Aspects of Particle Cosmology with an Emphasis on Baryogenesis

Home , Baryogenesis, Curvaton, Oliver Buchmueller

Aspects of particle cosmology with an emphasis on baryogenesis

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Graham Albert White

School of Physics, Monash University

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

in the

Department of Physics and Astronomy Monash University

Supervisor: Csaba Balazs

May 2017

1 Copyright Notices

Notice 1

Under the Copyright Act 1968, this thesis must be used only under the normal conditions of scholarly fair dealing. In particular no results or conclusions should be extracted from it, nor should it be copied or closely paraphrased in whole or in part without the written consent of the author. Proper written acknowledgement should be made for any assistance obtained from this thesis

Notice 2

I certify that I have made all reasonable efforts to secure copyright permissions for third-party content included in this thesis and have not knowingly added copyright content to my work without the owner’s permission.

2 Contents

1 Introduction 11

2 The Standard Model, the hierachy problem and cosmology 13 2.1 The Standard Model...... 13 2.1.1 Gauge symmetry ...... 14 2.1.2 Symmetry breaking and the Higgs sector...... 15 2.1.3 Standard Model Lagrangian ...... 15 2.1.4 The hierachy problem...... 16 2.1.5 Some cosmological shortcomings of the Standard Model . . 17 2.2 Supersymmetry...... 18 2.2.1 Superspace ...... 19 2.2.2 Non Interacting Supersymmetric Theories...... 20 2.2.3 Examples of supersymmetric theories ...... 21 2.2.4 A brief summary of supersymmetric cosmology ...... 22 2.3 The relaxion mechanism: another solution to the hierarchy problem 23 2.3.1 Declaration for thesis chapter 2...... 23 2.3.2 Published material for chapter 2: Naturalness of the relaxion mechanism ...... 26

3 Baryogenesis 53 3.1 Sakharov Conditions ...... 53 3.2 Introductory remarks for published material in thesis chapter 3 . . . 54 3.3 Declaration for thesis chapter 3...... 55 3.4 Published material for chapter 3: A pedagogical introduction to electroweak baryogenesis...... 57 3.5 A brief review of thermal ﬁeld theory...... 97 3.6 The effective potential at ﬁnite temperature ...... 98 3.6.1 Bubble nucleation...... 99 3.6.2 Transport equations...... 100

4 Solving transport equations during a cosmic phase transitions 101 4.1 Introductory remarks ...... 101 4.2 Declaration for thesis chapter 4...... 101 4.3 Published material for Chapter 4: General analytic methods for solving coupled transport equations: From cosmology to beyond . . 103

3 5 Solving bubble wall proﬁles 119 5.1 Introductory remarks ...... 119 5.2 Declaration for thesis chapter 5...... 119 5.3 Published material for chapter 5: Semi-analytic techniques for calculating bubble wall proﬁles...... 122

6 Particle cosmology in the NMSSM 133 6.1 Introductory remarks ...... 133 6.2 Declaration for thesis chapter 6...... 133 6.3 Published material for chapter 6: Baryogenesis, dark matter and inﬂation in the Next-to-Minimal Supersymmetric Standard Model . 136

7 Gravitational waves as a probe of vacuum stability via high scale phase transitions 163 7.1 Introductory remarks ...... 163 7.2 Declaration for thesis chapter 7...... 164 7.3 Published material for chapter 7: Gravitational waves at aLIGO and vacuum stability with a scalar singlet extension of the Standard Model166

8 Effective ﬁeld theory, electric dipole moments and electroweak baryogenesis 177 8.1 Introductory remarks ...... 177 8.2 Declaration for thesis chapter 8...... 178 8.3 Published material for chapter 8: Effective ﬁeld theory, electric dipole moments and electroweak baryogenesis ...... 180

9 Conclusions and future work 205

Bibliography 207

4 Abstract

Cosmology provides some compelling reasons to expect physics beyond the Standard Model on top of theoretical concerns such as the hierarchy problem and grand uniﬁcation. In this thesis I explore some of the intersection between particle physics and cosmology with an emphasis on electroweak baryogenesis. I will present original work on whether the relaxion mechanism solves the hierarchy problem, a pedagogical review of electroweak baryogenesis and present research on computational and phenomenological aspects of electroweak baryogenesis and cosmic phase transitions including gravitational wave signals. I also present a new testable paradigm for producing the baryon asymmetry of the Universe as well as a an application of effective ﬁeld theory to directly use experimental constraints to put bounds on CP violating operators.

5 List of research outputs

1. G. A. White A Pedagogical Introduction to Electroweak Baryogenesis Morgan & Claypool Publishers, IOC concise physics series. 2016 ISBN: 978-1-6817-4457-5

2. C. Balazs, G. White and J. Yue, JHEP 1703, 030 (2017) doi:10.1007/JHEP03(2017)030 [arXiv:1612.01270 [hep-ph]].

3. C. Balazs, A. Fowlie, A. Mazumdar and G. White, Phys. Rev. D 95, no. 4, 043505 (2017) doi:10.1103/PhysRevD.95.043505 [arXiv:1611.01617 [hep-ph]].

4. S. Akula, C. Balázs and G. A. White, Eur. Phys. J. C 76, no. 12, 681 (2016) doi:10.1140/epjc/s10052-016-4519-5 [arXiv:1608.00008 [hep-ph]].

5. A. Fowlie, C. Balazs, G. White, L. Marzola and M. Raidal, “Naturalness of the relaxion mechanism,” JHEP 1608, 100 (2016) doi:10.1007/JHEP08(2016)100 [arXiv:1602.03889 [hep-ph]].

6. G. A. White, “General analytic methods for solving coupled transport equations: From cosmology to beyond,” Phys. Rev. D 93, no. 4, 043504 (2016) doi:10.1103/PhysRevD.93.043504 [arXiv:1510.03901 [hep-ph]].

7. C. Balazs, A. Mazumdar, E. Pukartas and G. White, “Baryogenesis, dark matter and inﬂation in the Next-to-Minimal Supersymmetric Standard Model,” JHEP 1401, 073 (2014) doi:10.1007/JHEP01(2014)073 [arXiv:1309.5091 [hep-ph]].

8. J. P. Straley, G. A. White and E. B. Kolomeisky, “Casimir energy of a cylindrical shell of elliptical cross section,” Phys. Rev. A 87, no. 2, 022503 (2013) doi:10.1103/PhysRevA.87.022503 [arXiv:1403.3439 [cond-mat.stat-mech]]

9. G. A. White, “Higher entanglement and ﬁdelity with a uniformly accelerated partner via non-orthogonal states,” Phys. Rev. A 86, 032340 (2012) doi:10.1103/PhysRevA.86.032340 [arXiv:1210.3040 [quant-ph]].

10. M. Ramsey-Musolf, G. A. White and P. Winslow “Colour Breaking Baryogenesis,” Anticipated Febuary 2017.

11. S. Akula, C. Balazs and G. A. White, “Exploring baryogenesis in the 3 invariant NMSSM,” Anticipated Febuary 2017. Z

Names and Thesis Nature and % of student contributions of Publication Title Status Chapter contribution Monash student coaauthors 25% including coderiving analytic sections, writing Naturalness of the re- 2 Published some sections and editing N/A laxion mechanism the paper. Contributed to ideas throughout A pedagogical introduction to elec- 3 Published 100% (single author) N/A troweak baryogenesis General analytic methods for solving coupled transport 4 Published 100% (single author) N/A equations: From cosmology to beyond Semi-analytic tech- 70% including original niques for calculat- concepts and derivation 5 Published N/A ing bubble wall pro- as well as numerical im- files plementation. Baryogenesis, dark matter and 50% involved with nu- inflation in the merical implementation 6 Published N/A Next-to-Minimal as well as analytic deriva- Supersymmetric tion and writeup Standard Model Gravitational waves at aLIGO and vac- 50% involved with nu- uum stability with a merical implementation 7 Published N/A scalar singlet exten- as well as analytic deriva- sion of the Standard tion and writeup Model 60% including formulat- Effective field the- ing idea and plan for the ory, electric dipole project. Doing the nu- 8 moments and elec- Published merical calculations for N/A troweak baryogene- the BAU and deriving the sis major concepts and theo- rems in the paper.

8 Acknowledgements

I’d like to acknowledge my wife for putting up with me often being preoccupied with research and frequently working late nights and weekends. You are a very kind, patient, gentle and understanding wife and I know this PhD has been difficult for you at times. I would like to acknowledge Csaba Balazs for his fantastic supervision. He has a great way of talking you up in order to build your confidence. He initially surveyed my skill set and interests and chose a great subject in baryogenesis. He then helped me work through all messy literature on the subject before connecting me with one of the leaders of the field and giving me the confidence to apply for the prestigious American Australian Award. He also gave me the confidence to work independently and pursue ideas I had and convinced me that my work was strong enough to write a review book on the subject. I would like to thank Michael Ramsey-Musolf for accepting me as a guest scholar and teaching me so many subtle points on baryogenesis. I would then like to thank my many collaborators: Ernestas Pukartas, Andrew Fowlie, Liam Dunn, Luca Marzola, Jason Yue, Martti Raidal, Sujeet Akula, Peter Winslow, Anupam Mazumdar, Basem El-menoufi and Oleg Popov. I have learnt so much from so many of them and I would not be be able to write this thesis without them. I would in particular like to acknowledge Joseph Straley who offered me a very nice RA at a time when they were tricky to find. This allowed me to get a Masters and a paper which resulted in my acceptance into Monash University. Finally I would like to thank the funding agencies that have made this work possible. In particular I would like to thank the American Australian Association who awarded me the Keith Murdoch Fellowship, the J. L. William Fellowship and the Australian Post Graduate award. I would also like to acknowledge the Center of Excellence for Particle Physics at the Tera Scale (CoEPP) for funding my travel to multiple conferences and workshops.

Chapter 1

Introduction

Motivated by the hierarchy problem, theorists have proposed a landscape of models predicting new physics at the weak, or TeV, scale [1, 2, 3, 4, 5]. At the high energy frontier experiments will continue to probe weak and TeV scale physics both at the Large Hadron Collider (LHC) and, hopefully, at higher energy colliders [6]. The Standard Model has thus far survived incredible examination, well beyond the expectations of many. This can lead one to ask if any of the proposed solutions to the hierarchy problem are correct. A no answer would, surprisingly, shatter our notion of the naturalness of our Universe. Whereas a yes answer could lead to one of the richest discovery periods in the history of particle physics. On the other hand, cosmological data is already decisive in its rejection of the Standard Model. Of the four major pillars of particle cosmology - the baryon asymmetry, dark matter, dark energy and inflation - not one can be explained within the Standard Model. From a certain perspective, the Universe and everything in it including our existence is the data from a "particle accelerator" that reached energies one can never achieve here on Earth. The results are striking! The vast majority of the universe is made up of a combination of dark energy and dark matter - neither of which are accommodated within the Standard Model. Even the abundance of baryonic matter presents a puzzle [7]. The Universe is far more flat and the cosmic microwave background (CMB) is far too isothermal than one would expect and as such inflation has been proposed as necessary ingredient in our cosmic history [8]. However, inflation will washout any initial baryon asymmetry and the Standard Model Cabibbo–Kobayashi–Maskawa (CKM) matrix provides too feeble CP violation and the mass of the Higgs is too heavy to produce the baryon asymmetry of the Universe. To solve these puzzles, particle physics and cosmology have developed a rich dialogue of late. Electroweak precision studies at both the energy and precision frontier are beginning to make a dent in the parameter space where extra scalar fields can catalyze a strongly first order electroweak phase transition - a key ingredient in electroweak baryogenesis - and a 100 TeV collider might decisively answer this question [9]. Experiments that search for permanent electric dipole moments have improved in their sensitivity very rapidly and all but rule out some popular models of electroweak baryogenesis. Even gravitational waves at eLISA and aLIGO are probing the possibility of electroweak phase transition being strongly first order or multi-step [10, 11, 12].

11 For dark matter, gravitational lensing in the CMB[13] as well as studies of bullet clusters [14] indeed point to a particle solution to the dark matter problem. The DAMA experiment gives a striking signal [15] whereas the LUX and PANDA experiments are nearing the neutrino background in their search for WIMPs [16, 17]. As to inflation, searches for B modes in the CMB probe inflation, while improved precision in such searches will be able to probe physics at the inflaton scale by examining correlations in density fluctuations. Remarkably particle cosmology might even give insight to physics at the Grand Unification Theory (GUT) scale! [18] Finally, the precision frontier is also starting to give promising hints - from the flavour anomalies in LHCb and Belle [19, 20, 21, 22] to the anomalous results in g 2 experiments and perhaps these intriguing signals are cracks in the Standard Model− [23]. If these cracks widen they could also shed some light into our cosmological history. In other words just about every signal or negative search in any fundamental physical experiment has consequences for the interplay between particle physics and cosmology. It is a young field where many type of research whether it is formal progress, model building, phenomenology or numerical tool is possible. In this thesis I will look at particular aspects of the interplay between particle physics and cosmology with an emphasis on electroweak baryogenesis. I will begin with a review of the Standard Model and the hierarchy problem in chapter2 and its most popular solution - supersymmetry - before discussing a particle cosmology inspired solution known as the relaxion method where I will present some of my work (a copyright agreement limits how much I can put into this thesis). I will then focus more closely on baryogenesis in chapter3 presenting an overview of the subject including the review given in my book before discussing some technical and phenomenological work I have done. Specifically, I will present some work on how to analytically solve sets of coupled transport equations in chapter4 before applying these techniques to calculating bubble wall profiles in chapter5. I will then shift gears to focus more exclusively on phenomenology analysing the compatibility of the collider constraints on the NMSSM with particle cosmology constraints in chapter6 and then in chapter7 I look at gravitational wave signatures at aLIGO due to high scale phase transitions and how they are physically motivated by vacuum stability. I explore the use of effective field theory in baryogenesis in chpater8 before concluding in chapter9

12 Chapter 2

The Standard Model, the hierachy problem and cosmology

The Standard Model of particle physics is perhaps the most successful physical model ever tested [24]. It is of course known to be incomplete as it does not describe gravity. This fact requires the Standard Model to be modified at energy levels where gravitational effects become important; the so called “Plank scale". Unfortunately, physics at the Plank scale is potentially well beyond anything that can feasibly be tested on Earth in the foreseeable future. On the other hand, there is some evidence that the Standard Model may be incomplete at energies much lower than the plank scale. For example, numerous observations show that the Universe is much more massive than the (directly) observable matter of the Universe would suggest [25], the energy scale of the electroweak force in the Standard Model is in a region that requires incredible fine tuning [26], there is more matter than anti-matter in the Universe, and finally there is evidence that the Universe began with a period of incredibly rapid expansion known as "inflation" [8]. There was once hope that the Standard Model could explain the matter/antimatter asymmetry in the universe. However, the recent discover of a Higgs particle at 125 GeV is far too heavy for electroweak baryogenesis to be viable in the Standard Model. Further, there is little hope that any of the other problems can be solved within the Standard Model.

2.1 The Standard Model

For the sake of completeness, let us briefly review the Standard Model before commenting on aspects of its incompleteness. (I will ignore in this thesis the strong CP problem, the existence of neutrino masses, dark energy, quantum gravity and a number of intriguing hints in experimental data). The components of the Standard Model are fermions, scalar fields and gauge bosons which facilitate forces as well as interactions between the scalar and fermionic sectors. The fermions come in two varieties, quarks with non-zero baryon number and leptons with non-zero lepton number. Each fermion has three generations or flavours.

13 2.1.1 Gauge symmetry The Standard Model is most conveniently formulated in such a way where one introduces redundant degrees of freedom. Physically equivalent states are then related through gauge transformations. The simplest example of this is quantum electrodynamics (QED) where for example it can be convenient to express the electric field as the gradient of a scalar field as well as representing the magnetic field as the curl of a vector field. The cost of this is that one has introduced a redundancy in the theory. For the QED case, the theory is redundant under local phase transformations where fields transform ψ(x) eiα(x)ψ(x) . (2.1) → The generalization of this is to consider replacing such a local phase transformation with a general set of local transformations belonging to a Lie group.

Lie groups and the exponential map In general a continuous symmetry of Nature is described by a Lie group - a group that is also a smooth manifold. The smooth nature of the manifold ensures that the group indeed describes a continuous transformation. Using the fact that X2 eX = 1 + X + + ... (2.2) 2 and det[eX ] = eTr[X] (2.3) for X being any square matrix, one can generate elements of a group from the properties of the group. For example, special groups require that the determinant of all members of the group is equal to one. From the above equation I see that this is achieved for the set of matrices, X τ, that are traceless. It is well known that orthogonal groups have the property that∈ the transpose of a matrix is its inverse. A member of the orthogonal group is generated by the exponential of an antisymmetric matrix. Here, the set τ is known as the “Lie algebra" which generates the Lie group by the exponential map. Unfortunately the exponential of τ does not generate the entire group. However one can observe that the commutator between any two elements of a Lie algebra is itself a member of the algebra. For example, for a unitary group whose Lie algebra is the set of antihermitian matrices (or equivalently i multiplied by a hermitian matrix as is convention amongst physicists) one can see that

let A = [X1,X2] (2.4)

A† = (X X )† (X X )† (2.5) 1 2 − 2 1 = X†X† X†X† (2.6) 2 1 − 1 2 = X X X X = [X ,X ] = A τ. (2.7) 2 1 − 1 2 − 1 2 − ∈ Next we make use of the Barker-Campbell-Herstoff theorem, which states that 1 eAeB = exp[A + B + [A, B] + ....] (2.8) 2 where all terms in the unwritten sum are just functions of the commutators between A and B. Therefore if exp[A] and exp[B] are members of the Lie group, then G 14 exp[A] exp[B] . One can then use the set exp[αA] where A τ to completely map a finite× region∈ G of the group around the identity. It is customary∈ for α to be infinitesimal and this region to therefore also be infinitesimal but this need not be the case. If one can completely cover a region around the identity then one can use a series of products to generate the entire connected region of the group. The proof of this will not be repeated here. For an example consider the case of SU(2) which is one of the gauge groups of the Standard Model. Since it is a special group the Lie algebra generators must be traceless and Hermitian (since I will include a factor of i in the exponential). The 2 in SU(2) implies that the generating matrix are 2 2 matrices. Specifically they are the Pauli matrices ×

0 1 0 i 1 0 σ = , , . (2.9) i 1 0 i 0 0 1 − − It is easy to see that these matrices obey the commutation relations

[σi, σj] = 2iijkσk . (2.10)

Therefore the product of two elements of the SU(2) group is also an element of the group.

2.1.2 Symmetry breaking and the Higgs sector

The Standard Model gauge product group is SU(3)C SU(2)L U(1)Y , and its terms are known as colour, weak isospin and hypercharge group,× respectively.× It is difficult to give Standard Model particles a tree level mass without violating gauge symmetry. One mechanism to do this though is known as the Anderson-Higgs mechanism [27, 28] where a scalar field with gauge symmetries SU(2)L and U(1)Y acquires a vacuum expectation value (vev). Specifically, if the Higgs field has an effective potential V (H) = µ2 H2 + λ H 4 (2.11) − | | | | with H = (H+,H0)T, the potential has a minimum that is not invariant under SU(2)L and U(1)Y transformations. I am free to choose the direction of the vev within the Higgs internal space. Let us choose the vev v such that

H+ H = . (2.12) (h + h + v)/√2 r I Fermions acquire masses through Yukawa interactions where the Yukawa coupling times the vev are an effective mass term. Three linear combinations of gauge bosons acquire a mass through the kinetic term in the Lagrangian. Speciﬁcally, these particles are the W , and Z bosons. ±

2.1.3 Standard Model Lagrangian I now have all the necessary pieces to form the Standard Model Lagrangian

= + + + V (H) (2.13) LSM LKinetic Lgauge LYukawa 15 where V (H) is given in equation (2.11). The kinetic terms are given by

f¯Df/ + D HDµH (2.14) LKinetic ≡ µ Xf a a with Dµ ∂µ + igτ Aµ is the covariant derivative for the gauge group with Lie generators≡τ a. The covariant derivatives for the Higgs and left handed fermions contain gauge bosons for weak isospin (SU(2)L) whereas right handed fermions do not. The covariant derivative of all Standard Model fermions and the Higgs boson include gauge bosons for hypercharge (U(1)Y ) and finally the covariant derivative of both left and right handed quark fields have gauge bosons for colour (SU(3)C ). The T full set of fermions are three generations of quark doubles ((ULi,DLi) ) and their right handed counter parts as well as three generations of Lepton doublets ((νLi, eLi)) and the right handed partner of the three generations of the electrons (eRi). The gauge sector terms have the form 1 1 1 Tr[Ga Ga,µν] + Tr[Ba Ba,µν] + Y Y µν (2.15) Lgauge ≡ 4 µν 4 µν 4 µν with Gµν, Bµν and Yµν being the field strength tensors for the Standard Model gauge groups SU(3)C , SU(2)L and U(1)Y respectively. Generically the field strength tensors have the form a a 1 τ Gµν = [Dµ,Dν] (2.16) gs and similarly for the other gauge groups. Finally the Yukawa terms have the form

+ Y U¯ U H Y D¯ U φ− + Y U¯ D φ + Y D¯ D φ (2.17) LYukawa ≡ u L R 0 − u L R d L R d L R 0 where Y and Y are 3 3 matrices in ﬂavour space. d u × 2.1.4 The hierachy problem The Higgs boson should receive corrections to its mass from any new physics particles via radiative corrections. Suppose new physics appears at some scale Λ and couples to the Standard Model with some coupling gf,b, the mass of the Higgs acquires quadratic corrections of the form

g2 ∆m2 = c f,b Λ2 + (2.18) H f,b 16π2 ··· where cf,b = ( 2, 1) for fermions and bosons respectively [26]. One does have a tree level bare mass− term to cancel against the quadratic corrections. However, if the scale of new physics is n orders of magnitude above the weak scale, the ﬁrst 2n decimal places must cancel the loop corrections in order for the mass to be∼ weak scale. There is expected to be new physics at the Planck scale – a full 17 orders of magnitude higher than the weak scale! Aside from the stability of the Higgs mass, the weak scale value of the vacuum expectation value is also unstable because it receives tree level corrections from any other scalar particles that might acquire a vacuum expectation value. For example, suppose there is an additional scalar ﬁeld, S, that is a singlet under the Standard Model gauge group. In principle there can be many such scalar particles each with a vev at an arbitrary scale. Such a particle

16 2 2 couples to the Higgs via portal couplings ghsH S . Replacing S with its vev, this portal coupling because an effective correction to µ2. Again, it requires a ﬁne tuned cancellation for the Higgs vev to be weak scale (of course it is also possible that no new scalar singlets exist).

2.1.5 Some cosmological shortcomings of the Standard Model Outside the hierachy problem some of the most convincing reasons to look beyond the Standard Model are cosmological. Of particular interest to particle physicists are inﬂation, dark matter and baryogenesis.

Inflation One of the strongest pieces of evidence for the Big Bang cosmological model is that it correctly predicts cosmic microwave background radiation (CMB). However, measurements of the CMBR show that the universe is very much in thermal equilibrium. The troubling aspect of this is that in the expansion of the universe regions of the universe which one observes to be in thermal equilibrium are causally disconnected. This is known as the horizon problem. A solution was proposed by Alan Guth who noted that as the Universe expands, the Higgs field undergoes a phase transition as it cools [8]. If the Higgs field supercools, in a process similar to the supercooling of water, the Higgs field will not undergo the phase transition when the temperature falls below the critical temperature at which the change of phase normally occurs. During the phase of supercooling a large negative pressure drives the universe to expand exponentially fast. Then, once the phase transition occurs, latent heat is released into the Universe. This solves the horizon problem by allowing the Universe to obtain thermal equilibrium before the period of inflation occurs. Unfortunately, there are many problems with using the Higgs field as the driver of inflation. This motivates the need to look beyond the Standard Model to facilitate inflation.

