the lepton propagators which L 2 2 suppresses the cross section [42]. Second, σνγ(ˆs, Q ) increases with Q , due to the factor 4Q2/sˆ2, then becomes ﬂat when Q2 m2. Third, when Q2 is nearings ˆ (m + m )2, & ` − W ` T L an exponential cutoﬀ happens in both σνγ and σνγ, due to running out of phase space (s sˆ Q2 < (m + m )2). νγ ≡ − W `

68 σνA and discussion

The coherent (σνc) and diﬀractive (σνd) cross sections are then calculated with Eq. (3.10), by convolving the leptonic parts with the hadronic parts.

Figure 3.6 shows the cross sections. The features discussed in Sec. 3.3.1 mostly appear here (e.g., σ > σ > σ ). The threshold here is effectively set by E m2 /2Qeff , νeA νµA ντ A ν ∼ W max where Qeff is the effectively maximum Q of the form factors, which is 0.1 GeV for max ∼ 16O (coherent) and 1 GeV for nucleons (diffractive). Similarly, the typical momentum ∼ transfer, Q, for each E , is between m2 /2E and Qeff for both regimes. The sharp peak ν ∼ W ν max in σνγ (Fig. 3.4) is here smeared due to convolving with the form factors. The coherent cross sections ( Z2) are larger than diffractive ones ( Z), which is similar to Fig. 3.3, which ∝ ∝ uses EPA. For both regimes, the transverse part dominates, while the longitudinal part, as shown on the figure, is suppressed by Q2/sˆ. For the diffractive regime, the contribution ∼ from protons dominates, due to its electric form factor.

We tested the sensitivity of our cross section results to the choices of form factors. For

2 2 the coherent component, we also tried using a Gaussian form factor, e−Q /2a , which is

sometimes used for lighter nuclei. For 16O, we ﬁnd that the cross section is changed by

15% for E 105 GeV (where the inelastic component dominates anyway) and by 5% . ν ∼ . for E 106 GeV. For the diﬀractive component, we explored changing the vector mass in ν ∼ the form factor. For any reasonable change, the eﬀect on our calculated cross sections is negligible.

Figure 3.7 compares our results with previous ones from Ref. [38] (only the coherent regime was considered) and Refs. [39, 40] (EPA was used, and Pauli-blocking effect was not included for the diffractive component), and our “EPA + no Pauli blocking” results discussed above. Comparing to Refs. [39, 40], which uses a different EPA formalism, our

EPA result is close. Surprisingly, the result from the full calculation is only about half as large, for both coherent and diﬀractive regimes. This means that the EPA is still not valid at even such high-energy scales. The reason is that, as discussed in Sec. 3.4.2, the

2 T 2 T 2 nonzero Q suppresses the transverse cross section, σνγ(ˆs, Q ), compared to σνγ(ˆs, Q = 0)

69 Coherent and diffractive only 39 10 e C e C W 40 DI + 10 S ×

] 0 .01 1 41 V

e 10 G

+ 2 e

m 42 e c

10 [

E e, e,

/ 43 +

A 10

44 10

45 10 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 E [ GeV ]

Figure 3.8: Our coherent (solid lines) and diﬀractive (dashed lines) components of trident production cross sections on 16O. We show one typical channel for each category, i.e., CC, NC and CC+NC, to make the ﬁgure simple. For all the channels, see Appendix 3.8.4. Gray lines are for νe-induced W -boson production from Fig. 3.6, shown as a comparison. The corresponding antineutrino cross sections (i.e., obtained by CP-transforming the processes shown) are the same.

used in the EPA. The larger Q2, the larger the suppression. Physically, this is because,

although the incoming neutrino is very energetic, the charged particle that directly couples

to the photon may not be. The diﬀerence between the full calculation and the EPA in

the diﬀractive regime is larger than that in coherent regime is because the nucleon form

70 factor probes larger Q2 than the nuclear form factor, also because the Pauli-blocking effect suppresses the diffractive cross section. Another feature is that, for a specific Q2, the larger

2 2 the charged-lepton mass, the smaller the suppression, which is small when Q . m` . So the diﬀerence between full calculation and EPA is smaller for the muon and tau ﬂavors.

3.4.3 Trident production

T/L 2 Oﬀ-shell cross sections, σνγ (ˆs, Q )

The processes are the same as in Eq. (3.5), but again replacing the real photon, γ, by a

virtual photon γ∗. Same as above, we work in the CM frame, and both the phase space

term dPS3 and the leptonic matrix element are modified due to nonzero photon virtuality. The leptonic matrix element is modified due to the modification of the 4-momenta, the

details of which can be found in Appendix 3.8.3.

The phase space integration can be done by decomposing the three-body phase space

into two two-body phase spaces [41, 294]. The result is the same as Eq. (3.8), but replacing

s bys ˆ. The integration range is now ((m + m )2, sˆ Q2) for l, and νγ 1 2 − l l + Q2, sˆ Q2 + 2 Q2 (3.14) − − sˆ Q2 − for t. See Appendix 3.8.3 for details.

The major features due to the nonzero Q2 is the same as those of W -boson production

(Sec. 3.4.2).

σνA and discussion

Figure 3.8 shows the cross sections for the typical channels (for all channels, see Ap-

pendix 3.8.4). We start from Eν = 10 GeV, as below this energy, the cross sections have been shown in Refs. [42, 43]. Our results agree with theirs. Same as before, the threshold

here is eﬀectively set by E (m + m )2/2Qeﬀ , which is 0.1 GeV for 16O and 1 GeV ν ∼ 1 2 max ∼ ∼ for nucleons. The sharp peak in σνγ (Fig. 3.4) is smeared here due to convolving with the form factors. Other features and the physics are the same as those discussed in Sec. 3.3.2

71 for Fig. 3.4 and in Sec. 3.4.2 for Fig. 3.6.

3.5 Neutrino-nucleus cross sections: inelastic regime

In this section, we calculate the neutrino-nucleus cross sections, σνA, for W -boson and trident production in the inelastic regime, in which the partons of nucleons are probed.

3.5.1 Framework

The inelastic regime has two contributions, photon-initiated subprocess and quark-initiated subprocess [267].

The photon-initiated subprocess is that the hadronic coupling is through a virtual photon, which is similar to Sec. 3.4, but with larger photon virtuality Q2. Calculation of this subprocess involves the photon PDF, which describes the photon content of the nucleon

(e.g., Refs. [267, 295–302]). The photon PDF consists of elastic and inelastic components.

The elastic component corresponds to the diﬀractive regime of Sec. 3.4, and can be calculated from the nucleon electromagnetic form factors. The inelastic photon PDF consists of nonperturbative and perturbative parts, and the resonance region is included in the former [267, 300, 301]. For the W -boson production, this component was calculated by

Alikhanov [40]. For trident, this has never been considered.

The quark-initiated subprocess is that a quark of a nucleon is explicitly involved as an initial state of the scattering process. The propagator that couples to a quark can be photon,

W or Z boson. For W -boson production, this was mentioned in Alikhanov [40] (diagrams were also shown in its Fig. 8, plus another one from replacing the Z by a photon in the upper middle diagram) but not calculated. For trident production, this was calculated in

Ref. [41].

Those two subprocesses are at the same order though they may not seem to be, as the photon propagator to quark has an additional αEM. The reason is that the photon PDF has a factor of αEM implicitly [267]. A double-counting problem occurs if summing up the two subprocesses for the total

72 inelastic cross section. This is because the contribution from the photon propagator to the quark of the quark-initiated subprocess is already included in the inelastic photon PDF

(perturbative part) of the photon-initiated subprocess. We deal with this problem below.

For the PDF set, we use CT14qed [267, 303], which provides the inelastic photon, quark, and gluon PDFs self-consistently. The inelastic photon PDF of CT14qed is modeled as emission from the quarks using quark PDFs and further constrained by comparing with

ZEUS data on the DIS process ep eγ + X [304]. The quark PDFs are obtained by the → usual method and constrained by DIS and other data.

The reasons that we choose CT14qed are the following. First, it is the only PDF set that provides the inelastic component of photon PDF only. For the elastic part, we do not use the elastic photon PDF, which is obtained using EPA and does not include the neutron magnetic component form factor, as the treatment in Sec. 3.4 (diﬀractive regime) is better.

Second, it is also the only PDF set that provides the inelastic photon PDF for both proton and neutron. Finally, though the uncertainty of photon PDF is larger than the later ones by LUXqeD [300, 301] and NNPDF31luxQED [302], the central value is very close.

We use MadGraph (v2.6.4) [289] to do the calculation, which handles the PDFs and hard processes systematically. We remove kinematic cuts to get the total cross section of both processes. The model we choose in MadGraph is “sm-lepton masses”, which includes the masses of charged leptons, while the default “sm” does not. Moreover, this model uses diag(1, 1, 1) for the CKM matrix and ignores the masses of u, d, s, c quarks, which are good approximations for us. Note that for the initial-state neutrinos, which have only a left-handed chirality, we need “set polbeam1 = -100” (+100 for antineutrinos) to fully

polarize the beam, otherwise the cross section will be mistakenly halved. As a check of above

conﬁguration, we calculate the neutrino CCDIS cross sections and the result is consistent

with Refs. [24, 233, 237, 239, 305–307] within uncertainties.

3.5.2 W -boson production

For photon-initiated process, the diagrams are shown in Fig. 3.1. The factorization and

renormalization scales are chosen to be √sνγ. Our choice is consistent and has no ambiguity 73 Inelastic only

photon-initiated subprocess photon-initiated, protons only 39 photon-initiated, neutrons only 10 quark-initiated subprocess quark-initiated, photon propagators only quark-initiated, W & Z propagators only ]

1 C C D V IS e ×

G 0 .

2 1

m 40 c

[ 10

E /

41 10 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 3.9: Diﬀerent components of our inelastic neutrino-nucleus cross sections for W - boson production. Only νe is shown to keep the ﬁgure simple. For νµ and ντ , the photon- initiated cross sections are smaller (Fig. 3.10), while the quark-initiated cross sections are basically the same. See text for details.

p for both diagrams of Fig. 3.1 compared to (k p )2 (motivated by the ﬁrst diagram) − 1 − 1 p or (k p )2 (motivated by the second diagram). The result is only 10% larger than − 1 − 2 ' that using the default factorization and renormalization scales of MadGraph. Changing both scales to 2√sνγ or √sνγ/2 would increase or decrease the cross section by 15%. For quark- ∼ initiated subprocess, the diagrams can be found in Fig. 8 of Ref. [40], plus another one from

74 Inelastic only

Alikhanov, e 39 10 e C C D IS ]

1 0 .1 V e G

m 40 c

[ 10

e E D / i f , f t ra n c e t r iv e e h , o e C

41 10 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 3.10: Our inelastic neutrino-nucleus cross sections for W -boson production on 16O (solid lines), for all three ﬂavors. Also shown are previous results from Alikhanov [40] and, for comparison, coherent and diﬀractive cross sections of νe from Fig. 3.6. The corresponding antineutrino cross sections are the same.

replacing the Z by a photon in its upper middle diagram. We use the default factorization and renormalization scales of MadGraph, as using √sν,quark causes calculational problems.

Figure 3.9 shows the results for νe. (For νµ and ντ , the discussion below also ap- plies.) The photon-initiated subprocess is much larger than the quark-initiated process, because softer photons are favored (photons in the low-Q2 region). The contribution

75 Inelastic only Inelastic only

40 40 10 10

41 ] ] 41 10 1 10 1 V V e e 42 G G 10 2 2

m 42 m c c

10 [ [

43 10 E E / / + + A A e, e, + + 43 e, e, 10 + 44 + ee , , e e + 10 e e + + ee + + + e e 44 45 e ee e 10 0 1 2 3 4 5 6 7 8 10 0 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 E [ GeV ] E [ GeV ]

Figure 3.11: Our cross sections for trident production in the inelastic regime. Left: CC channels. Right: CC+NC and NC channels. Solid: the photon-initiated subprocess. Dashed: quark-initiated subprocess. The corresponding antineutrino cross sections are the same.

from protons is larger than that from neutrons, and their ratio is similar for both subprocesses. For the quark-initiated subprocess, also shown are the contributions from

photon-propagator diagrams 2 and from the W/Z-propagator diagrams 2, with the former | | | | being much larger than the latter. This indicates the relative importance, though the calculation of each component separately would break gauge invariance, especially above

E 108 GeV where it is not numerically stable and the mixings between photon and ν ∼ weak bosons are large. However, because the photon-propagator diagrams dominate the quark-initiated subprocess and, as mentioned in the last subsection, are already included in the photon-initiated process (inelastic photon PDF), we can ignore the quark-initiated subprocess in our calculation. (In other words, as long as the solid lines in Fig. 3.10 are much larger than the dot-dashed line in Fig. 3.9, the quark-initiated process can be ignored.) This also avoids the double-counting problem mentioned above. A more complete treatment that includes both subprocesses while avoiding double counting is beyond the scope of this work. One way is to use the W and Z PDFs [308–310], which may appear in future PDF sets.

76 Figure 3.10 shows the inelastic cross sections for all three ﬂavors. As before, σνeA >

σνµA > σντ A. Also shown are νe coherent and diffractive cross sections, for comparison. The inelastic cross section is the largest for most energies. Our result is smaller than the result of Alikhanov [40]. The difference is due to multiple reasons. First, we use much more up-to-date photon PDFs. Second, we use the dynamic scale √sνγ, which is more appropriate, while they used the fixed scale, mW . Third, we use MadGraph which does a full systematic calculation while they used the EPA.

3.5.3 Trident production

Figure 3.11 shows the cross sections for trident production processes in the inelastic regime.

The calculational choice is the same as that for W -boson production in the last subsection.

Different from before, here we separate CC channels (left) and CC+NC, NC channels (right) into two different panels. Our calculation is the first to include the nonperturbative part of the photon PDF for the trident production in which the resonance region is included. So here we start from Eν = 1 GeV. For CC channels, similar to those of W -boson production, the photon-initiated subprocess is much larger than the quark-initiated subprocess, and the latter is dominated by photon propagators to the quarks (not shown, for simplicity), which is already included by the former. Therefore, for the total inelastic cross section, we can also ignore the quark-initiated subprocess.

For CC+NC and NC channels, interestingly, the quark-initiated subprocess could be

(much) larger than the photon-initiated subprocess. The reason is that a pair of charged leptons can be split by a virtual photon emitted from the initial- or final-state quark on top of the NC DIS process, which does not happen in the CC channels as the two charged leptons have different flavors. (Radiative corrections through virtual W /Z bosons also exist but are suppressed by their large masses.) Therefore, the quark-initiated subprocesses of CC+NC and NC channels are enhanced compared to those of CC channels, and the lighter the final-state charged leptons, the larger the enhancement. For the total inelastic cross section of CC+NC and NC channels, we can sum the two subprocesses up, as the contribution from the double-counting region is negligible.

77 38 10 e

39 + C 10 CD IS × + 0.1 ] e e

1 W- 40 bos on V 10 pro e du ctio

G n Tridents (CC and CC+NC) 2 +

m e 41 ee , c e 10 , e e + [

+ ee e, Tridents (NC)

E + e, + /

A 42 e 10 + + + + ee e, e, + 43 10

44 10 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 E [ GeV ]

Figure 3.12: Our total cross sections (actually σνA/Eν) for W -boson and trident production on 16O. The colors and line styles are same as in Fig. 3.4( red, green, and blue lines are νe-, νµ-, and ντ -induced channels, respectively; solid lines are CC channels, and dashed lines are CC+NC channels; magenta dotted lines are NC channels, which depend on only the ﬁnal-state charged leptons). The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5) of Fig. 3.5. The corresponding antineutrino cross sections (i.e., obtained by CP-transforming the processes shown) are the same. See text for details.

Our result is diﬀerent from the DIS cross sections calculated in Ref. [41]. With limited details provided by Ref. [41], it is hard to trace the exact reason. In addition to the quark PDFs considered by Ref. [41], we also consider the photon PDFs, which has the nonperturbative part. For the quark-initiated process, we may include more diagrams, including the radiative-correction diagrams mentioned above.

78 0.25 + e + A e W A + A W + A + A W + A 0.20 Seckel e (iron)

0.15 Our e (iron)

0.10

0.05 Ratio to CCDIS cross section

0.00 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 3.13: Ratios of the W -boson production cross sections to those of CCDIS ((ν + ν¯)/2) [24]. Solid lines are for water/ice targets, dotted line for iron targets, and dashed lines are for the Earth’s averaged composition. Color assignment is in the legend. Also shown is the νe (iron) case of Seckel [38], much larger than ours.

3.6 Total cross sections of W -boson and trident production, and ratio to CCDIS

16 Figure 3.12 shows the total cross sections (σνA/Eν) with O for W -boson and trident production, which summarize our calculations in previous sections. We divide out Eν, the dominant trend, to highlight the deviations over the wide range of the x axis. Speciﬁcally, 79 the total cross sections are obtained by summing up the coherent, diﬀractive and inelastic components, for each interaction channel. For the W -boson production and trident

CC channels, the inelastic cross section is the photon-initiated subprocess, while for trident CC+NC and NC channels, it is from summing up both photon- and quark-initiated subprocesses. Reasons are discussed in Sec. 3.5.3.

We estimate the uncertainties of the W -boson and trident cross sections. For the coherent regime, it is about 6%, coming from higher order electroweak corrections (dominant) and nuclear form factors (see Ref. [43] for more details). For the diffractive regime, the nuclear effects lead to larger uncertainties. Our calculations include the Pauli-blocking factor derived from ideal Fermi gas model for nucleus and ignore other subdominant effects.

Therefore, to be conservative, we assume 30% uncertainty, following Ref. [43]. (Further details on the sensitivity to the choices of nuclear and nucleon form factors are given above.)

For the inelastic regime, the uncertainty mainly comes from choosing the factorization and renormalization scale, which is 15%, and mixing between photon and weak boson which ' matters more at higher energy, for which we give 25% to be conservative (Sec. 3.5). There ' is no study of the nuclear uncertainties of photon PDF, but they should be subdominant especially for light nucleus, like 16O, considering that of the quark PDFs are subdominant [311–314]. Their combination gives 30% for the inelastic regime. Combining the ' three regimes in quadrature, the uncertainty is estimated to be 15%. ' Figure 3.13 shows the ratios of the important channels to the neutrino CCDIS cross sec-

CCDIS CCDIS tions ((σν + σν¯ )/2) [24], for diﬀerent targets (right panel), including water/ice (for neutrino detectors), iron and the Earth’s averaged composition (for neutrino propagation).

The coherent cross sections ( Z2) for different isotopes are calculated in the same way ∝ as for 16O. For diffractive and inelastic cross sections, one can just rescale by atomic and mass numbers from 16O, as the cross section on single nucleon is independent of nucleus in above formalism (the Pauli-blocking factor in the diffractive cross section derived from ideal

Fermi-gas model [42, 43, 276] is nucleus independent). For hydrogen (1H) only, there is no coherent component, and the Pauli-blocking factor should not be included. The CCDIS cross sections for diﬀerent isotopes are calculated by multiplying their mass number by the

80 CCDIS cross section on isoscalar nucleon target [24]. The bias caused by the fact that some isotopes have a slightly diﬀerent number of protons and neutrons and by nuclear eﬀects on

PDFs is negligible.

The maximum ratios of W -boson production to CCDIS are 7.5% (ν ), 5% (ν ), ' e ' µ and 3.5% (ν ) on water/ice target, respectively. For trident production (not shown ' τ on the ﬁgure), the CC+NC and NC channels are large enough to matter, i.e., 0.75% ' (ν -induced), 0.5% (ν -induced), and 0.35% (ν -induced). This is a factor 0.1 of corre- e µ τ ' sponding W -boson production channels, as the dominant contribution at these energies are

from the W -boson production followed by W ν + ` (Sec. 3.3.2). These trident channels → ` and also hadronic decay of those W bosons could produce distinct signatures in the high-

energy neutrino detectors like IceCube [243], KM3NeT [258], and especially the forthcoming

IceCube-Gen2 [115]. In IceCube, there could already be a few W -boson production events.

IceCube-Gen2 will have much larger yields. The detectability and implications are detailed

in our companion paper [1]. (For clariﬁcation, the W -resonance enhancement part of the trident, which dominates above 104 GeV, is the W -boson production followed by leptonic ∼ decay of the W boson, so they would not be studied separately for detection.)

The maximum ratios of W -boson production to CCDIS are 14% (ν ), 10% (ν ), ' e ' µ and 7% (ν ) on iron, and 11% (ν ), 7.5% (ν ), and 5% (ν ) on the Earth’s ' τ ' e ' µ ' τ averaged composition. (The larger the charge number of a nucleus, the larger the ratio is, due to the coherent component Z2.) Trident CC and CC+NC channels (not shown) ∝ are 0.1 of these numbers, same as above. (As a comparison, the ratio for ν -iron case ' e by Seckel [38] is 25%, much larger than ours (14%).) This aﬀects the absorption rate of high-energy neutrinos when propagating through the Earth, which aﬀects the measurement of neutrino cross sections by IceCube [70, 71, 244–246].

3.7 Conclusions

The interactions of neutrinos with elementary particles, nucleons, and nuclei are a cornerstone of the Standard Model, and a crucial input for studying neutrino mixing, neutrino

81 astrophysics, and new physics. Above E 5 GeV, the neutrino-nucleus cross section ν ∼ is dominated by deep inelastic scattering, in which neutrinos couple via weak bosons to the quarks. However, additionally, there are two processes where the hadronic coupling is through a virtual photon: W -boson and trident production, the cross sections of which

increase rapidly and become relevant at TeV–PeV energies.

In this paper, we do a complete calculation of the W -boson and trident production

processes. We signiﬁcantly improve the completeness and precision of prior calculations.

We start by giving a systematic review of both processes, pointing out the improvements

that can/should be made compared to previous calculations (Sec. 3.2 and Table 3.1).

Our results can be put into three major categories. The neutrino-real photon cross sections for these two processes over a wide energy • range (Sec. 3.3). This sets the foundation for our neutrino-nucleus cross section

calculation and for discussing the underlying physics (Sec. 3.3). For trident pro-

duction, there are three diﬀerent categories of interaction channels (Eq. (3.5)):

CC, CC+NC, and NC, arising from diﬀerent groups of diagrams (Fig. 3.5). Inter-

2 estingly, for the CC and CC+NC channels, above sνγ = mW , the cross sections are enhanced by two orders of magnitude (Fig. 3.4). The reason is that the s-channel

like W -boson propagators in the trident diagrams (Fig. 3.5) can be produced on

shell (i.e., W -boson production in Fig. 3.1) and then decay leptonically. This indicates the uniﬁcation of these two processes.

A ﬁrst complete calculation of neutrino-nucleus W -boson production cross sec- • tions. For the neutrino-nucleus cross sections, we handle them in diﬀerent regimes:

elastic (Sec. 3.4; including coherent and diﬀractive) and inelastic (Sec. 3.5; includ-

ing both photon- and quark-initiated subprocesses). For W -boson production, we show that the previously used equivalent photon approximation [39, 40] is not good

for its cross section calculation. For the νe-induced channel, our results are about half of those in Refs. [39, 40] for both coherent and diﬀractive regimes, and 1/4 ∼ of Ref. [38] for coherent regime. The reasons for this are largely understood. A

82 signiﬁcant factor is the nonzero photon virtuality, Q2, which suppresses the cross

section (EPA assumes Q2 = 0 as used by previous works). Also, the Pauli-blocking

eﬀect suppresses the diﬀractive cross section. For the inelastic regime, we use an

up-to-date photon PDF and more reasonably dynamic factorization scale, √sνγ. Moreover, we do a ﬁrst calculation of the quark-initiated subprocess and ﬁnd that

they can be neglected below 108 GeV. '

A ﬁrst calculation of neutrino-nucleus trident production cross sections at TeV– • PeV energies. The full Standard Model is used in order to also study the TeV–

PeV behavior, compared to the four-Fermi theory used by previous work. The

equivalent photon approximation is also avoided. Moreover, we do a more careful

treatment of the inelastic regime, and, importantly, we for the ﬁrst time use the

inelastic photon PDF [267, 303] for trident calculation, which includes the reso-

nance region as a component of the nonperturbative part of the inelastic photon

PDF. More improvements are detailed in the previous sections.

These cross section are large enough to matter (Sec. 3.6). For a water/ice target, the W - boson production cross sections are 7.5% (ν ), 5% (ν ), and 3.5% (ν ) of CCDIS [24]. ' e ' µ ' τ (For the corresponding CC and CC+NC trident channels, they are 0.1 times the num- ' bers above.) This means these processes are detectable, or will be detectable, by high- energy neutrino detectors like IceCube [243], KM3NeT [258], and especially the forthcoming

IceCube-Gen2 [115], with distinct signatures [1]. For the iron target or the Earth’s averaged composition, the W -boson production cross sections are 14%/11% (νe), 10%/7.5% (νµ), and

7%/5% (ντ ) of CCDIS [24]. This aﬀects the absorption rate of high-energy neutrinos during propagation. Moreover, the DIS cross sections extracted from in-Earth absorption as seen by IceCube [70, 71, 244–246] contain a contribution from W-boson production. Note the fact that the cross section aﬀects the absorption rate exponentially may make these processes even more important than that shown by the numbers above.

This paper sets the theoretical framework and calculates the cross sections of these processes. In our companion paper [1], we discuss the phenomenological consequences, in-

83 cluding the effects mentioned above and other aspects of high-energy neutrino astrophysics that these processes make a difference, such as neutrino cross-section and spectrum measurement, flavor ratio determination, neutrino mixing, and new physics.

