Baryogenesis, Neutrinos and Dark Matter

STUDIES BEYOND THE STANDARD MODEL: BARYOGENESIS, NEUTRINOS AND DARK MATTER

A Dissertation

Presented to the Faculty of the Graduate School of Cornell University in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

by Gowri Kurup August 2020 c 2020 Gowri Kurup

Cornell University 2020

This dissertation studies aspects of three of the biggest unsolved problems with the Standard Model of Particle Physics: the matter-antimatter asymmetry in the universe, neutrino masses, and dark matter.

While the SM has been very successful in explaining most experimental results, it does not provide a satisfactory solution to the matter – anti-matter asymmetry problem. One of the simplest and most elegant solutions to this problem is electroweak baryogenesis. An addition to the Standard Model of a real, gauge-singlet scalar field, coupled via a Higgs portal interaction, can reopen the possibility of a strongly first-order electroweak phase transition and successful electroweak baryogenesis. In the first part of this work, we study a subset of such models and evaluate the bubble nucleation temperature throughout the parameter space where a first-order transition is expected. In addition, we also look at vacuum bubble wall velocity and how it impacts baryogenesis in these models. Another limitation of the SM is that it models the neutrinos as massless, even though experiments have shown that this is not true. The mass generation mechanism of neutrinos is still unknown, and popular see-saw models with Majorana masses have proven difficult to test. The Clockwork mechanism can be used to generate Dirac masses, and can explain the smallness of neutrino masses without introducing unnaturally small input parameters. In the second part of this thesis, we study the simplest Clockwork neutrino model, the ”uniform” clockwork, as well as a broader class of ”generalized” clockwork models. We derive constraints on such models from lepton-flavor violating processes, as well as precision electroweak fits. Further, we analyse simulated collider data and study the prospects for detection of the models at the LHC and future colliders.

The third and final part of this thesis is about Dark Matter. All the evidence we have for Dark Matter (DM) is through gravitational interactions, and its microscopic nature is one of the most urgent questions in theoretical physics. We present an unconventional model for the production of dark matter: the conformal freeze-in scenario. At the time when the dark sector is populated in the early universe, it is described by a strongly coupled conformal field theory and as such, cannot be described by particle states. As the universe cools, cosmological phase transitions in the standard model sector or loop induced deformations of the conformal field theory induce conformal symmetry breaking and confine- ment in the dark sector. One of the resulting dark bound states is stable on the cosmological time scales and plays the role of dark matter. BIOGRAPHICAL SKETCH

Gowri was born on October 26th, 1992, in Thiruvananthapuram, a city near the southern tip of India, to Narmada and Madhusoodhan Kurup. She did her schooling partly at the Holy Angel’s Convent, and mostly at Sarvodaya Vidyalaya in Thiruvananthapuram.

In 2010, Gowri joined the Indian Institute of Technology (IIT) - Bombay in Mumbai, as a freshman undergraduate. There, she took her ﬁrst steps in particle theory with Professor Ramadevi, who supervised her senior thesis. She graduated in 2014 with a bachelor’s degree in Engineering Physics with hon- ours, and a minor in Statistics. In the fall of 2014, Gowri joined the Department of Physics at Cornell Univer- sity as a graduate student. She pursued her research under the supervision and guidance of Professor Maxim Perelstein. After completing her doctorate, she will join the Rudolf Peierls Centre for Theoretical Physics at Oxford University as a postdoctoral research assistant in the fall of 2020. Besides physics, Gowri enjoys baking, cooking for friends, cryptic cross- words and word games.

iii Elsewhere is a negative mirror. The traveller recognizes the little that is his,

discovering the much he has not had and will never have.

—ITALO CALVINO , “Invisible Cities”.

iv ACKNOWLEDGEMENTS

Firstly, I would like to thank my advisor, Maxim Perelstein. I could not have asked for a more supportive and kind mentor, and the amount I have learned from him, in physics, life and career, is immeasurable. He was patient with me through all the ups and downs of research, and was someone I could always approach when I was stuck (and he always had an answer). I am grateful to have had the opportunity to be his student. I would also like to thank the other members of my committee, Yuval Grossman and Julia Thom-Levy for all the time they have given for me. I would like to thank Sungwoo Hong, whom I admire greatly for his work ethic and his expertise in physics. My graduate school experience improved markedly after he arrived at Cornell, and it was a joy to discuss physics with him. I would like to especially thank both Csaba Csaki and Yuval Grossman, for all that I learned from them, through courses and conversations. In addition,

Peter Lepage, Liam McAllister and Tom Hartman were excellent teachers who gave me a ﬁrm foundation in theoretical physics. I have received a lot of support from the physics department through Ka- terina Malysheva, Kacey Acquilano, Deb Hatﬁeld, Craig Wiggers, and Jenny Wurster, for which I am grateful. I was supported by a graduate fellowship for a part of my Ph.D. work, and I would like to thank the Cornell Graduate School for the same. I have grown as a physicist and as a person from interactions with many people in the physics community. At the risk of missing someone (apologies if I do), I would like to thank the following people. I am grateful to my fel- low high energy theory group members who have been wonderful friends and colleagues: postdocs, Sungwoo Hong, Matt Klimek and Gabe Lee; graduate

v students, Mehmet Demirtas, Geoffrey Fatin, Naomi Gendler, Manki Kim, Mijo Ghosh, Eliott Rosenberg, Ibrahim Shehzad, Namitha Suresh, Amirhossein Taj- dini, and also Fernanda Huller,¨ who will join us soon. I would also like to thank

Nima Afkhami-Jeddi, Jack Collins, Jeff Dror, Salvator Lombardo, Wee Hao Ng, Nic Rey-Le Lorier, John Stout, Ofri Telem and Yu-Dai Tsai who are no longer at Cornell. Ofri, Laura, and little Tomer deserve a special mention for being good friends and for Ofri’s encouraging words and career advice. Thanks also to Gabriele Rigo and Cem Eroncel¨ from Syracuse University, my friends at TASI 2018 (a fantastic experience), the Mainz 2019 summer school, the GGI 2017 win- ter school, the pre-SUSY workshop in 2019, and many others I met at various conferences. Further away from my research area, I am lucky to have friends from my graduate student cohort who were supportive, funny, and weird to an unfath- omable degree. So thank you, Soumyajit (Messi) Bose, Yi Xue Chong (and by extension, Sam Schultz), Alex Grant, Jihoon Kim, Jaeyoon Lee, Kevin Nangoi,

Albert Park, Meera Ramaswamy, Brian Schaefer, and Rahul Sharma. I am especially grateful to have had three people in my life, Mehmet Demirtas, Meera Ramaswamy and Soorya Suresh Babu, who have been a constant source of strength and support. Thank you for always being there for me, and for lis- tening and keeping my spirits up through all the difﬁculties in life and research. Last, but certainly not the least, I would like to thank my family for their un- wavering support and encouragement. Acha, Amma, Vallyamma and Chettan, I wouldn’t be here without you.

vi TABLE OF CONTENTS

Biographical Sketch ...... iii Dedication ...... iv Acknowledgements ...... v Table of Contents ...... vii List of Tables ...... ix List of Figures ...... x

1 Introduction 1 1.1 The Standard Model of Particle Physics ...... 1 1.2 Hints of New Physics ...... 4 1.3 Larger Issues ...... 5

2 Baryogenesis and the Origin of Matter 10 2.1 Baryon Asymmetry and Sakharov Conditions ...... 10 2.2 Electroweak Baryogenesis ...... 13

3 Dynamics of Electroweak Phase Transition in a Singlet-Scalar Exten- sion of the Standard Model 18 3.1 Introduction ...... 18 3.2 Setup ...... 21 3.3 Results ...... 25 3.4 Discussion ...... 32

4 A Tiny Review of Neutrinos 35 4.1 Introduction ...... 35 4.2 Neutrinos in Numbers ...... 37 4.3 The See-Saw Mechanism ...... 42

5 Clockwork Neutrinos 45 5.1 Introduction ...... 45 5.2 Uniform Clockwork Neutrino Model ...... 48 5.3 Experimental Constraints ...... 56 5.3.1 Lepton Flavor Violation ...... 57 5.3.2 Precision Electroweak Constraints ...... 59 5.4 Generalized Clockwork Neutrinos ...... 60 5.5 Collider Phenomenology ...... 65 5.6 Conclusions and Outlook ...... 72 5.A Perturbation Theory in p ...... 74 5.A.1 Eigenvalues ...... 74 5.A.2 Eigenvectors and Rotation Matrices ...... 76 5.B Lepton Collider Analysis Details ...... 82

vii 6 What is Dark Matter? 88 6.1 Evidence for the Existence of Dark Matter ...... 88 6.2 What We Know So Far ...... 91 6.3 Freeze-Out and Freeze-In of Dark Matter ...... 94

7 Conformal Freeze-In of Dark Matter 98 7.1 Introduction ...... 98 7.2 Particle Physics Framework ...... 99 7.3 Ultraviolet Theory ...... 104 7.4 Cosmological Evolution ...... 106 7.5 Dark Matter Phenomenology ...... 112 7.A Appendix A: Critical Dimension (d∗)...... 118 7.B Appendix B: Derivation of Relic Density ...... 120

Bibliography 125

viii LIST OF TABLES

4.1 Characteristics of various neutrino experiments, reproduced from [75]. SBL stands for Short Baseline and LBL stands for Long Baseline...... 39 4.2 Summary of neutrino oscillation parameters [77]. NO stands for Normal Ordering of neutrino masses, and IO stands for Inverted Ordering...... 40

5.1 Benchmark points (BPs) for the uniform clockwork model. First column: BP name. Next 4 columns: model input parameters. Last 3 columns: mass of the lightest CW neutrino; its coupling to weak current; and mass of the heaviest CW neutrino...... 65 5.2 Benchmark points (BPs) for the generalised clockwork model (LCW1). First column: BP name. Next 4 columns: model input parameters. Last 3 columns: mass of the lightest CW neutrino; its coupling to weak current; and mass of the heaviest CW neutrino...... 65 5.3 Cross sections of CW neutrino signatures at hadron and lepton colliders, before selection cuts (in fb). Acceptance cuts have been applied at the parton level: ∆R 0.4 for all visible object pairs; ≥ pT (`) 10 (20) GeV for hadron (lepton) colliders; pT (j) 20 ≥ ≥ GeV; ηj < 5; η` < 2.5...... 67 | | | | 5.4 Cut flow for the search for CW neutrino production in the 3`+E/T channel at the HL-LHC ( = 3 ab−1)...... 68 L 5.5 Center-of-mass energy and integrated luminosity required for a 3-sigma observation of the CW neutrino signal in electron- positron collisions...... 70 5.6 Cut flow table for the search for CW neutrinos at the U100 benchmark point, in e+e− collisions at √s = 250 GeV...... 72 5.7 Cut flow Table for generalized CW model G100. † The fact that we get 0 events is an artifact of low statistics of our background sample. When we estimate the signal significance, we used Pois- son statistics and used 3 Madgraph events, which corresponds to 5.4 actual events with the integrated luminosity shown. . . . . 84 5.8 Cut flow Table for generalized CW models: G300 and G750.... 85 5.9 Cut flow Table for uniform CW model for U100...... 86 5.10 Cut flow Table for uniform CW models: U400, U750, and U1000 87

7.1 Standard Model operators and corresponding mass gaps generated through different sources. Here, yi are the Yukawa couplings of the appropriate particles and sums are over ﬂavors in the quark/lepton sectors that the CFT operator couples to. . . . 103 7.2 Dominant modes of production in each portal...... 109

ix LIST OF FIGURES

2.1 Vacuum bubbles (left), and generation of baryon asymmetry at the bubble walls (right) [17]...... 16

3.1 Phase transition dynamics in the mS κ plane, with η = ηmin+0.1. − Region I (green): one-step strongly first-order transition; Re- gion II (yellow): two-step transition with strongly first-order electroweak-symmetry breaking step; Region III (red): no thermal phase transition (a would-be two-step transition, but bubbles fail to nucleate); Region IV (purple): same as red, with a would-be one-step transition; Region V (blue): second-order transition; Region VI (gray): no viable EWSB at zero temperature; Region VII (white): non-perturbative regime (η > 10). . . . . 26 3.2 Ratio S3/T , where S3 is the critical bubble action, for mS = 300 GeV and κ = 1.55 (red) and 1.54 (yellow). For both points, a two- step first-order transition is naively expected. In fact, thermal transition does not occur at κ = 1.55...... 27 3.3 Thermal potential at the critical temperature, along the line in field space connecting the EW-symmetric and broken vacua, for mS = 300 GeV, κ = 1.8, and two representative values of η, 2.0 (red) and 2.5 (yellow). For both points, a two-step first-order transition is naively expected. In fact, thermal transition does not occur at η = 2.0...... 29 3.4 Phase transition dynamics in the mS κ plane, with η = ηmin+2.5. − Same labeling and color code as in Fig. 3.1...... 30 3.5 Phase transition dynamics in the κ η plane, with mS = 300 GeV. − Same labeling and color code as in Fig. 3.1...... 31 3.6 Phase transition dynamics in the κ η plane, with mS = 300 GeV. − In region B (red) bubble walls accelerate to relativistic speeds and EWBG cannot occur, while in region A (blue) EWBG is possible...... 31 3.7 The height of the barrier between the EW-symmetric and broken vacua at zero temperature, in units of electroweak vev4. The dashed line indicates the boundary between the region where the thermal EWPT occurs (above the line) and where it does not occur (below the line)...... 32 3.8 Potential difference between the EW-symmetric and broken vacua at zero temperature, in units of electroweak vev4. The dashed line indicates the boundary between the region where the thermal EWPT occurs (above the line) and where it does not occur (below the line)...... 33

4.1 Allowed neutrino mass spectra: Normal Ordering (left) and In- verted Ordering (right)...... 40

x 4.2 Feynman diagram for neutrino-less double beta decay, a signature of lepton number violation. A Majorana mass term in the Lagrangian is required for this process to take place...... 42

5.1 Pictorial representation of the clockwork sector with right- handed zero mode. Single solid lines denote Dirac masses, while the double solid line denotes the Yukawa couplings involving the SM Higgs boson H. In the uniform CW model, mi m and ≡ qi q for all i...... 48 ≡ 5.2 Composition of the left-handed (left panel) and right-handed (right panel) mass eigenmodes in terms of the original clockwork fields in the uniform clockwork model with N = 15, m = v, q = 4.887, and y = 0.01...... 51 5.3 Spectrum of clockwork neutrinos in the uniform CW (left) and generalized Linear CW1 (center) and Linear CW2 (right) models. (For discussion of generalized CW models, see Section 5.4.) In all cases, N = 15, m = v, and y = 0.01; q = 4.887 (uniform), 0.76 (LCW1) and 0.73 (LCW2). The parameters were chosen so that −2 the pseudo-zero mode neutrino mass is mν = 8 10 eV in all · cases...... 54 5.4 Constraints on the parameter space of the uniform clockwork model from µ eγ and precision electroweak fits. Left panel: → normal (hierarchical) spectrum of active neutrinos. Right panel: degenerate spectrum of active neutrinos. Dashed/red lines indicate the mass of the lightest clockwork neutrino, while dot- dash/blue lines indicate the coupling of this state to the SM gauge currents...... 58 5.5 Composition of the left-handed (left panel) and right-handed (right panel) mass eigenmodes in terms of the original clockwork fields in the Linear CW 1 model (top line) and Linear CW 2 model (bottom line). In both cases, N = 15, m = v, and y = 0.01; q = 0.76 in the LCW1 model and 0.73 in the LCW2 model. . . . . 62 5.6 Top row: Constraints on the parameter space of the Linear CW 1 model from µ eγ and precision electroweak fits. Left panel: → normal (hierarchical) spectrum of active neutrinos. Right panel: degenerate spectrum of active neutrinos. Dashed/red lines indicate the mass of the lightest clockwork neutrino, while dot- dash/blue lines indicate the coupling of this state to the SM gauge currents. Bottom row: same, for the Linear CW 2 model. In the case of Linear CW 2 model, the masses and couplings of the heaviest clockwork state are plotted, since other clockwork states have strongly suppressed interactions with the SM. . . . . 64 5.7 Production of heavy clockwork modes k at hadron (left) and N lepton (right) colliders...... 66

xi 5.8 Distributions of CW neutrino production signal (blue) and background (orange) events in s12 (left panel) and s23 (right panel) at the LHC. The signal was simulated at the G100 benchmark point. 68 5.9 Distributions of CW neutrino production signal (blue) and background (orange) events in Mrec at the U100 (left panel) and G100 (right panel) benchmark points...... 69 5.10 Signal (blue) and background (orange) distributions for U100 + − model at a 250 GeV e e collider: M`jj (left) after pre-selection cuts, ∆Rjj (right) after M`jj cut, and M`ν (bottom) after ∆Rjj cut. 71

6.1 Rotation curves for NGC 6503, a dwarf spiral galaxy. Plot from [112]...... 88 6.2 Overlay of the weak lensing mass contours on the X-ray image of galaxy cluster 1E 0657–56. The gas bullet lags behind the DM subcluster. Plot from [113]...... 89 6.3 Broad overview of dark matter models and their mass ranges, along with search techniques and some current experimental anomalies. Mass ranges are approximate and can differ based on model-speciﬁc details. Figure from US Cosmic Visions 2017 [114]. 92 6.4 Comparison of freeze-out and freeze-on of particles in the early universe [4]. Relativistic species freeze out before exponential Boltzmann suppression kicks in, when the expansion of the universe becomes faster than the processes keeping them in equilibrium. This is not a viable scenario for DM, and the non- relativistic freeze-out and freeze-in lines show how relic abundance can be achieved successfully...... 95 6.5 Comparison of relic densities from freeze-out and freeze-on of particles and their dependence on couplings to the SM [115]. The quantity Ω is deﬁned as Ω ρ /ρ and Ω h2 0.1 gives the ≡ DM crit ∼ correct relic density...... 96

7.1 Diagrams that contribute to conformal symmetry breaking in the lepton portal. Blue circles indicate CFT operator insertions. The ﬁrst two induce an 2 term and the third induces an term OCFT OCFT below the weak scale...... 101 7.2 Loop induced diagrams that contribute to conformal symmetry breaking in the quark and gluon portals. Blue circles indicate CFT operator insertions. Both generate terms after elec- OCFT troweak symmetry breaking...... 101

xii 7.3 Top panel: Energy density in the CFT plasma or dark matter particles, as a function of the SM plasma temperature T , in the Higgs 8 17 portal scenario with Λ = 1.2 10 GeV and MU = 10 GeV. CFT × Bottom panel: Evolution of the CFT plasma temperature, as a function of T , for the same parameters. At all times, TD T , as required for the self-consistency of our calculations...... 111 7.4 Dark matter relic density contours (red) and observational/theoretical constraints, in the Higgs portal model. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is defined as ΛSM/4πv...... 113 7.5 Dark matter relic density contours (red) and observational/theoretical constraints, in the quark portal model with only the first generation of quarks. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is defined as ΛSM/4πv...... 114 7.6 Dark matter relic density contours (red) and observational/theoretical constraints, in the quark portal model with all generations of quarks. The same value of r has been used for comparison. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is defined as ΛSM/4πv.... 114 7.7 Dark matter relic density contours (red) and observational/theoretical constraints, in the gluon portal model. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is defined as ΛSM/4πv...... 115 7.8 Dark matter relic density contours (red) and observational/theoretical constraints, in the lepton portal model with coupling to only electrons. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is defined as ΛSM/4πv...... 116 7.9 Dark matter relic density contours (red) and observational/theoretical constraints, in the lepton portal model with coupling to all charged leptons. The same value of r has been used for comparison. Thick red line indicates parameters where the observed dark matter abundance is reproduced...... 116

xiii CHAPTER 1 INTRODUCTION

1.1 The Standard Model of Particle Physics

In 1930, when much of the current particle zoo was unknown, there was a heated debate among physicists about whether energy conservation is valid in beta decays, considering the continuous energy spectra that was observed.

Wolfgang Pauli postulated the existence of the neutrino, while Niels Bohr considered conservation laws to be statistical and not exact. Almost a century later, there are dozens of experiments devoted to studying neutrinos; physicists have determined the fine structure constant correctly to 14 significant digits; and the last particle that was predicted by the Standard Model, the Higgs boson, was discovered at a high energy particle collider. Particle physics has come a long way in the past century. It would not be an overstatement to call the Standard Model of Particle Physics the most successful theoretical model in the history of fundamental physics. It is quantitatively the most accurate quantum field theory that has been written down, having successfully predicted many observables to high precision and surviving years of experimental tests. Precision tests of the Standard Model have been conducted at colliders ranging from lepton colliders at just 45 GeV (the Stanford Linear Collider) and proton colliders at about 300 GeV (the Super Proton Synchroton, that discovered the W and Z bosons in 1983) to current proton colliders that can reach energies of 14 TeV (the Large Hadron Collider that discovered the Higgs boson in 2012).

One might be surprised that the model that explains almost everything we see in nature can be written down as a Lagrangian that covers a few lines at

1 most. To pin down the structure of the Standard Model (SM), all that is required is the gauge group and the matter content [1]. The SM has a single scalar ﬁeld and three generations of fermionic quarks and leptons that interact through three different forces - strong, weak and electromagnetic. Each force is associated with a gauge group, and gauge symmetry (with the proper representations for matter ﬁelds) dictates the structure of these interactions.

GAUGE GROUP: SU(3)C SU(2)L U(1)Y . • × × FERMIONIC MATTER CONTENT: All the SM fermions can be listed as be- • low, in terms of left-handed Weyl spinors. The notation used is (C,L)Y , and i stands for the ﬂavor index (i = 1, 2, 3).

c ¯ c ¯ c Qi : (3, 2)1/6; ui :(3, 1)−2/3; di :(3, 1)1/3; Li : (1, 2)−1/2; ei : (1, 1)1.

SCALAR CONTENT: The Higgs ﬁeld is an SU(2) doublet that sponta- • neously breaks the electroweak symmetry - SU(2) U(1)Y U(1)EM . × →

Writing all possible renormalizable gauge-invariant terms with these ingredients essentially gives the SM Lagrangian. A non-trivial condition on matter representations is that of anomaly cancellation. Any classical symmetry of the Lagrangian that is broken by quantum effects (or in other words, loop Feynman diagrams) is considered anomalous. An anomaly in a gauge symmetry signiﬁes a fundamental problem with the theory; it implies that the theory will be invalid at some scale that can be calculated from the anomaly, and unitarity and/or lo- cality will be lost at that scale. Thus, existence of gauge anomalies is evidence that the theory is merely a low energy effective ﬁeld theory, and must be replaced with an improved UV theory at cut-off scale. It is interesting to note that, despite our expectations that the SM will be replaced with a different theory at

2 some UV scale, it is a renormalizable theory with no anomalous gauge symmetries. In principle, the SM could be the ultimate theory, if not for its exclusion of gravity and the presence of a Landau pole in electromagnetic interactions at an even higher scale. Details of what the various issues that suggest the incom- pleteness of the SM will be discussed in Section 1.3

It is useful to examine the number of parameters that go into any theory. In the SM, there are a total of 19 parameters which can be classiﬁed as follows:

0 Gauge couplings: 3 parameters, g, g and gS, • Higgs sector: 2 parameters, v and λ, • Flavor sector: 13 parameters, including 3 lepton masses, and 10 physical • parameters in the quark Yukawa matrix,

Strong CP: 1 parameter, θQCD. •

One of the goals of theorists who study UV theories that can supercede the

SM, is to reduce the number of these parameters and make a more ‘predictive’ theory. This is especially needed in the ﬂavor sector, where the SM has no explanation for the large hierarchy in Yukawa couplings and the peculiar structure of the CKM matrix.