Dark matter It was first observed by Zwicky [29] that velocity dispersions in the Coma and Virgo galaxy clusters large than expected based on their visible mass. Later, Rubin and others confirmed the missing mass [30, 31, 32]. This has traditionally inspired two explanations: some modification of Newtons law at large distance, or some new form of matter that does not interact with light commonly referred to as dark matter. Several observations contribute substantially to the popularity of the dark matter explanation. First the behaviour of two colliding galaxy clusters known as the Bullet cluster is consistent with dark matter but inconsistent with the most popular modified gravity models. Second the scale of angular fluctuations in the CMB has a spectrum consisting of a peak for baryonic matter followed by successive peaks. This is predicted by dark matter but difficult to explain through modified gravity [33]. Finally there is the circumstantial evidence known as the so called "WIMP miracle" where if one assumes that the dark matter particle has a weak scale mass the right dark matter relic abundance is achieved for a weak scale cross section [34]. The Standard Model does not have a viable dark matter candidate so to explain this part of cosmology one needs to go beyond the Standard Model. Furthermore the

17 simplest extension of the Standard Model, massive neutrinos, can be a component of dark matter. However, they will not have enough mass density to explain all of dark matter.

Baryogenesis If one has a cosmic history that includes inﬂation, then one cannot have an asymmetry between baryons and anti-baryons as an initial condition as any initial baryon asymmetry would be washed out by inﬂation. Therefore one must produce it dy- namically.1 I will be primarily interested in electroweak baryogenesis which occurs during the electroweak phase transition [35]. As I will explain in later chapters the Standard Model fails to provide an explanation for this observed asymmetry.

2.2 Supersymmetry

The Higgs sector of the Standard Model is unique in that it describes the dynamics of a scalar quantum field. The superficial degree of divergence of a scalar field theory with a quartic interaction term is quadratic in the momentum cutoff which is a much more dramatic divergence than the rest of the Standard Model which diverges only logarthmically. However, since the masses of the weak gauge bosons (as well as the Higgs mass itself) depend upon the quadratically divergent Higgs mass, their masses should be of the order of the square of the momentum cutoff. Not only would this give masses well outside what is observed experimentally, the weak scale masses are strongly constrained to within about an order of magnitude of what is observed by the "unitarity bound" and the "perturbativity bound". The Standard Model is inconsistent outside this region. Strictly speaking it is actually possible within the Standard Model to achieve renormalized masses within these bounds as one has a free parameter, the bare mass, which one can fine tune to get the required values [26]. Much like the fact that once upon a time Geocentric models of the Universe needed to be fine tuned in order to be consistent with observation was actually a hint of new physics, the fine tuning required in the Standard Model may gave us hints at how to extend it [36] That the superficial degree of divergence is worse than logarithmic for a quartic scalar field theory is not unique. Quantum electrodynamics, for example, superficially diverges worse than logarithmically. However, both Lorentz and gauge symmetries conspire to reduce the actual degree of divergence. The fact that the Higgs field does not have a local gauge symmetry in essence means that the actual degree of divergence remains quadratic [24]. Naturally, if one could find a new symmetry of nature one could reduce the degree of divergence. Historically, there were many attempts to extend the Poincare group, in particular to find a symmetry group that would contain space time symmetries and internal symmetries in a single group. Ultimately all of these failed and in 1967, Coleman and Mandula proved a “no go” theorem that showed that you cannot extend the symmetries of the Standard Model in a non-trivial way [37]. The theorem was flawed however as it failed to take into account the spinor like nature of quantum particles [38]. Even still, through this

1Note that with arbitrarily large ﬁne tuning one can evade this claim.

18 came the realization that there is only one unique extension to spacetime, the theory now known as "supersymmetry".

2.2.1 Superspace Graded Lie groups and a violation of the Coleman-Mandula theorem The symmetry that describes supersymmetry is in fact not a Lie group but a “graded” Lie group [39]. A graded Lie group differs from a Lie group in that its associated Lie algebra is closed under anti-commutation rather than commutation. The full graded group is semi-simple and is described by the set of charges Pµ,Mµν,Q . These charges obey the following relations { }

[Pµ,Pν] = 0, (2.19) [M ,P ] = i (η P η P ) , (2.20) µν λ νλ µ − λµ ν [M ,M ] = i (η M η M µν νρ νρ µσ − νσ µρ− ηνρMησ + ηνσMνρ) , (2.21)

Qa,Qb = Qa† ,Qb† = 0, (2.22) { } { µ } Q ,Q† = σ P , (2.23) { a b} ab µ [Qa, P µ] = [Qa† ,Pµ] = 0, (2.24) b [Qa,Mµν] = (σµν)a Qb . (2.25)

From the above equations one can see that the entire algebra is closed under commutation and anti commutation. I therefore consider a set of inﬁnitesimal supersymmetry transformations generated by the exponential of the Lie algebra to follow the same recipes as for Lie groups eiαQ¯ 1 + iαQ¯ . (2.26) ≈

Although the charges Qa obey anticommutation relations, I can still use the Baker Campbell Horstoff theorem if the inﬁnitesimal parameter that they are multiplied by, α,¯ is a Grassmann valued number. Therefore if I can completely cover a region around the identity I can use products of inﬁnitesimal transformations to map the entire connected region of the graded Lie group. Since α¯ is Grassmann and I “begin” around the identity, the entire space that these transformations connect is a Grassmann valued space. Here and in Eq. (2.23) one sees the radical nature of supersymmetry. A supersymmetric transformation is something like the square root of a space time translation! Supersymmetry is something like a spinorial extension to space time such that one now has the usual 3+1 space time dimensions, xµ as well as four extra quantum dimensions, θi. Even stranger, the dimensions θi are Grassmann numbers that anticommute with each other and commute with ordinary numbers. It appears occasionally in the literature that “Superspace” is just a mathematical trick. However, as shown in the discussion above, if Eqs. (2.19-2.25) are the generators of the symmetries that describe our world then our world literally has four extra Grassmann valued dimensions. Of course I cannot directly observe these anti-commuting variables. However, one is convinced of curved space time because general relativity works and if supersymmetry works I will be convinced of anti commuting dimensions. Supersymmetry is a radical idea!

19 2.2.2 Non Interacting Supersymmetric Theories In quantum field theory, one considers functions of the ordinary space time variables, xµ. However, if the world is supersymmetric then I must consider functions of both the space time and the Grassmann coordinates. I begin with a scalar superfield µ (where by superfield I mean a function of x and θi). Any product of θ’s that contains more than four terms is then by definition zero. I can therefore write the most general scalar function as the sum i 1 Φ(ˆ x, θ) = i√2θγ¯ ψ (θγ¯ θ) + (θθ¯ ) S − 5 − 2 5 M 2 N 1 ¯ µ ¯ ¯ i + (θγ5γµθ)V + i(θγ5θ)[θ(λ + ∂ψ/ )] 2 √2 1 1 (θγ¯ θ)2[ ]. (2.27) − 4 5 D − 2S ˆ ˆ If Φ transforms like a scalar and Qa transforms like a spinor then QΦ transforms as a spinor. Clearly a transformation that multiplies or removes a θ to each term in Φˆ satisfies this condition. That is

e[iαQ¯ ] 1 + iαQ¯ ≈ ∂ = 1 + i α¯ + iα¯∂θ/ . (2.28) ∂θ¯ This set of transformations completely covers the Lie group around the identity and all other transformations that obey the correct transformation properties can be expressed as product of such infinitesimal transformations. Therefore Eq. (2.28) is sufficient to generate the entire set of supersymmetric transformations. Applying Q to Eq. (2.27) I can match powers of θ to get the following set of transformations for the components of the superfield

δ = i√2¯αγ ψ , (2.29) S 5 α γ α γ αV µ γ ∂/ α δψ = M i 5 N i µ 5 S , (2.30) − √2 − √2 − √2 − √2 δ =α ¯ λ + i√2∂ψ/ , (2.31) M δ = iαγ¯ λ + i√2∂ψ/ , (2.32) N 5 δV µ = iαγ¯ µλ + √2¯α∂µψ , (2.33) − 1 δλ = iγ αD [∂,/ γ ]V µα , (2.34) − 5 − 2 µ δ =α ¯∂γ/ λ . (2.35) D 5 In order to create a supersymmetric theory the action must be invariant under supersymmetric transformations. As seen in the earlier discussion it is sufficient for a theory to be invariant under infinitesimal supersymmetry transformations. Since I integrate our Lagrangian density over all space, a theory is still supersymmetric if the Lagrangian density transforms as a total derivative under supersymmetry transformations. Consider Eq. (2.35). One sees that the term of any Lagrangian, that is, the coefficient of (θγ¯ θ)2, will transform the correctD way. Setting λ = = 0 in 5 D 20 Eqs. (2.29-2.35) I see that the coefficient of the term + i also transforms as a total derivative. Such a field one calls a left chiral superfieldM forM reasons that will not be explained here and these coefficients are called “F” terms. The product of any left chiral superfields will also give a left chiral superfield that also has an F term. This is obvious from the additivity of the left chiral derivative (which again will not be explained here). I can project out F terms and terms by integrating over two and four Grassmann variables respectively. I can alsoD limit the types of F terms and terms that can appear in the Lagrangian if I insist that our theory is renormalizable.D A general noninteracting supersymmetric theory can then be written as

S = d4xd4θ + d4xd2θ (2.36) LD LF Z Z and from renormalizability one ﬁnds that the most general form of is LD N = ˆ† ˆ (2.37) LD Si Si n=1 X whereas the most general form of is LF 2 ∂f 1 ∂ f ¯ F = i i ψiPLψj L − ∂Sˆ ˆ F − 2 ∂Sˆ ∂Sˆ i i = i,j i j ˆ= X S S X S S 2 ∂f 1 ∂ f † + i † ψ¯ P ψ . ˆ Fi − 2 ˆ ˆ i R j i ∂Si ˆ= i,j ∂Si∂Sj ˆ S S = X X S S (2.38)

This master Lagrangian gives us an interesting phenomelogical prediction of supersymmetry. Consider D for the case of a single superﬁeld. The D term can be evaluated to be L

4 µ 1 d θ ˆ† Sˆ = ∂ †∂ + ψ¯∂ψ/ + † . (2.39) SL L µS S 2 F F Z I see that every particle has a corresponding superpartner! It turns out that this feature is how the quadratic divergences of the higgs masses are cancelled out. This justiﬁes our motivation for supersymmetry in the ﬁrst place.

2.2.3 Examples of supersymmetric theories Our master Lagrangian is still missing some key features of a theory that can plausibly describe anything to do with the real world. Such a theory would need spontaneous symmetry breaking to give particles a mass, they would also need supersymmetry breaking terms to ensure that the supersymmetric partners of Standard Model do not have the same mass as Standard Model particles (someone would have observed them otherwise!) and of course the fields in the theory would need to interact. Recall that to create an interacting theory in standard quantum field theories one makes the fields invariant under a space time dependent “gauge transformation”. To generalize this to supersymmetric theories one needs to make the gauge transformations superspace-time dependent. The result of this is that every gauge boson has a fermion

21 superpartner called a “gaugino” [39]. In order to make the masses of the superpartners much greater than their Standard Model partners I need to add mass terms to the Lagrangian for the superpartners. However, doing so breaks supersymmetry. Even though supersymmetry is a broken symmetry of nature in any theory that describes the real world, nonetheless the quadratic divergences that plagued the Standard Model are still canceled out if one merely adds mass terms to the Lagrangian [40]. Finally, electroweak symmetry breaking is done the same way as in the Standard Model, however, I have the added complication that to get a supersymmetric theory I require at least two Higgs bosons. This is another prediction of supersymmetry. A theory that is just like the Standard Model, save for the fact that it has two Higgs bosons and every particle has a heavier superpartner, is known as the “minimally supersymmetric Standard Model" (or MSSM). The second Higgs doublet is necessary for anomaly cancellation. Another model I will consider is the NMSSM to which the Lagrangian contains the superpotential [41] κ W = λ ˆHˆ Hˆ + ˆ3 S 1 · 2 3 S + y Qˆ Hˆ Uˆ c + y Qˆ Hˆ Dˆ c + y Lˆ Hˆ Eˆc (2.40) u · 2 d · 1 e · 1 ˆ ˆ where is a singlet under SU(3) SU(2) U(1), Hi are the superfields that generateS the two higgs fields, Qˆ corresponds× to× your quark doublet, Lˆ corresponds to the lepton doublet and Uˆ C , Dˆ C and EˆC correspond to the right handed fermion singlets. The 125 GeV Higgs mass is difficult to reproduce in the MSSM with light superpartners [42, 43, 44, 45, 46], consequently the baryogenesis window is extremely narrow in the MSSM [47]. I will therefore in this research be considering the NMSSM. I conclude with a comment on the large non conservation of baryon number in supersymmetric models which tends to sharply contradict observation due to strict bounds on the violation baryon and lepton number conservation. This is solved by proposing a new Z2 symmetry known as R parity which forbids tree level baryon and lepton number violation. Specifically, R parity is defined by

3B L+2s P = ( 1) − (2.41) R − where B,L and s are the baryon number, lepton number and spin respectively. All Standard Model particles has R parity of +1 whereas the super partners have R parity of 1. The result of this is that any interaction must conserve the number of supersymmetric− particles. This means that the lightest supersymmetric partner is completely stable.

2.2.4 A brief summary of supersymmetric cosmology Supersymmetry has incredible cosmological benefits. There are many new flat scalar directions in field space which can facilitate inflation [48, 49, 50, 51, 52, 53, 54, 55]. Under R parity the lightest supersymmetric partner is completely stable which provides a very good dark matter candidate [34, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. As for baryogenesis, supersymmetry naturally provides two mechanisms: during inflation one has the Affleck-Dine mechanism [66] which won’t be reviewed here (but is reviewed in my book). The other mechanism is electroweak baryogenesis. Supersymmetry facilitates this in a number of ways. First the presence of coloured

22 scalar particles (stops) in the plasma provides a strong source of friction that slows the expansion of bubble walls [67] – which is ideal for baryogenesis. Second the presence of new scalar particles that interact with the Higgs potentially catalyse a strongly ﬁrst order phase transition in contrast with the Standard Model [25]. Finally there are many new sources of CP violation through new coupling terms. I will examine how well electroweak baryogenesis is supported by simple supersymmetric models in later chapters.

2.3 The relaxion mechanism: another solution to the hierarchy problem

I previously discussed a dichotomy of explanations to explain fine tuning - the anthropic principle can be invoked if there is an anthropic reason why an observable is fine tuned otherwise one needs new physics. If the anthropic principle is invoked then one is conceding that something about our world cannot be explained by examining current local physics. There has been another very clever category of explanation that has been proposed recently where the anthropic principle is not invoked to explain the hierarchy problem but still the explanation is not found in current local physics as the explanation is cosmological. This proposal is known as the relaxion mechanism [5] where the Higgs couples to an axion field with a periodic potential. These periodic barriers form at the same time when electroweak symmetry breaking occurs and large Hubble friction during inflation prevents the Higgs vev from rolling past the first minimum. To determine whether this is indeed a solution to the hierarchy problem I invoke a rigorous Bayesian model comparison framework. The Bayesian framework has a fine tuning penalty and Occam’s razor automatically built in to the formalism. I find that, unfortunately, the simplest versions of relaxion method do not solve the hierarchy problem since one requires an unnaturally small Hubble parameter during inflation to allow periodic barriers to form. Remarkably this problem is so acute that the Standard Model turns out to be less fine tuned. Nonetheless relaxion is an interesting scenario and perhaps more exotic inflationary scenarios can overcome this difficulty.

2.3.1 Declaration for thesis chapter 2

Declaration by candidate In the case of the paper present contained in chapter 7, the nature and extent of my contribution was as follows: Publication Nature of contribution Extent of contribution Worked on analytic derivations throughout. 2 Wrote a number of sections and edited all sec- 25% tions. Contributed to discussions throughout. The following coauthors contributed to the work. If the coauthor is a student at Monash, their percentage contribution is given:

23 Author Nature of contribution Extent of contribution Involved in conceptual development and con- Csaba Balazs tributed to discussions throughout Developed the concept, worked on analytic derivations and implemented numerical scan. Wrote Andrew Fowlie and edited sections. Contributed to discussions throughout Heavily responsible for conceptual development. Luca Marzola Involved heavily in writeup. Contributed to discussions throughout. Martti Raidal Contributed to discussions throughout. The undersigned hereby certify that the above declaration correctly reﬂects the nature and extent of the candidate and co-authors’ contributions to this work.

2.3.2 Published material for chapter 2: Naturalness of the relaxion mechanism

26 JHEP08(2016)100 0. b,c | Springer QCD θ | June 24, 2016 August 1, 2016 : August 17, 2016 : : February 25, 2016 : and Martti Raidal Revised , Accepted b,c 10.1007/JHEP08(2016)100 Published Received doi: martti.raidal@cern.ch , Luca Marzola Published for SISSA by a [email protected] , Graham White, a [email protected] , . 3 Csaba Balazs, 1602.03889 a The Authors. c Beyond Standard Model, Higgs Physics