3.8 Supplemental Material

3.8.1 W -boson production: amplitudes in the Standard Model

The leptonic amplitudes of the diagrams of Fig. 3.1, in the unitarity gauge, are µ egW µ k/ p/2 + m` ν 5 L = u¯ (p1) γ − γ (1 γ )u (k) ν(W ) , (3.15a) 1 2√2 (k p )2 m2 − × − 2 − ` (k−p1)µ(k−p1)ν gµν 2 µ egW ν 5 − mW L = u¯ (p1) γ 1 γ u (k) 2 − 2√2 − × (k p )2 m2 − 1 − W h i gρλ(q + p )µ + gλµ(p p k)ρ + gµρ(k p q)λ (W ) , (3.15b) × 2 1 − 2 − − 1 − λ

where the m` and mW are masses of the charged lepton (p1) and the W boson (p2), respectively, and (W ) the polarization vector of the W boson, for which we have

4 X pµpν ν(W )ν(W ) = gµν + 2 2 . (3.16) i i − m2 i=1 W

The amplitudes above are used for both Sec. 3.3.1( WBP = (Lµ Lµ), where µ is the M µ 1 − 2 photon polarization vector) and Sec. 3.4.2(L µ = Lµ Lµ). 1 − 2

84 3.8.2 Trident production: amplitudes in the Standard Model

The leptonic amplitudes of the diagrams of Fig. 3.5, in the unitarity gauge, are

(p1+k2)α(p1+k2)β 2 eg gαβ m2 Lµ = W u¯ (k ) γβ 1 γ5 v (p ) − W 1 8 2 1 2 2 − − × (p1 + k2) mW + imW ΓW − µ p/2 /q + m2 α 5 u¯ (p2) γ − γ (1 γ )u (k1) , (3.17a) × (p q)2 m2 − 2 − − 2

(k1−p2)α(k1−p2)β 2 gαβ 2 µ egW α 5 − mW L = u¯ (p2) γ 1 γ u (k1) 2 8 − × (k p )2 m2 1 − 2 − W h i gγβ( p k k + p )µ + gβµ(k p q)γ + gµγ(q + p + k )β × − 1 − 2 − 1 2 1 − 2 − 1 2 (p1+k2)γ (p1+k2)δ gγδ 2 − mW δ 5 u¯ (k2) γ (1 γ )v (p1) , (3.17b) × (p + k )2 m2 + im Γ × − 1 2 − W W W

(k1−p2)α(k1−p2)β 2 eg gαβ m2 Lµ = W u¯ (p ) γα 1 γ5 u (k ) − W 3 4 2 1 2 2 − − × (k1 p2) mW − − β 5 /q p/1 + m1 µ u¯ (k2) γ (1 γ ) − γ v (p1) , (3.17c) × − (q p )2 m2 − 1 − 1

(k1−k2)α(k1−k2)β 2 5 eg 1 γ gαβ m2 Lµ = Z u¯ (k ) γα u (k ) − Z 4 4 2 2 2 1 2 2 − − × (k1 k2) mZ − − p/ /q + m2 µ 2 − β 1 2 1 5 u¯ (p2) γ γ + 2 sin θW + γ v (p1) , (3.17d) × (p q)2 m2 −2 2 2 − − 2

(k1−k2)α(k1−k2)β 2 5 gαβ 2 µ egZ α 1 γ − mZ L = u¯ (k2) γ u (k1) 5 − 4 2 − 2 × (k k )2 m2 1 − 2 − Z β 1 2 1 5 /q p/1 + m1 µ u¯ (p2) γ + 2 sin θW + γ − γ v (p1) . (3.17e) × −2 2 (q p )2 m2 − 1 − 1 µ µ Note that for L1 and L2 , the width term of the W propagator, imW ΓW , should be included. These are used for both Sec. 3.3.2( Tri = ((Lµ Lµ +Lµ) (Lµ +Lµ)), same as Eq. (3.7)) Mi µ 1 − 2 3 − 4 5 and Sec. 3.4.3(L µ = (Lµ Lµ + Lµ) (Lµ + Lµ)). i 1 − 2 3 − 4 5

85 T/L 2 3.8.3 Trident production: kinematics and phase space for σνγ (ˆs, Q )

Following Refs. [41, 291], we give more details of the kinematics and the three-body phase

T/L 2 space of σνγ (ˆs, Q ) for trident production, and derive the case for the virtual photon. The momenta are labeled in Fig. 3.5.

In the CM frame of the ν-A interaction and treating the two charged leptons together

(p p + p , i.e., the total momentum of `+ and `−), we can write the 4-momenta by ≡ 1 2 s + Q2 k = (1, sin θ, 0, cos θ) , (3.18a) 1 2√s − s + Q2 s Q2 q = − , sin θ, 0, cos θ , (3.18b) 2√s s + Q2 − s l k = − (1, 0, 0, 1) , (3.18c) 2 2√s − s + l s l p = , 0, 0, − , (3.18d) 2√s 2√s

where Q2 q2 is the photon virtuality, l p2, s s (k + q)2 (for simplicity, we use ≡ − ≡ ≡ νγ ≡ 1 s s in this section only), and θ is the angle of the incoming particles with respect to ≡ νγ the direction of p, which is chosen to be the z axis.

Deﬁne another variable,

l Q2(cosθ 1) + s + scosθ + s Q2(3 cosθ) + s scosθ t 2q (k k ) = − − − . (3.19) ≡ · 1 − 2 2s

This relation allows us to rewrite sin θ and cos θ in terms of t, which is Lorentz invariant,

86 then putting back into Eq.( 3.18), we obtain p ! Q2 + s (l + Q2 t)(lQ2 + s( 2Q2 s + t)) l(s Q2) + s(3Q2 + s 2t) k = , − − − , 0, − − , 1 2√s s l 2√s(l s) − − (3.20a) p ! s Q2 (l + Q2 t)(lQ2 + s( 2Q2 s + t)) l(s Q2) + s(3(Q2 + s 2t)) q = − , − − − , 0, − − , 2√s l s 2√s(s l) − − (3.20b) s l k = − (1, 0, 0, 1) , (3.20c) 2 2√s − l + s s l p = , 0, 0, − . (3.20d) 2√s 2√s

To ﬁnd the expression of p1 and p2, it is easier to go to the rest frame of p. We

do a Lorentz transformation to boost to this frame, using β = s+l , γ = s√+l for the s−l 2 sl transformation matrix,

2 p 2 2 2 2 2 ! 0 l + 2Q + s t (l + Q t)(lQ + s( 2Q s + t)) l lt + s(2Q + s t) k1 = − , − − − , 0, − − , 2√l s l 2√l(l s) − − (3.21a) p ! t 2Q2 (l + Q2 t)(lQ2 + s( 2Q2 s + t)) (2ls lt + 2Q2s st) q0 = − , − − − , 0, − − , 2√l l s − 2√l(l s) − − (3.21b) s l k0 = − (1, 0, 0, 1) , (3.21c) 2 2√s − p0 =(√l, 0, 0, 0) . (3.21d)

The situation can be further simpliﬁed if we work in the frame where q0 is along the z axis. So we need to do a rotation of above. The rotation angle, ηq, can be determined by

q0[2] q0[4] sin ηq = p , and cos ηq = p . (3.22) q0[2]2 + q0[4]2 q0[2]2 + q0[4]2

87 So, ﬁnally, we have   s 2 2 2 2 2 2 2 00  l + 2Q + s − t (l + Q − t)(lQ + s(−2Q − s + t)) l(4Q + 2s − t) + (2Q − t)(2Q + s − t)  k1 =  √ , − , 0,  , 2 2 2 r 2  2 l 4lQ + (t − 2Q ) t−2Q2  ( ) 2 −2l l + 4Q (3.23a) r ! t − 2Q2 1 (t − 2Q2)2 q00 = √ , 0, 0, + 4Q2 , (3.23b) 2 l 2 l

p 2 2 2 2 ! 00 s − l (l + Q − t)(lQ + s(−2Q − s + t)) (2ls − lt + 2Q s − st) k2 = √ , − , 0, − , (3.23c) 2 l p(t − 2Q2)2 + 4lQ2 2p(t − 2Q2)2 + 4lQ2 √ p00 = ( l, 0, 0, 0) . (3.23d)

In this frame, the p1,2 can be written as

00 00 00 00 00 00 p1 = E1, +ρ sin θ cos φ , +ρ sin θ sin φ , +ρ cos θ , (3.24a)

p00 = E , ρ sin θ00 cos φ00, ρ sin θ00 sin φ00, ρ cos θ00 , (3.24b) 2 2 − − −

where θ00 and φ00 are the angles with respect to the photon, q00, in the current frame, q 2 2 E1,2 = ρ + m1,2, and

l2 2l m2 + m2 + m2 m22 ρ2 = − 1 2 1 − 2 . (3.25) 4l

The three-body phase space can be done by decomposing it into two two-body phase spaces [41, 294] (each is independently Lorentz invariant),

dl dPS (k , p , p ) = dPS (k , p) dPS (p , p ) (3.26) 3 2 1 2 2π 2 2 2 1 2

with dΩ dPS (x , x ) = β(x , x ) (3.27) 2 1 2 1 2 32π2 which is frame independent, and s 2 2 2 2 2 2(x1 + x2) (x1 x2) β(x1, x2) = 1 2 + − 4 . (3.28) − (x1 + x2) (x1 + x2)

88 The dPS2(k2, p), in the CM frame, can be written as

dΩ s l dΩ dPS (k , p) = β(k , p) = − , (3.29) 2 2 2 32π2 2s 16π2 and dΩ d cos θdφ 1 s 2 = = dt , (3.30) 16π2 16π2 8π s + Q2 s l − which is Lorentz invariant, where the ﬁrst step uses the azimuthal symmetry of the system in the CM frame, and the second step uses Eq. (3.19).

The dPS2(p1, p2), in the rest frame of p, can be written as

dΩ00 dPS (p , p ) = β(p , p ) (3.31) 2 1 2 1 2 32π2 where β(p1, p2) can be derived using Eq. (3.28), i.e., r 2(m2 + m2) (m2 m2)2 β(p , p ) β(l) = 1 1 2 + 1 − 2 , (3.32) 1 2 ≡ − l l2

i.e., Eq. (3.9), and dΩ00 = d cos θ00dφ00 is the solid angle with respect to the photon in the

rest frame of p (c.f. Eq. (3.24)).

T/L 2 Putting above together, we get the phase space of the oﬀ-shell cross section, σνγ (ˆs, Q ), for the trident production,

1 1 dl dt dΩ00 1 1 dl dt dΩ00 dPS3 = 2 β(l) 2 , or dPS3 = 2 β(l) . (3.33) 2 (4π) 2π 2(sνγ + Q ) 4π 2 (4π) 2π 2ˆs 4π

This is the same as the phase space for the real photon case, Eq. (3.8), but replacing the s bys ˆ s + Q2. The integration range of l is now (m + m )2, s or νγ ≡ νγ 1 2 νγ (m + m )2, sˆ Q2. And the integration range of t can be obtain from Eq. (3.19), which 1 2 − gives

l l l + Q2, s + 2 Q2 , or l + Q2, sˆ Q2 + 2 Q2 . (3.34) νγ − s − − sˆ Q2 νγ − When Q2 = 0, all the above return to the on-shell photon case.

89 3.8.4 Trident production: coherent and diﬀractive cross sections for all channels

Figure 3.14 shows our elastic (coherent + diﬀractive components) cross sections for all trident channels.

39 Coherent + diffractive only 10 + e e W W + W + + 40 , , e e 10 + e, e, + e, e, +

] e e

1 + 41 e e

V +

e 10 ee

G +

2 +

m ee c

+ [

42 + e ee e E 10 / + A +

43 10

44 10 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 E [ GeV ]

Figure 3.14: Our elastic cross sections for all trident channels. We add the coherent and diffractive components together to simplify the figure. The colors and line styles are same as in Fig. 3.4( red, green, and blue lines are νe-, νµ-, and ντ -induced channels, respectively; solid lines are CC channels, and dashed lines are CC+NC channels; magenta dotted lines are NC channels, which depend on only the final-state charged leptons). The trident CC, NC, and CC+NC channels correspond to diagrams (1)–(3), (4)–(5), and (1)–(5) of Fig. 3.5. Gray dashed lines are the coherent and diffractive cross sections for W -boson production from Fig. 3.6, shown as a comparison. For antineutrinos, which have the same corresponding cross sections, take the CP transformation of the channel labels. See text for details.

90 Acknowledgments We are grateful for helpful discussions with Brian Batell, Eric Braaten, Mauricio Bus- tamante, Richard Furnstahl, Bin Guo, Liping He, Matheus Hostert, Junichiro Kawamura,

Matthew Kistler, Yuri Kovchegov, Gordan Krnjaic, Shirley Li, Pedro Machado, Aneesh

Manohar, Kenny Ng, Alexander Pukhov, Stuart Raby, Subir Sarkar, Juri Smirnov, Carl

Schmidt, Xilin Zhang, and especially Spencer Klein, Olivier Mattelaer, Sergio Palomares-

Ruiz, Yuber Perez-Gonzalez, Ryan Plestid, Mary Hall Reno, David Seckel, and Keping

Xie.

We used FeynCalc [292, 293], MadGraph [289] and CalcHEP [290] for some calculations.

This work was supported by NSF grant PHY-1714479 to JFB. BZ was also supported in part by a University Fellowship from The Ohio State University.

91 Chapter 4 W -boson and trident production in TeV–PeV neutrino observatories

Detecting TeV–PeV cosmic neutrinos provides crucial tests of neutrino physics and astrophysics. The statistics of IceCube and the larger proposed IceCube- Gen2 demand calculations of neutrino-nucleus interactions subdominant to deep-inelastic scattering, which is mediated by weak-boson couplings to nuclei. The largest such interactions are W -boson and trident production, which are mediated instead through photon couplings to nuclei. In a companion paper [2], we make the most comprehensive and precise calculations of those interactions at high energies. In this paper, we study their phenomenological consequences. We find that: (1) These interactions are dominated by the production of on-shell W -bosons, which carry most of the neutrino energy, (2) The cross section on water/iron can be as large as 7.5%/14% that of charged-current deep-inelastic scattering, much larger than the quoted uncertainty on the latter, (3) Attenuation in Earth is increased by as much as 15%, (4) W -boson production on nuclei exceeds that through the Glashow resonance on electrons by a factor of 20 for the best-fit IceCube spectrum, (5) The primary signals ' are showers that will significantly affect the detection rate in IceCube-Gen2; a small fraction of events give unique signatures that may be detected sooner.

The contents of this chapter were published in [1].

4.1 Introduction

The recent detections of TeV–PeV neutrinos by IceCube [28, 44, 117, 243, 259, 260] are a breakthrough in neutrino astrophysics. Though the sources of the diffuse flux have not been identified, important constraints on their properties have been determined [248, 249,

92 251, 256, 315–325]. In addition, there is a candidate source detection in association with a blazar ﬂare [55, 326]. The IceCube detections are also a breakthrough in neutrino physics.

By comparing the observed spectra of events that have traveled through Earth or not, the cross section can be measured at energies far above the reach of laboratory experiments [70,

71, 237, 244–246]. And many models of new physics have been powerfully limited by the

IceCube data [78–88, 112–114].

With new detectors — KM3NeT [258], Baikal-GVD [327], and especially the proposed

IceCube-Gen2 (about 10 times bigger than IceCube) [115] — the discovery prospects will be greatly increased, due to improvements in statistics, energy range, and ﬂavor information.

At high energies, neutrino-nucleus interactions are dominated by deep inelastic scattering

(DIS) mediated by weak-boson couplings to nuclei [231, 233]. For charged-current (CC) interactions, νe leads to a shower, νµ leads to a shower and a long muon track, and ντ leads to two showers that begin to separate spatially at 100 TeV [117]. For neutral-current ∼ (NC) interactions of all ﬂavors, showers are produced. Cherenkov light is produced by muon tracks and through the production of numerous low-energy electrons and positrons in showers.

With these coming improved detection prospects, new questions can be asked, including the role of subdominant interactions. We focus on those in which the coupling to the nucleus and its constituents is through a virtual photon, γ∗, instead of a weak boson [38–

43, 118, 119, 226, 266, 269, 272, 273, 276, 277, 280, 281]. The most important processes are

on-shell W -boson production, in which the underlying interaction is ν + γ∗ ` + W , and ` → trident production, in which it is ν + γ∗ ν + `− + `+. → 1 2 In a companion paper [2], we make the most comprehensive and precise calculations

of these cross sections at high energies. The cross section of W -boson production can

be as large as 7.5% of the DIS cross section for water/ice targets (and as large as 14%

for iron targets, relevant for neutrino propagation through Earth’s core) [2]. For trident

production, the most important channels are a subset of W -boson production followed by

leptonic decays [2]. To set a scale, IceCube has identiﬁed 60 events above 60 TeV in 7.5

years of operation [259, 260], so taking these subdominant processes into account will be

93 30 10 Glashow resonance ( e) 31 10 ]

2 32

m 10

c CCDIS

[

NCDIS 33 10 rod. on p ) -bos , / W , / 34 ( e/ e 10

35 10 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 4.1: Cross sections between neutrinos and 16O for W -boson production [2], compared to those for CCDIS [24], NCDIS [24], and the Glashow resonance (¯ν e− W −, taking into e → account eight electrons) [25].

essential for IceCube-Gen2. Moreover, the W -boson and trident events have complex ﬁnal states, which may allow their detection even sooner, in IceCube.

In this paper, we detail the phenomenological consequences of these processes. In

Sec. 4.2, we focus on their cross sections. In Sec. 4.3, we focus on their detectability.

We conclude in Sec. 4.4.

94 4.2 W -boson production cross sections and implications

In this section, we briefly review the total cross section for W -boson production (Sec. 4.2.1; details are given in our companion paper [2]), and present new calculations of the differential cross sections (Sec. 4.2.2). Then we talk about the implications, including the cross section uncertainties (Sec. 4.2.3) and the effects on neutrino attenuation in Earth (Sec. 4.2.4).

4.2.1 Review of the total cross sections

The nuclear production processes for on-shell W bosons are

ν + A `− + W + + A0 , (4.1a) ` → ν¯ + A `+ + W − + A0 , (4.1b) ` → where A and A0 are the initial and final-state nuclei and ` is a charged lepton. The neutrino- and antineutrino-induced processes have the same total and differential cross sections, but there is flavor dependence. The coupling to the nucleus and its constituents is through a virtual photon, γ∗ (contributions from virtual W and Z bosons are only important for

E > 108 GeV [2]). The process has a high threshold, E 5 103 GeV, due to the large ν ν ' × mass of the W boson, though much lower than the threshold for the Glashow resonance, which peaks at 6.3 PeV. Above threshold, the leptonic decays of the W boson (branching ' ratio 11% to each flavor) lead to the dominant contributions to trident production. ' The interactions happen in three different scattering regimes — coherent, diffractive, and inelastic — in which the virtual photon couples to the whole nucleus, a nucleon, and a quark, respectively. The corresponding cross sections are calculated separately and added to give the total cross section. For the coherent and diffractive regimes, we deal with the hadronic part in a complete way, which takes into account the photon virtuality, instead of using the equivalent photon approximation (as in, e.g., Refs. [39, 40]). Moreover, in the diffractive regime, we include the Pauli-blocking effects that reduce the cross section [42, 43, 270, 276].

For the inelastic regime, we point out that there are two subprocesses: photon-initiated and quark-initiated. For the former, we use the up-to-date inelastic photon PDF of proton and

95 32 10

34 6 E = 106 GeV 10 E = 10 GeV 33 10 e ] ]

5 GeV

2 E = 10 2 m m

c e c

e [ [

5 GeV 10

E = 10 W 35 E E

d 10 d / / d d e W 35 E E 10

36 10 36 10 3 2 1 0 1 2 3 4 5 6 4 5 6 10 10 10 10 10 10 10 10 10 10 10 10 10 E [ GeV ] EW [ GeV ]

Figure 4.2: Left: Differential cross sections for W -boson production in terms of the energy 5 of the charged lepton, shown for each neutrino flavor and two typical energies (Eν = 10 GeV 6 −1 and 10 GeV). The y axis is Edσ/dE = (2.3) dσ/d log10 E, matching the log scale on the x axis, so that relative heights of the curves at different energies faithfully show relative contributions to the total cross section. Right: Same, in terms of the energy of the W boson.

neutron [267, 303] and dynamical factorization and renormalization scales. For the latter, we do the ﬁrst calculation and ﬁnd that this sub-process can be neglected below 108 GeV. ' A key result is that our W -boson production cross section is smaller than that of previous work [38–40].

Figure 4.1 shows our W -boson production cross sections on 16O for diﬀerent neutrino

ﬂavors, along with other relevant processes. The width of the Glashow resonance is due to the intrinsic decay width of the W boson.

4.2.2 New results for the diﬀerential cross sections

For the differential cross sections, the most relevant results to detection are the energy distributions of the charged lepton (E`) and the W boson (EW ). The energy that goes to the hadronic part is negligible (see next paragraph). As above, the differential cross sections are calculated separately for the three regimes and summed. For the coherent and diffractive regimes, the phase-space variables we chose to calculate the total cross section in

96 Ref. [2] are not directly related to the energies of the ﬁnal states, so some transformations are needed; see Appendix 4.5.1. For the inelastic regime, following Ref. [2], we use MadGraph

(v2.6.4) [289] and analyze the event distributions in terms of the relevant quantities.

2 The energy that goes to the hadronic part, ∆Eh = Q /2mh (Appendix 4.5.1), is negligible compared to the detection threshold, which is 100 GeV for showers in IceCube. ∼ Here Q2 q2 is the photon virtuality; the hadronic mass, m , is the nuclear mass in ≡ − h the coherent regime and the nucleon mass in the diﬀractive and inelastic regimes. For the coherent and diﬀractive regimes, the corresponding nuclear and nucleon form factors are highly suppressed above Q 0.1 GeV and Q 1 GeV, respectively, which leads to ∼ ∼ ∆E (0.1)2/2/16 0.0003 GeV and ∆E (1)2/2/1 0.5 GeV. For the inelastic h . ' h . ' regime, although Q2 could be much larger, the cross section is still dominated by the low-

Q2 region (Q2 10 GeV2, i.e., ∆E 10/2/1 5 GeV) because the nonperturbative part . h . ' of the inelastic photon PDF [267, 300] dominates the cross section (see Sec. V.B of Ref. [2]).

Above is very diﬀerent the DIS, in which the energy transferred to the nucleus is 25%Eν on average [24, 231, 233, 237].

Therefore, E E + E is an excellent approximation for the coherent and diffractive ν ' ` W regimes and a good approximation for the inelastic regime. We checked this through the distribution of the sum of EW and E` , finding that this is nearly a delta function at Eν. Figure 4.2 shows the differential cross sections for the charged lepton (left) and W

5 6 boson (right), for each neutrino flavor and two typical energies, Eν = 10 and 10 GeV, summed over the contributions from all three scattering regimes. For the charged lepton, the differential cross section is relatively flat when plotted as Edσ/dE, which means that no specific energy range is particularly favored; the narrow bump near Eν does not contribute significantly to the total cross section. For the W boson, the differential cross section favors the highest possible energy, E E m . The differences between different flavors are W ' ν − ` due only to the charged lepton mass, m`, which sets the lower limit of the distribution. Therefore, when E m , the results for different flavors converge. This induces the ` ` opposite feature for the distribution of W boson, where the results for different flavors converge at lower energies.

97 Figure 4.3 shows the average energy for the charged lepton ` and the W boson for each

ﬂavor of initial neutrino. This is calculated as

R dσ E 1 dE E (E,Eν) h i = dE , (4.2) Eν Eν σ(Eν) where E = E` or EW . As can be expected from Fig. 4.2, the W boson is typically much more energetic than the charged lepton, except for when Eν is very large. In a crude approximation, at the main neutrino energies relevant to detection in IceCube or IceCube-

Gen2, all the neutrino energy goes to the W boson, with some dependence on neutrino

ﬂavor. Below, we provide more careful calculations.

4.2.3 Implication: Cross-section uncertainty

Figure 4.4 shows the ratios of the W -boson production to the neutrino CCDIS cross sections

CCDIS CCDIS ((σν + σν¯ )/2) [24]. We neglect the NCDIS cross section because it is smaller (see Fig. 4.1) and because the energy deposition is only 0.25E , which suppresses its ' ν importance [328]. What we show is most relevant for detection. For a water/ice target, the maximum ratios of W -boson production to CCDIS [24] are 7.5% (ν ), 5% (ν ), ' e ' µ and 3.5% (ν ). For an iron target or Earth’s average composition, the maximum ratios ' τ are 14%/11% (νe), 10%/7.5% (νµ), and 7%/5% (ντ ). This is more relevant to propagation (affected by both CCDIS and NCDIS) than detection (dominated by CCDIS); see Sec. 4.2.4 for details. The larger the charge number of a nucleus, the larger the ratio is. The coherent component is Z2, while the diffractive and inelastic components are Z, the same as for ∝ ∝ DIS. As noted, our results are significantly smaller than those of Seckel [38].

For the CCDIS cross section, the claimed uncertainties (from the parton-distribution functions) in 104–108 GeV are 1.5–4.5% in Ref. [24] and 1–6% in Ref. [237] (see also

Refs. [239, 307]). The impact of W -boson production is thus signiﬁcant and thus should be included in future calculations of neutrino-nucleus cross sections. Further, as IceCube has detected 60 events above 60 TeV (deposited energy) in the past 7.5 years [259, 260], this means that taking W -boson production into account is relevant for IceCube and essential for IceCube-Gen2, which would be about 10 times larger. This is detailed in Sec. 4.3.

98 1.0 0.9 ( ) 0.8 W e W ( ) 0.7 ( ) 0.6 W E

/ 0.5 E 0.4 ( ) 0.3 ( ) 0.2 ( e) 0.1

0.0 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 4.3: Average energy of the charged lepton (`) and W boson, divided by Eν, for each neutrino ﬂavor.

For future calculations of the cross-section uncertainties, aiming to reach the few-percent scale, we note some other corrections that should be taken into account. The DIS calculations are done at next-to-leading order in QCD, using corresponding parton distribution functions. However, as far as we are aware, next-to-leading order electroweak corrections [329–333] are not included. At the highest energies, nonperturbative electroweak cascades [334] may be a signiﬁcant eﬀect. Finally, there are other processes, such as tri-

99 charged lepton production [335–338], that become increasingly important at high energies.

In addition, in Ref. [339], Klein notes that going beyond assuming isoscalar nucleon targets and nuclear eﬀects on the parton distribution function should also be considered.

0.25 + e + A e W A + A W + A + A W + A 0.20 Seckel e (iron)

0.15 Our e (iron)

0.10

0.05 Ratio to CCDIS cross section

0.00 4 5 6 7 8 10 10 10 10 10 E [ GeV ]

Figure 4.4: Ratios of the W -boson production cross sections [2] to those of CCDIS ((ν + ν¯)/2) [24]. Solid lines are for water/ice targets, dotted for iron targets, and dashed for Earth’s average composition. Color assignments are noted in the legend. For comparison, we also show the νe (iron) result of Seckel [38], which is much larger than ours.

100 4.2.4 Implication: Attenuation in Earth

Starting in the TeV range, neutrinos may be signiﬁcantly attenuated while passing through

Earth. (For a path along an Earth diameter, τ = 1 at E 40 TeV.) Attenuation depends ν ' on the total CCDIS + NCDIS cross section, σ(Eν). Taking into account NCDIS increases the cross section by a factor 1.4 compared to CCDIS only [24, 231, 233, 237]. We ignore ' neutrino regeneration because of the steeply falling neutrino spectra.

The optical depth τ = Cσ, where C(cos θz) is the target number column density integrated along the line of sight, which depends on the zenith angle, θz. We use Earth’s average composition. The ﬂux is attenuated by a factor

A = e−τ(Eν , cos θz). (4.3)

The column density in the direction of the zenith angle θz is reasonably well known [340]. Though the change in the cross section due to W -boson production is not large, it aﬀects the argument of the exponential. For W -boson production, we calculate τ as the sum of results for the three regimes, taking into account that the targets in the coherent regime are nuclei, while in the diﬀractive and inelastic regimes they are nucleons.

Figure 4.5 (upper panel) shows the neutrino attenuation factor without (ADIS) and with

DIS+WBP (A ) W -boson production. For simplicity, we consider only νe, which has the largest such cross section (see Fig. 4.1).

Figure 4.5 (lower panel) shows the relative change to the attenuation factor,

1 ADIS+WBP/ADIS. For high energies and long paths through Earth, this can be − quite large, even though the W -boson production cross section is small compared to the CCDIS cross section. However, for A that is too small, the event rate would be too low to matter; accordingly, we use thin lines for where A 0.1. Even ≤ avoiding these regions, the change in A can be as large as 15%. This follows from

1 ADIS+WBP/ADIS = 1 exp( Cσ ) Cσ σ /σ τ σ /σ , which − − − WBP ' DIS × WBP DIS ' × WBP DIS is the multiplication of the optical depth and the cross section ratio, which 10%/1.4 7% ' ' for νe (from Fig. 4.4). The factor 1.4 roughly accounts for including NCDIS in addition to

101 CCDIS. For example, when A = 0.1, i.e., τ 2.3, 1 ADIS+WBP/ADIS 2.3 7% 15%, ' − ' × ' as above.

Interestingly, in IceCube’s through-going muon analysis, there was an unknown 2% deﬁcit of straight up-going events, compared with their Monte Carlo simulation [341]. Tak- ing W -boson production into account may explain this deﬁcit.

The Earth attenuation eﬀect allows a measurement of the neutrino-nucleus cross section at TeV–PeV energies in IceCube [70, 71, 237, 244–246]. The energy scales probed are far above those of laboratory experiments, for which the highest beam energies are 350 ' GeV [240–242]. In essence, the downgoing neutrino event rate depends on φσ, while the upgoing event rate depends on φσe−τ , and taking a ratio cancels the ﬂux and the detection cross section.