The following sections in this chapter will summarize a few pressing issues with the SM that is currently at the forefront of theoretical research in particle physics, and end with an overview of the structure of this dissertation.

3 1.2 Hints of New Physics

While the Standard Model has been extremely successful in many respects, there are quite a few issues that one can point out. Among the hints that the

SM maybe not be the end of the story, are the various “anomalies” that have cropped up over the last few years. It is important to point out that these are not discrepancies with statistical signiﬁcance reaching ‘discovery level’ or ﬁve standard deviations. A few of these are listed below:

1. Flavor anomalies: There are a few different discrepancies from theoretical calculations and experimentally observed values in the ﬂavor sector of the Standard Model [2, 3].

(a) The ratio of rates of B-meson decays to D mesons + τ or µ leptons (RD

and RD∗ ) has consistently exceeded SM expectations by 2 3 σ. ∼ − (b) The ratio of rates of B-meson decays to Kaons + muons or electrons

(RK and RK∗ ) has been less than the expected SM value of unity by 2.5 σ. ∼ 0 (c) The ratio K /K of CP violating amplitudes (through oscillations and decays) in Kaon decays to pions has had a 2.8 σ anomaly with SM ∼ lattice calculations, although systematic uncertainties on the theoret-

ical side make this contentious.

0 (d) The KOTO collaboration detected a few events in the KL π νν¯ → channel, far exceeding the expected SM branching ratio.

2. Muon g 2 experiments have measured the anomalous magnetic moment − of the muon to be 3 σ away from the SM value. ∼

4 3. The ATOMKI collaboration has detected anomalous decays of excited Be8 nuclei that suggest the possible existence of an MeV scale particle involved in the decay.

We do not know what the future holds, but one should not be surprised if some of these disappear with experiments taking more data in the coming years. But with any luck, there are hints of new physics hidden in them, and discoveries within our reach.

1.3 Larger Issues

Besides the ever-changing number of anomalies in experimental data, there are bigger, more structural problems with the Standard Model. It succeeds at doing what it claims to do, but falls very short in explaining or completely omits some rather important parts of the universe:

1. The most obvious omission is that of gravity. While the Standard Model

accounts for the electromagnetic, strong and weak forces, it completely neglects gravity. Unifying General Relativity and Quantum Field Theory is a notoriously difﬁcult task, but fortunately, we only need a quantum

theory of gravity to study extreme environments with very high energy density and curvature. The SM cannot explain processes in this case, and by excluding gravity, we know conclusively that the SM will break at or

below the Planck mass. Understanding quantum gravity is one of the most daunting and important tasks in modern physics, and there are multiple research directions currently being explored.

5 2. While not a problem with the model per se, gauge uniﬁcation is one of the goals of many beyond Standard Model theories. In the SM, the strong gauge group SU(3) is separate from the other gauge groups. Electroweak

theory uniﬁed electromagnetism and the weak force under one gauge group, while they are treated separately in low energy effective ﬁeld theories below electroweak symmetry breaking. Thus, a natural progression

for higher scale theories would be to unify the strong force as well, and many Grand Uniﬁed Theories (or GUTs) seek to do exactly this, by unifying the SM under a single group, such as SU(5).

3. The Standard Model omits neutrino masses completely. Neutrinos are extremely light, and for any process around the weak scale and above they

can be considered massless, and the SM works very well in this regime. Oscillation experiments in the last few decades have shown very clearly that the three generations of neutrinos have different masses. This goes

against the lepton family structure in the SM, where the neutrino gauge and mass eigenvectors coincide, and oscillations are forbidden for leptons. Additionally, their mass generating mechanism is not known. A more de-

tailed discussion is contained in Chapter 4.

4. The Standard Model does not include ‘Dark Matter’ (DM) or ‘Dark En-

ergy’. Dark energy is the driving force behind the accelerated expansion of the universe, and can be identiﬁed with a ‘cosmological constant’ in classical General Relativity. Dark matter is a mass of gravitationally interacting

matter that plays a fundamental role in the motion of stars and galaxies, and in the development of large scale structure in the universe [4]. Dark Matter comprises about 26% of the energy in the universe, and Dark En-

ergy covers about 69%, leaving a measly 5% for visible matter. When ∼

6 stated in terms of energies, the fact that the SM can only explain 5% of the universe highlights how limited our knowledge is. A detailed look at the evidence for and our understanding of dark matter can be found in

Chapter 6.

5. Another question that particle phenomenologists have been trying to

solve for many decades now, is the ‘Hierarchy Problem’ [5]. The crux of the problem lies in the fact that the mass of the Higgs boson is quadratically sensitive to UV scales: in other words, quantum corrections to the

Higgs mass grow as Λ2, where Λ is the Standard Model cutoff scale in an Effective Field Theory (EFT) perspective. If we assume the SM cutoff scale to be high ( 1 TeV ), one would expect this mass to be very high, but it has been measured to be at the weak scale and is only 125 GeV. Thus, ∼ either Nature is ﬁne-tuned and there is a miraculous cancellation between the bare mass and these quantum corrections, or the SM must be replaced

by a UV theory above the weak scale which tames these quantum corrections [5] or explains the cancellation with a symmetry [6].

6. The Standard Model also fails to explain why matter seems to dominate the visible sector of the universe over antimatter [7]. The symmetry between matter and antimatter is very weakly broken in the Standard Model

with one source of CP (charge conjugation and parity) violation: the CKM matrix. This is not enough and the SM does not contain the ingredients to meet the criteria for developing an excess of matter over antimatter in the

early universe to match current observations. Many extensions of the SM and possible solutions to this problem have been proposed, and Chapter 2 will discuss this more thoroughly.

7. Lastly, as discussed above, the SM has many parameters that need to be

7 experimentally measured. In addition, there is a wide range of masses from the neutrinos at eV scale to the top quark at about 173 GeV. There is no explanation for this large hierarchy, and the Yukawa couplings that

determine the masses are input parameters. The CKM matrix is also experimentally determined, with no explanation in the SM for its origin or why its diagonal values are much bigger than the off-diagonals. (A collection

of such questions form the so-called ‘ﬂavor puzzle’ [8].) The hope is that future uniﬁed theories would be more predictive with fewer parameters.

In addition, there are quite a few other intriguing signs of new physics, that are technically outside of the purview of the Standard Model. These are experimental results that conﬂict with simple extensions of the SM that seek to explain neutrino phenomena and cosmology. In the neutrino sector, there are several anomalies, the most signiﬁcant of which is the MiniBooNE measurement [9].

This is a 4.8 σ deviation from the standard three-neutrino setup constructed using solar and atmospheric neutrino oscillation data. (See Chapter 4 for details).

The SM is incorporated into a cosmological framework with an expanding universe and cold dark matter to form the ΛCDM model, often called the standard model of cosmology. A recent excess of electron recoil events in the

Xenon1T experiment could possibly point to the existence of non-baryonic matter with properties quite different from the simplest cold dark matter models. Further more, there has been a growing tension [10] between the Hubble expansion rate measured using supernovae and using the Cosmic Microwave Back- ground spectrum, which could be a result of new physics at recombination.

This dissertation studies three of the aforementioned problems: the matter- antimatter asymmetry, neutrino masses, and dark matter. Chapter 2 contains

8 an exposition of conditions and proposed solutions to obtain an excess of matter in the early universe. Chapter 3 explains work studying aspects of phase transitions and bubble nucleation in one such model: electroweak baryogenesis with a singlet scalar added to the SM. It is based on work done with Maxim Perelstein in [11]. A brief review of neutrino physics is contained in chapter 4, with chapter 5 studying a particular neutrino model (Clockwork Dirac neutrinos) in detail, based on work done in collaboration with Sungwoo Hong and Maxim Perelstein in [12]. Chapter 6 recaps what we know about dark matter and the evidence for it, in a fairly general and model-independent fashion. To conclude, chapter 7 studies a class of DM models that differ from conventional models with a particle picture, by studying dark sectors with strong coupling and conformal symmetry. It is based on continuing work done with Sungwoo

Hong and Maxim Perelstein, part of which is contained in [13].

9 CHAPTER 2 BARYOGENESIS AND THE ORIGIN OF MATTER

2.1 Baryon Asymmetry and Sakharov Conditions

The baryon asymmetry in our universe has been a long standing puzzle in theoretical physics. Our universe appears to be made of matter almost exclusively and the only antimatter particles that have been observed have been through pair production in accelerators and cosmic rays. This is puzzling, as we would expect a fairly symmetric distribution of matter and antimatter based on Stan- dard Model calculations (which would also mean that the rich and intricate matter structures in the universe would never have formed). One might wonder if there are macroscopic patches of antimatter in the universe, with just our local patch being matter-dominated. But in this case, there would be astrophysical gamma ray emissions from annihilation at the boundary of such patches, and such signals have not been observed.

The baryon asymmetry in the universe can be quantiﬁed as a dimensionless number by taking the ratio with the number density of photons in the universe currently. It has been measured to be,

nB n ¯ η − B 10−9. (2.1) ≡ nγ ∼

An early hypothesis to explain this, was that the asymmetry could be an initial condition of the universe. The problem with this idea is that we expect the universe underwent a period of inﬂation at early times, and any asymmetry present before inﬂation will be washed out to very dilute amounts. Thus, there needs to be a mechanism to produce baryon asymmetry after reheating.

10 In general, three conditions are necessary to create baryon asymmetry in the universe where none existed previously. These are commonly known as ‘Sakharov conditions’:

1. Baryon number violation,

2. Loss of thermal equilibrium in the early universe,

3. C and CP violation.

It is rather obvious that the baryon number symmetry should be broken in order to produce a baryon asymmetry. The Standard Model does not impose baryon number or lepton number symmetry, but they are accidental symmetries of the perturbative SM Lagrangian. However, the B + L current is anomalous, and this symmetry is broken by non-perturbative tunneling processes. Note, however, that B L is still conserved at the quantum level. The derivative of − the B + L current is as follows.

3g2 3 ∂ jµ = T rF a F˜µν a = ∂ Kµ , µ B+L 16π2 µν 16π2 µ µ µνρσ 2 with K = T r(FνρAσ + AνAρAσ) , (2.2) 3 where Fµν stands for the field strength tensor of the gauge fields that the particles couple to, g is the coupling strength and Aµ is the gauge field itself.

As the resultant is a total derivative, for Abelian gauge symmetries, the anomaly becomes zero. But for non-Abelian symmetries, it is possible to have ‘pure gauge’ vacuum states composed of large gauge transformations that do not asymptote to zero as r . These vacua are called ‘theta vacua’ and give → ∞ 1 µ rise to a non-zero anomaly. 16π2 K is essentially the Chern-Simons current and integrating it gives an integer corresponding to the ‘winding number’ of the

11 gauge conﬁguration. Thus, tunneling between theta vacua changes the winding number by integer values, and as a consequence break B + L symmetry. Because of the coefﬁcient of three and the fact that B L is conserved, these − tunneling processes produce ∆B = ∆L = 3. ±

The second condition of loss of thermal equilibrium is required to make the process of baryon asymmetry creation irreversible. If whatever process that cre- ates an asymmetry is in thermal equilibrium, the reverse process will wash out any excess baryons produced, since their rates are equal by the CPT theorem.

There are two natural ways to accomplish this. One is by out-of-equilibrium decays of heavy particles. When the temperature of the thermal bath in the early universe has fallen below the mass of some particle, its decays occur out of thermal equilibrium and the reverse process is highly suppressed. This is the case for example, in GUT baryogenesis, where heavy GUT scale bosons decay to produce baryon asymmetry. The other way is through ﬁrst-order phase transitions.

First-order phase transitions involve tunneling between vacua, as opposed to a smooth adiabatic transition like in the case of second order transitions. This sharp delineation of vacua and the resulting nucleation of vacuum bubbles generates loss of thermal equilibrium. Processes that can occur in the false vacuum can be forbidden in the true vacuum, and the expansion of the bubble as the ﬁeld tunnels to the true vacuum will prevent wash-out of any baryon asymmetry generated in the false vacuum. This is the key process in Electroweak Baryogenesis, which will be discussed in Section 2.2.

The third requirement for baryogenesis is that of C and CP violation. The violation of charge conjugation symmetry and CP symmetry ensures that the charge conjugate process or ‘anti-process’ of the production process under con-

12 sideration does not negate the produced baryon asymmetry. In general, theories with CP violation have complex phases in the Lagrangian interaction terms that cannot be rotated away by ﬁeld redeﬁnitions. In the SM, the CKM matrix has

a complex phase which contributes to CP violation. The θQCD term is another famous example of a CP violating term that could be included in the SM, but its coefﬁcient has been experimentally constrained to be extremely small (and its unnatural smallness deﬁnes the ‘strong CP problem’).

2.2 Electroweak Baryogenesis

It turns out that the Standard Model has all the ingredients necessary to meet the criteria outlined by Sakharov to produce an excess of matter over antimatter [14], although not to enough of a degree to explain the measured asymmetry. As explained in the previous section, non-perturbative tunneling processes violate baryon number symmetry in the SM. These processes are extremely suppressed at zero temperature, due to factors of e−2π/αW 10−65 and by Yukawa ∼ couplings. But at higher temperatures, there are ‘sphalerons’ that mediate transitions from one vacuum to the other. Sphalerons are saddle-point solutions to the ﬁnite temperature equations of motion that sit at the top of the energy barrier between theta vacua.

The rate of sphaleron processes are generically proportional to e−Esph/T , where Esph stands for the energy of the sphaleron conﬁguration. At higher temperatures, this exponential can become (1) and generate enough baryon O number violation to produce a macroscopic asymmetry. The sphaleron energy can be analytically computed in the regime where the temperature is below that

13 of the electroweak phase transition, and is proportional to the Higgs VEV. Below the weak scale, the rate can be shown to be, 3 Γsph Esph mW (T ) = (const) T 4e−Esph/T , V T T 8πv with Esph . (2.3) ' g

Above the electroweak temperature, however, these analytical formulae fail, and lattice computations need to be invoked. It has been shown that at high temperatures, the rate has the parametric form,

Γsph (const) α5 T 4, (2.4) V ' W where the dimensionless constant is about 25. ∼

Additionally, the SM also has CP violation, through the complex phase in ¨ the CKM matrix. However, the lowest order diagram that contributes to ¨CP¨ processes for baryogenesis, involving all three generations, is suppressed by a factor of 12 Yukawa couplings [15]. This results in a suppression factor of 10−20, which makes the amount of CP violation in the SM far too little to explain the baryon asymmetry in the universe.

Another bottleneck in setting up baryogenesis in the SM is the requirement of loss of thermal equilibrium. A possible scenario for loss of thermal equilibrium would be a ﬁrst order electroweak phase transition. But in the SM, the Higgs is too heavy for a ﬁrst order transition and current non-perturbative calculations point to a crossover transition during electroweak symmetry breaking [16].

All is not lost however, and these signs suggest that simple extensions of the

SM could be enough for baryogenesis to happen. Additional sources of CP vio-

14 lation are present in many such models, including the Minimal Supersymmetric Standard Model, and neutrino mass models with Majorana mass terms. Simi- larly, the nature of the electroweak phase transition can be modiﬁed by adding new interactions to the Higgs potential. One of the simplest examples of this, is the addition of a gauge-singlet scalar ﬁeld, S, coupled to the SM via a Higgs portal interaction. With the addition of a Z2 symmetry that stabilizes the particle

S, the Higgs potential is, 1 η V (H; S) = µ2 H 2 + λ H 4 + m2S2 + S4 + κS2 H 2. (2.5) − | | | | 2 0 4 | |

For large swathes of the parameter space in κ, η and m0, a ﬁrst order phase transition results.

With these modiﬁcations, baryogenesis is possible at the experimentally accessible and theoretically pleasing electroweak scale. The steps required for electroweak baryogenesis are as follows [17]. A pictorial representation is shown in Figure 2.1.

1. As the universe cools, a minimum develops in the Higgs potential with non-zero vacuum expectation value. Below the critical temperature (technically, below the nucleation temperature), vacuum bubbles nucleate and

grow. Inside the bubbles, the Higgs ﬁeld has a vacuum expectation value, while it is still in the false vacuum outside the bubbles.

2. Particles in the plasma outside the vacuum bubbles scatter with the bubble walls. The presence of CP violation in the theory favours the production

of CP and C asymmetries in the particle number densities.

3. Sphaleron processes produce baryon asymmetries, and are biased to produce more baryons than anti-baryons due to the CP asymmetry already

produced in the plasma.

15 Figure 2.1: Vacuum bubbles (left), and generation of baryon asymmetry at the bubble walls (right) [17].

4. The baryons produced outside the bubble walls are swept inside the bub-

bles as the vacuum bubbles expand. Inside the bubble, the sphaleron rate,

−Esph/T e is suppressed as its energy Esph is proportional to the Higgs ∼ VEV. Note that the temperature is at or below the weak scale after the

phase transition and the nucleation of bubbles. This prevents sphaleron processes from washing out the generated baryon asymmetry inside the vacuum bubble.

There are a few other possibilities for generating the baryon excess in the universe. An attractive mechanism is that of leptogenesis, which ties neutrino masses to the problem of matter-antimatter asymmetry [7]. The ﬁrst step in leptogenesis is the creation of a lepton asymmetry, as the name implies. This happens in the neutrino sector, as an excess of charged electrons would indicate a net electric charge in the early universe. This is achieved through out of equilibrium decays of heavy Majorana right-handed neutrinos. Such particles are a generic feature of see-saw models, which are brieﬂy reviewed in Chapter 4. As we have seen, sphaleron processes conserve B L and violate B + L. Thus, a − lepton asymmetry can be converted to a baryon asymmetry, and an excess in

16 lepton number, L, will bias sphalerons to produce an excess in baryon number, B. The downside of leptogenesis models lies in the difﬁculty of experimental veriﬁcation, a problem shared by the see-saw models that it invokes. In the see-saw mechanism, the heavy Majorana states tend to be far too heavy to be produced in colliders.

Another proposal for baryon asymmetry generation is the Affleck-Dine mechanism. In this type of baryogenesis, the key ingredient is a coherent scalar field that carries baryon and/or lepton numbers and couples to the inflaton.

Such scalar fields are naturally found in supersymmetric theories. During slow- roll inflation and reheating, decays of this scalar field can generate a net baryon excess. A favorable feature of this mechanism is that it can explain both baryonic and dark matter production simultaneously, which also serves as a reason for why the dark and baryonic matter energy densities are somewhat close in orders of magnitude. Unfortunately, the Affleck-Dine mechanism is also difficult to test experimentally, as the key processes happen at the inflation scale, which is too high for current experiments to access. At best, the discovery of supersymmetric particles at colliders or in dark matter experiments can serve as indirect evidence, depending on the nature of the Lightest Supersymmetric Particle (‘LSP’) [14].

17 CHAPTER 3 DYNAMICS OF ELECTROWEAK PHASE TRANSITION IN A SINGLET-SCALAR EXTENSION OF THE STANDARD MODEL

3.1 Introduction

In today’s Universe electroweak symmetry is broken, but at very high temperatures, which prevailed immediately after the Big Bang, the symmetry was restored. The transition between the symmetric and broken phases, the Elec- troweak Phase Transition (EWPT), occurred when the Universe was about a nanosecond old. Understanding the nature of this transition is an interesting question in its own right. It also has profound implications for understanding the origin of matter-antimatter asymmetry in the Universe: one of the most com- pelling explanations of this asymmetry, the Electroweak Baryogenesis (EWBG) scenario, is only possible if the EWPT is strongly ﬁrst-order [18]. (For a review, see [17].)

While it is at present not possible to recreate the EWPT in the lab, it has been suggested that measurements of properties of the Higgs boson can provide indirect information about the EWPT dynamics. In the Standard Model (SM), the EWPT is an adiabatic cross-over transition [19, 20, 21, 22]. A first-order transition is only possible in the presence of Beyond-the-SM (BSM) physics at the weak scale, with significant couplings to the Higgs sector. As a result, many models with first-order EWPT predict significant deviations of the Higgs couplings to gluons, photons, weak gauge bosons, and fermions that can already be tested at the Large Hadron Collider (LHC). In particular, supersymmetric models with stop-catalyzed first-order EWPT are already strongly disfa-

18 vored [23, 24, 25] (but not completely ruled out [26]). A broader variety of models will be probed by increasingly precise measurements of the Higgs couplings at the LHC and the proposed e+e− Higgs factories [27, 28, 29, 30]. A particularly direct and powerful probe of the EWPT dynamics is provided by the Higgs cubic self-coupling, which is predicted to have significant (& 20% or more) deviations from the SM in most models with a first-order EWPT [31, 32, 33]. This prediction can be conclusively tested at the 1 TeV upgrade of the International Linear Collider (ILC) [34, 35, 36] and the 100 TeV proton collider [37]. In addition, a strongly first-order EWPT may produce a potentially observable gravitational wave signature [38]. Complementarity between collider and gravitational wave signatures has been explored in Refs. [39, 40, 41, 42, 43].

One of the simplest extensions of the SM in which a ﬁrst-order EWPT is possible is a model with an additional gauge-singlet scalar ﬁeld, S, coupled to the SM via a Higgs portal interaction,

2 2 Vint = κ H S . (3.1) | | This is the only renormalizable interaction of S with the SM which is invariant under a Z2 symmetry, S S. This symmetry renders the S particle stable, → − and it may play the role of dark matter [44, 45, 46, 47]. It has been shown that this simple model can exhibit a strongly ﬁrst-order EWPT [48, 49, 50, 51, 52,

53, 54, 55]. The Z2 prohibits mixing between the doublet and singlet scalars, so that the 125 GeV Higgs particle has couplings to fermions and gauge bosons that are identical to the SM. Moreover, if mS > mh/2 and the decay h SS → is kinematically forbidden, the Higgs width is also unaffected. As a result, this model presents a difﬁcult case (sometimes dubbed a “nightmare scenario”) for tests of EWBG at future colliders, and it became an important benchmark for gauging their capabilities in this regard [56]. Some interesting recent work on

19 this benchmark model includes suggestions for additional observables that can help cover the relevant parameter space at a 100 TeV proton collider [56, 57], and an improved calculation of the thermal scalar potential [58].

An important aspect of the EWPT in this model, which has not yet been systematically taken into account in existing studies of future collider capabilities, is the dynamics of bubble nucleation during the transition. This study aims to

ﬁll this gap. In particular, we evaluate the bubble nucleation temperature, TN , throughout the parameter space relevant for EWSB and future colliders. We

find that TN is often significantly lower than the critical temperature Tc. In fact, in large regions of the parameter space, in particular those with a “two-step” EWPT (meaning that the S field acquires a vev before the Higgs field does), we

ﬁnd that bubbles do not nucleate at ﬁnite temperature at all, eliminating these regions as viable EWBG scenarios. In addition, if TN Tc, the large difference in the vacuum energies at the stable and metastable vacua can result in

“runaway” behavior of the bubble walls, which become highly relativistic [59]. This behavior is incompatible with the EWBG scenario. We identify the region of the parameter space (again primarily in the two-step regime) which suffers from this problem.1 The net result of the analysis is a signiﬁcant reduction of the parameter space with viable EWBG. We then comment on the implications of these additional constraints for the experimental probes of EWBG at future colliders. 1For recent studies of bubble-wall dynamics in this and similar models, see e.g. Refs. [60, 61, 62, 63].

