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Author/s: BALDES, IASON

Title: Early universe cosmology and the matter-antimatter asymmetry

Date: 2015

Persistent Link: http://hdl.handle.net/11343/55205

File Description: Early universe cosmology and the matter-antimatter asymmetry The University of Melbourne

Doctoral Thesis

Early Universe Cosmology and the Matter-Antimatter Asymmetry

Supervisors: Author: Assoc. Prof. Nicole F. Bell Iason Baldes and Prof. Raymond R. Volkas

Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy

in the

School of Physics The University of Melbourne

July 2015

Produced on archival quality paper Early Universe Cosmology and the Matter-Antimatter Asymmetry

by Iason Baldes

UNIVERSITY OF MELBOURNE Faculty of Science School of Physics Doctor of Philosophy

Abstract

The universe is made up almost exclusively of matter and not antimatter. Having made this observation, what can we say about the conditions of the first second of the universe after the big bang? The high temperatures present in that epoch allow the copious production of massive particles. These particles can create the matter- antimatter asymmetry which is thought to arise dynamically in the early universe. We first discuss a rarely studied way of generating the matter-antimatter asymmetry using collisions. We then investigate open questions in particle physics such as neutrino masses and the identity of dark matter in light of these considerations. Observable effects in particle colliders or proton decay are also investigated.

Chapter 1 provides an introduction to the baryon asymmetry of the universe. We begin by outlining the observational evidence showing that the universe has a baryon asym- metry. We then review the Sakharov conditions — the conditions required to generate a baryon asymmetry dynamically. We discuss various baryogenesis mechanisms, proposed in the literature, for generating the baryon asymmetry of the universe. We also intro- duce the Boltzmann equations which are used for calculating the asymmetry produced in standard model extensions. Possible related issues of the neutrino masses and dark matter density are also touched upon.

In Chapter 2 we consider the principles behind generating particle-antiparticle asym- metries using annihilations. This mechanism has not been extensively studied in the literature. In providing a general framework for generating asymmetries with annihila- tions we hope to point out the challenges and possibilities for further research in this area.

In Chapter 3 we apply what we have learned about asymmetry generation from anni- hilations to study a neutron portal baryogenesis scenario. We include the charge parity violating annihilations in our calculations. We discuss the formulation of the Boltzmann ii equations and solve these equations numerically. We show the annihilations play the dominant role over the decays in generating the baryon asymmetry in this scenario.

In Chapter 4 we change tack slightly and study the cosmological implications of two radiative inverse seesaw models. These models can explain the neutrino masses inferred from neutrino oscillations. They violate lepton number and can interfere with — or perhaps generate — the baryon asymmetry in the early universe. These models contain dark matter candidates and we discuss the phenomenology thereof. We discuss limits from colliders and other experiments.

In Chapter 5 we study extensions of the standard model involving exotic scalars. These scalars violate the baryon number and can therefore change the baryon asymmetry. We discuss limits from nucleon stability and cosmology. We also discuss the LHC phe- nomenology of such particles.

We then briefly conclude. Declaration of Authorship

This is to certify that:

i. the thesis comprises only my original work towards the PhD except where indicated in the Preface,

ii. due acknowledgement has been made in the text to all other material used,

iii. the thesis is fewer than 100 000 words in length, exclusive of tables, maps, bibli- ographies and appendices.

Signed:

Date:

iii Preface

This thesis comprises six main chapters. Chapter 1 is an original literature review. Chapter 2 is based on publication 1. Chapter 3 is based on publication 2. Chapter 4 is based on publication 3. Chapter 5 is based on publication 4. Chapter 6 is the conclusion. These publications were done in collaboration with Nicole F. Bell (publications 1–4), Alexander Millar (Publication 2), Kalliopi Petraki (publications 1–3) and Raymond R. Volkas (publications 1–4). While Nicole F. Bell, Kalliopi Petraki and Raymond R. Volkas are responsible for the original inspiration for these projetcs, all calculations, results and analyses presented within this thesis are my own work unless stated otherwise. Chapter 5, based on publication 4, deals with baryon number violating scalar particles. Such particles were also the topic of my MSc thesis. The following substantial additions and improvements have been made: (i) inclusion of an estimate of the loop integrals in the nucleon stability bounds, (ii) inclusion of the Boltzmann suppressed inverse decay rate which can have a substantial effect on the washout bound, (iii) corrected and updated calculation of the resonant production of diquarks at the LHC.

Publications

1. I. Baldes, N. F. Bell, K. Petraki, R. R. Volkas, “Particle-antiparticle asymmetries from annihilations,” Phys. Rev. Lett. 113 (2014) 181601, arXiv:1407.4566.

2. I. Baldes, N. F. Bell, A. Millar, K. Petraki, R. R. Volkas, “The role of CP violating scatterings in baryogenesis — case study of the neutron portal,” JCAP 1411 (2014) 041, arXiv:1410.0108.

3. I. Baldes, N. F. Bell, K. Petraki, R. R. Volkas, “Two radiative inverse seesaw mod- els, dark matter, and baryogenesis,” JCAP 1307 (2013) 029, arXiv:1304.6162.

4. I. Baldes, N. F. Bell, R. R. Volkas, “Baryon Number Violating Scalar Diquarks at the LHC,” Phys. Rev. D 84 (2011) 115019, arXiv:1110.4450. Acknowledgements

First of all, I thank my supervisors, Nicole Bell and Ray Volkas, for their support over many years during my MSc and PhD. Without their patience, belief in me and generosity with their time, I could not have completed this work. I thank Kalliopi Petraki for her patience and help during our collaboration.

I also thank my parents for their ongoing support and instilling in me a curiousity for intellectual pursuits. I thank Vesna for her support, patience, encouragement and unwavering enthusiasm for physics and science.

I should also mention the engaging and friendly atmosphere in CoEPP. Discussion with other students and postdocs helped me a great deal. In particular I would like to acknowledge Ahmad Galea, Benjamin Callen, Peter Cox, Stephen Lonsdale, Timothy Trott and Rebecca Leane for our discussions. Finally I especially thank Alex Millar for teaching me the Cutkosky rules.

v Contents

Abstract i

Declaration of Authorship iii

Preface iv

Publications iv

Acknowledgements v

Contents vi

List of Figures ix

Abbreviations xi

1 Introduction to the baryon asymmetry 1 1.1 Evidence for the baryon asymmetry ...... 1 1.1.1 The discovery of antimatter ...... 1 1.1.2 Problems with a baryon symmetric universe ...... 2 1.1.3 Big bang nucleosynthesis ...... 4 1.1.4 Cosmic microwave background ...... 4 1.2 Baryogenesis mechanisms ...... 6 1.2.1 Initial conditions ...... 6 1.2.2 Sakharov conditions ...... 6 1.2.3 Electroweak baryogenesis ...... 7 1.2.4 Leptogenesis ...... 9 1.2.5 Affleck-Dine baryogenesis ...... 13 1.2.6 Spontaneous baryogenesis ...... 13 1.2.7 Asymmetric Dark Matter ...... 14 1.2.8 Leptogenesis via collisions ...... 15 1.2.9 Baryogenesis from dark matter annihilation ...... 18 1.3 Boltzmann equations ...... 21 1.3.1 Thermodynamics and the collision integral ...... 21

vi Contents vii

1.3.2 S Matrix Unitarity and Time Reversal ...... 25 − 1.4 Conclusion ...... 28

2 Particle-antiparticle asymmetries from annihilations 30 2.1 Introduction ...... 30 2.2 Toy model ...... 31 2.3 Boltzmann equations ...... 38 2.4 Conclusion ...... 42

3 The role of CP violating scatterings in baryogenesis — case study of the neutron portal 44 3.1 Introduction ...... 44 3.2 Neutron portal ...... 46 3.2.1 Lagrangian ...... 46 3.2.2 Decays ...... 47 3.2.3 Scatterings ...... 49 3.3 Boltzmann equations ...... 51 3.3.1 Differential equations ...... 51 3.3.2 Chemical potentials ...... 52 3.3.3 Numerical solutions ...... 55 3.4 Constraints ...... 59 3.5 Conclusion ...... 60

4 The baryon asymmetry and dark matter in radiative inverse seesaw models 62 4.1 Introduction ...... 62 4.2 The inverse seesaw mechanism ...... 63 4.2.1 The generic mass matrix ...... 63 4.2.2 ISS and baryogenesis ...... 65 4.3 Law/McDonald radiative inverse seesaw ...... 67 4.3.1 Review of the model ...... 67 4.3.2 Constraints from BAU washout ...... 69 4.3.3 Dark Matter ...... 73 4.3.4 Concluding remarks on the Law/McDonald model ...... 82 4.4 Ma radiative inverse seesaw model ...... 83 4.4.1 Review of the model ...... 83 4.4.2 Constraints from BAU washout ...... 86 4.4.3 Resonant leptogenesis ...... 89 4.4.4 The ρ parameter ...... 90 4.4.5 LHC searches ...... 92 4.4.6 Dark matter ...... 94 4.4.7 Concluding remarks on the Ma model ...... 96 4.5 Conclusion ...... 97

5 The baryon asymmetry, nucleon stability and LHC searches for scalar diquarks 98 5.1 Introduction ...... 98 5.2 Baryon number violating scalars ...... 99 Contents viii

5.2.1 A catalogue of models ...... 99 5.2.2 The particular model: σ3.3σ3.3σ7.2 ...... 104 5.3 Washout of baryogenesis ...... 107 5.3.1 High Temperature Baryogenesis ...... 107 5.3.2 Low temperature baryogenesis ...... 108 5.4 Collider searches ...... 110 5.4.1 Approximately B conserving regime ...... 111 5.4.2 B violating regime ...... 112 5.5 Discussion ...... 114 5.6 Conclusion ...... 115

6 Conclusion 116

A Toy model cross sections and CP violation 119 A.1 Toy model cross sections and CP violation ...... 119 A.2 Thermally averaged decay rate ...... 123

B Neutron portal unitarity, cross sections, decay rates and CP violation124 B.1 Unitarity constraint for multiple quark generations ...... 124 B.2 Cross sections, decay rates and CP violation ...... 125

C Inverse seesaw cross sections 131 C.1 Double Higgs portal cross sections ...... 131 C.2 Cross Section for pp Z0 l+l− ...... 132 → → D Loop integrals and cross sections for baryon number violating diquarks134 D.1 Loop integrals for nucleon decay ...... 134 D.2 Resonant diquark production ...... 136

Bibliography 138 List of Figures

1.1 Tree and one-loop diagrams for the decay N lLΦ...... 11 → 1.2 Diagrams contributing to the CP violation in the leptogensis via collisions scenario...... 17 1.3 Diagrams leading to CP violating annihilations in the WIMPy baryogen- esis scenario ...... 20

2.1 Tree and one-loop diagrams for the annhilation Ψ1Ψ1 ff ...... 33 → 2.2 CP violation as a function of temperature in the toy model ...... 36 2.3 Numerical errors for the unitarity conditions in the toy model ...... 37 2.5 Example solution for the toy model Boltzmann equations ...... 41

3.1 Tree and loop level decays for X1 and X2 for the neutron portal . . . . . 47 3.2 CP violation in the X2 decays for the neutron portal ...... 48 3.3 Tree and loop level scatterings leading to CP violation in the neutron portal 50 3.4 Scattering CP violation as a function of temperature in the neutron portal 51 3.5 Example solution to the neutron portal Boltzmann equations ...... 55 3.6 Final baryon asymmetry as a function of the couplings in the neutron portal with MX2 = 100 TeV, MX1 = 90 TeV...... 57 3.7 Final baryon asymmetry as a function of the couplings in the neutron portal with MX2 = 100 TeV, MX1 = 50 TeV...... 57 3.8 Final baryon asymmetry as a function of the couplings in the neutron portal with MX2 = 100 TeV, MX1 = 10 TeV...... 58 3.9 Neutron-antineutron oscillation in the neutron portal ...... 60 3.10 Meson decay in the neutron portal ...... 60

4.1 Washout in the inverse seesaw ...... 65 4.2 Radiative mass generation in the Law/McDonald model ...... 69 4.3 Washout interactions in the Law/McDonald model ...... 70 4.4 Possible DM annihilation channels in the Law/McDonald model . . . . . 77 4.5 Required coupling for the observed DM abundance in the Law/McDonald model ...... 78 4.6 DM Nucleon scattering through t-channel Higgs exchange in the Law/M- cDonald model ...... 79 4.7 DM-Nucleon scattering cross section in the Law/McDonald model . . . . 80 4.8 Radiative mass generation in the Ma model ...... 85 4.9 Washout process in the Ma model ...... 86 4.10 Correction to the ρ parameter in the Ma model ...... 91 4.11 Limit on the Ma radiative ISS Z0 from ATLAS searches for dilepton res- onances...... 93

ix List of Figures x

4.12 Mass of the DM particle in the Ma model ...... 95 4.13 DM annihilation in the Ma model ...... 95

5.1 Tree level proton decay in the one scalar extension ...... 101 5.2 Proton decay in the one scalar extension with coupling to the third gen- eration ...... 101 5.3 Tree level n n¯ oscillation for the σ7σ3σ3 model ...... 104 − 0 0 5.4 Two loop nn π π decay for the σ7σ3σ3 model ...... 105 → 0 0 5.5 Two loop nn π π decay for the σ7σ3σ3 model ...... 105 → 5.6 Two loop n n¯ oscillation for the σ7σ3σ3 model with σ7 coupled to the − first and σ3 to the third generation quarks...... 106 5.7 BNV scattering for the σ7σ3σ3 model ...... 108 5.8 Limit on the coupling of the σ particles in order avoid washout ...... 109 5.9 Pair production of σ7σ¯7 through gluon-gluon fusion...... 110 5.10 BNV at the LHC dd σ¯7 σ3 + σ3...... 112 → → 5.11 Cross section for the process dd σ7 jj for pp collisions at √s = 8 TeV.113 → → Abbreviations

ADM Asymmetric Dark Matter BAU Baryon Asymmetry of the Universe BBN Big Bang Nucleosynthesis BNV Baryon Number Violating BSM Beyond the Standard Model CKM Cabibbo Kobayashi Maskawa CMB Cosmic Microwave Background CPT Charge Parity Time DM Dark Matter EFT Effective Field Theory EW Electroweak GIM Glashow Iliopoulos Maiani GUT Grand Unified Theory ISS Inverse Seesaw LHC Large Hadron Collider NNLO Next-to-Next-to-Leading Order PDFs Parton Distribution Functions PMNS Pontecorvo Maki Nakagawa Sakata QCD Quantum Chromodynamics SM Standard Model SUSY Supersymmetry UV Ultraviolet VEV Vacuum Expectation Value WIMP Weakly Interacting Massive Particle

xi Chapter 1

Introduction to the baryon asymmetry

1.1 Evidence for the baryon asymmetry

1.1.1 The discovery of antimatter

The Dirac equation famously predicts, for each fermion, the existence of a corresponding antifermion, with the same mass but opposite charge [1, 2]. No such particles had been observed in 1928 and this initially was thought to rule out the Dirac equation as the correct relativistic description of the electron. A few years later, in 1933, Anderson reported seeing positive particles with masses much smaller than the proton in cosmic rays [3]. These were eventually confirmed as being positrons: the antiparticle of the electron. The antiproton and antineutron were discovered at the Bevatron accelerator, more than twenty years later, in 1955 and 1956 respectively [4, 5].

The stability of the electron is guaranteed as it is the lightest particle carrying electric charge. In contrast one may, a-priori, think the proton could have a decay mode

p e+ + π0, (1.1) →

however, such decays have not been observed and stringent limits on the partial lifetimes have been set, e.g. τ 1.3 1034 years for the above decay mode [6]. This, historically, ≥ ×

1 Introduction. The baryon asymmetry 2 led to the introduction of the concept of conserved baryon number,

NB = Nb N , (1.2) − b

where Nb (Nb) represents the number of baryons (antibaryons).

In the standard model (SM), suppression of baryon number violation is understood to result from an accidental global symmetry present in the Lagrangian once all gauge invariant and renormalisable operators are written down. However, it was eventually realised that baryon number is violated at high temperatures in the SM, as we will see below.

These apparent symmetries between particles and antiparticles led to an obvious ques- tion. Why do we observe vastly more matter than antimatter in the Universe around us? This is still an open question today and is largely the topic of this thesis.

1.1.2 Problems with a baryon symmetric universe

Consider first a universe with equal number of baryon and antibaryons. At high tem- peratures, nucleon-antinucleon annihilation into pions

0 0 NN  π π ...  γγ..., (1.3)

keeps the nucleons in thermal equilibrium with the radiation bath. At temperatures below the nucleon mass the thermally averaged cross section for the annihilation into pions is σv 10−15 cm−3s−1, [7, 8] where v is the M¨ollervelocity [9, 10]. The reaction h i ≈ rate density scales as nN n σv . In approximate thermal equilibrium, we may replace N h i the nucleon densities with their equilibrium values, which for temperatures less than the nucleon mass are given by

m T 3/2 eq eq N −mN /T n = n gN e , (1.4) N N ≈ 2π where gN counts the degrees of freedom of N, MN is the mass of N and T is the temperature. In an expanding universe the number density of a particle absent any interactions will drop as R−3, where R is the scale factor. The annihilation (1.3) pro- ceeds efficiently until the reaction rate density drops below the dilution rate due to Introduction. The baryon asymmetry 3

expansion; 3HnN , where H = R/R˙ is the Hubble parameter. In a radiation dominated Friedmann–Lemaˆıtre–Robertson–Walker universe [11]

r 1/2 8πG 1.66g T 2 1 H = ρ = eff = , (1.5) 3 MP l 2t

−1/2 19 where MP l = G = 1.22 10 GeV is the Planck mass and ×  4  4 X Ti 7 X Ti g = g + g (1.6) eff i T 8 i T i=bosons i=fermions counts the effective radiation degrees of freedom (the sums run over the relativistic species). The nucleon annihilation rate drops below the expansion rate at temperatures T 20 MeV. If there is no separation of baryons from antibaryons in the early universe ≈ this implies a relic baryon to photon density ratio today of [12, 13]

n b 10−19. (1.7) nγ ≈

−10 This is almost ten orders of magnitude below the observed ratio nb/nγ 6 10 . In ≈ × −10 a baryon symmetric universe the ratio nb/nγ 6 10 is reached at temperatures ≈ × T 40 MeV, at which time the post-inflation causal volume H−3 contains a mass of ≈ baryons [14] −7 M 10 M , (1.8) ≈ where M is the solar mass. However, observations show a lack of characteristic gamma rays — expected from the boundaries of hypothetical matter and antimatter dominated regions — and hence baryons and antibaryons are necessarily separated at least over 12 cluster sizes of 10 M [8, 15, 16]. Hence a causal mechanism in the standard cosmology to separate the baryons from antibaryons in the early universe is ruled out [11, 14].

We therefore conclude that the observed baryon density is due to an asymmetry:

nb nb −10 ηB − 6 10 . (1.9) ≡ nγ ≈ ×

The symmetric component of baryon and antibaryons would then annihilate away due to the large nucleon-antinucleon annihilation cross section, leaving the small excess of baryons as the observed baryon density today. In calculations of the baryon asymmetry in baryogenesis, it is useful to normalise to the entropy density, s, rather than the photon Introduction. The baryon asymmetry 4 density. Expressed in such a way the baryon asymmetry is

nb nb nB −10 YB − 0.9 10 , (1.10) ≡ s ≡ s ≈ × as the entropy density today is s = 7.04 nγ [11].

1.1.3 Big bang nucleosynthesis

At time t 1 s or equivalently at temperatures T 1 MeV the universe is cool enough ∼ ≈ for protons and neutrons to fuse. Our knowledge of nuclear physics allows one to make predictions of the expected yield of different elements as a function of the baryon asym- metry. One can then compare with the observed abundances to deduce the baryon asymmetry. Care must be taken because isotopes may be created or destroyed by stars in the later universe [11, 17]. Typically observations of 2H, 3He, 4He and 7Li are used. Following such a procedure results in a conservative estimate of the baryon to photon density of [17] −10 −10 2.5 10 ηB 6 10 , (1.11) × ≤ ≤ ×

−10 −10 or alternatively expressed as a baryon-to-entropy density: 0.4 10 YB 0.9 10 . × ≤ ≤ × The concordance of the different isotope yields from BBN means the baryon asymmetry almost certainly came about before T 1 MeV. ≈

1.1.4 Cosmic microwave background

At time t 105 years the electrons and protons combined to form neutral Hydrogen ∼ (recombination) — the universe became transparent to light — and photons of the radiation bath underwent a last scatter. These photons then continued to red-shift and became the cosmic microwave background (CMB), accidentally discovered by Penzias and Wilson [18, 19], which we observe today.

The CMB consists of black body radiation with a temperature T = 2.73 K. It has been studied in detail by the COBE [20], WMAP [21] and Planck satellites [22], along with various ground based instruments such as the SPT [23]. Though isotropic down to the 10−5 level, the CMB does exhibit some fluctuations, i.e. regions hotter or colder than ∼ Introduction. The baryon asymmetry 5 the mean temperature. Taking the power spectrum of these anisotropies results in a characteristic pattern of peaks and troughs.

The power spectrum of the CMB can be explained in the ΛCDM framework: a universe consisting of the SM particles with a baryon asymmetry, collisionless cold dark matter (DM) and a cosmological constant. The ratio of baryons to DM crucially modifies the height of the peaks relative to the troughs. This is due to baryon acoustic oscillations: before recombination the baryons and photons are tightly coupled due to Thomson scat- tering while the baryons are drawn toward DM overdensities. The radiation pressure resulting from the photon-baryon coupling can expel baryons and photons out of over- densities. At recombination the photons decouple, the radiation pressure on the baryons ceases, and the photons become the CMB we observed [24]. The DM — coupled to the baryons through gravity — is also influenced by the behaviour of the baryon overden- sities [24]. The baryon and DM densities therefore affect the CMB power spectrum. Interpreted in the ΛCDM framework the CMB power spectrum allows one to extract a baryon density

2 Ωbh = 0.02264 0.0005 WMAP [21], (1.12) ± 2 Ωbh = 0.02207 0.00033 Planck [22], (1.13) ±

−1 −1 where h = (100/H0) km s MPc , H0 is the Hubble rate today, Ωb = ρb/ρc and 2 ρc = 3H /(8πG) is the critical density. There is some tension in the determination of h: WMAP finds h = 70.0 2.2 [21] while Planck finds h = 67.4 1.4 [22]. WMAP also ± ± provides a derived value [21]

−10 ηB = (6.19 0.14) 10 , (1.14) ± × which is broadly consistent with Planck and the value deduced from BBN. Alternatively, −10 the value is YB = (0.88 0.02) 10 , when expressed as a ratio to the entropy density. ± × Introduction. The baryon asymmetry 6

1.2 Baryogenesis mechanisms

1.2.1 Initial conditions

As we have seen so far, the universe we observe exhibits a baryon asymmetry. Aesthet- ically, we may prefer a universe which begins with nB = 0. This alone would perhaps be enough to consider baryogenesis mechanisms, i.e. scenarios which allow for the dy- namical generation of a baryon asymmetry. Another compelling reason to consider such mechanisms is the epoch of inflation we think the universe underwent at the earliest times [25–28]. Most notably, such an exponential expansion, with 50 60 e-folds of ∼ − growth, can help explain why apparently causally disconnected regions of the sky exhibit such uniform temperatures in the CMB. Inflation would also massively dilute any initial baryon number. We therefore consider baryogenesis to have occurred between the end of inflation and before BBN. This dilution is also why, in deriving Eq. (1.8), we considered only the post-inflation causal volume.

1.2.2 Sakharov conditions

Sakharov first listed the conditions required to generate a baryon asymmetry [29]. We list the conditions below as well as their possible manifestation in the SM and its extensions.

1. Baryon number violation. To proceed from a B = 0 state to a net B = 0 requires 6 the violation of B. No direct experimental evidence for B violation has come to light. However, B is violated in the SM itself: through the non-perturbative electroweak sphaleron process [30, 31]. These are thought to lead to copious B violation at temperatures above the EW phase transition. Though not yet exper- imentally verified, the sphalerons are theoretically non-controversial. Additional sources of B violation may, of course, be present in BSM physics as long as they do not conflict with experimental limits on e.g. proton decay.

2. Violation of C and CP. The interactions involving particles and antiparticles must be different for an asymmetry to be generated. This means both C and CP are violated. Both of these discrete symmetries are violated in electroweak sector of the SM: C by the fundemental structure of the theory [32], while CP was shown to be violated experimentally in 1964 [33]. However, the level of CP violation in Introduction. The baryon asymmetry 7

the SM is widely considered to be too small to generate the BAU [34]. But given these symmetries have been shown to be violated experimentally, it should be of no surprise if they were also violated by BSM physics.

3. Departure from thermal equilibrium. A self-consistent quantum field theory is necessarily invariant under CPT. This means the particle and antiparticle masses are the same and the energies of the microstates associated with the occupation numbers of particles and antiparticles are identical. In thermal equilibrium the system will occupy the same number of particle and antiparticle microstates and so an asymmetry can only develop if there is a departure from thermal equilibrium. In baryogenesis this is provided by the expansion of the universe.

Having discussed the general conditions necessary for a theory to generate the BAU dynamically; we now go on to look at some of the specific scenarios proposed for doing so in the literature.

1.2.3 Electroweak baryogenesis

Electroweak baryogenesis uses the B violation present in the standard model to generate the BAU. Baryon number is an accidental global symmetry of the SM Lagrangian. However, it is anomalous: it is a symmetry of the classical Lagrangian but does not remain a symmetry of the full quantum theory [35, 36].

In the SM the local gauge group is broken spontaneously by the Higgs mechanism [32, 37]:

SU(3)C SU(2)L U(1)Y SU(3)C U(1)EM. (1.15) ⊗ ⊗ → ⊗

At sufficiently high temperatures the effective mass parameter for the Higgs doublet has the opposite sign to the low temperature parameter. This means that, if the early uni- verse did indeed have a temperature T & 100 GeV, it was initially in the fully symmetric phase and then underwent a phase transition associated with the above breaking of the electroweak group. The non-perturbative electroweak sphalerons, topological transitions of the form X Vacuum QLiQLiQLilLi, (1.16) ⇐⇒ i=1,2,3 Introduction. The baryon asymmetry 8

where QLi (lLi) is the SM quark (lepton) doublet and i denotes generation, occur rapidly at temperatures above the electroweak phase transition [30]. These sphalerons — asso- ciated with the B anomaly — rapidly violate baryon plus lepton number, B + L, while preserving B L; however, no asymmetry can form unless there is a departure from − thermal equilibrium. At temperatures below the EW phase transition these effects are rapidly suppressed [31].

In a first order phase transition bubbles of broken phase exist surrounded with regions of unbroken phase. If the EW phase transition is first order; the BNV is switched off in the bubbles while proceeding rapidly outside. This provides the departure from equilibrium required for baryogenesis. CP violating collisions of SM particles with the bubble walls can then lead to a preference of particles to antiparticles falling into the bubbles of broken phase. This leads to a net baryon number inside the bubbles, where B violation has ceased, and no asymmetry outside, where any asymmetry is rapidly erased by the sphalerons [38].

To have a sufficiently strong first order phase transition in the SM, however, requires a Higgs mass [39]

Mh . 40 GeV, (1.17) which was ruled out by the LEP experiment [40]. In a crossover transition, predicted to occur in the SM with Mh = 125 GeV, the mass of the Higgs-like scalar observed at the LHC, no bubbles and hence no regions of net baryon number form [41]. Another challenge for electroweak baryogenesis is that the CP violation present in the SM is not large enough to produce the observed BAU [34]. Not enough particles compared to antiparticles would fall into the bubbles.

Successful models of EW baryogensis therefore extend the SM in order to achieve a first order phase transition and to increase the amount of CP violation [38]. The former is usually achieved by modifying the Higgs potential while the latter is achieved by introducing new CP violating interactions. EW baryogenesis provided a window for the BAU to be explained within the SM. The experimental measurements of the insufficient CP violation and the Higgs mass show we must go beyond the SM in order to explain the BAU. Introduction. The baryon asymmetry 9

1.2.4 Leptogenesis

The SM of particle physics does not formally contain gauge-singlet right-chiral Weyl degrees-of-freedom to pair with the neutral component of the lepton doublets. Neu- trinos would therefore remain massless after EW symmetry breaking. Observations of oscillations of solar [42, 43], atmospheric [44], accelerator [45, 46] and reactor neutri- nos [47, 48] have conclusively shown at least two of the neutrinos are massive. The neutrino mass eigenstates are not equal to the neutrino weak eigenstates. They are related by a transformation 3 X ∗ να = U νi , (1.18) | i αi| i i=1 where α = 1, 2, 3 denotes the weak eigenstate and i = 1, 2, 3 the mass eigenstate. One consequence of this misalignment of weak and mass eigenstates is that a propagating neutrino beam will undergo oscillations in which the proportion of the weak eigenstates in the beam changes with distance. The transformation encoding the misalignment can be parameterised by the PMNS matrix [49, 50]:

   −iδ     1 0 0 c13 0 s13e c12 s12 0 1 0 0                −iφa  U = 0 c23 s23  0 1 0   s12 c12 0 0 e 0  .     −    iδ −iφ 0 s23 c23 s13e 0 c13 0 0 1 0 0 e b − − (1.19) A recent universal fit finds the mass squared differences and mixing angles [51]

∆m2 = 7.50+0.18 10−5 eV2, (1.20) 21 −0.19 × ∆m2 = 2.473+0.070 10−3 eV2 (Normal), (1.21) 31 −0.067 × ∆m2 = 2.427+0.042 10−3 eV2 (Inverted), (1.22) 32 − −0.065 × +0.81 θ12 = 33.36−0.78, (1.23) +2.1 +1.3 θ23 = 40.0 50.4 , (1.24) −1.5 ⊕ −1.3 +0.44 θ13 = 8.66−0.46. (1.25)

The hierarchy — normal or inverted — and the Dirac phase, δ, are yet to be determined.