In present IceCube measurements of the cross section, the uncertainty is 35% when ' only the cross section normalization is checked (one wide energy bin for all data, plus assuming the shape of the standard model cross section) [70] or a factor of 4 when these ' assumptions are relaxed (several energy bins, no prior on the cross section shape) [71].

The energy ranges of both are comparable to where W -boson production is important.

However, the measurement uncertainties will decrease. In addition, in Ref. [70] the ratio of the measured cross section to DIS prediction is 1.3 0.45. The central value would be ± about 0.1 smaller if the contribution from W -boson production were included.

Last, attenuation effects also lead to slightly altered flavor ratios because the W -boson production cross sections are flavor dependent.

4.3 Detectability

In this section, we calculate the detection prospects. We focus on W -boson production.

The most important channels of trident production are a subset of W -boson production followed by leptonic decays [2]. We ﬁrst calculate the W -boson yields compared to those through the Glashow resonance (Sec. 4.3.1). Then, after a brief review of IceCube detection

(Sec. 4.3.2), we calculate the detectability of W -boson production from the shower spectrum

102 0 10

E = 105 GeV

1 6 7 10 10 GeV 10 GeV

Neutrino attenuation factor 2

S 10 I

D 0.2 A / P B W

+ 0.1 S I D A 0.0

1 1.0 0.8 0.6 0.4 0.2 0.0 Neutrino zenith angle, cos z

−σC DIS Figure 4.5: Upper: Neutrino attenuation factor, e , for νe in Earth. Dashed lines (A ) are for CCDIS and NCDIS without W -boson production. Solid lines (ADIS+WBP) include W -boson production. For attenuation factors below 0.1, the event rate is too low to use, which we denote by using thin lines. Lower: The relative change in the attenuation factor due to W -boson production.

(Sec. 4.3.3) and from unique signatures (Sec. 4.3.4).

103 4.3.1 Larger W -boson yields than Glashow resonance

The Glashow resonance (¯ν +e− W −) is well known for producing on-shell W bosons with e → a narrow feature in the cross section around Eν = 6.3 PeV. The maximum cross section is 10−30 cm2, a factor of about 100 larger than that of DIS. Once the intrinsic and detector ∼ energy resolution are taken into account, the eﬀects on the total event spectrum are less

dramatic but still important.

Surprisingly, we find that on-shell W -boson production is actually dominated by neutrino-nucleus interactions where the coupling to the nucleus is through a photon. The cross section is much smaller, but it involves all six neutrino flavors, and acts over a much wider energy range, in particular at lower energies, where the neutrino fluxes are much larger.

Figure 4.6 illustrates this. We multiply the cross sections by a power-law ﬂux,

1−α −2 −1 −1 Eν dΦ/dEν = (Eν/1 GeV) cm s GeV , with unit normalization, and plot results versus neutrino energy. We use α = 2.9, which matches the astrophysical neutrino spectrum from ﬁtting IceCube data [28, 260]. (Below, we calculate more realistic expectations for detection.) We multiply the ﬂux by a factor Eν so that the relative heights on the y axis faithfully display the relative numbers of events per logarithmic energy bin.

The yield from neutrino-nucleus W -boson production is a factor of 20 times that for ' the Glashow resonance. For α = 2.5 and 2.0, the factor is 3.5 and 0.5 respectively. ' ' Therefore, for TeV–PeV neutrino observatories, neutrino-nucleus W -boson production is the dominant source of on-shell W bosons unless the spectrum is very hard. This is a new and interesting physics point. When it comes to detection, the W bosons are not detected directly, due to their short lifetimes, and are instead detected by their decay products.

4.3.2 Review of detection in IceCube

We brieﬂy summarize the neutrino detection techniques used in IceCube and similar detectors [44, 116, 243, 342]. Neutrinos interact with nuclei and electrons, producing relativistic particles that emit Cherenkov light that is detected by photomultiplier tubes.

104 42 10

] Glashow resonance ( )

e 1 s

[

E d / 43

d 10 E × ) W-boson production E (

A (sum of 6 flavors) 44 10

3 4 5 6 7 8 10 10 10 10 10 10 E [ GeV ]

Figure 4.6: Relative W -boson yields due to W -boson production (νl + A l + 0 − − → W + A ) and the Glashow resonance (¯νe + e W ). We use dΦ/dEν = −2.9 −2 −1 −1 → (Eν/1 GeV) cm s GeV with unit normalization. The yield from W -boson production is 20 times that from the Glashow resonance, which can be seen by logarithmic ' integration of the peaks. The CCDIS (cyan, dashed) and NCDIS (magenta, dashed) cases are shown for comparison, though they do not produce on-shell W bosons.

A νe CCDIS event produces an electron that carries most of the neutrino energy and hadrons that carry the remainder. The electron initiates an electromagnetic shower of electrons, positrons, and gamma rays, with most of the Cherenkov emission coming from the most numerous low-energy but still relativistic charged particles. The hadrons initiate a hadronic shower that consists primarily of pions. The charged pions continue the hadronic

105 3 Conservative case 3 Optimistic case 10 10 C C 2 D 2 10 IS 10 , p p

e NCDIS, sume , of 6 flavors e NCDIS, sum of 6 flavors d d 1 e a 1 E nd E d 10 d 10 / / , N N

d 0 Glashow d 0 Glashow p p

e 10 e 10

d resonance d resonance

E W-boson production E W-boson production 1 1 10 (sum of 6 flavors) 10 (sum of 6 flavors)

2 2 10 10 0 0 10 10 B B 1 1

/ 10 / 10 S S

2 2 10 3 4 5 6 7 8 10 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 10 Edep [ GeV ] Edep [ GeV ]

Figure 4.7: Left: Shower spectrum (Upper) and detection significance of W -boson production (Lower) for the conservative case as regards identifying W -boson production events. Right: Same, but for the optimistic case. The main difference between the two cases is the change with the CCDIS channel. The shaded region below 60 TeV is below the IceCube threshold for cleanly identifying astrophysical neutrinos. We use 0.5 km3 as the approximate fiducial volume of IceCube [44], and assume 10 years of IceCube data. For E > 60 TeV, where W -boson production contributes 6 shower events, the cumulative dep ' detection significance should be 1.0σ for the conservative case and 3.2σ for the optimistic ' case. See text for details.

shower, but the neutral pions decay promptly, feeding the electromagnetic shower. The shower components induced by a νe event have high and comparable light yields, so that the total amount of Cherenkov light is proportional to Eν. Because of the light scattering in ice, a shower looks like a large ( 100 m), round blob, even though the shower is a narrow ∼ cigar-shaped blob of length 10 m. Forν ¯ , the total and diﬀerential cross sections are ∼ e slightly diﬀerent, but the detection principles are the same.

For ντ (andν ¯τ ) CCDIS events, the results can be very similar to those for νe (andν ¯e). The τ decay produces a hadronic shower that is displaced in time and position, though

these displacements start to become identiﬁable in IceCube only above 100 TeV [117]. ∼ At lower energies, ντ events are nominally indistinguishable from νe events. A way forward

could be possible using muon and neutron echoes [116]. For ντ (andν ¯τ ) events, the average

106 deposited energy is 20% less than E due to losses of neutrinos from τ leptonic decays. ' ν In addition, 17% of τ decays produce muon tracks, producing separable events.

All six ﬂavors of neutrinos cause NCDIS events that also produce showers. These appear identical to the other shower events above, though the energy deposition is typically only

0.25E . Because of the falling neutrino spectra, these events matter much less in the ' ν detection spectra [328].

For νµ (andν ¯µ) CCDIS events, the topologies are quite diﬀerent because the muon range is so long, already > 1 km at E 200 GeV. For events where the neutrino interaction µ ' is inside the detector (known as a contained-vertex or starting event), there is a hadronic

shower and a long muon track, which itself produces small showers along its length. Though

the muon is not contained, its energy can be estimated from its energy-loss ﬂuctuations, so

that the neutrino energy can be estimated. There can also be events where the neutrino

interacts far outside the detector, and only the throughgoing muon is detected. This enlarges

the eﬀective volume of the detector, but then only a lower limit on the neutrino energy can

be set.

Shower events are especially important because of the ability to faithfully reconstruct

the neutrino spectrum. The shower spectrum can be estimated as [87, 238, 315]

Z 1 dN ρiceVfidNA dΦ −τ(Eν , cos θz) Edep = 2π T d cos θz Eν (Eν) σ(Eν) e , (4.4) dEdep 18 −1 × dEν where E is the energy deposited in the detector from a shower, ρ 0.92 g cm−3 the dep ice ' 3 density of ice, Vfid = 0.5 km is the approximate fiducial volume of IceCube [44], NA the Avo- gadro number (ρiceVfidNA/18 gives the number of water targets), and T is the exposure time. For the neutrino flux, dΦ , we use that of Ref. [28], which includes both the atmospheric dEν

(dominated by νµ andν ¯µ) and astrophysical neutrino fluxes. The best-fit astrophysical flux, assuming 1:1:1 flavor ratios, is (2.46 0.8) 10−18(E/100 TeV)−2.92 GeV−1 cm−2 s−1 sr−1 ± × for each flavor (ν +ν ¯), which is consistent with a more recent result [260]. The σ is the

cross section between neutrino and water for diﬀerent interaction channels. For CCDIS

and NCDIS, we multiply the cross section on isoscalar nucleon targets [24] by 18, the mass

number for water. For the Glashow resonance, we multiply the cross section on electrons by

107 10, the charge number for water. For the attenuation factor, e−τ(Eν , cos θz), we use Ref. [343] with modiﬁcation to include the cross section for W -boson production. Once we obtain the shower spectra, we convolve them with a detector energy resolution of 15% [344].

4.3.3 Total shower detection spectrum

For the detection of W -boson production events, we consider two general scenarios. In this subsection, we consider ﬁnal states that contribute to the overall shower spectrum.

(The prospects for detection via track events are not favorable.) In the next subsection, we consider unique ﬁnal states that can be individually identiﬁed. We focus on IceCube [243], the largest detector for TeV–PeV neutrinos. Our results can be scaled to the proposed

IceCube-Gen2, whose instrumented volume is expected to be 10 times that of IceCube [115], and ﬁducial volume may be more, though with a higher energy threshold.

Table 4.1 shows the complex possibilities for W -boson production events, including pure shower, track, and other unique signatures, depending on the decay modes of the W boson and τ leptons. For the charged leptons from the initial interactions, their detectability depends on IceCube’s trigger threshold ( 100 GeV; note the analysis threshold of IceCube, ' 1 TeV, is less relevant because one would be searching for a lower-energy event in asso- ' ciation with a higher-energy event) [116]. Fig. 4.2 (left) is, up to an overall factor 2.3−1, the probability distribution in log E , and it is roughly ﬂat, with a median 100 GeV. 10 ` ∼ Therefore, we assume that half of the primary leptons are detectable and half not. In the

“Signatures” column of Table 4.1, we use “/” to distinguish the two cases. (The decaying

W ’s are always detectable.)

First, we calculate the change to the overall shower spectrum in the conservative case where ντ events appear as showers and where all showers are indistinguishable. We ignore events with an energetic muon track: ν CCDIS, ν CCDIS with τ µ, half of ν -induced µ τ → µ W -boson production, ν - and ν -induced W -boson production with W µ, τ µ, or e τ → → W τ µ, and Glashow resonance events with W µ or W τ µ. → → → → → Second, we calculate for an optimistic case where ντ CCDIS events are identiﬁable through a double-bang or double-pulse signature (above 100 TeV, this is becoming real- ∼ 108 istic with current technology [117]), and where electromagnetic and hadronic showers can be separated using echo techniques (this is not yet possible with IceCube, but it may be with IceCube-Gen2). Therefore, for CCDIS, we remove the remaining νe- and ντ -induced channels. For the W -boson production, we remove the channels that give tracks, pure EM showers (EM means electromagnetic), and other unique signatures (see Table 4.1).

For ν CC, E E , as both the ﬁnal state electron and hadrons produce showers. e dep ' ν For ν CC, E [ y + 0.7(1 y )] E 0.8E , where y 0.25, for both CCDIS and τ dep ' h i − h i ν ' ν h i ' NCDIS, is the average inelasticity, which is the fraction of neutrino energy transferred to the hadrons [231]. The factor 0.7 above is due to about 30% of the energy is taken away by neutrinos from τ decay. For all-ﬂavor NCDIS, E y E 0.25E . The ratios for dep ' h i ν ' ν CCDIS and NCDIS above are the similar to those used in Refs. [87, 116, 238].

For W -boson production, we use E E if the W decays hadronically and E dep ' ν dep ' 0.5Eν if the W decays leptonically. At the relevant energies, the W boson takes nearly all of the neutrino energy; even when it does not, the approximations here are good for the total energy deposition, because the charged lepton (e or τ) from the initial interaction deposits most of its energy. We make the same assumptions for W bosons produced via the Glashow resonance.

Figure 4.7 shows the total shower spectrum for the conservative (left) and optimistic

(right) cases for IceCube observations with T = 10 years (or 1 year of IceCube-Gen2 [115]).

W -boson production is subdominant, especially in the conservative case. The CCDIS events dominate, due to the large cross section and energy deposition. The NCDIS events are reduced in importance due the small energy deposition. The two-peaks feature of the

Glashow resonance is due to the leptonic and hadronic decays of W bosons.

The lower panels show the detection significance. For each bin, this is calculated by the number of W -boson production events divided by the square root of CCDIS+NCDIS events. The cumulative significance for detecting W -boson production, combining all the bins above E = 60 TeV, is 1.0σ for the conservative case and 3.2σ for the optimistic dep ' case, for 10 years of IceCube observations. The Glashow resonance events are not included because doing so would not appreciably affect the results.

109 Table 4.1: Diﬀerent ﬁnal state particles, signatures, corresponding fractions, and counts in IceCube. The counts are for greater than 60 TeV deposited energy and 10 years of IceCube observations (or 1 year for IceCube-Gen2). The numbers in the “Channel” column are the maximal ratios to the CCDIS cross section with water/ice. The numbers in “W decay” and “τ decay” columns are the branching ratios. For the “Final state” and “τ decay” columns, we omit the neutrinos; “h” means hadrons. The unique signatures are in boldface. The “/” divides the cases in which the charged lepton from the initial interaction is undetectable or detectable, which, to a good approximation, is half-half. The “Fractions” column shows the fraction of that row relative to the whole channel, which is the multiplication between the branching ratios of W and τ decay.

Final Channel W decay τ decay Signature Fraction Counts state

eνe, 11% e e Pure EM shower 11% 0.34 µνµ, 11% e µ Track without/with shower 11% 0.34 νe → eW e, 18% Pure EM shower 2.0% 0.06 (7.5% rel. τντ , 11% e τ µ, 17% Track without/with (displaced) shower 1.9% 0.06 to CCDIS) h, 65% Shower 7.2% 0.22 qq¯, 67% e h Shower 67% 2.08

eνe, 11% µ e Pure EM shower/Track with shower 11% 0.56 µνµ, 11% µ µ Single/Double tracks without shower 11% 0.56 e, 18% Pure EM shower/Track with (displaced) shower 2.0% 0.10 νµ → µW τντ , 11% µ τ µ, 17% Single/Double tracks without shower 1.9% 0.10 (5.0% rel. h, 65% Shower/Shower with (displaced) track 7.2% 0.36 to CCDIS) qq¯, 67% µ h Shower/Shower with track 67% 3.41 e, 18% Pure EM shower 2.0% 0.02 eνe, 11% τ e µ, 17% Pure EM shower/Track with (displaced) shower 1.9% 0.02 h, 65% Pure EM shower/Shower 7.2% 0.09 µ, 17% Single/Double tracks without shower 1.9% 0.02 µνµ, 11% τ µ e or h, 83% Track without shower/with (displaced) shower 9.1% 0.11 ντ → τW e e, 3% Pure EM shower 0.4% 0.004 (3.5% rel. µ µ, 3% Single/Double tracks without shower 0.3% 0.004 to CCDIS) τντ , 11% τ τ µ e/h, 29% Track without shower/with (displaced) shower 3.1% 0.04 h h/e, 65% Shower/Double bang 7.2% 0.09 e or h, 83% Shower 56% 0.69 qq¯, 67% τ h µ, 17% Shower/Shower with track 11% 0.14 Total counts 9.44

In summary, for Edep > 60 TeV and for 10 years of IceCube observations, W -boson production contributes 6 shower events, and the cumulative detection signiﬁcance should ' be 1.0σ for the conservative case and 3.2σ for the optimistic case. With 10 years of ' IceCube-Gen2, the counts would improve by a factor of 10 and the signiﬁcances by a ' factor √10 3.2. '

110 4.3.4 Unique signatures

Table 4.1 also shows the unique signatures (in boldface) that W -boson production could give in IceCube, including fractions and counts. The events that give unique signatures are from leptonic decays following W -boson production, therefore they are also trident events. The counts are calculated using Eq. (4.4). Some of them are background free compared to other standard model processes (DIS and the Glashow resonance). We focus on IceCube’s high- energy analysis, so the counts are for greater than 60 TeV deposited energy and also for 10 years of IceCube observations. For lower-energy analysis of IceCube, such as the medium- energy starting events [345] and the Enhanced Starting Track Event Selection [346], these unique signatures could also exist.

There could be “Double tracks” signatures, with one muon track from the initial interaction (µ or τ µ) and the other from the decay of W boson. See Table 4.1 for all the → contributing channels. The calculated counts are 0.34. Double tracks could also come ' from outside the detector, which would increase the counts. Some of the “Double tracks” events may have separation angles (θ) too small for them to be distinguished from a single track [273]. According to Refs. [347, 348], IceCube’s resolution for double tracks is about

150 m. Therefore, for the “Double tracks” traveling 1 km, as long as the cos θ . 0.99, they can be separated. Moreover, the “Double tracks” would be a background for dimuon-type new physics searches [348–350].

These “Double tracks” may also be identiﬁed because they are a subset of “track without shower” signatures. The no-shower feature is because the energy transferred to the hadronic part in W -boson production is mostly negligible (Sec. 4.2.2). The “track without shower” signatures are mostly background free, because in CCDIS the energy transferred to hadronic

5 part, for Eν > 10 GeV, is mostly above the IceCube threshold (100 GeV), according to the dσCCDIS/dy of Ref. [231]. Moreover, these signatures could also come from other channels of W -boson production (see Table 4.1 for details). The calculated total counts are 0.96. ' Interestingly, there is a “track without shower” candidate (Event 5) in the IceCube event list in Ref. [44] (arXiv version, page 15). It has no obvious shower activity, while all

111 seven other track events have prominent showers at their starting points. It is important to quantify the probability of this event coming from CCDIS. Event 5 deposited 71.4 TeV energy in IceCube, so the neutrino energy is 105 GeV. We can require that the hadronic ∼ energy be smaller than the full energy lost by muon in the initial 100 m of its path length.

This is conservative, as it would double the average energy deposited in the ﬁrst 100 m of the muon track compared to the second 100 m, which would be visible, unlike for Event 5.

According to Ref. [351], the corresponding energy loss of a 105 GeV muon would be about

5 4 TeV. From dσCCDIS/dy of Ref. [231] for Eν = 10 GeV, we can estimate the probability for having a hadronic energy smaller than a given value. For 100 GeV, it is 0.3%. For 4 ' TeV, it is 10%. Therefore, the probability for Event 5 being induced by CCDIS is small, ' even in conservative cases.

Moreover, “track with shower” events from W -boson production would also look diﬀer- ent from CCDIS in terms of the inelasticity distribution. For CCDIS, the dominant energy,

(1 y)E , goes to the track, with the smaller remainder, yE , going to the hadrons. For − ν ν those W -boson production events that are analyzed as CCDIS events with tracks, the non-

track energy comes primarily from the W -boson decay, and this is typically much larger

than the energy going to the track, in contrast to CCDIS. Therefore, W -boson production

events should be included in theoretical expectations of attempts to better measure the

νµ toν ¯µ ﬂux ratio and neutrino charged-current charm production [352]. This may also provide a way to detect W -boson production.

There could also be “Pure EM shower” signatures, where EM means electromagnetic.

The no-hadronic-shower feature is because the energy transferred to the hadronic part in

W -boson production is mostly negligible (Sec. 4.2.2). The major contributing channels are

W e following W -boson production induced by ν or by other ﬂavors with the initial → e charged lepton below the trigger threshold (see Table 4.1 for details). The calculated counts

are 0.82. This signature could be background free with the echo technique [116], same ' reason as for the “Track without shower” signature.

At last, there could also be the “Track + displaced shower” signatures (Table 4.1). The

calculated counts are 0.35. However, due to the short lifetime of the τ, it may be hard ' 112 to identify them. The ντ CC events with τ decay to muon will be a background. These unique signatures would help ﬂavor identiﬁcation. For example, the “Double track” and “Track without shower” signatures are dominated by νµ-induced W -boson production; the “Pure EM shower” are dominated by νe-induced W -boson production, and the “Track with displaced shower” is dominated by ντ -induced W -boson production. The unique signatures are also backgrounds for exotic signals.

4.4 Conclusions

It is time to study the role of subdominant neutrino-nucleus interactions for TeV–PeV neutrinos. These cosmic neutrinos provide essential probes for neutrino astrophysics and physics, and the statistics of IceCube and especially IceCube-Gen2 demand greater precision in the theoretical predictions used to interpret the data.

The most important subdominant processes not yet taken into account are those where the interaction of a neutrino with a nucleus and its constituents is through a virtual photon,

γ∗. These processes are W -boson production (ν + A l + W + A0) and trident production l → (ν + A ν + `− + `+ + A0). In a companion paper, we present the more comprehensive → 1 2 and precise calculations of these cross sections at high energies [2].

In this paper, we study the phenomenological consequences of these processes at TeV–

PeV energies for IceCube and related experiments. We have ﬁve major results:

1. These interactions are dominated by the production of on-shell W -bosons, which carry

most of the neutrino energy. The most important trident channels are a subset of W -

boson production followed by leptonic decays. The energy partition follows from the

calculation of the diﬀerential cross sections (Fig. 4.2) and the average energies (Fig. 4.3)

of the ﬁnal states. The lepton takes a modest amount of the neutrino energy and, in

stark contrast to DIS, the hadronic ﬁnal state takes almost none.

2. The cross section on water/iron can be as large as 7.5%/14% that of charged-current deep-

inelastic scattering, much larger than the quoted uncertainty on the latter. From Fig. 4.4,

the maximum ratios of W -boson production to CCDIS cross sections for water/ice targets

113 are 7.5% (ν ), 5% (ν ), and 3.5% (ν ). For iron targets, these are 14%, 10%, ' e ' µ ' τ ' ' and 7%. These are signiﬁcantly smaller than the early predictions of Seckel [38]. On ' the other hand, these ratios are much larger than the quoted uncertainties on the deep-

4 8 inelastic scattering cross section for Eν = 10 –10 GeV, which are 1.5–4.5% in Ref. [24] and 1–6% in Ref. [237]. We also point out other corrections to DIS that should be taken

into account for future calculations.

3. Attenuation in Earth is increased by as much as 15% due to these cross sections (Fig. 4.5).

They are also an inseparable part of the measured neutrino cross section. Though the

uncertainty of measured cross sections by IceCube is larger than the change in the cross

section due to W -boson production [70, 71], the measured uncertainties will decrease. In

addition, in Ref. [70] the ratio of the measured cross section to DIS prediction is 1.3 0.45. ± The central value would be about 0.1 smaller if the contribution from W -boson production

were included.

4. W -boson production on nuclei exceeds that through the Glashow resonance on electrons

by a factor of 20. From Fig. 4.6, the former produces on-shell W bosons 20 times ' ' more eﬃciently than the Glashow resonance if the neutrino spectrum index is 2.9, the

nominal value from ﬁtting IceCube data [28, 260]. This point was not previously known.

5. The primary signals are showers that will signiﬁcantly aﬀect the detection rate in IceCube-

Gen2; a small fraction of events give unique signatures that may be detected sooner. The

overall shower spectrum is changed by W -boson production. Based on the calculations

in Fig. 4.7, we show that this could be detected with 10 years of IceCube data above 60

TeV with signiﬁcance 1.0σ and 3.2σ for conservative and optimistic cases. In 10 years

of IceCube-Gen2, these would improve by a factor √10 3.2. We also note unique ' signatures that may be identiﬁed sooner, including with IceCube. (Though not explored

here, it would be interesting to consider their impact on detectors for ultra-high-energy

neutrinos.)

Since 2013, IceCube has opened the ﬁeld of high-energy neutrino astronomy, probing neutrino-nucleus interactions well above 1 PeV, far beyond the reach of laboratory experi-

114 ments. Now, only six years later, it is becoming important to take into account subdominant neutrino-nucleus interactions. This rapid progress hints at the discovery prospects of larger detectors, both for increased precision in probing astrophysics and the cross section as well as in searches for new physics.

4.5 Supplemental Material

4.5.1 dσνA/dE` and dσνA/dEW

In this section, we detail the calculations of the diﬀerential cross section for W -boson production in the coherent and diﬀractive regimes. For these two regimes, we use the formalism in Refs. [42, 266], and need to deal with the phase space by ourselves. For the inelastic regime, the phase space is handled with MadGraph [289].

In our companion paper [2], we use the center-of-momentum (CM) frame between the neutrino and the virtual photon, which is the most convenient for calculating the total cross section. In this frame, the 4-momentum can be easily written as

s + Q2 s + Q2 k = , 0, 0, , (4.5a) 1 2√s 2√s s Q2 s + Q2 q = − , 0, 0, , (4.5b) 2√s − 2√s

p = (E , 0, p sin θ, p cos θ) , (4.5c) 1 1 − − p = (E , 0, p sin θ, p cos θ) , (4.5d) 2 2 − −

where k1, q, p1, and p2 are the 4-momenta of the neutrino, virtual photon, charged lepton, and W boson, respectively, s = (k + q)2, Q2 = q2, and p = 1 − s (m + m )2 s (m m )2 1/2 /2√s. − l W − l − W However, to get the diﬀerential cross sections, dσνA/dE` and dσνA/dEW , we should

115 transform to the lab frame (nucleus-rest frame), in which

k1 = (Eν, 0, 0,Eν) , (4.6a)

0 0 q = q0, 0, q sin θq, q cos θq , (4.6b)

P = (mh, 0, 0, 0) , (4.6c)

P 0 = m q , 0, q0 sin θ , q0 cos θ , (4.6d) h − 0 − q − q

0 where P and P are the 4-momenta of initial and ﬁnal nucleus or nucleon, and mh is its p mass. It can be shown that q = Q2/2m , and q0 = (Q2/2m )2 + Q2. Note that the 0 − h h energy transferred to the hadronic part is ∆E = q = Q2/2m , which is small. h − 0 h The transformation from the neutrino-virtual photon CM frame to the lab frame can

be done by two boosts ﬁrst along the z axis and then the y axis (with Lorentz factors γz and γy), and then a rotation by x axis (cos ωq). It can be shown that,

2 2 (s + Q )(2Eνmh Q ) γz = − , (4.7a) p 2 2 8EνmhsQ (2Eνmh s Q ) s − − 2 2 1 2EνQ (2Eνmh s Q ) γy = 2 − − , (4.7b) (s + Q ) mh 1 cos ωq = . (4.7c) γy

Therefore, the energy of the charged lepton in the lab frame, E`, is,

Q2 4s Q2 s E1 2Eν + p cos θ Eν 2 2 2 2 2 − mh − s+Q − mh 4Eν Q 2Eν Q E` = + p sin θ 2 2 2 1 2√s (s + Q ) − mh(s + Q ) − 2 2 2 4Eν mhpQ cos θ sˆ(E1(Q 2Eν mh) + p cos θ(2Eν mh + Q )) = − − p 2 2mhsˆ sˆ Q s − 2 2Eν Q (2Eν mh sˆ) + p sin θ 2 − 1 , mhsˆ − (4.8) in which the second step is just rewriting s in terms ofs ˆ = 2(k q) = s + Q2, which is the 1 · phase-space variable for the cross-section calculation (see Eq. (10) of Ref. [2]). (In Eq. (4.5), we use s to make the expressions more symmetric.)