20 3.2 Setup

We supplement the SM with a real scalar ﬁeld S, uncharged under any of the

SM gauge groups, and impose a Z2 discrete symmetry, under which S S → − and all other ﬁelds are unchanged. The tree-level scalar potential has the form

1 η V (H; S) = µ2 H 2 + λ H 4 + m2S2 + S4 + κS2 H 2, (3.2) − | | | | 2 0 4 | | where H is the SM Higgs doublet. If µ2 < 0, there is an electroweak symmetry- breaking (EWSB) minimum at zero temperature, with H = (0, v/√2) and h i S = 0. Depending on parameters, the potential may also have an electroweak h i symmetry-preserving local minimum at H = 0 and S = s0. There are no h i h i stable minima with both H and S non-zero for any model parameters. The h i h i model is phenomenologically viable if the vacuum with H = 0 is the global h i 6 minimum of the potential, v 246 GeV and mh 125 GeV in this vacuum, and ≈ ≈ we restrict our attention to such parameters. This leaves three undetermined

(but constrained) parameters, m0, η, and κ.

The EWPT dynamics is determined by the effective finite-temperature potential Veff (T ), where T is temperature. Physically, Veff is the free energy density of space filled with constant, spatially homogeneous scalar fields:

ϕ Hbg = (0, ),Sbg = s , (3.3) √2 and all other ﬁelds set to zero. The effective potential has the form

Veﬀ (ϕ, s; T ) = V0(Hbg,Sbg) + V1(ϕ, s) + VT (ϕ, s; T ) , (3.4)

where V1 is the Coleman-Weinberg potential, and VT is the thermal potential [64,

21 65]. Both can be computed in perturbation theory. At the one-loop order,

Fi X gi( 1) V1(ϕ, s ) = − 64π2 i 2 h 4 mi (ϕ, s) 3 4 2 2 i mi (ϕ, s) log 2 mi (ϕ, s) + 2mi (ϕ, s)mi (v, 0) ; (3.5) × mi (v, 0) − 2

4 Fi X giT ( 1) VT (ϕ, s; T ) = − 2π2 i Z ∞ " r 2 !# 2 Fi 2 mi (ϕ, s) dx x log 1 ( 1) exp x + 2 (3.6) × 0 − − T where the sum runs over all SM and BSM particles in the theory, and gi, Fi and mi(ϕ, s) are the multiplicity, fermion number, and the mass (in the presence of background ﬁelds) of the particle i. The counterterms included in Eq. (3.5) ensure that the tree-level Higgs mass and vev in the present, zero-temperature

Universe are unchanged at one loop. The dominant contributions to V1 and VT typically arise from loops of the Higgs and singlet scalar themselves. In this case, the masses mi(h, s) are obtained by diagonalizing the scalar mass matrix:

1 m2 = m2 µ2 + (κ + 3λ)ϕ2 + (κ + 3η)s2 ∆ , (3.7) 1,2 2 0 − ± where ∆ = ((m2 + µ2 + (κ 3λ)ϕ2 + (3η κ)s2)2 + 16κ2ϕ2s2)1/2. We also include 0 − − contributions of the SM top quark and the electroweak gauge bosons, but ignore loops of other SM particles due to their small couplings to the Higgs. It is well known that light scalar- and gauge boson-loop contributions to VT suffer from an IR divergence. Certain classes of higher-loop contributions (so-called “daisy diagrams”) need to be resummed to obtain a good approximation for this object at T m, where m is the boson mass [66, 67]. This is achieved by employing the “ring-improved” version of VT , which is obtained from Eq. (3.6) by replacing the

2 2 zero-temperature masses mi(ϕ, s) with thermal masses, m m +Πi(T ), where i → i Πi is the one-loop two-point function at ﬁnite temperature. Recently, Ref. [58]

22 argued that in certain regions of parameter space, further classes of diagrams may need to be resummed. We do not include these effects in the present study, leaving such improvement for future work.

At high temperature, thermal loops generate positive mass-squared for both H and S fields, and the energetically favored configuration has zero background fields, (ϕ, s) = (0, 0). As the Universe cools and thermal masses decrease, this configuration becomes unstable and the fields develop expectation values, eventually ending up in the present vacuum, (ϕ, s) = (v, 0). This can occur in a number of ways. First, we distinguish between a “one-step” transition, in which the singlet field never develops an expectation value; and a “two-step” transition, (0, 0) (0, s) (v, 0). Secondly, each transition may be first-order or → → second-order. In the former case, two distinct local minima of Veff coexist over a range of temperatures. At high temperatures, the “symmetric” minimum is energetically preferred over the “broken” minimum. (In the case of the first step of a two-step transition, “symmetric” and “broken” refer to vacua with s = 0 and s = 0, both of which have unbroken electroweak symmetry.) The two minima 6 become degenerate at the critical temperature, Tc. As the Universe continues to cool, bubbles of the broken-minimum phase are nucleated. Nucleation probability per unit time per unit volume at temperature T is given by [68]

4 P T exp( S3/T ), (3.8) ∼ − where S3 is the action of a critical bubble. We use the CosmoTransitions code [69] to evaluate S3 numerically as a function of temperature. Nucleation temperature TN is the temperature at which the nucleation probability per Hub- ble volume becomes of order one; for electroweak phase transition, this corresponds to [68]

S3/TN 100. (3.9) ≈

23 In this discussion, we use this criterion to estimate TN explicitly throughout the model parameter space. (We assume TN = Tc for second-order transitions, since there is no metastable phase in that case.) Moreover, if a minimum with s = 0 6 develops, we evaluate the nucleation temperatures for both (0, 0) (0, s) and → (0, 0) (v, 0) transitions, to determine which one occurs ﬁrst. This provides → robust discrimination between one-step and two-step transitions. If the transition to EW-breaking vacuum is ﬁrst-order, EWBG scenario is viable only if the baryon asymmetry created at the expanding bubble wall is not washed out by sphalerons inside the broken phase. This requires

v(TN ) > 1, (3.10) TN where v(TN ) is the Higgs vev at the minimum of the effective potential at the temperature TN , i.e. at the time of the phase transition. There is some uncertainty as to the precise numerical criterion for baryon number preservation (see e.g. [70, 71]), with v/T thresholds between 0.6 and 1.4 quoted in the literature.

Varying the EWBG criterion within this range has no noticeable effect on the conclusions of our study, such as the phase diagrams presented below.

Another potentially important aspect of a ﬁrst-order EWPT is the velocity of the expanding bubble wall. The wall experiences outward pressure due to the difference in energy densities of the symmetric and broken vacua, Vvac(sym) − Vvac(br), where Vvac = V0 + V1. It also experiences pressure P from the thermal plasma of particles that it is moving through; since the particles are heavier in the broken phase than in the symmetric one, the effect of this pressure is to slow the wall down. The balance between these two forces determines whether the wall reaches a non-relativistic terminal velocity, or continues to accelerate until it becomes highly relativistic. In the latter case, electroweak baryogenesis cannot occur, since there is not enough time to generate the baryon-antibaryon

24 asymmetry in the region in front of the advancing bubble wall. Thus, to ﬁnd viable models of EWBG one must not only require a strongly ﬁrst-order EWPT, but also demand that the bubble wall does not reach vwall 1 [59]. Relativistic ∼ wall motion occurs if

Vvac(sym) Vvac(br) P > 0, (3.11) − − where the pressure P is calculated assuming vwall 1. This calculation was ∼ performed by Boedeker and Moore in Ref. [59], with the result

2 2 X 2 2 giTN mi (sym) P m (br) m (sym) J˜i , (3.12) i i 4π2 T 2 ≈ i − N where Z ∞ y2dy 1 J˜i(x) = . (3.13) p 2 2 0 y + x e√y +x + ( 1)Fi − We will apply the Bodeker-Moore (BM) criterion, Eq. (3.11), to further constrain the viable parameter space for EWBG. Note that Ref. [59] argued that if the BM criterion is satisfied, the walls will exhibit “runaway” behavior, continuing to accelerate indefinitely once they are relativistic. Very recently, the analysis has been refined to include the effect of transition radiation by charged particles crossing the bubble wall, with the result that the wall velocities are limited [72].

However, the newly established speed limit, γ 1/α, is still highly relativistic, ∼ so that the conclusions regarding viability of EWBG are unaffected.

3.3 Results

We performed a comprehensive scan of the model parameter space, (m0, κ, η). For each point in the scan with viable zero-temperature vacuum structure, we determine the transition history (one-step or two-step); critical temperature and

25 Figure 3.1: Phase transition dynamics in the mS κ plane, with η = ηmin+0.1. Re- − gion I (green): one-step strongly first-order transition; Region II (yellow): two- step transition with strongly first-order electroweak-symmetry breaking step; Region III (red): no thermal phase transition (a would-be two-step transition, but bubbles fail to nucleate); Region IV (purple): same as red, with a would-be one-step transition; Region V (blue): second-order transition; Region VI (gray): no viable EWSB at zero temperature; Region VII (white): non-perturbative regime (η > 10). transition order (for each step, in the case of two-step transition); nucleation temperature, for each first-order transition; and, in the case of first-order EWSB transition, whether or not the BM criterion is satisfied. The results are summarized in a series of two-dimensional slices through the parameter space,

Figs. 3.1, 3.4, 3.5, 3.6. For clarity, we trade the scalar mass parameter m0 for

2 2 1/2 the physical mass of the singlet scalar, mS = (∂ Vvac(v, 0)/∂S ) , in these plots.

26 The main new result is that in large parts of the parameter space where a naive criterion used in previous studies suggests a strongly ﬁrst-order electroweak phase transition, bubble nucleation in fact does not occur at any ﬁnite temperature, so there is no thermal phase transition at all. Instead, the system becomes trapped in the metastable state with unbroken electroweak symmetry, either at the origin or at (0, s0). Eventually, it may transition to the stable

EW-breaking vacuum by tunneling at T = 0, and such models may be viable descriptions of today’s Universe; however, they do not provide viable scenarios for electroweak baryogenesis. Any discussion of collider experiments required to test EWBG must take this constraint into account.

800

600

400 1.55

1.54

200

0 20 40 60 80 100 120

Figure 3.2: Ratio S3/T , where S3 is the critical bubble action, for mS = 300 GeV and κ = 1.55 (red) and 1.54 (yellow). For both points, a two-step ﬁrst- order transition is naively expected. In fact, thermal transition does not occur at κ = 1.55.

A striking example is provided by Fig. 3.1. Following Ref. [56], in this plot

4 4 we ﬁxed the singlet quartic coupling at η = ηmin+0.1, where ηmin = λm0/µ is the minimum value for which (v, 0) is the global minimum of the tree-level poten-

27 tial. Essentially the entire region where a two-step transition would be expected is eliminated due to failure to nucleate bubbles at any temperature. A two-step thermal phase transition can only occur in a very narrow sliver of parameter space at the bottom of this region, shown in yellow in Fig. 3.1. The reason is that in the two-step region, a large potential barrier between the EW-preserving and EW-breaking vacua is present at any temperature, down to T = 0. As a result, the critical bubble action S3 is limited from below, and if this limit is sufficiently large, the bubble-nucleation criterion (3.9) is never satisfied. This is illustrated in Fig. 3.2.2 In contrast, in the one-step region, there is no EW-preserving vacuum at T = 0 at tree level. This guarantees bubble nucleation at finite temperature, unless the couplings are very strong and loop corrections become important. Consequently, most of the one-step region survives this constraint.

The shape of the potential, and hence dynamics of bubble nucleation, depend on the singlet quartic coupling η as well as mS and κ. We find that for larger η, it is easier to find points in the two-step region where the thermal EWPT does occur, and is strongly first-order. The reason is that as η is increased, the critical temperature of the transition between the EW-symmetric and broken vacua increases, and both the height and the width of the potential barrier decrease; see Fig. 3.3. This makes tunneling between the two vacua easier, allowing a thermal phase transition to occur. The effect of varying η on the viable parameter space is illustrated in Figs. 3.4 and 3.5. Note, however, that even at large η, most of the two-step region is eliminated by the requirement of bubble nucleation at non-zero temperature.

2There is some uncertainty as to the precise numerical value of the right-hand side in Eq. (3.9). We use 100 in Figs. 3.1-3.6. We have checked that varying this threshold by 20% does not have a signiﬁcant effect on the phase diagrams. The reason is clear from Fig. 3.2: at the boundary between the regions with and without thermal phase transition, small changes in model parameters lead to large changes in the critical bubble action.

28 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 50 100 150 200

Figure 3.3: Thermal potential at the critical temperature, along the line in ﬁeld space connecting the EW-symmetric and broken vacua, for mS = 300 GeV, κ = 1.8, and two representative values of η, 2.0 (red) and 2.5 (yellow). For both points, a two-step ﬁrst-order transition is naively expected. In fact, thermal transition does not occur at η = 2.0.

Even if this requirement is satisfied, models in which the nucleation temperature TN significantly below the critical temperature Tc are likely to fail the BM criterion for relativistic bubble wall motion. This is because in this case, the symmetry-breaking vacuum would typically have a significantly lower vacuum energy at TN compared to the symmetric vacuum, resulting in a strong outward pressure on the bubble wall. To check this, we implemented the BM criterion,

Eq. (3.11), in our scans. The result, shown in Fig. 3.6, is consistent with expectations. The BM criterion eliminates a region bordering that where no thermal

EWPT occurs, since by continuity this is the region where TN is the lowest. This extra constraint must also be taken into account in the discussion of collider probes of EWBG.

To gain further insight into the reasons that the transition is not completed in

29 Figure 3.4: Phase transition dynamics in the mS κ plane, with η = ηmin + 2.5. − Same labeling and color code as in Fig. 3.1. large part of the parameter space, we plot the potential barrier height (Fig. 3.7), and the potential drop ∆V (Fig. 3.8) between the EW-symmetric and broken vacua at zero temperature. It is clear that the regions where the transition does not occur are characterized by large barrier heights and small potential drops.

This correlation can be used to identify regions where a thermal transition may not occur without a detailed calculation of a ﬁnite-temperature potential.

30 10

6 V II III VI 4

0 1.0 1.5 2.0 2.5 3.0

Figure 3.5: Phase transition dynamics in the κ η plane, with mS = 300 GeV. − Same labeling and color code as in Fig. 3.1.

8 B

6 Second Order PT A 4 No Thermal PT

2 Incorrect T=0 Vacuum 0 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3

Figure 3.6: Phase transition dynamics in the κ η plane, with mS = 300 GeV. In − region B (red) bubble walls accelerate to relativistic speeds and EWBG cannot occur, while in region A (blue) EWBG is possible.

31 10

0.002 6 Second Order PT

0.005 4 0.01

2 0.02 Incorrect T=0 Vacuum

0 1.6 1.8 2.0 2.2 2.4 2.6

Figure 3.7: The height of the barrier between the EW-symmetric and broken vacua at zero temperature, in units of electroweak vev4. The dashed line indicates the boundary between the region where the thermal EWPT occurs (above the line) and where it does not occur (below the line).

3.4 Discussion

We re-considered the dynamics of EWPT in a model with a singlet scalar ﬁeld

S coupled to the SM via a Z2-symmetric Higgs portal, Eq. (3.1). We found that the requirements of thermal EWPT (bubble nucleation at non-zero temperature) and non-relativistic bubble wall motion eliminate much of the parameter space that was previously thought to provide viable EWBG models. In particular, most of the parameter space where a two-step phase transition was thought to

32 10

0.02 6 0.025 Second Order PT 0.015

4 0.01

0.005 2

Incorrect T=0 Vacuum

0 1.6 1.8 2.0 2.2 2.4 2.6

Figure 3.8: Potential difference between the EW-symmetric and broken vacua at zero temperature, in units of electroweak vev4. The dashed line indicates the boundary between the region where the thermal EWPT occurs (above the line) and where it does not occur (below the line). occur, is now eliminated. The effect of the new requirements in the region where a one-step transition was expected is less signiﬁcant.

The model studied here has recently emerged as a useful benchmark for planning the physics program at future colliders. While absence of mixing between doublet and singlet states makes this model challenging to probe at the LHC, Ref. [56] argued that the proposed future facilities will be able to probe the EWBG scenario in this model conclusively. This can be achieved with a combination of Higgs cubic coupling measurements [32], direct Higgs portal searches

33 in channels such as pp V SS, qqSS [56, 57], and a very precise measurement → of σ(e+e− Zh) at electron-positron Higgs factories [73, 27, 74]. The new con- → straints considered here reduce the parameter space with viable EWBG, which should in principle make the colliders’ task easier. However, comparing the predictions for collider observables in Ref. [56] with the new constraints presented here indicates that the newly eliminated parts of the parameter space are the ones with the strongest collider signals. This should not be surprising, since for a given mS, our constraints place an upper bound on κ, while all collider observables deviate from the SM more with growing κ. (Note that T = 0 collider observables are independent of η up to the one-loop order, since η does not enter

Vvac in the present vacuum at one loop). Thus, the sensitivity goals established by previous studies as benchmarks for future colliders remain unchanged.

Our ﬁndings seem to indicate that in this model, a viable two-step ﬁrst-order transition occurs only in a rather special, narrow region of the parameter space, in effect requiring some degree of tuning between the model parameters. This may appear to make this scenario “unlikely”. However, it is important to remember that the parameters of the model may emerge from a more fundamental theory at higher energy scales, which may in fact correlate parameters that we treat as independent. Therefore, it would be incorrect to interpret our results as in any way reducing the motivation for an experimental program that will address the viability of EWBG in this model.

34 CHAPTER 4 A TINY REVIEW OF NEUTRINOS

4.1 Introduction

Neutrinos are some of the least understood particles in the Standard Model, mostly due to their extremely light masses and weak couplings. Detecting neutrinos is far more difﬁcult than other particles, and specialized detectors with very large target masses are required. In contrast, other leptons, quarks, gluons and photons are easily detected in colliders (even if distinguishing and identi- fying them is sometimes difﬁcult). Thus, most of our knowledge of neutrinos and their behaviour is from experiments outside of colliders, which are solely designed to study them. This chapter contains a description of the neutrino as a particle in the Standard Model, the discovery of neutrino masses and experiments that study neutrinos, followed by a review of the ‘See-Saw mechanism’, one of the more popular solutions to the ‘neutrino mass problem’ in the SM.

Neutrinos in the SM are singlets under SU(3)C and U(1)EM , and come in three ﬂavors, like the other leptons. They are part of the lepton doublets under

SU(2)L:       ν ν ν  e  µ  τ  LL,e =   ,LL,µ =   ,LL,τ =   . (4.1) e µ τ because of which the neutrinos couple to the weak charged and neutral currents [75].

g X µ − + CC = ν¯Llγ l W + c.c., (4.2) L −√ L µ 2 l g X µ − NC = ν¯Llγ νLlZµ + c.c. (4.3) L −2 cos θW l

35 In the SM, there are no right-handed neutrinos and hence there are no Dirac masses through the Higgs, as is the case for other fermions including the charged leptons. With this freedom, the mass eigenstates of neutrinos, being degenerate, can be expressed in the same basis as the gauge eigenstates. Thus, unlike in the quark case, the lepton sector does not have any ﬂavor mixing in the SM.

This, of course, turned out not to be true. Oscillation experiments in the past few decades have proven that mixing does indeed happen in the lepton sector, to a much greater degree than the quark sector. This directly implies that at least two flavors of neutrinos have masses, and the SM needs to be modified or extended to accommodate neutrino mass terms. As the evidence for neutrino masses is from flavor-mixing, it is indirect, and the absolute value of masses have not been measured. Measurements exist for mass-squared differences, and it is possible to set an upper bound on the sum of neutrino masses from cosmological considerations. As a result, the lightest neutrino state is allowed to be massless and still be consistent with data.

The neutrinos that are part of the gauge doublets are often called ‘active’ neutrinos, in order to distinguish them from extra states added in many neutrino mass models that do not couple to the gauge bosons directly. These extra states are often called ‘sterile’ neutrinos, and searches for steriles are ongoing.

Sterile neutrinos are a generic prediction of most models, and their possible discovery offers a window into the mechanism that generates the small masses of neutrinos.

An important feature of the Standard Model Lagrangian is that it respects some accidental global symmetries. Accidental symmetries are not imposed,

36 as implied by the name, and are a consequence of gauge symmetry and the particular matter content in the Lagrangian. In the case of the SM, these are baryon number, and individual lepton numbers:

GSM = U(1)B U(1)L U(1)L U(1)L . (4.4) × e × µ × τ

Breaking of these symmetries has measurable consequences, and is an experimental signature of extensions of the SM that break lepton number symmetries to add a mass term for the neutrinos.

4.2 Neutrinos in Numbers

Neutrino flavor oscillations are a direct signature of the neutrinos having non- degenerate masses, since it is the mismatch between gauge and mass eigenstates that enable oscillations. Just as the CKM matrix denotes flavor changing processes in the quark sector, a similar rotation matrix, dubbed the ‘PMNS matrix’, appears in the lepton sector. Due to the extremely low mass differences between neutrino flavors, this effect is not pronounced for the much heavier charged leptons. However, the neutrinos are extremely light and oscillations have very striking appearance or disappearance signatures in experiments that study them.

It is instructive to study the simplest scenario with only two neutrino ﬂavors, and the oscillation probability (in vacuum) in this case can be shown to be [76],

2 2 2 2 ∆m (eV ) L(km) P (να νβ) = sin 2θ sin 1.27 , (4.5) → Eν(GeV)

P (να να) = 1 P (να νβ), (4.6) → − →

37 where θ is the mixing angle that goes into the 2 2 rotation matrix in this setup × and ∆m2 = m2 m2. 1 − 2

This equation reveals a few things. First, measurements of the flux of initial and final neutrinos of particular flavors can be used to derive the corresponding oscillation probability, which can then be used to calculate the mass-squared differences. Thus, oscillations cannot be used to find the absolute masses of neutrinos. Second, the oscillation probability depends on the length or baseline of the experiment and the energy of the neutrino beam. This is very useful, as the two neutrino mass-squared differences currently known are of different scales, and experiments with different baselines and energies can study one or the other by maximizing the corresponding oscillation probability.

Neutrino oscillations have been studied using a wide variety of sources and detection techniques. The crucial requirements are to have high intensity sources and large detectors, as neutrinos interact very weakly with matter. De- tector designs generally tend to have large volumes of liquid, and neutrinos are detected using Cherenkov radiation or scintillation. Solid materials are sometimes also used, with the target material connected to phototubes to detect scin- tillations.

There are four types of experiments when classiﬁed by the type of neutrino source: solar, atmospheric, reactor and accelerator. Electron neutrinos are produced in large quantities in the sun through thermonuclear reactions. Detailed calculations of the ﬂux of these neutrinos have been completed, based on the

‘Solar Standard Model’. Over the last few decades, the Homestake, Kamiokande and Super-Kamiokande experiments have consistently measured a deﬁcit in electron neutrinos from the sun. This was called the ‘solar neutrino problem’

38 Table 4.1: Characteristics of various neutrino experiments, reproduced from [75]. SBL stands for Short Baseline and LBL stands for Long Baseline.

and was one of the first signs of non-zero neutrino masses. Atmospheric neutrinos are generated from the decays of pion and kaons produced from cosmic ray interactions. The fluxes of these neutrinos can be calculated based on details of cosmic rays, but uncertainties are a significant hurdle in this case. Accelerator neutrinos are generally produced by colliding protons with a target material, which produces mesons (pions and kaons) which then decay into neutrinos. Undecayed mesons and charged leptons produced in the process are stopped in the beam dump. Accelerator experiments offer more control, and the baseline and energy of neutrinos can be adjusted to optimize L/E to study particular oscillation scales. Nuclear reactors are strong sources of electron antineutrinos in the MeV energy range, that are produced from nuclear fission of heavy iso- topes. Reactor neutrino experiments study the disappearance of these antineutrinos through inverse beta decays. Characteristic values of the mass-squared differences studied in each type of experiment is shown in Table 4.1.