In the normal hierarchy the neutrino mass eigenstates follow the order mν1 < mν2 < mν3, while in the inverted hierarchy mν3 < mν1 < mν2, this leads to the different fits for the mass squared differences ∆m2 m2 m2 above. Furthermore it is not yet ij ≡ νi − νj Introduction. The baryon asymmetry 10 known whether neutrinos are Majorana, this may be established in neutrinoless double beta decay experiments [52], in which case two Majorana phases, φa and φb, will also have to be experimentally determined (in the case of Dirac neutrinos, these phases are unphysical, as they can be absorbed into the fields through a rephasing). The absolute mass scale is also not yet determined. However, it is constrained by cosmology [53], beta decay end point studies [54, 55] and neutrinoless double beta decay searches [56]. The most stringent constraints currently come from combinations of cosmological data, from which a recent study puts the limit on the sum of neutrino masses

X mνi 0.146 eV, (1.26) ≤ i at the 2σ C.L. [57]. While the most stringent limit from beta decay end point studies currently sets the constraint [58]

s X 2 2 mνe U m 2.12 eV. (1.27) ≡ | 1i| νi ≤ i

Assuming Majorana neutrino masses, current limits from neutrinoless double beta decay searches constrain the parameter,

X 2 mββ U mνi 0.14 0.38 eV, (1.28) ≡ 1i ≤ − i where the limit depends on the choice of nuclear matrix element [56].

A theoretically appealing way of explaining the neutrino masses is through the type-I seesaw mechanism [59–62],

∗ 1 c ∆ = λνijablLaiΦ NRj + MNijN NRj + H.c., (1.29) L b 2 Ri where the λνij are dimensionless Yukawa couplings, ab is the Levi-Civita symbol, lL (Φ) is the SM lepton (Higgs) doublet, the NR are SM gauge singlet fermions, i = 1, 2, 3 is a T generational index and N c CN dentotes the Lorentz covariant conjugate field, where ≡ C is the charge conjugation matrix [63]. (At least two NR are required to explain the mass squared differences, whether there are two or three such fields will not qualitatively change the discussion here.) The inclusion of the Majorana mass terms MNij means that lepton number is violated. We rotate to a basis in which MNij = MNi is diagonal Introduction. The baryon asymmetry 11

Figure 1.1: Tree and one-loop diagrams for the decay N lLΦ. →

without loss of generality. Once the Higgs gains a vacuum expectation value (VEV), T 0 Φ 0 = (0 vw/√2) , the Yukawa couplings result in Dirac mass terms, mDij = h | | i λνijvw/√2, giving mass to the SM neutrinos. The Majorona mass terms are allowed by the symmetries of the theory and lead to Majorana type neutrinos.

Upon diagonalisation of the mass matrices, in the well known limit MNk mDij, one  obtains mostly-active neutrinos with masses

2 mDij mνi , (1.30) ∼ MNk and mostly-sterile states with masses MNk. The mixing between the active and sterile ∼ states is of the order: m r m θ Dij νi . (1.31) ∼ MNk ∼ MNk

This relatively simple extension to the SM remarkably also allows for the generation of the BAU [64]. The decays of the heavy neutrinos violate CP and L: the resulting L asymmetry is rapidly reprocessed by the B +L violating sphalerons and one also obtains a net B asymmetry [65]. CP violation arises in the decays of the heavy states through interference of the tree level and loop level diagrams shown in Fig. 1.1.

At tree level the decay rate is given by

† (λλ )ii ΓNi = Γ(Ni Φl) + Γ(Ni Φl) = MN , (1.32) → → 8π i where the sum over decays to the three l generations has been performed and λν is the 3 3 matrix of Yukawa couplings in the Lagrangian (1.29). The CP violation is × parameterised in the following way:

Γ(Ni Φl) Γ(Ni Φl) i = → − → . (1.33) Γ(Ni Φl) + Γ(Ni Φl) → → Introduction. The baryon asymmetry 12

The CP violation in the decay of the Majorana neutrinos is found to be [66]

2 ! 1 1 X n † 2 o Mj i Im [λνλ ] f , (1.34) ' −8π † ν ij M 2 [λνλν]ii j i where  2 1 + x f(x) = √x + ln . (1.35) x 1 x − Note for quasi-degenerate heavy states, MNi MNj ΓNi, one obtains an additional | − | ∼ resonant enhancement to the CP violation which requires careful treatment of the reg- ulator [67].

The final asymmetry is determined by the amount of CP violation, the proportion of the L asymmetry converted into B by the sphalerons and any suppression of the asym- metry due to washout effects or the heavy neutrinos not reaching their fully populated equilibrium number densities. In the simplest scenario with a hierarchical spectrum

MN3,MN2 MN1, the decays of the N1 give essentially all of the final asymmetry.  In order to give a rough analytic estimate of the expected asymmetry it is useful to consider the washout parameter [68]

† Γ(N1 Φl) (λνλν)ii MP l mν K → = , (1.36) H(M ) 16π 1/2 m ≡ 1 1.66g T =M MN1 ∼ ∗ eff | N1 where we have used the seesaw formula (1.30), made the assumption mν mν and ∼ i defined 1/2 2 42geff T =MN1 vw −3 m∗ | = 2 10 eV. (1.37) ≡ MP l ×

The atmospheric mass difference, mν 0.05 eV > m∗, points towards the strong ∼ washout regime: K & 10. In this regime, provided the post inflation reheating tem- perature is high enough, the heavy neutrinos will come into thermal equilibrium and the final asymmetry is largely independent of initial conditions. The final asymmetry is then estimated to be [65, 68]

  C1 −4  mν  100 YB 10 1, (1.38) ∼ Kgeff T =M ∼ 0.05 eV geff T =M | N1 | N1 where C = 28/79 is a sphaleron conversion factor and geff T =M takes into account | N1 the other particles contributing to the entropy density but not to leptogenesis. Clearly, from Eq. (1.38), the observed asymmetry can be obtained in the strong washout regime Introduction. The baryon asymmetry 13

−6 provided that 1 & 10 . Similar estimates can be made for the weak and intermediate washout regimes [65, 68].

Using the calculation of the CP violation, Eq. (1.34), one can place a lower bound on −6 the heavy neutrino mass, by requiring sufficient CP violation (1 & 10 ) to explain the

BAU. In the case of strongly hierarchical Majorana neutrinos, MN3,MN2 MN1, this  translates to [69] 9 MN1 & 10 GeV. (1.39)

However, the basic leptogenesis framework can be extended to include oscillation, flavour and resonance effects, in order to relax the above bound [67].

1.2.5 Affleck-Dine baryogenesis

The scalar potential in SUSY scenarios generically contains many flat directions. The

flatness is lifted by the soft SUSY breaking scale msoft. The energy scale of inflation can

be much higher than the soft breaking scale, Λinf msoft, and the scalar partners to  the quarks and leptons can obtain large VEVs during inflation [70, 71]. Together with explicit violation of CP, arising from complex couplings, the VEVs of the scalar fields violate CP spontaneously. Baryon number is violated by the quartic or higher dimension operators. In such scenarios, the condensate of scalar quarks and leptons will therefore typically carry a large net baryon number [70, 71]. The scalar condensate eventually evaporates, typically by scattering with the thermal plasma [71], leaving us with the observed BAU.

1.2.6 Spontaneous baryogenesis

Spontaneous baryogenesis uses the spontaneous violation of CPT in order to generate the BAU [72, 73]. Consider the introduction of an operator of the form

1 µ ∆ = ∂µφJ , (1.40) L Λ B

µ where Λ is a UV-completion scale, φ is a scalar field and JB is the baryon current. As the universe cools, the scalar field may develop a slowly varying time derivative and the Introduction. The baryon asymmetry 14 above Lagrangian term may then be written as

1 ˙ 0 1 ˙ ∆ = φJ φnB, (1.41) L Λ B ≡ Λ which is a CPT violating interaction, which changes the effective chemical potential of baryons and antibaryons. As CPT is violated, the number density of particles and antiparticles may differ, even when in thermal equilibrium and without violation of CP [72]. Baryon number violation is still required in order to shift the initially B = 0 state to B = 0. If the baryon number violating (BNV) interactions drop out of 6 equilibrium before the interaction (1.41) switches off, the net nB remains frozen and forms the observed BAU [72].

Spontaneous baryogenesis may be considered a counter example to the second and third Sakharov conditions, which assumed CPT invariance. Note that the violation of CPT must occur spontaneously, i.e. can only come about from the expansion of the universe. The expansion is, of course, what leads to the out-of-equilibrium condition for the other baryogenesis scenarios.

1.2.7 Asymmetric Dark Matter

The DM density can accurately be determined from analysis of the CMB, it is found to be

2 ΩDM h = 0.1138 0.0045 WMAP [21], (1.42) ± 2 ΩDM h = 0.1196 0.0031 Planck [22]. (1.43) ±

Comparing to the baryon density, Eqs. (1.12) and (1.13), there is an apparent coinci- dence

ΩDM 5 Ωb 0.24. (1.44) ≈ ≈ Asymmetric DM models (ADM) are motivated largely by trying to explain this apparent coincidence [74–77]. This can be achieved by postulating the DM density is due to an asymmetry, nDM n , where nDM (n ) is the DM particle (antiparticle) number − DM DM density, in the same way as the baryon density is set by nB and efficient annihilation of the symmetric component. If the asymmetries can be related due to some high scale Introduction. The baryon asymmetry 15 physics process in the early universe,

nD nDM n nB, (1.45) ≡ | − DM | ∼ the apparent coincidence, ΩDM Ωb, can be explained provided the DM mass is of the ∼ order of the proton mass: mDM mp. Most ADM models focus on explaining nD nB. ∼ ∼ Relating the DM mass to the proton mass — itself set by the QCD scale — seems to be

more difficult to achieve [76]. So far it has been shown that the relation mDM mp can ∼ be explained in mirror matter models [78], through the use of infrared fixed points [79] and in certain Grand Unified Theories (GUTs) [80, 81].

The relation nD nB can be explained in a number of ways. In general, there is ∼ either a tranfer mechanism — which transfers an asymmetry created in either the dark matter or visible sector to the other — or the baryogenesis mechanism creates related asymmetries in both sectors directly. Typically this involves a dark baryon number,

BD, conserved at low temperatures in the DM sector but broken at high temperatures,

analogous to B L for the visible sector in e.g. leptogenesis scenarios. If BD and − B L are broken individually, while a linear combination of the two is conserved in the − overall theory, the asymmetries in BD and B L will necessarily be related. The usual − baryogenesis mechansims can be modified to incorporate ADM, e.g. particle decay [82], bubble nucleation such as in EW baryogenesis [83] or the Affleck-Dine mechanism [84].

1.2.8 Leptogenesis via collisions

Given that we will be be discussing baryogenesis from CP violating 2 2 collisions ↔ in Chapters 2 and 3 of this thesis, some remarks on baryogenesis mechanisms using CP violating collisions are now timely. We begin by describing the leptogenesis via collisions scenario detailed in Ref. [85]. Consider an extension of the SM with a hidden sector, with gauge group G0, so that the overall gauge symmetry of the theory is a direct product of the SM gauge group G with the hidden sector gauge group: G G0. A simple ⊗ example would be the mirror matter scenario in which G0 is isomorphic to the SM gauge 0 0 group: G ∼= G. Other choices for G are of course possible. Now if a hidden sector fermion, denoted l0, and a hidden sector scalar, denoted Φ0, form a gauge singlet, l0Φ0, heavy singlet neutrinos can play the role of messengers between the two sectors. The Introduction. The baryon asymmetry 16 interaction between the two sectors can be described by the Lagrangian

∗ 0 0 0∗ 1 c = λνijablLaiΦ NRj + λ l Φ NRj + MNijN NRj + H.c., (1.46) L b νij i 2 Ri

0 where the terms are the same as the type-I seesaw with the addition of the λνij Yukawa couplings of the hidden sector states. Let us again rotate to the diagonal basis for the heavy neutrinos: MNi = giMN , where MN parameterises the overall mass scale. At energy scales below MNi, the heavy singlet states can be integrated out and the essential physics can be explained by an effective field theory (EFT)

0 Aij Dij 0 0 Aij 0 0 0 0 = liljΦΦ + liljΦΦ + liljΦ Φ + H.c., (1.47) L 2MN MN 2MN

where the coupling constant matrices are A = λg−1λT , D = λg−1λ0T and A0 = λ0 g−1λ0T .

The mechanism works as follows. If the post inflation reheating temperature TR is

below the mass of the sterile states, TR < MNi, the Ni will not be produced and consequently not contribute to the BAU through decays as in the usual leptogenesis scenario. Furthermore assume an asymmetric reheating so that the hidden sector is 0 colder than the visible sector: T < T < MNi. The number densities are then initially underabundant in the G0 sector but start to be occupied through interactions of the form:

0 0 l + Φ l + Φ , l + Φ l0 + Φ0 , (1.48) → → 0 0 l + Φ l + Φ , l + Φ l0 + Φ0 . (1.49) → →

If the 2 2 interactions violate CP, more scatterings of antileptons than leptons into → the hidden sector can take place and a net B L can be produced. Diagrams leading − to CP violation in the collisions are depicted in Fig. 1.2.

The difference in the total cross section of l+Φ collisions into the hidden sector and l+Φ collisions into the hidden sector is what determines the final B L asymmetry. Through − the use of S-matrix unitarity and CPT, which we will discuss in detail in Sec. 1.3.2, the CP violation of the collisions into the hidden sector can be related to the CP violation in the collisions l + Φ l + Φ. The difference in this cross section and its CP conjugate → Introduction. The baryon asymmetry 17

Figure 1.2: Some of the diagrams contributing to the CP violation in the leptogensis via collisions scenario. is estimated to be 3Jsˆ ∆σ = 2 4 , (1.50) 32π MN where J = Im Tr[(h0†h0 )g−2(h†h)g−1(h†h)∗g−1] ands ˆ is the centre-of-mass energy { } squared. Taking into account the expansion of the universe once the inflaton begins to decay and the asymmetry can begin to grow, the final asymmetry is estimated to be

" # n2 ∆σ  T 3 1012 GeV4 Y = eq 2 10−8 J R , (1.51) B−L 9 4Hs ≈ × 10 GeV MN TR where neq is the equilibrium number density of a single radiation degree of freedom.

The out-of-equilibrium condition is satisfied as the two sectors are at different tempera- tures. For this mechanism to work it is therefore crucial for the above 2 2 interactions ↔ to be feeble enough not to bring the G0 sector into thermal equilibrium with the visible sector. If this were to occur any asymmetry would rapidly be erased due to washout pro- cesses. (Furthermore, if the interactions were to bring the two sectors into equilibrium, no further asymmetry can develop because the CP violation in the scattering of l + Φ into the G0 sector is balanced by CP violation with an opposite effect in l + Φ l + Φ → 0 0 collisions.) To avoid this fate requires the reaction rate, Γ neqσ(lφ l Φ ), to lie ≈ → below the expansion rate post reheating which yields a lower bound on the mass scale of the mediating neutrinos

 T 1/2 M 1012 GeV Q1/2 R , (1.52) N & 1 109 GeV Introduction. The baryon asymmetry 18

† where Q1 = Tr[D D]. Similarly, to obtain a sufficiently large asymmetry in the visible sector, one also wants washout processes of the form l+Φ l+Φ to be out of equilibrium. → This demands  1/2 12 1/2 TR MN 3 10 GeV Q , (1.53) & × 2 109 GeV

† where Q2 = Tr[A A]. Taking into account these lower bounds on M, one finds an upper limit on the produced asymmetry

   2 −8 TR 1 YB−L . 10 J 9 . (1.54) 10 GeV Max[Q1, 6Q2]

Hence the observed asymmetry can be obtained provided TR is sufficiently high.

1.2.9 Baryogenesis from dark matter annihilation

The paradigm of the WIMP miracle remains an appealing explanation for ΩDM . Dark matter particles, χ, with a weak scale coupling to the SM will come into thermal equi- librium with the SM through processes such as χ + χ SM, where SM is generally a ↔ multiparticle state consisting of SM particles.

The determination of the χ relic abundance proceeds similarly to the case, discussed above, regarding nucleon annihilations in a baryon symmetric universe. The annihilation 2 collision rate density scales as n σv , where nχ is the DM density and σv the velocity χh i h i averaged DM annihilation cross section. The term describing the dilution of nχ due to the expansion of the universe is given by 3Hnχ. The DM density follows the equilibrium eq density nχ until the expansion rate exceeds the annihilation rate which occurs when

nχ σv H. (1.55) h i ≈

After this occurs, the DM particle number remains essentially constant on cosmological time scales, being diluted only by the expansion of the universe.

The annihilation cross section required to match the observed ΩDM is approximately σv 10−26 cm3s−1. This is indicative of a weak scale cross section. Hence the term h i ≈ “WIMP miracle” — the DM density can be explained as a thermal relic with weak scale mass and cross section — exactly the area of parameter space we expect to discover new physics due to other motivations such as the hope of solving the hierarchy problem. Introduction. The baryon asymmetry 19

Many implementations of DM as a WIMP exist in the literature both in bottom up approaches (see e.g. Ref. [86]) as well as in broader contexts such as SUSY (see e.g Ref. [87]). Given the relatively simple picture of the WIMP miracle, it is of interest that concrete examples exist in which the DM annihilations themselves also lead to the

BAU [88–92]. While these scenarios do not necessarily lead to ΩDM ΩB, they do ∼ provide a framework to explain both densities in a more-or-less economical framework and represent a viable baryogenesis mechanism which can take place at relatively low, i.e (TeV), scales. O As an example let us describe a WIMPy baryogenesis mechanism proposed in Ref. [90]. The crucial terms in the Lagrangian are

1 c 1 2 2 1 c ij = mχχ¯ χ + mΨΨΨ¯ + m S + N¯ m Nj L 2 2 Sα α 2 i N 0 ¯ ¯˜ 0 ¯ ˜ + λe lLΦeR + λe ΨΦeR + λNi `ΦNi + λNi ΨΦNi (1.56) c ¯ + iλχα Sαχ¯ γ5χ + iλLα SαΨ`.

Here the terms in the first line describe the masses of the Majorana fermions χ and Ni,

the vector-like Dirac fermion Ψ and the pseudoscalars Sα. The second line gives the ∗ Yukawa couplings of the exotic fields with the SM Higgs Φ˜ iσ2Φ and SM leptons. ≡ The final line describes the interactions of the Sα. The exotic fields transform in the following way under the SM gauge group

χ (1, 1, 0),Ni (1, 1, 0) Ψ (1, 2, 1/2). (1.57) ∼ ∼ ∼ −

In addition the DM field χ is taken to be odd under a Z2, with the other fields even,

in order to guarantee its stablility. Note the Ni play the usual role of the right handed neutrinos in the type-I seesaw and can hence explain the neutrino masses [90].

The DM, χ, annihilates to the SM lepton doublet and the exotic vector-like lepton: χ + χ l + Ψ. CP violation in the annihilation process can lead to an preference of → l + Ψ states compared to l + Ψ. The diagrams leading to the CP violation are shown in Fig. 1.3. Introduction. The baryon asymmetry 20

χ l χ l l

S1 S1 S2

χ χ Ψ Ψ Ψ l χ l Ψ N S1 S2 Ψ Φ

χ l Ψ Φ

Figure 1.3: Top and bottom left: tree and loop diagrams leading to CP violating anni- hilations in the WIMPy baryogenesis scenario. Bottom right: lepton number violating decay channel for Ψ.

Once the Ψ are created via the annihilations they can decay via the modes

Ψ ΦeR, (1.58) →

Ψ ΦNi ΦΦl, (1.59) → →

Ψ ΦNi ΦΦl, (1.60) → → where we have also indicated the possibilities for the subsequent decay of the Ni. The overall process only violates L if the last decay chain occurs following DM annhilation. In order to generate a large asymmetry one typically wants the first decay mode to be suppressed compared to the final two. Obtaining a ratio ΓΨ→ΦeR /ΓΨ→ΦeR . 0.1, for mNi < mΨ, can be achieved with the choice of Yukawa couplings

s  m  m2 λ0 0.2 λ0 1 + Ni 1 Ni . (1.61) e . Ni 2 | | × × mΨ − mΨ

In order to achieve a sufficiently large B asymmetry, one also wants to suppress the washout processes mediated by the Sα

χ + l χ + Ψ, l + Ψ l + Ψ, l + l Ψ + Ψ, Ψ + l χ + χ. (1.62) ↔ ↔ ↔ ↔

Indeed the reason for introducing the Ψ in the first place was to allow such interactions to be suppressed during the freezeout of χ. This can be achieved by taking the Ψ particle Introduction. The baryon asymmetry 21 to be sufficiently massive, so that washout processes are Boltzmann suppressed during

χ freezeout, which will occur at temperatures T mχ/20. Numerical solutions show ∼ one can obtain the observed BAU with masses 0.5 mχ . mψ [89, 90]. Together with the kinematic constraint to allow the DM annihilation, χ + χ l + Ψ, this scenario → demands

0.5mχ mψ 2mχ. (1.63) . ≤ The mass of the field Ψ, charged under the SM gauge group, is therefore related to the DM mass in this model. It can be searched for in various experiments [90].

1.3 Boltzmann equations

1.3.1 Thermodynamics and the collision integral

Having reviewed the observational evidence for a baryon asymmetry and various baryo- genesis mechanisms which can produce the asymmetry dynamically, we now outline the theoretical techniques which will allow us to track the evolution of particle numbers and asymmetries in the early universe.

The number density of a species is related to the phase space density, f(E, µ), by

g Z n = X f(E, µ)d3p, (1.64) X (2π)3 where gX is a degeneracy factor which counts the internal degrees of freedom of the field, µ is the chemical potential, p is the momentum, E = pp2 + m2 is the energy. In kinetic equilibrium the phase space density is

1 f(E, µ) = , (1.65) e(E−µ)/T 1 ± where + ( ) is for fermions (bosons) and T is the temperature. −

−3 In an expanding universe with no collisions, a particle number density, nX , drops as R where R is the scale factor. Taking into account collisions the density evolves as

dn X + 3Hn = C(X), (1.66) dt X Introduction. The baryon asymmetry 22 where H = R/R˙ is the Hubble expansion rate and C(X) is the collision term, i.e. the rate of change of nX due to interactions with other particles. Let us imagine now such an interaction from state α to state β and its reverse process which changes X number by one unit. The collision term for this transition can be written as [11]

Z Z 4 X X  4 Cαβ(X) = ... dΠα1...dΠαndΠβ1...dΠβmδ pi pj (2π) − n 2 fβ1...fβm(1 fα1)...(1 fαn) (β α) (1.67) × ∓ ∓ |M → | 2o fα1...fαn(1 fβ1)...(1 fβm) (α β) , − ∓ ∓ |M → |

where is the matrix element, fψ is the phase space density of species ψ, the (1 fΨ) M ∓ are Pauli blocking (Bose enhancement) factors with (+) for fermions (bosons) and −

3 gψd pψ dΠψ = 3 (1.68) 2Eψ(2π) is the normalised volume element of the three momentum. We assume kinetic equilib- rium, i.e. a common temperature, T , throughout. We denote the equilibrium reaction rate density for a process α β as → Z Z 4 X X  4 W (α β) ... dΠα1...dΠαndΠβ1...dΠβmδ pi pj (2π) → ≡ − f eq ...f eq (1 f eq)...(1 f eq ) (α β) 2, (1.69) × α1 αn ∓ β1 ∓ βm |M → |

eq where fαi denotes the phase space density in the absence of a chemical potential.

To proceed note the relation

(Eψ−µψ)/T 1 fψ = fψe , (1.70) ∓

so that substituting in for each (1 fβj) one finds ∓

 m  X (Eβj µβj) fα1...fαn(1 fβ1)...(1 fβm) = fα1...fαnfβ1...fβmExp  −  . (1.71) ∓ ∓ T j=1

Similarly

" n # X (Eαi µαi) fβ1...fβm(1 fα1)...(1 fαn) = fα1...fαnfβ1...fβmExp − . (1.72) ∓ ∓ T i=1 Introduction. The baryon asymmetry 23

Under chemical equilibrium we have

X X µαi = µβj. (1.73) i j

Hence, chemical equilibrium and the delta function enforcing four momentum conserva- tion allows the replacement

fα1...fαn(1 fβ1)...(1 fβm) fβ1...fβm(1 fα1)...(1 fαn), (1.74) ∓ ∓ → ∓ ∓ under the integral sign in Eq. (1.67).

After making the replacement (1.74), it is clear that for matrix elements symmetric under T, (β α) = (α β) , one finds |M → | |M → |

W (α β) W (β α) = 0. (1.75) → − →

In a way, our derivation has proceeded in reverse, one could begin with assuming rapid interactions symmetric under T and show the phase space densities are necessarily of the canonical form. This follows from Boltzmann’s H theorem with suitable modifications to take into account Fermi and Bose statistics [93, 94].

For reasons of simplicity, one often makes an approximation of classical, i.e. Maxwell- Boltzmann statistics, in performing calculations. One drops all the Pauli blocking and Bose enhancement factors and the phase space densities become

  µψ Eψ f = Exp − . (1.76) Ψ T

eq Using Maxwell-Boltzmann statistics, the equilibrium number density, nαi, defined as the number density with µψ = 0, is

g M 2 T M  neq = αi αi K αi , (1.77) αi 2π2 2 T Introduction. The baryon asymmetry 24

where Mαi is the mass of the species and K2(x) is the modified Bessel function of the second kind of order two. The equilibrium reaction rate becomes

Z Z 4 X X  4 W (α β) = ... dΠα1...dΠαndΠβ1...dΠβmδ pi pj (2π) → − f eq ...f eq (α β) 2 × α1 αn|M → | neq ...neq vσ(α β) , (1.78) ≡ α1 αnh → i where vσ(α β) is the thermally averaged cross section. For a process ij kl, this h → i → can be calculated using the single intergral formula [95]

Z Λ2 ! gigjT √sˆ vσ(ij kl) = 4 eq eq pijEiEjvσK1 ds,ˆ (1.79) h → i 8π ni nj sˆmin T wheres ˆ is the centre-of-mass energy squared, pij is the initial centre-of-mass momentum,

K1(x) is the modified Bessel function of the second kind of order one and Λ is the effective theory cut-off. The lower terminal is

h 2 2i sˆmin = Max (mi + mj) , (mk + ml) , (1.80)

where mi is the mass of particle i; this represents the smallests ˆ allowed kinematically, the cross section is formally zero below this.

Note we explicitly write the upper terminal for this integral as the effective theory cut-off Λ. In an EFT, one expects new physical states and hence changing behaviour for the cross sections at and above this scale, i.e. the low energy theory breaks down and is no longer valid. If the temperature is close to this scale T Λ, it is therefore no longer ∼ appropriate to use the EFT. At lower temperatures T Λ these degrees of freedom  are no longer accessible, i.e. thermal fluctuations to energies Λ become rare. This is encoded in the above expression through the Bessel function, which has an exponential suppression ! " # √sˆ  πT 1/2 √sˆ K1 Exp (1.81) T ∼ 2√sˆ − T for √sˆ T . For T Λ, the reaction rate then becomes insensitive to the UV physics &  because the integral is effectively cut off by the exponential suppression before reaching √sˆ Λ, and hence it is also insensitive to the exact choice of the upper terminal. It ∼ is important to ensure one is in this regime when using these expressions. As we shall Introduction. The baryon asymmetry 25 see in Chapters 4 and 5, resonances at the scale Λ are typically negligible for T . Λ/40 (given H at temperatures T & 100 GeV).

For the non-equilibrium rate, one can factor out the chemical potentials from under the integral sign in the collision integral when using Maxwell-Boltzmann statistics. The non-equilibrium reaction rates are found using the appropriate re-weighting:

neq nα1...nαn W (α β) = eq eq W (α β). (1.82) → nα1...nαn →

A standard change of variable, using the relation of Eq. (1.5), is used to obtain the Boltzmann equations in a radiation dominated universe in terms of temperature rather than time. The total entropy is conserved in the absence of first order phase transitions in a radiation dominated universe and hence it is useful to normalise number densities to the entropy density Yψ nψ/s. The entropy density is ≡ 2π2 s = h T 3, (1.83) 45 eff where  3  3 X Ti 7 X Ti h = g + g (1.84) eff i T 8 i T i=bosons i=fermions counts the effective entropic degrees of freedom (the sums run over the relativistic species) [11]. The resulting differential equations are of the form,

!1/2 ! dY π M h T dg X = P l eff 1 + eff C(X) (1.85) 1/2 dT − 45 2 4geff dT s geff

where C(X) is the collision term for X. This form of the Boltzmann equations will be used to calculate the evolution of particle numbers and asymmetries in Chapters 2 and 3 of this thesis.