116 Then, with some approximations, s 2 2 2 Eν(E1 p cos θ) 4Eν Q 2EνQ E` − + p sin θ 2 1 ' √sˆ sˆ − mhsˆ − 2 1/2 EνE1 Eνp EνQ Q (4.9) = cos θ + p sin θ 2 + 4 sin θq √sˆ − √sˆ sˆ mh Eν (E1 p cos θ) . ' √sˆ −

For the above steps, in a word, we basically ignore the terms with Q2, motivated by the following. First, Q is much smaller than the scale at which the interaction happens. Speciﬁcally, the Q is highly suppressed above 0.1 GeV in the coherent regime ' (nuclear form factor) and 1 GeV in the diﬀractive regime (nucleon form factor), while ' √sˆ > m + m 80 GeV and E 4 103 GeV. Second, the energy scale at which the W ` ' ν & × W -boson production actually matters (ratio to CCDIS cross section is large; Fig. 4.4) is much higher than the threshold given above.

Therefore, following Eqs. (10)–(11) of Ref. [2],

2 2 Z sˆmax Z Qmax(ˆs) Z 1 Z sˆmax Z Qmax(ˆs) Z E`,max(ˆs) 2 2 √sˆ σνA = dsˆ dQ d cos θ dsˆ dQ dE` 2 ' 2 E p sˆmin Qmin(ˆs) −1 sˆmin Qmin(ˆs) El,min(ˆs) ν 2 Z E`,max Z sˆmax(El) Z Qmax(ˆs) √sˆ 2 dE` dsˆ dQ , ' E p 2 El,min ν sˆmin(El) Qmin(ˆs) (4.10)

from which we can get dσνA/dE`. In the equation above, the integrand is not shown. Here n o E = E (E + p)/√sˆ and E = Max E (E p)/√s,ˆ m are the upper and lower `,max ν 1 `,min ν 1 − ` limits of E`, obtained from Eq. (4.9).

The dσνA/dEW is obtained following a similar procedure.

Acknowledgments We are grateful for helpful discussions with Brian Batell, Tyce DeYoung, Francis Halzen,

Matthew Kistler, Shirley Li, Pedro Machado, Olivier Mattelaer, Kenny Ng, Yuber Perez-

Gonzalez, Ryan Plestid, Carsten Rott, Ibrahim Safa, Subir Sarkar, Juri Smirnov, Tianlu

Yuan, Keping Xie, and especially Mauricio Bustamante, Spencer Klein, Sergio Palomares-

117 Ruiz, Mary Hall Reno, and David Seckel.

We used FeynCalc [292, 293] and MadGraph [289] for some calculations.

This work was supported by NSF grant PHY-1714479 to JFB. BZ was also supported in part by a University Fellowship from The Ohio State University.

118 Chapter 5 First Detailed Calculation of Atmospheric-Neutrino Foregrounds for Super-Kamiokande Searches for the Diffuse Supernova Neutrino Background

The Diffuse Supernova Neutrino Background (DSNB) has not been detected yet, but there are excellent prospects for its discovery. Success will enable new tests of the core-collapse mechanism and the cosmic star-formation history. Super- Kamiokande (Super-K) is large enough to detect the DSNB signal, but it is obscured by detector backgrounds, especially due to atmospheric neutrinos (more exactly, a foreground, but hereafter we use the more common term). We calculate the rates and spectra of atmospheric-neutrino events in Super-K over a wide energy range, using contemporary inputs and quantifying the uncertainties. As a check, we find good agreement with Super-K muon and electron data in 100–104 MeV, the usual atmospheric data sample. As our main result, we also find good agreement with data in 16–90 MeV, the backgrounds to DSNB searches. This is the first detailed calculation in this energy range. Obtaining good agreement requires taking into account several subtle physical effects and uncertainties. In a companion paper, we calculate the details of how these background events register in Super-K, as well as new cuts to reduce them. Our results will benefit Super-K searches, especially once dissolved gadolinium is added to enable neutron detection. The contents of this chapter will be submitted for publication soon [353].

119 5.1 Introduction

The Diffuse Supernova Neutrino Background (DSNB) is the flux of all flavors of neutrinos and antineutrinos from massive-star core collapses in cosmic history [120]. The first detection and the eventual precision measurement of the DSNB will each be of great importance. Detecting a Milky Way supernova [53, 123, 354–357] will precisely measure one burst, whereas detecting the DSNB will probe the average neutrino emission per core collapse, including failed, optically dark collapses. In addition, the wait for a Milky Way supernova may be long, but the DSNB is always present. The most promising experiment for DSNB detection is Super-Kamiokande (Super-K) [45, 131, 133], a water-Cherenkov detector with a fiducial volume of 22.5 kton. The dominant detection channel is inverse beta decay (¯ν + p e+ + n) on free protons [129, 130], where the positron is detected by its e → Cherenkov light [128].

Figure 5.1 shows that the DSNB signal [29] is presently hidden by detector backgrounds.

Despite this, Super-K has set strong limits on the DSNB ﬂux [45, 131, 133], excluding some models. The underlying physics of these backgrounds is basically known, but not yet in detail. In the current DSNB analysis window, 16–90 MeV in electron total energy Ee, the dominant backgrounds are due to atmospheric neutrinos. The larger component is due to

the at-rest decays of invisible (sub-Cherenkov) muons, which are mostly produced by the

charged-current (CC) interactions of νµ +ν ¯µ. The smaller component is due to the CC

interactions of νe +ν ¯e. Overall, the rates of the DSNB signal and the atmospheric neutrino backgrounds in Super-K are 5 and 50 per year, respectively. Below 16 MeV, there are ∼ ∼ several types of backgrounds that we will focus on elsewhere.

In the near future, Super-K will be upgraded to SK-Gd with the addition of dissolved

gadolinium as 0.2% (by mass) gadolinium sulfate, Gd2(SO4)3 [57–59]. At present, a produced neutron will capture on a free proton, releasing a 2.2-MeV gamma ray [132], which is

hard to detect [131]. With gadolinium, which has a huge cross section for thermal-neutron

capture, the energy release is 8 MeV in a few gamma rays [132], which is easily detectable. ' The ability to tag neutrons will greatly suppress backgrounds, as the DSNB signal has a

120 Before new background cuts

1.4

] Reactor e,

1 spallation,

V 1.2

e solar e, M

atm. NC elastic 1 1.0 ) r y

n 0.8 Invisible muon decay, o t mostly CC k ( + )

5 .

2 0.6 2 (

[

e 0.4 ( e + e) CC E d / DSNB N 0.2 d

0.0 0 10 20 30 40 50 60 Ee [ MeV ]

Figure 5.1: Super-K’s present DSNB searches, which start at Ee = 16 MeV [45]. Note the green and red lines are in units of per MeV instead of per 4 MeV, to match the DSNB lines. The DSNB signal band is based on a Fermi-Dirac spectrum with a 6-MeV eﬀective (after neutrino mixing) temperature [29]. The backgrounds due to atmospheric neutrinos are based on our calculations (details in Fig. 5.5, and the text). Soon, added Gd will reduce these and other detector backgrounds.

neutron, but most backgrounds do not [57–59]. Further progress on reducing backgrounds will come from improved spallation cuts [134–136] and more sophisticated reconstruction of atmospheric neutrino neutral-current events [358]. The aim is to lower the analysis

121 threshold to 10 MeV, below which reactor neutrino backgrounds are overwhelming. ' This paper is the start of a program of work to understand and reduce atmospheric neutrino backgrounds for DSNB searches. This program has three aspects. First, to calculate the relevant atmospheric-neutrino backgrounds, which depend mainly on the fundamental physics of atmospheric-neutrino interactions in water and the properties of the produced particles. Here we focus on CC interactions and certain high-energy NC interactions (we leave low-energy NC events for future work). In Refs. [45, 131, 133], these backgrounds were modeled empirically, with free normalizations. In Ref. [57], the backgrounds were estimated. Here we bring much more sophisticated modeling and treatment of uncertainties. Second, to calculate the signatures of the detection products in Super-K and SK-Gd, which depend mainly on the passage of particles in matter [359] and the interplay with the detector physics of Super-K. Third, to identify places where uncertainties can be reduced through auxiliary data. Our program of work will be the ﬁrst comprehensive treatment of these topics.

In this paper, we focus on the ﬁrst aspect. In a companion paper [360], we focus on the second. In each, we cover parts of the third. Our results will be useful for Super-K, SK-

Gd, Hyper-Kamiokande [361], and other water Cherenkov detectors (e.g., ANNIE [362] and

WATCHMAN [363]). In addition to being a background for the DSNB, the atmospheric- neutrino interactions discussed here are interesting as signals. Our results suggest that careful work could lead to interesting new results on neutrino production physics [46], neutrino-nucleus interactions [138, 220–225], and neutrino mixing [364], though these topics are beyond our present scope.

In Sec. 5.2, we review the broad range of inputs and uncertainties needed for this work.

In Sec. 5.3, we demonstrate that our calculations reproduce Super-K atmospheric-neutrino data in 100–104 MeV, an important test. In Sec. 5.4 and Sec. 5.5, we demonstrate that our calculations also reproduce Super-K atmospheric data in 16–90 MeV, which, along with the detailed accounting of inputs and uncertainties, is our main result. In Sec. 5.6, we calculate the parent-particle spectra of the atmospheric-neutrino backgrounds, which provide important physical insights. We conclude in Sec. 5.7.

122 5.2 Framing the Problem

In this section, we review the expected DSNB signal and the observed backgrounds, then the ﬂuxes of atmospheric neutrinos, their interactions, and how those register in Super-

K. Though many of these points are known, this is the ﬁrst detailed synthesis of how to understand atmospheric-neutrino backgrounds for the DSNB.

5.2.1 Predicted DSNB signals

The DSNB is a guaranteed ﬂux, time-independent and isotropic, of all neutrino species.

Once detected, it will provide new information on the core-collapse mechanism as well as its cosmic rate. Moreover, as core collapses that directly produce black holes have neutrino signals comparable to or larger than those that produce successful supernovae [365, 366], the DSNB probes also the rate of failed supernovae [367, 368]. If the guaranteed DSNB ﬂux is not found, there must be surprising new physics or astrophysics.

We use the predictions of Ref. [29], which we believe have the most realistic ﬂuxes and uncertainties. (See also Refs. [369–377].) There are three major inputs for the DSNB signal [29, 120]: the neutrino emission per core collapse [378–380], the cosmic core-collapse rate [124], and the physics of detection. The ﬁrst is the primary observable, as it can only be measured by neutrino experiments; it is the most uncertain. The second can be determined by electromagnetic observations, and is relatively well known. The third is very well known.

As for neutrino mixing, we include it implicitly, as we consider only the eﬀective neutrino spectrum outside the supernova, after all mixing eﬀects have occured (for active neutrinos, no mixing occurs en route because neutrinos emerge from the dense matter as incoherent mass eigenstates). This spectrum, with mixing included, can be directly compared to the

SN 1987A data. It is a separate problem to relate the observed spectrum to the initial neutrino emission of the proto-neutron star.

The DSNB flux is obtained from a cosmological line-of-sight integral. The uncertainties due to cosmological parameters are negligible. The total flux per flavor is 10 cm−2 ∼ s−1 for nominal models [29, 120]. The dominant contribution arises from around redshift

123 z 1, due to the corresponding star-formation rate being 10 times the present rate. ∼ ∼ The best prospects for detecting the DSNB are forν ¯e in Super-K, due to the large cross section [129, 130] and the huge, low-background detector. The total cross section is σ(E ) ν ' 10−43 cm2(E 1.3 MeV)2, with the outgoing electron (we use electron to mean both ν¯e − electrons and positrons, unless we specify otherwise) carrying E E 1.3 MeV and e ' ν¯e − emitted near-isotropically.

5.2.2 Observed atmospheric neutrino backgrounds

In the present Super-K DSNB search window, 16–90 MeV in electron energy, there are two components, as shown in Fig. 5.1: a bump of known shape that peaks at 40 MeV and a ' slope that grows at higher energies. Together, these explain the data well, so that there is no indication of a DSNB signal or of other important backgrounds. The background rate is

50 events per year, compared to a signal rate of 5 events per year (as shown in Fig. 5.1, ∼ ∼ spectrum information provides additional power, though so far not enough). The low observed backgrounds [45, 131, 133] represent a signiﬁcant achievement. For comparison, the low-energy trigger rate is 10 Hz, the cosmic-ray muon rate is 2 Hz, and identiﬁed solar ∼ ∼ and atmospheric neutrino events are 25 per day. ∼ The bump component is due to electrons from muon decay at rest, produced with a delay of order microseconds, which is long compared to the time resolution of Super-K, which is a few nanoseconds. These muons are mostly produced by sub-GeV atmospheric νµ

andν ¯µ) interacting with oxygen and hydrogen nuclei [45, 131, 133]. In Sec. 5.4.1 we show that there is also a non-negligible contribution due to neutral-current (NC) interactions of

all-ﬂavor neutrinos up to a few GeV. These muons are produced with kinetic energy below

the Cherenkov threshold (Sec. 5.2.5), and hence are invisible in Super-K (visible muons

do not cause backgrounds for the DSNB signal). Once the muons decay, the isotropically

emitted electrons constitute a background for the DSNB. The spectrum shape is well known,

but its normalization is not known a priori (in the Super-K search; below, we predict it).

The slope component is due to promptly produced electrons from atmospheric (νe +ν ¯e) CC interactions. This component is much less important than previous one, and it can be

124 ﬁt from the 60–90 MeV data, which is important to properly normalize the bump at lower energies [45]. Neither its spectrum shape or normalization is known a priori (again in the

Super-K search; below, we predict these).

Until now, there were no detailed studies of the physics of these backgrounds, which have been characterized empirically by Super-K. By understanding the physics of these backgrounds, we can devise new ways to cut them.

5.2.3 Atmospheric neutrinos

To study the atmospheric neutrino backgrounds, we need to know the fluxes and their uncertainties. For the DSNB backgrounds, the most important energy range is Eν below a few GeV. Atmospheric neutrinos are created through cosmic-ray interactions in Earth’s atmosphere, which produce secondaries, including pions and kaons, which decay in flight, producing neutrinos [178, 381, 382]. Below a few GeV, neutrinos are mainly produced by the decays of pions (π+ µ+ + ν and its charge conjugate) and muons (from pion decay; → µ µ+ e+ + ν +ν ¯ and its charge conjugate). At low energies, the flux ratio (combining → e µ neutrinos and antineutrinos) is ν : ν : ν 1 : 2 : 0 before oscillations. Above several e µ τ ' GeV, the νe fraction decreases due to some muons reaching the ground before decaying.

For the atmospheric-neutrino ﬂuxes, we use those of Ref. [47] (HKKM2014) for Eν > 100 MeV, and Ref. [46] (FLUKA2005) for 10–100 MeV (these join reasonably well). The

uncertainties mostly follow those of the primary CR ﬂux and the hadronic secondary pro-

duction. For Eν > 100 MeV, Ref. [383], which is mostly relevant to HKKM2014, quotes the overall uncertainty as 7% in 1–10 GeV, but 20% at 0.3 GeV and 25% at 103 GeV. A ' & ' related paper, Ref. [384], which considers only νµ andν ¯µ, quotes 10–20% for 0.1–10 GeV and 40% at 103 GeV. For E < 100 MeV, the ﬂuxes and uncertainties are less well studied. ' ν FLUKA2005 quotes an uncertainty of . 25%. To be conservative, the uncertainties that we adopt are 25% for 10–100 MeV, 20% for 0.1–1 GeV, and 15% for 1–10 GeV. These ∼ ∼ ∼ should improve in the near future. A recent paper, Ref. [385], ﬁnds smaller uncertainties in

light of new cosmic-ray measurements, but it does not cover the full energy range we need.

Before entering Super-K, these neutrinos undergo mixing, which depends on energy

125 and baseline (related to the zenith angle of the neutrino direction). The dominant effects are due to maximal ν ν vacuum oscillations, which push the flavor ratios toward µ → τ ν : ν : ν 1 : 1 : 1. With increasing energy and baseline, Earth matter effects be- e µ τ ' come important. We calculate the full mixing effects using nuCraft [48], which is designed

for atmospheric-neutrino mixing in a three-neutrino framework. The uncertainties due to

oscillation parameters [386, 387] are smaller than those of the ﬂux.

5.2.4 Neutrino interactions

For the DSNB signal, up to a few tens of MeV, the most important interaction isν ¯ + p e → e+ + n on free (hydrogen) protons. Interactions with nuclei are suppressed by binding

eﬀects and interactions with electrons are suppressed by their small mass. The signal

detection cross section is well understood [129, 130]. Details about its signatures are noted

in Sec. 5.2.1.

For atmospheric neutrinos, which range from several tens of MeV to a peak in the GeV

range to still higher energies, the dominant interactions are with bound nucleons in oxygen.

(Gadolinium will be unimportant as a target, with mass density 10−3 and number density ∼ 10−4, both relative to oxygen.) At the simplest level, the two most important interactions ∼ are ν + 16O µ− + 15O+p andν ¯ + 16O µ+ + 15N+n, where these should be considered µ → µ → to be interactions with bound nucleons that are ejected as they are transformed by the CC

weak interaction.

In slightly more detail, neutrino-nucleus interactions in this energy range have two ele-

ments. First, a neutrino interacts with a single bound nucleon that has an initial momentum

due to its Fermi motion. Interactions with pairs of nucleons (also called short-range cor-

relations) are important but less common. Second, the struck (and often transformed)

nucleon(s), along with any other produced hadrons, travel through the nucleus, where they

may interact. This is called intranuclear hadron transport or ﬁnal-state interactions and it

complicates the whole picture of neutrino-nucleus interactions [388, 389]. The interaction

with the initial nucleon can be aﬀected by nuclear shadowing, and the availability of ﬁnal

states for the struck nucleon can be aﬀected by Pauli blocking.

126 The underlying physics of the initial neutrino nucleon interaction may be through quasielastic scattering (QE), as described just above, which is dominant for Eν . 1 GeV. Beyond there to E a few GeV, resonance production (RES), in which the nucleon is brieﬂy ex- ν ∼ cited to a Delta or other state, is dominant. At higher energies, deep inelastic scattering

(DIS), in which the neutrino interacts with quarks, becomes dominant. This simple description should be treated as deﬁning approximate guideposts, not sharp separations. For the atmospheric-neutrino backgrounds for the DSNB, the bump component is dominantly due to charged-current QE (CCQE) of νµ andν ¯µ, with some contribution from neutral-current

RES (NCRES) of all flavors. The slope component is primarily due to CCQE of νe andν ¯e. Neutrino-nucleus interactions are complex, not well understood, and hard to handle an- alytically. We use GENIE, which provides a comprehensive framework for neutrino-nucleus interactions [138–140]. GENIE takes into account the neutrino-nucleon/nucleus interaction vertices, nuclear effects, hadronization, final-state interactions, de-excitations of the final- state nucleus, and more, though all approximately. A GENIE comprehensive model includes specifications for all of these and has been tuned to reproduce experimental results on neutrino, electron, and hadron-nucleus scattering. Several comprehensive models are available.

The results are surely not perfect, but are good enough to guide our exploration of the physics of the interactions and how to improve cuts, and identify where new theory and data is needed.

The neutrino-nucleus interactions relevant for this work have large uncertainties that are not fully quantiﬁed, due to deﬁciencies on both the experimental and theoretical sides.

We assume overall uncertainties of 20% in this energy range, based on Refs. [390, 391]. In ∼ the Super-K DSNB searches [45], no uncertainty estimates are given because the interaction model is scaled to match the data. To further assess the uncertainties, we compare results using diﬀerent comprehensive models for GENIE (Sec. 5.4.1), ﬁnding that our choice of

20% is reasonable. These comparisons also help assess uncertainties on the numbers and ∼ kinematics of ﬁnal-state particles.

127 5.2.5 Physics of detection in Super-K

Super-K is a cylinder of diameter 39.3 m and height 41.4 m, filled with 50 ktons of ultrapure water [128]. The optically isolated inner detector has a mass of 32 kton and its fiducial volume has a mass of 22.5 kton. Super-K can detect Cherenkov light from relativistic charged particles via 11, 000 50-cm photomultiplier tubes (PMTs) that view the inner ∼ detector (the outer detector is viewed by a smaller number of smaller PMTs, providing an active veto layer). The inner-detector PMTs cover 40% of the detector wall, are sensitive ' to photons of wavelength 300–700 nm, and have a quantum efficiency of 0.1. ' ∼ To emit Cherenkov radiation, a charged particle’s speed must exceed the phase velocity of light in water, which is c/n, where n 1.33 is the refractive index of water and c is ' the speed of light in vacuum. This sets a theoretical threshold for the lowest-speed particle

that can be detected, β = 1 , which gives β 0.75, or a Lorentz factor γ 1.51. For th n th ' th ' electrons, muons, pions, and protons, respectively, this corresponds to kinetic energies of

0.26, 54, 72, and 481 MeV. In practice, to be detectable, the kinetic energies must be higher than this, as discussed below. The Cherenkov angle, θc, is deﬁned by cos θc = 1/nβ. For ◦ ultrarelativistic particles, θc = 42 is constant, while the angle is less for less-relativistic particles.

The pattern of PMT hits — ideally forming a clear ring-like pattern — from a particle

gives information on its type, position, energy, and direction. Its charge cannot be deter-

mined from the Cherenkov light alone. To set some scales, for relativistic electrons, the

light yield is about 6 detected photoelectrons per MeV, and the threshold for solar-neutrino

searches is 3.5 MeV (kinetic energy of recoil electron). For charged particles well above ∼ their respective thresholds listed above, the eﬃciency of detection is near-perfect and the

particle identiﬁcation based on ring characteristics is very good.

Not only relativistic charged particles are detectable. For example, while sub-Cherenkov

muons and charged pions are themselves invisible, their decay chains produce detectable

electrons; the nuclear captures of negative pions and muons can also lead to detectable

signals. And even though neutral particles are invisible, they can lead to detectable sig-

128 nals. Gamma rays can produce detectable electrons through Compton scattering and pair production. Neutral pions quickly decay to gamma rays. Neutrons ﬁnally get captured on nuclei, which produce gamma rays. Their inelastic interactions with nuclei, before capture, may also produce gamma rays. Finally, the initial production of sub-Cherenkov muons or charged pions can be accompanied by nuclear gamma rays or neutrons. In addition, unstable nuclei that later beta decay can be produced at the initial vertex or through charged-pion propagation. Further details will be discussed in our companion paper [360].

For DSNB searches, Super-K uses electron-like events with total energy 16 MeV <

Ee < 90 MeV [45]. The physical basis we use to separate DSNB signals from atmospheric neutrino backgrounds, dominating in this energy window, is that DSNBν ¯e events produce only one electron and one neutron (not presently detectable, but will be with gadolinium), while atmospheric-neutrino events can produce additional particles, including nuclear gamma rays. The DSNB event sample is defined by several key cuts. Events must be contained in the fiducial volume with no activity outside (noise reduction and fiducial volume cut). The Cherenkov radiation should be only one ring (multi-ring cut), which is fuzzy due to being caused by an electron as opposed to other particles (pion cut), and its angle should be 42◦ (cf. Cherenkov-angle cut). For the time structure, there should be only one peak ' (double-peak cut), with no other activity for tens of microseconds before or after (sub-event cut). These considerations, along with related ones for higher-energy events, help us define event classes to reproduce the Super-K data (Sec. 5.3, 5.4 and 5.5), and look for new ways to further reduce backgrounds [360].

5.3 Calculation Validation by Matching Super-K High- Energy Atmospheric Neutrino Data

In this section, we show that our calculations match well Super-K’s usual high-energy atmospheric-neutrino data sample [49]. Success is a precondition to producing accurate calculations for the low-energy data that cause the backgrounds for DSNB searches. For this section, we use the data from Super-K-I, spanning from April 1996 to July 2001 (1489

129 1 1 10 10 ] ]

1 1 r r s s

1 2 1 2

s 10 s 10

2 2 m m c c

[ [

E E d d / / 3 without oscillation 3 without oscillation d 10 d 10 e

E with oscillation E e with oscillation

0.5 × without oscillation 0.5 × e without oscillation

0.5 × with oscillation 0.5 × e with oscillation 1 0 1 2 1 0 1 10 10 10 10 10 10 10 E [ GeV ] E [ GeV ]

Figure 5.2: Our calculated fluxes of atmospheric neutrinos, without and with neutrino mixing, following Refs. [46–48]. Left: Results for νµ andν ¯µ. Right: Results for νe andν ¯e (note that the figure goes to lower energies; the notch is because we use different calculations above and below 0.1 GeV). Forν ¯ in both panels, we multiply by 0.5 for clarity.

days exposure). A later dataset [391] does not show the full charged-lepton spectra we need. These data include both electron-like and muon-like events in the 22.5-kton ﬁducial volume. The momentum resolution is about 2–3% at 1 GeV and decreases slowly with energy, being much smaller than the width of each bin. The signal eﬃciencies are 100%. ' We concentrate on data from 0.1–10 GeV, as data at higher energies, dominated by DIS, is less relevant for our focus on DSNB backgrounds.

For the initial atmospheric neutrino fluxes, we use HKKM2014 [47], the most complete calculation over a wide energy range. As the time period of Super-K-I almost spans from solar minimum to solar maximum in cycle 23, we use the average flux. To calculate three-flavor neutrino mixing with matter effects, we use nuCraft [48], which computes zenith-angle and energy-dependent mixing probabilities by direct numerical integration. We calculate fluxes at the location of Super-K and integrate over zenith angles. For mixing parameters, we adopt those of Ref. [386], a recent global fit.

Figure 5.2 shows our calculated atmospheric-neutrino ﬂuxes without and with mixing.

−1 The spectra are plotted as EdN/dE = (2.3) dN/d log10 E, matching the log scale on the

130 36 36 10 10 C C O C O C e C CC O C O 37 37 e 10 O CCQES 10

CC C H H C 38 38 e

] ] C 10 CC 10 C 2 2 H H e m m c c

[ [

39 39 10 10

40 40 10 10

41 41 10 1 0 1 10 2 1 0 1 10 10 10 10 10 10 10 E [ GeV ] E [ GeV ]

Figure 5.3: The GENIE cross sections for CC interactions with various interactions and targets as labeled. We only show the most important channels. Left: Results for νµ and ν¯ . Right: Results for ν andν ¯ (note that the ﬁgure goes to lower energies). Below 1 µ e e ∼ GeV, CCQE dominates, as shown for one speciﬁc channel in the left panel.

x-axes, so that relative heights of the curves at different energies faithfully show relative contributions to the integrated flux. The peaks near 0.1 GeV in neutrino energy reflect the peak near 1 GeV in cosmic-ray energy. At high energies, the spectra of νµ andν ¯µ follow the well-known power law for the parent cosmic rays, dN/dE E−2.7, while ν andν ¯ are ∼ e e steeper due to some muons reaching the ground before decaying. For νµ andν ¯µ mixing, the dominant effect is maximal vacuum mixing to ντ andν ¯τ , which depletes the original fluxes by a factor 2 (less at high energies). For ν andν ¯ mixing, the effects are modest, ' e e 2 because of the much longer oscillation length (due to small ∆m21) and small θ13. To compare to Super-K data, we could use their reconstructions of the neutrino spectra, e.g., those in Fig. 14 of Ref. [391], with which our calculations in Fig. 5.2 agree very well

(comparison not shown). However, comparing to the measured electron and muon spectra more directly tests the details of our calculations.