In particular, solar neutrinos and atmospheric neutrinos were used to determine the quantities ∆m2 and ∆m2 respectively. The sign of ∆m2 is known 21 | 31| 21 to be positive, when matter effects in the sun are appropriately included in the

2 calculation of oscillation probability. The sign of ∆m31 is unknown, and there are two possibilities: m3 > m1 and m1 > m3. These are the normal and inverted

39 Table 4.2: Summary of neutrino oscillation parameters [77]. NO stands for Nor- mal Ordering of neutrino masses, and IO stands for Inverted Ordering. mass orderings for the 3ν model, as shown in Fig. 4.1. While the differences between neutrino masses look stark in the scale shown in the diagram, it is important to remember that the absolute scale of neutrino masses is still unknown.

The only constraint on sum of masses is from cosmology, and current data allows for a quasi-degenerate spectrum, where neutrino masses are roughly the same order in mass, with small differences between them, either in the normal or inverted order. A summary of global analysis of neutrino oscillation parameters in the 3ν model [77] is shown in Table 4.2.

Figure 4.1: Allowed neutrino mass spectra: Normal Ordering (left) and Inverted Ordering (right).

A completely complementary approach to studying neutrinos is through

40 cosmology. Cosmological data such as the large scale structure of galaxies, the cosmic microwave background, type Ia supernovae, and big bang nucleosynthesis (BBN) are sensitive to neutrino masses and the number of neutrino species in the early universe. As neutrinos are relativistic, they free stream in the early universe and can damp density ﬂuctuations. This has subtle effects on large scale structure (LSS) and the cosmic microwave background (CMB). Cur- rently, combined Planck and Baryon Acoustic Oscillation (BAO) measurements give a bound on the sum of neutrino masses:

X mν < 0.13 eV (95%CL). (4.7)

The number of neutrino ﬂavors affects the abundances of light elements produced during BBN, and the effective (or active) number of species can also be constrained using this. Current limits are compatible with three active neutrinos, as is the case in the SM.

There are some caveats to such analyses, as they make certain assumptions about the neutrino models. For example, neutrinos are usually considered stable when calculating such bounds, and getting rid of this assumption can have signiﬁcant effects [78]. Many analyses also make the assumption that neutrinos are roughly degenerate when calculating sum of mass bounds from epochs after the neutrinos have become non-relativistic. This can lead to misleading or inconsistent results. Analysing cosmological data with model priors that are physically motivated (by oscillation experiments and theoretical considerations) can yield a more reliable bound [79].

41 4.3 The See-Saw Mechanism

Broadly, there are two classes of neutrino mass models: lepton number violating models and non-lepton number violating models. The latter contains only

Dirac mass terms, and thus requires right-handed neutrino states. The former category however, admits Majorana mass terms (with or without Dirac mass terms). As the SM only contains left-handed neutrinos, adding Majorana mass terms is a clear path forward. In this case, the mass term violates the accidental lepton number symmetry by two units, with neutrino-less double beta decay being one of the most striking experimental signatures of this breaking.

n p

W −

e− νe

νe

e− W −

n p

Figure 4.2: Feynman diagram for neutrino-less double beta decay, a signature of lepton number violation. A Majorana mass term in the Lagrangian is required for this process to take place.

The ‘see-saw’ mechanism is an elegant and simple model for neutrino masses, which extends the SM with mass terms that are consistent with current experimental bounds, and also offer an explanation as to why the neutrino masses are so small [76]. There are three types of see-saw models:

42 Type I: SM + heavy singlet fermions • Type II: SM + heavy triplet scalar • Type III: SM + heavy triplet fermions •

The general idea behind see-saw models, is to include heavy sterile states with Majorana mass terms and Dirac mass terms, and diagonalizing the neutrino mass matrix will then produce a hierarchy of scales which makes the SM neutrino masses tiny. For example, in type-I see-saw models with three sterile states, the following terms are added to the Lagrangian:

X X 1 ν¯i M ij N j + N¯ c i M ij N j + c.c., (4.8) L D R 2 R N R − L ⊃ i,j i,j where MD is a Dirac mass matrix (from Yukawa couplings to the Higgs), and

MN is a Majorana mass matrix for the sterile NR states. The sums are over the three ﬂavours.

c In the basis (νL,NR), the mass terms can be written as,     c 1 0 MD νL ¯ c     ν¯L NR . (4.9) − L ⊃ 2  T    MD MN NR

In the limit where the eigenvalues of MN are much heavier than the weak scale (and MD), this mass matrix can be diagonalized approximately as,     −1 T 2 0 MD MDMN MD 0 M T   −  D U   U   + 2 , (4.10) T ' O MN MD MN 0 MN where U is a rotation matrix composed of MD/MN and its transpose. To this order in MD/MN , the SM neutrino and heavy sterile neutrino masses are given by,

−1 T mν = Diag [ MDM M ]; mN = Diag [MN ]. (4.11) − N D

43 There are two features that make this setup an elegant solution to the neutrino mass problem. First, the light neutrino masses mν depend quadratically on MD, which when translated to a Higgs Yukawa coupling, implies that the masses are proportional to the square of the Yukawa couplings. Thus, making the Yukawa couplings smaller than one has a much stronger effect of decreasing the mass in the see-saw mechanism compared to the case of a pure Dirac mass

−1 term. Second, the factor of MN suppresses the masses and gives very light neutrinos. A natural choice for the scale of the heavy states is the GUT scale, which gives the correct order of magnitude for neutrino masses with weak scale Dirac masses (or in other words, (1) Yukawa couplings). O

However, there is a major drawback for see-saw models, and that is the dif-

ficulty of experimental verification. Because of the high scale of sterile states, they tend to be inaccessible to colliders. Lepton number violation in double beta decays would be a clear signal of the Majorana nature of neutrinos, but it is yet to be observed. It is also difficult to rule out see-saw models based on non-observation of neutrino-less double beta decays, as large swathes of the parameter space of see-saw models have expected signal strengths far beyond current experimental capabilities.

44 CHAPTER 5 CLOCKWORK NEUTRINOS

5.1 Introduction

Neutrino masses are at least six orders of magnitude smaller than the mass of the electron, and at least twelve orders of magnitude below the scale where all fermion masses are thought to originate, the electroweak scale. The most popular explanation for the smallness of neutrino masses is the see-saw mechanism.

While simple and theoretically attractive, this mechanism depends crucially on violation of lepton number symmetry. At this time, there is no experimental evidence that lepton number is violated, and it is a logical possibility that this symmetry is exact (or broken only by gravitational interactions). In this case, neutrino masses must be Dirac, and an alternative to see-saw is required to generate hierarchically small neutrino masses.

It is possible to generate small Dirac neutrino masses in models with extra dimensions of space. If the left-handed neutrino fields are localized to a brane, along with other fields charged under Standard Model (SM) gauge symmetries, while the right-handed neutrino field propagates in the bulk, the Dirac mass is suppressed by the geometric factor reflecting the small overlap between the left-handed and righ-handed wavefunctions. This idea has been realized in the context of large extra dimensions [80, 81], and in Randall-Sundrum setup [82].

Recently, a new mechanism for generating exponentially small couplings and masses has been proposed, the Clockwork (CW) Mechanism [83]. Among other applications of the Clockwork, it has been suggested that it can be used

45 to generate the observed neutrino masses without hierarchically small parameters in the Lagrangian of the theory. The right-handed neutrino emerges from a chain of four-dimensional fields with nearest-neighbor interactions in the theory space, while the left-handed neutrino is localized in the theory space. The right-handed neutrino zero-mode is exponentially suppressed at the site where the left-handed field resides, leading to an exponentially suppressed mass. The idea is similar to that of extra-dimensional models, and some CW models may be related to 5D constructions (with an appropriately chosen metric profile) by dimensional deconstruction [84]. In this discussion, we will use the four- dimensional point of view.

The goals of this chapter are to further develop the idea of CW mechanism for small neutrino masses, and to explore its phenomenological consequences. (For previous work on CW mechanism applied to neutrino masses, see [85, 86, 87, 88].) In Section 5.2, we present the simplest model that realizes the CW mechanism for neutrinos, which we call the uniform clockwork. This follows closely the model originally proposed in Ref. [83], but generalizes it to fully incorporate the three neutrino flavors of the SM, including flavor-mixing effects required by the observed neutrino oscillations. We also present analytic expressions for the spectrum and couplings of the “excited” CW neutrino states. These expressions are obtained within a perturbation theory in the parameter p, proportional to the Yukawa interaction which couples the SM to the CW sector. An- alytic perturbative expressions provide intuitive understanding of various phenomenologically important quantities. In Section 5.3, we discuss experimental constraints on this model from flavor-changing neutral current process µ eγ → and precision electroweak fits, and delineate the parameter space allowed by the existing data. We find that the excited CW neutrino states may have masses

46 around the weak scale, in the 100 GeV – 1 TeV range, without violating any constraints, with Yukawa couplings of order 10−2 10−1. In Section 5.4, we discuss − the “generalized” CW models, a generalization of the uniform model which allows for site-dependent Dirac masses in the clockwork sector. (An interesting example, the case of randomly drawn masses, has been previously considered in [89]; see also [90].) We give the general condition under which the generalized model produces a hierarchically small neutrino mass, and consider two explicit examples, “Linear Clockwork” models. Spectra and couplings of the excited CW neutrinos in these models are qualitatively different from the uniform CW case. As in the uniform case, we consider experimental constraints on the linear CW models, and find that in one of the linear CW models (LCW1) weak-scale excited neutrinos are allowed. Finally, in Section 5.5, we discuss collider phenomenology of the uniform CW and LCW1 models. We find that excited neutrinos can be produced at the LHC and the proposed next-generation electron-positron colliders with significant rates. We identify signatures of CW neutrino production at hadron and lepton colliders, and perform Monte Carlo studies of these signatures and their SM backgrounds. We find that with the current data set, the LHC does not yet have the sensitivity to this model. Some spectra with relatively light CW neutrinos can be probed at HL-LHC, but sensitivity decreases rapidly as the CW mass scale is increased. At lepton colliders, the situation is more promising, and the CW neutrinos can typically be discovered with realistic integrated luminosity as long as they are within the kinematic reach of a given collider. In Section 5.6, we summarize our findings and discuss possible directions for future studies of CW neutrino phenomenology.

47 χ0 χ1 χ2 χN 1 χN −

yv m1 m2 mN 1 mN · · · −

νL (mq)0 ψ1 (mq)1 ψ2 (mq)2 ψN 1 ψN − (mq)N 1 −

Figure 5.1: Pictorial representation of the clockwork sector with right-handed zero mode. Single solid lines denote Dirac masses, while the double solid line denotes the Yukawa couplings involving the SM Higgs boson H. In the uniform CW model, mi m and qi q for all i. ≡ ≡ 5.2 Uniform Clockwork Neutrino Model

The uniform clockwork neutrino model supplements the SM particle content with the following “clockwork ﬁelds”:

N left-handed Weyl fermion ﬁelds ψi, where i = 1 ...N; •

N + 1 right-handed Weyl fermions χj, where j = 0 ...N. •

All clockwork ﬁelds are singlets under the SM gauge groups. The La- grangian is N X † † cw = kin m ψi χi q ψi χ(i−1) + h.c. , (5.1) L L − i=1 − where m is the mass parameter (“clockwork mass”) and q is a dimensionless number of order one. We will assume q > 1, which, as we will see below, results in exponential suppression of neutrino mass. In this chapter, we will primarily consider m at the weak/TeV scale, motivated by the desire to accommodate the observed neutrino masses without introducing new scale hierarchies. In the uniform model, m and q are the same for each term in the sum; the model has a “translational” symmetry, i i + 1, broken only by the edge terms. The parti- → cle content and mass terms of the model are represented pictorially in Fig. 5.1,

48 where we represent each pair of left- and right-handed ﬁelds as a “site” (a gray circle), and each non-diagonal mass term as a “link” (a red line).

To incorporate three generations of neutrinos, we promote each of the clockwork fields to a flavor triplet, ψiα and χjα, α = 1 ... 3, and assume that all mass terms in Eq. (5.1) are diagonal in flavor space. With this assumption, the model has a global flavor SU(3)cw symmetry under which all clockwork fields trans- form in fundamental representation.

The clockwork sector is coupled to the SM through the Yukawa coupling connecting the “extra” right-handed clockwork fermion χ0 to the SM lepton doublet L:

αβ † Yuk = Y χ (H Lβ) + h.c. (5.2) L 0α · where H is the SM Higgs doublet, and Y is the matrix of Yukawa couplings.

The Yukawa coupling explicitly breaks the SU(3)cw SU(3)L flavor symmetry; × by construction, it is the only source of such breaking. In this sense, the model incorporates minimal flavor violation in the neutrino sector. This choice is motivated by non-observation of lepton flavor violating (LFV) processes, and is advantageous from the point of view of minimizing experimental constraints. We will work in a basis where the Yukawa matrix is diagonal:

Y = diag (y1, y2, y3). (5.3)

This assumption entails no loss of generality, since one can always perform flavor SU(3) rotations ψi Vcwψi, χj Vcwχj, and L VLL, to diagonalize → → → Y without affecting the Lagrangian in Eq. (5.1). Note that in this basis, lepton couplings to the SM W boson are not flavor-diagonal; their flavor structure is described by the usual PMNS matrix.

49 To understand the mass spectrum of the model, ﬁrst consider the limit Y = 0. Deﬁning “neutrino vectors”

T Ψ = (νL, ψ1, ψ2, . . . ψN ) ;

T X = (χ0, χ1, χ2, . . . , χN ) , (5.4) the mass term has the form Ψ†MX˜ + h.c., where the mass matrix is given by   0 0 0 0 0  ···     q 1 0 0 0 − ···    M˜ = m  0 q 1 0 0 . (5.5)    − ···   ......   ......    0 0 0 q 1 · · · − (Here and below, tildes indicate the Y = 0 limit.) The mass matrix can be diagonalized by a pair of unitary rotations, U˜L and U˜R, with the mass eigenstates ˜L N and ˜R given by N Ψ = U˜L ˜L,X = U˜R ˜R. (5.6) N N Translational symmetry of the model allows for exact, analytic diagonalization of the mass matrix. Since det M˜ = 0, there is a massless eigenstate, the zero- mode. The spectrum of massive modes is given by

1/2 2 kπ m˜ k = λ m, λk = 1 + q 2q cos , k = 1, 2, ,N. (5.7) k − N + 1 ···

Rotation of the right-handed ﬁelds to the mass eigenbasis is given by s q2 1 1 U˜ j0 = − , j = 0, ,N, (5.8) R q2 q−2N qN−j ··· − for the zero mode, and s ˜ jk 2 (N j)kπ (N j + 1)kπ UR = q sin − sin − , j = 0,...,N; k = 1, N, (N + 1)λk N + 1 − N + 1 ··· (5.9)

50 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14

Figure 5.2: Composition of the left-handed (left panel) and right-handed (right panel) mass eigenmodes in terms of the original clockwork ﬁelds in the uniform clockwork model with N = 15, m = v, q = 4.887, and y = 0.01. for the massive states. For the left-handed ﬁelds,

˜ 00 ˜ 0j ˜ j0 UL = 1, UL = UL = 0; r 2 jkπ U˜ jk = sin , j, k = 1, ,N. (5.10) L N + 1 N + 1 ···

In terms of the pictorial representation of the clockwork in Fig. 5.1, the massive left- and right-handed eigenmodes appear “delocalized”, mixing the ﬁelds at all sites in roughly equal measure. On the other hand, the zero mode is strongly localized. The left-handed part of the zero mode corresponds exactly to the ﬁeld

νL. The right-handed zero mode consists mainly of the ﬁeld χN , with rapidly decreasing admixtures from the ﬁelds located further to the left. In particular,

N the contribution of χ0 is suppressed by a factor of 1/q . (These features are illustrated in Fig. 5.2.) When the Yukawa coupling is turned on, the resulting

Dirac mass of the pseudo-zero mode is suppressed by the same factor, yielding an exponentially small neutrino mass for moderate values of q and N. In this way, the clockwork mechanism generates a small Dirac neutrino mass without small input parameters.

When a Yukawa coupling is present and the Higgs acquires a vev, the mass

51 matrix has the form   pα 0 0 0 0  ···     q 1 0 0 0 − ···    Mα = m  0 q 1 0 0 , (5.11)    − ···   ......   ......    0 0 0 q 1 · · · − where α = 1 ... 3 is the ﬂavor index, and we deﬁned

yαv pα = . (5.12) √2m

Here v = 246 GeV is the Higgs vev. The spectrum consists of N + 1 Dirac neutrinos for each ﬂavor:

j = ( L j, R j), j = 0 ...N (5.13) N N N where

Ψ = UL L,X = UR R, (5.14) N N and we suppressed the ﬂavor index. Since the Yukawa coupling explicitly breaks translational symmetry of the mass terms, it is no longer possible to obtain the spectrum and determine the rotation matrices UL and UR analytically. Numerical diagonalization can always be performed. However, it is also useful to obtain approximate formulas, valid in the situation when Yukawa coupling is small enough to be treated as a perturbation. The perturbation theory is developed systematically in Appendix 5.A. The lightest mass eigenstate is the pseudo-zero mode whose mass vanishes in the absence of the Yukawa. This state is identiﬁed with the experimentally observed (or “active”) neutrino. It has a mass (up to corrections of order p4)

2 1/2 2 N ! pα q 1 pα X Ck m0,α = m N 2 −−2N 1 , (5.15) q q q − 2(N + 1) λk − k=1

52 where α = 1 ... 3 labels the three active neutrino mass eigenstates (this index is not summed over when repeated), and

2 2q 2 Nkπ Ck = sin . (5.16) λk N + 1

yv As expected, we obtained m0 N , allowing to generate the observed neu- ∼ q trino mass scale with y 1, q a few, N 10. Note that at leading order in ∼ ∼ ∼ the perturbative expansion, the active neutrino masses are independent of m. The remaining N mass eigenstates, which we will call clockwork neutrinos, have masses (again up to corrections of order p4)

2 1/2 pα Ck mk,α = mλk 1 + , k = 1 ...N. (5.17) 2(N + 1) λk

Clockwork neutrino masses are of order m, typically around the weak/TeV scale. It can be easily seen from Eq. (5.7) that the spectrum consists of N states with masses in a band between (q + 1)m and (q 1)m (up to corrections of − order p2); a sample spectrum is shown in Fig. 5.3. Note that the width of the mass band is independent of N, so the states become more closely spaced with growing N: the splitting between neighboring clockwork mode masses is of order ∆m 2m/N. The perturbation theory developed in Appendix 5.A is valid ∼ only if the Yukawa shift in these masses is small compared to the splitting. This yields p 1 as a plausible condition for validity of the perturbation theory.

Rotations UL and UR that diagonalize the mass matrix have the form

ULα = U˜L (1 + ∆Lα) ,URα = U˜R (1 + ∆Rα) . (5.18)

For small p, the rotation matrices can be computed analytically. Up to correc-

53 3000 — — 2500 — — — — — — 2000 — — — — — — 1500 — — — — — — — — — 1000 — — — — — 500 — — — — 0 —

Figure 5.3: Spectrum of clockwork neutrinos in the uniform CW (left) and generalized Linear CW1 (center) and Linear CW2 (right) models. (For discussion of generalized CW models, see Section 5.4.) In all cases, N = 15, m = v, and y = 0.01; q = 4.887 (uniform), 0.76 (LCW1) and 0.73 (LCW2). The parameters −2 were chosen so that the pseudo-zero mode neutrino mass is mν = 8 10 eV in · all cases. tions of order p3, we obtain s 2 i0 0i 2 1 q 1 √Ci ∆Rα = ∆Rα = pα 2(N+1)− , i = 1 ...N; − − N + 1 q 1 λi − 2 p ij pα CiCj ∆Rα = , i, j = 1 . . . N, j = i; N + 1 λj λi 6 kk − ∆Rα = 0, k = 0 ...N, (5.19) for the right-handed rotation, and r i0 0i 1 Ci ∆Lα = ∆Lα = pα , i = 1 ...N; − N + 1 λi 2 s p ij pα λi CiCj ∆Lα = , i, j = 1 . . . N, j = i; N + 1 λj λj λi 6 − 2 ii pα Ci ∆Lα = , i = 1 ...N; −N + 1 2λi 2 N 00 pα X Ck ∆Lα = , (5.20) −N + 1 2λk k=1 for the left-handed rotation. We checked that UL and UR deﬁned by these formulas are unitary up to terms of order p3.

54 Phenomenology of the clockwork neutrino sector is controlled by its contribution to weak currents. The charged current Lagrangian is

+ µ+ CC = gW J + h.c., (5.21) L µ W where g is the SM weak coupling, and

N µ+ Vαβ µ X Vαβ µ 0j J = eαγ νLβ = e¯αγ (ULβ) PL jβ. (5.22) W √ √ 2 j=0 2 N

Here Vαβ is the standard PMNS matrix describing ﬂavor mixing in the neutrino

1−γ5 sector, and PL = 2 is the left-handed projector. For small p, we obtain

N ! µ+ Vαβ µ X 3 J = eLαγ PL κ0β 0β + κjβ jβ + (p ) , (5.23) W √ 2 N j=1 N O where

2 N pβ X Ck κ0β = 1 ; − N + 1 2λk k=1 s 1 Cj κjβ = pβ , j = 1 ...N. (5.24) − N + 1 λj

Physically, κ0 = 1 corresponds to a shift in the active neutrino charged current 6 coupling, while κj induce couplings of clockwork neutrinos to the SM electron and W boson. The ﬁrst effect occurs at (p2), while the second effect occurs at O (p). Both effects are ﬂavor-dependent. Note that κ2 + P κ2 = 1, as required O 0 j j by unitarity.

Neutral current (NC) interactions are described by

g µ NC = ZµJZ , (5.25) L cos θw where θw is the SM Weinberg angle, and

N µ 1 µ 1 X µ † j0 0k J = νLαγ νLα = jαγ (U ) (ULα) PL kα. (5.26) Z 2 2 N Lα N j,k=0

55 For small p, the active-neutrino NC has the form

N ! µ 1 µ 1 X 3 J = 0αγ PL η0α 0α + ηjα jα + h.c. + (p ), (5.27) Z 2 2 N N j=1 N O where

2 N pα X Ck η0α = 1 ; − N + 1 λk k=1 s 1 Cj ηjα = pα , j = 1 ...N. (5.28) − N + 1 λj

Physically, η0 = 1 corresponds to a shift in the coupling of active neutrinos to 6 the Z boson, while ηj terms induce off-diagonal couplings of the Z to an active and a clockwork neutrino.

5.3 Experimental Constraints

The uniform clockwork model has 6 parameters: m, q, N, and the three Yukawa couplings yα. (Equivalently, Yukawa couplings can be traded for parameters pα using Eq. (5.12).) Three combinations of these parameters correspond to active neutrino masses m0,α. Experimentally, only the two mass splittings have been measured so far: ∆m2 = m2 m2 = 7.2 10−5 eV2 and ∆m2 = m2 m2 = 21 0,2 − 0,1 · 32 0,3 − 0,2 2.5 10−3 eV2, while the overall mass scale is unknown. We will consider ± · two possibilities: the normal spectrum, with m0,1 = 0 and m0,3 m0,2; and the P degenerate spectrum, with m0,1 m0,2 m0,3 and m0,α = 0.2 eV [75]. The de- ≈ ≈ α generate spectrum corresponds to the largest values of active neutrino masses consistent with cosmology. These two choices correspond to two possible tex- tures in the Yukawa couplings: hierarchical y3 y2 y1 and quasi-degenerate y3 y2 y1. (In the case of inverted hierarchical spectrum, y2 y3 y1, ∼ ∼ ∼

56 clockwork phenomenology is similar to the normal spectrum case.) Once the spectrum is chosen, three combinations of the six parameters are ﬁxed. In this section, we will discuss experimental constraints on the remaining parameters.