1.3.2 S Matrix Unitarity and Time Reversal −

The relation (1.75), of course, does not hold for T violating theories. However, we can extract a weaker yet similar relation by considering S-matrix unitarity. For simplicity we will first consider the Maxwell-Boltzmann case and then generalise to full quantum statistics. Introduction. The baryon asymmetry 26

Consider an initial asymptotically free state, i , with given particle species, momenta, | i spins etc. The S-matrix takes this initial state and tells us the asymptotically free final state after the interaction: f = S i . (1.86) | i | i The dual space description of f is given by f = i S†, so that canonical normalisation | i h | h | of our states, 1 = f f = i S†S i , (1.87) h | i h | | i implies the first unitarity condition: S†S = 1 [9, 96]. Consequently, i = S† f , and | i | i using the normalisation condition,

1 = i i = f SS† f , (1.88) h | i h | | i one finds the second unitarity condition: SS† = 1 [9, 96]. The transition matrix T is related to the S-matrix by S 1+iT . Expanding out the first unitarity condition gives ≡

T †T = i(T † T ). (1.89) −

From the second unitarity condition one finds

TT † = i(T † T ), (1.90) − so that T †T = TT †. Considering the diagonal entries of these matrix products yields

X 2 X 2 Tni = Tin , (1.91) n | | n | | where the sum runs over all possible final states [97]. The matrix elements one uses to compute decay rates or scattering cross sections are related to the T matrix,

4 4 X X iTβα = (2π) δ ( pαi pβf ) i (α β), (1.92) − · M → i f where the δ function is simply enforcing four-momentum conservation. It is therefore clear that one also has

X X X X (α β) 2 = (β α) 2 = (β α) 2 = (α β) 2, (1.93) |M → | |M → | |M → | |M → | β β β β Introduction. The baryon asymmetry 27 where α is the CP conjugate of α and the sum runs over all possible states β and the second equality follows from CPT invariance. Using the replacement in Eq. (1.74), which for Maxwell-Boltzmann statistics reads

fα1...fαn fβ1...fβm, (1.94) →

and taking the sum over all possible final states one finds for the equilibrium reaction rates [98]

X X X X W (α β) = W (β α) = W (β α) = W (α β). (1.95) → → → → β β β β

Equation (1.95) means there must be a departure from thermal equilibrium for a baryon asymmetry to be produced (the third Sakharov condition). An exception is the sponta- neous baryogenesis scenario, in which CPT is violated spontaneously by the expansion of the universe, but the particles themselves remain in thermal equilibrium [72, 73].

We note that the same result holds for full quantum statistics. The collision term and phase space densities are modified to take into account quantum statistics [11], but the unitarity condition is also modified,

X 2 X 2 S˜β (α β) = S˜β (β α) , (1.96) |M → | |M → | β β

where S˜β = (1 fβ1)...(1 fβm) are the Pauli blocking or Bose enhancement factors [94, ∓ ∓ 99]. Taking the sum over β for the collision term one finds Eq. (1.95) also holds for full quantum statistics [100].

It is the unitarity of the S-matrix together with CPT invariance which elegantly ensures there are no spurious departures from the usual equilibrium phase space densities even if individual matrix elements are not invariant under time reversal [101, 102]. We will apply this unitarity constraint below so as to correctly relate the CP violation in the reaction rates which enter the Boltzmann equations.

Finally we mention another key result from S-matrix unitarity [94]. Using Eq. (1.90) one can express a T matrix element as

∗ X ∗ Tij = Tji + i TinTjn. (1.97) n Introduction. The baryon asymmetry 28

Taking the magnitude squared and following some rearranging one finds

" # 2

2 2 X ∗ X ∗ Tij Tji = 2Im Tji TinTjn + TinTjn . (1.98) | | − | | − n n

This shows that if one considers a perturbation theory about couplings λ, the lowest order term for the CP violation will occur at (λ3) and appear from the interference O between a tree level and loop level diagram — in which the intermediate loop states correspond to asymptotically free i.e. mass on-shell states [94].

1.4 Conclusion

In this chapter, we have discussed the evidence for the BAU. First of all, there is a lack of a characteristic gamma ray annihilation signal — showing that baryons and antibaryons are separated over distances larger than clusters. We have discussed why such a separation is not feasible in standard cosmologies. Strong evidence for a baryon asymmetry also comes from BBN and the CMB. Collectively these all point to a baryon −10 asymmetry YB 10 . ≈ Next we discussed the conditions required to generate the BAU dynamically. We pointed out the shortcomings of the SM in explaining the BAU. We are therefore required to consider theories beyond the SM in order to explain the BAU. Many different baryogen- esis mechanisms have been proposed in the literature and we discussed some of the most well known. We also briefly covered the possibly related issues of the neutrino masses and DM density.

We then went on to discuss thermodynamics and the Boltzmann equations — which will be used in the following chapters to track the evolution of particle number densities and particle-antiparticle asymmetries. Finally we discussed the relation between reaction rates which come about due to unitarity of the S-matrix and CPT. These unitarity relations will feature in our formulation of models which can generate asymmetries from particle-antiparticle annihilations in Chapters 2 and 3 below; they lead to the explicit manifestation of the third Sakharov condition in the Boltzmann equations. Introduction. The baryon asymmetry 29

We have also touched upon the neutrino masses and DM. These are further motivations for BSM physics and will play a central role — together with the BAU — in Chapter 4 of this thesis. Chapter 2

Particle-antiparticle asymmetries from annihilations

2.1 Introduction

As discussed in Chapter 1, in well known scenarios of baryogenesis, most famously leptogenesis [64, 94, 100, 103] and GUT baryogenesis [100, 103–105, 105–107], a matter- antimatter asymmetry is created by the out-of-equilibrium decay of a heavy particle. Similar mechanisms have also been applied to ADM scenarios [82]. The decays must be CP violating for a preference of matter to be created over antimatter. Furthermore, the asymmetry can only be created once the decaying particle has departed from thermal equilibrium, because S-matrix unitarity and CPT ensures no net preference for particle over antiparticle states can occur in equilibrium. Such scenarios have been studied extensively.

In contrast there has been much less focus on asymmetries created from annihilations. Again, due to the unitarity, one or more of the particles involved in the annihilation must go out of thermal equilibrium for an asymmetry to be generated [104–106]. This is what occurs in the WIMPy baryogenesis scenarios, discussed above, in which heavy neutral particles freeze out and become the DM density and at the same time create the BAU through their annihilations [88–92].

The WIMPy baryogenesis mechanism also explains the DM density, but with no asym- metry between DM particles and antiparticles. However, it may be possible to construct 30 Chapter 2. Asymmetries from annihilations 31 an ADM model in which such annihilations play a role. This research is a first step towards such a goal. An attempt to construct such a model was made previously [108]; however, the unitarity constraint was not properly taken into account. Asymmetry cre- ation during freeze-in using 2 2 annihilations, as in the leptogenesis-via-collisions ↔ scenario [85], has also been considered in the literature [85, 99, 109]. The effect of CP violating 2 2 annihilations has also been investigated in the context of both stan- ↔ dard and resonant leptogenesis [65, 110–113]. A baryogenesis model using CP violating annihilations was also considered in Ref. [114].

In this chapter we detail a general framework for models which seek to create particle- antiparticle asymmetries from annihilations. While certain aspects of such mechanisms are necessarily model dependent, other considerations, such as the unitarity relations and construction of the Boltzmann equations, are generic. Our focus in this chapter is on examining asymmetries from annihilations alone; in Chapter 3 we will examine scenarios in which decays and annihilations compete in creating the final asymmetry.

In order to study such a mechanism, we introduce a toy model involving the interactions between four fermions. We outline the Boltzmann equations for the model and show a non-zero source term develops when one or more of the species depart from equilibrium. We calculate the relevant thermally averaged cross sections and solve the Boltzmann equations numerically.

2.2 Toy model

Consider the interaction Lagrangian

1 c c 1 c c 1 c c = κ1Ψ Ψ1f f + κ2Ψ Ψ2f f + κ3Ψ Ψ1f f L 4 1 4 2 2 2 1 1 1 + λ Ψc Ψ Ψ Ψc + λ Ψc Ψ Ψ Ψc + λ Ψc Ψ Ψ Ψc + H.c., (2.1) 2 1 2 1 1 1 4 2 2 2 1 1 2 3 2 2 2 1

where the Ψ and f are Dirac fermions and the κi and λi are effective couplings with inverse dimension mass squared. Spinor indices are contracted between the first two and last two fermion fields in each of the terms. Chapter 2. Asymmetries from annihilations 32

The above Lagrangian violates the particle numbers associated with Ψ1,Ψ2 and f but preserves the linear combination

∆(Ψ1 + Ψ2 f). (2.2) −

We will show how these interactions will generate an asymmetry in the f sector and a related asymmetry in the Ψ sector, ∆(f) = ∆(Ψ1 + Ψ2), through 2 2 processes. ↔ The last three interaction terms of Eq. (2.1) break the particle numbers associated with

Ψ1 and Ψ2 individually but preserve ∆(Ψ1 + Ψ2). These latter interactions must be included to allow CP violation to arise in the interference between tree and loop level diagrams. Majorana masses are prohibited by the global symmetry of the Lagrangian

∆(Ψ1 + Ψ2 f) = 0. −

We assume f are in thermal equilibrium with the radiation bath and that Ψ1 and Ψ2 are coupled to the radiation bath only through their interactions in the above Lagrangian. The asymmetries are generated during the time when the Ψ particles are going out-of- equilibrium.

The above Lagrangian includes four physical phases in the couplings. To see this, note we have six couplings which may be complex, however some of these phases may be removed through a rephasing of the fields. The set of linear rephasing equations is of rank two, indicating four physical phases. In our choice of parameters below we will take κ1 and κ2 to be real without loss of generality.

CP violation arises in Ψ number changing interactions of the form ΨiΨj ff, in the → interference between tree level and one loop level diagrams, such as those depicted in Fig. 2.1. Chapter 2. Asymmetries from annihilations 33

Figure 2.1: Tree and one-loop diagrams for the annhilation Ψ1Ψ1 ff and Ψ1Ψ1 → → Ψ2Ψ2. The intermediate particles contribute to the CP violation when they go on-shell in the loop integral.

We parametrise the CP violating equilibrium reaction rate densities as

W (Ψ1Ψ1 ff) (1 + a1)A1, (2.3) → ≡

W (Ψ2Ψ2 ff) (1 + a2)A2, (2.4) → ≡

W (Ψ1Ψ2 ff) (1 + a3)A3, (2.5) → ≡

W (Ψ1Ψ1 Ψ1Ψ2) (1 + a4)A4, (2.6) → ≡

W (Ψ1Ψ1 Ψ2Ψ2) (1 + a5)A5, (2.7) → ≡

W (Ψ2Ψ2 Ψ2Ψ1) (1 + a6)A6, (2.8) → ≡

where the CP conjugate rates can be found by substituting ai ai and Ai Ai. For → − → example, in terms of the reaction rates, the CP violation a1 can be expressed as

W (Ψ1Ψ1 ff) W (Ψ1Ψ1 ff) a1 → − → . (2.9) ≡ W (Ψ1Ψ1 ff) + W (Ψ1Ψ1 ff) → → Chapter 2. Asymmetries from annihilations 34

Now consider the unitarity condition (1.95). Considering in turn the Ψ1Ψ1,Ψ2Ψ2,Ψ1Ψ2 and ff states yields

a1A1 + a4A4 + a5A5 = 0, (2.10)

a2A2 + a6A6 a5A5 = 0, (2.11) −

a3A3 a4A4 a6A6 = 0, (2.12) − −

a1A1 + a2A2 + a3A3 = 0. (2.13)

Note the unitarity condition for state ff, Eq. (2.13), is also demanded by Eqs. (2.10)– 2.12). We will see below that the CP violating rates calculated in terms of the underlying parameters of the Lagrangian do indeed respect these unitarity conditions.

We calculate the relevant cross sections and find the thermal averaged cross sections numerically by making use of the single integral formula, Eq. (1.79). The expressions for the cross sections can be found in Appendix A.

The CP violation in the annihilations at the cross section level i.e. the difference between the cross section and its CP conjugate, (σ σ), arises from the interference of tree and − loop level diagrams. It can be shown that the CP violation depends on the imaginary (absorptive) part of the interference [11]. An imaginary component arises when the loop diagram can be cut into two individual Feynman diagrams, with valid asymptotic initial and final states, i.e. composed of mass on-shell particles. This requires that (at least two of) the particles in the loop of the original diagram to be on-shell [97]. It is therefore possible to extract the imaginary component by judicious replacement of particle propagators with delta functions, in a process following Cutkosky [115]. This allows one to calculate the CP violation, (σ σ), terms of the underlying parameters of − the theory.

To find the CP violation in the equilibrium rates one must then take a thermal average,

Z Λ2 ! gigjT √sˆ v(σ(ij kl) σ(ij kl)) = 4 eq eq pijEiEjv(σ σ)K1 ds,ˆ (2.14) h → − → i 8π ni nj sˆmin − T where the definitions are the same as for Eq. (1.79). We evaluate (2.14) numerically and the results of such a calculation are depicted in Fig. 2.2. The CP asymmetries in the annihilations are temperature dependent. At high temperature, T Max[Mf ,M1,M2],  Chapter 2. Asymmetries from annihilations 35 one expects on dimensional grounds the CP symmetric thermally averaged cross sections to scale as σv κ2T 2, (2.15) h i ∼

where κ κi λi is the general scale of the relevant couplings. Similary, for T ∼ | | ∼ | |  Max[Mf ,M1,M2] and (1) phases, the CP violation is expected to scale as O

2 ai κT . (2.16) ∼

Note that in our example, a1 is necessarily suppressed on kinematic grounds at low temperatures (A1 does not depend on the heaviest particle mass, M2, while the CP violation is sensitive to M2 from the on-shell particles in the loop). The other CP violating parameters take values around

κ 2 ai Max[Mf ,M1,M2] , (2.17) ∼ 8π

at temperatures T Max[Mf ,M1,M2], as can be seen in Fig. 2.2. Note that naively  the CP violation will increase to unphysical values ai > 1, if we naively keep increasing | | T without taking into account the EFT cut-off at Λ κ−1/2. This is because our EFT ∼ is valid only for T Λ. In our examples plots we show temperatures up to T Λ/10. . ∼ The crucial process of freeze-out and asymmetry generation typically occurs at much lower temperatures, T Λ/1000, as we will see below in this and the next chapter, and ∼ hence we are well within the valid EFT regime at these crucial temperatures.1

We also check the unitarity conditions (2.10)–(2.13) are satisfied by our, part numerical, calculation of the CP violating reaction rates. This can be done by computing the percentage error associated with each unitarity condition

  aiAi ajAj akAk % Error = 100 ± ± , (2.18) aiAi + ajAj + akAk | | | | | | 1In practice when evaluating the numerical integrals for the reaction rates, the algorithm automat- ically used by Mathematica [116] for Λ → ∞, seemed to converge better than for finite choices, e.g. Λ = κ−1/2, particularly in terms of the errors in the CP violation (see Fig. 2.3). Choosing an integral cut-off Λ → ∞ rather than Λ = κ−1/2, we found differences of around 10% in the reaction rates at −1/2 −1/2 T = κ /10, and negligible differences ( 1%) at T . κ /100. Hence, we are well within the valid EFT regime when the crucial process of freeze-out and the majority of asymmetry generation occurs. This also allowed us to take Λ → ∞ in our Mathematica code for the numerical integrals, allowing us to factor out the couplings from the calculation of the reaction rates, meaning these only had to be performed once for each choice of the masses. Chapter 2. Asymmetries from annihilations 36

Figure 2.2: CP violation for parameters are set to Mf = 100 GeV, MΨ1 = 800 −13 −2 −i3π/4 iπ/3 GeV, MΨ2 = 2 TeV, κi = λi = 5 10 GeV , κ3 = e κ3 , λ1 = e λ1 , −iπ/6 | | −| iπ/| 4 × | | | |2 λ2 = e λ2 and λ3 = e λ3 . The diagonal dashed line corresponds to κi T | | | κ | 2 | | and the horizontal dashed line to 8π M2 . where the signs correspond to the relevant unitarity condition. The results of such a calculation are shown in Fig. 2.3.

Several additional CP even interactions play an important role. Washout interactions of the form Ψif Ψjf act to remove any asymmetry. These are denoted as →

W (Ψ1f Ψ1f) = W1, (2.19) →

W (Ψ2f Ψ2f) = W2, (2.20) →

W (Ψ1f Ψ2f) = W3. (2.21) →

Furthermore sufficiently rapid interactions of the form ΨiΨj ΨkΨl relate the chemical ↔ potentials of Ψ1 and Ψ2. These are also included in our numerical solutions below. These are denoted as

W (Ψ1Ψ1 Ψ2Ψ1) = Z1, (2.22) →

W (Ψ1Ψ2 Ψ2Ψ1) = Z2, (2.23) →

W (Ψ2Ψ2 Ψ1Ψ2) = Z3. (2.24) →

Finally we should consider possible decays of the toy model particles. We take the Ψ2 Chapter 2. Asymmetries from annihilations 37

0.4

0.2

0.0 Error %

-0.2

-0.4

10 100 1000 104 105 T GeV

Figure 2.3: Percentage error for the unitarity@ conditions.D The parameters are the same as for Fig. 2.2. The cancellation is never exact as the rates must be calculated using numerical integration. In particular note the glitch around T = 100 GeV. This does not lead to any spurious behaviour in the example solution to the Boltzmann equations presented below.

mass greater than the Ψ1 and f masses (MΨ2 > MΨ1,Mf ) and consider the decays of

Ψ2. The relevant Feynman diagrams are depicted in Fig. 2.4. A priori Ψ2 may have two decay channels

Γ(Ψ2 Ψ1ff) = (1 + γa)Γ2a, (2.25) →

Γ(Ψ2 Ψ1Ψ1Ψ1) = (1 + γb)Γ2b, (2.26) → where the γi parametrise the CP violation. The CP conjugate decay rates can be found by making the substitution γi γi. Unitarity implies → −

γaΓ2a + γbΓ2b = 0. (2.27)

Here we kinematically forbid the second decay channel, ensuring no CP violation is possible in the Ψ2 decays. This can be seen immediately from the unitarity condition.

Alternatively, one can see from the Feynman diagrams in Fig. 2.4, that if the Ψ2 → Ψ1Ψ1Ψ1 decay is kinematically forbidden, the intermediate particles in the Ψ2 Ψ1ff → loop diagram cannot go on-shell. Chapter 2. Asymmetries from annihilations 38

Ψ1 Ψ1 κ3 λ1

Ψ2 f Ψ2 Ψ1

f Ψ1

Ψ1 Ψ1

λ1 κ3 Ψ1 f Ψ2 Ψ2 f Ψ1 κ1 κ1∗

Ψ1 f f Ψ1

Figure 2.4: Tree and one-loop diagrams for the decays of Ψ2.

The remaining decay width is given by

2 5 κ3 (MΨ2) Γ = | | , (2.28) 2a 3072π3 where we have ignored the final state masses. We include the final state masses for the decay rate (see Appendix A.1) in our numerical solutions. The decay rate which appears in the Boltzmann equations below also includes a Lorentz factor suppression resulting from the thermal average (see Appendix A.2), we denote such decay rates with a superscript th.

2.3 Boltzmann equations

We can now write down the Boltzmann equations using the usual approximation of Maxwell-Boltzmann statistics. As discussed in Chapter 1, the use of Maxwell-Boltzmann statistics allows one to factor out the chemical potential of a species from the collision term. The nonequilibrium rate is then simply the equilibrium rate multiplied by the ratio of the number density to the equilibrium number density of the incoming particles. For notational clarity we define the ratio of the number density to the equilibrium number density as ni ni ri eq , ri eq . (2.29) ≡ ni ≡ ni Chapter 2. Asymmetries from annihilations 39

We assume f and f are in thermal equilibrium with the SM radiation bath so µf = µ . − f We find the Boltzmann equations for the number densities of Ψ1 and Ψ2, n1 and n2

respectively, and the asymmetries n∆1 n1 n and n∆2 n2 n in terms of the ≡ − 1 ≡ − 2 CP even and odd interaction rates. This results in a system of four coupled first order ordinary differential equations. The equations take the form

dn + 3Hn = (source terms) + (washout terms), (2.30) dt where the source terms can create an asymmetry once one or more species depart from equilibrium and ri = 1, while the washout terms drive the system towards equilibrium 6 and washout any asymmetries present.

The Boltzmann equation for n1 is

dn1 eq th h i h i h i + 3Hn1 = n Γ r2 r1rf rf + W1 r1rf r1rf + W3 r2rf r1rf dt 2 2a − − − h i h i h i + Z1 r2r1 r1r1 + Z2 r1r2 r1r2 + Z3 r2r2 r1r2 − − − h i h i h i + 2A1 rf rf r1r1 + A3 rf rf r2r1 + A4 r2r1 r1r1 − − − h i h i h i + 2A5 r2r2 r1r1 + A6 r2r2 r1r2 2a1A1 rf rf + r1r1 − − − h i h i a3A3 rf rf + r2r1 a4A4 r2r1 + r1r1 − − h i h i 2a5A5 r2r2 + r1r1 + a6A6 r2r2 + r1r2 . (2.31) −

The Boltzmann equation for n∆1 is

dn∆1 eq th h i = 3Hn∆1 + n Γ r2 r2 + r1rf rf r1rf rf dt − 2 2a − − h i h i + 2W1 r1rf r1rf + W3 r2rf r2rf + r1rf r1rf − − − h i h i h i + Z1 r2r1 r2r1 + 2Z2 r1r2 r1r2 + Z3 r2r1 r1r2 − − − h i h i + 2A1 rf rf rf rf + r1r1 r1r1 + A3 rf rf rf rf + r2r1 r2r1 − − − − h i h i + A4 r2r1 r2r1 + r1r1 r1r1 + 2A5 r2r2 r2r2 + r1r1 r1r1 − − − − h i h i + A6 r2r2 r2r2 + r1r2 r1r2 2a1A1 rf rf + rf rf + r1r1 + r1r1 − − − h i h i a3A3 rf rf + rf rf + r2r1 + r2r1 a4A4 r2r1 + r2r1 + r1r1 + r1r1 − − h i h i 2a5A5 r2r2 + r2r2 + r1r1 + r1r1 + a6A6 r2r2 + r2r2 + r1r2 + r1r2 . (2.32) − Chapter 2. Asymmetries from annihilations 40

The Boltzmann equation for n2 is

dn2 th eqh i h i h i + 3Hn2 = Γ n r1rf rf r2 + W2 r2rf r2rf + W3 r1rf r2rf dt 2a 2 − − − h i h i h i + Z1 r1r1 r2r1 + Z2 r1r2 r2r1 + Z3 r1r2 r2r2 − − − h i h i h i + 2A2 rf rf r2r2 + A3 rf rf r2r1 + A4 r1r1 r2r1 − − − h i h i h i + 2A5 r1r1 r2r2 + A6 r1r2 r2r2 2a2A2 rf rf + r2r2 − − − h i h i a3A3 rf rf + r2r1 + a4A4 r1r1 + r2r1 − h i h i + 2A5 r1r1 + r2r2 a6A6 r1r2 + r2r2 . (2.33) −

The Boltzmann equation for n∆2 is

dn∆2 th eqh i = 3Hn∆2 + Γ n r1rf rf r1rf rf + r2 r2 dt − 2a 2 − − h i h i + 2W2 r2rf r2rf + W3 r1rf r1rf + r2rf r2rf − − − h i h i h i + Z1 r2r1 r2r1 + 2Z2 r1r2 r2r1 + Z3 r1r2 r2r1 − − − h i h i + 2A2 rf rf rf rf + r2r2 r2r2 + A3 rf rf rf rf + r2r1 r2r1 − − − − h i h i + A4 r1r1 r1r1 + r2r1 r2r1 + 2A5 r1r1 r1r1 + r2r2 r2r2 − − − − h i h i + A6 r1r2 r1r2 + r2r2 r2r2 2a2A2 rf rf + rf rf + r2r2 + r2r2 − − − h i h i a3A3 rf rf + rf rf + r2r1 + r2r1 + a4A4 r1r1 + r1r1 + r2r1 + r2r1 − h i h i + 2a5A5 r1r1 + r1r1 + r2r2 + r2r2 a6A6 r1r2 + r1r2 + r2r2 + r2r2 . (2.34) −

The terms proportional to the ai are the source terms. By the application of the uni- tarity conditions (2.10)–(2.13) these terms can only generate asymmetries when the distribution of Ψ particles depart from equilibrium: ri = 1. The other collision terms 6 are washout terms, these drive the distributions toward their equilibrium values and the asymmetries toward zero.

We proceed to solve the Boltzmann equations numerically. We use the standard change of variable to obtain the Boltzmann equations in a radiation dominated universe in terms of temperature in the form given in Eq. (1.85). For the degrees of freedom which enter the Boltzmann equations through the radiation and entropy densities, we simply take Chapter 2. Asymmetries from annihilations 41

Figure 2.5: Example solution to the system of coupled Boltzmann equations with densities normalised to the entropy density Yψ nψ/s and shown evolving with tem- perature T , time proceeds right to left. Parameters≡ are set as in Fig. 2.2. The densities of Ψ1 and Ψ2 begin by tracking their equilibrium values at high temperature, at some stage the interactions freeze out and there is a departure from equilibrium. The Ψ2 eventually decay but Ψ1 is stable and hence Y1 is constant after freeze out. The asym- metries slowly grow from a high temperature and we are eventually left with stable asymmetries Y∆1 = Y∆f .

geff = heff = 100, corresponding to the order-of-magnitude of the SM degrees of freedom at high temperatures.

Having made the change of variable and calculated the reaction rates and CP violation, we solve the system of coupled Boltzmann equations using Mathematica [116]. An example solution is shown in Fig. 2.5.

To guard against spurious numerical artifacts possibly arising from departures from the unitarity conditions, as shown in Fig. 2.3, we use the unitarity conditions to re-express the source terms in terms of the three independent CP violating rates in our code. This removes one possibility of numerical errors in our solutions. The exercise results in negligible changes, below the percent level, in the calculated asymmetry.

The thermal history in this model proceeds as follows. At high temperatures the 2 2 ↔ annihilations keep Ψ1 and Ψ2 close to thermal equilibrium and only a small asymmetry can develop (due to the expansion term the particles are never exactly in equilibrium). The departure from equilibrium and hence the asymmetries increase as T decreases and Chapter 2. Asymmetries from annihilations 42

the reactions become less effective. At some point the Ψi effectively decouple and the overall asymmetry remains constant. In Fig. 2.5 this occurs around T 400 GeV. ≈ Crucial to obtaining an asymmetry (with a common T between sectors) is that at least some of the particles involved are massive: the decoupling of massless particles does not

lead to ri = 1. Numerically we find the maximum asymmetry is generated for decoupling 6 at T Mi. ∼

Eventually the heavier Ψ2 decay into Ψ1 and the final ∆(Ψ) asymmetry is stored in Ψ1.

Due to the different masses, couplings and phases, the asymmetries created in Ψ2 and

Ψ1 are different and hence the eventual ∆(Ψ) decays of Ψ2 do not washout the overall

asymmetry. We are left with an overall asymmetry ∆(Ψ) = ∆(Ψ1) = ∆(f).

Note that a large symmetric component of Ψ1 is still present: Y∆1 Y1. In a realistic | |  model, so as to not overclose the universe, the symmetric component should be annihi-

lated away. This can be achieved by introducing an interaction of the form Ψ1Ψ1 ff. → Alternatively Ψ1 and Ψ1 could eventually decay. The asymmetry can then be stored in the decay products. These could be regular baryons or if they make up the DM, and have a sufficiently large annihilation cross section to annihilate away the symmetric component, form asymmetric DM [117–119].

We have assumed kinetic equilibrium for the Ψi throughout. At high T this is a good approximation as the 2 2 interactions effectively transfer momentum between the ↔ Ψi and f. As we approach the decoupling point this approximation begins to breaks down [120–123]. This calculation can be further refined through the inclusion of depar- tures from kinetic equilibrium, full quantum statistics and thermal masses which could give (1) corrections to the final asymmetry [120–124]. O

2.4 Conclusion

In this chapter we have presented a generic setup for the generation of particle-antiparticle asymmetries from 2 2 processes, such as annihilations or scatterings. This is to be ↔ contrasted with the more well known scenario in which such asymmetries are generated via 1 2 out-of-equilibrium decays. → Chapter 2. Asymmetries from annihilations 43

Using a simple toy model as an example, we have explicitly outlined how the Boltzmann equations should be formulated, taking S-matrix unitarity and CPT invariance into account. We have shown that the CP violating annihilation rates, calculated with the help of the Cutkosky rules, do indeed satisfy the unitarity condtions. We have also presented an example numerical solution to the Boltzmann equations and explained its features.

Such techniques can be applied in calculation of particle-antiparticle asymmetries in models of ADM and baryogenesis. We will see an example of the latter in the next chapter. Chapter 3

The role of CP violating scatterings in baryogenesis — case study of the neutron portal

3.1 Introduction

As we have seen in the previous chapter, scatterings of particles in 2 2 type processes ↔ can violate CP and potentially produce asymmetries. In this chapter we will study such CP violating scatterings in a baryogenesis scenario by building upon what we have learned from the toy model.

The presence of CP violation in scatterings is not confined to models specifically con- structed to exploit this feature, such CP violation can also appear in baryogenesis- via-decay type scenarios. For example, scatterings of the heavy Majorana neutrinos in leptogenesis also violate CP [65, 110–113]. Detailed calculations show that such annhila- tions play a significant role in determining the baryon asymmetry for high temperatures, however, the final asymmetry is determined overwhelmingly by the decays in leptoge- nesis in the strong and intermediate washout regimes [110, 112]. (Due to cancellations between the signs of the high temperature and low temperature asymmetries, CP vio- lating scatterings can lead to (1) corrections to the final BAU in the weak washout O regime [112].)