The relaxion mechanism is a novel solution to the hierarchy problem. In this , [email protected] shatters the plausibility of the QCD relaxion model as it typically yields | QCD Rävala10, Tallinn 10143, Estonia Institute of Physics, UniversityRavila of 14c, Tartu, Tartu 50411, Estonia E-mail: [email protected] ARC Centre of Excellence forSchool of Particle Physics Physics, andWellington Astronomy, Monash Road, Melbourne, University, Victoria 3800,National Australia Institute of Chemical Physics and Biophysics, b c a θ Open Access Article funded by SCOAP Keywords: ArXiv ePrint: is favoured by huge Bayes-factorsHubble as parameter the is relaxion less models thanthat require the relaxion fine-tuning height models such of could that the solvecosmology the periodic the is barriers. hierarchy at problem, Thus, odds whilst we with find we their confirm that plausibility. their unconventional Constraints upon e.g., thefactors vacuum such energy that during| relaxion relaxation, however, models shrink are theFinally, only Bayes- we slightly augment favoured. ourinflationary models observables Including with from scalar-field the BICEP/Planck. inflation bounds We and find on consider that, measurements all of told, the Standard Model Abstract: first statistical analysis of theQCD relaxion mechanism, and we a quantify non-QCD the relaxion relativewhich plausibility model includes of an versus a the automatic penalty Standardbetween for Model the fine-tuning. weak with and Bayesian We Planck statistics, find scales, relaxion that models in are light favoured of by colossal the Bayes-factors. hierarchy Andrew Fowlie, Naturalness of the relaxion mechanism JHEP08(2016)100 is i a (1.1) h , and an h , 4 λh ] 11 + 1 φ ] that solves the hierarchy i 4 a – h 2 2 m − 8 φ f 10 cos i h h – 1 – 3 b 9 are coupling constants of dimension mass, m f − 4 2 h 7 and 17 6 b φ ], the relaxion mechanism cancels them with the vacuum i m 9 a h , 8 2 κ 7 3 Λ − 9 2 ∼ 11 µ 2 15 m = 14 1 ∼ , govern the weak scale via the scalar potential [ V ] recently proposed a relaxation mechanism [ 2 φ in relaxion models 1 µ | ] by utilising the dynamics of an axion-like field, dubbed the relaxion. In the 8 – QCD 5 θ 4.2.2 Calculation of (electroweak and QCD) observables 4.1.1 Calculation of observables 4.2.1 Relaxion physicality conditions | The ingenuity of the relaxion mechanism is that the dynamics of the relaxion field 5.1 Evidences 5.2 Observables 4.1 The Standard Model with scalar-field inflation 4.2 Relaxion models 3.1 3.2 Finite-temperature3.3 effects Baryon asymmetry where, because of quadraticcut-off corrections, we Λ, expect i.e., that the masses should be close to the ing supersymmetric particles [ expectation value (VEV) of a relaxion field. ensure a precise cancellation withoutWithin patent fine-tuning the of relaxion parameters or paradigm,axion-like initial interactions field, conditions. between a complex Higgs doublet, Graham et al. [ problem [ Standard Model (SM), the hierarchyweak problem scale. originates Whereas from supersymmetry quadratic cancels corrections them to with the new quadratic corrections involv- 6 Discussion and conclusions 1 Introduction 5 Bayesian analysis 4 Description of models 3 Analysis of relaxion potential Contents 1 Introduction 2 Bayesian fine-tuning JHEP08(2016)100 ] . . φ 3 12 (1.5) (1.2) (1.3) (1.4) ] utilised 0, however, 11 . This is the < ) 5 φ ( ]. In this method- is a dimensionless = 0, the barrier is 2 h i κ 22 m h , – h 15 πf = 0, breaking electroweak , we critique Jaeckel’s ap- + 2 i 6 2 φ h h , 0 → φ < ) , φ, φ i ( can be neglected. . a 2 h = 0 i h ¨ φ a m κ ]. In section 3 b h 2 . ∂φ m − ∂V P ) 14 m 2 , φ + ( µ M f λ 2 h ˙ 13 – 2 – 2 φ m ≡ & H − ) i i φ h h in the relaxion potential ( h q 0 otherwise. h + 3 2 h 2    . In the unbroken phase in which ¨ φ h m = πf i h h ] stress that it could require a fine-tuned inflationary sector if 11 and calculate their Bayesian evidences in section 4 is the VEV of the Higgs field, which is a function of the relaxion field i h h ]: is the Hubble parameter. If Hubble friction is substantial, the relaxion field could 10 H Ostensibly, the relaxion mechanism ameliorates fine-tuning associated with the weak We require, inter alia, that the relaxion field dissipates energy as it rolls or else it would Let us label the co-efficient of last decade in the context ofology, fine-tuning in in light supersymmetric of experimental models data [ in about a the model weak with scale a and BayesianWe inflation, evidence. describe we our We update further models our analyse belief — theinflation minimal relaxion relaxion — potential models in in section and section the SM augmented by scalar-field inflation. Unfortunately, there ismeasure no of consensus fine-tuning in ortheir high-energy about prominence. physics the In on logical earlier the work foundationsdeveloped appropriate to of judge a fine-tuning fine-tuning new in arguments, relaxion formalismcommon despite models, Barbieri-Giudice based Jaeckel et style on al. measures [ theirproach [ and intuition, instead whereas advocate Raidal a Bayesian et methodology, al. discussed numerous [ times over the where be in a slow-roll regime in which the acceleration scale, but Raidal etthe al. relaxion [ is the QCD axion because of constraints upon the Hubble parameter during this is ensured byLagrange equation Hubble for friction the relaxion —e.g. field a ref. originating [ term from the analogous expansion to of the a Universe friction see term in the Euler- Thus this mechanism could result in have sufficient kinetic energy to surmount the periodic barriers. In the relaxion paradigm, down and the relaxion fieldthe slowly rolls potential down is a linear suchsymmetry potential. that (EWSB) Once the and Higgs raising the fieldthe periodic acquires relaxion barrier. field a The in VEV, now-raisedresults a periodic in minimum. barrier a If traps weak the scale relaxion of field about cannot roll past a local minimum, it If the Higgs VEVfield is with non-zero, barriers the separated cosine by 2 term provides a periodic barrier for the relaxion coupling, and for convenience, such that the Higgs VEV may be written the VEV of a spurion field that breaks a shift symmetry JHEP08(2016)100 (2.2) (2.3) (2.4) (2.5) (2.1) ) is our prior ], and arguably M | 72 p – ( p 70 p with a brief discussion . d 6 Y — and ) p , M ) | ) , ) ) p D Prior odds D b a , ( | | ) ) p × ] and the principle of Occam’s razor b a M b M a · | | ) 22 M M M M ( ( – ( p ( D D ( ( P ) is a so-called likelihood function — the P P P 15 p p p . From the prior odds, we can calculate the M, ≡ | ≡ ≡ M M, D – 3 – | ( p D ( ] and cosmology see e.g. ref. [ Z p can be calculated by Bayes’ theorem and marginali- . 69 p – M ) = Prior odds , one begins by quantifying one’s relative degree of belief 26 b M ]. This methodology is becoming increasingly common in Bayes-factor | M Posterior odds 25 D – ( p and 23 a Posterior odds = Bayes-factor M Z ≡ ]. We briefly recapitulate the essential details. 73 ) is one’s prior belief in a model represents experimental data e.g., in this work data from BICEP/Planck. The M are the model’s parameters, ( D p P The likelihood function is uncontroversial as its form is dictated by the nature of an The Bayesian framework enables one to assign numerical measures to degrees of belief. experiment and it is a criticalform ingredient in of Bayesian the and frequentist prior statistics. density,we however, The pick remain role and contentious objective issues. priorsrespect In that rational as reflect constraints much from as our e.g., it knowledge symmetries. is or possible, ignorance about a parameter and sation, where probability density of ourdensity observed for data the given model’s parameters parameters probability densities, where the probability densitiesdences. in The question evidence are for known a as model Bayesian evidences or just evi- prior odds and the posterior odds are related by a so-called Bayes-factor: By applying Bayes’ theorem, it can be readily shown that the Bayes-factor is a ratio of posterior odds — one’sdata, relative degree of belief in the models updated with experimental where where in the models, prior to considering any experimental data. This is known as the prior odds, light of data seehigh-energy e.g. physics ref. see [ e.g.captures ref. the [ essence ofsee the e.g. hierarchy ref. problem [ [ To assess two models, of our findings. 2 Bayesian fine-tuning Bayesian statistics provides a logical framework for updating beliefs in scientific theories in first statistical analysis of a relaxion model. We close in section JHEP08(2016)100 ) ]. σ 22 (2.7) (3.1) (3.2) (2.6) – . The 15 φ , . max p i d φ . Q ih 4 a λ p )) h 1 2 λh d p , based on the fraction κ ( F + v Q + φ 2 R − i µ a . . v h ( − p 2 θ d 2 r m R i Q h = − h = ), p )) i d + p 1 h F ( 1.1 φ f max h i v Q /κ h 2 R h − vs. m v cos ( i θ h i p h ) is reminiscent of the Bayesian evidence if one R ]. Raidal et al. employed Barbieri-Giudice style and a 3 b – 4 – 3 b d h 11 = m m 2.6 Q 2 fκ v ) f − p ≤ 2 2 ) d i M p h V a ) = | ( h v Q = 0 result in a transcendental equation, p 2 V ) φ ( , that predicts a weak scale less than that observed: i φ/f p p ]. Whilst intuitive, such measures lack a logical foundation, a M · ≡ κm f h | i 4 ) κ , one finds that if there is a solution, it lies in the interval a 1 sin( 14 F p p , h ] includes a general discussion of parameterisation in the context of fine- ∂V/∂h − ( − 1 κ p 6 b 12 13 2 2 M, R µ m | where v q ( = p ] and by Raidal et al. [ max − = 0 and i R V 3 b 12 h m = ], for simplicity we consider only linear terms in the relaxion field = Z i ≤ h 74 ∂V/∂φ h min i , and omit a measure for the volume of parameter space, i.e., a prior. In other words, ≤ h h v h Curiously, Jaeckel’s measure in eq. ( Before closing, we briefly discuss attempts to quantify fine-tuning in a relaxion model We calculate Bayes-factors for the SM augmented with scalar-field inflation (SM + However, Jaeckel et al. [ min 1 i h h tuning. equations By graphing as in figure Let us further analyse the relaxion potential in eq. ( As in ref. [ The differences are thatscale, Jaeckel et al. pickby a following their step-function noses for and attempting the toet likelihood formulate al. fine-tuning for in create the a logical an weak manner, ersatz Jaeckel Bayesian evidence. 3 Analysis of relaxion potential considers measurements of thesation weak factors scale, for especially the if priors, one writes (unnecessary) normali- This measure contrasts with Barbieri-Giudice measuresparameter in space that it rather considers than aone’s a model’s choice entire single of point parameterisation in or it. measure for Jaeckel’s the measure, parameter however, space. depends on by Jaeckel et al. [ measures of fine-tuning [ though emerge inJaeckel intermediate et al. steps developed in a novelof a measure a of calculation electroweak model’s fine-tuning, of parameter space, the Bayesian evidence [ versus relaxion models. Thefactor final step to — find thatBayes-factor one’s of is updating posterior independent one’s odds of priorfor any — odds the prior with is parameters choices a of left — Bayes- the it to models is the in in question. reader. fact a That functional of is the not priors to say that a JHEP08(2016)100 (3.3) (3.4) (3.5) π 4 min i φ h ), marked by a ) = 1, π 3.2 3 φ/f max 0 i φ h < πf ) φ ( 2) ) must be real, 2 h , is a sharper bound for / ) = 1, m . 3.2 φ/f min 1 + 1 i π ) equals plus one, matching 2 10 h n -axis is shifted such that correct − h 10 3.1 2 i φ φ/f f | ≤ . If the square-root is imaginary, h = (2 2 2 i h i h i + h 6 b QCD a min h θ /κ h i | 2 ) . max 2 m m φ 1 i π h φ/f i φ a 3 b h h ≤ κm m > Correct Critical value of relaxion field sin( fκ Lower limit from quadratic root whenFirst sin( solution, Upper limit from next sin( 4 2 f 2 max i if i and 6 b a – 5 – φ h h 2 m 0 i 2 shifted by arbitrary amount a 3 b µ h κm 2 ). A necessary (though not sufficient) condition for m φ/f 4 i + (vertical brown dashed lines) close to the solution. EWSB m i h h a π f h f 2 min κ i ≈ (4 h π / h − 3 b λ min 2 i m h h ). The solution (red star) lies in the interval in eq. ( = i 3.1 min a i h π 2 φ κ − h ), that is, 1.2 Graphing the left-hand side (blue line) and right-hand side (green line) of the transcen- is the smallest integer such that π 3 in eq. ( 1 0 6 5 4 3 2 1 2 − occurs at small multiples of 2 n . If the barrier height is substantial, the lower bound reduces to the approximation − − i | h In other words, the relaxion mechanism ensures that the weak scale is independent h max i QCD h θ the maximum of themaximal. left-hand side, If required, and one the canIn improve subsequent this some point interval cases, with at piece-wise the expressions which positive byh the quadratic graphing. root, latter similar is to again thatfor for of quadratic corrections to the Higgs mass from a cut-off or unknown high-scale physics, where there are no solutions, otherwise,must there be are identified zero numerically. to Thelie four interval solutions between results inside the from the point recognising interval, at which that which a the solution right-hand must side of eq. ( This implies that solutions to the transcendental equation is that the root in eq. ( | is broken once the critical value of the relaxion field isand surpassed (vertical red dot-dashed line). Figure 1. dental equation in eq.brown ( pentagon and a greenside equals diamond. plus In one unusual (not cases, shown) may the be second a point sharper at bound. which The the right-hand JHEP08(2016)100 is 3 b , ] by (3.8) (3.9) (3.6) (3.7) m (3.11) (3.10) 78 max i h . Instead, GeV, and . π 9 1 ≤ h 10 is substantial, & ) and the second /f is simple see e.g. mod 2 i 2 min 2 f | φ i minima 3.1 h i m h /f h h i h κ QCD ) is as close to zero as φ inside this interval, but θ h | if otherwise, , i ≡ — as

h | η φ/f h fπ/λ. i 2 ··· a , i . h | QCD h + κ h θ π | i + GeV + 2 h to h √ 16 1 and π /κ 2 min − 2 − i − h 2 10 m h on ' π/ q 2 min i fπ i i a = 18 3 b ≤ h h a

− h h /f m – 6 – i λ , resulting in κ fκ 10 2 φ

··· max , | i |h ≡ h + QCD i × = Λ h | QCD h

2 min fπ θ i i ∼ κ | i ≤ h h . a h b √ h h = arcsin QCD i λ κ b 3 ]. Prima facie, the expression for | 2 θ m a | fm m h i ≤ h 79 a − ). Thus achieving a small weak scale requires a tiny spurion [ obtainable occurs at the minimum of the right-hand side of h 2 QCD 1 which minimises the right-hand side of eq. ( |

) where θ min 10 3.4 | i κ − h √ h QCD 10 θ and any corrections affect the position of | ]. (1 + ). arcsin arcsin m/ . 2 . Such a small coupling may be natural as it breaks a shift-symmetry see    | µ 77 3.1 min = , P i = in relaxion models h 76 M | QCD | θ ]; however, there may be issues due to the gauge symmetry at the basis of the | ≈ h 1 TeV in eq. ( , ≪ minima QCD 75 i i θ QCD ' ), such that, if there is a solution at that point, sin( h | a ], max θ h h | i m 3.1 h 80 and The minimum Unfortunately, if the relaxion is the QCD axion, we expect that barrier height h min η where line is never less than theof first line. The terms represented by the ellipses are higher powers possible. Thus we find that by utilising eq. ( eq. ( Numerically, however, this cannot be usedthere for is calculating a breakdown in numericalwe precision find in the expressions principal such solution as for 3.1 Let us investigate whether aexplaining relaxion model mightref. resolve [ the strong-CP problem [ where we impose anpick experimental limit onVEV, the QCD decaye.g. constant, ref. [ construction [ solving for the VEV of the Higgs field insideconnected the to interval QCD, may such be that unnecessary. The Higgs mass not the interval itself. The width of this interval is typically small such that numerically is independent of the Higgs mass, solving the hierarchy problem. In fact, the Higgs VEV is bounded by an expression that JHEP08(2016)100 , π + 1) and we n i arcsin 1 = h (2 h ≈ | ≈ ] demonstrated 84 ) originates from QCD φ/f θ | 3.10 ), which are reasonable for and not ). As the general expression . Eq. ( nπ 3.11 η 3.6 2 ≈ , such that min max i φ/f h from eq. ( -folds after EWSB. Furthermore, the flat would occur at the minimum of the right- e h | max κ i . If that minima occurs, however, outside the h – 7 – √ QCD , it is impossible. In that case, the right-hand h π i θ | h m/ h | ≈ ]. However, since this affects only the heights of the = 82 QCD θ and neglect all powers of | ] considers finite-temperature effects in an alternative relaxion mecha- minima i 81 h h ) if that minima occurred at 0 and not ): ]. If EWSB is delayed until a late time (corresponding to a lower 3.1 83 1.1 | ≈ Finite-temperature effects could, however, non-trivially affect the relaxion ) is monotonic inside the interval for the possible solutions for GeV. At such a high temperature, electroweak symmetry could be easily 2 QCD 10 3.1 θ | restored, with the effect of further hindering the viability of the model. regions of thetemperature zero-temperature effects. inflaton potential are stronglyWe modified find by that finite- theder reheating 10 temperature in our relaxion models is typically of or- Finite-temperature corrections tothe the potential effective (increasing potentialof the would EWSB gradients alter of [ thetemperature the shape after slopes), of reheating), possibly itthe delaying could requirement the constrain that onset when it inflation lasts at must least start 50 through Non-perturbative effects responsible for the inducedare effective temperature potential of dependent the [ relaxion barriers and not theirsubstantial, spacing it and is since unclear whether Hubble finite-temperatureof friction effects the would during impact relaxion inflation the mechanism. is viability typically We note, however, that ref. [ • • • 2 2. This is confirmed in our numerical analysis. 3.3 Baryon asymmetry We observe a significant baryonthat asymmetry generating this in asymmetry our — Universe. baryogenesis — Sakharov would require [ a departure fromnism. thermal Clearly all the mentionedconstraints finite-temperature on effects the relaxion have model thefollowing parameter projects. potential space, to to impose an further extent that will be quantified in 3.2 Finite-temperature effects In this paper andtemperature. in the literaturepotential so in far, eq. ( the relaxion mechanism was analysed at zero π/ consider the right-hand side evaluatedis at rather complicated, we applyphenomenologically the viable approximations points. in In eq.the ( fact, second phenomenologically regime in viable which points are always in considering that the minimumhand possible side of eq.such ( that interval for the possibleside solutions of for eq. ( We expand to first order in JHEP08(2016)100 (4.1) (4.2) (4.3) — in λ and ] also appear 2 µ 87 , , ]. Mixed inflation s 86 A 97 . Scenarios in which the σ and 3 ) s σ n , in a potential , σ r λ SM + | s ) , 4 σ 4 σ | ) ,A | σ s h σ s | λ λ ,A r, n + s ( + ]. Ultimately, we wish to find whether the 2 p 2 σ SM + · | 96 2 σ r, n | h ) ] after the electroweak phase transition which – ( | s – 8 – σ 2 m p 88 · µ 1 2 89 ,A , s = = 74 , and quartic coupling, h σ SM + SM) 2 σ V | | V , r, n . We denote this model by SM + m Z Z Z M M M ( ( ( -fold -folds of inflation occur during or immediately after EWSB ), a QCD relaxion model and a general relaxion model. For e model, the evidence approximately factorises into a factor for p p p e σ N σ ≈ = = Z violation, and baryon number violation. In the relaxion paradigm, -folds, CP e and C ]. Novel mechanisms that invoke multi-step phase transitions are also ruled out since 85 Note that in the SM + One could in principle have a multi-step phase transition that departed from the SM vacuum for 3 as the measurements areservables independent do not and affect model thewhich parameters weak results that scale in and affect quadratic vice-versa, corrections inflationary with to ob- the the exception inflaton of massbaryogenesis the and and cut-off, the later Λ, returned Higgs to mass. it but this somewhat undermines the motivation for relaxion models. the weak scale and a factor for the inflationary observables, and the number of mixed inflation, a canonical modelis of described scalar-field by inflation an see inflaton e.g. mass, ref. [ The SM Higgs sectorthe is SM described Higgs by potential, two bare Lagrangian parameters — and a cut-off at which the bare parameters are specified, Λ. We augment the SM with other relaxion models, seerelaxion e.g., mechanism ref. ensures [ thatversus the a SM. relaxion In model eachfrom model, is a all favoured cut-off, scalar-fields by Λ, receive quadratic which the corrections lies Bayesian to close evidence their4.1 to masses the Planck scale. The Standard Model with scalar-field inflation 4 Description of models We apply Bayesian model comparison toscalar-field three inflation models: (SM the + SM augmented with single-field to be incompatible with theby relaxion mechanism the because already of theare heavily further also constrained constraints exponentially implied dynamics VEVmeans of suppressed that [ the any subsequent inflaton. baryogenesisof scenario baryon and would Finally lepton have number to weakour violation. rely sphalerons conclusions As on about we a the shall different viability see, source of these difficulties inflation would in strengthen the present framework. however, the final 50and or inevitably wash-out so any potential netref. [ baryon number generated in thisthe process fields see must e.g. be ininflaton itself the could final generate the SM required vacuum baryon at asymmetry see the e.g. end ref. of [ inflation. equilibrium, JHEP08(2016)100 ] 99 and (4.7) (4.8) (4.9) (4.4) (4.5) (4.6) i -folds -folds e e σ . Inflation finishes σ ). In the first model, ) ) σ 1.1 . σ ( ) ( ]: i 00 . The number of σ V i V 98 ( σ 2 P π π 8 M ) = 1. The number of σ, f = 4 , ≡ d , σ , and the ratio of scalar to tensor 2 ( r ) | ) s 3 ) ) 2 ) σ σ n i σ i ( ( Λ ( 0 σ σ η ( β ( 0 V V V V + f | σ i 2 , σ π ) and µ ) and i Z 2 i P 3 σ such that σ ( π 2 2 ( P 128 – 9 – M λ η 8 g 2 f 3 6 V P 2 M σ − − 1 ) ) M s σ ' = σ ) + 2 ( i ( 0 = = 2 Z σ V s V ( H -fold M A e 6 N 2 P π − ]. We include a quadratic correction to the inflaton mass — 50. The spectral index, M 16 & = 1 ≡ . The normalisation of the scalar perturbations is arbitrary and 100 boson — which represents the weak scale — is calculated in the s s ) n n σ Z ( -fold e or N r , may be written to first order in the slow-roll parameters as see e.g. ref. [ r ], whereas in the second model, the relaxion is indeed the Peccei-Quinn axion. 78 is a loop factor. The QCD phase is an input parameter. β The mass of the , with numerical methods. f 4.2 Relaxion models We consider two relaxion models describedwe by do the potential not in identifyproblem eq. [ ( the relaxion with the Peccei-Quinn axion that solves the strong CP usual manner, where varies in the literature.experiment see For e.g. comparison ref.to [ with include a Planck dominant data, quantumare contribution we tree-level. to pick We fine-tuning solve that — forσ but of the otherwise the inflaton our field Planck formulas at the beginning and end of inflation, but cannot affect The normalisation of theHubble potential parameter, governs the amplitude of scalar perturbations and the For comparison with measurementsinflationary from observables Planck via the in so-called our slow-roll statistical parameters analysis, [ we calculate determines the inflatondesired field should at be theperturbations, beginning of inflation, 4.1.1 Calculation of observables where a prime indicates aonce derivative the with inflaton respect to fielddesired the before reaches inflaton inflation a field ends value (and in the case of the relaxion, after EWSB), JHEP08(2016)100 ]. 101 (4.14) (4.10) is achieved P M ≪ H . 4 . σ ) 3 σ / (barriers form) (4.12) λ (barrier heights) (4.16) 1 i 1 4 (vacuum energy) (4.11) , is not an input parameter. We, a h + 3 / 3 -fold 2 e σ 1 N , m m , b (classical beats quantum) (4.13) 1 3 σ (inflation lasts long enough) (4.15) m + , such that the cosmological constant vanishes ρ 2 -folds, e σ min( – 10 – 2 2 P m ]: 1 2 1 < M + ]. This introduces only four parameters: four couplings 2 = 0. Thus, low-scale inflation with σ 11 ) by the most general renormalisable single-field inflaton 3 3 ρ m . This implies that the potential must be fine-tuned such κ i 2 i 4 2 P 2 m 1.1 h a µ i 2 )+ h h m a M κ = 3 b b 2 2 h H r κ µ m V ,... ] with an inflaton field i ≪ & ∼ > < m σ ρ 11 h 3 2 f P ( . H < m 2 4 H P V M -fold µ e 2 i M a N H h κ = 0) = ≪ | σ ) ( i ). Because all models that we consider suffer from this fine-tuning problem, V σ = 0, as in Raidal et al. [ h ( σ ,... V i | The cosmological constant poses an infamous fine-tuning problem see e.g. ref. [ For a necessary epoch of low-scale inflation after the relaxion mechanism, we extend σ h , quantum corrections and contributions from spontaneous symmetry breaking i.e., ( 0 We assume that a first epochcosmological constant of later inflation cancelled is when provided the by Higgs the and slow-rolling relaxion relaxion fields fields, acquire and VEVs, and We assign zero likelihood to a point that violates the resulting condition, Graham et al. also list the conditions There are parameter points forfield which in the a back-reaction minimum. to EWSB Ifwould fails that be to were in trap the severe the case, disagreement relaxion for the with relaxion a observations. mechanism successful would Graham relaxion fail et mechanism and al. [ the list point conditions required V fine-tuning penalties fromtios the of cosmological Bayesian constant evidences.relaxion would We mechanism ensure approximately by that cancel applying the conditions in second on ra- epoch the of Hubble4.2.1 inflation parameter cannot during spoil inflation. Relaxion the physicality conditions that In almost all knownrequires models, extreme agreement with fine-tuningρ measurements between of a the cosmological bare constant cosmological constant in the Lagrangian, origin, in the potential —furthermore, the tune desired a dressed number vacuum of energy, in the vacuum, i.e., provided We suppose that pre-inflation multi-field dynamics dictate that inflation begins at the the relaxion potential inpotential eq. see ( e.g. ref. [ JHEP08(2016)100 . ]. 79 (4.17) (4.18) (Py)- -boson ]. The model. Z σ 11 MultiNest , approxi- | -boson [ -boson, ] for Monte- Z Z 01 in QCD . θ | 107 , the axion decay , a f 106 is an input parameter. | ]. QCD θ 108 | , we assign a likelihood of zero, since it model and relaxion models with ) (though delta-functions were first inte- i would reverse that transition, destroying σ h b h b 2.5 , was a product of at most five factors: g 4.2.1 1 = – 11 – Z H < m H > m M ]. 79 ), from which we calculated the mass of the 3.2 ). In the non-QCD relaxion model, ]. 1 3.1 is related to the height of the periodic barriers. The barriers result ], which utilises the nested sampling algorithm [ b This requires two ingredients: a likelihood function and a set of priors. m : a likelihood function for measurements of the mass of the : if a relaxion model (i.e., a point in a relaxion model’s parameter space) 105 4 – 102 : a likelihood function for the experimental lower-limit on : a likelihood function for the experimental upper-limit on [ (see section | QCD constant, approximated by a step-function see e.g. ref.theta [ mated by a step-function [ conditions violates physicality conditions in section would be in stark disagreement with observations. decay weak-scale In the SM, thisrelaxion is model, approximated this by is impossible,mass a as as delta-function there a and is function integrated no of by analytic the hand. expression Lagrangian for parameters. In the a θ | We utilised importance sampling, picked 1000 live points and a stopping criteria of 0 • • • • 4 We calculated Bayesian evidences forMultiNest our SM + Carlo integration in Bayesian evidencesgrated in by eq. hand). ( Our likelihood function, summarised in table and The calculations for the inflationary observables were identical to those in the SM+ 5 Bayesian analysis We calculated the VEVs ofbisecting the the Higgs and interval in relaxion eq. fields with ( numerical methods based on from a phase-transition to a QCDQCD condensate dynamics). (or a A condensate Hubble associatedthe with parameter barriers similar see non- e.g. ref. [ 4.2.2 Calculation of (electroweak and QCD) observables latter condition is unnecessary aswhether we a solve the solution potential exists. withinflation numerical To must, methods, avoid however, checking satisfy, destroying the periodic barriers, the second epoch of where, as before, that this epoch provides an acceptable Hubble parameter, as described in ref. [ JHEP08(2016)100 ; ; adding , reflecting | weak-scale decay QCD θ | -folds is already a , from Planck and e s A . All massive parameters 2 . 2 , the ratio of scalar to tensor s n . The main difference between the priors 2 because the number of Λ ] Dirac in SM, Gaussian in relaxion ∼ and BICEP/Planck measurements of inflationary ] Step-function 79 ] Gaussian -fold ] Step-function | 2 ] Step-function ] Gaussian e – 12 – m 79 109 N 110 [ 108 110 QCD ∼ ]. For simplicity, we neglect correlations amongst θ | 10 2 − µ ; and finally adding BICEP/Planck measurements in 0049 [ 034 [ 110 10 . . GeV [ , 0021 GeV [ 0 0 . 9 . 0 | 109 10 theta 12 at 95% [ . , 0 & | 094 QCD . a 9645 θ . f 1876 0 . QCD , and the amplitude of scalar perturbations, θ | r 91 ; adding a lower-bound on the axion decay constant, ) 3 : a likelihood for the spectral index, s | | A and inflaton masses — receive quadratic corrections from a cut-off, such that Z s a 10 2 r r < f n QCD . This enabled us to assess the individual impacts of the constraints. Likelihoods included in our Bayesian evidences for the scale of electroweak symme- M m θ theta decay | conditions Parameter Measurement Likelihood function ln(10 inflation weak-scale and Planck measurements and impose an upper-limit for the scalar-to-tensor ratio. inflation perturbations, BICEP measurements [ 2 By picking scale-invariant priors for the Lagrangian parameters, we implicitly acknowl- We picked uninformative scale-invariant priors for the dimensionful Lagrangian param- • µ an effective field theoryeliminated such approximately corners 18% may of be the considered prior inconsistent volume or in implausible. table edge This our lack of information aboutwere a sufficient reason possible to UV believe theory that that a might particular generate mechanism them. might generate If them, there imprint- — we expect that without fine-tuning for our QCD relaxionheight model is and related general to the relaxion QCDscale. model scale, is Furthermore, whilst we that in the assigned ina latter, zero the it relaxion prior former, is Lagrangian weight the no parameter to greater barrier with than corners about dimension of the mass parameter weak exceeded space the in cut-off which scale, Λ, as in inflation eters and cut-off because wea are shift-symmetry, ignorant and of a their linearlogarithmic scale, quantity. prior a Our for linear prior prior ranges for are summarised in table We applied the likelihoods incrementallyadding in five calculations peran model: upper only bound on observables. Table 1. try breaking, the axionobservables. decay Note constant, that we neglect statistical correlations in Planck measurements of inflationary JHEP08(2016)100 i ) and re- h QCD σ h 1 π π − π π π , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 4 , 4 , 10Λ , 4 , 4 , 4 4 4 40 40 50 40 50 40 4 4 4 , π , π , 10 100 100 100 20 40 − − i − − − − − − − − − − − − − − h QCD h 6 Λ − 1 − Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Log 10 Linear 50, 500 Linear 0 Linear 0 – 13 – | | σ i b b 2 3 3 2 2 1 2 σ 2 2 2 2 σ σ -fold a κ λ λ f µ µ λ λ Λ Λ e m m h m m m m QCD QCD m θ θ N | | SM + Parameter Prior QCD relaxion Priors for parameters in SM augmented with scalar-field inflation (SM + Non-QCD relaxion, as for QCD relaxion except grangian parameters at the GUTpriors. scale that must Likewise, be acknowledged theprinciple, in identification the ameliorate choice of the of fine-tuning a suitable the of UV-completion priors, the however, of scenario to-date the as wescale-invariant know priors. relaxion the of Bayes-factor model no is compelling could, a reason in functional to of pick anything other than dimensionful relaxion Lagrangian parameter exceeded a cut-off were assigned zero prioring weight. distinguishing correlations betweenof parameters, priors. it should In bemechanisms supersymmetric reflected such models, in as for one’s minimal instance, choice supergravity the knowledge (mSUGRA) of results particular in breaking relations between La- Table 2. laxion models. Masses are in Planck units. Furthermore, regions in parameter space in which a JHEP08(2016)100 ) 2 ). ), µ )). ) a (5.1) weak 4.14 4.14 5.1 3.10 ], and was . By doing 2 17 weak-scale 2 (see eq. ( GeV, the relaxion that, although we π/ 8 . This is similar to 2 30 | ≈ . We note that eq. ( 25 QCD θ . The preference for relaxion ) also impairs the credibility | results in a similar fine-tuning 28 ) dramatically impact the pref- − 2 , such that by chance 2 100 GeV. This, in essence, explains . 4.2.1 2 GeV) P . The latter results in approximately ∼ 2 Z . Thus tuning two scalar masses — 8 inflation P GeV) Z and relaxion models are summarised in M βM 8 M (10 M σ theta ∼ (10 ∼ , 2 in section | P 2 2 . m – 14 – 2 m βM QCD ] could favour relaxion models, as from eq. ( m ∼ θ 2 P | 2 2 94 ∼ µ µ 2 βM µ conditions , and the for a non-QCD relaxion model versus the SM is unaffected by 2 for relaxion models versus the SM. Thus, all data considered, the decay 30 − . . Note that lowering the quadratic corrections by supersymmetrizing the theta — in a relaxion model to be model is favoured by a Bayes-factor of at least about 10 and . We find that, considering only a measurement of the weak scale (i.e., 2 σ 3 m To further investigate this issue, we relaxed the Planck-scale cut-off, plotting evidence The final data-set of inflationary observables ( The preference for the QCD relaxion model is further damaged by measurements of the The physicality conditions ( as a function ofso, the cut-off we in wish the towere SM confirm the and that cut-off our our much QCD lowerpreviously QCD relaxion than found relaxion model the that in model Planck the figure would scale.scale relaxion be model cut-off, We favoured find was if versus in not the the figure favoured versus SM, cut-off the were SM lowered with a in Planck- each model to about 10 why the evidence for+ the SM conditions and relaxionSM models and are relaxion models similar,Bayes-factor see if might e.g. one ref. scale considers [ as only hand, the might cut-off help slightly squared. less, as Lowering it would the lower the Planck bounds mass, on scalar on masses the from eq. other ( The factors are inmass fact is approximately fine-tuned to the be fractionsand so of light versus parameter a space cut-off, penalty in as which tuning a a single scalar scalar mass such that factors of about 10 SM + results in an approximate limit of The preference of about 10 decay of the considered relaxionsuffers models. severe Low-scale fine-tuning inflation, required as in it the requires relaxion a paradigm, light scalar, and thus results in partial Bayes- in solving theassociated hierarchy with problem, their relaxion physicality conditions. models are hamstrungaxion by decay severe constant, fine-tuning zero preference for the QCD relaxion model as it predicts that the weak scale. erence for relaxion models.parameter spaces and The shrink the conditionsmodels Bayes-factors by wipe-out versus about a 10 the fraction SM almost of entirely the disappears. relaxion models’ In other words, despite their success The evidences and Bayes-factorstable for the SM + relaxion models are favouredfindings by for colossal the Bayes-factorsexpected, constrained of as about minimal the 10 supersymmetric SM with SM a versus Planck-scale cut-off the makes SM an egregious [ generic prediction for 5.1 Evidences JHEP08(2016)100 in 20 6 (5.2) Z GeV e.g., 8 16 conditions Evidence, , . a relaxion model. 12 QCD relaxion Model p versus the SM and d (b) 6 Λ (GeV) Y 10 ) 8 Relaxion. log weak-scale we plot the priors for the M and we approximated our | is negligible, confirming our 3 (b) Z p ( 10 M p − 4 the SM and )) 10 p (a) ( . Z |