5.3.1 Lepton Flavor Violation

As we saw in the previous section, the clockwork model entails ﬂavor- dependent shifts in the CC and NC couplings of SM leptons. Flavor-dependent couplings of SM leptons to massive clockwork neutrinos are also introduced. These effects induce lepton-ﬂavor violating (LFV) processes. The tightest experimental constraint1 on such effects is from the non-observation of the decay

µ eγ, whose branching ratio is currently constrained to be at most 4.2 10−13 → × at 90% c.l. [93].

In the clockwork model, the µ eγ branching ratio is → 3α Br(µ eγ) = 2 , → 8π |A| 3 N 2 X X ∗ 0j 2 mj,α = VµαV (ULα) F , (5.29) eα m2 A α=1 j=0 | | W where the loop function is given by [94, 95] 1 F (x) = 10 43x + 78x2 49x3 + 4x4 + 18x3 log x . (5.30) 6(1 x)4 − − − Within the small-p perturbation expansion developed in the previous section, the ﬁrst non-vanishing contribution to occurs at order p2. This contribution A can be conveniently written as

2 2 ∗ 2 ∗ 2 y3v Ve3Vµ3∆m32 Ve1Vµ1∆m21 = 2 −2 (m, q, N) , (5.31) A 2m · m0,3 ·F

1Currently, constraints from the decay µ eee and the µ e conversion are subdominant → → to µ eγ, but the situation may change with the next round of experiments [91, 92]. →

57 1 1

PEW μ →eγ

0 0 -3 μ →eγ 10 10-3

2.8 TeV 2.8 TeV 2.4 TeV -1 2.4 TeV -1 2 TeV 2 TeV 1.6 TeV 1.6 TeV 10-4 10-4 1.2 TeV 1.2 TeV 800 GeV 800 GeV

400 GeV 400 GeV -2 -2 100 GeV 100 GeV 10-5 10-5

-3 -3 200 400 600 800 1000 200 400 600 800 1000

Figure 5.4: Constraints on the parameter space of the uniform clockwork model from µ eγ and precision electroweak ﬁts. Left panel: normal (hierarchical) → spectrum of active neutrinos. Right panel: degenerate spectrum of active neutrinos. Dashed/red lines indicate the mass of the lightest clockwork neutrino, while dot-dash/blue lines indicate the coupling of this state to the SM gauge currents. where N 2 1 X Ck m λk (m, q, N) = F 2 F (0) . (5.32) F N + 1 λk m − k=1 W The expression in the square brackets is (0.1 1) depending on the assumed O − neutrino spectrum, while (0.1) for typical clockwork parameters. The F ∼ O experimental bound on µ eγ then roughly implies →

y3v 10−2. (5.33) m .

Either a mild hierarchy between v and m, with clockwork states around 10 TeV, or a Yukawa coupling of order 10−2, are necessary to satisfy this bound. In either case, the bound does not invalidate the original motivation for the clockwork model, since small parameters of the required size are by no means unusual in the SM. The second case, m v and y 10−2, is especially interesting from the ∼ ∼ phenomenological point of view, since the clockwork states are light enough to be produced at the LHC and the proposed lepton colliders. We will consider their collider phenomenology in Section 5.5.

58 Implications of non-observation of µ eγ for the model parameter space → are illustrated by Fig. 5.4. In these plots, we ﬁx the active neutrino spectrum (as discussed at the beginning of this section), and choose N = 20. We choose the clockwork mass m and the Yukawa coupling y3 as the remaining two degrees of freedom to describe the model parameter space, and present the constraints in terms of these parameters. In agreement with the intuition from Eq. (5.33), we observe that clockwork neutrinos at the weak scale, (100) GeV, can be consis- O −2 tent with the µ eγ constraint for moderately small Yukawas, y3 10 . → ∼

The µ eγ rate can also be computed without resorting to small-p perurba- → tion theory, by diagonalizing the mass matrix numerically and using Eq. (5.29). We ﬁnd that the constraints derived using this procedure are in excellent agreement with the results of a perturbative analysis. More generally, we ﬁnd that the small-p perturbation theory works well throughout the part of the parameter space allowed by the µ eγ constraint. →

5.3.2 Precision Electroweak Constraints

In the clockwork model, couplings of the active neutrinos to the SM gauge currents are shifted away from the SM values. This effect is described by shifts of

κ0 and η0 parameters away from 1, see Eqs. (5.24) and (5.28). Such shifts affect precision electroweak (PEW) fits: for example, κ0 = 1 modifies the lifetime of 6 the muon, while η0 = 1 modifies the invisible width of the Z boson. In general, 6 these shifts are flavor-dependent. In the case of normal active neutrino spectrum, the flavor-dependence is of the same order as the overall effect; in this situation, we expect that the LFV constraints such as µ eγ are much stronger →

59 than the ﬂavor-diagonal PEW constraints. On the other hand, in the case of degenerate spectrum, the ﬂavor-dependence in κ0 and η0 is small compared to their overall size. In this case, it is not a priori obvious whether LFV or PEW constraints would dominate.

To derive the PEW constraint, we used the three best-measured PEW observ-

0 ables (mZ , α and Γµ) as inputs to fix the underlying SM parameters (g, g and v), and performed a χ2 fit to the other PEW observables listed in Ref. [75]. Note that a shift in charged-current coupling κ0 affects the relation between Γµ and v; this effect was consistently taken into account in the fit. The 95% c.l bound on the clockwork parameter space imposed by the PEW fit is shown in Fig. 5.4. We conclude that even in the degenerate spectrum case, the LFV bounds on the model parameters are currently stronger than the PEW constraint.

5.4 Generalized Clockwork Neutrinos

The uniform clockwork model, proposed in Ref. [83] and developed in detail in Section 5.2, is only one representative of a much broader class of clockwork models that provide an exponentially small Dirac neutrino mass. As a more general example, consider a model with the same set of clockwork ﬁelds with diagonal and nearest-neighbor mass terms as before, but allow the nearest- neighbor (link) mass to vary along the clockwork chain. Using the same no-

60 tation as in Section 5.2, the mass matrix for each neutrino ﬂavor α is given by   pα 0 0 0 0  ···     q1 1 0 0 0 − ···    Mα = m  0 q 1 0 0 , (5.34)  2   − ···   ......   ......    0 0 0 qN 1 · · · − where qi are dimensionless parameters. If the Yukawa coupling is turned off, pα = 0, this mass matrix has zero determinant, and there is a massless zero- mode. The left-handed component of the zero mode is identical to νL. The right-handed component is a linear combination of the clockwork ﬁelds:

N X R0 = viχi, (5.35) N i=0 where v is the eigenvector of M corresponding to the zero eigenvalue:     0 0 0 0 v0          q1 1   v1  −         0 q 1   v  = 0. (5.36)  2   2   −     . ..   .   . .   .      qN 1 vN − Solving these linear equations iteratively yields

i Y vi = v0 qj . (5.37) j=1

The element v0 is unconstrained by the eigenvalue problem, but is ﬁxed by the normalization condition vT v = 1, which yields

1 1 v0 = < . (5.38) p 2 2 2 1 + q + (q1q2) + + (q1q2 qN ) q1q2 qN 1 ··· ··· ··· As long as all (or most of) qi’s are larger than one, the admixture of the ﬁeld χ0 in the right-handed zero-mode is suppressed “exponentially” (i.e. by the product

61 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14

Figure 5.5: Composition of the left-handed (left panel) and right-handed (right panel) mass eigenmodes in terms of the original clockwork ﬁelds in the Linear CW 1 model (top line) and Linear CW 2 model (bottom line). In both cases, N = 15, m = v, and y = 0.01; q = 0.76 in the LCW1 model and 0.73 in the LCW2 model.

of qi’s). When the Yukawa is turned on, the zero-mode acquires an exponentially suppressed mass: at leading order in the small-p expansion,

yαv m0,α = mpαv0 < . (5.39) √2q1q2 qN ··· The observed hierarchy between the weak scale and the neutrino masses can be generated, without introducing small or large parameters, for a broad variety of

qi choices. { }

Smallness of the right-handed zero mode at the SM site is the common feature of all such models. However, the properties of excited clockwork states can vary drastically depending on the model. In particular, we observed that in the uniform model, the clockwork spectrum consists of a band of states, centered at the clockwork scale m and separated by mq/N. These clockwork states are ∼

62 delocalized, mixing ﬁelds at all sites in roughly equal measure. In other clockwork models, these features may be quite different. As a concrete example, consider two Linear Clockwork models:

Linear CW 1 : qi = qi, i = 1 ...N;

Linear CW 2 : qi = q(N + 1 i), i = 1 ...N; (5.40) − where q > 1 is no longer required as long as Πqi 1. Sample spectra of CW neutrino modes in these two models are shown in Fig. 5.3; unlike the uniform model, the states are no longer confined to a relatively narrow mass gap, but instead are spread out in mass similar to traditional Kaluza-Klein theories (though unlike KK theories, the number of modes is finite). The composition of the right- handed components of mass eigenmodes in terms of the original clockwork fields in these models is illustrated in Fig. 5.5. In both cases, the zero mode is exponentially suppressed at the SM (leftmost) site, as expected. Contrary to the Uniform CW, each excited mode is to a good approximation localized at a single site. In the LCW1 model, the lightest clockwork mode is localized on the SM site, while in LCW2, the heaviest clockwork mode is localized on the SM site. These modes dominate the phenomenology, since couplings of all other modes to the SM are strongly suppressed.

Experimental constraints on the Linear CW models are shown in Fig. 5.6. In both cases, we ﬁxed N = 10, while three more parameters are ﬁxed by the choice of the active neutrino mass spectrum, as in the uniform model case. In the LCW1 model, constraints on the lightest clockwork state are somewhat weaker than in the uniform model: for the same mass, the allowed coupling of this state to the SM gauge currents is about one order of magnitude stronger in the LCW1 compared to the uniform model. The main reason for this is that in the LCW1

63 1 1

μ →eγ PEW

0 0 μ →eγ 10-2 10-2

3.2 TeV 3.2 TeV -1 -1 2.6 TeV 2.6 TeV 2 TeV 2 TeV 1.6 TeV 10-3 1.6 TeV 10-3 1.2 TeV 1.2 TeV 800 GeV 800 GeV -2 400 GeV -2 400 GeV 100 GeV 100 GeV 10-4 10-4 -3 -3 200 400 600 800 1000 200 400 600 800 1000

1 1 10-2 μ →eγ μ →eγ PEW 10-2

0 0 -3 10 10-3 35 TeV 30 TeV 30 TeV -1 25 TeV -1 25 TeV 20 TeV 20 TeV 15 TeV 10-4 15 TeV 10-4 10 TeV 10 TeV 5 TeV 5 TeV -2 -2 2 TeV 2 TeV

10-5 10-5

-3 -3 200 400 600 800 1000 200 400 600 800 1000

Figure 5.6: Top row: Constraints on the parameter space of the Linear CW 1 model from µ eγ and precision electroweak ﬁts. Left panel: normal (hier- → archical) spectrum of active neutrinos. Right panel: degenerate spectrum of active neutrinos. Dashed/red lines indicate the mass of the lightest clockwork neutrino, while dot-dash/blue lines indicate the coupling of this state to the SM gauge currents. Bottom row: same, for the Linear CW 2 model. In the case of Linear CW 2 model, the masses and couplings of the heaviest clockwork state are plotted, since other clockwork states have strongly suppressed interactions with the SM. model, the lightest state alone dominates the constraints, while in the uniform model, all clockwork states give comparable contributions, yielding a stronger constraint on each one. In the LCW2 model, the state that has a signiﬁcant coupling to the SM is at the top of the clockwork spectrum, and it tends to be quite heavy (in a few-TeV range) for models consistent with experimental constraints. As a result, LCW2 model is not an interesting target for TeV-scale collider phenomenology.

64 Name N q m y m1 g1 mN (GeV) (GeV) (GeV) U100 20 3.45 40 [0.0207, 0.0209, 0.0269] 98.6 0.0014 177.7 U400 16 5.00 100 [0.0541, 0.0546, 0.0704] 402.1 0.0011 598.6 U750 17 4.70 200 [0.0947, 0.0956, 0.1232] 743.9 0.0010 1137.5 U1000 18 4.40 310 [0.1362, 0.1376, 0.1773] 1059.5 0.0009 1670.6

Table 5.1: Benchmark points (BPs) for the uniform clockwork model. First column: BP name. Next 4 columns: model input parameters. Last 3 columns: mass of the lightest CW neutrino; its coupling to weak current; and mass of the heaviest CW neutrino.

Name N q m y m1 g1 mN (GeV) (GeV) (GeV) G100 10 2.60 40 [0.0178, 0.0180, 0.0232] 101.6 0.0182 1044.7 G300 11 2.05 160 [0.0373, 0.0377, 0.0485] 315.6 0.0122 3631.2 G750 11 2.20 360 [0.0811, 0.0819, 0.1055] 765.9 0.0109 8760.9

Table 5.2: Benchmark points (BPs) for the generalised clockwork model (LCW1). First column: BP name. Next 4 columns: model input parameters. Last 3 columns: mass of the lightest CW neutrino; its coupling to weak current; and mass of the heaviest CW neutrino.

5.5 Collider Phenomenology

We have established that current ﬂavor and PEW constraints do not preclude the possibility that the clockwork neutrino states are within kinematic reach of the LHC and future colliders currently under discussion. In this section, we will study the associated phenomenology.

For the collider study, we selected 4 benchmark points (BPs) in the uniform clockwork model, listed in Table 5.1, and 3 BPs in the generalized model LCW1, listed in Table 5.2. The lightest clockwork states at these BPs span the range between 100 GeV and 1 TeV, making them realistic targets for the current and near-future colliders. All BPs are allowed by the existing LFV and precision

65 ℓ ℓ

ν¯ j p e− Nk W Nk W ℓ j W Z

p ℓ e+ ν

Figure 5.7: Production of heavy clockwork modes k at hadron (left) and lepton N (right) colliders. electroweak constraints.

The simplest processes for heavy clockwork neutrino production involve s-channel exchange of electroweak gauge bosons, illustrated in Fig. 5.7.2 For hadron colliders, we focus on the W ∗ exchange process, since an additional charged lepton in the ﬁnal state improves observability of the signal. Once produced, k states promptly decay, with the dominant decay modes `W and N 0Z. (For k > 1, cascade decays involving intermediate CW modes may N be kinematically allowed, but the corresponding couplings are sub-dominant in p expansion, and branching ratios are small.) For our study, we focus on the charged-current decay `W . The uniform and generalized (LCW1) N → clockwork models were implemented in FeynRules [97, 98]. The signal and relevant backgrounds were simulated using MadGraph@aMC [99, 100]. The parton-level events are passed to Pythia8 [101] for hadronization and then to Delphes3 [102] to incorporate detector effects and jet reconstruction.

For hadron colliders, the signatures of CW neutrino production are 3` + E/T and 2` + 2j, corresponding to leptonic and hadronic W decays respectively.

2The pair-production process, which dominates for clockwork states in models explaining quark mass hierarchy [96], is strongly suppressed in this case since the relevant coupling involves two powers of the small parameter p.

66 + − pp 3` + E/T e e `νjj BP → → 14 TeV 100 TeV 250 GeV 500 GeV 3 TeV U100 0.66 4.2 7.4 12.8 3.9 G100 1.40 8.6 4.3 7.4 1.34 U400 3.0 10−3 0.032 – 0.81 7.6 × G300 0.014 0.12 – 5.8 6.1 U750 2.6 10−4 5.0 10−3 – – 5.9 × × G750 5.0 10−4 8.0 10−3 – – 7.9 × × U1000 5.0 10−5 1.7 10−3 – – 2.3 × × Table 5.3: Cross sections of CW neutrino signatures at hadron and lepton colliders, before selection cuts (in fb). Acceptance cuts have been applied at the parton level: ∆R 0.4 for all visible object pairs; pT (`) 10 (20) GeV for ≥ ≥ hadron (lepton) colliders; pT (j) 20 GeV; ηj < 5; η` < 2.5. ≥ | | | |

(Here ` = e or µ; we do not include taus in the analysis.) The trilepton signature has a slightly smaller rate but significantly lower backgrounds. Signal cross sections in this channel, before selection cuts, are listed in Table 5.3. These are total cross sections, summed over CW neutrino flavor and mode number k. Note that the structure of CW neutrino couplings ensures that two of the leptons form a same-flavor, opposite-charge (SFOC) pair, while the third lepton (from W decay) may be the same or opposite flavor. The main irreducible background to this search is pp WZ. LHC experiments have published searches → −1 in the 3` + E/T channel based on 35 fb integrated luminosity at √s = 13 TeV. Benchmark points U100 and G100 predict a few tens of signal events in this sample, before selection cuts. For all other benchmark points, cross sections are too small to get an appreciable number of events. The CMS collaboration’s search for sterile neutrino [103] was optimized for a signature similar to our CW neutrino, and is expected to have the best sensitivity. We have recast this search to estimate the limits on the CW neutrino model. We find that the benchmark points U100 and G100 are not ruled out by this search. Since U100 and G100 points provide nearly-maximal LHC signals consistent with LFV and PEW con-

67 G100- 14 TeV G100- 14 TeV Signal 0.25 Signal 0.4 Background Background 0.20 0.3

0.15

0.2 0.10 Normalized to one Normalized to one

0.1 0.05

0.00 0.0 0 50 100 150 200 250 300 0 50 100 150 200 250 300

s12 s23 Figure 5.8: Distributions of CW neutrino production signal (blue) and background (orange) events in s12 (left panel) and s23 (right panel) at the LHC. The signal was simulated at the G100 benchmark point.

LHC - 14 TeV ( = 3000 fb−1) U100 G100 L σ (fb) with parton-level cuts 0.66 1.39 # of signal events 1965 4180 Cuts: S (fb) BG (fb) S (fb) BG (fb) Pre-selection cuts 0.29 93 0.62 93 s12 < 80 GeV 0.13 13.5 0.38 13.5 s23 < 80 GeV 0.12 8.7 0.37 8.7 Mrec < 120 GeV – – 0.29 3.8 S/B 0.01 0.077 S/√B 2.3 8.2 S/√S + B 2.3 7.9

Table 5.4: Cut ﬂow for the search for CW neutrino production in the 3` + E/T channel at the HL-LHC ( = 3 ab−1). L straints in their respective models, we conclude that the current LHC constraints are not yet competitive with those discussed in Section 5.3.

Looking into the future, HL-LHC is expected to collect a = 3 ab−1 data set L at √s = 14 TeV.We have estimated the sensitivity of a simple search in the 3`+E/T channel with this data set. For this search, events are processed as follows. First, if there is only one SFOC lepton pair in the event, the leptons in this pair are labeled 1 and 2, while the remaining (opposite-ﬂavor) lepton is labeled 3. If there are multiple SFOC lepton pairs, the pair with the highest average pT is identi- | |

68 U100- 14 TeV G100- 14 TeV 0.20 Signal Signal 0.4 Background Background 0.15 0.3

0.10 0.2 Normalized to one Normalized to one 0.05 0.1

0.00 0.0 0 50 100 150 200 250 300 0 50 100 150 200 250 300

Mrec Mrec Figure 5.9: Distributions of CW neutrino production signal (blue) and background (orange) events in Mrec at the U100 (left panel) and G100 (right panel) benchmark points.

ﬁed with leptons 1 and 2, and the remaining lepton is labeled 3. Furthermore, labels 1 and 2 are assigned so that the leptons 2 and 3 form an opposite-sign pair. With this labeling, lepton 1 predominantly corresponds to the particle produced in association woth the , lepton 2 to the particle produced directly in N the decay `W , and lepton 3 to the particle produced in W decay. Invariant N → 2 2 masses of the opposite-sign lepton pairs, s12 = (p`1 + p`2) and s23 = (p`2 + p`3) , are a useful signal discriminant, peaking sharply around mZ in the background

(see Fig. 5.8 and Table 5.4). Further, neutrino four-momentum pν can be fully re- constructed3 using the conservation of transverse momentum and requirements

2 2 2 pν = 0 and (pν + p`3) = mW . It can then be used to calculate the mass of the CW

2 2 neutrino candidate, Mrec = (pν + p`2 + p`3) . In CW models with well-separated resonances, the signal appears as a sharp peak in this variable centered at the mass of the produced mode (or a series of peaks, if a number of modes can be produced with sizable rates). On the other hand, in CW models with closely- spaced resonances, the individual peaks are merged to a broader excess due to experimental resolution. This can be seen in Fig. 5.9, which compares the distri-

3A two-fold degeneracy is encountered when the quadratic W mass constraint is used to determine the z component of pν . We follow the algorithm from Ref. [104] to resolve this ambi- guity. If there are two real solutions, we choose the solution with the smaller absolute value of pz; if the solutions are complex, we use the real part as the pz.

69 BP U100 G100 U400 G300 U750 G750 U1000 √s, GeV 250 250 500 500 3000 3000 3000 −1 3σ, fb 220 50 4300 20 55 25 720 L Table 5.5: Center-of-mass energy and integrated luminosity required for a 3- sigma observation of the CW neutrino signal in electron-positron collisions.

butions in Mrec for uniform (U100) and generalized (G100) benchmark points.

The sharp nature of the peak in the latter model allows for further background suppression using a cut on Mrec. As a result, G100 model is easily discoverable at HL-LHC, while U100 may require a more reﬁned analysis to be tested; see

Table 5.4. Benchmark points with heavier remain inaccessible at the LHC, N even with the full HL-LHC data set, due to small production cross sections.

We have also studied the prospects for proposed future lepton colliders, such as the ILC [34, 105], CEPC [106], FCC-ee [107, 108] and CLIC [109, 110]. We analyzed three center-of-mass energies: √s = 250 GeV, 500 GeV, and 3 TeV. Signal cross sections for the 7 benchmark points, listed in Table 5.3, are in the

1 10 fb range in all cases where CW neutrinos are kinematically accessible. − This implies that (103 104) signal events would be collected with data sets O − envisioned at these colliders. Since hadronic backgrounds are much less of an issue at lepton colliders, we focus on hadronic W decays, i.e. the final state `νjj. SM backgrounds were simulated inclusively; the main irreducible background is e+e− W +W −, with one leptonic and one hadronic W decay. We find that → in all cases, a simple cut-based analysis is sufficient to observe the signal with realistic luminosities; see Table 5.5. Details of the analysis are discussed in Ap- pendix 5.B.

As an example, consider the case of U100 benchmark point, which as discussed above is non-trivial to observe at the LHC even with the full HL-

70 0.12 0.08 - Signal U100-250GeV Signal U100 250GeV Background 0.10 Background 0.06 0.08

0.06 0.04

0.04 omlzdt one to Normalized

omlzdt one to Normalized 0.02 0.02

0.00 0.00 0 50 100 150 200 250 300 0 1 2 3 Δ Mljj(GeV) Rjj

0.12 U100-250GeV Signal Background 0.10

0.08

0.06

0.04 omlzdt one to Normalized 0.02

0.00 0 50 100 150

Mlν(GeV)

Figure 5.10: Signal (blue) and background (orange) distributions for U100 + − model at a 250 GeV e e collider: M`jj (left) after pre-selection cuts, ∆Rjj (right) after M`jj cut, and M`ν (bottom) after ∆Rjj cut.