44 Chapter 3. CP violating scatterings in baryogenesis 45

The purpose of this chapter is to study the effects of CP violating scatterings on the generation of the baryon asymmetry more generally. We choose to study a neutron portal model in which the parameters are not restricted by having to explain low en- ergy neutrino data. Baryogenesis via CP violating decays in the neutron portal have been studied previously [125] (an ADM scenario was also discussed in Ref. [82]). CP violating scatterings in a neutron portal model were discussed in Ref. [108]; however, as mentioned previously, the unitarity constraint was not properly taken into account. We will extend the analysis of the neutron portal to include a full numerical treatment of the Boltzmann equations including CP violating scatterings. For previous work on a baryogenesis scenario using CP violating annihilations see Ref. [114].

Though we have chosen a simple neutron portal scenario as an example. Such effects are not constrained to this particular model and will play a role in baryogenesis scenarios more generally. Recently further work has appeared on the neutron portal operator as a baryogenesis mechanism, its possible links to neutrino masses, its UV completion and possible GUT origin [126, 127].1

In Sec. 3.2 we introduce the neutron portal Lagrangian and discuss the relevant decays and scatterings. In Sec. 3.3 we write down the Boltzmann equations for the evolution of the baryon asymmetry and show example numerical solutions. We show the CP violating scatterings play a dominant role in determing the final asymmetry in a large portion of the parameter space. We discuss the key qualitative differences that lead to CP violating scatterings being the dominant mechanism in this scenario but mostly negligible in leptogenesis. Before concluding we also discuss constraints on the model in Sec. 3.4. 1We also discussed possible UV completions in Publication 2, however, we do not include the dis- cussion in the present thesis as the work was primarily done by Alexander Millar. It suffices to say the operators discussed here can easily be UV completed through the addition of exotic coloured scalars (some of which are the focus of Chapter 5). Chapter 3. CP violating scatterings in baryogenesis 46

3.2 Neutron portal

3.2.1 Lagrangian

Consider the interaction Lagrangian [125]

c c ∆ =κ1ijkX1LuRi(dRj) dRk + κ2ijkX2LuRi(dRj) dRk + κ3ijuRiX1LX2LuRj L 1 1 + κ u X X u + κ u X X u + H.c., (3.1) 2 4ij Ri 1L 1L Rj 2 5ij Ri 2L 2L Rj where the κaijk are couplings with inverse dimensions of mass squared, uRi (dRi) is the right chiral up (down) type quark field of flavour i and the Xα are Majorana fermions.

We impose a global lepton number symmetry forbidding coupling of the Xα to the SM Higgs and lepton doublets. Spinor indices are contracted between the first two and last two fermion fields in each of the terms. We have suppressed colour indices. Due to the c c antisymmetric colour index and the identity Ψχ = χΨ the couplings κ1ijk and κ2ijk are necessarily antisymmetric in down type quark flavour.

Other operators in addition to the ones of Eq. (3.1) are allowed by the symmetries of

the theory e.g. dRiX1LX2LdRj. For simplicity we take such operators to be sufficiently suppressed compared to those of Eq. (3.1) as to be negligible. Even if present these would not qualitatively alter the outcomes below.

The decays of the heavier Xα — from now taken to be X2 — can be CP violating and if they occur out-of-equilibrium will lead to a baryon asymmetry. However, as we will see below, scatterings of the form uiXαL djdk also violate CP. We will study the effects → of this CP violation on the final asymmetry.

The above Lagrangian contains 28 physical phases if one considers couplings to all pos- sible flavour combinations. As a demonstration, and in order to simplify the analysis, we will restrict ourselves to considering only couplings to the first generation up quark and second and third generation down quarks, s and b. From now on we will suppress flavour indices on the above couplings. After rephasing we are left with two physical phases. For simplicity we set iπ/2 κ1 = e κ1 , (3.2) | | Chapter 3. CP violating scatterings in baryogenesis 47

u X1

Xα s X2 u

b u

u X 2 X 1 s

u b

Figure 3.1: Tree and loop level decays for X1 and X2. The particles in the loop for the lower diagram can go on-shell: interference between this and the tree diagram leads to CP violation in the X2 decay. The X2 in the loop in the analogous X1 decay diagram cannot go on-shell and consequently there is no CP violation in the X1 decay.

while we set the other phase (of κ3) to zero. This choice for the phases leads to the largest CP violation, if the final asymmetry exceeds the observed one, the asymmetry

can always be reduced by decreasing the κ1 phase. The other couplings are taken to be real.

3.2.2 Decays

The tree level decay rate Γ(Xα usb) + Γ(Xα usb) is → →

2 5 κα (MXα) Γ = | | , (3.3) αA 512π3 where we have ignored the masses of the final state particles. X2 has an additional decay channel, X2 X1uu, with rate →

2 5 κ3 (MX2) Γ = | | , (3.4) 2B 1024π3 where we have again ignored the masses of the final state particles (the full integral expression for massive X1 can be found in Appendix B.2).

CP violation in the decays of X2 arises from the interference between the tree and one loop diagrams depicted in Fig. 3.1. We parameterise the CP violation in the X2 decay Chapter 3. CP violating scatterings in baryogenesis 48 in the following way

1 Γ(X2 usb) = (1 + D)Γ2A, (3.5) → 2 1 Γ(X2 usb) = (1 D)Γ2A. (3.6) → 2 −

The CP violation can be calculated in terms of the underlying parameters of the theory by using the Cutkosky rules to extract the imaginary component of the interference term [97, 115]. The results are shown in Fig. 3.2 and further details can be found in Appendix B.2. From dimensional arguments one expects

10-5

10-6

10-7 È D Ε È 10-8

10-9

10-10 0 20 40 60 80 100

MX1 TeV

−14 Figure 3.2: CP violation in the decay of X2@, withDMX2 = 100 TeV and κa = 10 −2 κa 2 GeV , as a function of MX1. The horizontal dashed line corresponds to D = 16π MX2.

∗ ∗ 1 Im[κ1κ2κ3] 2 κ 2 D 2 MX2 MX2, (3.7) ∼ 16π κ2 ∼ 16π | |

in the massless limit for X1, where the second relation follows for couplings of a similar

magnitude (κa) κ and an order one phase. This is confirmed by our detailed O ∼ calculation, as can be seen in Fig. 3.2.

One must also take into account CP violation in the scattering process usb usb → mediated by an X2 with the real intermediate part of the scattering subtracted [94]. We define this rate as

W (usb usb) = (1 + OS)WOS, (3.8) → Chapter 3. CP violating scatterings in baryogenesis 49

where the CP conjugate can be found by taking OS OS. The unitary condition → − (1.95) — taking usb as the initial state and summing over all possible final states — enforces the relation: 1  W =  neqΓ . (3.9) OS OS 2 D 2 2A

The CP violating rate, (3.9), balances the CP violation in the decays when X2 is in thermal equilibrium [94] and is included in our Boltzmann equations below. This is entirely analogous with real intermediate subtraction of the heavy Majorana neutrinos in leptogenesis, which must also be done to ensure the correct unitary behaviour in equilibrium [94, 124]. The CP symmetric washout rate, W (usb usb), mediated by → 4 off shell X1 and X2, is (κ ) and are negligible for the parameters we are interested in O below, these have been omitted from the Boltzmann equations.

3.2.3 Scatterings

Next we turn our attention to the scatterings. CP violation arises due to the interfer- ence between the tree and one-loop diagrams such as those depicted in Fig. 3.3. We parameterise the relevant CP violating equilibrium collision terms as

W (u + X1 s + b) = (1 + 1)W1, (3.10) →

W (u + X2 s + b) = (1 + 2)W2, (3.11) →

W (u + X1 u + X2) = (1 + 3)W3, (3.12) →

where the CP conjugate interaction rate can be found by making the substitution

i i. These interactions will contribute to the baryon asymmetry. The unitar- → − ity conditions for these interactions give2

1W1 + 2W2 = 0, (3.13)

1W1 + 3W3 = 0, (3.14)

2W2 3W3 = 0. (3.15) − 2The unitarity condition for couplings to all possible quark flavour combinations is discussed in appendix B.1. Chapter 3. CP violating scatterings in baryogenesis 50

Xα s X1 X2

u b u u

X2 X1 s

u u b

Figure 3.3: Example tree and loop level scatterings leading to CP violation. Further loop diagrams exist with e.g. X1 and X2 interchanged.

CP conserving (to the order we are working) collision terms are also present; we denote them in the following way

W (s + X1 u + b) = W4, (3.16) →

W (s + X2 u + b) = W5, (3.17) →

W (X1 + X1 u + u) = W6, (3.18) →

W (X2 + X2 u + u) = W7, (3.19) →

W (X1 + X2 u + u) = W8. (3.20) →

Similary to the toy model, for T MX2, the CP violation scales as 

2 i κT . (3.21) ∼

While for temperatures T MX2, the CP violating parameters take values close to 

κ 2 i M , (3.22) ∼ 16π X2 except for 1, which is suppressed kinematic grounds at low temperatures, as can be seen in Fig. 3.4. Chapter 3. CP violating scatterings in baryogenesis 51

Figure 3.4: CP violation in the scatterings with MX2 = 100 TeV, MX1 = 50 TeV, −14 −2 κa 2 κa = 10 GeV . The horizontal dashed line corresponds to  = 16π MX2 and the 2 diagonal dashed line corresponds to  = κaT . Note the kinematic suppression of 1 | | at T . MX2.

3.3 Boltzmann equations

3.3.1 Differential equations

The simplifying approximation of Maxwell-Boltzmann statistics allows one to factor out the chemical potential from the overall reaction rate density. Let us again use the definitions nΨi µΨ  nΨi µΨ  rΨi eq = Exp , rΨi eq = Exp . (3.23) ≡ nΨi T ≡ nΨi T The non-equilibrium reaction rate for the process ij kl is then given by W neq(ij → → kl) = rirjW (ij kl). The Boltzmann equations can then be conveniently expressed → 3 using this notation. The Boltzmann equation for nX1 is given by

dnX1 1 th eq h i th eq h i = 3HnX1 + Γ n (rurdrd + rurdrd) 2rX1 + Γ n rX2 rX1 dt − 2 1A X1 − 2B X2 − h i h i + W1 rdrd + rdrd rX1ru rX1ru + W3 rX2ru + rX2ru rX1ru rX1ru − − − − h i h i h i + 2W4 rdru + rdru rX1rd rX1rd + 2W6 1 rX1rX1 + W8 1 rX2rX1 − − − − h i + 3W3 rdrd rdrd + rX2ru rX2ru . (3.24) − − 3We will see below that the three flavours of right chiral down type quarks have the same chemical potentials in the temperature ranges we are interested in. This allows us to set rd = rs = rb in the following equations. Chapter 3. CP violating scatterings in baryogenesis 52

The Boltzmann equation for nX2 is given by

dnX2 1 th eq h i th eq h i = 3HnX2 + Γ n ((rurdrd + rurdrd) 2rX2 + Γ n rX1 rX2 dt − 2 2A X2 − 2B X2 − h i h i + W2 rdrd + rdrd rX2ru rX2ru + W3 rX1ru + rX1ru rX2ru rX2ru − − − − h i h i h i + 2W5 rdru + rdru rX2rd rX2rd + 2W7 1 rX2rX2 + W8 1 rX2rX1 − − − − 1 th eq h i h i + DΓ n (rurdrd rurdrd) + 3W3 rdrd rdrd + rX1ru rX1ru . 2 2A X2 − − − (3.25)

The Boltzmann equation for the baryon-minus-lepton number, nB−L, is given by

dnB−L 1 th eq h i 1 th eq h i = 3HnB−L + Γ n rurdrd rurdrd + Γ n rurdrd rurdrd dt − 2 1A X1 − 2 2A X2 − h i h i + W1 rX1ru rX1ru + rdrd rdrd + W2 rX2ru rX2ru + rdrd rdrd − − − − h i h i + 2W4 rX1rd rX1rd + rdru rdru + 2W5 (rX2rd rX2rd + rdru rdru) − − − − h i + 3W3 rX1ru + rX1ru rX2ru rX2ru − − 1 th eq h i + DΓ n 2rX2 (rurdrd + rurdrd) . (3.26) 2 2A X2 −

The terms proportional to 3 and D are the source terms coming from scatterings and decays respectively. These lead to the generation of a baryon asymmetry once there is a departure from thermal equilibrium. Note we have used the unitarity conditions to express the source terms in a form in which one explicitly sees they vanish when all

rΨi = 1 (thermal equilibrium). The other terms are the washout terms: these drive the solutions back to the equilibrium values.

3.3.2 Chemical potentials

The chemical potentials of the quarks and leptons depend on the baryon and lepton asymmetries. Here we are interested in determining the chemical potentials of the right chiral up and down type quarks. We consider two regimes, above and below the elec- troweak phase transition. We approximate the transition to occur instantaneously at a

temperature TEW at which point we take the SU(2) sphalerons to have switched off and the SM quarks and leptons to have gained masses through the Higgs mechanism. Chapter 3. CP violating scatterings in baryogenesis 53

3.3.2.1 Above TEW

In determining the chemical potentials in terms of nB−L in this regime one takes into account (i) the SU(3) sphalerons, (ii) the SU(2) sphalerons, (iii) the SM Yukawa inter- actions, (iv) conservation of the weak hypercharge. The analysis is the same as for the standard leptogenesis scenario [128]. The chemical potential of the right chiral up type quark is given by 10 nB−L µuR = . (3.27) −79 T 2 The right chiral down type quark chemical potential is given by

38 n µ = B−L . (3.28) dR 79 T 2

The baryon asymmetry is related to nB−L

28 n = n . (3.29) B 79 B−L

The chemical potentials of the up and down type quarks are therefore determined in terms of nB−L in this temperature regime. The final nB in this temperature regime is used as an initial condition for the Boltzmann equations at lower temperatures.

3.3.2.2 Below TEW

Below the electroweak phase transition the electroweak sphalerons have switched off and the weak hypercharge is no longer conserved. However, lepton number is now conserved and the overall electric charge must still vanish. The fields present are listed in Table 3.1. The baryon number asymmetry is approximately given by [128]

T 2 n = B, (3.30) B 6 where B = 2Nuµu + 2Ndµd and Nu (Nd) is the number of relativistic up (down) type quark flavours. The lepton number asymmetry is approximately given by [128]

T 2 n = L, (3.31) L 6 Chapter 3. CP violating scatterings in baryogenesis 54

Field Relativistic degrees of freedom chemical potential u 6 Nu µu × d 6 Nd µd × e 2 Ne µe × ν 1 Nν µν × Table 3.1: SM fermionic fields and relativistic degrees of freedom for temperatures below the electroweak phase transition. We set Nν = 3 and use a simple step function to model Nu, Nd and Ne, as degrees of freedom are removed from the plasma.

where L = 2Neµe + 3µν and Ne is the number of relativistic charged lepton generations. The chemical potentials are determined using the following constraints.

Net electric charge vanishes. Once the temperature drops below the mass of a field • its number density is very rapidly suppressed, if the field is in chemical equilibrium, as is the case for the SM fermions. We approximate this by only counting fields with mass below the temperature in contributing to the net charge of the plasma. This yields a constraint

2Nuµu Ndµd Neµe = 0. (3.32) − −

Rapid interactions involving W boson exchange give the relation •

µu + µe µd µν = 0. (3.33) − −

YL nL/s is constant after TEW . This gives the constraint • ≡

L 2Neµe µ = − , (3.34) ν 3

where L is computed using the lepton asymmetry YL at TEW .

Combining these three constraints gives the up quark chemical potential

    −1 1 1 1 3Nu Nu µu = L + B + + 1 + + + 2Nu . (3.35) 3 2Nd 2Ne × Ne Nd

The down type chemical potential can neatly be expressed as

B 2Nuµu µd = − . (3.36) 2Nd Chapter 3. CP violating scatterings in baryogenesis 55

Figure 3.5: Example solution to the Boltzmann equations with MX2 = 100 TeV, −16 −2 MX1 = 50 TeV, all κa = 10 GeV and number densities expressed in ratios to entropy YΨ nΨ/s. The baryon asymmetry with CP violation only in scatterings (de- ann ≡ dec cays) YB (YB ) is also shown. CP violating scatterings are the dominant asymmetry generation mechanism in this example.

In this temperature regime the chemical potentials of the quarks can therefore be de- termined in terms of nB and the net lepton number nL at the sphaleron switch-off temperature.

3.3.3 Numerical solutions

We solve the Boltzmann equations numerically using Mathematica [116]. This involves evolving the three coupled ordinary differential equations from a suitably high tem- perature to a temperature at which the B violation is no longer effective. The initial temperature is chosen to be high enough so the 2 2 interactions rates are well above ↔ the expansion rate. The initial conditions correspond to equilibrium distributions for 4 X1 and X2 and zero nB−L. At the electroweak phase transition the sphalerons switch

off and the lepton number remains constant. We then proceed to track nB rather than

nB−L using the value of nB obtained at TEW as an initial condition.

4Any initial B − L would be erased by the on-shell interactions involving the new particles at the UV completion scale — even below this scale the effective 2 ↔ 2 interactions are rapid enough to make the analysis independent of initial conditions so long as the post-inflation reheating temperature of the universe is sufficiently high. Chapter 3. CP violating scatterings in baryogenesis 56

Finally a dilution factor is applied to the final YB if the X2 and X1 particles are suffi-

ciently long lived. This dilution factor comes about because the X2 and X1 can come to dominate the energy density of the early universe. The universe switches from a

radiation to a matter dominated epoch until the X2 and X1 decay. The decays of these particles then produces a large amount of entropy — slowing down the rate at which the universe is cooling — and lead to a departure from the assumption of constant entropy in the early universe. The dilution factor can be approximated as [129]

 Y M  d = Max 1.8h1/4 α T fo Xα , 1 , (3.37) S eff | 1/2 (ΓαMP l)

where Yα T fo is the density to entropy ratio of the decaying particle at freeze out and | Γα is its decay rate. Being more abundant at freezeout and having a longer lifetime, X1 tends to dominate the dilution factor.

For the effective radiation degrees of freedom, geff , we use the values given in Ref. [130]. The effective entropic degrees of freedom are the same for the temperature range of

interest here, heff = geff (the two differ only after neutrino decoupling).

An example solution to the Boltzmann equations is shown in Fig. 3.5. The cosmolog- ical history of the universe proceeds in the a similar way to the toy model. At high

temperature the 2 2 interactions are rapid, keeping the nX1 and nX2 close to their ↔ equilibrium values. Due to the expansion of the universe the particles are never exactly

in equilibrium. As the temperature decreases the interaction rates drop and rX1, rX2

and YB continue to increase. The size of the source term will depend not only on the

CP violation but also on how far away the temperature is from MX2 and MX1; in the

massless limit rX1 = rX2 = 1 even in the absence of interactions (assuming a common

T ). Eventually, at Γα H, the decays take over. Excess X2 and X1 decay away and ∼ the X2 decays also contribute to the baryon asymmetry. The freezeout temperature is determined by the coupling size; numerically we find the maximum asymmetry for

freezeout at T Mα, i.e. before the number densities of X1 and X2 become Boltzmann ∼ suppressed.

The CP violation in the decays and the scatterings increases κ. However, as κ ∼ increases, the freeze out of the 2 2 interactions occurs closer and closer to the decay ↔ temperature, i.e. at Γα H, and the particles are kept closer to thermal equilibrium. ∼ Chapter 3. CP violating scatterings in baryogenesis 57

10-9 10-10 10-11 È

B -12

Y 10 È 10-13 10-14 10-15 10-18 10-17 10-16 10-15 10-14 10-13 -2 Κa GeV

@ D Figure 3.6: Left: the final baryon asymmetry as a function of the couplings (all set equal) κa, the masses have been set to MX2 = 100 TeV and MX1 = 90 TeV. The obseved value of the baryon asymmetry is indicated by a gray horizontal line and is −16 −2 −15 −2 reached for couplings in the range 10 GeV . κa . 10 GeV . Also shown are sca dec the final asymmetries YB (YB ) calculated with CP violation only in the scatterings (decays). Right: same as in the left plot but on a double logarithmic scale.

10-9 10-10 10-11 È

B -12

Y 10 È 10-13 10-14 10-15 10-18 10-17 10-16 10-15 10-14 10-13 -2 Κa GeV

@ D Figure 3.7: Same as in Fig. 3.6 but with MX1 = 50 TeV. The decays now play a more significant role but the final baryon asymmetry cannot match the observed value for any choice of couplings κa.

The resulting effect on YB is shown in Figs. 3.6 – 3.8 in which the final asymmetry first increases as the couplings increase but eventually becomes suppressed.

Also shown in Figs. 3.6 – 3.8 is the domination of the CP violating scatterings in de-

terming the final YB for the majority of the parameter space. This is because the CP violation due to the scatterings — which scales as κT 2 — can be relatively high at the freezeout temperature. This balances the relatively small departure from equilib- rium just prior to freezeout, when the 2 2 interactions generate the majority of the ↔ asymmetry, compared with the departure from equilibrium at Γ2 H, when the decays ∼ contribute to the asymmetry. The CP violation in the decays is typically much smaller than the CP violation in the scatterings at freeze out — at least for couplings small Chapter 3. CP violating scatterings in baryogenesis 58

10-9 10-10 10-11 È

B -12

Y 10 È 10-13 10-14 10-15 10-18 10-17 10-16 10-15 10-14 10-13 -2 Κa GeV

@ D Figure 3.8: Same as in Fig. 3.6 but with MX1 = 10 TeV. Note the sign of the asym- metry can be changed by changing the sign of the CP violating phase. The horizontal gray line now indicates the magnitude of the observed asymmetry but with opposite sign. The CP violation in the decays is now close to maximal but the scatterings still −14 −2 dominate for κa . 10 GeV .

enough for a significant departure from equilibrium to take place. As MX1 is decreased the CP violation in decays becomes more important. The reasons are twofold: smaller

MX1 results in a greater D and in larger washout effects after X2 freezes out for | | kinematic reasons.

Note that the scaling of the CP violation with temperature is different in leptogenesis. The dimensionless couplings in leptogenesis mean the CP violation in scatterings will not grow as strongly with temperature, as it does here, but remain mostly constant [65, 110– 113]. This explains why the CP violating scatterings can play a crucial role in the neutron portal but only a negligible role in the strong and intermediate washout regimes in leptogenesis in determining the final YB [110, 112]. Note also the possible (1) O corrections to YB in weak washout leptogenesis are due to a relative cancellation between the asymmetry created at high temperature by the scatterings and at low temperature by the decays [112]. In the neutron portal, in contrast, the asymmetries can be many orders of magnitude apart, i.e. Y sca Y dec . | B |  | B | As for the toy model, the precision of these calculations can be improved by tak- ing into account departures from kinetic equilibrium, quantum statistics and thermal masses [120–124]. The corrections are expected to be at most (1), as in the case for O leptogenesis. Chapter 3. CP violating scatterings in baryogenesis 59

3.4 Constraints

Constraints on the neutron portal have previously been discussed in Ref. [125]. The decay of relic particles after t 1 s can disrupt big bang nucleosynthesis (BBN) [131– ∼ 133]. Considering the X1 lifetime one finds a constraint [125]

 5/2 1 TeV −18 −2 κ1 & 10 GeV . (3.38) MX1

Similarly from the X2 lifetime one requires [125]

 5/2 1 TeV −18 −2 κ2 or κ3 & 10 GeV , (3.39) MX2 where we have ignored the final state MX1 mass. The κ3 bound becomes more stringent as MX1 is increased.

c c Operators of the form κXαLdRuRdR or κXαLdRQLQL, where Xα is Majorana, are constrained by limits on neutron-antineutron oscillations (see Fig. 3.9). The oscillation period is estimated as [125]

   −13 −2 2  6 8 MXα 10 GeV 250 MeV τn−n 3 10 s , (3.40) ∼ × × 1 TeV κ ΛQCD where we have approximated the nuclear matrix element with ΛQCD. Comparison to the 8 experimental limit from bound neutrons τn−n 2.4 10 s [134] (or from free neutrons ≥ × 7 τn−n 8.6 10 s [135] — which has smaller theoretical uncertainty) shows broad ≥ × compatibility in the parameter range of interest. The operators we considered, however, couple only off-diagonally in down quark flavour. The oscillation period is therefore further suppressed and certainly does not pose any problems for the parameter choices we have been interested in above. Similar conclusions hold for the loop induced meson mixing operators [125].

c c Flavour off-diagonal operators such as κXαLuRsRbR + κXαLuRsRdR will lead to meson decays such as B+ π+K0 (see Fig. 3.10). The contribution to the branching ratio is → estimated as

1 TeV4  κ 4 Br (B+ π+K0) 10−38 , (3.41) X −12 −2 → ≈ × MXα 10 GeV Chapter 3. CP violating scatterings in baryogenesis 60

u u

Xα d d

d d

Figure 3.9: Neutron-antineutron oscillation induced by operators of the form c c κXαLdRuRdR or κXαLdRQLQL.

b d s

Xα u u d

+ + 0 c Figure 3.10: Meson decay B π K due to the operators κXαLuRsRbR + c → κXαLuRsRdR. which is far below the experimental observation Br(B+ π+K0) = (2.3 0.07) 10−5 → ± × [136–138]. The most important constraints on this scenario therefore come from BBN and neutron-antineutron oscillations.

3.5 Conclusion

In this chapter we studied the effects of CP violation in 2 2 interactions for baryo- ↔ genesis. As a case study we have taken a neutron portal model in which two Majorana fermions X1 and X2 are coupled to the neutron operator. In order to find the final baryon asymmetry, we calculated all the relevant decay rates and 2 2 scattering rates ↔ and solved the Boltzmann evolution equations numerically. From dimensional grounds 2 the CP violation in the decays scales as κMX2 where κ is the order-of-magnitude of the 2 relevant couplings. The CP violation in the scatterings scales as κMX2 at low temper- 2 atures but grows as κT for T MX2. Consequently CP violating 2 2 scatterings  ↔ can play a crucial role at high temperature even when the departure-from-equilibrium is relatively small. Indeed for many areas of the parameter space the CP violating scat- terings play a dominant role in determining the final baryon asymmetry. This is to be contrasted with leptogenesis — in which the CP violating scatterings do not scale as strongly with temperature — and have only a negligible effect on the final asymmetry outside of the weak washout regime [65, 110–113]. We also discussed constraints on Chapter 3. CP violating scatterings in baryogenesis 61 the model from experiments and BBN. The techniques learned can be applied to other baryogenesis scenarios in order to take into account the, possibly dominant, corrections due to CP violating scatterings. Chapter 4

The baryon asymmetry and dark matter in radiative inverse seesaw models

4.1 Introduction

In the previous two chapters we have studied how the baryon asymmetry can arise through CP violating scatterings. We now take a slight change in direction and dis- cuss two models which can explain the neutrino masses. The neutrino masses can be explained as arising from the three minimal tree level UV completions of the Weinberg operator: the type-I [59–62], type-II [139–144] or type-III seesaws [145].

Lepton number is violated in the seesaw models and so the introduction of such BSM physics can also have implications for the BAU. The most famous example is of course leptogenesis in the type-I seesaw [64], which we have described in Chapter 1. The scale of new physics can a-priori be quite broad in the standard seesaw scenarios. For example, the new physics scale for the type-I seesaw may range from sub-electroweak up to the GUT scale [146].

It is therefore of interest to consider alternatives which may be more directly accessible experimentally. The inverse seesaw (ISS) explains the neutrino masses with new physical states at the (TeV) scale [147, 148]. The mass parameters at this scale are actually L O

62 Chapter 4. Radiative inverse seesaw models 63 conserving, while another small L violating parameter, which we shall denote µ, leads to small but non-zero light neutrino masses. Clearly small µ is technically natural as µ 0 enhances the symmetry of the theory. Another further possibility is to have µ → actually be a radiatively generated parameter, which brings us to the radiative inverse seesaw models [149–153].

In this chapter we will study the cosmological history of two such radiative ISS models, one with explicit violation of B L [152], the other with spontaneous B L viola- − − tion [149]. (The choices are also motivated for being UV complete and not relying on fine tuning, Ref. [153] already discussed the DM phenomenology of their radiative ISS model.) We first discuss their compatibility with baryogenesis. We derive constraints on the models by demanding they do not washout the BAU. The baryogenesis temperature may lie either above and below the ISS scale, and we derive constraints for both possi- bilities. We also investigate whether resonant leptogenesis is possible using the radiative ISS degrees of freedom.

These models also contain DM candidates and so we proceed to investigate their com- patibility with the observed DM density and limits on the DM parameter space. In this way we can explore the viable parameter space of these models and learn how they may be detected or ruled out.

This chapter is organised as follows. In Sec. 4.2 we introduce the ISS mechanism and discuss the natural suppression of L violating scatterings in the ISS [154]. In Sec. 4.3 we introduce the Law/McDonald radiative ISS [152], derive constraints from washout and study its DM phenomenology. In Sec. 4.4 we repeat the analysis for the Ma radiative ISS [149].