0 M

Z QCD

0.08 0.00 0.40 0.32 0.24 0.16 . This illustrates that a relaxion model could Evidence, log θ | − , that is, Z 10 – 15 – 2 M Z conditions (log δ 8 Evidence, Z and Standard Model ) = 6 M | Z Λ (GeV) SM. 10 M weak-scale 4 log . This little-hierarchy problem could scotch the Bayes-factor of 10 (a) (log p SUSY 2 M versus the SM with Planck-scale quadratic corrections. Including low-scale The evidence as a function of the cut-oﬀ, Λ, in , the Bayes-factor favours our QCD relaxion model by 10 31 σ m 0

decay With a cut-oﬀ allowed to be as low as 10 TeV, considering

Z GeV, but not the hierarchy problem by itself. Hence, in isolation, our QCD relaxion

0.0 0.3 0.2 0.1 0.5 0.4 Evidence, 8 -boson mass in the SM and our QCD relaxion model that result from the non-informative measurement with a DiracPlanck function. scale, the Whereas relaxion theless model than SM the results favours Planck in a scale,the considerable weak resolving QCD probability the scale axion, hierarchy mass the close problem. posterior atexpectations. to We probability scales find the that much that if the relaxion is priors for Lagrangian parameters in table This illustrates their genericequivalent predictions to for the the Bayesian weak evidence scale. if our This data would were be log numerically favour of the supersymmetrized relaxion model. 5.2 Observables To illustrate the resolution ofZ the hierarchy problem, in figure and about 10 inflation in e.g., a supersymmetrized relaxionmass model, however, might necessitate an inflaton model could be significantlythe favoured. little-hierarchy problem In in other a words, supersymmetric10 the model relaxion with soft-breaking mechanism masses maymodel at solve about cannot improve fine-tuning compared to the SM. Figure 2. The evidence includes be significantly favoured ifby the supersymmetrizing cut-off the were SM lowered and from relaxion model. the Planck The scale evidences to are plotted about in 10 arbitrary units. JHEP08(2016)100 | 20 QCD θ | -boson against 3 81 28 Z ) − 6 − − 53 77 25 27 33 25 − iv − − − − − 10 10 10 10 − ). A Bayes- 10 10 10 ), ( ≪ ≪ ≪ by the a QCD relaxion 30 0 decay (b) 39 5 3 (GeV) − 1 10 1 10 45 42 2 inflation Z 10 10 − − 1 10 10 M 10 ≪ ≪ 10 10 10 ) physicality conditions in ≪ ≪ Relaxion. ≪ -10 ii log , for the SM augmented with P (b) ), ( 5 7 6 6 39 the SM and Prior pdf − − − − − -20 10 10 10 10 (a) QCD relaxion Model weak-scale -30

2 2 0.01 0.00 0.05 0.04 0.03 0.02 32 pdf Prior 34 28 30 2 − − − − − − 1 10 10 10 10 -boson mass in 20 – 16 – Z and partial Bayes-factors, -boson mass ( Z B of the 4 4 16 34 30 30 − − − 10 ) constraints on the axion decay constant ( 10 10 ) the i iii weak-scale +=conditions +=decay +=theta +=inflation 12 ), ( ), a relaxion toy-model and a QCD relaxion toy-model. We apply data (GeV) ) ) 10 σ Z σ σ ) ) 10 M SM. σ σ GeV 10 , Bayes-factors, 10 · 8 Z log (a) SM + SM + GeV 10 relaxion) 1 10 relaxion) 10 GeV 10 · / / · / / ) SM + SM + conditions σ / / ) BICEP/Planck measurements of inﬂationary observables ( 4 v Prior pdf Data-set Prior distribution of log (SM + (relaxion) Evidences, Standard Model (relaxion (relaxion Z Z (QCD relaxion) P B ) and ( 0 (QCD relaxion (QCD relaxion (QCD relaxion (QCD relaxion Z

P B P B

0.24 0.16 0.08 0.00 0.40 0.32 ro pdf Prior theta Figure 3. model including nogreater data. than that The in densitythe the weak at SM. scale the This with correct illustrates respect that to weak the the scale relaxion SM. in mechanism The improves the densities fine-tuning relaxion are of plotted model in is arbitrary much units. of all data consideredchange in thus relative far. plausibility ofA A two ratio partial models of Bayes-factor in greater is light thanimportant of one a findings incrementing indicates ratio in the that of blue: data amass, by Bayes-factors, relaxion but that a toy-model indicating that relaxion single is once the toy-models data-set. favoured.relaxion all are toy-models. We constaints highlight favoured are our by included, most about that preference 10 is reversed to about 10 Table 3. scalar-field inflation (SM + incrementally in five data-sets:relaxion ( models ( ( factor is a ratio of evidences, indicating the change in relative plausibility of two models in light JHEP08(2016)100 . ), s = s A 300 , as 3 3 A , we . In with b 4 P P m m H M M and and s s ∼ ≫ 240 n n ≪ 4 2 , , , it is applied. H r r m H (GeV) ) . H < v b (b) λ H ( 180 f , for . Again, it must be | 3 is avoided. H < m ) = 4.2 m 4 i P | is rather complicated. In 120 σ Posterior pdf M h f = ( With | ∼ V 2 Hubble parameter, . ) , for a general renormalisable in- (b) s i m 60 | H A σ h ( = V | and General inflaton potential with is applied. Consequently, the Bayes- 1 s b model. 0 m n | σ , r

0.0030 0.0015 0.0000 0.0075 0.0060 0.0045 otro pdf Posterior )), is not applied; whereas in H < m ). Other than by extreme ﬁne-tuning of the 4.17 σ – 17 – λ 20 (4 , the posterior shows preference for / (eq. ( 4 2 b . On the other hand, consider that all dimensional 4a , the constraint that the Hubble parameter is less than the m b 16 m , where the potential is an order-four polynomial. The (a) 4 P . (GeV) ) = b i H < m , though in this case the function M ≪ )), but with Planck/BICEP measurements of H σ 4 P )). In 10 h 12 ( H M ≪ 4.17 V H < m 4.10 ∼ ) i 4 2 σ we plot the posterior distribution of the Hubble parameter 8 h m (eq. ( ( ) 4 b λ V ( Without f Posterior pdf Hubble parameter, log (a) 4 Posterior distribution for the Hubble parameter, ) = i H < m σ h General inﬂaton potential ( 0 V

This can be understood analytically. As discussed in section To illustrate the ﬁne-tuning required in the second epoch of inﬂation in a relaxion = 0. By dimensional analysis, the result must be that

0.8 0.0 2.4 1.6 3.2 otro pdf Posterior 1 6 Discussion and conclusions We constructed models that utilised aet relaxation al. mechanism recently proposed tomodels by solve Graham involves the solving hierarchy a problem. transcendental equation with Unfortunately, numerical finding methods. the We weak presented scale in relaxion there is noparameters chance are that equal tothat the mass-squared term, any case, one must still require fine-tuning such that mass-squared term is typically Planckian assolution it involves received roots a quadratic ofm correction. a The general cubic equation.fact, the For result simplicity, insteadmass-squared is (which consider that might not that agree with Planck/BICEP measurements of In the unconstrained caseexpected, in resulting figure in afactor tiny significantly evidence favours once inflation in the SM + must have that scenario, in figure and without constraint thatbarriers, the Hubble parameter is less than the height of the periodic Figure 4. flationary potential (eq. ( height of the periodicBoth barriers, plots include Planck/BICEP measurements of JHEP08(2016)100 , as explained P M . severely impair the | H QCD and BICEP/Planck mea- θ | | in favour of the SM augmented QCD θ | 25 2 if the relaxion is the QCD axion. π/ . This resulted in so-called Bayesian s | ≈ A if one considered only the weak scale. Once QCD θ 30 and | , in order to prevent inflation from destroying s P – 18 – n M , r ). The resulting Bayes-factors are then of order unity. ≪ in their favour, the same Bayes-factors are scotched by 5.1 H 30 , confirming that | QCD θ | with respect to the non-QCD relaxion model. This stems from a constraint and the accompanying discussion. 25 4 To conclude, we also remark that our results stand in the absence of a clear UV- Thus, whilst the analysed relaxion models indeed solve the hierarchy problem lead- Constraints upon the QCD relaxion decay constant and We found that the Bayes-factors — ratios of Bayesian evidences that indicate how We performed the first statistical analysis of a relaxion model by scanning relaxion completion for the consideredthe relaxion origins models of and the thatinformative parameters the priors. in lack the of If relaxion informationdifferent Lagrangian regarding such choice was of a encoded priors UV-completion that in were could the re-establish available, adopted the it un- plausibility would of possibly the relaxion dictate framework. a inflation, ultimately leading to awith Bayes-factor scalar-field of inflation. about 10 We anticipate, furthermore,genesis that and detailed thermal consideration effects of baryo- (includingtroweak the symmetry) disastrous would possibility further of damagethat reheating the the restoring plausibility required of elec- unconventional relaxion cosmology models is and the Achilles’ conclude heel of the relaxion mechanism. in figure ing to Bayes-factors of aboutconstraints 10 upon parameters in the relaxion potential and the Hubble parameter during about 10 upon the Hubble parameterparameter during must inflation: be fine-tuned withinthe to the periodic relaxion barriers frameworkpotentials in the supported the by Hubble current relaxion observations generically potential. predict that In contrast, the polynomial inflationary in the SM, as demonstrated in eq. ( plausibility of theBICEP/Planck QCD demolish relaxion that model, ofwe the whereas find surviving, that inflationary non-QCD the relaxion observables SM model. augmented measured with In by scalar-field this inflation is regard, favoured by a Bayes-factor of versus the SM by a colossalwe factor included of about physicality 10 conditionsfactors upon were inflation decimated to during aboutQCD relaxation, 100 relaxion however, for model. the the The Bayes- non-QCDmechanism physicality relaxion results, condition in model on fact, and in the about a energy fine-tuning 1 density penalty for similar during to the the that relaxion for the hierarchy problem evidences for our relaxion modelsner, we augmented calculated with Bayesian scalar-field evidences inflation. for In the a SM similar augmented man- with scalar-field inflation. one ought to update one’s relative prior belief in two models — favoured relaxion models for a lower bound on models’ parameter spaces with thesurements nested of sampling algorithm, the considering weak datasurements scale, from of mea- the inflationary axion observables decay constant, an analytic expression for an interval bounding the weak scale and an analytic expression JHEP08(2016)100 ] ]. , D 19 SPIRE (2004) Relaxion IN ]. ]. 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Baryogenesis

I have argued that particle cosmology provides some of the most compelling reasons to believe that there is physics beyond the Standard Model. Baryonic matter makes up a small fraction 4.9% of the Universe’s energy budget meaning that 95% of the Universe is made up of stuff everyone is yet to understand [33]. However, as I will argue, from a cosmological perspective no one even understands the baryonic component. So 100% of the Universe is unexplained! Indeed, the three pillars of the Standard Model of particle cosmology – inflation, the production of dark matter and the generation of a baryon asymmetry – all require physics beyond the Standard Model. That being said, some striking confirmations of the Standard Model of cosmology gives a strong argument that its the Standard Model of particle physics that needs modification rather than cosmology. The cosmic baryon abundance can be measured through two completely independent methods. First through baryon acoustic oscillations in the cosmic microwave background and second through the Deuterium abundance which is very sensitive to the baryon abundance. It is convenient to express the cosmic baryon abundance as a fraction of the entropy density as this gives a number that does not change as the Universe expands [68] n Y = B = 8.59 0.11 . (3.1) B s ± This agreement is a triumph of modern cosmology as it is highly non-trivial for two independent measures to agree. One sees very few anti-baryons in the Universe. The few experiments observe is consistent with secondary production [69]. This baryon asymmetry is much too large to be a consequences of some initial condition as inflation will dilute any initial asymmetry. Specifically if one has a Planckian energy density pre-inflation, the maximum baryon asymmetry one can have assuming constant entropy is about five orders of magnitude too small.

3.1 Sakharov Conditions

Well before all this was known, it was shown by Sakharov that any mechanism that spontaneously produced the BAU would have to satisfy three conditions [70, 71] Violation of baryon (and lepton) number conservation, • C and CP violation, • 53 Departure from equilibrium. •

The first condition can be satisfied either through electroweak sphalerons, explicit B violation occurring in grand unified theories or through anomalous currents in Cherns Simons gravity. Electroweak sphalerons have the added advantage of converting P violating electroweak processes into C violating processes. This in conjunction with CP violating three body couplings (Yukawa or tri-scalar) can satisfy the second Sakharov condition. Satisfying the third option is where there has been the most variety in suggestions – from inflation, to the decay of heavy particles to a cosmic phase transition. One can mix and match different mechanisms for satisfying these three conditions and arrive at the major paradigms for generating the BAU. For example combining heavy particle decay (heavy neutrinos) mixed with electroweak sphalerons and neutrino sector CP violation is the essential ingredients of leptogenesis. I will focus our attention on electroweak baryogenesis which utilizes electroweak sphalerons during the electroweak phase transition. Electroweak baryogenesis has the significant advantage of being testable as the new physics it requires is generally weak scale. The Standard Model in principle has all three ingredients – electroweak sphalerons provide a source of baryon number violation at high temperature, the CKM matrix is a source of CP violation and as the Universe cools the Higgs field acquires a vev in the electroweak phase transition. However, on a quantitative level the Standard Model fails to fulfil the last two Sakharov conditions. For a Higgs mass of 125 GeV, the electroweak phase transition does not proceed by bubble nucleation so the departure from equilibrium is not dramatic enough to facilitate enough baryon production. Furthermore the CKM matrix provides too feeble a source of CP violation.

3.2 Introductory remarks for published material in thesis chapter 3

Electroweak baryogenesis is a subject that has lacked a comprehensive reference that is self contained and includes everything (although some very fine reviews exist that include some very useful information see [72, 73]). As such I wrote a book that covers enough to take one from a graduate level to being able to perform state of art calculations focusing on the vev insertion approach.1. I partially include this book here. Unfortunately IOP does not give permission for more than two chapters to be shared so I will also presented abbreviated summary of some of the content throughout this section focusing on the background most needed for later chapters in this thesis. This review contains a self contained summary on the baryon number violation in the Standard Model, finite temperature quantum field theory, phase transitions in quantum field theory, the derivation and solution to transport equations using the closed time path formalism, details of relevant thermal parameters such as thermal masses, widths, diffusion coefficients etc., and electric dipole moments.

1for a nice explanation of the WKB approach see [74, 75] and the references therein

54 3.3 Declaration for thesis chapter 3

Declaration by candidate In the case of the paper present contained in chapter 3, the nature and extent of my contribution was as follows: Publication Nature of contribution Extent of contribution 3 Single author paper 100% The undersigned hereby certify that the above declaration correctly reﬂects the nature and extent of the candidate and co-authors’ contributions to this work.