Neutrinos are detected through their interaction products, for which we focus on muons and electrons. The momentum spectrum of a particle f can be calculated by summing interaction channels over neutrino species, ν, and targets, T (oxygen and hydrogen nuclei,

131 Total SK data SK data 3 3 Total 10 sys. uncert. 10 sys. uncert. +O n n +O i i e b b r r e e p +O p s s t t n n e+O e e v v e 2 e 2 f f o 10 o 10 r +H r e e b b e+H m m u +H u N N e+H

1 1 10 1 0 1 10 1 0 1 10 10 10 10 10 10 p [ GeV ] pe [ GeV ]

Figure 5.4: Our calculated results (lines, as labeled) for charged-lepton spectra induced in Super-K by atmospheric neutrinos, compared their measured data (points with statistical uncertainties only) [49]. Left: Muons, combining FC single-ring, multi-ring and PC data. Right: Electrons, single-ring FC data only, which dominates at lower energies, so at high energies (shaded region), some disagreement is expected. Overall, the agreement is well within the estimated theoretical uncertainties, which are dominated by the systematics (shown as the green arrow in each panel), at the level of a few tens of percent.

and electrons),

Z dNf X dΦ = ∆t NT dEν dΩz (Eν, cos θz, φz) dpf dEν × νT →f

dσνT →f Posc(Eν, cos θz) (Eν, pf ) (5.1) dpf where ∆t is the exposure time (for this section only, 1489 days), NT the number of targets in the fiducial volume (N = 7.5 1032, N = 1.5 1033, N = 7.5 1033), and Ω (cos θ , φ ) O × H × e × z z z is the solid angle. The dΦ/dEν is the initial atmospheric neutrino flux, Posc(Eν, cos θz) the oscillation probabilities, and their convolution over the zenith angle, cos θz, gives the neutrino flux after mixing. The dσνT →f /dpf is the differential cross section for a interaction channel. The geometry of Super-K is not considered as the propagation length of the produced particles are much shorter than the detector scale. Only muons can travel an appreciable distance, i.e., 5 m (E/1 GeV). We take into account nearly all of the effects '

132 by directly comparing to the Super-K data classes, as described next.

The contributing interaction channels depend on the data samples. For muons, we focus on fully contained (FC) single-ring ( 180 MeV–10 GeV momentum) and multi-ring ' ( 600 MeV–30 GeV visible energy events) muon-like events, plus partially contained (PC) ' data (& 300 MeV visible energy; mostly muons). For electrons, we use only the FC single- ring events (> 100 MeV momentum). In this energy range, these classes include nearly all of the events [49].

We simulate the neutrino interactions with GENIE (v2.12.0) [140]. For this section,

we use the default model of GENIE, which includes the following: 1) Nuclear model for the

nucleon targets: a relativistic Fermi gas model (RFG) modiﬁed by Bodek and Ritchie to

incorporate short-range nucleon-nucleon correlations [392]. Other important nuclear eﬀects

are also included, including Pauli blocking, shadowing, anti-shadowing, EMC, de-excitation,

etc. 2) For the neutrino-nucleon interaction: the Llewellyn-Smith model [220] for CCQE, the

Rein-Sehgal model [393] for (CC and NC) RES, and the Paschos-Yu model [394] for DIS. 3)

For hadronization, the AGKY model. 4) For ﬁnal-state interactions, the INTRANUKE/hA

model is used. Further details are given in Refs. [138–140].

Figure 5.3 shows the most important CC cross sections. To better show the increase of

the cross sections with energy, we show σ instead of the often-used σ/E . Below 1 GeV, ν ∼ QE dominates; starting at few GeV, RES dominates because the thresholds for resonances

are surpassed; and above several GeV, DIS dominates because the nucleons are resolved.

The most important target is oxygen, due to its large number of nucleons compared to

hydrogen (while electrons are more numerous, their cross sections are suppressed). At low

neutrino energies,ν ¯ interactions on protons become more important because they are not suppressed by nuclear-binding eﬀects; the ν interaction on protons has a high threshold

because there is no QE contribution. Antineutrinos generally have smaller cross sections

than neutrinos, in part due to the cancellation between the vector and axial components.

Figure 5.4 shows our calculated charged-lepton spectra for Super-K. For muon data (left

panel), we have combined the three types, which are dominated by FC single-ring events

below a few GeV and by the other types above. For electron data (right panel), it is only the

133 published single-ring events, which dominate at lower energies. Therefore, at high energies

(shaded region), our calculation is larger than the data but this is expected. For GENIE results, we keep all the CC events from neutrino interactions in the ﬁducial volume. (In the Super-K data, contamination from NC events are small [49].) This matches the data choice, especially at lower energies. For the consistency with previous plots, we present counts per bin in log10 E. The basic shape of the spectra is a peak, due to the convolution of the falling spectrum (Fig. 5.2) and the rising cross section (Fig. 5.3). Overall, especially near the peaks, the agreement is excellent, well within the uncertainty of a few tens of percent which is expected from that of the ﬂux and cross section. At the lowest energies in both spectra, our results are slightly smaller than the Super-K data, but this is where the uncertainties are largest, as discussed in detail in Sec. 5.2.4.

The results of this section thus show that the framework we use — flux, mixing, total and differential cross sections, and detector response — adequately reproduces Super-K data over 0.1–10 GeV in charged-lepton momentum. As a test of the cross-section ∼ dependence, we tried GENIE’s EffSFTEM model set (effective spectral-function nuclear model with transverse enhancement), finding nearly the same results. The results of this section thus increase our confidence in the results of the next section, where we continue to lower energies.

5.4 New Results on Super-K Low-Energy Atmospheric data: Invisible-Muon Component

In this and the next section, we detail our calculations of Super-K (low-energy) atmospheric neutrino backgrounds for DSNB searches. This section focuses on the invisible-muon component, the larger of the two. These events come from the decay of muons at rest, and hence are isotropic. Our simulated dataset is calculated with GENIE [138–140], supplemented by

FLUKA [141, 142] simulations to take into account the propagation of secondaries.

We try to understand all aspects of the underlying physics and to reproduce the data, pointing out uncertainties in the procedure. Until now, this had never been done in detail. In

134 Ref. [45], Super-K made a careful search for the DSNB in the range 16 MeV < Ee < 90 MeV, combining the data of Super-K-I (1497 days exposure), II (794 days exposure) and III

(562 days exposure). These data, after various cuts optimized for DSNB searches, are dominated by atmospheric neutrino backgrounds. This is what we refer to as the new low- energy atmospheric-neutrino data, because it has not been analyzed as a signal until now.

We mainly compare with Super-K-I data, which has the only published signal eﬃciency [45], which is about 90% above 24 MeV, and lower at lower energies.

The decay electrons from invisible muons follow the Michel spectrum, with a small distortion due to µ− always undergoing atomic capture (mostly on oxygen) and decaying in orbit. The total decay spectrum is well measured by Super-K using stopped cosmic-ray muon data. When counting Michel electrons, we always mean in this energy range. For the invisible-muon component, only the total number of decay electrons matters, while for the smaller (νe +ν ¯e) CC component, both the spectrum and the total number matter.

5.4.1 Theoretical calculations

Our calculational framework is based on the following,

Z dNf X dΦ = ∆t NT dEν dΩz (Eν, cos θz, φz)Posc(Eν, cos θz) dpf dEν νT →f

dσνT →f (Eν, pf ) EvtCls Corr. Det. (5.2) × dpf ⊗ ⊗ ⊗

The leftward of the ﬁrst “ ” is the same as Eq. (5.1), used for the Super-K high-energy at- ⊗ mospheric neutrino data. The complications come from the following three terms. “EvtCls” is the event class deﬁned from physically interpreting Super-K analysis cuts, with details in Sec. 5.4.1.“Corr.” is the physical corrections that need to be made, with details in

Sec. 5.4.1.“Det.” is the corrections for the detector eﬀects, mainly the Cherenkov thresholds, with details in Sec. 5.4.1. For brevity, we omit writing the energy-dependent signal eﬃciency [45], though we take it into account.

135 Deﬁne event classes for GENIE results

We define the event classes of our GENIE simulation results (“EvtCls” in Eq. (5.2)) in terms of our physical interpretations of Super-K analysis cuts. Compared to the high- energy data, the low-energy data have many more cuts due to the difficulties of the DSNB search, for which the signal consists of a single electron in the right energy range and no other detectable activity. For defining the background sample, the most important particles to keep track of are charged leptons, all types of pions, and nuclear gamma rays. We cut other particles like relativistic protons, but they are rare, < 1%.

To start, we cut νµ CC events with the muon above its theoretical Cherenkov threshold (p 120.2 MeV). A muon that has Cherenkov radiation will trigger the detector, typically µ ' giving a coincident event with the second event 2 µs later, clearly distinguishable from a ∼ DSNB signal event (e.g., sub-event cut). We call this step a “naive” calculation.

In addition to the invisible muon and its decay electron, there could be pion(s) produced in the neutrino interaction. We ignore all those events, because of the following. First, for a π0, it always decays to two gamma rays and is easy to be cut. The two gamma rays are mostly emitted in nearly opposite directions, due to the modest boosts, and pair-produce electrons. This could give two or more rings (e.g., multi-ring cut). Second, a π+ or a

π− above the Cherenkov threshold has Cherenkov radiation from itself and from its decay

muon then electron, which makes them easy to be cut. Third, a π+ below the Cherenkov

threshold is also easy to be cut due to the electron from its decay chain. At last, a π− below

the Cherenkov threshold mostly undergoes atomic capture then nuclear capture [395, 396].

Though this sometimes does not have strong signature, the total amount of such π− events

are small.

Moreoever, there is a considerable fraction of events with nuclear gamma rays. They

are from de-excitations of remnant nuclei, which may be left in an excited state after the

neutrino interactions kick out one or more nucleons. These events may be cut due to their

“2-peak” timing features (sub-event cut) [45], since these gamma rays are prompt after

neutrino interactions, whereas the Michel electrons are delayed by the muon lifetime. We

136 do not know the eﬃciency of cutting these gamma rays by Super-K (γ), as there is no documented information about this. In our calculation, we show results for both γ = 0% and γ = 100%.

More about GENIE choices

As in Sec. 5.3, we use the default set of models of GENIE (v2.12.0). For this and next section, CCQE matters most, for which GENIE uses the Llewellyn-Smith model [220] with an empirical meson-exchange current model for 2p/2h.

For the nuclear gamma rays, GENIE’s model is based on Refs. [397, 398] and Br 50%. γ ' Ref. [397] is a theoretical calculation based on the nuclear shell model, and it gives the nuclear gamma-ray energies and probabilities for the p1/2, p3/2, and s1/2 states, respectively, for which one proton or neutron hole is made. The predicted Brγ is close to the generally assumed canonical value, 50%. Ref. [398] is an experimental measurement for the gamma- ' ray energies and probabilities for one proton hole in the s1/2 state. A more recent calculation also gives Br 50% [399]. γ ' However, neutrino interactions at few hundred MeV could kick out more than one nucleon due to short-range correlations and FSI. This has large uncertainties on both the theoretical and experimental sides. Later, after comparing our step-by-step calculations with Super-K data, we discuss Brγ and γ more.

Physical corrections

There are three physical corrections (“Corr.” in Eq. (5.2)) that need to be made.

First, µ− capture. In contrast to µ+, which always decay after stopping, µ− always undergo atomic capture, mostly on oxygen, after losing most of their energy [395, 396].

Then 21% of them undergo nuclear capture, and the products cannot mimic a DSNB ' signal. To account for this, we reduce the overall number of µ− by 21%, which only ' aﬀects the νµ+O (and H) CC channels. Second, the contribution from NC invisible π+ channels. A π+ below the Cherenkov

threshold will mostly decay to an invisible muon. Therefore, if a neutrino interaction

137 Table 5.1: Our predicted numbers of decay electrons (16 MeV < Ee < 90 MeV) from invisible muons, using GENIE’s default model set. The “Naive” calculation is defined in Sec. 5.4.1, and its improvement to the “Standard” calculation is defined in Sec. 5.4.1. We build on the latter by including the “Coulomb” corrections (Sec. 5.4.1) and also the “Threshold” corrections (Sec. 5.4.1). We show results for two sets of assumptions about nuclear gamma rays, favoring the second set. All calculations include signal efficiency and other detector effects. Numbers in boldface are bottom-line numbers.

Br = 50%, = 0% Br = 50%, = 100% Interaction channel γ γ γ γ Naive Standard Coulomb Threshold Standard Coulomb Threshold

νµ+O CC 166 94 125 172 50 66 90 ν¯µ+O CC 41 34 24 36 16 11 17 νµ+H CC 5 0 0 0 0 0 0 ν¯µ+H CC 14 14 12 17 14 12 17 Total CC 227 142 161 225 80 90 125 NC π+ 49 46 57 28 26 33 Total 227 191 207 282 108 116 158 Total/Super-K(146) 1.55 1.31 1.42 1.93 0.74 0.79 1.08

Table 5.2: Same as Table 5.1, but using the GENIE’s EﬀSFTEM model set.

Br = 50%, = 0% Br = 50%, = 100% Interaction channel γ γ γ γ Naive Standard Coulomb Threshold Standard Coulomb Threshold

νµ+O CC 157 114 153 202 59 80 106 ν¯µ+O CC 41 34 23 35 17 12 18 νµ+H CC 5 0 0 0 0 0 0 ν¯µ+H CC 14 14 12 17 14 12 17 Total CC 261 161 188 254 89 104 141 NC π+ 48 45 55 25 24 29 Total 261 209 233 309 114 128 170 Total/Super-K(146) 1.79 1.43 1.6 2.12 0.78 0.88 1.16

produces a invisible π+ and no other visible particles, it would mimic the DSNB signal.

These events come from all neutrino ﬂavors interacting with oxygen or hydrogen, and we call the bulk of them the NCπ+ channel. As noted in Sec. 5.4.1, π− undergo nuclear capture

before decay, so they do not mimic a DSNB signals. There are countereﬀects where some

π+ interact before decay and some π− decay in ﬂight, but they are negligible. We call the

calculations to this point a “standard” calculation.

Third, Coulomb distortion eﬀects, which are not included in the GENIE’s default models.

138 The Coulomb distortion happens when the outgoing charged particles from neutrino interactions feel the Coulomb field of the nucleus. For negatively charged particles, the Coulomb attraction lowers their momentum while increasing their production amplitude, and the opposite for positively charged particles. The distortion effects increase as the charged-particle energy decreases or the charge number of the nucleus (Z) increases, and are important for this work. Widely used Coulomb correction methods are the Fermi function and effective momentum approximation (EMA) [400, 401]. However, the former only works well for electrons below 10 MeV, and the latter only for scattering of ultrarelativistic electrons on ∼ nuclei.

Here we use the modiﬁed EMA method [402], which works well for both muons and electrons at the relevant energies. (It should also work well for charged pions, as they have the similar masses to muons.) In this method, the nucleus is approximated as a uniformly charged sphere with radius R and the neutrino is assumed to interact at the center. Therefore the outgoing charged particles initially have an electrostatic potential of

V = 3Zα/2R. This potential induces a shift of the total energy of the charged particle ± and a rescaling of the scattering amplitude.

We include the Coulomb correction for all CC and NCπ+ channels. For the latter, which are mostly from the decay of ∆ inside the nucleus, we ignore the distortion on these intermediate states due to their heavy mass and short lifetime. Moreover, we do not apply the correction to the scattering amplitude on these π+ because they are not directly from neutrino interactions. Overall, the correction increases the number of invisible muons of

+ the νµ+O CC channel and decreases those of theν ¯µ+O(H) CC and NCπ channels.

Detector eﬀects on the Cherenkov threshold

1 In above calculations, the theoretical Cherenkov threshold, βth = n , is used for the charged particles. However, if a charged particle does not have enough energy above the threshold, it will not produce enough Cherenkov photons to trigger the detector. Therefore, a practical

Cherenkov threshold should be calculated.

For Super-K, this is related to the sub-event cut of their analysis, in which they reduced

139 muons just above the theoretical Cherenkov threshold [45]. Accordingly, such an event can be cut as long as the muon makes a super-low energy (SLE) trigger (before decay).

The SLE trigger has a threshold of 17–24 photoelectrons in the inner detector, where the precise value depends on analysis period. We use the lowest number, 17, as it is believed that even a lower number of photoelectrons may still have chance to be identiﬁed (private communication with Micheal Smy).

We need to convert the number of photoelectrons to the corrected threshold on pµ. First, we assume the number of photoelectrons equals the number of PMTs hit, as the total number of PMTs in Super-K inner detector, > 104, is much larger. Then considering the coverage and quantum eﬃciency of Super-K PMTs, this corresponds to about 340 Cherenkov photons with wavelength 300–700 nm emitted by the charged particle. Using Eq. (4.4) of

Ref. [403], we get the correct thresholds for charged particles, which are used hereafter.

This mainly matters for muons ( 145 MeV) and π± ( 187 MeV). The conversion above ' ' is consistent with those from practical Super-K detector simulations (private communication with Chenyuan Xu).

Another nuclear model, EﬀSFTEM

We try another set of models of GENIE, EﬀSFTEM, to give a sense of the uncertainties of

the invisible-muon yield due to neutrino-interaction modeling. There are two diﬀerences

from the default model set. First, EﬀSFTEM uses an eﬀective spectral function nuclear

model [404]. Second, instead of meson exchange current model, it uses transverse enhance-

ment model (TEM) [405].

5.4.2 Summary of predictions and uncertainties of the invisible-muon component

Table 5.1 and Table 5.2 show our predicted numbers of decay electrons from invisible muons,

for the default and EﬀSFTEM model sets of GENIE respectively. For each table, we show

the results from the major calculational steps as represented by diﬀerent columns from left

to right.

140 2.0 SK-I data ]

1 V e M 1.5 1 r y

) Total n o

t 1.0 k

Inv. decay 5 . 2 2 (

[

0.5 e

E e + e CC d / N d 0.0 0 10 20 30 40 50 60 70 80 90 Ee [ MeV ]

Figure 5.5: Our final calculation (last column of Table 5.1) of atmospheric neutrino backgrounds compared to Super-K-I data [45] (note the different axis scales compared to Fig. 5.1). Even though the data and curves are shown with 4-MeV steps, we have converted them to units of per 1 MeV to match Fig. 5.1. The two components (dashed) and their sum (solid) are shown. Overall, our final calculation — an absolute prediction, not a fit — agrees well with Super-K data.

Overall, the contributions from the four CC channels basically follow their corresponding cross sections (Fig. 5.3), especially for the “naive” calculation (Sec. 5.4.1).

From our “naive” (Sec. 5.4.1) to our standard calculation (Sec. 5.4.1), we mainly remove • the events with pion(s) and take into account µ− capture. For oxygen channels, this

141 removes a large fraction of the events. For νµ+H CC, all the events are removed due to

no CCQE events. Forν ¯µ+H CC, all the events survive because CCQE dominates.

The NCπ+ channels (Sec. 5.4.1) are included starting from the second column. Overall, • they contribute 20%. This is an important contribution, not fully noticed before. ' The Coulomb correction (Sec. 5.4.1) increases the ν +O CC channel by 30% and • µ ' decreases theν ¯ +O CC andν ¯ +H CC channels by 30% and 15% respectively. The µ µ ' changes due to both energy shift and amplitude rescaling are in the same direction

because the invisible muons follow an increasing spectrum. For the CC channels overall,

combining all of the above, the Coulomb correction increases the yield by 15%. For ' NCπ+ channels, it decreases the yield by 5% because of having only the energy shift ' eﬀect.

The Cherenkov threshold correction, from 120 to 135 MeV for p and from 159 • ' µ ' to 187 MeV for p + , increases the numbers by 35%. The number is sensitive to the π ' threshold because of the increasing muon spectrum.

We show both the cases of removing all the events with nuclear gamma rays ( = 100%) • γ and removing none of them (γ = 0%). The former reduces about half of the events (Sec. 5.4.1).

The above features apply for both Table 5.1 for default model set and Table 5.2 for the

EffSFTEM model set. Comparing these two tables, the EffSFTEM model set increases the numbers by 10%. ' Here we estimate the uncertainties of our predictions. The uncertainty of atmospheric neutrino flux is about 20%, as discussed in Sec. 5.2.3. For the uncertainty of neutrino ∼ interactions, in Sec. 5.2.4 the number 20% is concluded. Moreover, in Sec. 5.4.1, by ∼ comparing two different sets of models of GENIE, a variation of 10% is shown, consistent ' with above. Here we use 20% to be conservative. For the uncertainty of the Cherenkov ∼ threshold correction, due to a lack of information about their analysis, an uncertainty of

20% is assigned, which is comparable to the amount that it increases the total number. ∼

142 Adding these uncertainties in quadrature gives a total uncertainty of 35%. This is large ∼ enough to cover the uncertainties due to Coulomb correction, etc. Therefore, our ﬁnal calculation gives 160 56 events. ±

5.4.3 Comparison with data and implication for nuclear gamma rays

Figure 5.5 shows the comparison to Super-K data. The spectrum of the decay electrons

(Michel spectrum plus distortion due to µ− decay on the atomic orbit) we use is from Super-

K measurement, which naturally includes the detector resolution eﬀects. The number of those detected decay electrons, from Super-K’s likelihood ﬁt, is 146 (Super-K-I), with about

10% uncertainty [45]. This number is used to compare with our calculation in Table 5.1 and 5.2. Our final calculation, 160 56, agrees very well. ± For the nuclear gamma branching ratio (Brγ) and cut efficiency (γ), though they both have large uncertainties, comparing our final calculation with Super-K data gives important insights. The consistency indicates that the Br 50% 100% = 50% that we use γ × γ ' × is a good choice. Since we do not expect the true Brγ to be very different from 50%, this means that Super-K should have already done an efficient cut on the atmospheric neutrino backgrounds with nuclear gamma rays. This can also be seen in Table 5.1 and 5.2; if we assume γ = 0% the prediction would be about two times Super-K’s measurement.

5.5 New results on Super-K Low-Energy Atmospheric data:

(νe +ν ¯e) CC component

In this section, we discuss our calculation of the spectrum and uncertainties for the (νe +ν ¯e) CC component of the atmospheric-neutrino background, the smaller of the two components.

These interactions produce a single primary electron. While these events have some direc-

tionality in principle, the low statistics make this hard to use.

Our calculation is similar to that for the invisible muon component of Sec. 5.4, with a

few diﬀerences. Here the atmospheric neutrino ﬂuxes below 100 MeV are important, for

which we use the FLUKA2005 calculations [46]. For the event class, we assume γ = 0%.

143 0.25 ] ]

1 0.006 1 V V e e 0.20 Total CC M M 0.005

1 1 r r y y

1 1 0.15 0.004 ) ) n n o o t t k k

0.003

5 5 . . 0.10

+O CC 2 2 2 2 ( (

0.002 [ [

+O CC

E 0.05 E d d / + / 0.001

N H N

d C d C 0.00 0.000 100 200 300 400 500 0 500 1000 1500 2000 2500 3000 3500 4000 E [ MeV ] E [ MeV ]

Figure 5.6: Distribution of parent atmospheric neutrinos relevant to the invisible muons in our ﬁnal calculation (Sec. 5.4.1). Left: (ν +ν ¯ ) CC component ( 80% of total). Right: µ µ ' NC π+ component ( 20% of total). '

While nuclear gamma rays are produced, they are emitted promptly and nearly immediately

produce secondary electrons through Compton scattering. The distance and time separa-

tions between the primary and secondary electrons are below the resolution of Super-K.

The total energy of the secondary electrons is about 5 MeV, much less than that of the

primary electron (16–90 MeV), making it hard to form multi-peak or multi-ring features,

especially for higher primary electron energies. We comment below on identifying the sec-

ondary electrons through distinct Cherenkov rings. There are no NC channels that produce

appreciable numbers of electrons in this energy range.

The uncertainties are signiﬁcantly larger than for the invisible muon component (for

those, see Sec. 5.4.2). As shown in the next section, Sec. 5.6, here the relevant neutrino

energies are lower. The νe +ν ¯e ﬂux arises from a further step in the decay chains, i.e., from π µ e decay instead of π µ decay, which increases uncertainties, in part due → → → to the fraction of muons that hit the ground and how they decay. In addition, the lower

the neutrino energy, and the greater the diﬀerence between neutrino and lepton energy,

the less well the nuclear models work, as they are based on the Fermi-gas approximation.

Quantitatively, we expect total uncertainties of 45%, combining the uncertainties from '

144 0.200 0.200 +O CC +O CC ] e ] e

1 1 0.175 e+O CC 0.175 e+O CC

V +H CC V +H CC

e e e e

M Total M Total

0.150 0.150 1 1 r r y y 0.125 0.125 1 1 ) ) n n

o 0.100 o 0.100 t t k k

5 5 . .

2 0.075 2 0.075 2 2 ( (

[ [

e 0.050 e 0.050 E E d d / /

N 0.025 N 0.025 d d 0.000 0.000 0 100 200 300 0 100 200 300

E e [ MeV ] E e [ MeV ]

Figure 5.7: Distribution of parent atmospheric neutrinos relevant to the (νe+¯νe) CC component in our ﬁnal calculation (Sec. 5.5). Left: Results for 16 MeV < Ee < 55 MeV. Right: Results for 55 MeV < Ee < 90 MeV. These comprise about 45% and 55%, respectively, of the events in 16 MeV < Ee < 90 MeV. The boundary of 55 MeV is roughly where the invisible-muon and electron components of the background cross in Fig. 5.5.

ﬂux (25%), cross section (30%), Coulomb and Cherenkov threshold correction (20%), larger than 35% for the invisible-muon component. We view this as a minimum estimate because ' we are extrapolating the cross section uncertainties which, in reality, might be larger.

Our predicted spectrum using the default GENIE model set matches the shape of the measured data shown in Fig. 5.5 but is about 40% higher. For the EﬀSFTEM model set, it is about 60% higher. These are reasonably near the (minimum) uncertainties estimated above,

45%. (We focus our comparisons in the energy range 60–90 MeV, where only the electron ' component of the background is important.) This excess is hinted at in Fig. 5.4, where our prediction is 30% higher than the data for the electron component near 100 MeV, whereas it is only 15% higher than the data for the muon component near 200 MeV. While the muon component has a known spectrum (muon decay at rest), the electron component does not.

However, the fact that the shapes of our two calculations agree with each other and with data suggests that we can treat the spectrum shape as known. For our ﬁnal calculation, we scale down the predicted spectrum to match the data. Although this technique is not preferred, the change is within the uncertainties.

145 Figure 5.5 shows our ﬁnal calculation for the (νe +ν ¯e) CC component, including the change in normalization. The spectrum increases with energy because both the ﬂux

(Fig. 5.2) and cross section (Fig. 5.3) do. As veriﬁed by our calculations, the electron component of the background is subdominant in the DSNB energy range. It may be even smaller, as γ could be nonzero, with Super-K cutting some events where the ring from a secondary electron is recognizable despite its low energy due to being emitted at a large angle relative to that of the primary electron. A more detailed study will be done in future work.

5.6 Parent Neutrino Distributions

In this section, we calculate the parent neutrino distributions relevant to the Super-K low- energy data. We focus on our final calculation with the GENIE default model set — including all the physical and detector effects we calculate separately — as this agrees best with Super-K data. These parent-neutrino distributions are needed to determine how to best focus work on reducing uncertainties. The shapes of these distributions are primarily due to the competition between decreasing fluxes and cross sections at low energies and the difficulty of producing the DSNB-mimicking final states at high neutrino energies. The shape of the differential neutrino-nucleus cross section also plays a role.