4.2 The inverse seesaw mechanism

4.2.1 The generic mass matrix

The ISS [147, 148] is generically realised with the additional terms in the Lagrangian

ab † 1 c 1 c ∆ = λν lLaΦ NR + MRNRSL + µ(SL) (SL) + µ˜(NR) (NR) + H.c., (4.1) L b 2 2 Chapter 4. Radiative inverse seesaw models 64

where Φ (1, 2, 1/2) is the usual SM Higgs doublet, lL (1, 2, 1/2) is the SM lepton ∼ ∼ − doublet, and NR and SL are singlets under the SM gauge group. We leave the details of additional symmetries and symmetry breaking which results in the above terms [155] † c unspecified for now (a field redefinition ensures the absence of the lLΦ (SL) term). After T Φ gains a vacuum expectation value (VEV) Φ = (0 , vw/√2) and breaks the EW h i symmetry SU(2)L U(1)Y U(1)EM, the neutral-fermion mass matrix is (assuming ⊗ → one generation):

   c  0 mD 0 (νL) 1       = c ¯     , (4.2) νL NR SL mD µ˜ MR  NR  L 2     c 0 MR µ (SL)

where mD λνvw/√2. Diagonalisation of the mass matrix for µ, µ˜ mD,MR leads to ≡  a light neutrino mass: 2 mD mν 2 µ . (4.3) ≈ MR c The heavy Majorana neutrinos N1 and N2, superpositions of mostly (SL) and NR, have mass eigenvalues µ MN MR . (4.4) ≈ ± 2 (Theµ ˜ dependence enters at higher order.) Due to the small mass splitting, they form a pseudo-Dirac pair. The mixing between the light and heavy states is approximately

θ mD/MR. In the limit µ 0 the light neutrinos become massless at tree level, ' → although loop corrections give finite contributions to their masses unless both µ, µ˜ 0 → [156]. In this limit one also recovers a global lepton number symmetry in the Lagrangian, hence small choices of µ andµ ˜ are technically natural. This is the small parameter generated radiatively in the ISS models we return to below.

This discussion can be generalised to a realistic three generation case. The entries in the mass matrix (4.2), are promoted to 3 3 matrices, and the light neutrino mass matrix × is then to first order: −1 T −1 T Mν = MDMR µ(MR ) MD. (4.5)

Various studies on the collider phenomenology [157, 158], flavour violating processes [159–162], and neutrinoless double beta decay [163, 164] of the ISS have appeared in the literature. Chapter 4. Radiative inverse seesaw models 65

Figure 4.1: ∆L = 2 process which can wash out the baryon asymmetry at high temperatures. Both heavy mass eigenstates N1 and N2 contribute, but destructive interference for small mass splittings µ MN crucially limits the washout rate. 

4.2.2 ISS and baryogenesis

Before the EW phase transition, the non-perturbative sphaleron processes rapidly wash out any existing B + L asymmetry, while they preserve B L. However, if there exist − other rapid L violating processes, then both B and L will be driven to zero. In the ISS, the SM lepton number is violated by the parameters µ andµ ˜. Provided that these parameters are small enough, the B asymmetry is not washed out.

Let us examine the possible washout of a B asymmetry created, by processes unrelated

to neutrinos, at a temperature above the heavy neutrino mass scale, TBG > MN , where

TBG is the temperature of baryogenesis. In this case, the inverse decays to on-shell heavy

Majorana neutrinos lL + Φ Ni are not Boltzmann suppressed. Such inverse decays → ∗ followed by decays Ni lL + Φ can change the lepton number by two units. This → washout process is shown in Fig. 4.1.

It was found in Ref. [154] that in the ISS such washout processes can be naturally suppressed, and we review the findings here. For simplicity we take from now onµ ˜ = 0, as this parameter does not enter the neutrino mass eigenvalues to first order, and focus on µ instead. The more in-depth discussion of Ref. [154] allows both parameters to be non-zero with no major differences to the discussion here.

For typical mass choices the heavy neutrinos can decay into a component of the Higgs doublet and a lepton N Φ+lL. For heavy neutrino masses well above the Higgs mass, → MN Mh, the decay rate is  λ2 Γ = ν M , (4.6) D 8π N Chapter 4. Radiative inverse seesaw models 66

and for temperatures T > MN one includes a Lorentz factor (MN /T ), which arises as an approximation of the thermally averaged decay rate. In the type-I seesaw, the effectiveness of washout is well captured by the washout parameter,

ΓD K , (4.7) H ≡ T =MN

1/2 2 where H g T /MP l is the expansion rate of the universe [11]. Typically washout ∼ eff processes are ineffective for K . 1, while any net B is erased for K & 1.

In the ISS, however, the small mass splitting between the heavy pseudo-Dirac neutrinos leads to destructive interference in the scattering process depicted in Fig. 4.1. This leads to a suppressed washout rate,

2 2 µ Keff = Kδ , (4.8) Γ H ≡ D T =MN

where δ µ/ΓD [154]. As expected, in the limit of lepton number conservation, µ 0, ≡ → the heavy neutrinos form a Dirac pair, there is complete destructive interference in the ∆L = 2 scattering process, and washout of the baryon asymmetry does not occur.

Requiring Keff . 1 to avoid washout translates into a bound

 3/2 MN µ < λν 6 keV. (4.9) 1 TeV ×

For λν = 0.1, MN = 1 TeV, light neutrino masses mν = 0.1 eV are explained by µ = 330 eV (cf. Eq. (4.3)). So the light neutrino masses can be explained while avoiding washout of the BAU. Alternatively, one can express the above bound in terms of the mixing angle 1/3  5/6  1/6 −2  mν  1 TeV 100 θ & 1.4 10 . (4.10) × 0.1 eV MN geff To satisfy current experimental constraints (e.g. from universality of the weak interac- tion and rare leptonic decays) requires roughly, for MN above the EW scale, a mixing angle θ (10−2) [160, 165]. (The usually stringent constraint on the active-sterile . O mixing angle from neutrinoless double beta decay [166] is greatly weakened due to the small lepton number violation in the ISS [163, 164].) Future experimental tests at the θ 10−3 level can therefore help test the parameter space corresponding to washout ∼ avoidance [167]. Chapter 4. Radiative inverse seesaw models 67

For baryogenesis temperatures below the heavy neutrino scale, TBG < MN , the number density of heavy states is Boltzmann suppressed. This means the thermally averaged ∗ rate of the net washout process lL+Φ Ni lL+Φ is suppressed by both a Boltzmann → → factor and the small µ. In this regime one is even safer from washout.

What about baryogenesis itself in the ISS? The obvious candidate is resonant leptogene- sis [168]. The resonance itself is due to an almost degenerate pair of heavy neutrinos and exactly such pairs appear in the ISS. Studies have shown resonant leptogenesis is indeed possible in the ISS framework [169], and it is even possible to link this with the DM abundance [170]. It should be noted, however, that the calculation of the CP violation in the resonant regime has been under debate in the literature [67].

To explain the smallness of µ we now turn to models where it is generated radiatively. Because of the smallness of µ, washout of the BAU is suppressed in ISS models. However, to generate a finite value of µ radiatively requires the addition of new fields and L- violating interactions. We will see the effects of these new fields and interactions on washout of the BAU, and study the DM candidates in the radiative ISS models. We study the Law/McDonald [152] and Ma radiative ISS models in turn [149].

4.3 Law/McDonald radiative inverse seesaw

4.3.1 Review of the model

The Law/McDonald radiative ISS model [152] employs explicit violation of B L, so − that processes other than scattering through the heavy neutrinos crucially contribute to washout of the BAU. The model introduces an additional abelian gauge symmetry to the standard model gauge group

G = GSM U(1)d . (4.11) ⊗

All the SM particles are neutral under this new charge. The following exotic fermions are introduced (with convention QEM = I3 + Y ):

ER,L (1, 3, 1)(0) NR,L (1, 1, 0)(1). (4.12) ∼ − ∼ Chapter 4. Radiative inverse seesaw models 68

Along with the SM Higgs Φ (1, 2, 1/2)(0), the following scalars are introduced ∼

ξ (1, 3, 1)( 1), η (1, 1, 0)(1), χ (1, 1, 0)(2). (4.13) ∼ − − ∼ ∼

The following Yukawa interactions and bare Dirac mass terms are allowed

0 c 0 c ∆ = +yElLΦER + hξELξNR + h ERξNL + hχNRN χ + h NLN χ − L ξ R χ L (4.14) +MEELER + MN NLNR + H.c.

The VEV pattern is chosen to be

0 0 φ = vw/√2 = 174 GeV, ξ , η = 0, χ = vχ/√2 = 0 . (4.15) h i h i h i h i 6

The tree level mass matrix for the neutral fermions is then

   c  0 yEvw/√2 0 (νL) 1       = 0 c 0    0  . (4.16) νL (ER) EL yEvw/√2 0 ME  ER  L 2     0 c 0 ME 0 (EL)

Note NL,R are heavy Majorana fermions decoupled from the above matrix, and we have not yet taken into account the radiative corrections which will generate the terms

0 c 0 0 c 0 µ˜(ER) ER and µ(EL) EL after symmetry breaking. The scalar potential is given by:

† h 2 † † † † i VS = (Φ Φ) µΦ + λφ(Φ Φ) + λφξ(ξ ξ) + ληφ(η η) + λφχ(χ χ)

† h 2 † † † i + (ξ ξ) µξ + λξ(ξ ξ) + λξη(η η) + λξχ(χ χ)

† h 2 † † i + (η η) µη + λη(η η) + ληχ(χ χ) (4.17)

† h 2 † i + (χ χ) µχ + λχ(χ χ)  1  + λ ξΦΦη + µ ηηχ† + H.c. . ξφη 2 ηχ

The SM lepton number is violated explicitly by the combination of Eq. (4.17) and

Eq. (4.14), but a Majorana mass term for the EL and ER fields is forbidden due to 0 c 0 hypercharge. After χ and Φ gain VEVs, a radiative mass term µ(EL) EL is generated, 0 c 0 as shown in Fig. 4.2. A mass termµ ˜(ER) ER is also generated radiatively. The radiative mass is approximately 2 2 4 2 1 hξhχλξφη vwvχ µ 2 6 µηχ, (4.18) ≈ 16π 120 ΛISS Chapter 4. Radiative inverse seesaw models 69

χ h i φ φ h i h i φ φ h i h i 0 0 EL EL NR NR χ h i

Figure 4.2: Radiative mass generation in the Law/McDonald model.

where ΛISS MN ,Mη,Mξ is the scale of the new physics, usually taken to be (TeV), ∼ ∼ O and one may choose µηχ (TeV). This radiatively generated µ is exactly the small ∼ O parameter required in the mass matrix for the ISS. The light neutrinos masses are then

2 2 2 !  2  6 yEhξhχλξφη vχ 1 TeV  µηχ  mν −5 0.1 eV . (4.19) ≈ 10 ME ΛISS 1 TeV ×

4.3.2 Constraints from BAU washout

Previously we have seen that in the ISS framework, the washout of the BAU is naturally suppressed due to the smallness of the lepton-number violating parameter µ. In the Law/McDonald radiative ISS model, µ is in fact generated only after EW symmetry breaking and cannot itself cause washout. However, the SM lepton number is explicitly violated by the couplings which eventually generate µ and give rise to the radiative ISS. The interactions of Eqs. (4.17) and (4.14) can then lead to washout. Here, we investigate under what conditions the BAU is not washed out.

The B L of the SM remains a good symmetry if any of the following sets of parameters − vanishes

0 0 yE , µηχ , λξφη , hξ, h , hχ, h . (4.20) { } { } { } { ξ} { χ}

(Note that ME has to be large since ER,L are charged under the SM gauge group.) The BAU can change only if interactions which involve all of the above couplings are efficient. We show such a sequence of interactions in Fig. 4.3. In order to preserve the BAU, we require that the interactions mediated by at least one of these sets of Chapter 4. Radiative inverse seesaw models 70

lL lL ER

N L Φ ER Φ∗ ξ Φ∗ NL

χ Φ ξ∗ η Φ∗

Φ η∗ η

Figure 4.3: An example of a sequence of interactions present in the early universe which shifts the lepton number. couplings are out of equilibrium in the period between baryogenesis and the EW phase transition. We discern two cases, depending on the scale of baryogenesis: TBG > ΛISS and TEW < TBG < ΛISS.

TBG > ΛISS •

1. The Yukawa couplings of Eq. (4.14) induce 2-body decays of the heaviest ∗ particle participating in each operator, e.g. NR EL + ξ . Assuming these → decays are kinematically allowed, they proceed with thermally averaged decay rate 2   hj Mj Mj ΓD , (4.21) ≈ 16π T

where Mj ΛISS is the mass of the decaying particle, hj is the relevant ∼ Yukawa coupling, and from now on Lorentz factors, such as (Mj/T ), ap- pearing in decay rates are understood to be present only for temperatures above the mass of the decaying particle. To prevent washout one requires 1/2 2 (nj/nl)ΓD H g T /MP l, for TEW < T < TBG, where nj (nl) is the . ∼ eff number density of species j (leptons). The strongest constraint arises from the

latest time the density of the exotic particles is still significant: at T ΛISS. ≈ This implies

1/2  3  1/4  1/2 1/2 ΛISS −7  geff  ΛISS hj . 16πgeff 10 , (4.22) MP l ≈ 100 TeV Chapter 4. Radiative inverse seesaw models 71

where geff are the relativistic degrees of freedom at T Mj ΛISS. ∼ ∼ 0 0 If this condition holds for yE, or hξ and hξ, or hχ and hχ, the lepton-number violating decays and inverse decays induced by these couplings are rare, and the BAU remains frozen.

2. The thermally averaged rate of the decay mode χ ηη is → 2 µηχ µχ  ΓD χ→ηη . (4.23) ≈ 32πµχ T

Hence the lepton number of the universe is approximately conserved if

1/2  3  1/4  3/2 1/2 ΛISS  geff  ΛISS −4 µηχ . 32πg 10 GeV. (4.24) MP l ≈ 100 1 TeV

3. The cross section for the scattering process η∗ + ξ∗ Φ + Φ is approximately → 2 λξφη 1 σ 2 2 , (4.25) ∼ 16π (T + ΛISS)

and the corresponding scattering rate is Γ n(T ) σ. In the relativistic regime, ∼ Γ T , and hence increases with temperature slower the Hubble parameter. ∝ In the T < ΛISS regime, Γ becomes Boltzmann suppressed. Thus, for suffi- ciently small coupling,

 g 1/4  Λ 1/2 λ 10−7 eff ISS , (4.26) ξφη . 100 1 TeV

this process is never in equilibrium.

4. Lepton-number violating processes may also occur via some of the ISS degrees

of freedom propagating off-shell. For example the scattering process NR + ∗ χ NR EL + ξ has an approximate cross section, → →

2 2 (hξhχ) T σ 2 2 2 , (4.27) ∼ 8π (T + MN )

which leads to a bound

 g 1/4  Λ 1/2 (h h ) 10−7 eff ISS . (4.28) ξ χ . 100 1 TeV Chapter 4. Radiative inverse seesaw models 72

Satisfying at least one of the bounds of Eqs. (4.22), (4.24), (4.26) and (4.28)

ensures no washout of the BAU, if baryogenesis has occured at temperatures TBG >

ΛISS. However, as can been seen from Eq. (4.19), these bounds would imply that the contribution to the active neutrino masses generated via this mechanism is insignificant.

TEW < TBG < ΛISS • The ISS degrees of freedom remain in thermal and chemical equilibrium even at temperatures much below the ISS scale, due to their gauge, Yukawa and scalar

interactions. However, at TBG < ΛISS, their density is Boltzmann suppressed. This implies that the exotic leptons carry only a small fraction of the net lepton number of the universe.

Consider for example again the 2-body decay rate of the ISS degrees of free-

dom via the Yukawa couplings of Eq. (4.14). The decay rates at TBG < ΛISS 2 are Γj h Mj/16π, where as before Mj is the mass of the decaying parti- ≈ j cle and hj is the Yukawa coupling that causes the decay. The lepton asym- 3 metry will be approximately conserved if Γjnj Hnl, where nl T is the  ∼ lepton-number density, carried mostly by the light relativistic SM leptons, and 3/2 nj (MjT/2π) exp( Mj/T ) is the number density of the non-relativistic ex- ≈ − otic leptons.

For Yukawa couplings hj (1), this yields the constraint ∼ O

ΛISS & 42 , (4.29) TBG

(where we set Mj ΛISS). The interaction considered in the case (ii) gives a ∼ similar constraint, while the scattering rates in cases (iii) and (iv) are sufficiently suppressed for ΛISS & 25. (4.30) TBG

We therefore uncover an important feature of this model: baryogenesis has to occur below the ISS scale. If it occurs before the EW phase transition, the ISS scale has to be sufficiently high, as seen by Eq. (4.29). If it occurs after the EW phase transition, no limit on ΛISS applies from considering the survival of the BAU today. Note that in this model, µ can only be generated after EW symmetry breaking (see Eq. (4.18)), Chapter 4. Radiative inverse seesaw models 73 because the exotic fermions which gain a radiative Majorana mass carry hypercharge. This precludes resonant leptogenesis within this model, as leptogenesis must occur before the sphalerons switch off. It may, however, be possible that another of the ISS fields generates the asymmetry, with washout being suppressed by having the masses of at least one of the other exotic fields at a sufficiently high scale above TBG. The construction of such a scenario, ensuring the satisfaction of the Sakharov conditions [29] and generation of the observed BAU, is beyond the scope of this work.

4.3.3 Dark Matter

In the Law/McDonald model the scalars, ξ and η, and the fermions, NR and NL, are odd under an accidental Z2 symmetry. The lightest of these is a DM candidate. As- sume for now that the lightest state is in the scalar sector (we will briefly discuss the fermionic case below). The neutral components of ξ and η mix and the masses of the real and imaginary components split. The general mass squared matrix in the basis   Re(η) Re(ξ0) Im(η) Im(ξ0) is

 2 2  Mη + µηχvχ/√2 λξφηvw/4 0 0    2 2   λξφηvw/4 Mξ 0 0    , (4.31)  2 2   0 0 Mη µηχvχ/√2 λξφηvw/4  − −  2 2 0 0 λξφηv /4 M − w ξ

where we have taken λξφη and µηχ to be real and positive without loss of generality as one can absorb the phase by redefining the fields, and

2 2 2 2 M µ + ληφv /2 + ληχv /2, (4.32) η ≡ η w χ 2 2 2 2 M µ + λξφv /2 + λξχv /2. (4.33) ξ ≡ ξ w χ

The lightest of the eigenstates forms the dark matter. If the admixture of ξ0 in the DM

field is small (which is the case for Mξ Mη) interactions with the Z boson will be  suppressed. Scattering of DM with SM particles via the Z or Z0 is also suppressed due to the mass splitting between the imaginary and real components of the DM field. (Gauge bosons interact off diagonally with the real and imaginary components of scalar fields.) Annihilation and scattering are dominated by the interactions with the Higgs fields, Chapter 4. Radiative inverse seesaw models 74 and for simplicity we will work in this regime. We then have a scalar Higgs portal DM candidate which has been studied extensively in the literature [86, 171–173]. A possible difference from the standard Higgs portal arises when the heavier Higgs boson is light enough to influence the DM annihilation cross section [174–178]. We will explore this possibility in greater detail below in the context of the Law/McDonald model, though the discussion is largely applicable to generic Higgs portal models with an EW singlet Higgs mixing with the SM Higgs.

4.3.3.1 Double Higgs portal DM

We denote our (real scalar) DM candidate η, from now, and take its mass to be

2 2 2 2 M = µ + ληφv /2 + ληχv /2 µηχvχ/√2, (4.34) DM η w χ − where µηχ > 0. One can choose parameters for the scalar potential (4.17) so Φ, and χ T gain the following VEVs: Φ = (0 , vw/√2) , and χ = vχ/√2. The VEVs satisfy the h i h i following equations:

1 µ2 + λ v2 + λ v2 = 0, (4.35) χ 2 φχ w χ χ 1 µ2 + λ v2 + λ v2 = 0. (4.36) φ 2 φχ χ φ w

To ensure that the potential is bounded from below and that (Φ, χ, η) = (vw/√2, vχ/√2, 0) is a minimum requires:

λφ, λχ, λη > 0, (4.37)

2 λφχ < 4λφλχ, (4.38)

2 MDM > 0, (4.39) p ληχ > 2 ληλχ, (4.40) − p ληφ > 2 ληλφ. (4.41) −

2 2 An additional condition for symmetry breaking is µχ < 0, or µφ < 0, and to ensure 2 there is no deeper minimum (at least at zero temperature) one can take µη > 0. The above conditions are necessary but not sufficient to ensure the potential is bounded from Chapter 4. Radiative inverse seesaw models 75 below. An additional sufficient condition is

( 2 ) 1 (4λφληχ 2λφχληφ) 2 λη > − 2 + ληφ . (4.42) 4λφ 4(4λφλχ λ ) − φχ

The squared mass matrix for the physical Higgs bosons φ and χ is given by

    2 1   2λφvw λφχvwvχ φ = φ χ     . (4.43) L 2 2 λφχvwvχ 2λχvχ χ

One then finds the mass eigenstates (mH mh), ≥ q 2 2 2 2 2 2 2 m = λχv + λφv (λχv λφv ) + (λφχvwvχ) , (4.44) h χ w − χ − w q 2 2 2 2 2 2 2 m = λχv + λφv + (λχv λφv ) + (λφχvwvχ) . (4.45) H χ w χ − w

The mass eigenstates are superpositions of φ and χ,

      h cos θ sin θ φ   =     , (4.46) H sin θ cos θ χ − where the mixing angle θ is given by the following,

λ v v tan θ = φχ w χ . (4.47) 2 2 q 2 2 2 2 λχv λφv + (λχv λφv ) + (λφχvwvχ) χ − w χ − w

Note h (H) couples to SM particles the same way as the SM Higgs but with an additional factor of cos θ ( sin θ) at the vertex. We identify h with the Higgs-like particle discovered − at the LHC.

In the limit where the H is very heavy the important DM interactions proceed through the SM-like Higgs, and one in effect returns to the standard Higgs portal scenario. If H is lighter, however, one now has annihilation to SM particles and interactions with quarks proceeding through both h and H propagators. We term this a double Higgs portal and we shall compare the two scenarios below. Similar ideas have been explored previously in the literature: Refs. [174–176] considered two Higgs doublet models, while Ref. [177] considered fermionic DM with an exotic SM singlet Higgs. We consider scalar DM with an exotic singlet Higgs, a scenario which has been analysed previously in Ref. [178]. One important difference is that we focus more on the phenomenology of high DM masses, Chapter 4. Radiative inverse seesaw models 76 while the focus previously has been on the lower mass range MDM . 150 GeV.

In terms of the Higgs mass eigenstates one finds the following interactions with the DM

1 2 ∆ = η (ληφvw cos θ + ληχvχ sin θ µηχ sin θ)h − L 2 − 1 2 + η ( ληφvw sin θ + ληχvχ cos θ µηχ cos θ)H 2 − − 1 2 + η (ληχ cos θ sin θ ληφ cos θ sin θ)hH (4.48) 2 − 1 + η2(λ sin2 θ + λ cos2 θ)H2 4 ηφ ηχ 1 + η2(λ cos2 θ + λ sin2 θ)h2. 4 ηφ ηχ

To reduce the number of parameters, and allow one to gain an appreciation of the

phenomenology of the double Higgs portal, we study a special case where ληχ = ληφ λ ≡ 0 0 and µηχ = λ µ , where we will vary λ but keep µ > 0 and fixed. The interaction of | | ηχ ηχ the Higgs bosons with the DM then simplifies to

1 0  2 ∆ = vw cos θ + vχ sin θ Sign[λ] µ sin θ λη h − L 2 − · ηχ 1 0  2 + vχ cos θ vw sin θ Sign[λ] µ cos θ λη H (4.49) 2 − − · ηχ 1 1 + λη2H2 + λη2h2. 4 4

To see the effect of the second Higgs boson, we present two cases. Case A is where the heavy Higgs boson is taken to be heavy enough and the mixing negligible, so it is effectively decoupled. In this limit one obtains the standard Higgs portal. For case B,

the double Higgs portal, we choose some parameters, namely MH = 500 GeV, vχ = 1 0 TeV, µηχ = 2 TeV, and examine the change in phenomenology for different choices of the Higgs mixing parameter sin θ.

Furthermore, given the above choice of parameters, we have calculated λχ, λφ, and λφχ as functions of sin θ. One finds that condition (4.38) is satisfied for the entire range of the mixing angle ( π/2 θ π/2), and that µ2 < 0 so that symmetry breaking can − ≤ ≤ χ proceed.

As the heavy Higgs boson enters loop corrections to the W and Z boson propagators, a constraint on sin θ comes from the oblique parameters T , S, and U. By following

the analysis of Refs. [178–180] one finds that for MH = 500 GeV the mixing angle is Chapter 4. Radiative inverse seesaw models 77

Figure 4.4: Possible DM annihilation channels. The diagrams on the lower line give only a small correction and are not included in the calculations. constrained to be sin θ 0.43 (4.50) | | ≤ by EW precision observations at 95% C.L. (assuming the other exotic particles give only negligible contributions.)

4.3.3.2 DM relic abundance

One can then calculate the annihilation cross section of DM into standard model par- ticles. The relevant Feynman diagrams are shown in Fig. 4.4. To obtain the velocity averaged cross section, one should integrate over the velocity distribution. However a good approximation is obtained with,

σv σv sˆ=4M 2 . (4.51) h i ≈ | DM

One can then simply compare to the required annihilation cross section, for the observed relic abundance [181], which for DM masses greater than 10 GeV is approximately [182],

−26 3 σv DM 2.2 10 cm /s. (4.52) h i ≈ ×

2 2 As can be seen from Eq. (4.34), MDM gains a contribution λvχ/2. For small DM masses 2 λ < 0 is therefore necessary, given our choice vχ = 1 TeV and µη > 0. Some degree of 2 fine tuning of µ is of course required for MDM λvχ. η  Chapter 4. Radiative inverse seesaw models 78

Figure 4.5: Required coupling for the correct DM abundance for different values of sin θ and Case A (single Higgs portal) for comparison. Parameters have been chosen so 0 that λ ληφ = ληχ, µ = 2 TeV, MH = 500 GeV, and vχ = 1 TeV. ≡ ηχ

We have calculated the required coupling λ ληφ = ληχ, with the constraint λ < 0. ≡ (The relevant expressions for the cross sections can be found in Appendix C.1.) The results for different choices of the mixing parameter are shown in Fig. 4.5. The observed relic abundance can be achieved with values of λ 0.1 for most of the DM mass ∼ − range. Note the falls in the required coupling at the points where 2MDM = Mh/H , as the annihilation cross section is enhanced by an s-channel resonance of the Higgs boson propagator.

We have also checked that the constraints of Eqs. (4.40)–(4.42) can be satisfied with perturbative choices of the quartic coupling λη (except for small regions of parameter space around sin θ 0.05, where for low DM masses λ > 1). ≈ | |

4.3.3.3 Direct detection

DM particles will scatter off nuclei through t-channel Higgs boson exchange (see Fig. 4.6). The spin independent DM-nucleon scattering cross section for our DM candidate η is given by 2 4 2  2 SI λ MNucfN A B σDM−Nuc = 2 2 + 2 , (4.53) 4π(MDM + MNuc) mH mh Chapter 4. Radiative inverse seesaw models 79

Figure 4.6: DM Nucleon scattering through t-channel Higgs exchange. The effective coupling of the Higgs with the Nucleon is indicated.

where MNuc 0.94 GeV is the nucleon mass, fN parametrizes the effective Higgs- ' nucleon coupling, and the vertex factors are

0 n v µ o A = sin θ χ cos θ sin θ Sign[λ] 2 ηχ cos θ , (4.54) − vw − − · vw 0 n v µ o B = cos θ cos θ + χ sin θ Sign[λ] 2 ηχ sin θ . (4.55) vw − · vw

SI Note that uncertainties in fN result in roughly a factor of 5 uncertainty in σ ∼ DM−Nuc [183]. We use a value of fN = 0.3 [184–187].

SI Having calculated λ, we then find σDM−Nuc for different values of sin θ. The results are presented in Fig. 4.7. The XENON100 limit from 255 live days of data taking [188], the LUX limit from 85 live days of data taking [189] and projected XENON1T exclusions for 2017 in the case of no WIMP-nucleon scattering [190], are plotted for comparison.

We see that MDM & 100 GeV is still largely viable, but that much of the remaining parameter space can be ruled out by XENON1T.

Note that due to different interferences the scattering rate can be above or below the zero mixing case. A minimum is reached for sin θ = 0.052, which is exactly where −  A B  2 + 2 = 0, (4.56) mH mh so that the DM-nucleon cross section is zero. Similar interference was noted in the case of a two Higgs doublet extension of the Higgs portal in Ref. [175]. Chapter 4. Radiative inverse seesaw models 80

Figure 4.7: DM-Nucleon scattering cross section for case A and case B with differ- ent values of sin(θ), the upper limits from XENON100 and LUX and the projected sensitivity for XENON1T in 2017.

4.3.3.4 Higgs signals at colliders

The double Higgs portal can be tested in a number of ways at colliders [175, 176, 191, 192], some of them common with the usual single Higgs portal [180, 183, 193, 194]. Mixing with the exotic Higgs will reduce the production of the SM-like Higgs,

2 SM σpp→h = cos θ σ . (4.57) · pp→h

The total width will also be reduced by a factor of cos2θ, however, the SM-width for

mh = 125 GeV, roughly 4 MeV, is already far below current experimental resolution. The discovery of a SM-like Higgs boson at the LHC [195, 196] allows us to constrain this scenario. The observed signal strength over the SM expectation and 1σ uncertainty is [197, 198]:

+0.08 σ/σSM = 1.00 0.09 (stat.) (theo.) 0.07 (syst.) (CMS), (4.58) ± −0.07 ± +0.14 σ/σSM = 1.3 0.12 (stat.) (syst.) (ATLAS). (4.59) ± −0.11

Combination of the results put a limit on the universal suppression of all channels so that σ/σSM & 0.7 at the 95 % C.L. level [199] (for earlier studies see [200–202]). This Chapter 4. Radiative inverse seesaw models 81 translates into a limit sin θ 0.54, (4.60) | | ≤ if one assumes no invisible decays of the 125 GeV resonance [199]. (An alternative, more conservative, treatment of the theoretical uncertainty in σSM modifies the above bound to be sin θ 0.7 instead [203].) Equation (4.60) is comparable to the constraint | | . from EW precision observables, Eq. (4.50), however the limit on the mixing angle from measurement of the signal strength is generally applicable and not limited to our choice

MH = 500 GeV.