3.4 Published material for chapter 3: A pedagogical introduction to electroweak baryogenesis

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Triplet seesaw model: from inflation to asymmetric dark matter and leptogenesis Chiara Arina A Pedagogical Introduction to Electroweak Baryogenesis

A Pedagogical Introduction to Electroweak Baryogenesis

Graham Albert White ARC Centre of Excellence for Particle Physics at the Tera-scale, School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800 Australia and Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800 Australia

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ISBN 978-1-6817-4457-5 (ebook) ISBN 978-1-6817-4456-8 (print) ISBN 978-1-6817-4459-9 (mobi)

DOI 10.1088/978-1-6817-4457-5

Version: 20161101

IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 40 Oak Drive, San Rafael, CA, 94903 USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK For Samuel White

Contents

Preface x Acknowledgements xi Author biography xii

1 Introduction 1-1 References 1-4

2 The Sakharov conditions 2-1 References 2-2

3 Baryon number violation in the Standard Model 3-1 3.1 The axial anomaly 3-1 3.1.1 Dimensional regularization 3-2 3.1.2 The Fujikawa method 3-4 3.1.3 Baryon and lepton number violation 3-7 3.2 The Chern–Simons form, baryon number violation, and the 3-8 winding number 3.3 Winding number and non-abelian gauge groups 3-10 3.4 Solitons and instantons 3-12 3.5 The sphaleron 3-15 References 3-21

4 Phase transitions 4-1 4.1 Closed time path formalism 4-3 4.2 A brief review of the effective potential at zero temperature 4-8 4.3 The effective potential at finite temperature 4-12 4.3.1 The Standard Model example 4-15 4.3.2 Issues of gauge invariance 4-16 4.3.3 Daisy re-summation 4-19 4.3.4 The sphaleron rate at high temperature 4-19 4.4 The bounce solution 4-21 4.5 Analytic techniques for the single field case 4-22 4.5.1 Single field approximation 4-22 4.5.2 Developing ansatz solutions 4-24

vii A Pedagogical Introduction to Electroweak Baryogenesis

4.6 Path deformation method 4-26 4.7 Perturbative method 4-28 4.7.1 Observations on convergence 4-31 4.7.2 A numerical example 4-32 4.7.3 Wall width and variation in β 4-33 4.8 Baryon washout condition 4-34 References 4-36

5 CP violation 5-1 References 5-4

6 Particle dynamics during a phase transition 6-1 6.1 Particle current divergences and self-energy 6-1 6.2 Transport coefficients and sources 6-4 6.2.1 A biscalar with VEV insertions 6-5 6.2.2 Dirac fermions with VEV insertions 6-11 6.2.3 Chiral fermions with VEV insertions 6-13 6.2.4 Triscalar and Yukawa interactions 6-15 6.2.5 Four-body interactions 6-22 6.3 Local equilibrium approximations 6-23 6.4 Gauge and supergauge equilibrium 6-23 6.5 Fast rate approximations 6-24 References 6-25

7 Plasma and bubble dynamics 7-1 7.1 Imaginary time formalism 7-1 7.2 Diffusion coefficients 7-2 7.3 Thermal widths 7-6 7.4 Thermal masses 7-7 7.5 Bubble wall velocity 7-8 References 7-12

8 Transport equations 8-1 8.1 The MSSM under supergauge equilibrium 8-1 8.2 Solution using fast rates, diffusion approximation, and ultrathin 8-3 wall approximations 8.3 Solution without fast rates 8-5

viii A Pedagogical Introduction to Electroweak Baryogenesis

8.4 Deriving the analytic solution 8-5 8.5 Beyond ultrathin walls 8-10 References 8-11

9 The baryon asymmetry 9-1 References 9-2

10 A brief phenomenological summary 10-1 References 10-3

11 Other mechanisms for producing the baryon asymmetry 11-1 11.1 Leptogenesis 11-2 11.2 Affleck–Dine 11-3 11.3 Using inflation 11-4 References 11-6

12 Discussion and outlook 12-1 References 12-2

ix Preface

We present a mostly self-contained pedagogical review of the theoretical background to electroweak baryogenesis as well as a brief summary of some of the other prevailing mechanisms for producing the asymmetry between matter and antimatter using the minimal supersymmetric Standard Model as a pedagogical tool whenever appropriate. This book covers an in-depth look at baryon number violation in the Standard Model, the necessary background in ﬁnite temperature ﬁeld theory, plasma dynamics, and how to calculate the out-of-equilibrium evolution of particle number densities throughout a phase transition.

x Acknowledgements

A lot of what I learnt about Baryogenesis was from Michael Ramsey Musolf and Csaba Balazs. I would also like to acknowledge the post-doctorates and students that answered many questions of mine especially Peter Winslow and Huaike Guo. I would also like to acknowledge David Paganin, Jonathan Kozaczuk, Giancarlo Pozzo, Kaori Fuyuto and Andrew Fowlie for useful comments regarding this work. Finally I would like to acknowledge my wife for enduring many nights of me working late during this project.

xi Author biography

Graham Albert White

Graham White grew up in north Queensland, Australia. He developed a love of physics after reading some popular science books as a teenager and decided he wanted to become a physicist shortly after. Graham White is a Doctoral Student at Monash University studying Baryogenesis and holds a masters degree from University of Kentucky and has studied Baryogenesis at the Amherst Center for fundamental interactions (University of Massachusetts Amherst) as a guest scholar. He currently lives in Melbourne with his wife and son.

xii IOP Concise Physics A Pedagogical Introduction to Electroweak Baryogenesis

Graham Albert White

Chapter 1 Introduction

The origin of the baryon asymmetry of the Universe (BAU) is one of the deepest short-comings of our understanding of particle physics, as it cannot be explained within the Standard Model. For instance, one cannot simply set the baryon asymmetry as an initial condition as this would be washed out by inflation1.On the other hand, coinciding estimates of the baryon asymmetry using different techniques are a triumph of modern cosmology. The baryon asymmetry can be estimated by the deuterium abundance and from the cosmic microwave background (CMB), where the relative sizes of Doppler peaks in the temperature anisotropy are sensitive to the BAU. These two methods give the overlapping estimates of the baryon to entropy ratio2 [2–4] ⎧ (7.3 2.5) 10−11 , BBN nn n ⎪ ±× BB− ¯ B⎨ 11 YB (9.2 1.1) 10− , WMAP (1.1) = s ≈=s ⎪ ±× 11 ⎩(8.59±× 0.11) 10− , Planck. This remarkable overlap in the estimates of the BAU from light element abundances (particularly deuterium) and baryon acoustic oscillations are shown in figure 1.1 and figure 1.2 respectively. Reproducing this estimate using particle physics makes one of the three pillars of the Standard Model of particle cosmology, the other two being inflation [5] and dark matter [6]. Like inflation and dark matter, it requires at least some additions to the Standard Model. Furthermore, like the other two pillars of the Standard Model of particle cosmology, there are a very large variety of models to explain this peculiar fact about our Universe. Two of the most elegant explanations

1 For a recent attempted exception to this, albeit a ﬁne-tuned one, see [1]. 2 Sometimes the baryon asymmetry is compared to the photon density, nBB/7.04nYγ ≈ , rather than the entropy. However during the early Universe many particles are expected to be in thermal equilibrium making

YB more convenient. doi:10.1088/978-1-6817-4457-5ch1 1-1 ª Morgan & Claypool Publishers 2016 A Pedagogical Introduction to Electroweak Baryogenesis

Figure 1.1. The abundances predicted by the Standard Model of Big Bang nucleosynthesis (BBN) [7] for 4He, D, 3He, and 7Li. Here the bands show the range for the 95% confidence level and the boxes indicate the light element abundances—the smaller boxes show 2.75σ statistical errors; the larger boxes 2.75σ statistical and systematic errors. The wide band indicates the BBN concordance range, whereas the vertical narrow band indicates the baryon asymmetry measured via the CMB given at the 95% confidence level. Reproduced from [2]. are leptogenesis [9] and the Affleck–Dine mechanism [10]. Unfortunately both tend to be well out of reach of the particle colliders of today and of the foreseeable future. The focus of this review will be electroweak baryogenesis, which is the term for any mechanism that produces the matter–antimatter asymmetry during the electroweak phase transition. Such a scenario requires physics beyond the Standard Model that must couple relatively strongly to Standard Model particles and have masses that are not too far above the weak scale. Therefore, unlike Affleck–Dine baryogenesis or leptogenesis, electroweak baryogenesis has the tantalizing prospect of being tested, at least indirectly, by weak and TeV scale searches at the large hadron collider (LHC). Apart from testability, electroweak baryogenesis has the attractive feature that coincides the breaking of the symmetry between particles and anti-particles with the spontaneous breaking of the one symmetry we know to be broken—electroweak symmetry. Unfortunately the literature on this exciting subject is somewhat opaque to newcomers. There are some very nice pedagogical introductions to small parts of the theoretical foundations of baryogenesis scattered throughout the literature if one digs hard enough3. However, the study of baryogenesis is arguably too decoupled

3 For recent reviews see [11–14].

1-2 A Pedagogical Introduction to Electroweak Baryogenesis

Figure 1.2. ΩM −ΩΛ constraints due to CMB, baryon acoustic oscillations, and Supernova Cosmology Project Union2.1 SN constraints including SN systematic errors. Reproduced from [8]. from the rest of high energy physics given its promising phenomenological implications and the fundamental nature of the question it attempts to answer. Furthermore, the techniques one needs to learn to research in the ﬁeld of electroweak baryogenesis have a large cross-over with other calculations in particle cosmology—including other models of producing the baryon asymmetry such as leptogenesis. This primer will therefore give a mostly self-contained introduction to the ﬁeld of electroweak baryogenesis, assuming the reader has a graduate level knowledge of

1-3 A Pedagogical Introduction to Electroweak Baryogenesis

particle physics, including dimensional regularization, some basic path integral techniques, the computation of amplitudes at tree and loop level, the Standard Model Lagrangian, and Big Bang cosmology, as well as a rudimentary knowledge of effective field theory and the minimal supersymmetric Standard Model (MSSM),4 which we will use as a pedagogical tool where appropriate. There are of course many candidates for producing the BAU, but it is my hope that the theoretical foundations given in this book should make learning such mechanisms significantly more tractable. References [1] Krnjaic G 2016 Can the baryon asymmetry arise from initial conditions? arXiv: 1606.05344 [2] Eidelman S et al 2004 (Particle Data Group Collaboration) Review of Particle Physics Phys. Lett. B 592 1 [3] Spergel D N et al 2003 (WMAP Collaboration) First year wilkinson microwave anisotropy Probe (WMAP) observations: determination of cosmological parameters Astrophys. J. Suppl. 148 A16 [4] Ade P A R et al 2014 Planck 2013 results. XVI. cosmological parameters Astron. Astrophys. 571 A16 [5] Bassett B A, Tsujikawa S and Wands D 2006 Inflation dynamics and reheating Rev. Mod. Phys. 78 2 [6] Feng J L 2010 Dark matter candidates from particle physics and methods of detection Annu. Rev. Astron. Astrophys. 48 495–545 [7] Cyburt R H et al 2008 An update on the big bang nucleosynthesis prediction for 7Li: the problem worsens J. Cosmol. Astropart. Phys. JCAP11(2008)012 [8] Suzuki N et al 2012 The Hubble Space Telescope Cluster Supernova Survey. V. Improving the dark-energy constraints above z > 1 and building an early-type-hosted supernova sample 756 85 [9] Fukugita M and Yanagida T 1986 Baryogenesis without grand unification Phys. Lett. B 174 1 [10] Affleck I and Dine M 1985 A new mechanism for baryogenesis Nucl. Phys. B 249 361 [11] Morrissey D E and Ramsey-Musolf M J 2010 Electroweak baryogenesis New J. Phys. 14 12 [12] Cline J M 2012 Baryogenesis arXiv: 0609145(hep-ph) [13] Riotto A and Trodden M 1999 Recent progress in baryogenesis Annu. Rev. Nucl. Part. Sci. 49 46 [14] Trodden M 1999 Electroweak baryogenesis Rev. Mod. Phys. 71 5 [15] Kuroda M 1999 Complete lagrangian of MSSM arXiv: 9902340(hep-ph) [16] Csaki C 1996 The minimal supersymmetric standard model (MSSM) Mod. Phys. Lett. A 11.08 599–613

4 If you lack knowledge of the MSSM see [15,16] for an introduction.

1-4 IOP Concise Physics A Pedagogical Introduction to Electroweak Baryogenesis

Graham Albert White

Chapter 3 Baryon number violation in the Standard Model

The baryon number is a classically conserved quantity in the Standard Model as can be demonstrated by Noether’s theorem or even a cursory look at the available interactions. This intuition breaks down when one quantizes the Standard Model, due to so-called anomalous processes, which will be described in this chapter. That some fundamental symmetries may not survive the process of quantization is highly counter-intuitive. This is not just a feature of field theory. It was shown in [1, 2] that even standard quantum mechanics with certain potentials—specifically a delta function or a 1/r2 potential—have anomalous violations of conservation laws. The key anomaly of interest is the so-called chiral or axial anomaly. This anomaly, combined with the fact that the SU(2) gauge symmetry is a symmetry of left-handed particles only, will conspire to fulfil Sakharov’s baryon number violation condition.

3.1 The axial anomaly Let us warm up, however, by considering the simplest possible fermionic ﬁeld theory, a single massless, free fermion ﬁeld with a U(1) gauge symmetry. Such a theory has a global phase symmetry. Less obvious is that such a model’s Lagrangian is also invariant under the following transformation

i ψ → e.θγ5 (3.1) From these symmetries we can of course use Neother’s theorem to derive conserved currents. The Noether currents associated with these two symmetries are

Jμ = ψγ¯ μ ψ (3.2)

J5μ = ψγγ¯ 5 μ ψ, (3.3)

doi:10.1088/978-1-6817-4457-5ch3 3-1 ª Morgan & Claypool Publishers 2016 A Pedagogical Introduction to Electroweak Baryogenesis

Figure 3.1. The famous triangle diagram responsible for anomalous quantum violations of classical chiral symmetry. The circle denotes a γ5 on the vertex.

μ μ which implies Ward identities ∂ Jμ = 0 and ∂ J5μ = 0. Obviously we expect the product of conserved currents to also be a conserved quantity. Therefore we should not even need to bother to calculate the divergence of the following quantity:

−Γi(,,)0μνλx12 x x 3 =〈 ∣ TJ() 5 μ ()()()0. xJxJx 3 ν 1 λ 2 ∣ 〉 (3.4) Amazingly this quantity is not zero. This remarkable result is known as the ‘triangle anomaly’ [3, 4] due to the fact that the Feynman diagram related to this amplitude has three vertices, as shown in ﬁgure 3.1. In the author’s opinion the most effective way to learn the axial anomaly is through direct calculation. We present a review of what we believe is probably the simplest derivation in the literature using operator methods with dimensional regularization, followed by the Fujikawa method which is a path integral derivation.

3.1.1 Dimensional regularization Let us calculate this example using the ever-familiar dimensional regularization. A potential ambiguity that can arise in such a calculation is that there is no clear way to generalize the matrix γ5 to 4 − 2ϵ dimensions although various proposals exist [5]. We follow the approach of [6], which demonstrated that no knowledge of any of γ5 properties is needed to calculate the triangle anomaly1. The axial current to one-loop order due to the triangle anomaly is given by

dDpkdD μ iqx· Mμνλ 0()0Jx5 ==+e()(),ApAkνλ q k p . (3.5) ∫ (2ππ )D (2 )D

The above amplitude relates to the two diagrams in figure 3.1, and the second diagram is obtained from the first by merely interchanging (p,)ν and (k, λ). Hence, it suffices to compute one diagram

dDllk⎧ l lp⎫ Mμνλ(,kp ) ie2 ⎨ μλν− + ⎬, (3.6) (1) =− D γγ5 22γ γ 2 ∫ (2)(π ⎩ lk−+ )iϵ l + i(ϵ lp++ )iϵ ⎭

1 Part of this calculation has some cross-over with the calculation in [7].

3-2 A Pedagogical Introduction to Electroweak Baryogenesis

where the curly brackets denote a trace operation and the photon momenta are on-shell. Although we introduced this section stating that we are considering a massless fermion model, we will use a mass to regularize the integrals and take the 2 limit m → 0 at the end. Let us integrate over loop momentum and contract with qμ. The result we can write as the sum of two terms

MAμνλ νλ B νλ i(,)(,),qkpkpμ (1) =+ (3.7) where ⎡ ⎛ ⎞⎤ ie2 1 1−x 1 Aνλ ⎢ Δ ⎥ (,kp )=−dd x y ln ⎜ ⎟ 32πϵμ2 ∫∫0 0 ⎣ ⎝ 2 ⎠⎦ (3.8) αλ ν ×+−−+−{qxkypqypxkγγ55 γ γγα ()()(1 ) γ (1 ) λαν αλ ν ×−γγγγα qypxk γγγ5 () − γγα} and

ie2 1 1−x Bνλ (,kp )=− dd x y 16π 2 ∫∫0 0 (3.9) λν {}qxγγγ5()()()(1)−− kypxkypxk − +− (1) yp × . Δ In the above we have deﬁned the parameters

2211 Δ=−qxy + m, = −γπ +ln 4 . (3.10) ϵϵ The tensor Aνλ contains a divergent piece. If we employ the identities

αμ μαμλν νλμ μλν γγγα =−(2 + 2), ϵγ γγγγγα =− 2 γγγ + 2 ϵγγγ (3.11) along with the cyclic property of traces, we can reduce this tensor to the form

ie2 Aνλ ⎡ λν νλ νλ λν div. =−−+⎣4{pqγγ5555 } 4 kq{} γγ 2 pq {} γγ 2{ kq γγ } 96πϵ2 (3.12) νλ ⎤ +−2.gkp{}γγ55 pk⎦

So all traces with four Dirac matrices have already canceled and we are left with a series of terms that are zero in four dimensions. However, we need not make an assumption about the value of these traces. The term multiplying 1/ϵ is antisymmetric under the interchange (p,)ν and (k, λ) so it cancels when the two diagrams are added, giving AAνλ λν div.(,kp )+= div. (, pk ) 0. (3.13)

3-3 A Pedagogical Introduction to Electroweak Baryogenesis

fi Aνλ O The nite part of comes from the ()ϵ part of equation (3.11) and the ln Δ part in equation (3.8). The result in the massless limit reads e2 lim⎣⎡AAνλ (kp , ) λν ( pk , )⎦⎤ λναβ pk. (3.14) +=m 0 2 ϵ α β ϵ→0 = 6π Note that the above tensor is infrared finite, so the inclusion of a mass regulator was a convenient calculation tool. Moving to Bνλ(,kp ),wefind that the potentially singular terms cancel identically so we can take the limit D → 4 and write e2 lim [BBνλ (kp , ) λν ( pk , )] λναβ pk. (3.15) +=m = 0 2 ϵ α β ϵ→0 3π Putting everything together yields the well-known, but non-zero result

e2 dDpkdD μναβ iqx· 00∂·J5 = ϵ e()().ApAkpkνλα β (3.16) 2π 2 ∫ (2ππ )D (2 )D

Thus we have have found by explicit calculation the anomalous violation of chiral current conservation. We have chosen a particular way of performing the calculation that makes it maximally clear that the anomaly is not some ill effect of how we regulate our integrals. Indeed the anomaly is experimentally demonstrated in 0 predicting the correct rate for pion decay πγ→ γ. Furthermore, as remarked in the previous section, it has been shown to happen in ordinary quantum mechanics and the above calculation has been repeated with multiple different regularization schemes. Nonetheless we present one more derivation in the next subsection, since it naturally organizes the anomalies in the Standard Model such that it is straightforward to show that lepton and baryon currents have anomalous currents.

3.1.2 The Fujikawa method Since the anomaly is independent of the regularization scheme we will calculate the anomalies using the path integral method with a hard cutoff regularization scheme. The regularization free nature of the anomaly is more opaque in this scheme, so I will assume that the reader understood the results of the previous section where things were more clear. The cost of this opaqueness will be justiﬁed by the simplicity of the calculation that follows. Let us begin by returning to the anomalous chiral current

5 Jμ = ψγγ¯ 5 μ ψ. (3.17)

As in the previous section, the triangle diagram anomalously violates the classical conservation law derived through Noether’s theorem. This time our fermions interact with Standard Model gauge bosons. Path integral techniques are far more efﬁcient and systematic than trying to ﬁnd all anomalous operators and hoping you have not missed any. The path integral’s approach begins with simply adding the current to the Lagrangian [8, 9]

3-4 A Pedagogical Introduction to Electroweak Baryogenesis

⎡ c⎤ id4xaJ3 μ Wa⎣ , A⎦ [d ][d ]e∫ QCD− μ 5 . (3.18) μ λ = ∫ ψψ¯ ﬁ We then vary W with respect to aμ by an in nitesimal function ∂μ β in order to test whether W is invariant under such a change. Using the deﬁnition of functional derivatives

⎡ bb⎤ ⎡ ⎤ δβ[lnWWaA ]=−∂− ln⎣ ()()μμ ,μ ⎦ ln⎣ WaAμ , μ ⎦ δ ln W d4x =− μ ∫ δa (3.19) 4 μ = id∫ xJ¯5 ∂μβ 4 μ δβ[lnWxJx ]=− i∫ d ∂μ ¯5 ( ).

μ So if δ[lnW ]= 0, ∂μJ5 = 0 and the chiral current is conserved. To test for an anomaly one determines if the statement

⎣⎡ cc⎦⎤ ⎣⎡ ⎦⎤ Waμμ−∂β, Aλ = Waμ , Aλ (3.20) holds true. Let us begin by seeing if there is a change of variables that takes our primed Lagrangian back to the original form. That is

aaμ a LALAJLAQCD′ ()()ψψ′,,¯ ′≡μμQCD ψψ ′ ,,¯ ′−∂→()μ β5 QCD () ψψ ,,¯ μ .(3.21)

The appropriate set of transformations is

−iβγ5 ψβγψψ′=()1i −5 ∼ e (3.22)

−iβγ5 ψψ¯ ′= ¯ ()1i−∼ βγψ5 ¯ e. (3.23) This can be veriﬁed by direct substitution in which one ignores all terms O()β2 and uses the anti-commutivity of the gamma matrices. The only other modiﬁcation to make to our function W is the measure

[dψψ ][d¯ ]→′ [d ψ ][d ψ¯ ′ ] (3.24) in which we will pick up a Jacobian determinant J . We will later verify that J is ψ independent of and ψ¯ so we can take it out of the integral

⎡ a⎤ id4xL a J μ Wa⎣ , A⎦ [d ][d ] J e ∫ QCD− μ 5 μμ−∂βψψμ =∫ ′ ¯ ′ 4 μ idxLQCD aμ J5 (3.25) =′J ∫ [dψψ ][d¯ ′ ]e ∫ − ⎡ a⎤ = JW⎣ aμ,. Aμ ⎦

3-5 A Pedagogical Introduction to Electroweak Baryogenesis

Therefore we can write the variation of ln W exclusively in terms of the Jacobian

⎡ aa⎤ ⎡ ⎤ δβ[lnWWaA ]=−∂− ln⎣ ()()μμ ,μ ⎦ ln⎣ WaAμ , μ ⎦ ⎡J aa⎤ ⎡ ⎤ (3.26) =−ln⎣ Wa()μ , Aμ ⎦ ln⎣ Wa ()μ , Aμ ⎦ J = ln . Our Jacobian is of Grassman valued functions so it is given by the inverse of the functional determinant of our transformations

J ii1βγ55 βγ = Det [e e ]− . (3.27) We can use a simple identity, DetCC= exp[Tr ln ], to write this determinant in terms of a trace

J 2iTrβγ5 = e.− (3.28) This determinant diverges so we need to introduce a regulator. Let us ﬁrst separate the divergent part of the trace

Tr Tr d4xx , (3.29) βγ5 =′∫ βγ5 where the primed trace is over internal degrees of freedom. To regularize the integral we rewrite it as

⎛ ⎞2 ⎜⎟D 4 −⎝ M ⎠ (3.30) Trβγ5 =′ lim Tr dxx βγ5 e x . M→∞ ∫ The square in the regulator can be expanded

⎛ D ⎞2 11⎛ ⎞ 1 ⎜ ⎟ ⎜ μμνa a ⎟ 2 μ gFλσ()DgF σ . (3.31) ⎝MM⎠ =∂∂+2 ⎝ 4 3 μν⎠ ≡M 2 +·3

ﬁ Here the de nition of σ · F is given implicitly. To calculate the trace of the regulator we introduce the complete set of states

d d ⎛ D ⎞2 d ppd ⎜⎟ Tr lim Tr d4x ′ xp pe−⎝ M ⎠ p p x βγ5 =′ d 22d ∣ 〈 ∣βγ5 ∣ ′〉 ′∣ M→∞ ∫ (2ππ ) (2 ) d 2 d p ipD gF M2 lim Tr d4x e−+()μ μ +·3σ ) (3.32) =′ d βγ5 M→∞ ∫ (2π ) d 4 d p pM22 D 2 g F2 ipDM2 lim Tr d x ee−+·+·()3σ . =′ d βγ5 M→∞ ∫ (2π )

We will now Taylor expand the right-hand side up to O(1/M 4 ) keeping in mind that each factor of p2 is of order O()M 2

3-6 A Pedagogical Introduction to Electroweak Baryogenesis

d p2 ⎡ 4 d p 1 2 Tr lim Tr d x e1M2 ⎢ DgF βγ5 =′d βγ5 ⎣ −2 () +·3 σ M→∞ ∫ (2π ) M

1 2 2 2 {}DgF3σ 4 pD ++·−·+2M 4 ()3M 6 pDpDD2 g F pDD2 g FpD ×·( ·{} +3σσ ·+·{} +3 · · 4 ⎤ (3.33) DgFpDpD2 σ ()pD4⎥ ++···+{}3 ) M 8 ·⎦ iM 4 ⎡ gFσ lim Tr i d4x ⎢1 3 · =′βγ5 2 ×−⎣ 2 M→∞ ∫ (4π ) M ⎛ ⎞⎤ 11 1 ⎡ ⎤2 1 ⎡ ⎡ ⎤⎤ ⎜ ()gFσσ2 ⎣ DD, ⎦ ⎣ DD,,⎣ F⎦⎦⎟⎥ . +·++M 4 ⎝ 2 3 12 μν6 μ μ ·⎠⎦

The only trace that survives is the one with a product of two σ. Recalling that

μν αβ μναβ Trγσ5 σ=− 4i ϵ , (3.34) it is then easy to take the trace over the internal degrees of freedom to show that ⎡ 3 S ⎤ J 4 α μν =−exp⎢ i d xβ FF˜μν⎥. (3.35) ⎣ ∫ 8π ⎦ We therefore have

μ 3αs μν ∂=μJFF5 ˜μν. (3.36) 8π This effective operator generates the amplitude given in equation (3.15) which was derived using operator techniques and dimensional regularization.