Figure 5.6 shows the results for the invisible-muon component. The dominant part is from CC interactions in 150–350 MeV, principally the νµ + O channel. Theν ¯µ + H turns on at a lower energy than the others due to the lack of a high nuclear threshold. There is also a 20% component from NC interactions that produce invisible π+; the parent neutrinos are mostly from 500 MeV–2 GeV due to the high energy threshold for resonance production.

Figure 5.7 shows the results for the (νe +ν ¯e) CC component. Most of the parent neutrinos are from 20–200 MeV, principally the νe + O channel. The relative contributions from diﬀerent channels are similar to the CC component of the invisible muons.

For both background components, the most striking point is that the neutrino energies are much higher than the measured electron energies. (For the DSNB signal, these are sep-

146 arated by only 1.3 MeV.) For the invisible-muon component, this is because the electrons ' are produced through muon decay at rest, which forgets the initial muon energies. For the electron component, this is because the neutrino interaction rate only becomes appreciable at high enough neutrino energies; the low observed electron energies arise through the tail of the diﬀerential cross section.

The fact that the background parent neutrino energies are high opens opportunities to cut those events through the other secondary particles that are produced but which are not yet detected. For example, the DSNB-induced neutrons are mostly < 1 MeV, but atmospheric neutrino induced neutrons could be as energetic as 100 MeV, and will travel much further [57]. These points will be reported on in detail in our companion paper [360].

5.7 Conclusions

The ﬁrst detection of the DSNB is of great importance, as it will provide a new test of the neutrino emission per core collapse and the cosmic core-collapse rate. Super-K is big enough to have collected 100 DSNB interactions in its 20 years of operation, but they ∼ & are presently obscured by detector backgrounds. Beginning in 2020, Super-K is adding dissolved gadolinium, which will greatly reduce these backgrounds. Even so, theoretical work is needed to better understand the physical origin of these backgrounds and how to cut them further.

In this paper, we perform the ﬁrst comprehensive calculations of the dominant atmospheric-neutrino backgrounds to DSNB detection. We begin in Sec. 5.2 by systematically reviewing the inputs to the signal and background calculations. Using this framework, in Sec. 5.3, we show that we can reproduce the Super-K high-energy atmospheric neutrino data in detail. With this validation, we tackle the harder calculation of the low-energy atmospheric neutrino data and the theoretical predictions for the uncertainties. In

Sec. 5.4.1, we show that we can reproduce the invisible-muon component of the background within uncertainties if we take into account several subtle physical and detector eﬀects. In

Sec. 5.5, we show the same for the (νe +ν ¯e) CC component, which is the smaller of the two

147 components. In Sec. 5.6, we calculate the parent-neutrino energy spectra for these detector backgrounds. Because these neutrino energies are relatively high, this indicates paths to cut these events through other particles produced in the interactions.

This paper is thus the ﬁrst step towards our goal of revealing the DSNB signal events in Super-K. In our next paper, Ref. [360], we calculate in detail new ways to cut the backgrounds based on the physical insights of the present paper. These results will apply not only to Super-K, but also to SK-Gd [57–59] and Hyper-Kamiokande [361].

Acknowledgments We thank the GENIE collaboration for providing a comprehensive framework for neutrino- nucleus interactions [138–140], especially Costas Andreopoulos, Steven A. Dytman and

Gabriel N. Perdue for assistance and discussions. This research also makes use of FLUKA, which simulates particle transport in matter [141, 142]. We are grateful for helpful comments from Shirley Li, Kenny Ng, Michael Smy, Mark Vagins, Chenyuan Xu, and Guanying Zhu.

BZ and JFB were supported by NSF grant PHY-1714479. In the later part of this work,

BZ was supported by a Neutrino Theory Network grant provided through Fermilab.

148 Bibliography

[1] Bei Zhou and John F. Beacom. W -boson and trident production in TeV–PeV neutrino

observatories. Phys. Rev. D, 101(3):036010, 2020.

[2] Bei Zhou and John F. Beacom. Neutrino-nucleus cross sections for W-boson and

trident production. Phys. Rev. D, 101(3):036011, 2020.

[3] A. Albert et al. Constraints on Spin-Dependent Dark Matter Scattering with Long-

Lived Mediators from TeV Observations of the Sun with HAWC. Phys. Rev. D,

98:123012, 2018.

[4] A. Albert et al. First HAWC Observations of the Sun Constrain Steady TeV Gamma-

Ray Emission. Phys. Rev. D, 98(12):123011, 2018.

[5] Qing-Wen Tang, Kenny C.Y. Ng, Tim Linden, Bei Zhou, John F. Beacom, and An-

nika H.G. Peter. Unexpected dip in the solar gamma-ray spectrum. Phys. Rev. D,

98(6):063019, 2018.

[6] Tim Linden, Bei Zhou, John F. Beacom, Annika H.G. Peter, Kenny C.Y. Ng, and

Qing-Wen Tang. Evidence for a New Component of High-Energy Solar Gamma-Ray

Production. Phys. Rev. Lett., 121(13):131103, 2018.

[7] Bei Zhou, Kenny C. Y. Ng, John F. Beacom, and Annika H. G. Peter. TeV Solar

Gamma Rays From Cosmic-Ray Interactions. Phys. Rev. D, 96(2):023015, 2017.

[8] Bei Zhou, Yun-Feng Liang, Xiaoyuan Huang, Xiang Li, Yi-Zhong Fan, Lei Feng, and

149 Jin Chang. GeV excess in the MilkyWay: The role of diﬀuse galactic gamma-ray

emission templates. Phys. Rev. D, 91(12):123010, 2015.

[9] Bin Liu, Wei Chen, Yun-Feng Liang, Bei Zhou, Hao-Ning He, Pak-Hin Thomas Tam,

Lang Shao, Zhi-Ping Jin, Yi-Zhong Fan, and Da-Ming Wei. GRB 131231A: Implica-

tions of the GeV emission. Astrophys. J. Lett., 787:L6, 2014.

[10] Bei Zhou, Xiang Li, Tao Wang, Yi-Zhong Fan, and Da-Ming Wei. Fast radio bursts

as a cosmic probe? Phys. Rev. D, 89(10):107303, 2014.

[11] Xiang Li, Bei Zhou, Hao-Ning He, Yi-Zhong Fan, and Da-Ming Wei. Model-dependent

estimate on the connection between fast radio bursts and ultra-high energy cosmic

rays. Astrophys. J., 797(1):33, 2014.

[12] Yun-Feng Liang, Bei Zhou, Hao-Ning He, Pak-Hin Thomas Tam, Yi-Zhong Fan, and

Da-Ming Wei. High energy emission of GRB 130821A: constraining the density pro-

ﬁle of the circum-burst medium as well as the initial Lorentz factor of the outﬂow.

Astrophys. J., 781(2):74, 2014.

[13] Yi-Zhong Fan, P.H.T. Tam, Fu-Wen Zhang, Yun-Feng Liang, Hao-Ning He, Bei

Zhou, Rui-Zhi Yang, Zhi-Ping Jin, and Da-Ming Wei. High energy emission of GRB

130427A: Evidence for inverse Compton radiation. Astrophys. J., 776:95, 2013.

[14] U.F. Katz and Ch. Spiering. High-Energy Neutrino Astrophysics: Status and Per-

spectives. Prog. Part. Nucl. Phys., 67:651–704, 2012.

[15] Rebecca K. Leane, Kenny C. Y. Ng, and John F. Beacom. Powerful Solar Signatures

of Long-Lived Dark Mediators. 2017.

[16] A. A. Abdo et al. Fermi-LAT Observations of Two Gamma-Ray Emission Components

from the Quiescent Sun. Astrophys. J., 734:116, 2011.

[17] D. Seckel, Todor Stanev, and T. K. Gaisser. Signatures of cosmic-ray interactions on

the solar surface. Astrophys. J., 382:652–666, 1991.

150 [18] Zhen Cao. LHAASO: Science and Status. Frascati Phys. Ser., 58:331, 2014.

[19] Huihai He. Design highlights and status of the LHAASO project.

[20] A. Albert et al. Science Case for a Wide Field-of-View Very-High-Energy Gamma-Ray

Observatory in the Southern Hemisphere. 2 2019.

[21] M.U. Nisa, J.F. Beacom, S.Y. BenZvi, R.K. Leane, T. Linden, K.C.Y. Ng, A.H.G. Pe-

ter, and B. Zhou. The Sun at GeV–TeV Energies: A New Laboratory for Astroparticle

Physics. 3 2019.

[22] https://www.nbi.ku.dk/english/research/experimental-particle-physics/

icecube/astroparticle-physics/.

[23] Carlo Giunti and Chung W. Kim. Fundamentals of Neutrino Physics and Astro-

physics. 2007.

[24] Amanda Cooper-Sarkar, Philipp Mertsch, and Subir Sarkar. The high energy neutrino

cross-section in the Standard Model and its uncertainty. JHEP, 08:042, 2011.

[25] V. Barger, Lingjun Fu, J. G. Learned, D. Marfatia, S. Pakvasa, and T. J. Weiler.

Glashow resonance as a window into cosmic neutrino sources. Phys. Rev., D90:121301,

2014.

[26] James Madsen. Ultra-High Energy Neutrinos. In 38th International Symposium on

Physics in Collision, 1 2019.

[27] Christian Haack and Christopher Wiebusch. A measurement of the diﬀuse astro-

physical muon neutrino ﬂux using eight years of IceCube data. PoS, ICRC2017:1005,

2018.

[28] Claudio Kopper. Observation of Astrophysical Neutrinos in Six Years of IceCube

Data. PoS, ICRC2017:981, 2018.

[29] Shunsaku Horiuchi, John F. Beacom, and Eli Dwek. The Diﬀuse Supernova Neutrino

Background is detectable in Super-Kamiokande. Phys. Rev., D79:083013, 2009. 151 [30] Kenny C. Y. Ng, John F. Beacom, Annika H. G. Peter, and Carsten Rott. First

Observation of Time Variation in the Solar-Disk Gamma-Ray Flux with Fermi. Phys.

Rev., D94(2):023004, 2016.

[31] A. U. Abeysekara et al. Observation of the Crab Nebula with the HAWC Gamma-Ray

Observatory. 2017.

[32] M. Aguilar et al. Electron and Positron Fluxes in Primary Cosmic Rays Measured

with the Alpha Magnetic Spectrometer on the International Space Station. Phys.

Rev. Lett., 113:121102, 2014.

[33] F. Aharonian et al. Probing the ATIC peak in the cosmic-ray electron spectrum with

H.E.S.S. Astron. Astrophys., 508:561, 2009.

[34] F. Aharonian et al. The energy spectrum of cosmic-ray electrons at TeV energies.

Phys. Rev. Lett., 101:261104, 2008.

[35] David Staszak. A Cosmic-ray Electron Spectrum with VERITAS. PoS,

ICRC2015:411, 2016.

[36] Matthew D. Kistler and Hasan Yuksel. New Constraints on the Highest-Energy

Cosmic-Ray Electrons and Positrons. 2009.

[37] Markus Ahlers and Kohta Murase. Probing the Galactic Origin of the IceCube Excess

with Gamma-Rays. Phys. Rev., D90(2):023010, 2014.

[38] D. Seckel. Neutrino photon reactions in astrophysics and cosmology. Phys. Rev. Lett.,

80:900–903, 1998.

[39] I. Alikhanov. The Glashow resonance in neutrinophoton scattering. Phys. Lett.,

B741:295–300, 2015.

[40] I. Alikhanov. Hidden Glashow resonance in neutrino-nucleus collisions. Phys. Lett.,

B756:247–253, 2016.

152 [41] Gabriel Magill and Ryan Plestid. Neutrino Trident Production at the Intensity Fron-

tier. Phys. Rev., D95(7):073004, 2017.

[42] Peter Ballett, Matheus Hostert, Silvia Pascoli, Yuber F. Perez-Gonzalez, Zahra

Tabrizi, and Renata Zukanovich Funchal. Neutrino Trident Scattering at Near De-

tectors. JHEP, 01:119, 2019.

[43] Wolfgang Altmannshofer, Stefania Gori, Justo Martn-Albo, Alexandre Sousa, and

Michael Wallbank. Neutrino Tridents at DUNE. 2019.

[44] M. G. Aartsen et al. Evidence for High-Energy Extraterrestrial Neutrinos at the

IceCube Detector. Science, 342:1242856, 2013.

[45] K. Bays et al. Supernova Relic Neutrino Search at Super-Kamiokande. Phys. Rev.,

D85:052007, 2012.

[46] G. Battistoni, A. Ferrari, T. Montaruli, and P. R. Sala. The atmospheric neutrino

ﬂux below 100-MeV: The FLUKA results. Astropart. Phys., 23:526–534, 2005.

[47] M. Honda, M. Sajjad Athar, T. Kajita, K. Kasahara, and S. Midorikawa. Atmospheric

neutrino ﬂux calculation using the NRLMSISE-00 atmospheric model. Phys. Rev.,

D92(2):023004, 2015.

[48] Marius Wallraﬀ and Christopher Wiebusch. Calculation of oscillation probabilities of

atmospheric neutrinos using nuCraft. Comput. Phys. Commun., 197:185–189, 2015.

[49] Y. Ashie et al. A Measurement of atmospheric neutrino oscillation parameters by

SUPER-KAMIOKANDE I. Phys. Rev., D71:112005, 2005.

[50] K. Hirata et al. Observation of a Neutrino Burst from the Supernova SN 1987a. Phys.

Rev. Lett., 58:1490–1493, 1987.

[51] R. M. Bionta et al. Observation of a Neutrino Burst in Coincidence with Supernova

SN 1987a in the Large Magellanic Cloud. Phys. Rev. Lett., 58:1494, 1987.

153 [52] E. N. Alekseev, L. N. Alekseeva, I. V. Krivosheina, and V. I. Volchenko. Detection of

the Neutrino Signal From SN1987A in the LMC Using the Inr Baksan Underground

Scintillation Telescope. Phys. Lett., B205:209–214, 1988.

[53] Kate Scholberg. Supernova Neutrino Detection. Ann. Rev. Nucl. Part. Sci., 62:81–

103, 2012.

[54] M. G. Aartsen et al. Observation of High-Energy Astrophysical Neutrinos in Three

Years of IceCube Data. Phys. Rev. Lett., 113:101101, 2014.

[55] M. G. Aartsen et al. Neutrino emission from the direction of the blazar TXS 0506+056

prior to the IceCube-170922A alert. Science, 361(6398):147–151, 2018.

[56] Robert Stein et al. A high-energy neutrino coincident with a tidal disruption event.

2020.

[57] John F. Beacom and Mark R. Vagins. GADZOOKS! Anti-neutrino spectroscopy with

large water Cherenkov detectors. Phys. Rev. Lett., 93:171101, 2004.

[58] Chenyuan Xu. Current status of SK-Gd project and EGADS. J. Phys. Conf. Ser.,

718(6):062070, 2016.

[59] Luis Labarga. The SuperK-gadolinium project. PoS, EPS-HEP2017:118, 2018.

[60] Kenny C. Y. Ng, John F. Beacom, Annika H. G. Peter, and Carsten Rott. Solar

Atmospheric Neutrinos: A New Neutrino Floor for Dark Matter Searches. Phys.

Rev., D96(10):103006, 2017.

[61] C. A. Argelles, G. de Wasseige, A. Fedynitch, and B. J. P. Jones. Solar Atmospheric

Neutrinos and the Sensitivity Floor for Solar Dark Matter Annihilation Searches.

JCAP, 1707:024, 2017.

[62] Joakim Edsjo, Jessica Elevant, Rikard Enberg, and Carl Niblaeus. Neutrinos from

cosmic ray interactions in the Sun. JCAP, 1706:033, 2017.

[63] M. Masip. High energy neutrinos from the Sun. Astropart. Phys., 97:63–68, 2018. 154 [64] M. G. Aartsen et al. Searches for neutrinos from cosmic-ray interactions in the Sun

using seven years of IceCube data. 2019.

[65] Y. Fukuda et al. Evidence for oscillation of atmospheric neutrinos. Phys. Rev. Lett.,

81:1562–1567, 1998.

[66] Q. R. Ahmad et al. Direct evidence for neutrino ﬂavor transformation from neu-

tral current interactions in the Sudbury Neutrino Observatory. Phys. Rev. Lett.,

89:011301, 2002.

[67] Q. R. Ahmad et al. Measurement of day and night neutrino energy spectra at SNO

and constraints on neutrino mixing parameters. Phys. Rev. Lett., 89:011302, 2002.

[68] S. Fukuda et al. Determination of solar neutrino oscillation parameters using 1496

days of Super-Kamiokande I data. Phys. Lett., B539:179–187, 2002.

[69] K. Abe et al. Search for Diﬀerences in Oscillation Parameters for Atmospheric Neu-

trinos and Antineutrinos at Super-Kamiokande. Phys. Rev. Lett., 107:241801, 2011.

[70] M. G. Aartsen et al. Measurement of the multi-TeV neutrino cross section with

IceCube using Earth absorption. Nature, 551:596–600, 2017.

[71] Mauricio Bustamante and Amy Connolly. Extracting the Energy-Dependent

Neutrino-Nucleon Cross Section above 10 TeV Using IceCube Showers. Phys. Rev.

Lett., 122(4):041101, 2019.

[72] G. L. Fogli, E. Lisi, A. Marrone, and A. Palazzo. Global analysis of three-ﬂavor

neutrino masses and mixings. Prog. Part. Nucl. Phys., 57:742–795, 2006.

[73] R. N. Mohapatra and A. Y. Smirnov. Neutrino Mass and New Physics. Ann. Rev.

Nucl. Part. Sci., 56:569–628, 2006.

[74] Frank T. Avignone, III, Steven R. Elliott, and Jonathan Engel. Double Beta Decay,

Majorana Neutrinos, and Neutrino Mass. Rev. Mod. Phys., 80:481–516, 2008.

155 [75] Stephen F. King and Christoph Luhn. Neutrino Mass and Mixing with Discrete

Symmetry. Rept. Prog. Phys., 76:056201, 2013.

[76] X. Qian and P. Vogel. Neutrino Mass Hierarchy. Prog. Part. Nucl. Phys., 83:1–30,

2015.

[77] Andr de Gouva. Neutrino Mass Models. Ann. Rev. Nucl. Part. Sci., 66:197–217,

2016.

[78] F. Cornet, Jose I. Illana, and M. Masip. TeV strings and the neutrino nucleon cross-

section at ultrahigh-energies. Phys. Rev. Lett., 86:4235–4238, 2001.

[79] Paolo Lipari. CP violation eﬀects and high-energy neutrinos. Phys. Rev., D64:033002,

2001.

[80] John F. Beacom, Nicole F. Bell, Dan Hooper, Sandip Pakvasa, and Thomas J. Weiler.

Decay of High-Energy Astrophysical Neutrinos. Phys. Rev. Lett., 90:181301, 2003.

[81] Tao Han and Dan Hooper. Eﬀects of electroweak instantons in high-energy neutrino

telescopes. Phys. Lett., B582:21–26, 2004.

[82] Tao Han and Dan Hooper. The Particle physics reach of high-energy neutrino astron-

omy. New J. Phys., 6:150, 2004.

[83] Dan Hooper, Dean Morgan, and Elizabeth Winstanley. Lorentz and CPT invariance

violation in high-energy neutrinos. Phys. Rev., D72:065009, 2005.

[84] M. C. Gonzalez-Garcia, F. Halzen, and M. Maltoni. Physics reach of high-energy and

high-statistics icecube atmospheric neutrino data. Phys. Rev., D71:093010, 2005.

[85] Kenny C. Y. Ng and John F. Beacom. Cosmic neutrino cascades from se-

cret neutrino interactions. Phys. Rev., D90(6):065035, 2014. [Erratum: Phys.

Rev.D90,no.8,089904(2014)].

[86] Kunihto Ioka and Kohta Murase. IceCube PeVEeV neutrinos and secret interactions

of neutrinos. PTEP, 2014(6):061E01, 2014. 156 [87] Mauricio Bustamante, John F. Beacom, and Kohta Murase. Testing decay of astro-

physical neutrinos with incomplete information. Phys. Rev., D95(6):063013, 2017.

[88] Pilar Coloma, Pedro A. N. Machado, Ivan Martinez-Soler, and Ian M. Shoe-

maker. Double-Cascade Events from New Physics in Icecube. Phys. Rev. Lett.,

119(20):201804, 2017.

[89] M. Ajello et al. Impulsive and Long Duration High-Energy Gamma-ray Emission

From the Very Bright 2012 March 7 Solar Flares. Astrophys. J., 789(1):20, 2014.

[90] M. Ackermann et al. High-energy Gamma-Ray Emission from Solar Flares: Sum-

mary of Fermi Large Area Telescope Detections and Analysis of Two M-class Flares.

Astrophys. J., 787:15, 2014.

[91] M. Ackermann et al. Fermi-LAT Observations of High-energy Behind-the-limb Solar

Flares. Astrophys. J., 835(2):219, 2017.

[92] Douglas J. Rodgers, David J. Lawrence, William C. Feldman, and Patrick N. Pe-

plowski. Neutrons and energetic charged particles in the inner heliosphere: Measure-

ments of the MESSENGER Neutron Spectrometer from 0.3 to 0.85 AU. Journal of

Geophysical Research (Space Physics), 120(2):841–854, February 2015.

[93] David J. Lawrence, Patrick N. Peplowski, William C. Feldman, Nathan A. Schwadron,

and Harlan E. Spence. Galactic cosmic ray variations in the inner heliosphere from

solar distances less than 0.5 AU: Measurements from the MESSENGER Neutron

Spectrometer. Journal of Geophysical Research (Space Physics), 121(8):7398–7406,

August 2016.

[94] Jonathan L. Feng, Jordan Smolinsky, and Philip Tanedo. Detecting dark mat-

ter through dark photons from the Sun: Charged particle signatures. Phys. Rev.,

D93(11):115036, 2016.

[95] Chiara Arina, Mihailo Backovic, Jan Heisig, and Michele Lucente. Solar γ-rays as a

Complementary Probe of Dark Matter. 2017. 157 [96] Carl Niblaeus, Ankit Beniwal, and Joakim Edsjo. Neutrinos and gamma rays from

long-lived mediator decays in the Sun. JCAP, 1911:011, 2019.

[97] Joseph F. Dolan and G. G. Fazio. The gamma-ray spectrum of the sun. Reviews of

Geophysics, 3(2):319–343, 1965.

[98] Laurence E. Peterson, Daniel A. Schwartz, R. M. Pelling, and David McKenzie. The

Upper Limit Solar Gamma-Ray Spectrum to 10 Mev. Journal of Geophysical Research,

71:5778, December 1966.

[99] H. S. Hudson. Proc Gamma Ray Observatory Workshop (Greenbelt: Goddard Space

Flight Center), 4:351, 1989.

[100] https://ocw.mit.edu/courses/nuclear-engineering/ 22-611j-introduction-to-plasma-physics-i-fall-2003/lecture-notes/

chap2.pdf.

[101] D. J. Thompson, D. L. Bertsch, D. J. Morris, and R. Mukherjee. Energetic gamma

ray experiment telescope high-energy gamma ray observations of the Moon and quiet

Sun. Journal of Geophysical Research, 102(A7):14735–14740, July 1997.

[102] Elena Orlando and Andrew W. Strong. Gamma-ray emission from the solar halo and

disk: a study with EGRET data. Astron. Astrophys., 480:847, 2008.

[103] Igor V. Moskalenko, Troy A. Porter, and Seth W. Digel. Inverse Compton scattering

on solar photons, heliospheric modulation, and neutrino astrophysics. Astrophys. J.,

652:L65–L68, 2006. [Erratum: Astrophys. J. 664, L143 (2007)].

[104] Elena Orlando and Andrew Strong. Gamma-rays from halos around stars and the

Sun. Astrophys. Space Sci., 309:359–363, 2007.

[105] S. Agostinelli et al. GEANT4: A Simulation toolkit. Nucl. Instrum. Meth., A506:250–

303, 2003.

[106] http://geant4.cern.ch/support/proc_mod_catalog/physics_lists/. 158 [107] M. Aguilar et al. Precision Measurement of the Proton Flux in Primary Cosmic

Rays from Rigidity 1 GV to 1.8 TV with the Alpha Magnetic Spectrometer on the

International Space Station. Phys. Rev. Lett., 114:171103, 2015.

[108] K. A. Olive et al. Review of Particle Physics. Chin. Phys., C38:090001, 2014.

[109] M. Aguilar et al. Precision Measurement of the Helium Flux in Primary Cosmic

Rays of Rigidities 1.9 GV to 3 TV with the Alpha Magnetic Spectrometer on the

International Space Station. Phys. Rev. Lett., 115(21):211101, 2015.

[110] Elena Orlando and Andrew Strong. StellarICs: Stellar and solar Inverse Compton

emission package. 2013.

[111] Elena Orlando and Andrew W. Strong. A software package for stellar and solar

inverse-Compton emission: Stellarics. Nucl. Phys. Proc. Suppl., 239-240:266–269,

2013.

[112] Jaime Alvarez-Muniz, Jonathan L. Feng, Francis Halzen, Tao Han, and Dan Hooper.

Detecting microscopic black holes with neutrino telescopes. Phys. Rev., D65:124015,

2002.

[113] Dan Hooper, Dean Morgan, and Elizabeth Winstanley. Probing quantum decoherence

with high-energy neutrinos. Phys. Lett., B609:206–211, 2005.

[114] Jordi Salvado, Olga Mena, Sergio Palomares-Ruiz, and Nuria Rius. Non-standard

interactions with high-energy atmospheric neutrinos at IceCube. JHEP, 01:141, 2017.

[115] Erik Blaufuss, C. Kopper, and C. Haack. The IceCube-Gen2 High Energy Array.

PoS, ICRC2015:1146, 2016.

[116] Shirley Weishi Li, Mauricio Bustamante, and John F. Beacom. Echo Technique to

Distinguish Flavors of Astrophysical Neutrinos. Phys. Rev. Lett., 122(15):151101,

2019.

159 [117] Juliana Stachurska. First Double Cascade Tau Neutrino Candidates in IceCube and

a New Measurement of the Flavor Composition. In 36th International Cosmic Ray

Conference (ICRC 2019) Madison, Wisconsin, USA, July 24-August 1, 2019, 2019.

[118] Shao-Feng Ge, Manfred Lindner, and Werner Rodejohann. Atmospheric Trident Pro-

duction for Probing New Physics. Phys. Lett., B772:164–168, 2017.

[119] Rhorry Gauld. Precise predictions for multi-TeV and PeV energy neutrino scattering

rates. 2019.

[120] John F. Beacom. The Diﬀuse Supernova Neutrino Background. Ann. Rev. Nucl. Part.

Sci., 60:439–462, 2010.

[121] S. M. Adams, C. S. Kochanek, J. R. Gerke, and K. Z. Stanek. The search for failed

supernovae with the Large Binocular Telescope: constraints from 7 yr of data. Mon.

Not. Roy. Astron. Soc., 469(2):1445–1455, 2017.

[122] G. G. Raﬀelt. Stars as laboratories for fundamental physics. 1996.

[123] Kei Kotake, Katsuhiko Sato, and Keitaro Takahashi. Explosion mechanism, neutrino

burst, and gravitational wave in core-collapse supernovae. Rept. Prog. Phys., 69:971–

1144, 2006.

[124] Andrew M. Hopkins and John F. Beacom. On the normalisation of the cosmic star

formation history. Astrophys. J., 651:142–154, 2006.