If the DM is light enough, the Higgs can decay invisibly to two DM particles. The invisible partial decay width of the Higgs (h ηη) is given by →

1/2 2 2 ! λ  0 2 4mη Γ = vw cos θ + vχ sin θ Sign[λ] 2µηχ sin θ 1 2 . (4.61) 32πmh − · − mh

This leads to a large invisible branching fraction Br(h ηη) 0.99 for light DM masses → & MDM 10 GeV, except for a very narrow region around a mixing angle sin θ = 0.049 . − where the effective coupling of the hηη term is zero. This helps to rule out such light DM candidates as the direct detection experiments lose sensitivity for low mass DM candidates.

If the mixing angle is large enough, the heavy state could also be detected. For a Higgs mass of 500 GeV, CMS data currently sets a 95% C.L. [204]:

σ/σSM 0.21. (4.62) '

The most stringent limit on the mixing angle will come from assuming 2MDM > MH , so there is no invisible decay H ηη. There is also an additional complication of H hh → → decays [205], however, for the parameters we have chosen the relevant branching ratio turns out to be negligible. The limit on the mixing angle from high mass Higgs searches is then sin θ 0.46. (4.63) | | ≤

Thus the most stringent limit on the mixing angle in case B with MH = 500 GeV comes from considering EW precision observables, Eq. (4.50), while the most widely applicable constraint comes from measurements of the signal strength of the 125 GeV resonance, Chapter 4. Radiative inverse seesaw models 82

Eq. (4.60), as no assumptions about the mass of the heavier Higgs or VEV have to be taken as an input.

4.3.3.5 Fermionic DM

Alternatively one may consider fermionic Higgs portal DM. The important terms are then c 0 c ∆ = hχχNR(NR) + h χNL(NL) + MN NLNR + H.c., (4.64) − L χ

c where the lightest mass eigenstate of the NL, (NR) admixture is the DM candidate

(protected by the same accidental Z2 as in the scalar DM case). In such a scenario

DM is constrained to be very heavy from direct detection experiments, MDM & 2 TeV, unless the DM mass is at one of the two Higgs resonances, there is parity violation, or a Sommerfeld enhancement [177]. So while we have studied the scalar DM case above, either scalar or fermionic DM is viable in the Law/McDonald model. Further generalizations taking into account interactions with the Z0 in the fermionic DM scenario are also possible, but we will not pursue the details here.

4.3.4 Concluding remarks on the Law/McDonald model

We have seen that in the Law/McDonald model, baryogenesis is severely constrained. The explicit lepton number violation will tend to wash out any baryon asymmetry created at a high scale. However at temperatures roughly a factor of 25-40 below the ISS mass scale, Boltzmann suppression of the thermally averaged reaction rates means that the ISS fields will not contribute to washout (though not necessarily all of the ISS fields must have such high masses). Unfortunately the small L violating parameter is only generated after EW symmetry breaking, meaning that resonant leptogenesis is not possible in the Law/McDonald radiative ISS.

We found that the Law/McDonald model can easily accommodate the observed DM abundance through the Higgs portal. Also, in a simple extension, an additional Higgs state can also contribute to the DM annihilation cross section and DM nucleon scattering cross section. Apart from narrow regions of parameter space, either on resonance points for the annihilation rate, or near complete destructive interference for the DM nucleon Chapter 4. Radiative inverse seesaw models 83 scattering cross section, these models can for the most part be tested by the XENON1T experiment.

4.4 Ma radiative inverse seesaw model

4.4.1 Review of the model

We now consider the Ma radiative ISS model [149]. An important difference in this model is that B L is a protected symmetry down to the ISS scale. First let us review − the main features of this model. It employs an additional U(1) which originates from a GUT with the following breaking pattern

SO(10) SU(5) U(1)χ → ⊗ (4.65) SU(3)C SU(2)L U(1)Y U(1)χ. → ⊗ ⊗ ⊗

The exotic charge is given by

Qχ = 5(B L) 4Y = 5(B L) + 4I3L 4QEM . (4.66) − − − −

There are two scalars, η1 and η2, that transform under U(1)χ as per

Qχ(η1) = 1,Qχ(η2) = 2, (4.67) and are neutral with respect to the SM gauge group. These will be used for the radiative

ISS and to break the new U(1)χ. To allow the usual Yukawa interactions,

ab † ab † = λdQL ΦdR + λu QLaΦ uR + λelL ΦeR + λν lLaΦ NR + H.c., (4.68) L · b · b the SM Higgs doublet must carry a U(1)χ charge,

Φ = (φ+, φ0) (1, 2, 1/2, 2). (4.69) ∼ −

The fermion multiplets of SO(10) contain a right handed neutrino, NR (1, 1, 0, 5), ∼ − per generation of SM fermions. To these are added more SM neutral fermions with Qχ given by

1 S3L 3, 4 S2L 2, 5 S1L 1, (4.70) × ∼ − × ∼ × ∼ − Chapter 4. Radiative inverse seesaw models 84 where the required number of copies has been indicated to form an anomaly-free set. The following Yukawa terms involving the exotic scalars are permitted

† c c † c ∆ = fR3NRS3Lη + f23(S3L) S2Lη1 + f12(S2L) S1Lη + f11(S1L) S1Lη2 + H.c. (4.71) L 2 1

The scalar potential is given by

2 † † 2 † 2 † † V = (Aη1η2 + H.c.) + λ11(η1η1) + λ22(η2η2) + λ12(η1η1)(η2η2) 2 † † 2 † † † † + µΦΦ Φ + λΦ(Φ Φ) + λ1Φ(Φ Φ)(η1η1) + λ2Φ(Φ Φ)(η2η2) (4.72) 2 † 2 † + µ1(η1η1) + µ2(η2η2),

where A is a coupling constant with dimension of mass. The U(1)χ symmetry is spon- taneously broken by

η2 vχ/√2 = 0, η1 = 0. (4.73) h i ≡ 6 h i Before radiative corrections are taken into account the neutral fermion mass matrix is given by

   c  0 mD 0 0 (νL)     1   mD 0 MN 0   NR  c     = νL (NR) S3L S1L     , (4.74) L 2    c  0 MN 0 0  (S3L)      c 0 0 0 M1 (S1L)

where mD = λνvw/√2, MN = fR3vχ/√2, and M1 = √2f11vχ. Note the S2L are massless at tree level, and that the scalar interaction term,

† ∗ † † ∗ † † Aη1η1η + A η η η2 Aη1η1vχ/√2 + A η η vχ/√2, (4.75) 2 1 1 → 1 1 induces a mass splitting between the mass of the real (mR) and mass of the imaginary

(mI ) components of η1, 2 2 m m = 2√2Avχ, (4.76) R − I where we set A to be real and positive without loss of generality as any complex phase can be absorbed into the fields. Chapter 4. Radiative inverse seesaw models 85

Figure 4.8: Radiative mass generation for S2 and S3 in the Ma model.

The exchange of Re(η1) and Im(η1) then give one-loop radiative Majorana masses to the S2L as can be seen in Fig. 4.8:

2  2  2  2  2  (f12) M1 mR mR mI mI M2 = ln ln . (4.77) 16π2 m2 M 2 M 2 − m2 M 2 M 2 R − 1 1 I − 1 1

Similarly one obtains two-loop radiative Majorana masses for the S3L

2  2  2  2  2  (f23) M2 mR mR mI mI µ M3 = ln ln , (4.78) ≡ 16π2 m2 M 2 M 2 − m2 M 2 M 2 R − 2 2 I − 2 2 which are exactly the small parameters required in the mass matrix of Eq. (4.74) for the ISS mechanism.

Note η1 and S2L are odd under an accidental Z2 symmetry, while all the other particles

are even. We will see below, that for typical choices of the parameters, S2L is lighter

than η1 as its mass arises radiatively and therefore S2L is the DM candidate. One can find various approximate forms for the radiative masses depending on the sizes of various 2 2 2 parameters. If we take m m m = 2√2Avχ,M1, Eq. (4.77) can be approximated R  R − I as 2 √2(f12) AvχM1 M2 2 2 , (4.79) ≈ 8π mI while for natural choices of the parameters m2 , m2 M 2, one obtains from Eq. (4.78): R I  2

2 2 2 2 (f12) (f23) A vχM1 µ M3 4 4 . (4.80) ≡ ≈ 32π mI

To give a numerical example, for vχ M1 A (10 TeV), and mI = 50 TeV, one ∼ ∼ ∼ O 2 2 2 finds M2 (f12) 7 GeV and M3 (f23) (f12) 5 MeV (in very close agreement to ≈ × ≈ × what the exact formulas give). Chapter 4. Radiative inverse seesaw models 86

Figure 4.9: Washout process involving the exotic fermions and scalars. Also shown are the Yukawa interactions which link this washout process to the SM sector. Exchange of the real and imaginary components of η1 leads to destructive interference. Not shown are the interactions involving f12 and f11 which ensure S2L cannot be assigned non-zero SM lepton number.

4.4.2 Constraints from BAU washout

Until U(1)χ is broken, B L is an exact symmetry of the Ma model, and generation or − 1 washout of the BAU will not take place. The VEV vχ breaks Qχ = 5(B L) 4Y , − − and hence the B L as defined above. Baryogenesis can then take place, either through − resonant leptogenesis, or some other suitable baryogenesis scenario with TBG Tχ vχ. ≤ ∼ However, as shown in Fig. 4.9, the ISS fields can now also wash out the asymmetry.

A new conserved global B L can be defined, even with η2 = 0, provided any of the − h i 6 following parameters vanishes:

λν, fR3, f23, f12, f11, A. (4.81)

If the interactions mediated by one of the above parameters are out of equilibrium the BAU is safe. We first examine what values the parameters must take so as to not wash out the BAU. In Sec. 4.4.3 we examine the possibility for resonant leptogenesis, where the lightest pseudo-Dirac pair generates the BAU.

1This constraint could be circumvented through sequestration of positive and negative B − L charge into the visible sector and a hidden sector as in asymmetric DM models [74–77]. In the context of the model studied here, it would require introducing additional fields and interactions, such that the B − L symmetry encompasses two low-energy global U(1) symmetries – the (B − L)V carried by the SM fields and a dark baryon number BD. The construction of such a model in an SO(10) GUT is beyond the scope of this work. Chapter 4. Radiative inverse seesaw models 87

TBG ΛISS vχ • ≈ ≈

1. The decays and inverse decays NR lL + Φ are controlled by λν. The decay ↔ rate is approximately λ2M M  Γ ν N N . (4.82) ≈ 8π T

Requiring nN Γ . nlH, where nN (nl) is the NR (lepton) number density, leads to a washout avoidance condition:

 g 1/4  M 1/2 λ 10−7 eff N . (4.83) ν . 100 10 TeV

2. The mixing between the NR and S3L states is controlled by fR3 = √2MN /vχ. ∗ Scattering Φ + lL NR/S3L η + S2L has an approximate cross section → → 1

2 2 (λνf23) MN σ 2 2 2 . (4.84) ∼ 8π (T + MN )

The scattering rate Γ σT 3 is inefficient compared to H provided ∼

  1/4  1/2 −6 1  geff  ΛISS fR3 . 10 , (4.85) λνf23 100 10 TeV

and for such values washout is avoided.

3. The Yukawa couplings f23 and f12 induce two body decays of the heaviest particle involved in the relevant interaction (such decays are kinematically

allowed given S2L, which has a radiative and hence negligible mass compared to the other particles, can appear in the final state). We estimate the decay rate as (f )2M M  Γ i j j , (4.86) ≈ 16π T

where fi denotes the relevant Yukawa coupling and Mj the mass of the heav- iest particle. This leads to a washout avoidance condition of

 g 1/4  Λ 1/2 f 10−6 eff ISS . (4.87) i . 100 10 TeV Chapter 4. Radiative inverse seesaw models 88

∗ c 4. Scattering processes S2L + η S1/S S2L + η1 can be suppressed with 1 → 1 → small f11 = M1/√2vχ. We estimate the cross section to be:

4 2 (f12) M1 σ 2 2 2 . (4.88) ∼ 8π (T + M1 )

This gives a washout avoidance condition

 2 1/2  1/2 −7 1  geff  ΛISS f11 . 10 . (4.89) f12 100 10 TeV

5. Washout may proceed through decays and inverse decays η1 provided they are kinematically allowed. However, for a small enough A, and hence small

enough mass splitting between Re(η1) and Im(η1), destructive interference leads to washout suppression. We estimate the decay rate as

2 2 ( f23 + f12 )mI mI Γ | | | | , (4.90) ≈ 16π T

and the effective washout parameter to be

 Γ  2 Keff = δ , (4.91) H T =mI ·

where δ = (mR mI )/Γ. One finds this translates into a washout avoidance − condition

1/4  3/2 −4 p 2 2  geff  ΛISS A 10 GeV f23 + f12 . (4.92) . | | | | 100 10 TeV

If such decays are not kinematically allowed, washout can instead proceed

through a scattering process mediated by off-shell η1. We estimate the cross section as

4  2 ( f23 + f12 ) 1 1 2 σ | | | | 2 2 2 2 T (4.93) ∼ 128π T + mR − T + mI 2 4 ! ( f23 + f12 ) 2√2Avχ 2 = | | | | 2 2 2 2 T . (4.94) 128π (T + mR)(T + mI )

This translates into a washout avoidance condition

 2 1/4  3/2 −2 1  geff  ΛISS A . 10 GeV . (4.95) f23 + f12 100 10 TeV | | | | Chapter 4. Radiative inverse seesaw models 89

Satisfying any of the above washout bounds means the BAU is safe. However, similarly to the Law/McDonald model, such small choices for the couplings ruins the ISS mechanism.

TBG < ΛISS • So as to ensure the preservation of the BAU, while allowing for the ISS mechanism, we may suppress washout by taking one or more of the exotic particle masses to

be higher than TBG. This ensures that at TBG the number density of that particle will be sufficiently Boltzmann suppressed for it to not play a role in washout. For decay processes the requirement on the mass scale is

ΛISS & 42, (4.96) TBG

while for scattering processes the constraint is relaxed to roughly

ΛISS & 25. (4.97) TBG

This is again similar to the washout in the Law/McDonald model. For TBG < TEW the bounds of Eqs. (4.96) and (4.97) do not apply as the sphalerons are no longer active.

4.4.3 Resonant leptogenesis

The lightest pseudo-Dirac pair of neutrinos, Nα, can generate the BAU in resonant leptogenesis if the mass splitting results in sufficient CP violation. There is an interplay between the CP violation and washout, which both depend on µ, the mass splitting of the Nα pair. There has been controversy in the literature regarding the estimate of the CP violation. Let us begin with the estimate of Ref. [154] and return to the possible issues and alternative estimates afterwards. Reference [154] found the observed BAU −5 can be generated with δ = µ/Γα 10 , where Γα is the width of the Nα and µ the ≈ mass splitting of the pair, heavy neutrino mass MNα (1) TeV, and Yukawa coupling ∼ O −2 λν 5 10 . ≈ ×

The scalar mass mI can be taken to a sufficiently high scale so that the other interactions

of the ISS do not interfere with the generation of the BAU. Let us take mI 50 TeV and ∼ Chapter 4. Radiative inverse seesaw models 90

vχ A M1 10 TeV, then MN (1) TeV can be achieved with fR3 0.1. Note ∼ ∼ ∼ ∼ O ∼ that solutions of the Boltzmann equations taking into account decay and scattering processes show the BAU is generated at TBG MNα/10, which allows the out-of- ≈ 0 equilibrium condition for Nα to be met even with a relatively low Z mass: MZ0 &

2MNα [169].

−5 −5 Choices of δ = 10 , λν = 0.05, and MNα = 2 TeV imply: µ = 0.2 10 GeV. This is × compatible with a radiative origin in the Ma model as, for these parameters, Eq. (4.80) 2 2 −3 −2 gives µ (f12) (f23) 5 10 GeV. The light neutrino mass is mν 10 eV. ≈ × × ≈

In the δ 1 regime, the CP asymmetry of Nα decays is given in Ref. [154] as  λ2 µµ˜  ν sin α, (4.98) ' −16π (µ2 +µ ˜2 + 2µµ˜ cos α) where α is a physical phase of the ISS mass matrix andµ ˜ is the Majorana mass c −3 µ˜NR(NR) . In this model one typically expectsµ ˜ 10 µ, due to suppression by ∼ additional Yukawa couplings in the radiative generation ofµ ˜, which allows for  10−8. ∼ The washout factor is Keff 6. Taking into account a sphaleron reprocessing and dilu- ≈ −2 tion factor, the final baryon number density to photon number density is, ηB 10  , ∼ | | −10 and can easily be made to match the observed value ηB 6 10 [22] in this scenario. ≈ × However, note the estimate of the CP violation, Eq. (4.98), has been a source of con- troversy in the literature. In particular, note that  does not tend to zero in the L conserving limit µ, µ˜ 0. This may indicate a possible issue with the regulator used → in deriving the result. The differing estimates of the CP violation have recently been discussed in Ref. [67] — careful consideration of where the approximations used may break down indicate the maximum CP violation may well occur for Γα µ [67]. This ∼ would mean the Ma radiative ISS would not achieve sufficient CP violation in order to explain the BAU.

4.4.4 The ρ parameter

In the Ma model, the Higgs doublet responsible for EW symmetry breaking also carries

U(1)χ charge. This leads to a tree level correction to the Z boson mass, and consequently Chapter 4. Radiative inverse seesaw models 91

Figure 4.10: Correction to the ρ parameter in the Ma model. Different choices for the exotic gauge coupling are given in terms of the weak hypercharge coupling constant −4 −3 gY 0.37. The 2σ limits 4 10 < ρ 1 < 10 are indicated by the horizontal dashed≈ lines. A positive (negative)− × correction− to ρ 1 is indicated by a solid (dashed) line. Solid (dashed) lines below 10−3 ( 4 10−4 )− represent allowed parameter space. The peaks correspond to where the| mass| | eigenstate× | associated with the Z boson changes from m+ to m−. to the ρ parameter, 2 MW ρ 2 2 , (4.99) ≡ MZ cos θW where the Weinberg angle is defined in terms of the weak isospin, g, and weak hyper- charge, gY , coupling constants as tan θW gY /g. Experimental observations give a ≡ value ρ 1 = 4+3 10−4 [206]. The masses of the neutral gauge bosons are given by − −4 ×

2 ( " 2 # 2 vW 2 2 2 vχ m± = g + gY + 16gχ 1 + 2 8 vW s )   v2 2 v2 2 2 2 χ 2 2 2 χ g + gY + 16gχ 1 + 2 64gχ(g + gY ) 2 . (4.100) ± vW − vW

The correction to the ρ parameter is shown in Fig. 4.10. Two parameter regimes are allowed: either vχ vw, in which case MZ0 MZ or gχ gY , in which case MZ0     MZ . The two regimes differ in whether m+ or m− corresponds to the Z boson. As can be seen in Fig. 4.10 the first decoupling regime requires vχ 7.7 TeV, given gχ gY , in & ∼ order to remain within the 2σ limit of the ρ parameter. The second decoupling regime requires gχ gY /100 and vχ 5 TeV. Note that in the limit vχ 0 one obtains a . . → Chapter 4. Radiative inverse seesaw models 92 massless exotic gauge boson, however, there is still a correction to the Z boson mass due to the U(1)χ charge of Φ. This can be seen in Fig. 4.10, in which non-negligible corrections to ρ persist for sufficiently large gχ, even in the vχ 0 limit. →

Given the GUT origin of the model, we consider choices of gχ . gY /100 unattrac- tive. Furthermore the interactions of the resultant light Z0 — which interacts with the SM quarks and leptons — are constrained by observations of parity violating Caesium transitions [207–209]:  M 0  g 10−4 Z . (4.101) χ . 1 GeV

This means the DM annihilation cross section through the Z 0 is too weak (following a similar anaysis as in Sec. 4.4.6.2) and would lead to overclosure of the universe. We therefore only consider the vχ vw regime from here on.  The Z Z0 mixing angle is given by − 8g (g sin θ + g cos θ ) tan 2φ = χ Y W W , (4.102) Z g2 + g2 16g2 (1 + v2 /v2 ) Y − χ χ w

2 −3 so one finds φZ (vw/vχ) 10 given vχ 7.7 TeV. We will see the consequences | | ' . & of this for our DM candidate below.

So far we have dealt with mixing coming from the mass matrix. As the theory contains particles charged under both U(1)Y and U(1)χ, kinetic mixing between the respective field strength tensors will be generated radiatively. The full mass matrix for the Z and Z0 bosons, taking into account both kinetic and mass mixing, can then be found [210]. By following this procedure we have checked that including such a term makes no qualitative difference to our discussion above. Furthermore one typically expects radiatively gener- ated kinetic mixing to be small and therefore negligible for mixing in this model [211].

4.4.5 LHC searches

We also consider limits on the Z0 from the LHC. The most stringent limits come from searches for dilepton resonances [212–214]. The limit from the combination of e+e− and µ+µ− events is shown in Fig. 4.11. We have approximated the acceptance factor as being the same as for the benchmark sequential SM Z0. We use the central value of the next-to-next-to-leading order (NNLO) MSTW 2008 parton distribution functions Chapter 4. Radiative inverse seesaw models 93

Figure 4.11: Limit on the Ma radiative ISS Z0 from ATLAS searches for dilepton res- onances (l = e, µ combination). The production cross section times branching fraction into ll for three different coupling choices is shown.

(PDFs) [215]. In calculating the signal cross section we have assumed three generations 0 of S2L fermions to which the Z decays invisibly. The expression for the cross section can be found in Appendix C.2. An estimate of the higher QCD corrections (K factor) is also included. We use an enhancement of K = 1.16, as found for the ATLAS Z0 analysis for dilepton resonances of 2 and 3 TeV [213], the results do not depend strongly on this choice.

As can be seen in Fig. 4.11, a choice of e.g. gχ = 0.1 leads to a lower bound of MZ0 & 2.7 −3 TeV. Given that MZ0 2gχvχ, this corresponds to φZ 10 . Larger gχ means ' | | . stronger collider limits on MZ0 . However, ATLAS and CMS currently only provide limits for Z0 resonances up to 3.5 TeV [212–214], and a simple extrapolation is not possible as the limit on the signal strength is rising for high mass regions (due to acceptance effects). For our purposes it suffices that the bound on the mixing angle is approximately −3 φZ 10 . | | . In Fig. 4.11 we have only used the ATLAS limit. The CMS e+e− and µ+µ− limits have not yet been officially combined into l+l− and are expected to give similar constraints. Chapter 4. Radiative inverse seesaw models 94

4.4.6 Dark matter

4.4.6.1 DM candidate mass

In the Ma radiative ISS, there exists an accidental Z2 symmetry under which only η1

and S2L transform. The lightest of these species is stable and forms a DM candidate. By examining the scalar potential in Eq. (4.72) we see that the masses of the real and

imaginary components of η1 after symmetry breaking are given by

2 2 2 2 2 mR = (µη1 + λ1φvw/2 + λ12vχ/2) + √2Avχ (4.103)

2 2 2 2 2 m = (µ + λ1φv /2 + λ12v /2) √2Avχ, (4.104) I η1 w χ −

where A > 0 and for reasons of naturalness one typically expects µη1 vχ A. The ∼ ∼ fermion S2L gains a radiative Majorana mass, M2, at 1-loop order, and one typically expects

mI M2, (4.105) 

if fine-tuning is absent. To provide a sense of the typical values for M2, we have have p p plotted M2 as a function of Avχ, with M1 = Avχ for different choices of mI in

Fig. 4.12. One can see that as one moves away from the fine tuned limit mI = 0, the radiative mass M2 decreases to be in the range

−2 −3 2 M2 (10 10 )(f12) vχ. (4.106) ≈ −

4.4.6.2 Cold DM scenario

Given the discussion of the ρ parameter, one requires vχ vw. We will discuss the DM  relic abundance in light of this constraint here. Let us assume the S2L species have a −8 thermal abundance in the early universe, as will be the case unless gχ 10 gY . .  There is furthermore an absence of any interactions of the form S2LS2L(σ) where σ

represents any of the scalar fields in the theory. As S2L is charged only under U(1)χ, it can annihilate only through the Z0, and due to mixing of the gauge bosons, through the Z boson [216] (see Fig. 4.13). Chapter 4. Radiative inverse seesaw models 95

2 p p Figure 4.12: M2/(f12) as a function of Avχ, with M1 = Avχ, for different choices of mI .

Figure 4.13: Annihilation process of S2L which determines its relic density.

So as not to overclose the universe one requires the annihilation cross section to be larger −9 −2 than σv DM 1.9 10 GeV [182]. However annihilation through the Z boson is h i ≈ × 0 suppressed by the small Z Z mixing angle. For example, consider the 2M2 MZ − . regime. It suffices for our purposes to use a simple estimate for the annihilation cross section: sin2 φ G2 M 2 σv Z F 2 , (4.107) h i ≈ 2π

0 where φZ is the Z Z mixing angle and GF is the Fermi constant. Requiring σv − h i & 2 σv DM , and substituting for the mixing angle φZ (vw/vχ) , one finds h i ' p  2 2π σv DM vχ M2 & h i . (4.108) GF vw

Given the constraint from the ρ parameter, vχ & 7 TeV, this demands M2 & 7.5 TeV, Chapter 4. Radiative inverse seesaw models 96 which contradicts our original assumption of 2M2 . MZ . One also finds a too small cross section in the 2M2 & MZ regime.

0 Annihilations occuring through the Z are suppressed by the large mass MZ0 . So anni- hilations through the Z0 are also too weak.

Finally we note that the cross section will be much higher at a resonance point, where

2M2 MZ or 2M2 MZ0 [216], and this could avoid overclosure. However, this ' ' corresponds to a fine tuning of the DM mass.

From this discussion we see that for a simple cold DM scenario the S2L DM candidate overcloses the universe if we demand no fine tuning. One would have to go to more convoluted DM scenarios such as warm dark matter to perhaps find viable regions of parameter space. The simple cold DM case is ruled out.

4.4.7 Concluding remarks on the Ma model

The Ma radiative ISS model has several positive features for baryogenesis and washout avoidance. Resonant leptogenesis may be accommodated — modulo the possible issues with the estimates of the CP violation — as the mass splitting of the heavy neutrinos is generated before the EW phase transition, allowing the interplay between heavy neutrino decays and sphalerons crucial for leptogenesis. Also, washout due to the other ∆L interactions required for the radiative ISS can be suppressed with suitable choices of mass parameters.

On the other hand we have also seen a drawback of the Ma model with respect to cold DM. We found that the constraints from measurements of the ρ parameter translated into DM annihilations through the Z and Z0 gauge bosons which are too weak to avoid overclosure. These simple cold DM scenarios are therefore ruled out for the Ma radia- tive ISS (apart for possibly small areas of parameter space corresponding to resonant enhancements). One would therefore have to go to more convoluted scenarios such as warm dark matter to find a suitable DM scenario. Chapter 4. Radiative inverse seesaw models 97

4.5 Conclusion

Motivated originally by how new physics explanations for the baryon asymmetry, dark matter, and neutrino oscillations can overlap with each other, we began by looking at the properties of the inverse seesaw. We saw how the inverse seesaw mechanism offers various advantages, namely a scale low enough for direct experimental tests, and the possibility of resonant leptogenesis, or at least a suppressed washout rate if some other mechanism is responsible for the baryon asymmetry.

The inverse seesaw mechanism does comes with a cost: the small lepton number violating parameter that must enter the mass matrix. Despite its technical naturalness, there may be a more appealing mechanism to generate it. One of the options is a radiative origin. We therefore continued our investigation by focusing on two radiative inverse seesaw models.

It was found in the case of the Law/McDonald radiative inverse seesaw model, the DM abundance is easily explained with the Higgs portal mechanism, or the double Higgs portal. The advantages of the inverse seesaw with respect to baryogenesis, however, are lost due to the extra fields present which break lepton number explicitly with their interactions. In case of the Ma radiative inverse seesaw model, the situation may perhaps be said to be reversed: it may accommodate resonant leptogenesis (depending on which calculation of the CP violation is correct), but does not feature a simple cold dark matter candidate. Chapter 5

The baryon asymmetry, nucleon stability and LHC searches for scalar diquarks

5.1 Introduction

A simple approach to considering the possibilities for BSM physics is to consider ex- tending the SM scalar sector. An obvious possibility is the existence of exotic scalars, charged under the SM gauge group, which couple to the SM quarks and leptons. Such particles may very well also have quantum numbers allowing for the violation of the global B and L symmetries. In this chapter we examine baryon number violation in leptoquark and diquark models. Though we take a bottom-up approach, listing all the possibilities given the SM fields and gauge group, these scalars may also have a GUT origin [217, 218]. We shall determine constraints on these models from nucleon stability and washout of the baryon asymmetry and also determine whether such BNV could be detected at the LHC.

The addition to the SM of a particle with both diquark and leptoquark couplings leads to tree level proton decay. For scalars with only diquark or leptoquark like couplings, but not both, interactions between two or more scalars can still lead to BNV and are constrained from nucleon stability [219–221]. However, the additional interactions in- volved suppress the size of the effect. We will derive constraints by considering such

98 Chapter 5. Scalar Diquarks 99 interactions, together with dominant couplings to the third generation of SM fermions, which can further suppress any low energy BNV signal and hence lower the allowable energy scale of the BNV physics.