3.1.3 Baryon and lepton number violation At this stage we have rederived the chiral anomaly using functional techniques and taking into account the gauge symmetries of the Standard Model. The SU(2) gauge bosons couple only to left-handed particles so the chiral anomaly interferes with lepton and baryon number conservation. Consider the classically conserved currents 1 J μ ψγμ()1. γ5 ψ (3.37) LR, = 2 ¯ ∓ Let us ﬁrst couple the left-handed current to the Standard Model. Following the procedure as before we consider the term W[,,,]aABGμμμμμ+∂β where μ L =++∂LaSM ()μμβ JL . This can be related to the term W[,aABGμμμμ , , ]by the transformations

−iβγ5 ψβγψψ′=()1i −5 ∼ e (3.38)

3-7 A Pedagogical Introduction to Electroweak Baryogenesis

−iβγ5 ψψ¯ ′= ¯ ()1i−∼ βγψ5 ¯ e. (3.39) μ Wethencanusethemethodoutlinedbeforetoderiveanexpressionfor∂μJL μ 22 2 2 with associated regulator exp{[(∂+μ gAμ + g ′ Bμμ + gs G )γ ] / M } ≡ exp[ DL / M ]. Similarly we can couple the right-handed current to our Lagrangian. The transformation required to bring us back to our original W function is

+iβγ5 ψβγψψ′=()1i +5 ∼ e (3.40)

+iβγ5 ψψ¯ ′= ¯ ()1i+∼ βγψ5 ¯ e . (3.41) Our Jacobian will acquire an extra minus sign. This time the associated regulator is μ 22 2 2 exp{[(∂+μ gAμ + gs Gμ )γ ] / M } ≡ exp[ DR / M ]. Following the previous calculation one ﬁnds for quarks

μμ μ 3αS μν ∂−≡∂=μ []JJLR μ J5 GG˜μν (3.42) 8π

μ 3αW μν ∂=μJBBL ˜μν. (3.43) 8π When one performs the calculation for leptons there is only the anomalous left- handed current which is equal to one third the above equation since one does not perform a trace over colour. Noting that the baryon number of a quark is 1/3 we can write the exact symmetry

μ μ ∂=∂μBLμ . (3.44)

3.2 The Chern–Simons form, baryon number violation, and the winding number We have now demonstrated that both lepton and baryon number are violated in the Standard Model through these strange anomalous currents. In this subsection we will try and find the sort of field configuration that violates baryon and lepton number. This is one part of the background theory which can quickly get unnecessarily formal, so we will be taking an approach to make things only as formal as is useful to understanding the subject at the level needed to do research in this field2. Let us first make some manipulations to our effective action. Recall that

2 We do not follow any particular reference as they often lack details in the areas we need and are too formal or too detailed in areas orthogonal to this analysis. Nonetheless, some useful resources for more information are [10–12].

3-8 A Pedagogical Introduction to Electroweak Baryogenesis

g2 μ μμναβ4 ⎣⎡ ⎦⎤ ∂=∂=μLBμ dTrxFFϵ μν αβ 32π 2 ∫ 2 g 4 μναβ ⎡ aaabc bc⎤ =∂∂−∂+dTrxAAfAAϵ μμ⎣ ν ν μμν⎦ 32π 2 ∫ ⎡ a ade d e⎤ ×∂⎣ αβAAfAAeta −∂ βαα + b ⎦ 2 ⎡ ⎤ g 4 μναβ 2i =∂∂−dTrxAAAAAϵ μναβναβ⎢ ⎥ (3.45) 8π 2 ∫ ⎣ 3 ⎦

abc ade μναβ ⎡⎡ a b⎤⎡ de⎤⎤ + ffϵνTr⎣⎣ AAμαβ , ⎦⎣ AA⎦⎦ 2 ⎡ ⎤ g 4 μναβ 2i =∂∂−dTrxAAAAAϵ μναβναβ⎢ ⎥ 8π 2 ∫ ⎣ 3 ⎦ g2 ⎡ 1 ⎤ Tr⎢A d A AAA⎥. =∧+∧∧2 ⎣ ⎦ 8π ∫S3 3 Note we used the Jacobi identity in the above derivation. Also in the last line we have used the divergence theorem and wrote the result in a compact notation using wedge products. If you are unfamiliar with this notation rest assured we will use them sparingly. The integrand here is known as the ‘Chern–Simons action’ as it is the surface integral of the Chern–Simons form [13]. The field configurations that will be relevant are finite-energy solutions, so we require them to be well-behaved at infinity

⎛ 1 ⎞ Aggi −1 O⎜ ⎟. (3.46) μμ→∂ +⎝r2 ⎠

That is, we require that in some gauge the field vanishes at a rate faster than 1/r2 up to ‘pure gauge’ contributions. Field configurations that do not satisfy the finite- energy condition make no contribution to the functional integral so they can be ignored. We note that the integral of the Chern–Simons form is the volume integral of a total divergence. We can then use Gauss’ divergence theorem to turn this into an integral over the surface at spatial infinity. If we enter the pure gauge part into equation (3.45) we obtain g2 ⎡ 2i4 ⎤ μ μναβ ⎢ 21− − 1 −−−111⎥ ∂=μL dTriSμϵ gg ∂ναβ() ∂∂− gg gggggg ∂ μ ∂ α ∂ β 8π 2 ∫ ⎣ 3 ⎦ g2 ⎡ 2 ⎤ dTrSμναβ ⎢ gg−1 ggg −− 11 g gggggg −−− 111⎥ =−∂∂∂−∂∂∂2 μϵ ⎣ να()() β μ α β⎦ 8π ∫ 3 (3.47) g2 ⎡ 2 ⎤ μναβ −−−111 −−− 111 =dTrSμϵ ⎢ gggggg ∂∂∂−∂∂∂μαβ gggggg μαβ⎥ 8π 2 ∫ ⎣ 3 ⎦ g2 μναβ ⎣⎡ −−−111⎦⎤ =∂∂∂dTrSμϵ ggggggμαβ , 24π 2 ∫

3-9 A Pedagogical Introduction to Electroweak Baryogenesis

where in the second line we used the identity

111 ∂==∂xx10()()gg−−− =∂ x g g +∂ g ( x g ) . (3.48) The form of this is very interesting. Consider the volume element for the internal group

1 ⎡ 111⎤ dΩ=V Trddd.⎣()()()gg−−− ∧ gg ∧ gg ⎦ (3.49) 24π 2 ∫

So the particle current divergence is proportional to the volume integral over the internal group pulled back onto Euclidean space–time. In other words it is proportional to the number of times the ﬁeld winds around the group at the three- dimensional surface at spatial inﬁnity. If the integral of the Chern–Simons form is non-zero we therefore denote it with a ‘non-zero’ Chern–Simons number due to the connection with the winding number.

3.3 Winding number and non-abelian gauge groups The kind of field configuration that violates baryon number conservation is one with a non-zero ‘Chern–Simons number’, which we argued in the previous section is satisfied if the winding number is non-zero. In this section we will make things a little more concrete. The winding number is a concept from topology. As we only need surface level understanding of the concept we will only give a few examples to build intuition toward the subject. First consider the case where the internal space of a gauge field is a simple circle. Now consider the following mapping:

(i ) gννθ()θ = e . (3.50)

The winding number is deﬁned as how often this function winds around the circle. It is easy to show that the following equation is the winding number:

i 2π ddθθg−1 /d. (3.51) 2π ∫0 What is obvious also from explicit calculation is that the product of two mappings gives a winding number that is the sum of the winding numbers of each mapping. That is, if the winding number of one function, f ()θ is ν1 and the winding number of g()θ is ν2, then the winding number of fg()()θθis ν12+ ν . This just follows from the properties of exponential functions. However, let us prove this in a way that may seem needlessly complicated for this simple example, but will be useful for the more complicated case we are interested in. First we consider a small change in the winding number

δδλg = i( ). (3.52)

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It is easy to show that δν = 0, 1 i 2π d()gg− − δ νδν+=dθδ()gg + dθ 2π ∫0 dθ 21π ⎛ −− 1⎞ 1 ⎜ dggdδλ ⎟ =+ν δg −i (3.53) 2π ∫0 ⎝ dθ dθ ⎠ 1 2π dδλ =+ν dθ 2π ∫0 dθ = ν. Thus we have shown that the winding number is invariant under continuous transformations. So we see that the winding number of the product of two functions is just the sum of the two winding numbers. Now consider the case where the gauge ﬁeld has a local SU(2) symmetry. In the previous section we claimed that the winding number for this group is given by 1 νμαβ ⎣⎡ −−−111⎦⎤ ν =∂∂∂dTrSνϵ ggggggμαβ , (3.54) 24π 2 ∫ which we will prove to be the winding number in a way completely analogous to the case of a circle. It is straightforward by direct substitution to take a conﬁguration 4 with winding number of unity, g =·σ x/r (where of course σ = 12), and show that it indeed gives one when inserted into the above equation. So we can prove it is the winding numbing by demonstrating that winding number of two maps is the sum of their individual winding numbers using the same techniques we used for the circle. Once again we begin with considering a small change in the winding number. Recall the identity that

11 ()∂=−∂xxgg−− g() g , (3.55) which we can use to write

νμαβ ⎣⎡ −1 aa⎦⎤ δν=∂∂∂∫ d σν ϵ Trμαβgg δλ ( x ) τ . (3.56) This vanishes upon integration by parts. It is then trivial to see that our expression for ν is indeed the winding number as one can then continuously deform our expression for g such that it is equal to one for the upper part of the hypersphere, and deform another mapping of winding number one such that it is equal to one for the lower part of the hypersphere. The winding number of the product of these two mappings is then 2. A similar process can produce a mapping gν with winding number ν. So what does this mean? If we consider the SU(2) gauge fields in four dimensional Euclidean space and track their value as we walk around spatial infinity, if the field winds around its internal group at least once in this walk then baryon and lepton number is violated. So to understand the process of baryon number violation we will then seek to understand these topologically interesting processes. Specifically we will find in the following sections that we are interested in

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lumps of ﬁeld, from the perspective of four-dimensional Euclidean space, called instantons, and unstable temporally localized lumps called sphalerons.

3.4 Solitons and instantons In the previous sections we demonstrated that field configurations that violate baryon or lepton number must be topologically non-trivial—that is they must have non-zero Chern–Simons number. In this section we will demonstrate that topologically interesting field configurations generically are associated with quasi-particles. Let us begin with the simplest example. Consider a simple scalar field in 1 + 1 dimensions with degenerate minima. For simplicity and clarity we will set a bunch of parameters in this theory to one to avoid them cluttering our equations as we only wish to discuss some qualitative features of the theory. The ‘uncluttered’ version of the Lagrangian for 1 + 1-dimensional scalar field theory is ⎛ ⎞22⎛ ⎞ 1 dϕϕ1 d 1 2 3 ⎜ ⎟ ⎜ ⎟ ()ϕ2 1. (3.57) =−−−2 ⎝ dtx⎠ 2 ⎝ d ⎠ 4 Now consider finite-energy solutions to the action that are stationary in time (that is we set the d/dϕ t term to zero). Without putting pen to paper we can already make remarks about the types of solutions we will find. The key to this is knowing that if the solution is finite then the field must take the value of one of the minima at plus and minus spatial infinity. This gives us two types of solution, a field that has the fi same minima at ±∞ and a eld that goes from one minimum at −∞ to another at fi +∞. The second class of solutions is topologically non-trivial. Consider a speci c example of the second type of solution to the classical equations of motion ⎡ ⎤ ϕ(xxx )=± tanh⎣ ( − 0 ) 2⎦ . (3.58) The energy profile of this solution is a highly localized lump of field. This can be seen if we write the energy density of the solution

1 2⎡ ⎤ ϵ()xxxsech⎣ (0 ) 2 ⎦ (3.59) =−2 which we depict in ﬁgure 3.2. This is known as a classical lump or a soliton. It is a quasi-particle whose existence derives from a degeneracy in the ground state. Although the solution given above is stationary, we can easily boost to a moving frame to obtain moving soliton solutions. Recalling that under a Lorentz transformation, ϕϕ()xx→Λ ( ), we can write the following class of moving solitons ⎡ ⎛ ⎞⎤ ⎢ 1 ⎜()xx−−0 vt⎟⎥ ϕ(xt , )= tanh . (3.60) ⎣⎢ ⎝ 2 ⎠⎦⎥ 2 1 − u

In the Standard Model we have a very large degeneracy in the ground state since one can have different winding numbers. Field conﬁgurations that are topologically interesting in an analogous way to the soliton described above will also look like

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Figure 3.2. Depiction of a soliton in a single spatial dimension for the simple case of two degenerate minima at ±1. The energy profile is highly localized which justifies its status as a quasi-particle. localized lumps in the field except in all four Euclidean dimensions. These lumps are known as instantons since they are localized in time. If you reverse the Wick rotation, returning to 3 + 1 dimensions, these instantons look like tunneling processes [14]. Having more dimensions and a more complicated degeneracy means we cannot immediately write down the instanton solution as we did with the soliton. In the case where we have SU(2) gauge symmetry, it helps to first show that the instanton is a solution to the classical equations of motion—it is a local minimum of the action. First take the field strength tensor and make use of the trivial identity ⎡ ⎤ 44μν μν αβ dTrxFFμν = dTr x⎣⎢ FFFFμν ˜˜αβ ⎦⎥ ∫∫ (3.61) 4 ⎡ μν ⎤ ⩾ ∫ dTrxFF⎣ μν ˜ ⎦ . This means that

8π 2 dTr4xFFμν ν (3.62) ∫ μν ⩾ g2 fi with the equality being satis ed for FF= ˜. To construct an ansatz for an instanton we have the conditions that the winding number is 1, the configuration is finite- energy and the solution is a local extrema of the action. For simplicity let us ignore the gauge coupling constant by setting it to 1 and write the ansatz

21− Afrgxgxμμ=∂i()(),() (3.63)

2 where f is a radial function that goes to 1 as r →∞ faster than 1/r . We can substitute this ansatz into the classical equations of motion (or equivalently the ﬁ equation FF= ˜)to nd

22 2 2 2 fr()−=−′ fr () rf () r , (3.64)

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which is satisﬁed for r2 fr2 . (3.65) ()= 22 r + ρ The instanton is then [15, 16] ()xa2 −1 ⎣⎡ ⎦⎤ − Axμμ()()()=−∂− g x a gx a ()xa22ρ −+ (3.66) ρ2 ()x 1 . ϕ =+ 2 ()xa−

That Aμ()x winds around the group once is clear from the fact that it is a radial fi function times the pure gauge form for the ν = 1 gauge con guration. This can also be seen by explicitly putting the above solution into the Chern–Simons form (3.47). The above instanton tunnels between two neighboring vacua and in the process produces three leptons and three baryons. If all particles are Standard Model this means nine quarks and three leptons. Calculating the energy density of this solution once again shows a clump of field highly localized in space and time, as can been seen in figure 3.3. The amplitude for an instanton rate at zero temperature can be found by simply inserting the instanton solution into the path integral and making a saddle-point approximation. Doing so one finds that the instanton amplitude is proportional to 22 173 exp[−∼ 8π /g ] 10− [14] which is an incredible suppression! So at zero temperature it cannot possibly be important to explain the baryon asymmetry. Finally let us see how the ground state of the Standard Model can be thought of as a periodic – fi potential [17 19]. Let an eigenstate of de nite winding number be denoted by ∣n〉.

Figure 3.3. The energy proﬁle of an instanton as a function of r2 is also highly localized. This is the motivation behind thinking of the instanton as a quasi-particle in Euclidean space. Here the parameter ρ = 1.

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Figure 3.4. Standard Model ground state is actually periodic with a potential height of Esph. Each degenerate minimum corresponds to a different Chern–Simons number. A shift to the right from one minimum to another is the instanton, whereas the local maxima are the sphalerons. Each degenerate ground state has a different Chern–Simons number, NCS. Instantons move one to the right from one ground state to another with a higher Chern–Simons number whereas anti-instantons move one to the left.

The conjugate variable to n will then be a phase variable. An eigenstate of phase can be written as

∞ in θ = ∑ e.θ n (3.67) n=−∞ Let us consider the term 〈′∣θθexp [ −HT ] ∣〉. We of course have calculated the term 22 〈nHTnnHTnATg +1 ∣ exp [ − ] ∣〉−〈 + 1 ∣ ∣〉≈ exp[ − 8π / ]. We can make the approximation where we ignore multiple tunneling events and consider only single ﬁ tunneling transitions. De ning E()(θδθ−′=〈′∣∣ θ ) θH θ〉 we have ⎡ 2 ⎤ −8π EEA()θ 0 2 exp⎢ ⎥ cosθ . (3.68) =− ⎣ g2 ⎦

The ground state can be approximated as a periodic potential with a potential height proportional to the tunneling amplitude as depicted in figure 3.4. Let us conclude the section by summarizing what we have learned so far about baryon number violation in the Standard Model: • Classical conservation laws can be anomalously broken. • Triangle anomalies in the Standard Model result in an anomalous violation of both baryon and lepton number conservation. • The field configurations that anomalously violate baryon/lepton number are ones with a non-zero Chern–Simons number. This means they wrap around the internal space at the surface at spatial infinity in Euclidean space. • Fields with integer winding number at infinity are solutions to the classical equations of motion. • Fields with integer winding number look like localized lumps in Euclidean space.

3.5 The sphaleron Let us now try and bootstrap another topologically interesting quasi-particle that is a good candidate for baryon number violation. Speciﬁcally, the quasi-particle we will discuss is the sphaleron whose etymology is from the Greek word that means to ‘go down’, because it is an unstable particle [20, 21]. In the previous section we presented the instanton with a winding and Chern–Simons number of 1. It turns out

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that the instanton is less important than the sphaleron for our purposes because sphaleron rates become unsuppressed at high temperatures. As such we give it far closer attention. Physically, the instanton corresponds to tunneling through a barrier whereas a sphaleron corresponds to classically passing over the barrier. A sphaleron is an unstable particle corresponding to an instantaneous moment in time and is a solution to the classical equations of motion. It is due to its importance in the logical framework of this field that we give it such deep attention. It should be noted that we only focus on the lowest energy electroweak sphaleron before touching on the SU(3) sphaleron, since the lowest energy sphaleron will dominate. Of course we note that there is no formal proof that we are aware of that the sphaleron we present is indeed the lowest energy but it is widely believed to be so. To begin our bootstrap let us start with insisting that the Higgs field, which is an SU(2)L doublet, is well-behaved at infinity. Well-behaved in this case means the covariant derivative must vanish

Drμϕ →→∞0as . (3.69) This gives the condition

igA a a . (3.70) ∂=μϕτϕμ ∞ ∞ ∞ In this case we keep the gauge coupling constant as we will eventually calculate the sphaleron rate. We want ∂ϕ/0∂=t and we want a topologically non-trivial solution. So let us set to zero along with and map the space–time surface onto A0∣∞ d/dϕ t∣∞ ∞ — the internal space by setting Ai to be proportional to the right invariant one form also called the Maurer–Cartan form

⎛ 0 ⎞ ϕ∞ ⎜ ⎟ (3.71) =Ω⎝v 2 ⎠ and

i ∞ − 1 A ().i − (3.72) i = g ∂Ω Ω

We will use Ω instead of g here and throughout for an element of the gauge group to avoid confusion with the coupling constant. The Hermitian conjugate of the above is a solution. Both of these solutions can be verified by direct substitution. This is the first requirement of a sphaleron: that it reduce to the Maurer–Cartan form at infinity in order to be a finite-energy solution. Fields with integer winding number are known to be (multi-)instantons. The only other topologically interesting cases that can potentially violate baryon number are the cases where we have a non- integer and non-zero Chern–Simons number. It will turn out that sphalerons have a Chern–Simons number of 1/2. However, the Chern–Simons number was shown in the previous section to be related to the winding number, which is forced to be an integer. So it is something of a surprise that a non-integer Chern–Simons number is possible given its close relationship to the winding number. Indeed the

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Chern–Simons number of any vacuum state must be an integer so a non-integer Chern–Simons number means it must continuously deform from one vacuum state to another. To complete our bootstrap we explicitly make use of the Pauli matrices and make ﬁ our elds cover the internal group as we cover the spatial sphere at r =∞while passing through zero somewhere in space via use of a radial function. Therefore a suitable ansatz is ⎛ 0 ⎞ ϕ∞ hr() ⎜ ⎟ (3.73) =Ω⎝v 2 ⎠ and

i()fr ∞ − 1 A ()i − (3.74) i = g ∂Ω Ω with JG JG x σ . (3.75) Ω= · r

Also ffr≡ ()and h(r) are radial functions chosen to solve the equations of motion subject to the boundary conditions h(0)==f (0) 0 and h()∞=f () ∞=1. Let us now complete the motivation for studying closely this quasi-particle over the instanton, by demonstrating that these processes become unsuppressed when electroweak symmetry is restored at high temperature. We apologize to the reader for slightly skipping ahead in the flow of our logical argument by using a small amount of finite temperature field theory from the later parts of this primer. The reader should regard the form of the next few equations as divinely given and then return to this section after reading the later sections on finite temperature quantum field theory (QFT). The sphaleron rate can be found by substituting the above solution into the finite temperature version of the path integral. The high temperature sphaleron rate is given by ω − NN ETsph Γ=sph()()eVV rot trκ Δ , (3.76) 2π where Δ=ETSsph/() 3 ϕsph , where

⎡ ⎤ 1 321 a a S3⎣ϕϕ⎦ d xDi WW V(). ϕ(3.77) sph =++T ∫ 4 ij ij ﬁ Here Wij are the eld strength tensors for Ai,Wij≡∂ iAA j −∂ j i. With some effort we can reduce this to ⎛ ⎞ ⎡ ⎤ ΔEsph 4()πλvT S3⎣ϕ ⎦ B⎜ ⎟ (3.78) sph ≡ T = gT ⎝ g2 ⎠

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with

⎡ 2 ⎛ ⎞ ∞ ⎛ ⎞ λ ⎢ ⎜df ⎟ 8 22 B⎜ ⎟ =+−d4r ff(1 ) ⎝ g2 ⎠ ∫0 ⎣ ⎝drr⎠ 2 (3.79) 2 ⎛ ⎞2 ⎤ rhd 1 λ 2 ⎜ ⎟ hf22(1 ) rh22()1,⎥ +−+−2 ⎝ dr ⎠ 4 g2 ⎦