LHC data set. At electron-positron colliders, √s = 250 GeV is sufﬁcient to produce CW neutrinos at the U100 benchmark point. Fig. 5.10 shows distributions of signal and background events in three variables that are useful

p 2 for signal/background discrimination, M`jj = (p` + pj1 + pj2) , ∆Rjj, and

p 2 M`ν = (p` + pν) . Here pν is the neutrino 4-momentum, reconstructed from the three-dimensional missing momentum supplemented with the condition mν = 0. The signal distribution in M`jj results from a number of peaks, corresponding to different CW modes, merged into a continuous “hump” due to experimental smearing effects. The larger values of ∆Rjj for the signal events are due to the fact that for a 100 GeV CW neutrino, the W boson is almost exactly at rest in the lab frame. Finally, the distributions in M`ν reﬂect that fact that almost all `ν pairs in the background come from a single W decay, while in the

71 U100 √s = 250 GeV, = 2000 fb−1 L Cuts S (fb) BG (fb) Parton-level cuts 7.4 3200 Pre-selection cuts 3.2 1650 M`jj [70, 140] GeV 2.3 160 ∈ ∆Rjj [1.8, 3.5] 1.9 91 ∈ M`ν [60, 90] GeV 1.3 43 ∈ S/B 0.03 S/√B 9.0 S/√S + B 8.9

Table 5.6: Cut ﬂow table for the search for CW neutrinos at the U100 benchmark point, in e+e− collisions at √s = 250 GeV. signal this is not the case. A series of cuts in these three variables, summarized in Table 5.6, is sufﬁcient for a 3-sigma observation of the signal with about 220 fb−1, ignoring systematic errors. The required integrated luminosity is far below the 2 5 ab−1 projected at the proposed e+e− Higgs factories. With the full − projected data sets, such colliders can perform detailed measurements to un- cover the nature of the signal. For example, the shape of the signal distribution in M`jj can be used to distinguish the CW neutrino tower from a single massive sterile neutrino state appearing in other models. We will study the details of this measurement in future work.

5.6 Conclusions and Outlook

In this chapter, we investigated complete, fully realistic models which produce small Dirac neutrino masses without unnaturally small parameters using the clockwork mechanism. The main results can be summarized as follows:

72 In the uniform clockwork model, a perturbation theory was developed • and applied to obtain approximate analytic expressions for quantities of phenomenological interest, such as excited CW neutrino masses and cou-

plings;

Experimental constraints on the uniform model from ﬂavor-changing de- • cay µ eγ and precision electroweak ﬁts were calculated. It was found → that CW neutrinos can have masses in the 100 GeV – 1 TeV range, within reach of the LHC and proposed lepton colliders, with neutrino Yukawa

couplings of order 10−1 10−2; − It was shown that the uniform clockwork model is only one representa- • tive of a much more general class of models that implement the clockwork mechanism for neutrino masses. Phenomenology of two sample generalized CW models was studied;

Collider signatures of CW neutrinos in the uniform and a generalized CW • models were studied using Monte Carlo simulations of signal and back-

ground. It was found that at the LHC, models with light ( 100 GeV) CW ∼ neutrinos can be discovered using the 3` + E/T signature, although the integrated luminosities required are larger than what has been collected so

far. Lepton colliders will be able to discover the CW neutrinos as long as they are within their kinematic range.

In the future, we would like to extend the analysis of this chapter in several directions. First, as already mentioned, it would be interesting to understand whether and how collider experiments can distinguish the CW model from a more traditional model with a single heavy neutrino. Second, throughout this chapter we chose the CW mass scale m to be in the neighborhood of

73 the weak scale. This is motivated by simplicity and by our interest in collider phenomenology, but is by no means required by the models themselves. (Note that the light neutrino mass, Eq. (5.15), is to leading order completely independent of m.) It would be interesting to investigate phenomenologically a broader range of the CW scales, which may entail constraints and signatures different from the ones considered here.

5.A Perturbation Theory in p

In this Appendix, we will compute eigenvalues and eigenvectors to the clockwork mass matrix after including the Yukawa coupling to the Standard Model sector at the zeroth site, for the uniform clockwork model of Section 5.2. They are already known exactly for the y = 0 case where the clockwork modes are completely decoupled from the SM. For the y > 0 case, we can compute them

√yv to leading order using perturbation theory in p = 2m as shown below. In the following, matrices and vectors with tildes are unperturbed (with y = 0) and those without tildes are perturbed. For the eigenvalues, we use λ¯i to refer to the perturbed eigenvalues, and λi are quantities deﬁned in Eq. (5.7) which are the eigenvalues of the unperturbed squared matrix.

5.A.1 Eigenvalues

If we consider M †M and its eigenvectors, the unperturbed (right) eigenvectors (i) ˜ ji are given by w˜j = UR , where the superscript (i) denotes the mode number, while the subscript j is the vector component. On including the Yukawa

74 coupling, the perturbation to this matrix has zeroes in all entries except for (δM 2)00 = p2:   p2 + q2 q 0 0  − ···    M †M  q 1 + q2 0 0 =  − ···  2  . . . . . m  ......   . . . .   0 0 q 1 · · · −   q2 q 0 0   2  − ···  p 01×N  2     q 1 + q 0 0   = − ···  + (5.41)  . . . . .    ......     . . . .     0N×1 0N×N 0 0 q 1 | {z } · · · − δM 2 | {z } 2 M˜ †M˜ m unperturbed m2

Thus, to ﬁrst order in perturbation theory in p, the shift in eigenvalues can be

† written as follows (note that λi are the eigenvalues of the squared matrix M˜ M˜ and the actual masses of the neutrinos will be mi = m√λi):

2 (i) δM (i) 2 0i 2 δλi = w˜ w˜ = p (U˜ ) (5.42) m2 R

This gives the results in Eq. (5.15) and (5.17). The only difference is the (p3) O term in Eq. (5.15). This term requires a higher order calculation using the determinant form of the eigenvalue equation, which we do not expand on here.

These results can be cross-checked by considering the unperturbed left- handed eigenvectors as well, but the calculation in this case is not as straightforward, because the perturbation to M˜ M˜ † has more matrix entries, with some at (p). The leading order ( (p)) perturbation calculation in the left-handed O O case results in zero deviation in eigenvalues, since the ﬁrst non-trivial deviations appear at (p2). This necessitates a second-order calculation. We have O

75 performed this calculation and conﬁrmed explicitly that MM † and M †M have identical eigenvalues.

5.A.2 Eigenvectors and Rotation Matrices

Once the eigenvalues are known, it is relatively straightforward to calculate the perturbed eigenvectors, and thus the rotation matrices that diagonalize the perturbed mass matrix. For both left and right eigenvectors, it is more convenient to do this in the basis in which the unperturbed clockwork mass matrix is diag- ˜ † † ˜ onal. Thus, we consider the eigenvalue equations for the matrices UL(MM )UL ˜ † † ˜ for the left-handed case, and UR(M M)UR for the right-handed case. We will refer to right eigenvectors as w(i) and left eigenvectors as v(i) in this basis. The rotation matrices in our convention have the eigenvectors as its column vectors:

ij ˜ ij (j) UL = ( UL(I + ∆L)) = (vo )i (5.43) ij ˜ ij (j) UR = ( UR(I + ∆R)) = (wo )i (5.44)

where I is a unit matrix, and vo and wo are the eigenvectors in the original clockwork basis.

In the ‘clockwork-diagonal’ basis (where the p = 0 mass matrix is diagonal),

(i) the unperturbed eigenvectors are columns of the identity matrix, i.e. v˜j = (i) w˜j = δij. Thus, we write the perturbed eigenvectors as,

ij (j) (I + ∆L) = vi (5.45)

ij (j) (I + ∆R) = wi (5.46)

(i) (i) We will solve for v and w in the rest of this section. ∆L and ∆R are related

76 to v(i) and w(i) as mentioned above, and these are the quantities shown in Sec- tion 5.2, in Eqs. (5.19) and (5.20).

77 Left eigenvectors:

In the basis in which the unperturbed clockwork mass matrix is diagonal, the eigenvalue equation for MM † can be written as follows:    0     p2 W †        λ1     (i)   + λ¯iIN+1 v = 0 (5.47)  .      ..    −         W 0N×N  λN where λi is the unperturbed eigenvalue and λ¯i is the exact eigenvalue. W is a column vector obtained from rotating the perturbation terms in the matrix to this basis. Its terms are given by r r 2p2q2 Njπ p2 Wj = sin = λjCj − N + 1 N + 1 − N + 1

Decomposing the eigenvalues and eigenvectors into sums of unperturbed and perturbed parts, we have,

λ¯i = λi + ∆i (5.48)

(i) (i) (i) (i) vj =v ˜j + δvj = δij + δvj (5.49)

where from the previous section, we know the values of ∆i = δλi.

The unperturbed parts in the eigenvalue equation give a zero on multiplying

78 δij. Thus, the equation becomes,   0 λi  −     λ1 λi   −  (i)   δv  ...      λN λi −    p2 W †       (i) (i) +   δλiIN+1 (˜v + δv ) = 0 (5.50)   −     W 0N×N

This simpliﬁes to the following equations:

2 (0) PN (0) i = 0, j = 0 :(p ∆0)(1 + δv ) + Wkδv = 0 (5.51) − 0 k=1 k (0) (0) (0) i = 0, j = 0 : λjδv + Wj(1 + δv ) ∆0δv = 0 (5.52) 6 j 0 − j (i) 2 (i) PN (i) i > 0, j = 0 : λiδv + Wi + (p ∆i)δv + Wkδv = 0 (5.53) − 0 − 0 k=1 k (i) (i) i > 0, j = i : Wiδv ∆i(1 + δv ) = 0 (5.54) 0 − i (i) (i) (i) i, j > 0, j = i :(λj λi)δv + Wjδv ∆iδv = 0 (5.55) 6 − j 0 − j

On solving these equations, dropping higher order terms and plugging in the appropriate δλi when necessary, we get the following results:

(0) Wj vj = , j > 0 (from Eq. (5.52)) (5.56) − λj

(i) (0) Wi v0 = vi = , i > 0 (from Eq. (5.54)) (5.57) − λi (i) WiWj vj = , j > 0, j = i (from Eq. (5.55)) (5.58) λi(λi λj) 6 − These results are consistent with equations Eq. (5.51) and Eq. (5.53) up to (p2).4 O

The equations above do not constrain the deviations in the diagonal ele-

(i) ments of the left-handed vectors, δvi , as all terms containing them enter only 4In showing the consistency with Eq. (5.51), one may need the following identity, which can

79 at higher order. These can be obtained from the normalization of the vectors, or in other words, from the unitarity of the left-handed rotation matrix. To do so, it is important to note that the deviations in (left) eigenvectors are not of the same order. The pseudozero mode eigenvector has deviations of (p), while heavier O 2 mode eigenvectors have deviations of (p ). To start with, the unitarity of UL O implies

† † ∆L + ∆L + ∆L∆L = 0 (5.60)

2 3 Expanding ∆L = p 1 + p 2 + (p ) then implies that O O O

† 2 † † p( + 1) + p ( + 2 + 1) + = 0 (5.61) O1 O O2 O O1O ··· † † † + 1 = 0 ; + 2 + 1 = 0. (5.62) ⇒ O1 O O2 O O1O

In terms of eigenvector deviations, this implies,

δv(0) + δv(i) = 0, for i = 0 (5.63) i 0 6 δv(i)+δv(j) + δv(0)δv(0) = 0, for i, j = 0 (5.64) j i i j 6 N (0) (0) X (0) 2 δv0 +δv0 + (δvi ) = 0 (5.65) i=1

(i) (0) The correction to vi and v0 can thus be determined using the unitarity of the rotation matrix, and we get,

1 2 δv(i) = δv(0) , for i = 0 (5.66) i −2 i 6 N (0) 1 X (0) δv = (δv )2 (5.67) 0 2 i − i=1 be proven using Eqs. (1.447.3) and (1.353.3) of Ref. [111].

N 2 jNπ 2 X sin N+1 N + 1 1 q 1 = 2 1 2N 2 −−2N . (5.59) λj 2q − q q q j=1 −

80 In summary, to leading order, the left eigenvectors are given by,

2 N (0) p X Ci v = 1 (5.68) 0 N + 1 2λ − i=1 i s 2 (0) (i) p Ci vi = v0 = (5.69) − N + 1 λi 2 (i) p Ci vi = 1 (5.70) − N + 1 2λi 2 r p (i) p λj CiCj vj = (5.71) N + 1 λi λi λj −

Right eigenvectors:

As seen before, the perturbation matrix in the right-handed case (i.e. M †M) is simpler than in the left-handed case. In the ‘clockwork-diagonal’ basis, we get the following eigenvalue equation:

† (i) (D + zz λ¯iIN+1)w = 0 − where D is the diagonal matrix of unperturbed eigenvalues = diag(λ0, λ1 ) ··· and z is a vector given by z = U˜ † (p, 0, )T . In terms of previously deﬁned R ··· quantities, the column vector z simpliﬁes to s r !T q2 1 p2 p z = p − , Ci q2(N+1) 1 N + 1 − Using a perturbative expansion as in the case of the left eigenvectors, the eigenvalue equation becomes,

† (i) (i) (D + zz (λi + ∆i)IN+1)(w ˜ + δw ) = 0 − (i) † (i) (i) (D λiIN+1)δw + (zz ∆iIN+1)(w ˜ + δw ) = 0 ⇒ − −

The jth row of this equation gives,

N (i) X (i) (i) (λj λi)δw + zjzk(δik + δw ) ∆i(δij + δw ) = 0 (5.72) − j k − j k=0

81 (i) where the substitution w˜j = δij has been made.

The corrections to the eigenvalues ∆i are known from previous calculations

2 (i) and can be written in terms of z as ∆i = zi . Solving for δwj for various cases as before, we get the following:

(i) wi = 1 (5.73) s 2 p (0) (j) z0zj 2 1 q 1 Cj wj = w0 = = p 2(N+1)− (5.74) − − λj − N + 1 q 1 λj − 2 p (i) zizj p CiCj wj = = . (5.75) λi λj N + 1 λi λj − −

5.B Lepton Collider Analysis Details

In this Appendix, we present a brief summary of lepton collider analysis for all seven benchmark points introduced in Section. 5.5. Choices of theory parameters for each of the benchmarks are given in Table. 5.1 and Table. 5.2. The relevant process at lepton colliders is shown as a Feynman diagram in Fig. 5.7. Considered center-of-mass energies and integrated luminosities are √s = 250 GeV ( = 2000 fb−1), 500 GeV ( = 4000 fb−1), and 3 TeV ( = 2000 fb−1). L L L

We imposed pT,` > 20 GeV and pT,j > 20 GeV cuts for both signal and background events simulations. We then imposed the following cuts on the fully processed (i.e. after hadronization (via Pythia8) and detector effects (via

82 Delphes3)), data as our event selection cuts (called “pre-selection cuts”):

N` 1,Nj 2 ≥ ≥

η` < 2.5, ηj < 2.5 | | | |

∆R`j > 0.4, ∆Rjj > 0.4 (5.76)

pT,` > 20(1 + r) GeV, pT,j > 20(1 + r) GeV,

where N`(j) denotes the number of leptons (jets) in the event and 0 < r < 1 parametrizes smearing by detector effects. The size of r will primarily be determined by the jet E resolution of the detector, and typically for lepton colliders it will be (5)%. We, however, will use 25% for most of our analysis to make ∼ O a conservative estimation. The only exception will be for G100 and U100 where the majority of signal events are distributed near the low pT regime and hence we use 5% to secure enough signal events.

We summarize the list of kinematic cuts and their efﬁciencies in Table 5.7 and Table 5.8 for generalized CW models and in Table 5.9 and Table 5.10 for uniform models. We denote the jet with highest pT as j1.

83 G100 – √s = 250 GeV √s = 500 GeV √s = 3000 GeV – ( = 2000 fb−1) ( = 4000 fb−1) ( = 2000 fb−1) L L L Cuts S (fb) BG (fb) S (fb) BG (fb) S (fb) BG (fb) Parton-level cuts 4.3 3184 7.4 1693 1.3 449 Pre-selection cuts 1.4 1650.0 2.2 877.8 0.01 166.3 M`jj [85, 102]GeV 0.7 13.4 – – – – ∈ ∆Rjj [1.8, 4] 0.7 6.4 – – – – ∈ η` [0.5, 1.4] 0.4 1.9 – – – – | | ∈ Mjj [68, 95]GeV 0.3 0.7 – – – – ∈ M`jj [40, 105]GeV – – 1.8 4.1 – – ∈ Mjj [67, 90]GeV – – 1.2 1.5 – – ∈ M`jj [60, 110]GeV – – – – 0.008 0.06 ∈ M`ν [0, 500]GeV – – – – 0.008 0.02 ∈ pT,` [0, 50]GeV – – – – 0.008 0.006 ∈ † ∆R`j [0, 0.9] – – – – 0.008 0 1 ∈ S/B 0.5 0.8 2.8 S/√B 18.9 61.8 6.5 S/√S + B 15.3 46.1 3.3

Table 5.7: Cut ﬂow Table for generalized CW model G100. † The fact that we get 0 events is an artifact of low statistics of our background sample. When we estimate the signal signiﬁcance, we used Poisson statistics and used 3 Madgraph events, which corresponds to 5.4 actual events with the integrated luminosity shown.

84 – G300 G750 – √s = 500 GeV √s = 3000 GeV √s = 3000 GeV – ( = 4000 fb−1) ( = 2000 fb−1) ( = 2000 fb−1) L L L Cuts S (fb) BG (fb) S (fb) BG (fb) S (fb) BG (fb) Parton-level cuts 5.8 1693 6.1 449 7.9 449 Pre-selection cuts 3.4 780.9 1.2 155.8 1.9 155.8 M`jj [290, 335]GeV 2.5 79.9 – – – – ∈ pT,` [100, 250]GeV 2.0 18.0 – – – – ∈ pT,j [70, 150]GeV 1.7 9.1 – – – – 1 ∈ ∆R`jj [0.8, 1.5] 1.5 4.8 – – – – ∈ ET [0, 85]GeV 1.3 3.3 – – – – ∈ M`jj [250, 340]GeV – – 1.1 11.0 – – ∈ ηj1 [1.6, 2.5] – – 1.0 4.2 – – | | ∈ ET [0, 150]GeV – – 0.7 1.4 – – ∈ M`jj [720, 820]GeV – – – – 1.5 10.7 ∈ pT,j [140, 450]GeV – – – – 1.3 4.7 1 ∈ ηj [0.6, 2.5] – – – – 1.0 1.5 | 1 | ∈ M`ν [0, 510]GeV – – – – 0.6 0.5 ∈ S/B 0.4 0.5 1.4 S/√B 43.7 26.3 42.5 S/√S + B 37.2 21.4 27.5

Table 5.8: Cut ﬂow Table for generalized CW models: G300 and G750.

85 U100 – √s = 250 GeV √s = 500 GeV √s = 3000 GeV – ( = 2000 fb−1) ( = 4000 fb−1) ( = 2000 fb−1) L L L Cuts S (fb) BG (fb) S (fb) BG (fb) S (fb) BG (fb) Parton-level cuts 7.4 3184 12.8 1693 3.9 449 Pre-selection cuts 3.2 1650.0 5.3 877.8 0.09 166.3 M`jj [70, 140]GeV 2.3 163.3 – – – – ∈ ∆Rjj [1.8, 3.5] 1.9 91.5 – – – – ∈ M`ν [60, 90]GeV 1.3 43.0 – – – – ∈ M`jj [50, 150]GeV – – 4.3 30.8 – – ∈ ηj [0.7, 2.5] – – 3.4 17.9 – – | 1 | ∈ η` [ 1.4, 1.4] – – 2.3 6.9 – – ∈ − M`jj [80, 160]GeV – – – – 0.08 1.0 ∈ pT,j [100, 400]GeV – – – – 0.06 0.5 1 ∈ pT,` [0, 100]GeV – – – – 0.04 0.2 ∈ η` [ 2, 2] – – – – 0.02 0.04 ∈ − S/B 0.03 0.3 0.6 S/√B 9.0 54.5 5.2 S/√S + B 8.9 47.3 4.1

Table 5.9: Cut ﬂow Table for uniform CW model for U100.

86 – U400 U750 U1000 – √s = 500 GeV √s = 3000 GeV √s = 3000 GeV √s = 3000 GeV – ( = 4000 fb−1) ( = 2000 fb−1) ( = 2000 fb−1) ( = 2000 fb−1) L L L L Cuts S (fb) BG (fb) S (fb) BG (fb) S (fb) BG (fb) S (fb) BG (fb) Parton-level cuts 0.8 1693 7.6 449 5.9 449 2.3 449 Pre-selection cuts 0.5 780.9 3.0 155.8 1.2 155.8 0.4 155.8 M`jj [380, 460]GeV 0.4 285.1 – – – – – – ∈ pT,j [90, 250]GeV 0.3 116.6 – – – – – – 1 ∈ pT,` [190, 240]GeV 0.1 10.1 – – – – – – ∈ M`jj [400, 600]GeV – – 2.7 29.1 – – – – ∈ ηj1 [1.4, 2.5] – – 1.9 5.1 – – – – | | ∈ ET [0, 150]GeV – – 1.6 2.7 – – – – ∈ M`jj [740, 1020]GeV – – – – 1.0 26.6 – – ∈ pT,j [150, 600]GeV – – – – 0.8 10.9 – – 1 ∈ M`ν [0, 800]GeV – – – – 0.7 5.4 – – ∈ ET [0, 130]GeV – – – – 0.6 3.4 – – ∈ η` [ 2.2, 2.2] – – – – 0.4 1.5 – – ∈ − ηj [0.4, 2.5] – – – – 0.4 0.8 – – | 1 | ∈ M`jj [1050, 1450]GeV – – – – – – 0.3 24.2 ∈ pT,j [150, 700]GeV – – – – – – 0.3 11.3 1 ∈ M`ν [0, 650]GeV – – – – – – 0.2 3.8 ∈ ET [0, 100]GeV – – – – – – 0.2 2.2 ∈ η` [ 2.2, 2.2] – – – – – – 0.09 0.7 ∈ − S/B 0.01 0.6 0.4 0.1 S/√B 2.9 43.5 18.0 5.0 S/√S + B 2.8 34.4 14.9 4.7

Table 5.10: Cut ﬂow Table for uniform CW models: U400, U750, and U1000

87 CHAPTER 6 WHAT IS DARK MATTER?

6.1 Evidence for the Existence of Dark Matter

The term ‘Dark Matter’ refers to any non-baryonic matter that has (so far) been detected gravitationally in the universe. There is a large and growing body of evidence that the visible matter (i.e. the Standard Model particles) that we know and understand comprise only a small fraction of the total matter content of the universe. This includes observations across various distance scales in the universe, from galactic to cosmological scales.

Figure 6.1: Rotation curves for NGC 6503, a dwarf spiral galaxy. Plot from [112].

One of the earliest detections of dark matter was through galactic rotation curves, as shown in Fig. 6.1. Using the visible matter seen in a galaxy, the velocity of stars in the galactic system can be estimated. But the measured rotational velocities far exceeded the predicted values, which implied either that our un-

88 derstanding of gravity at galactic scales is incorrect, or that there exists invisible matter in the galaxy that exerts a gravitational pull on the visible matter.

Since then, there have been multiple observations that lend credence to the latter hypothesis. On larger scales, gravitational lensing observations of galaxy clusters have strongly implied the existence of dark matter. The mass of a galaxy cluster between earth and a bright distant galaxy can be estimated using the distortion in the light from this galaxy due to the gravitational lensing effect of the galaxy cluster. Galaxy cluster masses have been measured to exceed their estimated visible matter content consistently.

Figure 6.2: Overlay of the weak lensing mass contours on the X-ray image of galaxy cluster 1E 0657–56. The gas bullet lags behind the DM subcluster. Plot from [113].

Arguably, the clearest lensing signature is that of the so-called ‘bullet cluster’ [112]. The visible and dark matter content of two colliding galaxy clusters behave very differently during the collision. Visible matter particles interact

89 with each other and are slowed signiﬁcantly, while dark matter is not slowed much and passes through the collision. This separation of dark and visible matter can be clearly observed through lensing techniques, as shown in Fig. 6.2.