Since the diquarks and leptoquarks all carry electromagnetic and colour charge, the gauge interactions are too strong for the requisite departure from thermal equilibrium needed for baryogenesis to take place unless the new scalars have masses (TeV) [222],  O or they also interact with a gauge singlet [223–225]. In this chapter we are interested in the phenomenology of these models at lower scales, and we will limit ourselves to deriving constraints from washout of baryogenesis, rather than using the additional interactions as a source for the baryon asymmetry.

This chapter is organised as follows. In Sec. 5.2 we review the possible scalar leptoquarks and diquarks and their BNV interactions. We examine the constraints from nucleon stability on one of the interactions, considering couplings to first and third generation quarks. In Sec. 5.3 we analyse the effect such an interaction can have on baryogenesis, showing stringent limits apply for models which break the global B L symmetry, if − they are not to washout the BAU. In Sec. 5.4 we consider LHC searches for the exotic scalars and discuss whether the detection of such BNV physics is at all possible at the LHC.

5.2 Baryon number violating scalars

5.2.1 A catalogue of models

The models studied in this chapter are those introduced in Ref. [220]. (The introduction of such interactions leads to charge quantisation immediately from the classical structure of the theory [219, 220].) Here we use the same notation for the list of all posssible scalar leptoquarks and diquarks. To begin with, we take the SM fermions and right-handed neutrinos to transform in the usual way under the SM gauge group SU(3)C SU(2)L ⊗ ⊗ U(1)Y (with convention Q = I3 + Y/2)

lL (1, 2, 1), eR (1, 1, 2), νR (1, 1, 0), ∼ − ∼ − ∼ (5.1) QL (3, 2, 1/3), uR (3, 1, 4/3), dR (3, 1, 2/3). ∼ ∼ ∼ − Chapter 5. Scalar Diquarks 100

The possible scalar leptoquarks and diquarks have the same quantum numbers as quark and lepton bilinears. The possible fields are therefore

c c c σ1.1 QL(lL) uR(eR) dR(νR) (3¯, 1, 2/3) ∼ ∼ ∼ ∼ c σ1.2 QL(lL) (3¯, 3, 2/3) ∼ ∼ σ2 QLeR uRlL (3¯, 2, 7/3) ∼ ∼ ∼ − c c σ3.1 QL(QL) uR(dR) (3, 1, 2/3) ∼ ∼ ∼ − c σ3.2 QL(QL) (3, 3, 2/3) ∼ ∼ − c c σ3.3 QL(QL) uR(dR) (6¯, 1, 2/3) ∼ ∼ ∼ − c σ3.4 QL(QL) (6¯, 3, 2/3) ∼ ∼ − (5.2) c σ4 uR(νR) (3¯, 1, 4/3) ∼ ∼ − σ5 dRlL QLνR (3¯, 2, 1/3) ∼ ∼ ∼ − c σ6.1 uR(uR) (3, 1, 8/3) ∼ ∼ − c σ6.2 uR(uR) (6¯, 1, 8/3) ∼ ∼ − c σ7.1 dR(dR) (3, 1, 4/3) ∼ ∼ c σ7.2 dR(dR) (6¯, 1, 4/3) ∼ ∼ c σ8 dR(eR) (3¯, 1, 8/3). ∼ ∼

Note the leptoquarks in Eq. (5.2) carry B = 1/3, and the diquarks B = 2/3. As − − we are breaking this symmetry, the following particles then carry identical quantum numbers c σ1.1 = σ (3¯, 1, 2/3) 3.1 ∼ c σ1.2 = σ (3¯, 3, 2/3) 3.2 ∼ (5.3) c σ4 = σ (3¯, 1, 4/3) 7.1 ∼ − c σ6.1 = σ (3, 1, 8/3). 8 ∼ − Such particles with both leptoquark and diquark couplings lead to baryon number vio- c lation. For example, introducing the σ σ1.1 = σ (3¯, 1, 2/3) scalar the Lagrangian ≡ 3.1 ∼ is extended to include the terms

c c ∆ = λ1.1(eR) σuR + λ3.1(uR)σd + H.c., (5.4) L R

where λ1.1 and λ3.1 are dimensionless coupling constants, and generational and colour indices have been suppressed. Such interactions lead to tree level proton decay as shown

in Fig. 5.1. If we take both Yukawa couplings to the SM fermions to be λ λ1.1 λ3.1, ≡ ≈ Chapter 5. Scalar Diquarks 101

u u

u u σ

d e+

Figure 5.1: Proton decay, p π0e+, in a one exotic scalar extension with coupling to the first generation. →

u u

u s b σ t W W

+ d t τ ν

Figure 5.2: Proton decay, p K+ν¯, in a one exotic scalar extension with coupling to the third generation. → we obtain the following estimate of the proton decay rate on dimensional grounds [220]

4 5 ! λ ΛQCD Γ 4 , (5.5) ∼ O Mσ where Mσ is the mass of the exotic scalar and we have estimated the contribution from the nuclear matrix element as being ΛQCD. Given the bound on the partial lifetime for p π0e+ is τ > 1.3 1034 years [6], this translates to a limit on the mass of → ×

16 Mσ λ 10 GeV. (5.6) & ×

If we instead take the dominant coupling to be to the third generation, proton decay now proceeds through the diagram in Fig. 5.2. Including an estimate of the loop integrals (see Appendix D.1 for details), the decay rate is estimated as

4 8 5     2 λ g ΛQCD Vub Vtd Vts MbMτ Mt Mt Γ 4 | || || 2 | ln ln , (5.7) ∼ Mσ Mt Mb Mτ Chapter 5. Scalar Diquarks 102

where g 0.65 is the SU(2)L coupling constant. Given the limit on the partial lifetime ≈ for p K+ν¯, τ > 2.3 1033 years [226], this translates to a limit on the mass of → ×

11 Mσ λ 10 GeV. (5.8) & ×

While we have considered a UV complete interactions here, it is also possible to take an effective field theory approach and use the nucleon stability constraints to put stringent limits on all dimension six BNV operators (the lowest dimension effective BNV operators possible) involving the higher generations, meaning that BNV interactions are unlikely to be observed at the LHC [227]. If some unknown GIM-like cancellation mechanism were to suppress the nucleon instability, however, BNV interactions involving the top quark may become accessible at the LHC [228].

Instead of searching for a GIM-like cancellation mechanism, we concentrate on BNV that proceeds through multiple heavy scalars and derive the most conservative limits on the couplings to the SM, which come from interactions with the third generation quarks and leptons. Each additional scalar of mass M suppresses the amplitude of the nucleon decays by a factor of M 2, so the nucleon stability experiments will put less stringent conditions on M. (The effective operators would then correspond to dimension nine or twelve, instead of the dimension six operators considered in Refs. [227, 228].)

Hence from now on, we restrict our attention to the scalars which are not subject to such a stringent mass bound, namely: σ3.3, σ3.4, σ5, σ6.2 and σ7.2. Baryon number is then broken by the introduction of two scalars and an interaction between them. The possible scalar-scalar interactions, without the particles of Eq. (5.3), are listed below. The ∆B = 1 list is

σ5, ρ σ5σ5σ5ρ → (5.9) σ5, Φ σ5σ5σ5Φ, → where Φ is the SM Higgs doublet and ρ represents a new Higgs like scalar ρ (8, 2, 1). ∼ The ∆B = 2 list is

σ3, σ7 σ3.3σ3.3σ7.2 → σ3.4σ3.4σ7.2 (5.10) → σ6, σ7 σ6.2σ7.2σ7.2. → Note that the models of Eq. (5.9) conserve B L while the models of Eq. (5.10) break − B L. This is an important distinction when it comes to consideration of the interaction’s − Chapter 5. Scalar Diquarks 103 effects on baryogenesis. Breaking baryon number using these scalar-scalar interactions leads to less stringent bounds from nucleon stability.

The σ particles in Eq. (5.2), are either scalar diquarks or scalar leptoquarks. The usual constraints, from low energy experiments, on the couplings of diquarks and leptoquarks to standard model fermions then apply. For example, from consideration of atomic parity violation of Caesium, leptoquarks coupled to the right handed first generation fermions are constrained by M > λ 2 TeV [229]. For further constraints on the leptoquarks (in × particular on products of different couplings) see Refs. [229–232]. Products of diquark couplings are also constrained from limits on flavour changing neutral currents (FCNC) and meson-antimeson oscillations [233]. These limits, however, do not apply in the limit where all but one of the couplings vanish.

There are also constraints on the masses of leptoquarks and diquarks from collider experiments. From searches of pair produced leptoquarks the best limits for the three generations are [234–237]

MLQ > 1005 GeV 1st generation (jjee),

MLQ > 1070 GeV 2nd generation (jjµµ), (5.11) MLQ > 740 GeV 3rd generation (bbττ),

MLQ > 634 GeV 3rd generation (ttττ),

where we have indicated the final state searched for with j =jet and quoted the limit for 100% branching fraction into the relevant final state. Diquarks can also be produced at colliders and we will return to their collider phenomenology in Sec. 5.4.

Considering the lists of interactions, (5.9) – (5.10), we choose to study the σ3.3σ3.3σ7.2 interaction in greater detail, due to its interesting phenomenology. First of all, it involves diquarks, which can be produced more favourably than leptoquarks in hadron colliders.

Secondly, σ3.3 couples to an up and down type quark. This means the possibility of top quarks in the final state, which is a clearer signal than generic jets. For simplicity, we

examine the SU(2)L singlet, σ3.3, rather than the SU(2)L triplet, σ3.4. Chapter 5. Scalar Diquarks 104

d d

d σ7 d

σ3 σ3 u u

Figure 5.3: Tree level n n¯ oscillation for the σ7σ3σ3 model with the scalars coupled to the first generation quarks.−

5.2.2 The particular model: σ3.3σ3.3σ7.2

For the reasons given above, in the remainder of this chapter we shall focus on the

σ3.3σ3.3σ7.2 model. The SM Lagrangian is extended to include the terms

c c ∆ = λ3(dR) uRσ3.3 + λ7(dR) dRσ7.2 + µσ3.3σ3.3σ7.2 + H.c., (5.12) L where λ3, λ7 are dimensionless coupling constants and µ has dimension of mass. Gen- erational and colour indices have been suppressed. We will primarily be interested in the Mσ7 > 2Mσ3 hierarchy, especially for phenomenology at the LHC, which we study in Sec. 5.4.

The couplings λ3 and λ7 are constrained from low energy experiments measuring FCNC and neutral meson oscillations. For example if the coupling to the third generation bb quarks λ7 = 0.1, the coupling to the second generation quarks is constrained to be ss −3 0 0 λ < 10 , for Mσ = 1 TeV or the B B oscillation frequency would exceed the 7 7 s − s experimental value [233, 238, 239].

For couplings to the first generation quarks, there is also a stringent constraint from neutron-antineutron (n n¯) oscillations (see Fig. 5.3). Using dimensional analysis, the − rate of oscillation is estimated as

2 6 ! µλ7λ3ΛQCD Γ(n n¯) 2 4 . (5.13) − ∼ O Mσ7 Mσ3 Chapter 5. Scalar Diquarks 105

d u

u b u σ3 b t d σ7

d t σ3 b u b u

d u

0 0 Figure 5.4: Two loop nn π π decay for the σ7σ3σ3 model with the scalars coupled to the third generation quarks.→

d t

u b σ3 u

d u u σ7

d u u

σ3 u u b

t d

0 0 Figure 5.5: Two loop nn π π decay for the σ7σ3σ3 model with the scalars coupled to the third generation quarks.→

The lower limit from experiment for the oscillation time is currently around τ > 108s

[134, 135]. Taking λ λ3 λ7, Mσ Mσ Mσ , and µ λMσ, we obtain a bound ≡ ≈ ≡ 3 ∼ 7 ∼

4/5 6 Mσ λ 10 GeV. (5.14) & ×

A similarly stringent constraint from dinucleon decay has previously been calculated [220].

We have set µ λMσ as this ensures the two decay modes of σ7 have similar branching ∼ Chapter 5. Scalar Diquarks 106

d d

σ d t 7 t d

σ3 σ3 u b b u

Figure 5.6: Two loop n n¯ oscillation for the σ7σ3σ3 model with σ7 coupled to the − first and σ3 to the third generation quarks. ratio. In the limit µ 0, the BNV aspect of this model is switched off and no constraints → from nucleon decay or washout exist. A small µ, however, makes the branching ratio for

σ¯7 σ3 + σ3 insignificant. Detecting any BNV interaction at colliders then becomes → even more unlikely.

The constraint on Mσ, however, changes when the dominant coupling is taken to be to the third generation SM fermions. The most stringent constraint then comes from nn π0π0 (see Figs. 5.4 and 5.5), which has a partial lifetime τ > 3.4 1030 years [240]. → × Estimating the loop integrals (see Appendix D.1), and then using dimensional analysis and taking µ λMσ, the double neutron decay rate is estimated to be ∼

8 8 4 16 23 ! λ Vub Vtd g Λ | | | | QCD Γ 8 4 10 . (5.15) ∼ O MW Mt Mσ

This translates into a limit on the mass of

4/5 Mσ λ 0.1 GeV. (5.16) & ×

The simplest n n¯ diagram now involves four loops, so we expect it to be even more − suppressed. If the particles couple predominantly to the third generation, we see that nucleon stability constraints do not preclude BNV from occurring at LHC energies.

Finally, let us mention an intermediate case where the σ7 couples to the first generation and the σ3 to the third. The limit on the mass again comes from n n¯ oscillations (see − Fig. 5.6). The oscillation rate is estimated as

 2 4 2 2 ! 2 Mb g Vtd Vub 6 Γ(n n¯) µλ7λ3 | 2| | 4 | ΛQCD . (5.17) − ∼ O Mt Mσ7 Mσ3 Chapter 5. Scalar Diquarks 107

This translates as a limit on the mass of

4/5 Mσ λ 1 TeV, (5.18) & × which is again within reach of the LHC.

5.3 Washout of baryogenesis

5.3.1 High Temperature Baryogenesis

If baryogenesis were to occur before the temperature, T , of the universe reached the mass scale of the σ particles, as would be the case in some high temperature leptogenesis scenario, the creation and decay of the σ particles in a ∆(B L) = 0 sequence can − 6 erase the matter-antimatter asymmetry. For example, in the Mσ7 > 2Mσ3 case such a sequence could proceed on-shell in the following way

d + d σ7 (5.19) →

σ7 σ3 + σ3 (5.20) →

σ3 u¯ + d.¯ (5.21) →

If the rate of one of these steps, Γ, is less than the expansion rate of the universe, H, the ∆(B L) = 0 process will not be occurring rapidly enough for washout [103, 241, 242]. − 6 If we again take λ λ3 λ7 and µ λMσ we see to avoid washout we require ≡ ≈ ∼

  1/2 2 2 Mσ geff T Γ λ Mσ < H , (5.22) ∼ T ∼ MP l

Mσ where the Lorentz factor, T , is understood to be present for T > M. Remembering that this is required to hold for all T > Mσ (below which the initial inverse decay will be Boltzmann suppressed), this translates into a washout avoidance condition

 g 1/4  M 1/2 λ 10−8 eff σ . (5.23) . 100 1 TeV

Note this is a more stringent constraint than the limit from n n oscillations, Eq. (5.14), − for (TeV) masses. So washout of high temperature leptogenesis could take place with O Chapter 5. Scalar Diquarks 108

t

b σ3 b σ7 t

σ3 b

b

Figure 5.7: BNV scattering for the σ7σ3σ3 model the addition of such fields, even with couplings only to the first generation (which are the most constrained from nucleon stability).

The addition of such fields with couplings greater than the above constraint, would mean any existing asymmetry in the quark sector above the mass scale of the σ particles will be removed by their BNV interactions.

5.3.2 Low temperature baryogenesis

Now we examine the scenario where baryogenesis occurs at a temperature below the mass scale of the σ particles, which may e.g. be the case in electroweak baryogenesis.

The inverse decay dd σ¯7 is now Boltzmann suppressed →  3/2   2 Mσ Mσ Γ λ Mσ Exp − . (5.24) ∼ T T

To avoid washout we require this rate to be less than H at the temperature of baryoge- nesis, TBG, giving the bound

!1/2  M  g1/2T 7/2 λ Exp σ eff BG . (5.25) . T 5/2 BG MP lMσ

Note the effect of the exponential suppression. Setting TBG = 100 GeV and Mσ = 1 −7 TeV, we obtain a bound of λ . 10 . Due to there being only one order of magnitude

difference between Mσ and TBG, the inverse decay is not yet sufficiently suppressed for Chapter 5. Scalar Diquarks 109

Figure 5.8: Limit on the coupling, λ, of the σ particles in order avoid washout in the σ7σ3σ3 model for different choices of the baryogenesis temperature. Couplings above the relevant lines would washout the BAU. The limit is a combination of Eqs. (5.23), (5.25) and (5.27), but with the full numerical expressions for the Boltzmann suppression and Lorentz factor for Eq. (5.25). the on-shell sequence to be insignificant, and thus the stringent bound on λ applies. In contrast, TBG = 100 GeV and Mσ = 4.5 TeV yields λ . 0.1.

Washout can also proceed through the off-shell BNV scattering process depicted in Fig. 5.7. The rate is estimated as

µ2λ6T 11 Γ 12 . (5.26) ∼ Mσ

When compared to H, and again taking µ λMσ, this translates into the washout ∼ avoidance condition

 g 1/16  M 5/4  T 9/8 λ 10−1 eff σ BG . (5.27) . 100 1 TeV 100 GeV

We summarise the limits on the coupling from washout, Eqs. (5.23), (5.25) and (5.27) in Fig. 5.8. On the left hand side of the plot one sees the TBG Mσ limit, Eq. (5.23), ≥ applies. For TBG slightly below Mσ, the exponential suppression in Eq. (5.25) is not suf- ficient to prevent washout and a stringent limit on the coupling still applies. Eventually, Chapter 5. Scalar Diquarks 110

b

σ3 t σ g 7

σ3 b σ7 b t σ7 g

b

Figure 5.9: Pair production of σ7σ¯7 through gluon-gluon fusion. Theσ ¯7 decays into ¯ ¯ ¯¯ σ3σ3 resulting in a t¯b t¯b bb final state. The bb tb tb final state from the σ7σ¯7 pair production would occur in the same amount. at Mσ & 45TBG, the exponential suppression in Eq. (5.25) means the limit from washout avoidance comes from the off-shell processes, Eq. (5.27). For these larger masses, we can have λ 0.1 1 while still satisfying constraints from washout avoidance and nucleon ∼ − stability.

5.4 Collider searches

Given the stringent limits from nucleon stability, one does not typically expect to find evidence for BNV physics at the LHC accessible energy scale. Indeed it is only by assuming dominant coupling to the third generation quarks for at least one of the scalars involved in the BNV interactions that we were able to successfully lower the allowable energy scale of BNV physics. The hierarchy required in the couplings to different quark generations may be seen as problematic due to the renormalisation group running of couplings. Having acknowledged this issue — let us put it aside for now and examine whether the detection of such BNV physics is at all possible at the LHC — even with the most favourable “tuned” choices for the couplings.

As we have seen, washout of baryogenesis can be avoided in two ways. Either the couplings controlling the BNV are small, ensuring an approximate B symmetry even at the Mσ scale, or baryogenesis occurs at a temperature far enough below Mσ for the σ mediated BNV processes to be out of equilibrium. We refer to the two cases as the Chapter 5. Scalar Diquarks 111 approximately B conserving and the B violating regimes respectively and discuss their LHC phenomenology in turn.

5.4.1 Approximately B conserving regime

In the limit of µ 0, we effectively turn off the BNV interaction and return to the → standard diquark phenomenology. The production of the exotic scalars can take place either through the Yukawa coupling with the quark fields or by gluon-gluon fusion (see Fig. 5.9). In the limit of µ 0, the Yukawa coupling to the first generation may then → be large enough for the qq σ jj signal to be significant. Such diquark production → → has been studied in Refs. [238, 243, 244]. Experimental limits on such dijet resonances have been set [245, 246]. Gluon-gluon fusion depends only on the colour charge. For a unity branching ratio into two jets, the LHC at √s = 14TeV with = 100 fb−1 L of data, can discover such pair produced colour sextet scalars with masses below 1050 GeV [247, 248].

−5 If µ λMσ, the washout avoidance condition implies a lifetime cτ > 10 m, so the ∼ observation of displaced vertices may occur. Note that with such long lifetimes the σ −7 particles would hadronise before decay. Given the coupling to quarks is small, λ . 10 , resonant production is suppressed and we are left with only pair production through gluon-gluon fusion. If we assume the hierarchy Mσ7 > 2Mσ3 , we must only concern ourselves with the branching ratio forσ ¯7 σ3 + σ3 and to which generation quarks → the σ particles decay. If the coupling was to the first two generations, it would not be possible to extract the existence of a BNV process, due to the generic nature of the jets.

Reference [249] provides a plot of the pair production cross section of colour sextet scalars as a function of the particle mass. Pair production of the scalars declines from 6pb ≈ for Mσ = 500 GeV to 60 fb for Mσ = 1 TeV at √s = 14 TeV. Taking Mσ > 2Mσ , ≈ 7 3 branching ratio r forσ ¯7 σ3σ3, and coupling of the scalars to the third generation → quarks we have the final states of pair produced σ7σ¯7 in Table 5.1 (see Fig. 5.9 for an example of the decay chain).

One could then use the semi-leptonic decay to distinguish top from anti-top quarks in the final state. This would reduce the signal by a factor of (2/9) for every top quark required to decay to a lepton. But we are still left with many b jets, the charge of which Chapter 5. Scalar Diquarks 112

Final state Fraction of total bb¯b¯b (1 r)2 − bb tb tb r(1 r) − t¯¯b t¯¯b ¯b¯b r(1 r) − t¯¯b t¯¯b tb tb r2

Table 5.1: Final states from pair production of σ7σ¯7 given branching ratio r for σ¯7 σ3σ3, and coupling of the scalars to the third generation quarks. → t

d σ3 σ 7 b t

σ3 d

b

Figure 5.10: BNV at the LHC dd σ¯7 σ3 + σ3. → → is even more difficult to extract. Furthermore due to the initial B = 0 σ7σ¯7 state, there are the same number of quark and antiquark jets over a many event average. Such a BNV signal is therefore unlikely to be detected conclusively at the LHC.

Now to the case λ 0 and µ λMσ. The phenomenology does not change much, →  except for large enough µ, the decay width σ7 σ3 +σ3 becomes large and σ7 may then → not decay at a displaced vertex. The σ3 will still decay into jets at a displaced vertex.

5.4.2 B violating regime

In this regime, both the coupling λ and µ may be large, provided TBG is sufficiently low and the nucleon decay constraint is respected. As we have shown in Sec. 5.2.2, for couplings µ/Mσ λ 1 and Mσ 1 TeV, the nucleon decay constraint can only be ∼ ∼ ∼ evaded if at least one of the diquarks has only negligible couplings to the first quark generation. The pair production signal is similar to that of the previous regime, albeit with no displaced vertices.

Quantitative differences to the previous regime may, however, arise in the resonant production of σ7. The production occurs through dd σ¯7, and there may also be a → Chapter 5. Scalar Diquarks 113

Figure 5.11: Estimate of the cross section times branching fraction times acceptance for the process dd σ7 jj at √s = 8 TeV and comparison to the limits for dijet → → resonances from ATLAS and CMS [246, 250]. We assume two decay channels for σ7, the dd and σ3σ3 final states. As an example we have assumed equal branching fractions for the two channels.

significant branching fractionσ ¯7 σ3 + σ3 (see Fig. 5.10). Note we require σ7 to couple → to the first generation to ensure sufficient resonant production, which in turn requires

dominant coupling of σ3 to the third generation to evade the nucleon stability constraints.

For purposes of illustration we choose values λ7 = 0.5, 0.1 and 0.03 and branching ratio r(σ ¯7 σ3σ3) = 0.5. Furthermore we assume σ3 decays to the third generation quarks. → These choices are consistent with the nucleon stability limit, Eq. (5.18), albeit rather favourably chosen.

We use the Breit-Wigner approximation for scattering through narrow resonances to calculate the cross section for dd σ7. The details can be found in Appendix D.2. We → use the central NNLO MSTW 2008 PDFs [215] to calculate the collider cross section. The cross section is compared to the LHC limits for dijet resonances in Fig. 5.11 [246, 250]. The acceptance factor for isotropic decays into hard jets is estimated as A ≈ 0.6 [246, 250]. A K-factor enhancement is also included. We use K = 1.22, the lower value for the range K 1.22 1.32 suggested in Ref. [243]. ≈ − As can be seen from Fig. 5.11, the LHC limits on dijet resonances are constraining the Chapter 5. Scalar Diquarks 114 parameter space of this model. The BNV signal — excess ttbb events over ttbb events — would also be present and could in principle be searched for at the LHC. However, given the special way the couplings have been chosen, this probably does not warrant a dedicated search.

Finally we point out that if λ exceeds the washout constraints, detection of such inter- actions could be used to rule out certain high temperature baryogenesis mechanisms, without having to probe their possibly high energy scales, Λ 1 TeV, directly. 

5.5 Discussion

Let us summarise the overall phenomenological picture for such BNV models. First the approximately B conserving regime. For the diquark interactions of Eq. (5.10), B L − is broken so for such a low mass, one of the couplings must be small for washout of baryogenesis to be avoided. This forces us to rely on pair production for any sizable BNV process to be occurring at the LHC, making detection of the BNV interaction very difficult. Even without a clear BNV signal, multiple scalars may be discovered through the four jet pair production signal, or in the case µ 0, the dijet signal if λ is large → enough.

Once the properties of the scalars begin to emerge, one would infer the possibility of a gauge-invariant cubic interaction between the recently discovered scalars. Thus the possibility of BNV in this sector of particles would be hinted at. The precision study of such an interaction would be left to further experimental work.

A clearer BNV signal at the LHC may occur in the B violating regime, through resonant production, albeit for a very specific choice of parameters. If such a choice is not realised we are again left with signals of multiple scalars. For masses greater than around 1 TeV, only the dijet signal governed by the Yukawa coupling to first generation quarks is accessible at the LHC, pair production being too small for such masses.

If we instead examine the B L conserving interactions of Eq. (5.9) we are left with − leptoquark instead of diquark scalars. The washout constraint then does not apply, but being leptoquarks, these again would not be produced singly in large numbers at hadron colliders (pair production through gluon-gluon fusion is still possible). That is, even in Chapter 5. Scalar Diquarks 115 an optimistic scenario with EM strength Yukawa coupling, the production mechanisms ∗ − + − − g + d d σ¯5b + e and pp γ + d + X e e + d + X e +σ ¯5b + X only → → → → → result in a cross section of 3.4 fb [251] and 10 fb [252, 253] respectively for Mσ = 1 TeV at √s = 14 TeV at the LHC. But due to the nucleon stability constraints this coupling is expected to be small. So we expect BNV leptoquarks to only be pair produced in large numbers in a hadron collider (not produced on their own). This again results in a B = 0 initial state, making detection of the BNV interaction difficult.

5.6 Conclusion

We have examined an extension of the standard model involving diquarks with renormal- isable terms. We have shown that the nucleon stability constraints are greatly weakened if the exotic scalars are coupled to the third generation quarks. The diquark-diquark interaction violates B L, so a further constraint can be derived from washout. To avoid − washout of high scale baryogenesis, we are left with very stringent constraints on at least one of the couplings no matter to which generation quarks the scalars couple. Although the new scalars can be produced at the LHC, these stringent coupling constraints greatly inhibit any clear identification of such BNV interactions at the LHC.

If the heaviest diquark has a mass higher than Mσ & 45TBG, the most stringent con- straint again comes from nucleon stability rather than washout avoidance. If the scalar couples to the SM via the third generation of quarks, the couplings may be quite large, allowing for certain BNV processes to occur copiously at LHC energies. However, we have seen that the couplings must be carefully tuned for this possibility to occur. In ei- ther case, even if no BNV interaction is detected conclusively, the exotic particles would show up in other ways in the experiments. The discovery of multiple scalars would be interesting in itself, and once an overall picture emerged, the possibility of BNV in this sector of particles could be postulated. Chapter 6

Conclusion

The theory of antimatter is one of the triumphs of quantum field theory. However, the baryon asymmetry observed on a cosmological scale requires an extension of the SM. In this thesis we have studied high energy extensions of the SM that could explain the BAU or other unexplained observations such as the nature of DM or the origin of the neutrino masses. We have seen throughout the intimate connection between particle physics and cosmology — as the relevant energy scale of particle physics increases, the new states and interactions can play a significant role at higher temperatures, i.e. earlier times after the big bang, than have so far been probed.

In Chapter 1 we introduced the evidence for the baryon asymmetry: the CMB power spectrum, primordial element yields in BBN and the absence of high energy gamma rays from regions of baryon-antibaryon interactions. We discussed baryogenesis models which have been proposed in the literature. EW baryogenesis in the SM has been ruled out by the LEP Higgs searches. This has led to the need for a BSM explanation. One of many options is the CP violating decays of heavy particles such as occurs in leptogenesis.

In Chapter 2 we considered a toy model to study CP-violating annihilations, rather than decays, as an avenue for producing particle-antiparticle asymmetries. This acts as a proof-of-principle for creating such asymmetries. It also allowed for investigation of how such asymmetries would form in general. Furthermore, the calculation of the CP violating rates showed these obeyed the unitarity relations derived from the S-matrix — an elegant example of quantum field theory in action.