2 where rgvTx≡∣∣() . Note that when v()T = 0the rate is no longer exponentially suppressed. We will show in later chapters that electroweak symmetry is indeed restored at very high temperatures, meaning that at some stage in our cosmic history sphalerons, and therefore baryon number violation, were unsuppressed! This is the key feature of sphalerons which makes them attractive as a source of baryon number violation at very high temperature when electroweak symmetry is restored. There is one remaining thing we have to show to conclude our discussion of sphalerons in terms of showing that we indeed have a non-zero Chern–Simons number. Recall that the Chern–Simons form is ⎡ 2 ⎤ Tr⎢W d W WWW⎥ (3.80) ⎣ ∧+3 ∧∧⎦ where for A0 = 0, the above can be written

Tr [WW∧= d ] ϵϵijk WWW i()() ∂−∂=−∂−∂ j k j j ijk i W j j WW i k (3.81) a b c Tr [WWW∧∧ ] =−ϵϵijk abc WWWi j k and the Chern–Simons number for the sphaleron is

g2 Q(Sphaleron) d,30xK (3.82) = 2 32π ∫tt= 0 where K0 is the Chern–Simons form. Recall our ﬁeld conﬁguration,

2if W aσ a −1d (3.83) i =−g Ω Ω

∂Ω where Ω is given above and d is the external derivative dxi. Remarkably the Ω ∂xi form of f will not affect the Chern–Simons number as long as the boundary conditions given above are met. It is easy to see that the above equation reduces to

a 2f Wxd iiabbϵ x . (3.84) i =−g

We cannot calculate the Chern–Simons number in this form as the derivation of the Chern–Simons number assumes that our ﬁeld goes to zero faster than 1/r. Let us therefore perform a gauge transformation U exp[i (r )x ], where ()r is a =Θ·2 σ r Θ

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radial function that goes to zero at the spatial origin. Recall that a gauge transformation for non-abelian ﬁelds has the form ⎛ ⎞ a ⎜−2f ⎟ 1 gWσϵσaaikka U †−xUi()i UU . (3.85) i → ⎝ r2 ⎠ +∂ It is straightforward to show that in this gauge the sphaleron solution looks like xx a 2 ia ABC gW Aϵδiab x b B() ia r x i x a ai ai ai, (3.86) i =+−+Θ′≡++r2 where the prime denotes a radial derivative and 1 ⎡ ⎤ AAr() ⎣ 1 2fr ( ) cos( ) 1⎦ and ≡=2 () − Θ− r (3.87) 1 ⎡ ⎤ BBr() ⎣()12()sin().fr ⎦ ≡=r3 − Θ Such abbreviations may be disorientating but we will continue to use them throughout because it becomes far too cumbersome to write out every term a explicitly. The sphaleron now has the correct asymptotic behavior. Since W i is spherically symmetric the integral over angular variables in the calculation is trivial allowing us to rewrite the Chern–Simons number of the sphaleron as

2 g 20 Q(Sphaleron) = rrKd. (3.88) 8π ∫tt= 0 Using above equations and verifying this with equation (3.45) we can write ⎛ ⎞ 0 ⎜ a a 2g a b c⎟ KFWϵϵϵijk ijk abcWWW . (3.89) =−⎝ ij k + 3 i j k⎠

Let us ﬁrst calculate the ﬁrst term in the Chern–Simons density. Due to our gauge transformation it consists of three terms

a a a ϵijkFWij k ≡++(), Aak B ak C ak Wk (3.90) which in turn are deﬁned by the three terms we deﬁned before Aia, Bia and Cia as follows ⎡ AA⎤ Aak≡∂−∂ϵ ijk⎣ i ja j ia⎦ etc. (3.91)

With a bit of effort one can simplify the expressions for (ABCak,,) ak ak , reducing them down to the simple form

2 2 gA []4rxxAAδδka k a ka ak =−′+r (3.92) gBak =′+2[ϵiak x i B r 3] B

3-19 A Pedagogical Introduction to Electroweak Baryogenesis

i gCak =−2ϵiak x Θ′ . (3.93) a We now contract the above with W k which gives a sum of nine terms It is easy to show that these terms are

23B 2 A gAka ka =′4 BAr 2 C gAka ka = 0 gAka ka =Θ′4 A 8ABr2 + (3.94) 2222ABC gBka ka=−4[ Br ′ + 3 BAr ] gB ka ka = 0 gB ka ka = 0 222ABC 2 gCka ka=Θ′400. A r gC ka ka = gC ka ka =

So we have, in summary, the ﬁrst term in the Chern–Simons density calculated as

22 gr a a 1 54411 1 4 ϵijkFij Wk =′++ BA r ABr r A[3] B ′++Θ′ r B A r . (3.95) 8π 2ππ2 π π

2 a b c ﬁ Next we have the term g ϵϵijk abcWWWi j k . Let us rst calculate the product of two sphalerons contracted with the permutation symbols

a b ABCABC⎣⎡ ⎦⎤ ϵϵijk abcWWi j =++++ ϵabc ϵ ijk[ ai ai ai ] bj bj bj . (3.96)

It is useful to organize the terms into a matrix

⎛AA AB AC⎞ ⎜ ai bj ai bj ai bj⎟ 2 BA BB BC g ϵϵijk abc⎜ ai bj ai bj ai bj ⎟ ⎜ ⎟ ⎝ CAai bj CB ai bj CC ai bj ⎠

⎛ 2 ⎞ ⎜ 20xxAkc−Θ′ Aϵ kbcb x ⎟ (3.97) 22 2 ⎜ 02xxBrkc BΘ′()δ kc r − xx kc⎟ = ⎜ ⎟ 2 ⎝−Θ′AxBϵδkbc b Θ′() kc r − xx k c 0 ⎠ ≡ M.

Contracting the last sphaleron gives the following

2 a b c gWWWMMϵϵijk abc i j k =++[()()2()2()]11kc 22 kc M 23 kc + M 13 kc ABC (3.98) ×++[]kc kc kc ABC =++[][],μντkc kc kc kc ++ kc kc where we have again used the abbreviation

μντkc ≡+{(MM11 )kc ( 22 ) kc }, kc ≡ 2( M 23 ) kc , kc ≡ 2( M 13 ) kc . (3.99)

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We then have ⎛ ⎞ ⎛ ABC⎞ 22⎡ 22⎤ ⎜μμμkc kc kc kc kc kc⎟ ⎜ 002Θ′rA⎣ + rB⎦⎟ ACC⎜ ⎟ ⎜νννkc kc kc kc kc kc ⎟ = 24 , (3.100) ⎜ ⎟ ⎜ 04BrΘ′ 0 ⎟ ⎝τττkcABC kc kc kc kc kc ⎠ ⎝ 22 ⎠ 40Θ′Ar 0 so we can therefore write

22 gr a b c 1 421 62 ϵϵijk abcWWWi j k =Θ′+ rA rB Θ′. (3.101) 24π 4ππ4 In total we have

1 ⎡ 544 Q=−d2 r⎣ BA ′ r + 4 ABr − 2 r A [ B ′ r + 3] B (3.102) 4π ∫tt= 0

24262⎤ +Θ′+Θ′+42Ar rA 2. rB Θ′⎦ Simplifying the above equation we can then can the following solvable integral for the Chern–Simons number

1 Θ∞() Q(sphaleron) =ΘΘ−Θd[ cos] 2 (0) π ∫Θ (3.103) 1 = 2 as required. So let us summarize. The sphaleron is an unstable quasi-particle due to local- ization in time and it must continuously go between vacuum states. It is a ﬁnite- energy solution so it is well-behaved at the spatial boundary. The simplest solution is then derived by mapping the external space onto the internal space and coupling to a radial function that gives the desired properties and is determined by insisting that the sphaleron be a solution to the classical equations of motion. Such a sphaleron would be a local extremum of the action for any gauge group with an SU(2) subgroup. It has the nice property that the sphaleron rate is unsuppressed when electroweak symmetry is restored. It also has the unusual property that the Chern– Simons number is fractional and shown to be to equal 1/2. That it is non-zero means it facilitates baryon number non-conservation.

References [1] Coon S A and Holstein B R 2002 Anomalies in quantum mechanics: the 1/r2 potential Am. J. Phys. 70 513 [2] Holstein B R 2014 Understanding an anomaly Am. J. Phys. 82 591 [3] Adler S 1969 Axial-vector vertex in spinor electrodynamics Phys. Rev. 177 2426

3-21 A Pedagogical Introduction to Electroweak Baryogenesis

[4] Bell J S and Jackiw R 1969 A PCAC puzzle: πγγ0 → in the omega-model Nuovo Cimento A 60 47 [5] Jegerlehner F 2000 Facts of life with gamma (5) arXiv: 0005255(hep-th) [6] El-Menouﬁ B K and White G A 2015 The axial anomaly, dimensional regularization and Lorentz-violating QED arXiv: 1505.01754(hep-th)

[7] Ferrari R 2014 Managing γ5 in dimensional regularization and ABJ anomaly arXiv: 1403.4212 [8] Fujikawa K 1979 Path-integral measure for gauge-invariant fermion theories 42 18 [9] Donoghue J F, Golowich E and Holstein B R 2014 Dynamics of the Standard Model vol 35 (Cambridge: Cambridge University Press) [10] Rajaraman R 1982 Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory (North Holland: Amsterdam) [11] Coleman S 1988 Aspects of Symmetry: selected Erice lectures (Cambridge: Cambridge University Press) [12] Manton N and Sutcliffe P 2004 Topological Solitons (Cambridge: Cambridge University Press) [13] Chern S and Simons J 1974 Characteristic forms and geometric invariants Ann. Math. 99 48–69 [14] ’t Hooft G 1976 Symmetry breaking through Bell-Jackiw anomalies Phys. Rev. Lett. 37 8 [15] Wilczek F 1976 Inequivalent embeddings of SU(2) and instanton interactions Phys. Lett. B 65 160 [16] Belavin A A, Polyakov A M, Schwartz A S and Tyupkin Y S 1975 Pseudoparticle solutions of the Yang–Mills equations Phys. Lett. B 59 79 [17] Jackiw R and Rebbi C 1976 Vacuum periodicity in a Yang–Mills quantum theory Phys. Rev. Lett. 37 172 [18] Callan C G, Dashen R F and Gross D J 1976 The structure of the gauge theory vacuum Phys. Lett. B 63 334 [19] Polyakov A M 1997 Quark Conﬁnement and Topology of Gauge Theories B 120 429 [20] Manton N S 1983 Topology in the Weinberg-Salam theory Phys. Rev. D 28 8 [21] Klinkhamer F R and Manton N S 1984 A saddle-point solution in the Weinberg-Salam theory Phys. Rev. D 30 10

3-22 3.5 A brief review of thermal ﬁeld theory

In this section I give a brief review of finite temperature quantum field theory referring the reader to my book for a more in depth discussion. It is inappropriate to use zero temperature quantum field theory in a far from equilibrium setting of cosmic phase transition. I begin by deriving the closed time path contour [76, 77, 78, 79, 80, 81]. Recall the equilibrium density matrix of a quantum system [82] e βH ρ(0) = − . (3.2) βH Tre− Here the null argument of the density matrix refers to some time t after which the system departs from equilibrium. Time dependence can be introduced in the same way as ordinary quantum mechanics

ρ(t) = U(t, 0)ρ(0)U †(t, 0) (3.3) with the time evolution operator deﬁned as

t i dt00H(t00) U(t, t0) = T e− t0 . (3.4) R Note then that the density matrix itself can be written in terms of unitary operators U(T iβ, T ) ρ(0) = − . (3.5) TrU(T iβ, T ) − The time dependent evolution of an operator is then

TrU(t, 0)U(T iβ, T )U(0, t)A A(t) = − (3.6) h i TrU(T iβ, T ) − TrU(T iβ, T )U(T,T 0)U(T 0, t)AU(t, T ) = − (3.7) TrU(T iβ, T )U(T,T )U(T ,T ) − 0 0 This suggests that for a ﬁnite temperature out of equilibrium system I should use the generating functional of the form

Z[β, C ] = TrU (T iβ, T )U (T,T 0)U (T 0,T ) . (3.8) J JC − JC JC Let us take T . The above equation then implies that I take a time contour from to the time→t ∞infinitesimally above the real axis, drop infinitesimally below the real−∞ axis and then travel back to before making a trip perpendicular to the real line by an amount iβ. The result when−∞ calculating two point propagators is that there are now twice as many. One can have both fields on the top half of the time contour, both on the bottom, or one on each. Using causality, hermiticity and unitarity [83], one can constrain the form of these propagators. For example, scalar propagators take the form

T < 1 ∆ (p) ∆ (p) p2 M 2+i 0 < T¯ = − 1 ∆ (p) ∆ (p) 0 p2 M 2 i − − 1 1 0 Θ( p0) 2πi nB( p0 ) + − − | | 1 1 Θ(p0) 0 (3.9)

97 whereas the fermion propagators are

p+m ST (p) S<(p) / 0 = p2 M 2+i S<(p) ST¯(p) − p/+m 0 p2 M 2 i ! − − 1 1 0 Θ( p0) 2πi(p/ + m) nF ( p0 ) + − − . − | | 1 1 Θ(p0) 0 − (3.10)

For our purposes it is enough to know just the form of these propagators.

3.6 The effective potential at ﬁnite temperature

The Higgs potential receives temperature corrections which eventually lead to the deepest minimum being one where electroweak symmetry is intact [84]. It is at this temperature where there is an electroweak phase transition. In this brief section I apply what I just derived for describing finite temperature field theory to derive the finite temperature corrections to a simple scalar theory

m2 λ V = φ2 + φ4. (3.11) 2 4! It is easiest to calculate the derivative of the effective potential at one loop with respect the mass [85, 86, 87]. This means calculating a single diagram and I can insert our finite temperature propagators. In fact there is only one propagator to consider since the external legs must be on the positive time contour ∂∆V 1 d4p i = + 2πn (ω )δ(p2 m2) . (3.12) ∂m2(φ) 2 (2π)4 p2 m2(φ) + i B p − Z − The key observation here is that the one loop correction to the effective potential is a sum of the usual zero temperature piece and the finite temperature piece. Integrating over the delta function I can write the finite temperature piece I have ∂∆V d3p 1 T = n (ω) . (3.13) ∂m2(φ) (2π)3 2ω B Z The above equation can be integrated to give the thermal bosonic corrections to the effective potential in terms of the bosonic thermal function JB (forthcoming). Similarly the thermal corrections to the effective potential due to fermions can be written in terms of the fermionic thermal function JF . In total the thermal correction to the effective potential to one loop is

T 4 m2 T 4 m2 ∆V = J f + J b , (3.14) T 2π2 F T 2 2π2 B T 2 Xf Xb with

2 ∞ 2 √x2+z2 JB(z ) = dxx ln 1 e− , (3.15) 0 − Z 2 ∞ 2 √x2+z2 JF (z ) = dxx ln 1 + e− . (3.16) 0 Z 98 These thermal functions have a high temperature expansion π4 π2z2 πz3 J (z2) + , B ≈ −45 12 − 6 7π4 π2z2 J (z2) . (3.17) F ≈ 360 − 24 So ﬁnite temperature effectively give a temperature dependence to quadratic and cubic couplings in the effective potential. If the parameters are chosen right, the Higgs potential could evolve such that there remains a barrier between the true and false vacua as the Universe cools. In such a situation the phase transition would proceed through bubble nucleation [88, 89, 90, 91]. This is precisely the far from equilibrium setting one needs to catalyse baryon production in the EWBG mechanism.

3.6.1 Bubble nucleation The phase transition occurs when a 1/e fraction of the Hubble volume is in the new phase. This is known as the nucleation temperature. If I denote the tunneling probability as p(t) then via dimensional analysis I can write the relation

p(t)t4 1 . (3.18) ∼ I can also write the temperature dependent tunneling probability as follows

4 S /T p(T ) T e− E (3.19) ≈ where SE is the three dimensional Euclidean action which, assuming spherical symmetry, has the form

2 ∞ 1 ∂φ S = 4π r2dr i + V ( φ ) . (3.20) E 2 ∂r { i} 0 " i # Z X Assuming the false vacuum is located at the origin in ﬁeld space, the bounce satisﬁes the boundary conditions dφ i = 0, φ ( ) = 0 (3.21) dr i ∞ and of course is the solution to the classical equations of motion ∂2φ 2 ∂φ ∂V i + i = . (3.22) 2 ∂r r ∂r ∂φi Using the relation between time and temperature

45 M T 2t = p (3.23) 16π3 g r √ ∗ one can derive the condition for the nucleation temperature

SE TN = 170 4 ln 2 ln g∗ (3.24) T − 1GeV − N which for the electroweak phase transition satisﬁes S /T 140. E N ≈ 99 3.6.2 Transport equations Within the closed time path formalism, divergences of particle number currents can be related to the same particles self energies through the manipulation of Dyson– Schwinger equations. Systematically doing so results in a network of coupled transport equations which have the general form [92, 93, 94]

∂ J µ = Γijµ + SCP . (3.25) µ i − i i j X Let us begin with the Dyson-Schwinger equations

G˜(x, y) = G˜0(x, z) + d4w d4zG˜0(x, w)Σ(˜ w, z)G˜(z, y), Z Z G˜(x, y) = G˜0(x, z) + d4w d4zG˜(x, w)Σ(˜ w, z)G˜0(z, y), (3.26) Z Z where Σ(˜ ) is a matrix of self energies. To make use of these equations I hit both · equations with the Klein Gordon operators x and y respectively. I then subtract µ each equation from the other taking the limit of x y. The left hand side gives ∂µJ whereas the right hand side gives a matrix where all→ the elements are equivalent. The result is

x0 ∂ J µ = dz d3zTr [Σ>(x, z)G<(z, x) G<(x, z)Σ(z, x) µ − 0 − Z−∞ Z Σ>(x, z)G<(z, x) + G>(x, z)Σ<(z, x)] . (3.27) −

100 Chapter 4

Solving transport equations during a cosmic phase transitions

4.1 Introductory remarks

The behaviour of particle densities during a cosmic phase transition is described by a series of quantum transport equations which are a set of coupled non-linear inhomogeneous differential equations. These equations are generally either numerically solved or they are approximately solved using an assumption where certain interactions occur sufﬁciently fast that these interactions are in local equilibrium. However, it has been demonstrated that using this approximation can yield a baryon asymmetry two orders of magnitude different to the value acquired numerically [93]. As such there is ample motivation for a more reliable analytic approach. In this work I take the strategy of approximating the transport equations by a set of coupled linear differential equations solved in two regions. These linear differential equations are solved (semi-)analytically and then these solutions are matched at the phase boundary such that they satisfy continuity conditions. One can then take this solution and add a perturbation to give the true solution. Expanding everything in the transport equations to ﬁrst order one can then derive a new set of coupled linear inhomogeneous differential equations which can also be solved. This sets up a perturbative series. I compare our results to a purely numeric calculation from [95].

4.2 Declaration for thesis chapter 4

Declaration by candidate In the case of the paper present contained in chapter 4, the nature and extent of my contribution was as follows:

Publication Nature of contribution Extent of contribution 4 Single author 100% The undersigned hereby certify that the above declaration correctly reﬂects the nature and extent of the candidate and co-authors’ contributions to this work.

101

4.3 Published material for Chapter 4: General analytic methods for solving coupled transport equations: From cosmology to beyond

103 PHYSICAL REVIEW D 93, 043504 (2016) General analytic methods for solving coupled transport equations: From cosmology to beyond

G. A. White* ARC Centre of Excellence for Particle Physics at the Tera-scale, School of Physics and Astronomy, Monash University, Victoria 3800, Australia (Received 23 November 2015; published 4 February 2016) We propose a general method to analytically solve transport equations during a phase transition without making approximations based on the assumption that any transport coefficient is large. Using a cosmic phase transition in the minimal supersymmetric standard model as a pedagogical example, we derive the solutions to a set of 3 transport equations derived under the assumption of supergauge equilibrium and the diffusion approximation. The result is then rederived efficiently using a technique we present involving a parametrized ansatz which turns the process of deriving a solution into an almost elementary problem. We then show how both the derivation and the parametrized ansatz technique can be generalized to solve an arbitrary number of transport equations. Finally we derive a perturbative series that relaxes the usual approximation that inactivates vacuum-expectation-value dependent relaxation and CP-violating source terms at the bubble wall and through the symmetric phase. Our analytical methods are able to reproduce a numerical calculation in the literature.