[125] Yu Seon Jeong, Sergio Palomares-Ruiz, Mary Hall Reno, and Ina Sarcevic. Probing

secret interactions of eV-scale sterile neutrinos with the diﬀuse supernova neutrino

background. JCAP, 1806:019, 2018.

[126] Basudeb Dasgupta, Ranjan Laha, and Anupam Ray. Neutrino and positron con-

straints on spinning primordial black hole dark matter. 2019.

[127] Cyril Creque-Sarbinowski, Jeﬀrey Hyde, and Marc Kamionkowski. Resonant Neutrino

Self-Interactions. 2020. 160 [128] Y. Fukuda et al. The Super-Kamiokande detector. Nucl. Instrum. Meth., A501:418–

462, 2003.

[129] P. Vogel and John F. Beacom. Angular distribution of neutron inverse beta decay,

anti-neutrino(e) + p —¿ e+ + n. Phys. Rev., D60:053003, 1999.

[130] Alessandro Strumia and Francesco Vissani. Precise quasielastic neutrino/nucleon

cross-section. Phys. Lett., B564:42–54, 2003.

[131] H. Zhang et al. Supernova Relic Neutrino Search with Neutron Tagging at Super-

Kamiokande-IV. Astropart. Phys., 60:41–46, 2015.

[132] https://www.nndc.bnl.gov/.

[133] M. Malek et al. Search for supernova relic neutrinos at SUPER-KAMIOKANDE.

Phys. Rev. Lett., 90:061101, 2003.

[134] Shirley Weishi Li and John F. Beacom. First calculation of cosmic-ray muon spallation

backgrounds for MeV astrophysical neutrino signals in Super-Kamiokande. Phys.

Rev., C89:045801, 2014.

[135] Shirley Weishi Li and John F. Beacom. Spallation Backgrounds in Super-Kamiokande

Are Made in Muon-Induced Showers. Phys. Rev., D91(10):105005, 2015.

[136] Shirley Weishi Li and John F. Beacom. Tagging Spallation Backgrounds with Showers

in Water-Cherenkov Detectors. Phys. Rev., D92(10):105033, 2015.

[137] Kevin James Kelly, Pedro AN Machado, Ivan Martinez Soler, Stephen J Parke,

and Yuber F Perez Gonzalez. Sub-GeV Atmospheric Neutrinos and CP-Violation

in DUNE. Phys. Rev. Lett., 123(8):081801, 2019.

[138] C. Andreopoulos et al. The GENIE Neutrino Monte Carlo Generator. Nucl. Instrum.

Meth., A614:87–104, 2010.

161 [139] Costas Andreopoulos, Christopher Barry, Steve Dytman, Hugh Gallagher, Tomasz

Golan, Robert Hatcher, Gabriel Perdue, and Julia Yarba. The GENIE Neutrino

Monte Carlo Generator: Physics and User Manual. 2015.

[140] http://www.genie-mc.org/.

[141] Alfredo Ferrari, Paola R. Sala, Alberto Fasso, and Johannes Ranft. FLUKA: A multi-

particle transport code (Program version 2005). 2005.

[142] T. T. Bhlen, F. Cerutti, M. P. W. Chin, A. Fass, A. Ferrari, P. G. Ortega, A. Mairani,

P. R. Sala, G. Smirnov, and V. Vlachoudis. The FLUKA Code: Developments and

Challenges for High Energy and Medical Applications. Nucl. Data Sheets, 120:211–

214, 2014.

[143] H. Nishino et al. Search for Proton Decay via p —¿ e+ pi0 and p —¿ mu+ pi0 in a

Large Water Cherenkov Detector. Phys. Rev. Lett., 102:141801, 2009.

[144] K. Abe et al. Search for proton decay via p νK+ using 260 kilotonyear data of → Super-Kamiokande. Phys. Rev., D90(7):072005, 2014.

[145] K. Abe et al. Search for proton decay via p e+π0 and p µ+π0 in 0.31 → → megatonyears exposure of the Super-Kamiokande water Cherenkov detector. Phys.

Rev., D95(1):012004, 2017.

[146] K. Abe et al. The Search for n n¯ oscillation in Super-Kamiokande I. Phys. Rev., − D91:072006, 2015.

[147] K. Choi et al. Search for neutrinos from annihilation of captured low-mass dark matter

particles in the Sun by Super-Kamiokande. Phys. Rev. Lett., 114(14):141301, 2015.

[148] C. Kachulis et al. Search for Boosted Dark Matter Interacting With Electrons in

Super-Kamiokande. Phys. Rev. Lett., 120(22):221301, 2018.

[149] K. Abe et al. Indirect Search for Dark Matter from the Galactic Center and Halo

with the Super-Kamiokande Detector. 5 2020. 162 [150] E. J. Schneid et al. EGRET observations of X-class solar ﬂares. Astronomy and

Astrophysics Supplement Series, 120:299–302, December 1996.

[151] M. Ackermann et al. High-energy Gamma-Ray Emission from Solar Flares: Sum-

mary of Fermi Large Area Telescope Detections and Analysis of Two M-class Flares.

Astrophys. J., 787:15, 2014.

[152] Melissa Pesce-Rollins, Nicola Omodei, Vahe’ Petrosian, Wei Liu, Fatima Rubio

da Costa, Alice Allafort, and Qingrong Chen. First detection of >100 MeV gamma

rays associated with a behind-the-limb solar ﬂare. Astrophys. J., 805(2):L15, 2015.

[153] Douglas J. Rodgers, David J. Lawrence, William C. Feldman, and Patrick N. Pe-

plowski. Neutrons and energetic charged particles in the inner heliosphere: Measure-

ments of the MESSENGER Neutron Spectrometer from 0.3 to 0.85 AU. Journal of

Geophysical Research: Space Physics, 120(2):841–854, 2015.

[154] David J. Lawrence, Patrick N. Peplowski, William C. Feldman, Nathan A. Schwadron,

and Harlan E. Spence. Galactic cosmic ray variations in the inner heliosphere from

solar distances less than 0.5 AU: Measurements from the MESSENGER Neutron

Spectrometer. Journal of Geophysical Research: Space Physics, 121(8):7398–7406,

2016.

[155] J. R. Jokipii. Propagation of cosmic rays in the solar wind. Reviews of Geophysics

and Space Physics, 9:27–87, 1971.

[156] Marius Potgieter. Solar Modulation of Cosmic Rays. Living Rev. Solar Phys., 10:3,

2013.

[157] D. J. Thompson, D. L. Bertsch, D. J. Morris, and R. Mukherjee. Energetic gamma

ray experiment telescope high-energy gamma ray observations of the moon and quiet

sun. Journal of Geophysical Research: Space Physics, 102(A7):14735–14740, 1997.

[158] Robert W. Atkins et al. Milagrito: A TeV air shower array. Nucl. Instrum. Meth.,

A449:478–499, 2000. 163 [159] G. Aielli et al. Layout and performance of RPCs used in the Argo-YBJ experiment.

Nucl. Instrum. Meth., A562:92–96, 2006.

[160] K. Hibino et al. A New project to search for high-energy gamma-ray point sources in

Tibet. Nucl. Phys. Proc. Suppl., 10B:219–227, 1989.

[161] M. Amenomori et al. Probe of the Solar Magnetic Field Using the Cosmic-Ray Shadow

of the Sun. Phys. Rev. Lett., 111(1):011101, 2013.

[162] Olivia Enriquez-Rivera and Alejandro Lara. The Galactic cosmic-ray Sun shadow

observed by HAWC. PoS, ICRC2015:099, 2016.

[163] Sami K. Solanki, Bernd Inhester, and Manfred Schussler. The solar magnetic ﬁeld.

Rept. Prog. Phys., 69:563–668, 2006.

[164] Eric Priest. Magnetohydrodynamics of the Sun. Cambridge University Press, 2014.

[165] Thomas Wiegelmann, Julia K. Thalmann, and Sami K. Solanki. The magnetic ﬁeld

in the solar atmosphere. The Astronomy and Astrophysics Review, 22(1):78, 2014.

[166] L. J. Gleeson and W. I. Axford. Solar Modulation of Galactic Cosmic Rays. Astrophys.

J., 154:1011, 1968.

[167] Ilias Cholis, Dan Hooper, and Tim Linden. A Predictive Analytic Model for the Solar

Modulation of Cosmic Rays. Phys. Rev., D93(4):043016, 2016.

[168] P. Bobik et al. Systematic Investigation of Solar Modulation of Galactic Protons

for Solar Cycle 23 using a Monte Carlo Approach with Particle Drift Eﬀects and

Latitudinal Dependence. Astrophys. J., 745:132, 2012.

[169] Shoko Miyake and Shohei Yanagita. Eﬀects of the tilted and wavy current sheet

on the solar modulation of galactic cosmic rays. In 29th International Cosmic Ray

Conference (ICRC 2005) Pune, India, August 3-11, 2005, 2006.

[170] A. U. Abeysekara et al. Sensitivity of the High Altitude Water Cherenkov Detector

to Sources of Multi-TeV Gamma Rays. Astropart. Phys., 50-52:26–32, 2013. 164 [171] M. Ambrosio et al. The Observation of upgoing charged particles produced by high-

energy muons in underground detectors. Astropart. Phys., 9:105–117, 1998.

[172] S. R. Kelner, Felex A. Aharonian, and V. V. Bugayov. Energy spectra of gamma-

rays, electrons and neutrinos produced at proton-proton interactions in the very high

energy regime. Phys. Rev., D74:034018, 2006. [Erratum: Phys. Rev. D79, 039901

(2009)].

[173] Norman Hodgson Baker and Stefan Temesv´ary. Tables of convective stellar envelope

models. Institute for Space Studies, Goddard Space Flight Center, National Aeronau-

tics and Space Administration, 1966.

[174] J. E. Vernazza, E. H. Avrett, and R. Loeser. Structure of the Solar Chromosphere. Ba-

sic Computations and Summary of the Results. The Astrophysical Journal, 184:605–

632, September 1973.

[175] A. A. Abdo et al. Fermi Large Area Telescope Observations of the Cosmic-Ray

Induced gamma-ray Emission of the Earth’s Atmosphere. Phys. Rev., D80:122004,

2009.

[176] M. Ackermann et al. Inferred Cosmic-Ray Spectrum from Fermi Large Area Telescope

-Ray Observations of Earths Limb. Phys. Rev. Lett., 112:151103, 2014.

[177] Carsten Rott, Jennifer Siegal-Gaskins, and John F. Beacom. New Sensitivity to Solar

WIMP Annihilation using Low-Energy Neutrinos. Phys. Rev., D88:055005, 2013.

[178] T. K. Gaisser and M. Honda. Flux of atmospheric neutrinos. Ann. Rev. Nucl. Part.

Sci., 52:153–199, 2002.

[179] Michael Kachelriess, Igor V. Moskalenko, and Sergey S. Ostapchenko. Nuclear en-

hancement of the photon yield in cosmic ray interactions. Astrophys. J., 789:136,

2014.

[180] H. S. Ahn et al. Discrepant hardening observed in cosmic-ray elemental spectra.

Astrophys. J., 714:L89–L93, 2010. 165 [181] Y. S. Yoon et al. Cosmic-Ray Proton and Helium Spectra from the First CREAM

Flight. Astrophys. J., 728:122, 2011.

[182] E. Atkin et al. First results of the cosmic ray NUCLEON experiment. 2017.

[183] I. V. Moskalenko, S. Karakula, and W. Tkaczyk. The Sun as the source of VHE

neutrinos. Astron. Astrophys., 248:L5–L6, 1991.

[184] I. V. Moskalenko and S. Karakula. Very high-energy neutrinos from the sun. J. Phys.,

G19:1399–1406, 1993.

[185] G. Ingelman and M. Thunman. High-energy neutrino production by cosmic ray in-

teractions in the sun. Phys. Rev., D54:4385–4392, 1996.

[186] G. L. Fogli, E. Lisi, A. Mirizzi, D. Montanino, and P. D. Serpico. Oscillations of solar

atmosphere neutrinos. Phys. Rev., D74:093004, 2006.

[187] T. Tanaka et al. An Indirect Search for WIMPs in the Sun using 3109.6 days of

upward-going muons in Super-Kamiokande. Astrophys. J., 742:78, 2011.

[188] M. G. Aartsen et al. Search for dark matter annihilations in the Sun with the 79-string

IceCube detector. Phys. Rev. Lett., 110(13):131302, 2013.

[189] C. Rott, T. Tanaka, and Y. Itow. Enhanced Sensitivity to Dark Matter Self-

annihilations in the Sun using Neutrino Spectral Information. JCAP, 1109:029, 2011.

[190] M. Ackermann et al. Measurement of separate cosmic-ray electron and positron spec-

tra with the Fermi Large Area Telescope. Phys. Rev. Lett., 108:011103, 2012.

[191] M. Aguilar et al. First Result from the Alpha Magnetic Spectrometer on the Inter-

national Space Station: Precision Measurement of the Positron Fraction in Primary

Cosmic Rays of 0.5–350 GeV. Phys. Rev. Lett., 110:141102, 2013.

[192] Shoji Torii. Calorimetric electron telescope mission. Search for dark matter and nearby

sources. Nucl. Instrum. Meth., A630:55–57, 2011.

166 [193] Jin Chang. Dark Matter Particle Explorer: The First Chinese Cosmic Ray and Hard

γ-ray Detector in Space. Chin. J. Spac. Sci., 34(5):550, 2014.

[194] Walter Heitler. The quantum theory of radiation. Courier Corporation, 1954.

[195] V. Berezinsky, A. Z. Gazizov, and S. I. Grigorieva. On astrophysical solution to

ultrahigh-energy cosmic rays. Phys. Rev., D74:043005, 2006.

[196] Kristoﬀer K. Andersen and Spencer R. Klein. High energy cosmic-ray interactions

with particles from the Sun. Phys. Rev., D83:103519, 2011.

[197] S. Karakula, G. Kociolek, I. V. Moskalenko, and W. Tkaczyk. Gamma-rays from

point Galactic sources. Astrophys. J. Suppl., 92:481–485, 1994.

[198] Luis A. Anchordoqui, John F. Beacom, Haim Goldberg, Sergio Palomares-Ruiz,

and Thomas J. Weiler. TeV gamma-rays from photo-disintegration/de-excitation of

cosmic-ray nuclei. Phys. Rev. Lett., 98:121101, 2007.

[199] Luis A. Anchordoqui, John F. Beacom, Haim Goldberg, Sergio Palomares-Ruiz, and

Thomas J. Weiler. TeV γ− rays and neutrinos from photo-disintegration of nuclei in

Cygnus OB2. Phys. Rev., D75:063001, 2007.

[200] Kohta Murase and John F. Beacom. Very-High-Energy Gamma-Ray Signal from

Nuclear Photodisintegration as a Probe of Extragalactic Sources of Ultrahigh-Energy

Nuclei. Phys. Rev., D82:043008, 2010.

[201] Igor V. Moskalenko and Andrew W. Strong. Anisotropic Inverse Compton Scattering

in the Galaxy. The Astrophysical Journal, 528(1):357–367, 2000.

[202] J. Chang et al. An excess of cosmic ray electrons at energies of 300-800 GeV. Nature,

456:362–365, 2008.

[203] Oscar Adriani et al. An anomalous positron abundance in cosmic rays with energies

1.5-100 GeV. Nature, 458:607–609, 2009.

167 [204] Stefano Profumo. Dissecting cosmic-ray electron-positron data with Occam’s Razor:

the role of known Pulsars. Central Eur. J. Phys., 10:1–31, 2011.

[205] J. Hinton, S. Funk, R. D. Parsons, and S. Ohm. Escape from Vela X. Astrophys. J.,

743:L7, 2011.

[206] O. Adriani et al. The cosmic-ray electron ﬂux measured by the PAMELA experiment

between 1 and 625 GeV. Phys. Rev. Lett., 106:201101, 2011.

[207] Segev BenZvi, Daniel Fiorino, Zigfried Hampel-Arias, and Mehr Un Nisa. Towards a

Measurement of the e+e− Flux above 1 TeV with HAWC. PoS, ICRC2015:248, 2016.

[208] Tijana Prodanovic, Brian D. Fields, and John F. Beacom. Diﬀuse gamma rays from

the galactic plane: probing the gev excess and identifying the TeV excess. Astropart.

Phys., 27:10–20, 2007.

[209] Hugo Ayala Solares, C. Michelle Hui, and Petra Hntemeyer. Fermi Bubbles with

HAWC. PoS, ICRC2015:749, 2016.

[210] John Pretz. Highlights from the High Altitude Water Cherenkov Observatory. PoS,

ICRC2015:025, 2016.

[211] John Pretz. Limit on an Isotropic Diﬀuse Gamma-Ray Population with HAWC. PoS,

ICRC2015:820, 2016.

[212] Kun Fang, Bing-Bing Wang, Xiao-Jun Bi, Su-Jie Lin, and Peng-Fei Yin. Perspective

on the cosmic-ray electron spectrum above TeV. 2016.

[213] Marco Cirelli, Gennaro Corcella, Andi Hektor, Gert Hutsi, Mario Kadastik, Paolo

Panci, Martti Raidal, Filippo Sala, and Alessandro Strumia. PPPC 4 DM ID: A Poor

Particle Physicist Cookbook for Dark Matter Indirect Detection. JCAP, 1103:051,

2011. [Erratum: JCAP1210,E01(2012)].

[214] Jatan Buch, Marco Cirelli, Galle Giesen, and Marco Taoso. PPPC 4 DM secondary: A

168 Poor Particle Physicist Cookbook for secondary radiation from Dark Matter. JCAP,

1509(09):037, 2015.

[215] Kohta Murase and John F. Beacom. Constraining Very Heavy Dark Matter Using

Diﬀuse Backgrounds of Neutrinos and Cascaded Gamma Rays. JCAP, 1210:043, 2012.

[216] Arman Esmaili and Pasquale Dario Serpico. Are IceCube neutrinos unveiling PeV-

scale decaying dark matter? JCAP, 1311:054, 2013.

[217] Brian Feldstein, Alexander Kusenko, Shigeki Matsumoto, and Tsutomu T. Yanagida.

Neutrinos at IceCube from Heavy Decaying Dark Matter. Phys. Rev., D88(1):015004,

2013.

[218] M. Actis et al. Design concepts for the Cherenkov Telescope Array CTA: An advanced

facility for ground-based high-energy gamma-ray astronomy. Exper. Astron., 32:193–

316, 2011.

[219] https://agenda.infn.it/contributionDisplay.py?contribId=24&confId=

12038.

[220] C. H. Llewellyn Smith. Neutrino Reactions at Accelerator Energies. Phys. Rept.,

3:261–379, 1972.

[221] Y. Hayato. NEUT. Nucl. Phys. Proc. Suppl., 112:171–176, 2002.

[222] O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich,

A. B. Larionov, T. Leitner, J. Weil, and U. Mosel. Transport-theoretical Description

of Nuclear Reactions. Phys. Rept., 512:1–124, 2012.

[223] O. Lalakulich, K. Gallmeister, and U. Mosel. Neutrino Nucleus Reactions within the

GiBUU Model. 2011. [J. Phys. Conf. Ser.408,012053(2013)].

[224] T. Golan, J. T. Sobczyk, and J. Zmuda. NuWro: the Wroclaw Monte Carlo Generator

of Neutrino Interactions. Nucl. Phys. Proc. Suppl., 229-232:499–499, 2012.

169 [225] J. A. Formaggio and G. P. Zeller. From eV to EeV: Neutrino Cross Sections Across

Energy Scales. Rev. Mod. Phys., 84:1307–1341, 2012.

[226] Wolfgang Altmannshofer, Stefania Gori, Maxim Pospelov, and Itay Yavin. Neutrino

Trident Production: A Powerful Probe of New Physics with Neutrino Beams. Phys.

Rev. Lett., 113:091801, 2014.

[227] Enrico Bertuzzo, Sudip Jana, Pedro A. N. Machado, and Renata Zukanovich Funchal.

Dark Neutrino Portal to Explain MiniBooNE excess. Phys. Rev. Lett., 121(24):241801,

2018.

[228] Francesco Capozzi, Shirley Weishi Li, Guanying Zhu, and John F. Beacom. DUNE

as the Next-Generation Solar Neutrino Experiment. 2018.

[229] Pedro AN Machado, Ornella Palamara, and David W Schmitz. The Short-Baseline

Neutrino Program at Fermilab. Ann. Rev. Nucl. Part. Sci., 69, 2019.

[230] Thomas K. Gaisser, Francis Halzen, and Todor Stanev. Particle astrophysics

with high-energy neutrinos. Phys. Rept., 258:173–236, 1995. [Erratum: Phys.

Rept.271,355(1996)].

[231] Raj Gandhi, Chris Quigg, Mary Hall Reno, and Ina Sarcevic. Ultrahigh-energy neu-

trino interactions. Astropart. Phys., 5:81–110, 1996.

[232] John N. Bahcall. Gallium solar neutrino experiments: Absorption cross-sections,

neutrino spectra, and predicted event rates. Phys. Rev., C56:3391–3409, 1997.

[233] Raj Gandhi, Chris Quigg, Mary Hall Reno, and Ina Sarcevic. Neutrino interactions

at ultrahigh-energies. Phys. Rev., D58:093009, 1998.

[234] K. Langanke, G. Martinez-Pinedo, P. von Neumann-Cosel, and A. Richter. Super-

nova inelastic neutrino nucleus cross-sections from high resolution electron scattering

experiments and shell model calculations. Phys. Rev. Lett., 93:202501, 2004.

170 [235] Alfredo G. Cocco, Gianpiero Mangano, and Marcello Messina. Probing low energy

neutrino backgrounds with neutrino capture on beta decaying nuclei. JCAP, 0706:015,

2007.

[236] Takashi Yoshida, Toshio Suzuki, Satoshi Chiba, Toshitaka Kajino, Hidekazu Yoko-

makura, Keiichi Kimura, Akira Takamura, and Dieter H. Hartmann. Neutrino-

Nucleus Reaction Cross Sections for Light Element Synthesis in Supernova Explosions.

Astrophys. J., 686:448–466, 2008.

[237] Amy Connolly, Robert S. Thorne, and David Waters. Calculation of High Energy

Neutrino-Nucleon Cross Sections and Uncertainties Using the MSTW Parton Distri-

bution Functions and Implications for Future Experiments. Phys. Rev., D83:113009,

2011.

[238] Ranjan Laha, John F. Beacom, Basudeb Dasgupta, Shunsaku Horiuchi, and Kohta

Murase. Demystifying the PeV Cascades in IceCube: Less (Energy) is More (Events).

Phys. Rev., D88:043009, 2013.

[239] Chien-Yi Chen, P. S. Bhupal Dev, and Amarjit Soni. Standard model explanation of

the ultrahigh energy neutrino events at IceCube. Phys. Rev., D89(3):033012, 2014.

[240] William Glenn Seligman. A Next-to-Leading Order QCD Analysis of Neutrino - Iron

Structure Functions at the Tevatron. PhD thesis, Nevis Labs, Columbia U., 1997.

[241] M. Tzanov et al. Precise measurement of neutrino and anti-neutrino diﬀerential cross

sections. Phys. Rev., D74:012008, 2006.

[242] M. Tanabashi et al. Review of Particle Physics. Phys. Rev., D98(3):030001, 2018.

[243] M. G. Aartsen et al. A combined maximum-likelihood analysis of the high-energy

astrophysical neutrino ﬂux measured with IceCube. Astrophys. J., 809(1):98, 2015.

[244] Dan Hooper. Measuring high-energy neutrino nucleon cross-sections with future neu-

trino telescopes. Phys. Rev., D65:097303, 2002.

171 [245] E. Borriello, A. Cuoco, G. Mangano, G. Miele, S. Pastor, O. Pisanti, and P. D.

Serpico. Disentangling neutrino-nucleon cross section and high energy neutrino ﬂux

with a km3 neutrino telescope. Phys. Rev., D77:045019, 2008.

[246] Spencer R. Klein and Amy Connolly. Neutrino Absorption in the Earth, Neutrino

Cross-Sections, and New Physics. In Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013):

Minneapolis, MN, USA, July 29-August 6, 2013, 2013.

[247] P. Padovani and E. Resconi. Are both BL Lacs and pulsar wind nebulae the as-

trophysical counterparts of IceCube neutrino events? Mon. Not. Roy. Astron. Soc.,

443(1):474–484, 2014.

[248] Irene Tamborra, Shin’ichiro Ando, and Kohta Murase. Star-forming galaxies as the

origin of diﬀuse high-energy backgrounds: Gamma-ray and neutrino connections, and

implications for starburst history. JCAP, 1409:043, 2014.

[249] Kohta Murase, Yoshiyuki Inoue, and Charles D. Dermer. Diﬀuse Neutrino Intensity

from the Inner Jets of Active Galactic Nuclei: Impacts of External Photon Fields and

the Blazar Sequence. Phys. Rev., D90(2):023007, 2014.

[250] Carsten Rott, Kazunori Kohri, and Seong Chan Park. Superheavy dark mat-

ter and IceCube neutrino signals: Bounds on decaying dark matter. Phys. Rev.,

D92(2):023529, 2015.

[251] Rhorry Gauld, Juan Rojo, Luca Rottoli, Subir Sarkar, and Jim Talbert. The prompt

atmospheric neutrino ﬂux in the light of LHCb. JHEP, 02:130, 2016.

[252] Kohta Murase, Dafne Guetta, and Markus Ahlers. Hidden Cosmic-Ray Accelerators

as an Origin of TeV-PeV Cosmic Neutrinos. Phys. Rev. Lett., 116(7):071101, 2016.

[253] Kohta Murase, Ranjan Laha, Shin’ichiro Ando, and Markus Ahlers. Testing the

Dark Matter Scenario for PeV Neutrinos Observed in IceCube. Phys. Rev. Lett.,

115(7):071301, 2015. 172 [254] Mauricio Bustamante, John F. Beacom, and Walter Winter. Theoretically palatable

ﬂavor combinations of astrophysical neutrinos. Phys. Rev. Lett., 115(16):161302, 2015.

[255] Atri Bhattacharya, Rikard Enberg, Mary Hall Reno, Ina Sarcevic, and Anna Stasto.

Perturbative charm production and the prompt atmospheric neutrino ﬂux in light of

RHIC and LHC. JHEP, 06:110, 2015.

[256] Keith Bechtol, Markus Ahlers, Mattia Di Mauro, Marco Ajello, and Justin Vanden-

broucke. Evidence against star-forming galaxies as the dominant source of IceCube

neutrinos. Astrophys. J., 836(1):47, 2017.

[257] M. Kadler et al. Coincidence of a high-ﬂuence blazar outburst with a PeV-energy

neutrino event. Nature Phys., 12(8):807–814, 2016.

[258] S. Adrian-Martinez et al. Letter of intent for KM3NeT 2.0. J. Phys., G43(8):084001,

2016.

[259] Juliana Stachurska. IceCube High Energy Starting Events at 7.5 Years – New Mea-

surements of Flux and Flavor. EPJ Web Conf., 207:02005, 2019.

[260] Austin Schneider. Characterization of the Astrophysical Diﬀuse Neutrino Flux with

IceCube High-Energy Starting Events. In 36th International Cosmic Ray Conference

(ICRC 2019) Madison, Wisconsin, USA, July 24-August 1, 2019, 2019.

[261] E. Fermi. On the Theory of the impact between atoms and electrically charged

particles. Z. Phys., 29:315–327, 1924.

[262] C. F. von Weizsacker. Radiation emitted in collisions of very fast electrons. Z. Phys.,

88:612–625, 1934.