This observation not only bolsters the argument for existence of dark matter, but also suggests two signiﬁcant properties of dark matter: (i) dark matter interacts mostly gravitationally with visible matter; any other interaction, if it exists, is very feeble, and (ii) there are strong constraints on dark matter self-interactions.

Dark matter plays a signiﬁcant role in the ΛCDM model, often called the

‘Standard Model of cosmology’. In the ΛCDM model (where CDM stands for Cold Dark Matter), it is defined as a fluid that has zero pressure and interacts only gravitationally with other components of the universe. The ΛCDM model makes robust predictions for both the CMB spectrum and the matter power spectrum. The large scale structure of the universe that we see today is sourced by early universe density fluctuations in the dark matter fluid. Density fluctuations with a size less that the horizon size H−1 grow with time, while ‘super- horizon’ modes are frozen. Over-densities in dark matter form gravitational wells that attract baryonic matter, which also interacts directly with the photon bath. Relativistic species at matter-radiation equality can damp these density fluctuations through free-streaming, while cold dark matter is required to re- produce the large scale structure we observe. Thus, the statistical distribution of galaxy clusters provides evidence for the existence of cold dark matter. The interplay between photon pressure and gravitational pressure of baryons gives rise to Baryon Acoustic Oscillations in the matter power spectrum, which gives more information about the fraction of baryonic matter content in the early universe. Additionally, as baryons interact strongly with photons, the CMB spectrum is correlated with the matter power spectrum, and the fact that the ΛCDM

90 model prediction matches the measured spectrum very well further strengthens the argument for dark matter.

6.2 What We Know So Far

From the various observations described in the previous section, a few general properties of dark matter can be inferred [4]:

1. It comprises about 26% of the energy content of the universe, while 5% is visible matter. As implied by the name, it behaves as matter and redshifts as such in the universe currently.

2. Other than gravitational interactions, dark matter interacts very weakly with visible matter, if at all. It cannot emit photons in the galaxies and

galaxy clusters we observe today. Additionally, at the time of structure formation, its interaction with the baryon and photon bath should be primarily gravitational in order to be consistent with the observed matter

power spectrum.

3. Dark matter self-interactions are strongly constrained by bullet cluster ob-

servations and the matter power spectrum. Thus, it should be ‘collision- less’ on large scales — relatively strong self interactions are allowed on short scales, but long range self-interactions (from the presence of a cou-

pling to a ‘dark radiation’ bath) are forbidden. This is because long range self-interactions can lead to ‘dark acoustic oscillations’ in the matter power spectrum similar to baryons, and will also be inconsistent with bullet clus-

ter observations.

91 Figure 6.3: Broad overview of dark matter models and their mass ranges, along with search techniques and some current experimental anomalies. Mass ranges are approximate and can differ based on model-speciﬁc details. Figure from US Cosmic Visions 2017 [114].

4. Dark matter should be ‘cold’. DM particles should be non-relativistic (i.e. their temperature should be below their mass) during structure formation

in the universe. If this is not the case, density perturbations in the early universe can get washed out and large scale structure, as we see it today, would not be formed.

5. Considering our current observations of dark matter, it should be stable with lifetimes of the order of at least the age of the universe.

There is a wide array of theoretical models that can satisfy these properties, and also explain how dark matter was produced in the early universe. The masses and spins of dark matter particles can also vary widely between these

92 models. Although a vast range of masses are possible, some limits can be set from general considerations.

On the most massive end, dark matter can be made of composite objects of

4 5 masses as high as 10 10 M (where M stands for the mass of the sun). This − limit is set by the existence of dwarf galaxy halos of that order, and having dark matter be composed of heavier particles would be inconsistent with galactic dynamics. Primordial black holes are popular candidates for super-massive dark matter, and current theoretical limits allow at least some part of dark matter to be PBHs.

On the least massive end, dark matter particles can be as light as 10−22 eV (if it is a boson). Such light particles can be considered as a classical ﬁeld, as the occupation number of dark matter particles will be very high. In this scenario, dark matter is produced non-thermally and can be cold even when the mass is very light. The lower bound on mass can be easily understood from studying the de-Broglie wavelength of the dark matter. For masses around 10−22 eV, λdB will be (1 kpc). Lighter particles will have wavelengths above the O kiloparsec scale, and can no longer be conﬁned to galactic halos. Additional constraints apply if dark matter particles are fermions. Fermions have a maxi- mum phase space density as a direct consequence of Pauli’s exclusion principle, and cannot have arbitrarily high occupation numbers. This gives a bound of m (100 eV). DM & O

Further limits can be set if dark matter was ever in thermal equilibrium with the Standard Model bath in the early universe. For example, a bound on ‘warm dark matter’ can be derived from requiring dark matter to be non-relativistic during large scale structure formation. In this case, we require mDM & 5 keV.

93 6.3 Freeze-Out and Freeze-In of Dark Matter

A large fraction of dark matter models fall under the category in which dark matter is produced through interaction with an initial thermal bath of SM particles in the early universe, and the current DM relic density is set by the mass and interaction strength of the DM particle in the model.

There are two distinct and antithetical possibilities that are popular in this class of models: freeze-out and freeze-in. In broad terms, the freeze-out mechanism starts with a bath of DM particles in thermal equilibrium with the SM, which dilutes through annihilation of DM into lighter SM particles as temperature of the universe falls below the DM mass, and the relic density is set when the expansion of the universe overcomes the rate of this annihilation process.

Freeze-in, on the other hand, involves starting with a negligible dark matter number density in the early universe, and producing DM particles from the SM bath until the temperature of the universe falls much below the mass and production is exponentially suppressed. The appealing characteristic of both scenarios is that the relic abundance is independent of UV parameters such as the reheating temperature of the universe after inﬂation. Production of dark matter is dominated by IR physics, and the ﬁnal relic abundance can be calculated solely from the masses and couplings to the SM of the DM particle.

In the case of freeze-out, the annihilation rate into lighter particles increases with the coupling of the dark matter to the Standard Model. Thus, the relic abundance is inversely proportional to the square of the coupling strength, as a higher coupling strength implies that more of the DM can annihilate away before freeze-out. On the other hand, in the freeze-in mechanism, the production

94 Figure 6.4: Comparison of freeze-out and freeze-on of particles in the early universe [4]. Relativistic species freeze out before exponential Boltzmann suppression kicks in, when the expansion of the universe becomes faster than the processes keeping them in equilibrium. This is not a viable scenario for DM, and the non-relativistic freeze-out and freeze-in lines show how relic abundance can be achieved successfully.

rate is proportional to the square of the coupling, and hence relic abundance is directly proportional to it. Since freeze-in requires the dark sector to not be in thermal equilibrium with the SM bath, the coupling in this case has to be very small. In the case of freeze-out, couplings are required to be much larger in order to annihilate away excess dark matter density (see Fig. 6.5).

A useful framework to discuss DM production and relic density is using the quantity Y n /sγ, known as the yield. Here, n is the number density of ≡ DM DM dark matter particles and sγ is the entropy of the photon bath at any given time. It is a convenient measure to use, as it remains constant in time after the rate of expansion of the universe has exceeded that of any number changing processes.

In other words, after freeze-out/freeze-in, the current relic density can easily be

95 Figure 6.5: Comparison of relic densities from freeze-out and freeze-on of particles and their dependence on couplings to the SM [115]. The quantity Ω is deﬁned as Ω ρ /ρ and Ω h2 0.1 gives the correct relic density. ≡ DM crit ∼ recovered as follows.

ρDM = mDM nDM = mDM Yfo/fi sγ,0. (6.1)

Using simple arguments, the parametric dependence of the yield in the freeze-out and freeze-in mechanisms can be written as follows,

1 mDM 2 Mpl Yfo 2 ; Yfi λ , (6.2) ∼ λ Mpl ∼ mDM where Mpl is the Planck mass, and λ refers to the coupling strength to the Stan- dard Model. This relationship between couplings and relic density is summarised in Fig. 6.5. Generically, couplings in the freeze-in scenario are 10 12 ∼ − orders of magnitude below that of freeze-out models, making experimental detection more difﬁcult.

There are other candidates as well, of which, a famous example is that of axions produced through the misalignment mechanism. In this case, many constraints such as the warm dark matter bound can be evaded. However, freeze-in and freeze-out models remain popular as they are completely calculable using

96 IR physics. For example, in the misalignment mechanism, the initial ﬁeld value is a UV quantity, which affects the cosmological history of the axionic ﬁeld.

97 CHAPTER 7 CONFORMAL FREEZE-IN OF DARK MATTER

7.1 Introduction

The microscopic nature of dark matter is one of the central open questions in fundamental physics which cannot be addressed within the Standard Model (SM). Many theoretical ideas have been suggested, and an extensive experimental effort is under way to test some of the proposals [114]. While the precise nature of the dark matter sector varies greatly among the proposed models, all of them postulate that dark matter consists of point-like particles (e.g. WIMPs or axions), their bound states (e.g. dark atoms), or particle-like extended objects (monopoles, Q-balls, etc.), both today and throughout its cosmological history. However, viable extensions of the SM exist in which new physics sectors do not contain spatially localized particle-like excitations at all [116]. A well-known example is a conformal ﬁeld theory (CFT) [117, 118, 119], where scale invariance precludes the existence of stable ﬁnite-size states. In this letter, we show how dark matter can arise from a new physics sector which is described by a CFT throughout most of its cosmological history.

An immediate objection to the idea of dark matter made out of CFT “stuff” is that conformal invariance dictates that the energy density of such stuff redshifts like radiation (ρ a−4), rather than non-relativistic matter (ρ a−3), as the Uni- ∝ ∝ verse expands. However, in any phenomenologically viable model, conformal invariance is at most approximate and must be broken to some degree. In particular, any interactions of the CFT sector with the non-conformally-invariant

SM inevitably break the symmetry. Generically, such effects induce a “gap”

98 mass scale, below which the sector is no longer conformal and its spectrum consists of spatially localized particle degrees of freedom. Below, we will discuss a scenario in which dark matter production in the early Universe occurs at temperatures above the gap scale, so that throughout the production process the dark sector can be well approximated by a CFT. At the same time, the gap scale, which is induced by cosmological phase transitions in SM, is sufﬁciently large so that the dark sector behaves as non-relativistic matter during CMB decoupling, structure formation, and today, as required by observations.

7.2 Particle Physics Framework

We extend the SM by postulating a dark sector, whose ﬁelds do not carry SM gauge charges. The dark sector is assumed to be invariant under the conformal group. It is coupled to the SM via

λCFT int = D SM CFT , (7.1) L ΛCFT O O where is a gauge-invariant operator consisting only of SM ﬁelds, is OSM OCFT an operator within the dark-sector CFT, and ΛCFT is the energy scale where the CFT is replaced by its ultraviolet (UV) completion. We will consider the regime of small Wilson coefﬁcient λ 1, where the conformal symmetry breaking CFT introduced by Eq. (7.1) can be treated as a (technically natural) small perturbation. A UV completion of the CFT that naturally generates λ 1 is discussed CFT below. If and have scaling dimensions d and d, respectively, then OSM OCFT SM D = d + d 4. (7.2) SM − The dark-sector CFT may be strongly coupled, resulting in large anomalous dimensions and non-integer d.

99 A simple and predictive scenario for CFT breaking in the infrared (IR) is to consider SM operators with = 0, which automatically triggers such hOSM i 6 breaking through the interaction term in Eq. (7.1), if is relevant, i.e. d < 4. OCFT In this scenario must be a scalar operator, and CFT unitarity then requires OCFT d 1. Some obvious choices for , which will be covered in this chapter, are: ≥ OSM

Higgs portal: = H†H (d = 2); • OSM SM † Quark portal: = HQ qR (d = 4); • OSM L SM µν Gluon portal: = G Gµν (d = 4). • OSM SM

For these portals, = 0 in the infrared (IR), due to the Higgs vacuum hOSM i 6 expectation value (VEV) and the QCD chiral condensate. Conformal symmetry is broken at a scale Mgap, and simple dimensional analysis can be used to estimate this scale: 1 4−d λCFT SM Mgap hO i (7.3) dSM +d−4 ∼ ΛCFT

There exists an entirely different class of gauge invariant Standard Model operators, that do not have a non-zero VEV, = 0, in the infrared. In this hOSM i 6 case, the mass gap can only be generated through loops. Conformal symmetry can be broken through the appearance of relevant 2 terms induced through OCFT SM loops. The generated 2 operator should have dimension d < 4, and in OCFT the large-N limit, this corresponds to having d 2. The mass can also be OCFT . generated through loop induced couplings to operators that do have a VEV in the IR, like those mentioned previously. An example of such an operator that will be studied in this chapter is:

† Lepton portal: = HL `R (d = 4). • OSM SM

100 An important consideration in mass gap calculations is to include all sources of conformal symmetry breaking. Often there are competing effects when there are multiple loop induced relevant deformations. For example, in the case of the lepton portal, both 2 and terms are generated through SM loops. OCFT OCFT Additionally, 2 terms can be generated from a loop with just the leptons, OCFT as well as a two-loop diagram with the Higgs included. The one-loop diagram appears only below the weak scale, when the Higgs can be set to its VEV. The various diagrams are shown in Fig. 7.1.

l+ l+ l+

h h h

l− l− l− Figure 7.1: Diagrams that contribute to conformal symmetry breaking in the lepton portal. Blue circles indicate CFT operator insertions. The ﬁrst two induce an 2 term and the third induces an term below the weak scale. OCFT OCFT

In the case of the quark and gluon portals, there is a straighforward VEV induced mass gap from the respective operators, but additional deformations are caused by loop induced couplings to the Higgs, as shown in Fig. 7.2.

h g q

h h q

q¯ h g Figure 7.2: Loop induced diagrams that contribute to conformal symmetry breaking in the quark and gluon portals. Blue circles indicate CFT operator insertions. Both generate terms after electroweak symmetry breaking. OCFT

101 When multiple diagrams generate a certain operator ( or 2 ), these OCFT OCFT diagrams can be summed to find the coefficient of the relevant operator and the mass gap can be extracted with dimensional analysis. When both and OCFT 2 operators are generated, the operator that induces the higher mass gap OCFT breaks conformality first, and the theory is assumed to be confined below this scale. In other words, it is assumed that these deformations are independent competing effects, and whichever generates the larger breaking is considered to dominate the mass gap, and the other is neglected. As we will see, this is a reasonable approximation due to the hierarchy between these scales in the cases we consider.

Since is a relevant operator, an additional symmetry must be invoked OCFT to avoid its appearance in the CFT Lagrangian, which would lead to incalculable IR breaking of the CFT that would generically dominate the effect of Eq. (7.1).

This may for example be a 2 symmetry under which is odd, explicitly Z OCFT broken only by the interaction with the SM. However, the operator product expansion (OPE) of generally contains singlet scalar operators, which OCFT ×OCFT are even under 2 [120]. Numerical CFT bootstrap provides an upper bound Z on the dimension of the lowest singlet scalar operator in the OPE [121, 120]. Re- quiring that no 2-even relevant operators are generated implies d > 1.61 [121]. Z Note that this bound is model-dependent and may be avoided if a larger symmetry is used, or if operator coefﬁcients are ﬁne-tuned.

Even with discrete symmetry, loops involving SM particles will induce IR breaking of the CFT. For example, in the case of Higgs portal,

2 λCFT ΛSM δ CFT D 2 CFT , (7.4) L ∼ ΛCFT 16π O

where ΛSM is the scale where quadratic divergence in the Higgs loop is cut off,

102 through Loop-induced 2 OSM OCFT OCFT OCFT through H†H hOSM i 1 1 4−d 2−d † λCFT 2 λCFT v H H d−2 v — d−2 m ΛCFT ΛCFT h

1 1 1 4−d P 2 2 4−d 2 2 2−d † λCFT 3 λCFT ( i yqi ) v ΛSM λCFT ΛSM v HQLqR d vΛQCD d 2 d 2 + ΛCFT ΛCFT 16π ΛCFT 16π 4π

1 1 1 4−d (P y2 ) v2Λ2 4−d 4 2−d µν λCFT 4 λCFT i qi SM λCFT ΛSM G Gµν d ΛQCD d 2 2 d 2 ΛCFT ΛCFT (16π ) ΛCFT 16π

1 1 P 2 2 4−d 2 2 2−d † λCFT ( i y`i ) v ΛSM λCFT ΛSM v HL `R — d 2 d 2 + ΛCFT 16π ΛCFT 16π 4π

Table 7.1: Standard Model operators and corresponding mass gaps generated through different sources. Here, yi are the Yukawa couplings of the appropriate particles and sums are over ﬂavors in the quark/lepton sectors that the CFT operator couples to.

for example by compositeness or supersymmetry. This effect is subdominant to the breaking due to the Higgs VEV as long as ΛSM . 4πv, the usual condition for naturalness of the weak scale. In the rest of this chapter the cutoff scale is

parametrized as ΛSM = α 4πv where α is a number between 1/4π and 4π and v is the Higgs VEV.

For all four portals, the gap scales corresponding to each kind of deformation are summarised in Table 7.1. In every case, the deformation due to electroweak symmetry breaking with the Higgs getting a non-zero VEV dominates. The hierarchy between weak and QCD scales implies that even for other operators that develop their own vacuum expectation values, the loop-induced coupling to

H†H causes conformal symmetry breaking at a higher scale than the tree-level

2 VEV-induced breaking. In general, deformations are also sub-dominant, OCFT due to different scaling in d, and loop factors. It is to be noted that this is not a

103 general statement for all possible values of λCFT and ΛCFT . In particular, in the parameter space where relic density can be achieved via freeze-in, 2 defor- OCFT mations are subleading to deformations for the operators studied here. OCFT

Once the conformal symmetry is broken, the spectrum consists of particle- like excitations with masses Mgap which can be thought of as bound states of ∼ the original CFT degrees of freedom. We assume that one of these excitations is stable on cosmological time scales, for example, due to a discrete symmetry. This is the particle that will play the role of dark matter (DM). Regarding the DM particle mass, we will consider two possibilities. One is that the DM particle is

a generic bound state, with mass mDM = Mgap (up to order-one factors). The second one is that the DM particle is a pseudo-Goldstone boson (PGB) of an approximate global symmetry spontaneously broken at Mgap, similar to pions in QCD. In this case, m Mgap is natural, with the DM mass dictated by the DM amount of explicit symmetry breaking.

7.3 Ultraviolet Theory

The strongly-coupled, conformally invariant dark sector of our model can arise from a weakly-coupled, asymptotically free theory in the UV. This can be a simple SU(Nc) gauge theory with NF fermion flavors i, which flows towards a Q Banks-Zaks (BZ) fixed point in the infrared [122]. The theory becomes strongly coupled, and approximately conformal, at a scale ΛCFT. The interaction with the SM starts out as 1 int = SM BZ , (7.5) dU L MU O O

104 where BZ is a gauge-invariant operator in the dark-sector gauge theory, with a O scaling dimension dBZ. Since dU = dSM+dBZ 4 > 0, this interaction is irrelevant, − and MU Λ is required for consistency. At the scale ΛCFT, this interaction CFT term is matched to the one in Eq. (7.1), with

dU ΛCFT λCFT 1. (7.6) ∼ MU

† For example, we may consider the quark bilinear operator BZ = i, with O Qi Q dBZ = 3. Our scenario requires that this operator acquire a large anomalous dimension, γBZ (1), at the IR ﬁxed point. Such large anomalous dimensions, ∼ O with the sign consistent with our scenario, have been observed in lattice studies of SU(3) gauge theory with Nf = 10 [123] and Nf = 8 [124, 125, 126, 127, 128], as well as in analytic scheme-independent calculations at higher orders in perturbation theory [129, 130].

We note that in the case of Higgs portal for the special value d = 2, the theory we consider bears some superﬁcial resemblance to the “scale-invariant” dark sectors consisting of massless, weakly-coupled ﬁelds, considered in previous studies (see for example [131, 132]). However, the physics is completely different: our dark sector is strongly coupled, the operator has a non- OCFT perturbatively large anomalous dimension (which just happens to be an integer at this special point), and is genuinely (not just classically) conformally invariant.

With the exception of the IR breaking of conformal symmetry, the ﬁeld theory model considered here is identical to Georgi’s “unparticle” framework [116, 133], with SM coupling to a scalar CFT operator. (For earlier work on cosmology with unparticles, see [134, 135, 136, 137, 138].) In other words, the dark sector behaves as an unparticle at energy scales above Mgap and below ΛCFT.

105 7.4 Cosmological Evolution

We assume that after the end of inflation, the inflaton decay reheats the SM sector to a temperature TR, but the dark sector is not reheated due to absence of a direct coupling to the inflaton. As the Universe expands and cools after reheating, collisions and decays of SM particles gradually populate the dark sector. Assuming Mgap TR < ΛCFT, this process occurs via production of CFT stuff (“unparticles”). The dark sector cannot be described by Boltzmann equations, since the concept of particle number density is not applicable in the CFT. However, since the dark sector has many degrees of freedom and they interact strongly among themselves, it will be in a spatially isotropic thermal state. Rotational symmetry dictates that the energy-momentum tensor of this state has the form T µν = diag(ρ , P , P , P ), while conformal in- CFT − CFT − CFT − CFT 1 variance further requires PCFT = 3 ρCFT . The CFT energy density is given by

4 ρCFT = ATD, (7.7)

where TD is the temperature of the CFT sector, and A is an order-one model- dependent constant. We will study a scenario where TD T at all times, where T is the SM plasma temperature; at the same time, TD > Mgap during the period when the dark sector is populated, so that the CFT description is appropriate.

On the SM side, the particle number is well-deﬁned and the Boltzmann equations have the usual form, with collision terms describing the loss of SM particles due to annihilations (SM+SM CFT) and decays (SM CFT), and → → their creation due to inverse processes. The evolution of the SM energy density ρSM follows from the Boltzmann equations:

dρSM + 3H(ρ + P ) = ΓE(SM CFT) + ΓE(CFT SM) , (7.8) dt SM SM − → →

106 where H is the Hubble expansion rate, and ΓE are energy transfer rates per unit volume. In our scenario, the CFT sector will always remain at densities far below equilibrium with the SM, and ΓE(CFT SM) can be safely neglected. → The energy transfer rate from SM to CFT is given by

X ΓE(SM CFT) = ninj σ(i + j CFT)vrelE → i,j h → i P + ni Γ(i CFT)E , (7.9) i h → i where the sums run over all SM degrees of freedom coupled to the CFT. The cross sections and decay rates can be evaluated using the technique of Georgi [116, 133]. For example, with the Higgs portal, the Higgs decay contribution is given by

2 2 2(d−1) fdλCFT v mh T nh Γ(h CFT)E = 2d−4 K2(mh/T ) (7.10) h → i ΛCFT where mh is the Higgs boson mass, K2(x) is the modiﬁed Bessel function of the second kind, and, Γ(d + 1/2) f = 2−2dπ1/2−2d . d (Γ(d 1)Γ(2d)) −

The annihilation contribution (when T mh) is given by 2d+1 2 2 d T nh σ(hh CFT) vrel E = λCFT 2d+1 2d−4 . (7.11) h → i 2(2π) ΛCFT The CFT sector is populated at the time when the energy density is dominated

1 by relativistic SM matter, PSM = 3 ρSM , so that SM and CFT energy densities redshift in the same way. The total energy of the two sectors can only change due to work done against the expansion of the Universe: d (ρ + ρ ) + 4H (ρ + ρ ) = 0. (7.12) dt CFT SM CFT SM Subtracting Eq. (7.8), we ﬁnd that the CFT energy density evolves according to

dρCFT + 4Hρ = ΓE(SM CFT) , (7.13) dt CFT →

107 with the initial condition ρCFT = 0 at T = TR.