116 Conclusion 117

In Chapter 3 we applied what we had learned from the toy model to a neutron portal baryogenesis scenario. First of all, this allowed us to link the CP-violating collisions to the SM fields. Secondly it allowed us to investigate a scenario with both CP-violating collisions and CP-violating decays. We derived the Boltzmann equations for the evolu- tion of the asymmetries and solved these numerically for the first time in the literature. The CP-violating collisions were also included for the first time in the neutron portal and it was shown that these actually provide the dominant contribution to the final baryon asymmetry for much of the parameter space.

Crucial to this was the scaling with the temperature for the CP-violating scatterings. The CP violation grows with temperature up to the scale of the UV cutoff. This was contrasted to what occurs in leptogenesis — in which the CP violation in the scatterings does not grow much larger than the CP violation in the decays. As such distinctions can be made from simple dimensional analysis, it should be possible to determine, from now on, in which baryogenesis scenarios the CP-violating scatterings warrant further investigation.

In Chapter 4 we investigated the cosmological history of two — the Law/McDonald and Ma — radiative inverse seesaw models. These models can explain the neutrino masses with a relatively low scale. The fundamental origin of the masses may then be within direct experimental reach. As is often the case for lepton number violating neutrino mass models, the new physics also has an effect on early universe cosmology. Obviously such models can have an effect on the BAU. We discussed the new washout introduced by the radiative ISS and showed that these models will generally erase any net B created at or above the ISS scale. The Ma radiative ISS possibly allows resonant leptogenesis. The Law/McDonald model on the other hand only gives a mass splitting for the pseudo- Dirac pair after EW symmetry breaking and hence a different baryogenesis mechanism is required.

These models also contain DM candidates — neutral particles stable under Z2 sym- metries — the lightest of which could give a non-negligible contribution to the matter density of the universe today. It was shown that the Ma model would tend to overclose the universe if the DM candidate is produced in the usual thermal way. The simplest possibility for the DM in the Ma model is therefore ruled out. A more convoluted mech- anism is required to make this model work. The Law/McDonald model offered a simple Conclusion 118

DM scenario in which the SM and an exotic Higgs would act as portals to the dark sector. Much of the parameter space can be tested by the ongoing LUX and upcoming XENON1T detectors. Possible signals at the LHC were also discussed.

In Chapter 5 we discussed baryon number violating scalars. The allowed parameter space of such particles is heavily constrained by nucleon stability. The most conservative bounds are obtained by considering dominant couplings to the third generation of SM fermions. Deriving such bounds requires the estimate of loop integrals involving W bosons. Models in which B violation proceeds through interactions between multiple exotic scalars also allow the scale of new physics to be lowered. However, such couplings will tend to erase the B asymmetry in the early universe. Baryogenesis would then need to take place below the mass scale of the exotic scalars. We also investigated the LHC phenomenology of such models.

Overall then we have considered BSM extensions and their interplay with cosmology. We have seen how such models can help explain the unresolved puzzles of particle physics and cosmology and also how they are constrained by observations. We hope that future experimental and observational progress can bring the two fields — particle physics and cosmology — even closer together. Appendix A

Toy model cross sections and CP violation

We provide further details regarding the cross sections, CP violation and decay rates for the toy model studied in Chapter 2.

A.1 Toy model cross sections and CP violation

A.1.1 Ψ1Ψ1 ff →

We denote energies as a + b c + d and the initial (final) momentum as pi (pf ). The → CP symmetric cross section is given by

2 κ1 pf h 2 2 ih 2 2i EaEbσv = | | EaEb + p M EcEd + p M . (A.1) 8π√sˆ i − Ψ1 f − f

The difference in cross sections is given by

  ∗ ∗ 1 ∗ ∗ EaEb(σ σ)v = Im[κ λ κ3]g(MΨ1,MΨ2) + Im[κ λ κ2]g(MΨ2,MΨ2) − 1 1 2 1 2 pf h 2 2 ih 2 2i EaEb + p M EcEd + p M , (A.2) × 2π√sˆ i − Ψ1 f − f

119 Appendix A. Toy model cross sections and CP violation 120 where the loop integral contributes

s " # 2(m2 + m2) (m2 m2)2 1 2 i j i j g(mi, mj) = sˆ (mi + mj) 1 + − . (A.3) −8π − − sˆ sˆ2

Note the cross section goes as (coupling)2 while the CP violation goes ass ˆ (coupling)3. × Hence below the new physics scale the CP violation will not reach unphysical values, i.e.

σ σ − < 1 fors ˆ below the UV scale. (A.4) σ + σ

A remark as to the nature of this CP violation is in order. Upon thermal averaging, one obtains a T dependence for the CP violation as discussed in Chapters 2 and 3. This T dependence has its origin in the energy dependence of the cross sections and is not due to a finite- T statistical effect. In particular we have used Maxwell-Boltzmann statistics consistently throughout. Hence the CP violation discussed here is not due to incorrectly using quantum statistics for the external states and Maxwell-Boltzmann statistics for the internal states, which can lead to spurious behaviour for CP violating rates [254]. Note also that the CP violation persists even in the T 0 limit, apart from those processes → in which it is necessarily suppressed due to the particles in the loop being more massive than the external particles (see Figs. 2.2 and 3.4). Hence the CP violation discussed here cannot be a finite-T statistical effect and is a fundamentally different behaviour to that discovered in soft leptogenesis scenarios in Refs. [255–257] and pointed out as spurious in Ref. [254].

A.1.2 Ψ2Ψ2 ff →

The CP symmetric tree level cross section is

2 κ2 pf h 2 2 ih 2 2i EaEbσv = | | EaEb + p M EcEd + p M . (A.5) 8π√sˆ i − Ψ2 f − f Appendix A. Toy model cross sections and CP violation 121

The difference in cross sections is given by

  1 ∗ ∗ EaEb(σ σ)v = Im[κ λ2κ1]g(MΨ1,MΨ1) + Im[κ λ3κ3]g(MΨ1,MΨ2) − 2 2 2 pf h 2 2 ih 2 2i EaEb + p M EcEd + p M . (A.6) × 2π√sˆ i − Ψ2 f − f

A.1.3 Ψ1Ψ2 ff →

The CP symmetric tree level cross section is

2 κ3 pf h 2 ih 2 2i EaEbσv = | | EaEb + p MΨ1MΨ2 EcEd + p M . (A.7) 8π√sˆ i − f − f

The difference in cross sections is given by

  ∗ ∗ ∗ EaEb(σ σ)v = Im[λ1κ1κ ]g(MΨ1,MΨ1) + Im[λ κ2κ ]g(MΨ2,MΨ1) − 3 3 3 pf h 2 ih 2 2i EaEb + p MΨ1MΨ2 EcEd + p M . (A.8) × 4π√sˆ i − f − f

A.1.4 Ψ1Ψ1 Ψ2Ψ1 →

The CP symmetric tree level cross section is

2 λ1 pf h 2 2 ih 2 i EaEbσv = | | EaEb + p M EcEd + p MΨ1MΨ2 . (A.9) 4π√sˆ i − Ψ1 f −

The difference in cross sections is given by

  ∗ ∗ EaEb(σ σ)v = Im[λ1λ λ3]g(MΨ2,MΨ2) + Im[λ1κ1κ ]g(Mf ,Mf ) − 2 3 pf h 2 2 ih 2 i EaEb + p M EcEd + p MΨ1MΨ2 . (A.10) × 2π√sˆ i − Ψ1 f −

A.1.5 Ψ1Ψ1 Ψ2Ψ2 →

The CP symmetric tree level cross section is

2 λ2 pf h 2 2 ih 2 2 i EaEbσv = | | EaEb + p M EcEd + p M . (A.11) 8π√sˆ i − Ψ1 f − Ψ2 Appendix A. Toy model cross sections and CP violation 122

The difference in cross sections is given by

  1 ∗ ∗ ∗ EaEb(σ σ)v = Im[λ2κ1κ ]g(Mf ,Mf ) + Im[λ2λ λ ]g(MΨ1,MΨ2) − 2 2 1 3 pf h 2 2 ih 2 2 i EaEb + p M EcEd + p M . (A.12) × 2π√sˆ i − Ψ1 f − Ψ2

A.1.6 Ψ2Ψ2 Ψ2Ψ1 →

The CP symmetric tree level cross section is

2 λ3 pf h 2 2 ih 2 i 4EaEbσv = | | EaEb + p M EcEd + p MΨ1MΨ2 . (A.13) π√sˆ i − Ψ2 f −

The difference in cross sections is given by

  ∗ ∗ ∗ ∗ EaEb(σ σ)v = Im[λ κ2κ ]g(Mf ,Mf ) + Im[λ λ λ2]g(MΨ1,MΨ1) − 3 3 3 1 pf h 2 2 ih 2 i EaEb + p M EcEd + p MΨ1MΨ2 . (A.14) × 2π√sˆ i − Ψ2 f −

A.1.7 Ψ2 Ψ1ff decay →

The decay rate is given by

! 5 2 Z xmax     2  2 κ3MΨ2 x MΨ1 MΨ1 Mf Γ2a = | 3| dx 1 + 2 x 128π xmin 2 − MΨ2 MΨ2 − MΨ2 − !−1 !1/2 M 2 M 2 1 x + Ψ1 x2 4 Ψ1 × − MΨ2 − MΨ2 "  2  2  2#!1/2 MΨ1 Mf Mf gp 1 + x, , , × MΨ2 − MΨ2 MΨ2 (A.15) where 2 2 2 gp[x, y, z] = x + y + z 2xy 2yz 2xz (A.16) − − − Appendix A. Toy model cross sections and CP violation 123 and the terminals are

  MΨ1 xmin = 2 , (A.17) MΨ2  2  2 MΨ1 Mf xmax = 1 + 4 . (A.18) MΨ2 − MΨ2

A.2 Thermally averaged decay rate

The decay rates which appear in the Boltzmann equations are thermally averaged [65],

K (M/T ) Γth = 1 Γ, (A.19) K2(M/T ) where M is the mass of the decaying particle, Γ is the decay rate in the rest frame of the decaying particle and Kn(x) is the modified Bessel function of the second kind of order n. Appendix B

Neutron portal unitarity, cross sections, decay rates and CP violation

We provide further details of the unitarity constraint and cross sections for the neutron portal studied in Chapter 3.

B.1 Unitarity constraint for multiple quark generations

Here we discuss the unitarity constraint in the case of couplings to all possible quark

flavours and show that at least two Xα are required to obtain a non-zero source term from the CP violating annihilations.1 We denote the equilibrium reaction rates as

W (Xα + di uj + dk) = (1 + αijk)Wαijk, (B.1) →

W (Xα + di Xβ + dj) = (1 + ζαiβj)Zαiβj, (B.2) →

W (ui + dj uk + dl) = (1 + aijkl)Aijkl, (B.3) →

where the CP conjugate can be found by taking αijk αijk, ζαiβj ζαiβj or → − → − aijkl aijkl. Note we have also included quark flavour changing interactions in order → − 1 For clarity, we only consider initial states Xα + di here; the arguments can easily be generalised to include Xα + ui initial states.

124 Appendix B. Neutron portal unitarity, cross sections, decay rates and CP violation 125 to make the argument more general. The unitarity constraint then yields

X X ζαiβjZαiβj + αijkWαijk = 0, (B.4) βj jk X X ajkmlAjkml + αijkWαijk = 0. (B.5) ml αi

Now consider the case of only one Majorana Xα = X1. (The argument may be easily

modified for Dirac X1 and the conclusions remain unchanged.) Taking rd = rs = rb and

ru = rc = rt the Boltzmann equation for the baryon asymmetry (excluding the decay terms) is

dnB X h i + 3HnB = W1ijk (1 1ijk)(rdru + rX1rd) (1 + 1ijk)(rdru + rX1rd) . dt − − ijk (B.6)

Now consider only the source term

X h i 1ijkW1ijk rdru + rX1rd + rdru + rX1rd . (B.7) − ijk

A departure from equilibrium due to the expansion of the universe gives rX1 = 1 and 6 ru = rd = ru = rd = 1 and the source term may be written as

X X X X (2 + 2rX ) 1ijkW1ijk = (2 + 2rX ) ζ1i1jZ1i1j = 0, (B.8) − i jk i j where the first equality follows from Eq. (B.4) and the second as ζαiαj = ζαjαi. Hence − the generation of an asymmetry from annihilations with only a single Xα is not possible.

B.2 Cross sections, decay rates and CP violation

B.2.1 Cross sections

Here we collect the results of our calculations for the cross sections used in the above analysis. The sums over initial and final colours have been performed. The cross sections

have been calculated in the centre-of-mass frame. Ea and Eb (Ec and Ed) denote initial (final) state energies of the particles in the order listed. The initial momentum is denoted Appendix B. Neutron portal unitarity, cross sections, decay rates and CP violation 126

pi and the final momentum pf . The centre-of-mass energy is √sˆ.

u + Xα s + b → 2 3 κα pf  2  2  EaEbσv = | | EaEb + p EcEd + p (B.9) 2π√sˆ i f

s + Xα u + b → 2   3 κα pf 1 2 2 EaEbσv = | | EaEbEcEd + p p (B.10) 2π√sˆ 3 i f

X2 + u X1 + u → 2   3 κ3 pf 2 2 4 2 2 Re[κ3κ3] EaEbσv = | | 2EaEbEcEd + EcEdpi + EaEbpf + pi pf + 2 MX2MX1EbEd 8π√sˆ 3 κ3 | | (B.11)

X2 + X1 uu → 2   3 κ3 pf 2 2 2 Re[κ3κ3] 2 EaEbσv = | | 2EaEbEcEd + pi pf 2 MX2MX1(EcEd + pf ) (B.12) 8π√sˆ 3 − κ3 | |

Xα + Xα uu → 2   3 κα+3 pf 2 2 2 2 2 EaEbσv = | | 2EaEbEcEd + p p M (EcEd + p ) . (B.13) 8π√sˆ 3 i f − Xα f

B.2.2 Cross sections — CP violation

CP violation for uR + X1 uR + X2: → This CP violation arises from the interference of two tree level diagrams (corresponding to different continuous fermion lines) with a loop level diagram.

( 3pf ∗ h 2ih 2 i EaEb(σ σ)v = g1(ˆs, ms, mb) 2Im[κ3κ1κ ] EaEb + p EcEd + p − π√sˆ 2 i f ) ∗ ∗ + Im[κ3κ1κ2]MX1MX2EbEd , (B.14) Appendix B. Neutron portal unitarity, cross sections, decay rates and CP violation 127 where

s " # 2(m2 + m2) (m2 m2)2 1 2 2 i j i j g1(ˆs, mi, mj) = sˆ (m + m ) 1 + − , (B.15) 32π − i j − sˆ sˆ2

CP violation for uR + X1L s + b: → This CP violation arises from the interference of a tree level diagram with two different loop level diagrams.

6pf ∗ ∗ h 2ih 2 i EaEb(σ σ)v = Im[κ κ2κ ]g1(ˆs, mu,MX2) EaEb + p EcEd + p − π√sˆ 1 3 i f 3pf ∗ h 2 i Im[κ κ2κ3]Ebg2(ˆs, mu,MX2)MX1MX2 EcEd + p , (B.16) − π√sˆ 1 f

where

s " # 2(m2 + m2) (m2 m2)2 1 2 2 i j i j g2(ˆs, mi, mj) = sˆ + m m 1 + − . (B.17) −32π√sˆ i − j − sˆ sˆ2

CP violation for uR + X2 sR + bR: → This CP violation arises from the interference of a tree level diagram with two different loop level diagrams.

6pf ∗ h 2ih 2 i EaEb(σ σ)v = Im[κ κ1κ3]g1(ˆs, mu,MX1) EaEb + p EcEd + p − π√sˆ 2 i f 3pf ∗ ∗ h 2 i Im[κ κ1κ ]Ebg2(ˆs, mu,MX1)MX1MX2 EcEd + p . (B.18) − π√sˆ 2 3 f

B.2.3 Decay rates

B.2.3.1 Γ(X1L usb) →

Ignoring the final state masses the decay width is

2 5 Γ1A κ1 M = | | X1 , (B.19) 2 1024π3 so Γ1A is the sum over the partial widths Γ(X1L usb) + Γ(X1L usb). → → Appendix B. Neutron portal unitarity, cross sections, decay rates and CP violation 128

B.2.3.2 Γ(X2 usb) →

Ignoring the final state masses the decay width is

2 5 Γ2A κ2 M = | | X2 , (B.20) 2 1024π3 so Γ2A is the sum over the partial widths Γ(X2 usb) + Γ(X2 usb). → →

B.2.3.3 Γ(X2 X1Luu) →

For this decay we take into account the mass of X1L. In integral form the width is given by

2 5 Z Z  3 κ3 MX2 ΓX2B = | | dxa dxb xa(1 + µ12 xa) + xb(1 µ12 xb) 512π3 − − − 2  2Re[κ3] MX1 + 2 (1 µ12 xb) , (B.21) κ3 MX2 − − | |

2 where µ12 (MX1/MX2) and the limits of integration are ≡

1  p 2  1  p 2  [2 xa] x 4µ12 xb [2 xa] + x 4µ12 , (B.22) 2 − − a − ≤ ≤ 2 − a −

2√µ12 xa 1 + µ12. (B.23) ≤ ≤

B.2.4 CP violation in the X2 decay

This CP violation arises from the interference of a tree level diagram with two different loop level diagrams. The tree level amplitude for X2 usb with momenta pi, pa, pb, pc → respectively is given by

∗h ih i T = iκ u(pa)Lu(pi) u(pc)Lv(pb) . (B.24) M − 2 Appendix B. Neutron portal unitarity, cross sections, decay rates and CP violation 129

Here L (R) denotes the left (right) projection operator. The first loop diagram has amplitude

h i Z 4 Tr R(p/t + k/)k/ ∗ ∗h ih i d k L1 = κ3κ1 u(pa)Lu(pi) u(pc)Lv(pb)   , M (2π)4 2 2 2 2 [pt + k] M k m − X1 − u (B.25) where pt = pi pa. The second loop diagram has amplitude − h i Z 4 v(pi)Rkv/ (pa) ∗ h i d k L2 = κ1κ3MX1 u(pc)Lv(pb)   . (B.26) M (2π)4 2 2 2 2 [pt + k] M k m − X1 − u

The total CP violation can be expressed as:

D = γ1 + γ2, (B.27)

∗ where γ1 (γ2) comes from the interference between and L1 ( L2). MT M M

∗ B.2.4.1 Contribution from L1 MT M

5 Z 1−µ12   ∗ ∗ MX2 2 MX2xa γ1Γ2A = 3Im[κ2κ1κ3] 4 dxaxa(1 xa)g3 , (B.28) 512π 0 − 2 where

q 2 2 2 2 Z 1 x0(x + x0 x + E 2x0Ecθ + M x0Ecθ) 0 0 − X1 − g3(E) = dcθ q (B.29) −1 2 2 2 x + E 2x0Ecθ + M + x0 Ecθ 0 − X1 − and

2 2 (MX2 2MX2E MX1) x0 = − − . (B.30) 2(MX2 E Ecθ) − −

∗ B.2.4.2 Contribution from L2 MT M

4 Z 1−µ12   ∗ MX1MX2 MX2xa γ2Γ2A = 3Im[κ2κ1κ3] 4 dxaxa(1 xa)g4 , (B.31) 512π 0 − 2 Appendix B. Neutron portal unitarity, cross sections, decay rates and CP violation 130 where

Z 1 x2E(1 + c ) g (E) = dc 0 θ (B.32) 4 θ 2 2 2 1/2 −1 (x 2x0Ecθ + E + M ) + x0 Ecθ 0 − X1 − and the expression for x0 is given in Eq. (B.30). Appendix C

Inverse seesaw cross sections

We provide the formulas used for the cross sections of DM annihilation and resonant Z0 production in Chapter 4.

C.1 Double Higgs portal cross sections

2 We give the cross sections fors ˆ = 4mη used to estimate the required coupling λ. The interference effects from diagrams involving s-channel h and H exchange can be neatly expressed by defining

2 2 A B P (s) 2 + 2 , (C.1) | | ≡ (s m ) + iΓH mH (s m ) + iΓhmh − H − h where Γh(H) is the width of the SM-like (exotic) Higgs boson and the vertex factors A and B are given in Eqs. (4.54) and (4.55).

C.1.1 Cross section for ηη hh →

2 s 2 λ mh σv = | | 2 1 2 . (C.2) 64πmDM − mDM

C.1.2 Cross section for ηη HH →

Same as Eq. (C.2), but with mh mH . →

131 Appendix C. Inverse seesaw cross sections 132

C.1.3 Cross section for ηη ff → 3/2 2 m2 ! NC λ 2 2 f σv = | | P (4mDM ) 1 2 , (C.3) 4π | | − mDM where f denotes a SM fermion with mass mf and number of colours NC .

C.1.4 Cross section for ηη W +W − →

2 4  2 2! s 2 λ MW 2 2 1 mDM MW σv = | 2 P (4mDM ) 1 + 1 2 2 1 2 . (C.4) 4πmDM | | 2 − MW − mDM

C.1.5 Cross section for ηη ZZ →

Same as for Eq. (C.4) but with MW MZ and an additional overall factor of 1/2. →

+ C.2 Cross Section for pp Z0 l l− → →

Fermion: uL uR dL dR eL eR νL S2L 0 QZ0 /gχ: 1 -1 1 3 -3 -1 -3 2

Table C.1: Charges of the fermions contributing to the Z0 width.

To find the production cross section for the Z0 we follow the procedure given in Ref. [258]. For completness we have provided the key formulas in this section. We begin by writing the covariant derivative for the neutral current as

µ µ µ µ 0µ DN.c. = ∂ + iQγA + iQZ Z + iQZ0 Z . (C.5)

Note we have absorved the coupling constants into the charges. The charges for the relevant fermions are given in Table C.1. The interaction Lagrangian may be written as

0f 0f 0µ Z0 = fγµ(g g γ5)fZ , (C.6) L V − A Appendix C. Inverse seesaw cross sections 133 where the vector and axial-vector couplings are given by

0f 1 g = [Q 0 (f ) + Q 0 (f )], (C.7) V 2 Z L Z R 0f 1 g = [QZ0 (fL) QZ0 (fR)]. (C.8) A 2 −

The parton level cross section is given by

0q 2 0q 2 0l 2 0l 2 0 + − sˆ ( gV + gA )( gV + gA ) σ(qq Z l l ) = | | | 2| 2 | | 2 2| | , (C.9) → → 36π (ˆs M 0 ) + M 0 Γ 0 − Z Z Z where the width of the Z0 can be calculated using

N M 0 0 0 Γ(Z0 ff) = C Z ( g f 2 + g f 2), (C.10) → 12π | V | | A | where NC are the number of colours. In calculating the total width we sum over all three generations and assume four copies of S2L per generation. The cross section at a pp collider is given by

Z 1 Z 1 0 + − X σ(pp Z l l ) = dx1dx2[fq(x1,Q)fq(x2,Q) + fq(x2,Q)fq(x1,Q)] → → q 0 0

0 + − p σ(qq Z l l , sˆ = x1x2 sˆpp), × → → (C.11)

where fq(x, Q) is the PDF of q carrying momentum fraction x at momentum transfer p scale Q and sˆpp is the total centre-of-mass energy of the pp collider. We have checked our procedure by comparing to the sequential SM calculation presented in Ref. [212]. Appendix D

Loop integrals and cross sections for baryon number violating diquarks

We provide further details for the calculation of the loop factors for the nucleon decay diagrams and the resonant diquark production in Chapter 5.

D.1 Loop integrals for nucleon decay

To estimate the rate of the process in Fig. 5.2 we first evaluate the contribution from one of the fermion lines forming a loop with the W boson. Using relations for the spinors such as u = Cv¯T (see appendix G.4 in Ref. [259]), and ignoring the external momenta we find the Feynman amplitude for such a loop

Z d4k gµν  k/ + m   k/ + m  v¯ g V γνR b λ R t g V γµLu, (D.1) (2π)4 k2 m2 w ub k2 m2 3 k2 m2 w td − W − b − t where L(R), is the left (right) chiral projection operator. This can be simplified to

Z d4k 1 1 1  (4λ V V g2 m m )[¯vLu] . (D.2) 3 td ub w b t (2π)4 k2 m2 k2 m2 k2 m2 − w − b − t

134 Appendix D. Loop integrals for nucleon decay 135

Taking the leading term of the integral,

  1 mt 2 2 ln , (D.3) 8π mt mb we see this fermion line contributes a factor to the overall amplitude of

  1 mb mt 2 ln v¯Lu. (D.4) 2π mt mb

Then ignoring the external momenta, we find the dependence on the propagator masses for the overall diagram. The matrix element is simply estimated to be the mass of the neutron, with exponent set to restore the correct dimension of mass to the decay rate. This yields the rate of Eq. (5.7).

To estimate the rate of the process in Fig. 5.5 we note the integral around the loops has the form

Z d4a  1 1 1 1 (2π)4 a2 m2 a2 m2 a2 m2 a2 m2 − w − t − b − σ3 n Z d4k 1 1 1 1 1 o . (D.5) × (2π)4 k2 m2 k2 m2 k2 m2 k2 m2 (k + a)2 m2 − W − t − b − σ3 − σ7

Focusing first on the integral over k, and introducing the Feynman parameters xi, we rewrite this integral in the following form

Z 1 Z 1−x1 Z 1−x2−x1 Z 1−x3−x2−x1 4! dx1dx2dx3dx4 0 0 0 0 Z 4 n 2 2 2 d k (k + ax1) + a x1(1 x1) (1 x1 x2 x3 x4)m × − − − − − − t −5 2 2 2 2 o x1m x2m x3m x4m + i . (D.6) − σ7 − W − σ3 − b

(We have reintroduced the i,  > 0 term in the denominator of the propagators.) After

changing variable to l = k + ax1 this becomes a standard integral which evaluates to

2i Z 1 Z 1−x1 Z 1−x2−x1 Z 1−x3−x2−x1 − 2 dx1dx2dx3dx4 (4π) 0 0 0 0 n 2 2 a x1(1 x1) (1 x1 x2 x3 x4)m × − − − − − − t −3 2 2 2 2 o x1m x2m x3m x4m + i . (D.7) − σ7 − W − σ3 − b Appendix D. Loop integrals for nucleon decay 136

Now writing a2 = a2 a2, the denominator is zero for 0 − i

n 1 2 a0 = [mt (1 x1 x2 x3 x4) ± x1(1 x1) − − − − − 1/2 2 2 2 2 2o 0 + x1m + x2m + x3m + x4m ] + a i . (D.8) σ7 W σ3 b i ∓

This means any singularities occur in the second or fourth quadrant on the a0 plane, 2 2 allowing us to Wick rotate so that aE0 = ia0 and a = a . E0 − Equation (D.5) now becomes

Z 4 2 d aE  1 1 1 1 − (4π)2 (2π)4 2 2 2 2 2 2 a + m2 aE + mw aE + mt aE + mb E σ3 Z 1 Z 1−x1 Z 1−x2−x1 Z 1−x3−x2−x1 dx1dx2dx3dx4 × 0 0 0 0 2 2 a x1(1 x1) + (1 x1 x2 x3 x4)m ×{ E − − − − − t 2 2 2 2 −3 + x1m + x2m + x3m + x4m . (D.9) σ7 W σ3 b }

Now this is in the form R daf(a)g(a), where g(a) represents the integral over the Feyn- man parameters, with f(a), g(a) positive definite. The integral over the Feynman pa- rameters, g(a), is maximal for a = 0. If g(a) A for all a, and f(a), g(a) are positive ≤ definite R daf(a)g(a) R daf(a)A. Applying this to Eq. (D.9), we find the leading term ≤ of this integral to be  1   1  2 2 2 2 2 , (D.10) mσ3 mt mt mσ3 mσ7 where the second bracket is the leading term of g(a = 0), and the first the remaining integral R daf(a), and we have ignored factors of i, π, and logarithms of (1). Using a O similar technique of dimensional analysis as outlined before we again obtain the rate of Eq. (5.15).

D.2 Resonant diquark production

Consider the coupling of σ7 to first generation down type quarks

c = λ7σ7(d )dR + H.c. (D.11) L R Appendix D. Loop integrals for nucleon decay 137

If we explicitly include the Clebsch-Gordan coefficients — contracting the 3 dimensional representations of SU(3) to form the six dimensional representation — the Lagrangian becomes

h c c c i = λ7 σ7rr(d )dRr + σ7bb(d )dRb + σ7gg(d )dRg L Rr Rb Rg h i √ c c c + 2λ7 σ7rb(dRr)dRb + σ7rg(dRr)dRg + σ7bg(dRb)dRg + H.c., (D.12) where subscripts indicate colours and we have used the identity Ψcχ = χcψ.

Consider now the decay σ7rr dRrdRr. Summing the two possibilities for contracting → the operator with the final state gives an overall amplitude

= 2iλ7[u(pa)Lv(pb)], (D.13) M where pa and pb are final state momenta. Taking into account the identical final state particles one finds a decay width

2 λ7 Γ(σ7rr dRrdRr) = | | Mσ7. (D.14) → 8π

Now consider the decay σ7rb dRrdRb, the amplitude is a factor of √2 smaller than → Eq. (D.13) because only one contraction is possible. While the decay rate is the same as for Eq. (D.14), because we integrate over all angles for the distinct final state particles.

Hence the decay rate into quarks, σ7 dRdR, is independent of colour and given by → Eq. (D.14).

We approximate the parton level cross section using a Breit-Wigner approximation [260]. Taking into account the various statistical factors arising from spins and colours the cross section is given by

8π Γ(σ7 dRdR)Γ(σ7 final) σ(dd σ7 final) = → → , (D.15) → → 3 (s M2 )2 + M2 Γ2 − σ7 σ7 σ7 where we find the total decay rate using the (assumed) branching fraction of σ7 σ3σ3. → Bibliography

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