[263] E. J. Williams. Nature of the high-energy particles of penetrating radiation and status

of ionization and radiation formulae. Phys. Rev., 45:729–730, 1934.

[264] M.A. Kozhushner and E.P. Shabalin. Production of lepton particle pairs on a coulomb

center. Soviet Journal of Experimental and Theoretical Physics, 41:949, 1961. 173 [265] E. P. Shabalin. The µ+µ− and e+e− Pair Production Cross Sections for Neutrinos

Scattered by Nuclei. Soviet Journal of Experimental and Theoretical Physics, 16:125,

1963.

[266] W. Czyz, G. C. Sheppey, and J. D. Walecka. Neutrino production of lepton pairs

through the point four-fermion interaction. Nuovo Cim., 34:404–435, 1964.

[267] Carl Schmidt, Jon Pumplin, Daniel Stump, and C. P. Yuan. CT14QED parton distri-

bution functions from isolated photon production in deep inelastic scattering. Phys.

Rev., D93(11):114015, 2016.

[268] T. D. Lee and Chen-Ning Yang. THEORETICAL DISCUSSIONS ON POSSIBLE

HIGH-ENERGY NEUTRINO EXPERIMENTS. Phys. Rev. Lett., 4:307–311, 1960.

[269] T. D. Lee, Chen-Ning Yang, and P. Markstein. Production Cross-section of Inter-

mediate Bosons by Neutrinos in the Coulomb Field of Protons and Iron. Phys. Rev.

Lett., 7:429–433, 1961.

[270] J. S. Bell and M. J. G. Veltman. Intermediate boson production by neutrinos. Phys.

Lett., 5:94–96, 1963.

[271] J. S. Bell and M. J. G. Veltman. Polarisation of vector bosons produced by neutrinos.

Phys. Lett., 5:151–152, 1963.

[272] R. W. Brown. Intermediate boson. i. theoretical production cross-sections in high-

energy neutrino and muon experiments. Phys. Rev., D3:207–223, 1971.

[273] R. W. Brown, R. H. Hobbs, and J. Smith. Intermediate boson. ii. theoretical muon

spectra in high-energy neutrino experiments. Phys. Rev., D4:794–813, 1971.

[274] G. Arnison et al. Experimental Observation of Isolated Large Transverse Energy Elec-

trons with Associated Missing Energy at s**(1/2) = 540-GeV. Phys. Lett., B122:103–

116, 1983.

174 [275] M. Ageron et al. ANTARES: the ﬁrst undersea neutrino telescope. Nucl. Instrum.

Meth., A656:11–38, 2011.

[276] J. Lovseth and M. Radomiski. Kinematical distributions of neutrino-produced lepton

triplets. Phys. Rev., D3:2686–2706, 1971.

[277] K. Fujikawa. The self-coupling of weak lepton currents in high-energy neutrino and

muon reactions. Annals Phys., 68:102–162, 1971.

[278] K. Koike, M. Konuma, K. Kurata, and K. Sugano. Neutrino production of lepton

pairs. 1. Prog. Theor. Phys., 46:1150–1169, 1971.

[279] K. Koike, M. Konuma, K. Kurata, and K. Sugano. Neutrino production of lepton

pairs. 2. Prog. Theor. Phys., 46:1799–1804, 1971.

[280] R. W. Brown, R. H. Hobbs, J. Smith, and N. Stanko. Intermediate boson. iii. virtual-

boson eﬀects in neutrino trident production. Phys. Rev., D6:3273–3292, 1972.

[281] R. Belusevic and J. Smith. W - Z Interference in Neutrino - Nucleus Scattering. Phys.

Rev., D37:2419, 1988.

[282] D. Geiregat et al. First observation of neutrino trident production. Phys. Lett.,

B245:271–275, 1990.

[283] S. R. Mishra et al. Neutrino tridents and W Z interference. Phys. Rev. Lett., 66:3117–

3120, 1991.

[284] T. Adams et al. Evidence for diﬀractive charm production in muon-neutrino Fe and

anti-muon-neutrino Fe scattering at the Tevatron. Phys. Rev., D61:092001, 2000.

[285] M. Anelli et al. A facility to Search for Hidden Particles (SHiP) at the CERN SPS.

2015.

[286] M. Antonello et al. A Proposal for a Three Detector Short-Baseline Neutrino Oscil-

lation Program in the Fermilab Booster Neutrino Beam. 2015.

175 [287] R. Acciarri et al. Long-Baseline Neutrino Facility (LBNF) and Deep Underground

Neutrino Experiment (DUNE). 2015.

[288] F. J. P. Soler. nuSTORM: Neutrinos from Stored Muons. In Proceedings, Topical Research Meeting on Prospects in Neutrino Physics (NuPhys2014): London, UK,

December 15-17, 2014, 2015.

[289] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao,

T. Stelzer, P. Torrielli, and M. Zaro. The automated computation of tree-level and

next-to-leading order diﬀerential cross sections, and their matching to parton shower

simulations. JHEP, 07:079, 2014.

[290] Alexander Belyaev, Neil D. Christensen, and Alexander Pukhov. CalcHEP 3.4 for

collider physics within and beyond the Standard Model. Comput. Phys. Commun.,

184:1729–1769, 2013.

[291] M. I. Vysotsky, I. V. Gaidaenko, and V. A. Novikov. On lepton pair produc-

tion in neutrino nucleus collisions. Phys. Atom. Nucl., 65:1634–1642, 2002. [Yad.

Fiz.65,1676(2002)].

[292] R. Mertig, M. Bohm, and Ansgar Denner. FEYN CALC: Computer algebraic calcu-

lation of Feynman amplitudes. Comput. Phys. Commun., 64:345–359, 1991.

[293] Vladyslav Shtabovenko, Rolf Mertig, and Frederik Orellana. New Developments in

FeynCalc 9.0. Comput. Phys. Commun., 207:432–444, 2016.

[294] http://hitoshi.berkeley.edu/233B/phasespace.pdf.

[295] M. Gluck, Cristian Pisano, and E. Reya. The Polarized and unpolarized photon

content of the nucleon. Phys. Lett., B540:75–80, 2002.

[296] A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne. Parton distributions

incorporating QED contributions. Eur. Phys. J., C39:155–161, 2005.

176 [297] Richard D. Ball, Valerio Bertone, Stefano Carrazza, Luigi Del Debbio, Stefano Forte,

Alberto Guﬀanti, Nathan P. Hartland, and Juan Rojo. Parton distributions with

QED corrections. Nucl. Phys., B877:290–320, 2013.

[298] A. D. Martin and M. G. Ryskin. The photon PDF of the proton. Eur. Phys. J.,

C74:3040, 2014.

[299] L. A. Harland-Lang, V. A. Khoze, and M. G. Ryskin. Photon-initiated processes at

high mass. Phys. Rev., D94(7):074008, 2016.

[300] Aneesh Manohar, Paolo Nason, Gavin P. Salam, and Giulia Zanderighi. How bright

is the proton? A precise determination of the photon parton distribution function.

Phys. Rev. Lett., 117(24):242002, 2016.

[301] Aneesh V. Manohar, Paolo Nason, Gavin P. Salam, and Giulia Zanderighi. The

Photon Content of the Proton. JHEP, 12:046, 2017.

[302] Valerio Bertone, Stefano Carrazza, Nathan P. Hartland, and Juan Rojo. Illuminat-

ing the photon content of the proton within a global PDF analysis. SciPost Phys.,

5(1):008, 2018.

[303] https://hep.pa.msu.edu/cteq/public/index.html.

[304] S. Chekanov et al. Measurement of isolated photon production in deep inelastic ep

scattering. Phys. Lett., B687:16–25, 2010.

[305] Martin M. Block, Loyal Durand, and Phuoc Ha. Connection of the virtual γ∗p cross

section of ep deep inelastic scattering to real p scattering, and the implications for N

and ep total cross sections. Phys. Rev., D89(9):094027, 2014.

[306] Carlos A. Argelles, Francis Halzen, Logan Wille, Mike Kroll, and Mary Hall Reno.

High-energy behavior of photon, neutrino, and proton cross sections. Phys. Rev.,

D92(7):074040, 2015.

177 [307] Valerio Bertone, Rhorry Gauld, and Juan Rojo. Neutrino Telescopes as QCD Micro-

scopes. JHEP, 01:217, 2019.

[308] Christian W. Bauer, Nicolas Ferland, and Bryan R. Webber. Standard Model Parton

Distributions at Very High Energies. JHEP, 08:036, 2017.

[309] Christian W. Bauer and Bryan R. Webber. Polarization Eﬀects in Standard Model

Parton Distributions at Very High Energies. JHEP, 03:013, 2019.

[310] Bartosz Fornal, Aneesh V. Manohar, and Wouter J. Waalewijn. Electroweak Gauge

Boson Parton Distribution Functions. JHEP, 05:106, 2018.

[311] K. Kovarik et al. nCTEQ15 - Global analysis of nuclear parton distributions with

uncertainties in the CTEQ framework. Phys. Rev., D93(8):085037, 2016.

[312] Hamzeh Khanpour and S. Atashbar Tehrani. Global Analysis of Nuclear Parton

Distribution Functions and Their Uncertainties at Next-to-Next-to-Leading Order.

Phys. Rev., D93(1):014026, 2016.

[313] Rong Wang, Xurong Chen, and Qiang Fu. Global study of nuclear modiﬁcations on

parton distribution functions. Nucl. Phys., B920:1–19, 2017.

[314] Kari J. Eskola, Petja Paakkinen, Hannu Paukkunen, and Carlos A. Salgado. EPPS16:

Nuclear parton distributions with LHC data. Eur. Phys. J., C77(3):163, 2017.

[315] Matthew D. Kistler and John F. Beacom. Guaranteed and Prospective Galactic TeV

Neutrino Sources. Phys. Rev., D74:063007, 2006.

[316] John F. Beacom and Matthew D. Kistler. Dissecting the Cygnus Region with TeV

Gamma Rays and Neutrinos. Phys. Rev., D75:083001, 2007.

[317] Kohta Murase, Todd A. Thompson, Brian C. Lacki, and John F. Beacom. New Class of

High-Energy Transients from Crashes of Supernova Ejecta with Massive Circumstellar

Material Shells. Phys. Rev., D84:043003, 2011.

178 [318] Kohta Murase and Kunihito Ioka. TeVPeV Neutrinos from Low-Power Gamma-Ray

Burst Jets inside Stars. Phys. Rev. Lett., 111(12):121102, 2013.

[319] Matthew D. Kistler, Todor Stanev, and Hasan Yksel. Cosmic PeV Neutrinos and the

Sources of Ultrahigh Energy Protons. Phys. Rev., D90(12):123006, 2014.

[320] Kohta Murase, Markus Ahlers, and Brian C. Lacki. Testing the Hadronuclear Origin

of PeV Neutrinos Observed with IceCube. Phys. Rev., D88(12):121301, 2013.

[321] Markus Ahlers and Francis Halzen. Pinpointing Extragalactic Neutrino Sources in

Light of Recent IceCube Observations. Phys. Rev., D90(4):043005, 2014.

[322] Matthew D. Kistler and Ranjan Laha. Multi-PeV Signals from a New Astrophysical

Neutrino Flux Beyond the Glashow Resonance. Phys. Rev. Lett., 120(24):241105,

2018.

[323] I. Bartos, M. Ahrens, C. Finley, and S. Marka. Prospects of Establishing the Origin

of Cosmic Neutrinos using Source Catalogs. Phys. Rev., D96(2):023003, 2017.

[324] Imre Bartos, Doga Veske, Azadeh Keivani, Zsuzsa Marka, Stefan Countryman, Erik

Blaufuss, Chad Finley, and Szabolcs Marka. Bayesian Multi-Messenger Search Method

for Common Sources of Gravitational Waves and High-Energy Neutrinos. 2018.

[325] Mauricio Bustamante and Markus Ahlers. Inferring the ﬂavor of high-energy astro-

physical neutrinos at their sources. Phys. Rev. Lett., 122(24):241101, 2019.

[326] M. G. Aartsen et al. Multimessenger observations of a ﬂaring blazar coincident with

high-energy neutrino IceCube-170922A. Science, 361(6398):eaat1378, 2018.

[327] A. D. Avrorin et al. The prototyping/early construction phase of the BAIKAL-GVD

project. Nucl. Instrum. Meth., A742:82–88, 2014.

[328] John F. Beacom and Julian Candia. Shower power: Isolating the prompt atmospheric

neutrino ﬂux using electron neutrinos. JCAP, 0411:009, 2004.

179 [329] Roger J. Barlow and Stephen Wolfram. Electromagnetic Radiative Corrections to

Deep Inelastic Neutrino Interactions. Phys. Rev., D20:2198, 1979.

[330] R. Petronzio. Electromagnetic Radiative Corrections to Deep Inelastic (anti)-neutrino

Experiments. In Neutrino physics at accelerators, pages 140–148, 1978.

[331] K. P. O. Diener, S. Dittmaier, and W. Hollik. Electroweak radiative corrections to

deep inelastic neutrino scattering: Implications for NuTeV? Phys. Rev., D69:073005,

2004.

[332] A. B. Arbuzov, D. Yu. Bardin, and L. V. Kalinovskaya. Radiative corrections to

neutrino deep inelastic scattering revisited. JHEP, 06:078, 2005.

[333] K. P. O. Diener, S. Dittmaier, and W. Hollik. Electroweak higher-order eﬀects and

theoretical uncertainties in deep-inelastic neutrino scattering. Phys. Rev., D72:093002,

2005.

[334] V. Berezinsky, M. Kachelriess, and S. Ostapchenko. Electroweak jet cascading in the

decay of superheavy particles. Phys. Rev. Lett., 89:171802, 2002.

[335] B. C. Barish et al. Observation of Trimuon Production by Neutrinos. Phys. Rev.

Lett., 38:577–580, 1977.

[336] A. C. Benvenuti et al. Observation of a New Process with Trimuon Production by

High-Energy Neutrinos. Phys. Rev. Lett., 38:1110, 1977.

[337] T. Hansl et al. Characteristics of Trimuon Events Observed in High-energy Neutrino

Interactions. Nucl. Phys., B142:381–409, 1978.

[338] Vernon D. Barger, T. Gottschalk, and R. J. N. Phillips. Radiative and Hadronic

Production of Trimuons in Neutrino Collisions. Phys. Rev., D18:2308, 1978.

[339] Spencer R. Klein. Probing high-energy interactions of atmospheric and astrophysical

neutrinos. 2019.

180 [340] Adam M. Dziewonski and Don L. Anderson. Preliminary reference earth model.

Physics of the Earth and Planetary Interiors, 25(4):297 – 356, 1981.

[341] Joeran Stettner. Measurement of the Diﬀuse Astrophysical Muon-Neutrino Spectrum

with Ten Years of IceCube Data. In HAWC Contributions to the 36th International

Cosmic Ray Conference (ICRC2019), 2019.

[342] Markus Ahlers and Francis Halzen. High-energy cosmic neutrino puzzle: a review.

Rept. Prog. Phys., 78(12):126901, 2015.

[343] Aaron C. Vincent, Carlos A. Argelles, and Ali Kheirandish. High-energy neutrino

attenuation in the Earth and its associated uncertainties. JCAP, 1711(11):012, 2017.

[JCAP1711,012(2017)].

[344] M. G. Aartsen et al. Energy Reconstruction Methods in the IceCube Neutrino Tele-

scope. JINST, 9:P03009, 2014.

[345] M. G. Aartsen et al. Atmospheric and astrophysical neutrinos above 1 TeV interacting

in IceCube. Phys. Rev., D91(2):022001, 2015.

[346] Manuel Silva and Sarah Mancina. Measurement of the Diﬀuse Muon Neutrino Fluxu-

sing Starting Track Events in IceCube. In 36th International Cosmic Ray Conference

(ICRC 2019) Madison, Wisconsin, USA, July 24-August 1, 2019, 2019.

[347] R. Abbasi et al. Lateral Distribution of Muons in IceCube Cosmic Ray Events. Phys.

Rev., D87(1):012005, 2013.

[348] Sandro Kopper. Search for Neutrino Induced Double Tracks as an Exotic Physics

Signature in IceCube. PoS, ICRC2015:1104, 2016.

[349] Ivone Albuquerque, Gustavo Burdman, and Z. Chacko. Neutrino telescopes as a direct

probe of supersymmetry breaking. Phys. Rev. Lett., 92:221802, 2004.

[350] Ivone F. M. Albuquerque, Gustavo Burdman, and Z. Chacko. Direct detection of

supersymmetric particles in neutrino telescopes. Phys. Rev., D75:035006, 2007. 181 [351] Dmitry Chirkin and Wolfgang Rhode. Muon Monte Carlo: A High-precision tool for

muon propagation through matter. 2004.

[352] M. G. Aartsen et al. Measurements using the inelasticity distribution of multi-TeV

neutrino interactions in IceCube. Phys. Rev., D99(3):032004, 2019.

[353] Bei Zhou and John F. Beacom. In preparation.

[354] T. Totani, K. Sato, H. E. Dalhed, and J. R. Wilson. Future detection of supernova

neutrino burst and explosion mechanism. Astrophys. J., 496:216–225, 1998.

[355] Amol S. Dighe and Alexei Yu. Smirnov. Identifying the neutrino mass spectrum from

the neutrino burst from a supernova. Phys. Rev., D62:033007, 2000.

[356] Basudeb Dasgupta, Amol Dighe, Georg G. Raﬀelt, and Alexei Yu. Smirnov. Multiple

Spectral Splits of Supernova Neutrinos. Phys. Rev. Lett., 103:051105, 2009.

[357] L. Hudepohl, B. Muller, H. T. Janka, A. Marek, and G. G. Raﬀelt. Neutrino Signal

of Electron-Capture Supernovae from Core Collapse to Cooling. Phys. Rev. Lett.,

104:251101, 2010. [Erratum: Phys. Rev. Lett.105,249901(2010)].

[358] Linyan Wan. First Measurement of Atmospheric Neutrino Neutral-Current Quasi-

Elastic Interactions for Supernova Relic Neutrino Search at Super-Kamiokande.

https://indico.desy.de/indico/event/18342/session/35/contribution/125.

[359] C. Patrignani et al. Review of Particle Physics. Chin. Phys., C40(10):100001, 2016.

[360] Bei Zhou and John F. Beacom. In preparation.

[361] K. Abe et al. Hyper-Kamiokande Design Report. 2018.

[362] I. Anghel et al. Letter of Intent: The Accelerator Neutrino Neutron Interaction

Experiment (ANNIE). 2015.

[363] M. Askins et al. The Physics and Nuclear Nonproliferation Goals of WATCHMAN:

A WAter CHerenkov Monitor for ANtineutrinos. 2015.

182 [364] Orlando L. G. Peres and A. Yu. Smirnov. Oscillations of very low energy atmospheric

neutrinos. Phys. Rev., D79:113002, 2009.

[365] Ken’ichiro Nakazato, Kohsuke Sumiyoshi, Hideyuki Suzuki, and Shoichi Yamada. Os-

cillation and Future Detection of Failed Supernova Neutrinos from Black Hole Forming

Collapse. Phys. Rev., D78:083014, 2008. [Erratum: Phys. Rev.D79,069901(2009)].

[366] Cecilia Lunardini. Diﬀuse neutrino ﬂux from failed supernovae. Phys. Rev. Lett.,

102:231101, 2009.

[367] C. S. Kochanek, J. F. Beacom, M. D. Kistler, J. L. Prieto, K. Z. Stanek, T. A. Thomp-

son, and H. Yuksel. A Survey About Nothing: Monitoring a Million Supergiants for

Failed Supernovae. Astrophys. J., 684:1336–1342, 2008.

[368] Amy Lien, Brian D. Fields, and John F. Beacom. Synoptic Sky Surveys and the

Diﬀuse Supernova Neutrino Background: Removing Astrophysical Uncertainties and

Revealing Invisible Supernovae. Phys. Rev., D81:083001, 2010.

[369] Lawrence M. Krauss, Sheldon L. Glashow, and David N. Schramm. Anti-neutrinos

Astronomy and Geophysics. Nature, 310:191–198, 1984. [,674(1983)].

[370] G.S. Bisnovatyi-Kogan and Z.F. Seidov. Supernovae, Neutrino Rest Mass, and the

Middle-Energy Neutrino Background in the Universe. In 11th Texas Symposium on

Relativistic Astrophysics, pages 319–327, 1982.

[371] S. E. Woosley, J. R. Wilson, and R. Mayle. Gravitational Collapse and the Cosmic

Antineutrino Background. The Astrophysical Journal, 302:19, Mar 1986.

[372] Tomonori Totani, Katsuhiko Sato, and Yuzuru Yoshii. Spectrum of the supernova relic

neutrino background and evolution of galaxies. Astrophys. J., 460:303–312, 1996.

[373] R. A. Malaney. Evolution of the cosmic gas and the relic supernova neutrino back-

ground. Astropart. Phys., 7:125–136, 1997.

183 [374] D. H. Hartmann and S. E. Woosley. The cosmic supernova neutrino background.

Astropart. Phys., 7:137–146, 1997.

[375] M. Kaplinghat, G. Steigman, and T. P. Walker. The Supernova relic neutrino back-

ground. Phys. Rev., D62:043001, 2000.

[376] S. Ando, K. Sato, and T. Totani. Detectability of the supernova relic neutrinos and

neutrino oscillation. Astropart. Phys., 18:307–318, 2003.

[377] Cecilia Lunardini and Irene Tamborra. Diﬀuse supernova neutrinos: oscillation eﬀects,

stellar cooling and progenitor mass dependence. JCAP, 1207:012, 2012.

[378] Alessandro Mirizzi, Irene Tamborra, Hans-Thomas Janka, Ninetta Saviano, Kate

Scholberg, Robert Bollig, Lorenz Hudepohl, and Sovan Chakraborty. Supernova Neu-

trinos: Production, Oscillations and Detection. Riv. Nuovo Cim., 39(1-2):1–112, 2016.

[379] Irene Tamborra, Lorenz Huedepohl, Georg Raﬀelt, and Hans-Thomas Janka. Flavor-

dependent neutrino angular distribution in core-collapse supernovae. Astrophys. J.,

839:132, 2017.

[380] Irene Tamborra and Kohta Murase. Neutrinos from Supernovae. Space Sci. Rev.,

214(1):31, 2018.

[381] Thomas K. Gaisser. Atmospheric neutrino ﬂuxes. Phys. Scripta, T121:51–56, 2005.

[382] Roger Wendell and Kimihiro Okumura. Recent progress and future prospects with

atmospheric neutrinos. New J. Phys., 17(2):025006, 2015.

[383] Morihiro Honda, T. Kajita, K. Kasahara, S. Midorikawa, and T. Sanuki. Calculation

of atmospheric neutrino ﬂux using the interaction model calibrated with atmospheric

muon data. Phys. Rev., D75:043006, 2007.

[384] G. D. Barr, T. K. Gaisser, S. Robbins, and Todor Stanev. Uncertainties in Atmo-

spheric Neutrino Fluxes. Phys. Rev., D74:094009, 2006.

184 [385] Justin Evans, Diego Garcia Gamez, Salvatore Davide Porzio, Stefan Sldner-Rembold,

and Steven Wren. Uncertainties in Atmospheric Muon-Neutrino Fluxes Arising from

Cosmic-Ray Primaries. Phys. Rev., D95(2):023012, 2017.

[386] Francesco Capozzi, Eleonora Di Valentino, Eligio Lisi, Antonio Marrone, Alessandro

Melchiorri, and Antonio Palazzo. Global constraints on absolute neutrino masses and

their ordering. Phys. Rev., D95(9):096014, 2017.

[387] S. Gariazzo, M. Archidiacono, P. F. de Salas, O. Mena, C. A. Ternes, and M. Tr-

tola. Neutrino masses and their ordering: Global Data, Priors and Models. JCAP,

1803(03):011, 2018.

[388] Tomasz Golan, Cezary Juszczak, and Jan T. Sobczyk. Final State Interactions Eﬀects

in Neutrino-Nucleus Interactions. Phys. Rev., C86:015505, 2012.

[389] S. A. Dytman and A. S. Meyer. Final state interactions in GENIE. AIP Conf. Proc.,

1405:213–218, 2011.

[390] B. Aharmim et al. A Search for Neutrinos from the Solar hep Reaction and the Diﬀuse

Supernova Neutrino Background with the Sudbury Neutrino Observatory. Astrophys.

J., 653:1545–1551, 2006.

[391] E. Richard et al. Measurements of the atmospheric neutrino ﬂux by Super-

Kamiokande: energy spectra, geomagnetic eﬀects, and solar modulation. Phys. Rev.,

D94(5):052001, 2016.

[392] A. Bodek and J. L. Ritchie. Further Studies of Fermi Motion Eﬀects in Lepton

Scattering from Nuclear Targets. Phys. Rev., D24:1400, 1981.

[393] Dieter Rein and Lalit M. Sehgal. Neutrino Excitation of Baryon Resonances and

Single Pion Production. Annals Phys., 133:79–153, 1981.

[394] E. A. Paschos and J. Y. Yu. Neutrino interactions in oscillation experiments. Phys.

Rev., D65:033002, 2002.

185 [395] L. I. Ponomarev. Molecular structure eﬀects on atomic and nuclear capture of mesons.

Ann. Rev. Nucl. Part. Sci., 23:395–430, 1973.

[396] D. F. Measday. The nuclear physics of muon capture. Phys. Rept., 354:243–409, 2001.

[397] H. Ejiri. Nuclear deexcitations of nucleon holes associated with nucleon decays in

nuclei. Phys. Rev., C48:1442–1444, 1993.

[398] K. Kobayashi et al. Detection of nuclear de-excitation gamma-rays in water Cherenkov

detector. Nucl. Phys. Proc. Suppl., 139:72–76, 2005. [,72(2005)].

[399] Artur M. Ankowski, Omar Benhar, Takaaki Mori, Ryuta Yamaguchi, and Makoto

Sakuda. Analysis of γ-ray production in neutral-current neutrino-oxygen interactions

at energies above 200 MeV. Phys. Rev. Lett., 108:052505, 2012.

[400] M. Traini, Sylvaine Turck-Chieze, and A. Zghiche. Deep Inelastic Electron Scattering

in the Distorted Wave Born Approximation: An Analytic Approach. Phys. Rev.,

C38:2799–2812, 1988.

[401] C. Giusti and F. D. Pacati. Electron distortion in quasifree (e,e?p) reactions. Nucl.

Phys., A473:717–735, 1987.

[402] Jonathan Engel. Approximate treatment of lepton distortion in charged current neu-

trino scattering from nuclei. Phys. Rev., C57:2004–2009, 1998.

[403] John F. Beacom and Sergio Palomares-Ruiz. Neutral current atmospheric neutrino

ﬂux measurement using neutrino proton elastic scattering in Super-Kamiokande.

Phys. Rev., D67:093001, 2003.

[404] A. Bodek, M. E. Christy, and B. Coopersmith. Eﬀective Spectral Function for

Quasielastic Scattering on Nuclei. Eur. Phys. J., C74(10):3091, 2014.

[405] A. Bodek, H. S. Budd, and M. E. Christy. Neutrino Quasielastic Scattering on Nuclear

Targets: Parametrizing Transverse Enhancement (Meson Exchange Currents). Eur.

Phys. J., C71:1726, 2011. 186 [406] J. Alcaraz et al. Protons in near earth orbit. Phys. Lett., B472:215–226, 2000.

[407] T. Sanuki et al. Precise measurement of cosmic ray proton and helium spectra with

the BESS spectrometer. Astrophys. J., 545:1135, 2000.

[408] Sadakazu Haino et al. Measurements of primary and atmospheric cosmic - ray spectra

with the BESS-TeV spectrometer. Phys. Lett., B594:35–46, 2004.

187