With minor simplifying assumptions, such as ignoring the masses of colliding SM particles and the temperature dependence of the effective number of

SM degrees of freedom g∗, Eq. (7.13) can be solved analytically (see Appendix

B for details). The qualitative behavior of ρCFT with temperature is dictated by the dimension d of the operator . We deﬁne the critical dimension, d∗ as OCFT follows,

d∗ = 9/2 dSM. (7.14) −

For d above the critical dimension d∗ (see Appendix A for details), most of the

CFT energy is produced at high temperatures (close to TR), by pair-annihilations of SM particles; the decay contribution, if present, is subdominant.1 On the other hand, for d < d∗, contributions from both pair-annihilations and decays (if present) grow with decreasing T . The resulting CFT density is IR dominated and can be calculated without knowledge of UV quantities such as TR, as in the freeze-in scenario of Ref. [140, 115]. (For a review of variations of freeze-in models and phenomenology, see [141]). We will focus on this case for the rest of the chapter. We call this scenario Conformal Freeze-In, or COFI.

The critical dimensions are d∗ = 5/2 for the Higgs portal, and d∗ = 3/2 for the quark portal and the lepton portal. For the quark and lepton portals, pair annihilations start at the weak scale, and hence, in the UV-dependent regime, relic density depends on the weak scale and not on the reheating temperature.

Therefore, the relic density in these scenarios can be calculated without dependence on details of inﬂation-era cosmology even above the critical dimension.

For the gluon portal, the critical dimension is given by d∗ = 1/2, and by

1For earlier work on freeze-in in the UV sensitive regime, see for e.g. [139].

108 Dominant production mode SM Temperature OSM Higgs Portal H CFT T Tweak → . Quark Portal qq¯ CFT Λ < T < Tweak → QCD Gluon Portal gg CFT Λ < T < Λ → QCD CFT + − Lepton Portal ` ` CFT m` < T < Tweak → Table 7.2: Dominant modes of production in each portal.

unitarity, the CFT operator dimension can never be lower than d∗. Thus, production is always UV-dominated, and changing the reheating temperature of the universe can affect the relic density.2

The dominant dark sector energy density production modes for each portal are described in Table 7.2. For all portals, production of CFT energy effectively ceases soon after the SM temperature drops below the mass of the particle coupled to the CFT. In the Higgs portal, Higgs pair annihilations end at T mh, ∼ and Higgs decays to the dark sector occur right below the weak scale. There is also a residual quark/gluon-CFT interaction below the weak scale, induced through integrating out the Higgs, but numerically, its effect is subdominant. For the lepton portal, production continues until the temperature reaches the mass of the corresponding ﬂavor of lepton, T m`. In the quark and gluon ∼ portal cases, pair annihilations end at mq for heavy quarks and at T Λ for ∼ QCD light quarks and gluons, since QCD conﬁnement occurs at this scale and free quarks and gluons no longer exist in the thermal bath.

For the quark and lepton portal models, we consider two interesting scenar-

2However, this scenario is still somewhat predictive, due to the powers of the reheating temperature, TR, being low compared to that of other dimensionful factors such as ΛCFT . For many orders of magnitude in chosen TR, the correct relic density is produced for very similar gap mass scales.

109 ios - first, we consider the case in which the dark sector couples democratically to all flavors of quarks/leptons; second, we assume that the CFT operator couples only to the first generation (u and d quarks; electrons). In the former case, heavy fermions in the loop determine the mass gap, while in the latter, only the first generation fermions can run in the loop, and the mass gap is generated at a lower scale due to Yukawa suppression. The fermions in the first generation contribute the most to production of CFT-sector energy in both cases, as pair annihilations of light particles continue till lower energy scales than heavier particles that are no longer in the thermal bath below their mass. The two scenarios give different mass gap scales for which the correct relic abundance is produced, as shown in Figs. 7.5, 7.6, 7.8, and 7.9.

It is worth noting that in the quark and lepton portals, pair annihilations can only occur after electroweak symmetry breaking, i.e. production starts as the SM temperature drops to the weak scale. In theory, in these two portals, there are contributions from hff¯ CFT and hf f +CFT, where f refers to quarks → → or leptons depending on the SM operator used. These are subdominant due to phase space suppression, and have been neglected in the calculations in this chapter.

−4 After the end of production, ρCFT redshifts as a , until the CFT temperature drops to TD Mgap. At that time, conformal symmetry is broken and a con- ∼ ﬁning phase transition takes place. We assume that all of the energy stored in the CFT sector is transferred to DM particles, on a time scale short compared to

Hubble at that time. If m Mgap, the dark matter energy density will con- DM tinue to redshift as radiation until its temperature drops below mDM , after which it behaves as non-relativistic matter.

110 -3 ρD ∼  10-5

-4 ρD ∼ 

10-7

Decay

10-9 Annihilation

-11 10 mDM TR TD ~m DM TSM ~Λ QCD TSM ~m h

TSM 1000 TD

0.001

0.001 1 1000

Figure 7.3: Top panel: Energy density in the CFT plasma or dark matter particles, as a function of the SM plasma temperature T , in the Higgs portal scenario 8 17 with Λ = 1.2 10 GeV and MU = 10 GeV. Bottom panel: Evolution of the CFT × CFT plasma temperature, as a function of T , for the same parameters. At all times, TD T , as required for the self-consistency of our calculations.

With these assumptions, the relic density can be estimated analytically (see Appendix B). For example, for the Higgs portal, the relic density is dominated by the Higgs decay contribution, and is given by

 1/4  3 −9/2 " 3d # A f g∗ (6− 2 ) h m i d (Mgap/mh) Ω h2 = 0.1 DM   , (7.15) DM 1 MeV  10−5  10−12

111 where g∗ 100 is the number of relativistic SM degrees of freedom at the weak ∼ scale. An example of the evolution of CFT/DM energy density, for the Higgs portal scenario and parameters that provide the observed DM relic density, is shown in Fig. 7.3. This and all ﬁgures below are based on full numerical solutions of Eq. (7.13) (which is in good agreement with Eq. (7.15) for the Higgs portal scenario).

The lower panel of Fig. 7.3 demonstrates that the CFT plasma remains at temperatures well below those of the SM plasma, TD T , as required for the self-consistency of our calculations. In fact, it can be shown that this is automatically the case for parts of the parameter space where the correct relic density can be achieved through the freeze-in process.

7.5 Dark Matter Phenomenology

Since the dark matter mass and its interaction strength are related through the breaking mechanism described above, the COFI scenario is remarkably predictive. In particular, there is a nearly universal relationship between the dark matter relic density and the gap scale, with only a mild dependence on other parameters. In the case of the Higgs portal, the observed relic density is reproduced for Mgap 1 10 MeV, while for the gluon portal, it is ∼ − Mgap 0.1 10 MeV. For the quark and lepton portals, the correct relic den- ∼ − sity is achieved for Mgap 0.1 1 MeV in the case of ﬁrst generation coupling, ∼ − and Mgap 1 10 MeV for democratic coupling. The parameters that pro- ∼ − duce the correct relic density for each of the models are given in Figs. 7.4, 7.5,

7.6, 7.7, 7.8 and 7.9. Note that the plots show the dark matter mass, given by

112 2.4

2 2.2 — ΩDMh

Self Int. 2.0 0.01 0.1 10

1.8

1.6 Bootstrap

1.4 "Naturalness"

1.2

1.0 1 10 100 1000 104

Figure 7.4: Dark matter relic density contours (red) and observational/theoretical constraints, in the Higgs portal model. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is deﬁned as ΛSM/4πv.

m = r Mgap, and not the mass gap itself. The parameter r is chosen between DM × 0.1 and 0.01, so that the self-interaction constraint can be avoided as much as possible while achieving the correct relic abundance.

As mentioned previously, there is a caveat for the gluon portal scenario - SM CFT production is UV-dominated and happens predominantly around → the reheating temperature. However, all is not lost, since a factor of 106 increase in reheating temperature is required to decrease the mass gap by a factor of 10. (See 7.B for relic density formulae). This means that for large ranges of reheating

5 temperature (TR 1 TeV 10 TeV), the conclusion regarding mass gap scale ∼ − holds. For higher reheating temperatures, the mass gap falls by about one order

113 2.0

Warm DM Figure 7.5: Dark matter 1.8 2 — ΩDMh relic density contours (red) and observational/theoretical con- 1.6 straints, in the quark Bootstrap portal model with only the ﬁrst generation of Self Int. quarks. Thick red line 1.4 indicates parameters

0.01 0.1 10 where the observed dark matter abundance 1.2 SN1987A is reproduced. α is deﬁned as ΛSM/4πv.

1.0 0.01 0.10 1 10 100 1000

2.0 Figure 7.6: Dark matter relic density contours

1.8 2 — ΩDMh (red) and observational/theoretical constraints, in the quark portal model 1.6 Bootstrap with all generations of quarks. The same value of r has been 1.4 used for comparison. Thick red line indicates 0.01 0.1 10 parameters where the observed dark matter 1.2 abundance is reproduced. α is deﬁned as SN1987A ΛSM/4πv. 1.0 1 10 100 1000 104

of magnitude for every 5 6 orders in magnitude in TR. ∼ −

114 2.0

0.01 0.1 10 1.8 2 — ΩDMh

1.6 Bootstrap

1.4

1.2 Self Int. SN1987A

1.0 0.01 0.10 1 10 100 1000 104

Figure 7.7: Dark matter relic density contours (red) and observational/theoretical constraints, in the gluon portal model. Thick red line indicates parameters where the observed dark matter abundance is reproduced. α is deﬁned as ΛSM/4πv.

If the DM particle is a generic bound state of a strongly-coupled theory with m Mgap, its elastic self-scattering interaction cross section can be estimated DM ∼ as σ 1/ 8πM 2 . This is far too large, in all portal scenarios, to be consistent self ≈ gap with bounds from galaxy clusters such as the Bullet cluster [113, 4]. We therefore consider the case where the DM is a PGB, with mass ratio r = m /Mgap 1. DM The self-scattering cross section scales as

6 6 mDM r σself 8 = 2 , (7.16) ∼ 8πMgap 8πMgap where we assumed that self-scattering is mediated by states with masses Mgap ∼ (e.g. the counterparts of the ρ meson of QCD), and derivative couplings of the

PGB have been taken into account. Modest values of r 0.01 0.1 are sufﬁcient ∼ −

115 2.0

Warm DM Figure 7.8: Dark matter 1.8 2 — ΩDMh relic density contours (red) and observational/theoretical 1.6 constraints, in the lep- Bootstrap ton portal model with coupling to only elec- Self Int. trons. Thick red line 1.4 indicates parameters

0.01 0.1 10 where the observed dark matter abundance 1.2 is reproduced. α is deﬁned as ΛSM/4πv.

1.0 0.001 0.010 0.100 1 10 100

2.0

Figure 7.9: Dark matter relic density contours 1.8 2 — ΩDMh (red) and observational/theoretical constraints, in the 1.6 lepton portal model Bootstrap with coupling to all charged leptons. The same value of r has 1.4 been used for compar-

0.01 0.1 10 ison. Thick red line indicates parameters 1.2 where the observed dark matter abundance is reproduced.

1.0 1 5 10 50 100 500

116 to avoid the self-interaction bounds for parameters with viable ΩDM . This is illustrated in Figs. 7.4, 7.5, 7.6, 7.7, 7.8 and 7.9.

Another important phenomenological constraint is that the DM should be cold, i.e. remain non-relativistic during structure formation [142]. This constraint is somewhat weaker than in the case of SM sterile neutrinos, m & 5 keV, because the CFT sector is colder than the SM. Nevertheless, for the quark and lepton portals with couplings to the ﬁrst generation, the warm DM bound rules out a part of the parameter space; see Figs. 7.5, and 7.8. For the scenario with couplings to all generations, the DM mass tends to be higher 3 so that it redshifts as matter for a longer period of time than the case with couplings only to the ﬁrst generation. This means that at the end of production, the amount of

CFT-sector energy density in this scenario is smaller than that of the other, making the temperature of the CFT sector even colder, and the warm DM bound weaker. For the other portals, the DM is heavier and this bound is irrelevant.

There are many experimental and observational constraints on the strength of CFT/DM coupling to the SM. These include LHC searches for unparticles produced in qq¯ annihilations [143, 144] in the quark portal scenario, or Higgs decays [145, 146] in the Higgs portal scenario; LEP bounds on e+e− γ+ → E/T [147] in the lepton portal case; bounds on invisible meson decays involving unparticles from the B-factories [148, 149, 150]; supernova SN1987a energy loss and stellar cooling due to unparticle or DM emissions [151, 134, 152]; mod- iﬁcation of the ionization history due to energy injection by late-time DM annihilations [153, 154, 155, 156]; diffuse X- and Gamma-ray backgrounds [157]; and spectral distortion of the CMB blackbody distribution by early energy injec-

3The DM mass and mass gap are higher in the democratic coupling case, because the heavy fermions in the loop lead to larger breaking and thus, the mass gap is higher for a given coupling.

117 tion [158]. We checked that our scenarios are consistent with all these bounds, due to a highly suppressed coupling between the dark sector and the SM. Note also that the Big Bang Nucleosynthesis (BBN) bound on the number of new light degrees of freedom [159] does not apply, because TD T at the time of BBN.

On the theoretical side, there are several consistency conditions that may further constrain the parameter space, as shown in the ﬁgures for each portal. The “naturalness” bound in the Higgs portal plot stems from requiring that if

TR < ΛCFT , then MU < MPl. “Bootstrap” condition, d > 1.61, is explained in section 7.2. Both these bounds are model-dependent, and may be modiﬁed or eliminated by the choice of a UV completion of the CFT and symmetry charge assignment for , respectively. OCFT

We conclude that for all four models studied, the conformal freeze-in (COFI) dark matter scenario is easily viable, consistent with all observational and theoretical constraints. This scenario contains a novel DM candidate, a CFT bound state, whose mass is predicted to be in the experimentally interesting sub-MeV range. The lower bound on the DM mass is about a keV, where the warm dark matter constraint becomes insurmountable.

7.A Appendix A: Critical Dimension (d ) ∗

Here, we derive the critical dimension (d∗) of the CFT operator, below which dark sector energy production is IR-dominated and independent of early Uni- verse parameters such as the reheating temperature.

In the examples considered in this chapter, SM + SM CFT and SM → →

118 CFT processes are the only relevant modes of production. Others, such as 3 SM CFT processes, are disfavored strongly by phase space suppression → and can be ignored. In the case of Standard Model decays, production is always UV-insensitive, as the bulk of it occurs when the temperature is of the same order as the mass of the SM particle. On the other hand, production from 2 CFT processes may occur primarily either in the UV or in the IR, depend- → ing on the dimension of the CFT operator coupled to the Standard Model.

Consider the following coupling, as shown in Eq. (7.1):

λCFT int = (D−4) SM CFT , (7.17) L ΛCFT O O where D = d + dSM as before.

We will calculate the critical dimension for the case of 2 CFT processes as → follows. From dimensional analysis, it is easy to relate the collisional term in the Boltzmann equation for two SM particles to the SM temperature and the CFT

2(D−4) scale, since the cross-section under consideration will have a factor of 1/ΛCFT . Without any dimensionless factors, we get

2 6 T 2D−9 nSM T , σ v E 2(D−4) ∼ h i ∼ ΛCFT T 2D−3 ΓE(SM CFT) Λ2D−8 . (7.18) ⇒ → ∼ CFT

The dark sector energy density can be estimated by integrating the Boltzmann equation (Eq. (7.13)) with the added constraint that there is no dark sector energy above the reheating temperature. Thus, for temperatures below reheating

(T < TR),

−η −η 4 T TR ρCFT T − with η = 9 2(d + d ). (7.19) ∼ η − SM

For η > 0 d < d∗ = 9/2 dSM, the reheating temperature term will be ⇒ −

119 negligible due to its negative exponent, and most of the dark sector energy density will be produced at lower temperatures. Thus, it will be IR-dominated, and relic density will not depend strongly on UV parameters such as the reheating temperature. On the other hand, for d > d∗, the reheating temperature plays a relevant role: energy density production peaks at TR, and then dilutes due to the expansion of the Universe.

In the Higgs portal case, we checked that, as long as d < d∗, Higgs decay is the dominant process in production of CFT energy density. Above the critical dimension however, this is not necessarily true, as the scattering contribution may dominate for a sufﬁciently high reheating temperature.

7.B Appendix B: Derivation of Relic Density

In this section, we show a brief derivation of Eq. (7.15), that relates observed dark matter relic density to parameters in the theory in the Higgs portal. In addition, the computation for Eq. (7.10) is shown in more detail. Using the same procedure, analytical results for relic density can be computed for all portals considered in this chapter, and the results for other portals are summarised at the end without going into technical details.

In the Higgs portal case, as mentioned before, below the critical dimension d = 5/2, dark matter production is dominated by the Higgs decay process. At ∗ temperatures below the electroweak phase transition, the effective interaction between the dark sector and the SM becomes,

λCFT v int = D h CFT . (7.20) L ΛCFT √2 O

120 The energy transfer rate through this process is given by Eq. (7.10) and can be computed as follows: ZZ 4 4 2 nh Γ(h CFT) E = dΠhdΠCFTfh(2π) δ (ph P )Eh . (7.21) h → i − |M|

Here and below, P = pCFT is the momentum carried by the dark sector. The phase space for the CFT sector is chosen to be identical to that of “unparticles” as prescribed by Georgi in [116]. Using Georgi’s notation, we have,

nh Γ(h CFT) E h → i ZZ 3 4 2 λ2 d ~ph d P −βEh 4 4 2 d−2 v CFT = 3 4 e (2π) δ (ph P ) Ad (P ) Eh 2d−4 (2π) 2Eh (2π) − 4 ΛCFT A v2 λ2 Z 3 q d CFT 2 d−2 d ~ph 2 2 = 2d−4 (mh) 3 exp( ~ph + mh), 4ΛCFT 2(2π) | | (7.22) where, 16π5/2 Γ(d + 1/2) Ad = . (7.23) (2π)2d Γ(d 1)Γ(2d) −

Setting p = ~ph and simplifying gives | | A v2 λ2 (m2 )d−2 Z q d CFT h 2 dp 2 2 nh Γ(h CFT) E = 2d−4 4πp 3 exp( β p + mh) h → i 4ΛCFT 2(2π) − A v2 λ2 (m2 )d−2 Z q d CFT h 2 2 2 = 2 2d−4 p dp exp( β p + mh). 32π ΛCFT − (7.24)

The integral represents a Bessel function of the second kind. Additionally, in

2 our notation, fd = Ad/16π . Thus, on simplifying, we get,

2 2 2(d−1) fdλCFT v mh T nh Γ(h CFT)E = 2(d−4) K2(mh/T ). (7.25) h → i ΛCFT The CFT energy density at any point in time (as a function of the Standard Model bath temperature) can be obtained by integrating the Boltzmann equation given in Eq. (7.13). To get a simple estimate, it sufﬁces to do this calculation

121 in the relativistic approximation where the Higgs is assumed to be massless and is described by a Maxwell-Boltzmann distribution. The process roughly starts around the electroweak scale v and continues till the SM temperature reaches ∼ the Higgs mass.

In the relativistic approximation (i.e., taking the limit mh 0 in the thermal → average calculation), the energy transfer rate in this process is given by,

2d−4 2 2 mh 3 nh Γ(h CFT)E = 2fd λCFT v 2d−4 T . (7.26) h → i ΛCFT We integrate the Boltzmann equation with this collisional term, ignoring the temperature dependence of g∗ for now, and enforcing the condition that decays are inactive above the electroweak scale. Thus, we have,

2 2d−4 3 2M∗fdλCFT mh 4 v ρCFT (T ) = p T 3 1 , (7.27) 3 g∗(T )v ΛCFT T −

3/2 where M∗ = 3√5/(2π ) Mpl, comes from the deﬁnition of Hubble as H =

2 √g∗ T /M∗.

At T mh, as the Higgs falls out of the thermal bath, this process becomes ∼ exponentially suppressed, and further production of dark sector energy can be neglected for this analysis. The energy density present in the dark sector then

−4 redshifts like radiation (ρ a ) until its temperature TD becomes comparable ∝ to the mass of the dark matter candidate. After this point, it redshifts like matter (ρ a−3) as required. ∝

Thus, 2 2d 3 2M∗fdλCFT mh v ρCFT (mh) = p 2d−4 3 1 , (7.28) 3 g∗(mh)v ΛCFT mh − and

2 2d−4 3 2M∗fdλCFT g∗(Tm) mh v 4 ρCFT (Tm) = 3/2 3 1 Tm, (7.29) 3(g∗(mh)) v ΛCFT mh −

122 where Tm is the SM temperature at which the dark sector temperature drops to the mass of the dark matter candidate. From Eq. (7.7), we know that at this

4 temperature, ρCFT = A mDM , and thus, the relic density is given by 3 4 g∗(T0)T0 ρDM(T0) = A mDM 3 , (7.30) g∗(Tm)Tm where T0 is the current CMB temperature. Additionally, from Eq. (7.29), Tm is given by, −1 " 2 2d−4 3 # 4 4 2M∗fdλCFT g∗(Tm) mh v Tm = A mDM 3/2 3 1 (7.31) 3(g∗(mh)) v ΛCFT mh − Using Eq. (7.31) in Eq. (7.30) gives the relic density of dark matter from the

Higgs portal in terms of other parameters in the theory.

Note that we use g∗(T0) g∗(Tm) (1). This is a reasonable approx- ∼ ∼ O imation, as both temperatures are below the QCD scale. g∗(mh), denoted as

v3 just g∗ below, is approximately (100). We also replace m3 1 (1) for O h − → O this order-of-magnitude estimate. Additionally, we substitute Mgap in the equa-

tion instead of λCFT and ΛCFT using the mass gap equations. Taking the ratio of

ρDM(T0) and the present critical energy density gives Eq. (7.15):

 1/4   (6− 3d )  3 −9/2 Mgap 2 2 A f g∗ Ω h h m i d mh DM = DM     . (7.32) 0.1 1MeV  10−5   10−12 

This simple estimate is in good agreement with the results of numerical integra- tion of Eq. (7.13).

Following the same procedure, the relic density can be calculated for each of the other three portals. These equations are given below, neglecting deriva- tives of g∗, but keeping all scales intact. For the quark and lepton portals, coupling to all ﬂavors equally is assumed. The analytical formulae for the scenario with only ﬁrst generation couplings can be easily derived by replacing

123 the heavy fermion masses in the equations with that of light fermions (appropriately summed).

Quark Portal:

2 3/4 6−3d/2 1/4 3 M∗ 4d(d 1) 2d−3 2d−3 ρ (T0) = m M A T − (v Λ ) DM DM gap 0 α4v4m2 (2d 3)(2π)2d+1 − QCD top − (7.33)

Gluon Portal:

2 2 3/4 6−3d/2 1/4 3 M∗ 144d (d 1)(d + 2) 2d−1 ρ (T0) = m M A T − T DM DM gap 0 α4v4m4 (2d 1)(2π)2d−1 R top − (7.34)

Lepton Portal:

2 3/4 6−3d/2 1/4 3 M∗ 4d(d 1) 2d−3 2d−3 ρ (T0) = m M A T − (v m ) DM DM gap 0 α4v4m2 (2d 3)(2π)2d+1 − e τ − (7.35)

As in the Higgs portal case examined previously in this appendix, these analytical estimates are in good agreement with the numerically integrated results shown in Figs. 7.4, 7.5, 7.6, 7.7, 7.8 and 7.9.

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