DOI: 10.1103/PhysRevD.93.043504

I. INTRODUCTION as there are particle species in a given model so one may be left with a daunting number of coupled inhomogeneous The phase history of our Universe is unknown as it differential equations to solve. Therefore one usually depends on the full content of the scalar potential [1]. The proceeds either by solving the system numerically or by many possibilities have inspired a multitude of mechanisms using a set of approximations to reduce the number of to produce the observed asymmetry between particles and QTEs to a single one which is then solved in two regions antiparticles during a phase transition [2–5]. The observed with the solutions matched at the phase boundary. The baryogenesis of the universe (BAU) is estimated to be [6] former method is very cumbersome and the latter method involves using a set of fast rate approximations to reduce 7 3 2 5 10−11 nB ð . . Þ × ; BBN the set of coupled transport equations down to one [see e.g., YB ¼ ¼ : ð1Þ s ð9.2 1.1Þ × 10−11; WMAP [16,17] and the references therein for this treatment in the minimal supersymmetric standard model (MSSM)]. However, recently it has been shown that in the case of Although the methods derived in this work have general the MSSM with R-parity conservation, the assumption of application, we will be working most closely with the fast Yukawa and triscalar rates is infrequently justified [18]. electroweak-baryogenesis picture. In the electroweak bar- Furthermore, the practice of solving the equations in two yogenesis picture, the BAU is produced during the electro- regions effectively approximates the VEV profile with a weak phase transition [7]. To avoid electroweak sphalerons step function and ignores any effects of a thick bubble wall, from washing out any produced electroweak baryogenesis an assumption we refer to in this paper as the “ultrathin wall one requires this phase transition to be strongly first order approximation.” In this work we show how to analytically [8]. The particle dynamics become non-Markovian during go beyond these approximations by deriving an exact the far-from-equilibrium conditions that occur during a solution to a general set of QTEs. We begin by considering cosmic phase transition, rendering equilibrium quantum the set of QTEs that govern the MSSM during an field theory inadequate as a tool to analyze the behavior of electroweak phase transition derived under the assumption particle number densities throughout. One instead is of local supergauge equilibrium and null number densities required to use the closed time path (CTP) formalism – for weak bosons. This is admittedly not an exact system to [9 14]. The usual treatment involves deriving a set of begin with but it is useful as an example before we coupled quantum transport equations (QTE) involving CP- generalizing our method to larger systems of coupled conserving relaxation terms and CP-violating sources transport equations with multiple CP-violating source derived from scattering amplitudes evaluated using the terms. We also show how to analytically derive an CTP formalism [15,16]. There are in general as many QTEs expansion that goes beyond the ultrathin wall approximation. Moreover, we show that once one knows the form of *[email protected] the solution—which has a general form—one can more

2470-0010=2016=93(4)=043504(14) 043504-1 © 2016 American Physical Society G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) quickly derive the solution using a parametrized ansatz μ H T Q H C=P ∂μH −Γ Γ − − S : 5 ¼ H þ Y þ H~ ð Þ whose parameters are determined by direct substitution. kH kT kQ kH The layout of our paper is as follows. In Sec. II we use MSSM under supergauge equilibrium as an illustrative Our strategy will involve writing the above equation in example of how to derive an analytic solution to a set of what we call “cascading form” where the first equation is a QTEs without using a fast rate approximation. In Sec. III function of just two number densities, the second a function we show how to rederive this solution quickly using a of three and the third also a function of three number parametrized ansatz. We then generalize our methods to an, densities as well as the CP-violating source. In our example in principle, arbitrary set of transport equations with it is easy to write this set of equations in such a form, the multiple CP-violating sources in Sec. IV, before showing only complication being the existence of two space-time- how to go beyond the ultrathin wall approximation in dependent source terms. To overcome this, notice that both Sec. V. Finally in Sec. VI we compare our analytic solution sources have the same space-time dependence up to a to what one might get using the fast rate approximation for overall constant of proportionality one set of parameters as well as by numerically calculating the lowest-order correction to the ultrathin wall regime; our C=P 1 C=P conclusions are then presented in Sec. VII. S~ ¼ S ~ : ð6Þ t 2a H

II. THE MSSM: AN ILLUSTRATIVE EXAMPLE It is therefore straightforward to take a set of linear combinations that are in cascading form by taking the The transport equations for the MSSM under the combinations ð3Þþð4Þ, ð1 þ aÞ × ð3Þþð1 − aÞ × ð4Þþð5Þ assumption of supergauge equilibrium [19] as well the and 2 × ð3Þþð4Þþð5Þ. If we use the standard diffusion μþ ≈0 assumption that W have been derived many times approximation before (see for instance [16] and the references therein). Furthermore, under supergauge equilibrium, the transport μ 2 ∂μJ ¼ v n_ − D ∇ n ð7Þ equations have the same structure as a 2 Higgs doublet W n model up to wino-Higgsino-vacuum interactions. Thus, we just quote the result here. Under the assumption of super- and make an assumption about the geometry of the bubble gauge equilibrium we only need to write coupled transport wall, we can reduce the problem to a one-dimensional one. equation for the following number densities: Shifting to the bubble wall rest frame using the variable z ≡ jvwt − xj, we can write the set of coupled transport equations as a set of differential equations in z. A strategy Q ¼ nt þ nb þ nt~ þ n~ L L L bL for a solution becomes immediately apparent when the T n n~ equations are written in cascade form. The first equation is ¼ tR þ tR a differential operator acting on T and Q which can be 0 − − − 0 H ¼ nHþ þ nH nH nH u u d d solved for T (or equivalently Q) by treating Q and its

0 − − − 0 derivatives as a inhomogeneous source. Thus we can write nH~ þ þ nH~ nH~ nH~ : ð2Þ u u d d the second equation in terms of H and Q which can also be The set of coupled differential equations are solved in a similar way so that H is a function of Q. The third and final equation is then an equation for Q and the CP-violating source which can be solved via the usual μ þ T Q − T Q ∂μT ¼ Γ þ − Γ − methods. M k k M k k T Q T Q − Γ T − H − Q A. Deriving the analytic solution Y k k k T H Q Let us rewrite our linear combinations of transport 2Q T 9ðQ þ TÞ i Γ − C=P equations in terms of implicitly defined coefficients aXj þ SS þ þ St~ ð3Þ kQ kT kB where X ∈ fQ; T; Hg is the field index, j ∈ f1; 2; 3g is the equation number and i ∈ 0; 1; 2 is the power of the f g derivative in z, μ þ T Q þ T Q ∂μQ ¼ −Γ þ − Γ − M k k M k k T Q T Q i ∂i i ∂i 0 aQ1 Q þ aT1 T ¼ ð8Þ T H Q þ ΓY − − kT kH kQ i ∂i i ∂i i ∂i 0 aQ2 Q þ aT2 T þ aH2 H ¼ ð9Þ 2Q T 9ðQ þ TÞ − 2Γ − − SC=P SS þ t~ ð4Þ i ∂i i ∂i i ∂i Δ kQ kT kB aQ3 Q þ aT3 T þ aH3 H ¼ ðzÞ: ð10Þ

043504-2 GENERAL ANALYTIC METHODS FOR SOLVING COUPLED … PHYSICAL REVIEW D 93, 043504 (2016) − i ∂i In the above equation and throughout this paper we of T ¼ aQ1 k ð17Þ course use Einstein’s summation convention for repeated indices. The first equation can be used to solve for T in terms of Q if one uses the method of variable coefficients i i Q ¼ a 1∂ k: ð18Þ and treats the part of the equation involving Q and its T derivatives as an inhomogeneous source for T. The result of this is It is a trivial check to verify that these solutions indeed Z solve the first transport equation. Substituting these equa- X z i 1 1 κ −κ ∂ Q tions into Eq. (8) we have a differential equation that is now z y i − β T ¼ 2 e e aQ1 i dy i a function of k and H only, which means we can use the a 1 κ∓ − κ ∂y T same tricks, ð11Þ 0 i ∂i i ∂i i ∂i where the integration constants have to be zero as explained ¼ aQ2 Q þ aT2 T þ aH2 H ð19Þ in Appendix B. We neglect them for the rest of the derivation as they clutter notation and detract from the i j − i j ∂iþj i ∂i main derivation. The above equation has the problem that if ¼ðaQ2aT1 aT2aQ1Þ k þ aH2 H: ð20Þ we used it to eliminate T in subsequent equations we would be left with a set of equations that are a mixture of an The solution for H is of course integral and differential equations. We wish to have a pure differential equation at the end. To achieve this we use a κ Z X z z iþj 1 e −κ j j ∂ k series of variable changes. The derivation is somewhat y i − i H ¼ 2 e aQ2aT1 aT2aQ1 i dy: κ∓ −κ ∂y nontrivial so we give extra detail in Appendix A. Here we aH2 sketch out the main points of the calculation. We begin with ð21Þ the following change of variables: Z z −κ We can make the exact same changes of variables as before h ¼ e yQdy ð12Þ to solve this in terms of l (which is analogous to k in the solution to the first equation), from which we can eliminate the integral and the exponent in Eq. (11) at the cost of having T now defined in terms of X4 two functions. These functions are related via the identity − δ i j − i j ∂n H ¼ iþj−nðaQ2aT1 aT2aQ1Þ l ð22Þ n 0 κ −κ ¼ 0 ð − þÞz 0 hþ ¼ e h−: ð13Þ

To remove the exponential outside of the former integral we i ∂i k ¼ aH2 l ð23Þ use an additional change of variables

κ j ¼ e zh : ð14Þ where the Kronecker delta was added to make the structure of the solution more transparent and we include the sum We are not quite done; we would like to now write over n to make its limits obvious. Subbing our solution for everything in terms of a single variable. This can be k into the equations for Q and T gives achieved by either of the following variables: Z X4 i j i j n z H ¼ − δ − ða a − a a Þ∂ l k ¼ eκ∓z e−κ∓yj dy; ð15Þ iþj n Q2 T1 T2 Q1 n¼0 X4 − δ i j ∂n from which we can relate both j to k through the equation T ¼ iþj−naQ1aH2 l n¼0 0 4 j ¼ k − κ∓k: ð16Þ X δ i j ∂n Q ¼ iþj−naT1aH2 l: ð24Þ We now have a single variable k and it is straightforward to n¼0 derive expressions relating T and Q to derivatives in k.For the sake of convenience let us rescale k to remove the Equation (10) is now defined completely in terms of l and denominator and write the source,

043504-3 G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) 6 X for the broken and symmetric phase respectively; the values Δ δ i j k − i j k ðzÞ¼ iþjþk−nðaT1aH2aQ3 aQ1aH2aT3 xi can be derived from the equation n¼0 − i j k i j k ∂n −1 aT1aQ2aH3 þ aQ1aT2aH3Þ l x~ ¼ M d:~ ð29Þ X6 abc i j k n j−1 δ − ϵ a a a ∂ l ≡ α ¼ iþjþk n Ta Hb Qc The matrix M is given by Mij i where j doubles as an n¼0 exponent and an index which both go from 1 to 6 and X6 6 d~ ≡ ½0; …; 1=a T. Finally the integration constants β and ≡ an∂nl: ð25Þ l i l y are determined by the boundary conditions as follows. n¼0 i Since we began with a set of three second-order differential The use of the permutation symbol arises from the fact that equations for three densities we require that each of these i 0 number densities be continuous at the bubble wall along aH1 ¼ . The above is a straightforward inhomogeneous differential equation which can be solved using the stan- with their derivative. We also require the number densities dard method of variable constants. The solution is to vanish at ∞. All exponents therefore have to be positive definite in the symmetric phase. Therefore 6 Z X z α −α iz iyΔ − β l ¼ xie e ðyÞdy i ð26Þ yi ¼ 0 ∀ γi ≤ 0: ð30Þ i¼1 To ensure number densities go to zero in the broken phase in the broken phase and we have a condition on all positive definite exponents Z X6 ∞ γ −α iz β iyΔ ≡ ∀ α ≥ 0 l ¼ yie ð27Þ xi i ¼ xi dye ðyÞ Ii i : ð31Þ 0 i¼1 Finally our continuity conditions need to be met. The in the symmetric phase. Here the exponents αi and γi are the roots of the equations continuity conditions are not met by requiring l and its derivatives to be continuous at the phase boundary even X6 though this will give the right number of conditions. Instead nαn 0 0 0 0 al ¼ we require that H; T; Q; H ;T and Q are continuous at n¼0 z ¼ 0. As an example consider the case when the last three 6 X γi exponents are greater than 0 as well as the first three αi nγn 0 al ¼ ; ð28Þ exponents. The continuity conditions can be met when the n¼0 following equation is satisfied: T ð x4β4 x5β5 x6β6 x1β1 x2β2 x3β3 y1 y2 y3

0 1−10 1 100000000 I1 B C B C B 010000000C B I2 C B C B C B 001000000C B C B C B I3 C B C B C B A α4 A α5 A α6 A α1 A α2 A α3 A γ1 A γ2 A γ3 C B 0 C B Qð Þ Qð Þ Qð Þ Qð Þ Qð Þ Qð Þ Qð Þ Qð Þ Qð Þ C B C B C B C ¼ B α4AQðα4Þ α5AQðα5Þ α6AQðα6Þ α1AQðα1Þ α2AQðα2Þ α3AQðα3Þ γ1AQðγ1Þ γ2AQðγ2Þ γ3AQðγ3Þ C B 0 C B C B C B α α α α α α γ γ γ C B 0 C B ATð 4Þ ATð 5Þ ATð 6Þ ATð 1Þ ATð 2Þ ATð 3Þ ATð 1Þ ATð 2Þ ATð 3Þ C B C B C B C B α4A ðα4Þ α5A ðα5Þ α6A ðα6Þ α1A ðα1Þ α2A ðα2Þ α3A ðα3Þ γ1A ðγ1Þ γ2A ðγ2Þ γ3A ðγ3Þ C B 0 C B T T T T T T T T T C B C @ A @ 0 A AHðα4Þ AHðα5Þ AHðα6Þ AHðα1Þ AHðα2Þ AHðα3Þ AHðγ1Þ AHðγ2Þ AHðγ3Þ 0 α4AHðα4Þ α5AHðα5Þ α6AHðα6Þ α1AHðα1Þ α2AHðα2Þ α3AHðα3Þ γ1AHðγ1Þ γ2AHðγ2Þ γ3AHðγ3Þ 0 1 0 1 −1 I1 13 3 0 B × C B C B ~ α ~ γ C B I2 C ≡ @ AXð Þ AXð Þ A B C: ð32Þ @ I3 A α~ α α~ γ ð AÞXð Þ ð AÞXð Þ 0

043504-4 GENERAL ANALYTIC METHODS FOR SOLVING COUPLED … PHYSICAL REVIEW D 93, 043504 (2016) It is now straightforward to derive the solutions to T, H be turned into a polynomial by multiplying through and Q with the denominators

6 Z i i i i i i X z 0 ¼ a 3∂ Q þ A ðαÞa 3∂ T þ A ðαÞa 3∂ H α −α Q T T H H α iz iyΔ − β H ¼ AHð iÞxie e ðyÞdy i X6 i¼1 i j k i j k ↦ δ − ða 1a 2a 3 − a 1a 2a 3 6 Z iþjþk n T H Q Q H T X z 0 α −α n¼ T ¼ A ðα Þx e iz e iyΔðyÞdy − β T i i i − i j k i j k ∂n αz i¼1 aT1aQ2aH3 þ aQ1aT2aH3Þ e 6 Z 6 X z X α −α α α iz iyΔ − β ≡ n∂n z Q ¼ AQð iÞxie e ðyÞdy i ð33Þ al e i¼1 n¼0 0 ¼ anαn: ð36Þ with known functions l Solving this gives the set of homogeneous solutions. X4 − δ i j − i j αn Repeating the above steps in the symmetric phase AH ¼ iþj−nðaQ2aT1 aT2aQ1Þ γz with ansatz X ¼ AxðγÞe gives the solutions in n¼0 that phase. X4 − δ i j αn (iii) Finally, solve the inhomogeneous equation AT ¼ iþj−naQ1aH2 0 n¼ X6 X4 i j k i j k 0 ¼ δ − ða a a − a a a δ i j αn iþjþk n T1 H2 Q3 Q1 H2 T3 AQ ¼ iþj−naT1aH2 : ð34Þ n¼0 n¼0 − i j k i j k ∂n aT1aQ2aH3 þ aQ1aT2aH3Þ lðzÞ: ð37Þ III. FAST REDERIVATION WITH A PARAMATERIZED ANSATZ The inhomogeneous solution is then trivially found to be The above derivation was cumbersome even for the 6 Z X z simple example we used and yet the solution was relatively α −α X A α x e iz e iyΔ y dy − β 38 simple. Now that we know the form of the solution, it is far ¼ Xð iÞ i ð Þ i ð Þ i¼1 more efficient to start with this form of the solution as a parametrized ansatz and use the differential equations to in the broken phase and work out the parameters of the ansatz. This is achieved by the following simple steps. Let the ansatz for number X6 γiz density X that solves the homogeneous version of our X ¼ AXðγiÞyie ð39Þ αz 1 differential equations be X ¼ AxðαÞe . To find these i¼ functions, Ax, as well as α do the following: in the symmetric phase as before. The values for xi can be (i) To determine the functions AxðαÞ substitute our solutions into the first two equations to get relations found by substituting Eq. (38) into the inhomogeneous between these functions. Doing so gives equation and insisting that the coefficients of any derivatives of ΔðzÞ are zero and the coefficient of ΔðzÞ on the left- − i αi hand side is 1. This gives the familiar expression aQ1 AT ¼ j AQ a αj −1 T1 x~ ¼ M d~ ð40Þ − i αi i αi ðaQ2 AQ þ aT2 ATÞ A ¼ : ð35Þ −1 T H j αj with M ¼ αj and d~ ¼½0; …; 0; 1=a6 as before. The aH2 ij i l integration constants yi and βi can be found as above by This differs from our earlier expression. However insisting that Q, T and H are well behaved at ∞ and all this just amounts to a rescaling of xi in our final three functions as well as their derivatives are continuous at expression. It is a trivial exercise to reverse this the bubble wall. The solutions for βi and yi are the same as rescaling and verify that the expressions AX are the in the previous section so we do not repeat it here. same as before. Remarkably, the above method of solving the set of αz (ii) Substitute our functions AxðαÞe into the third differential equations analytically is at least as fast as using equation with the CP-violating source switched an approximation that assumes the strong sphaleron and off. The result is a rational polynomial which can Yukawa rates are fast in order to reduce our set of transport

043504-5 G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) equations to a single differential equation which one then The solution to this equation is of course solves. X 1 1 κ H −ai ∂im e z ¼ l2 þ 2 κ − κ IV. GENERALIZATIONS aH2 ∓ Z z In this section we discuss how far generalizations of the −κ y 1 × e Δ1ðyÞdy − β ; ð47Þ above procedure can go. The generalization of our procedure is trivial to any case where one has only one CP- β 0 violating source and one can manipulate the transport with ¼ for reasons given in Appendix B. This solution equations into the cascading form feeds into the third equation which obtains a more complicated inhomogeneity as a result, D1½X1;X2¼0 i i a 3∂ m þ fðΔ1Þ¼Δ2; ð48Þ D2½X1;X2;X3¼0 ð41Þ m

. with . Z X 2 −1 z 0 ða Þ κ −κ 1 DN−1½X1;X2;X3 XN¼ Δ H i κi z yΔ − β fð 1Þ¼ aH3 e e 1ðyÞdy κ∓ − κ DN−1½X1;X2;X3 XN¼ΔðzÞ: ð42Þ iþ1 i 2 0 þ a κ Δ1ðzÞþa Δ ðzÞ : ð49Þ One can just use the parametrized ansatz approach defined H3 H3 1 in the previous section to very quickly derive the solution. Much more daunting are the cases where one either has The solution now has a very similar structure to before, multiple CP-violating sources or one cannot manipulate the Z transport equations into the above form. Fortunately in this X z α −α iz iy Δ − Δ − β section we will show that in principle an analytical solution m ¼ xie e ½ 2ðyÞ fð 1Þdy i ð50Þ can always be found even in the presence of such i complications and the parametrized ansatz approach gen- α α β erally reduces the difficulty of problems greatly. with xi, i, AXð iÞ and i determined as before. Thus in principle one can use the parametrized ansatz approach to simplify the problem here as well so long as one is careful CP A. Multiple -violating sources in deriving what to replace ΔðzÞ with. In the MSSM there are several CP-violating sources. Handling the extra CP-violating sources, however, was B. Other equation structures done by simply noting that they are all proportional to The equation structures that we encountered in the vðxÞ2β_ðxÞ. This situation becomes far more nontrivial in the MSSM were not the only possible structures that could case where there are numerous CP-violating source terms arise. In this brief subsection we would like to briefly proportional to different functions of z. For example the consider other possible equation structures. Consider an NMSSM has source terms that are a function of the equation where the linear terms are missing, singlet’s VEV profile, vSðzÞ, as well as the Higgs profiles vuðzÞ and vdðzÞ. Consider the set of transport equations 2 2 ∂X1 ∂ X1 ∂X2 ∂ X2 from the previous section with an extra CP-violating term, a1 þ a1 þ a1 þ a1 ¼ 0: ð51Þ 1j ∂z 1j ∂z2 2j ∂z 2j ∂z2 i i i i a 1∂ Q þ a 1∂ T ¼ 0 ð43Þ Q T The procedure is subtly different from before. The first step i ∂i i ∂i i ∂i Δ is the same as before where we solve for X1 by treating the aQ2 Q þ aT2 T þ aH2 H ¼ 1ðzÞð44Þ derivatives of X2 as inhomogeneous source,

i i i i i i ∂ ∂ ∂ Δ 1 aQ3 Q þ aT3 T þ aH3 H ¼ 2ðzÞ: ð45Þ a j1 1 − 2 z Z a a 1j 2 j1 z 2 y ∂ ∂ e a 1 X2 1 X2 If we solve the first transport equation as before the second X1 − e 1j a a : ¼ 2 2j ∂ þ 2j ∂ 2 ð52Þ equation becomes a1j z z

Δ i j − i j ∂iþj i ∂i Just as before we aim to remove the exponentials and the 1ðzÞ¼ðaQ2aT1 aQ1aT2Þ l þ aH2 H integral using a series of changes of variables. This time, i ∂i i ∂i ¼ al2 l þ aH2 H: ð46Þ though, our variables changes are

043504-6 GENERAL ANALYTIC METHODS FOR SOLVING COUPLED … PHYSICAL REVIEW D 93, 043504 (2016)

1 a Z 1j XN z 2 y a dX2 hðzÞ¼ e 1j dy ð53Þ X1 ¼ g1jXj þ f1 dy j¼2 . . 1 a . ¼ . Z 1j z − 2 y a jðzÞ¼ e 1j hðyÞð54Þ XN Xi ¼ gijXj þ fi j¼iþ1 which leaves us with a very similar form for the solution . . (after a suitable rescaling), . ¼ .

XN−1 ¼ gN−1;NXN þ fN−1: ð62Þ 1 0 1 00 X1 ¼ a2jj þ a2jj ð55Þ 1 This gives us a total of 2 NðN − 1Þ parameters to use. We − 1 0 − 1 00 can also take linear combinations of the N − 1 transport X2 ¼ a1jj a1jj : ð56Þ equations that do not contain the CP-violating source to 1 − 1 Another important extension is to consider the case where meet another 2 NðN Þ condition. Unfortunately this 3 − 1 − 2 we cannot reduce the first equation to a differential brings us short of the required 2 ðN ÞðN Þ conditions. equation involving only two number densities. In the In reality this analysis is probably unrealistically pessi- simplest case our first equation is a differential operator mistic for two reasons. First, while the initial set of acting on three number densities, transport equations indeed can have all N fields appearing in the relaxation terms of all equations, there are only two i i i i i i derivative terms initially in each transport equation. Thus it a1 ∂ X1 þ a2 ∂ X2 þ a3 ∂ X3 ¼ 0 ð57Þ j j j may be possible to make careful manipulations that evade the arguments given above. Second, the equations tend to in which case we can reduce this to an equation involving have a lot more structure than our naive analysis lets on. For only two number density by defining auxillary functions example a number of equations might have the same fðzÞ and gðzÞ as follows: relaxation term on the right-hand side so the combination of two transport equations with this term might remove X1ðzÞ¼g12X2ðzÞþg13ðzÞX3 þ fðzÞð58Þ 0 many coefficients, aij, for the price of one. Nevertheless it is useful to consider an unrealistically pessimistic scenario X2ðzÞ¼g23X3ðzÞþgðzÞð59Þ where all conditions need to be met. The lowest number of densities that are not guaranteed a path that converts our if a0 ≠ 0 and QTEs into a cascading form in this overly pessimistic kj scenario is 5. We then have the transport equations

X1ðzÞ¼g13ðzÞX3 þ fðzÞð60Þ k ∂k 0 ∀ ≤ 4 aji Xj ¼ i ð63Þ

k ∂k Δ X2ðzÞ¼g23X3ðzÞþgðzÞð61Þ aj5 Xj ¼ ð64Þ

with ak ≠ 0 ∀ i; j and k. We will now sketch out how a otherwise. One can then choose gij such that they cancel the ij i ∂i solution can be derived. Let us define terms a3j X3 in the above transport equation. We therefore have two techniques to convert a set of transport equations 3 to cascading form—taking linear combinations of equa- X3ðzÞ¼g34X4ðzÞþg35ðzÞX5 þ f1ðzÞð65Þ tions and using the method as described immediately 4 above. X4ðzÞ¼g45X5ðzÞþf1ðzÞð66Þ

in such a way as to remove X5 from the first equation. C. Arbitrary number of equations Throughout this argument the subscript on f will denote the Consider the case where we have N transport equations number of times a variable has been changed and the and a single CP-violating source. In general there are superscript will denote the original field that was replaced. 3 4 2 ðN − 1ÞðN − 2Þ conditions to satisfy to reorganize the Repeating this process to remove f1 and then the resulting 3 equations into a cascading form. The most general exten- f2 allows us to write the first equation in terms of two linear 1 sion of the use of auxiliary functions is to define the combinations of number densities which can be denoted f3 2 following: and f3. We can then solve this equation to write both in

043504-7 G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) −1 X terms of the variable k following the same steps as outlined i i A2 ¼ A ðαÞa α in previous sections. The second through fifth equations i αi j j1 a21 j≠2 now involve four variables only. One can again use the −1 X i i same trick to eliminate one number density from the second A3 A α a α ¼ i αi jð Þ j2 equation leaving us with something in the form (and we are a32 j≠3 only interested in a general form as we will use a para- −1 X i i A4 A α a α metrized ansatz approach shortly) ¼ i αi jð Þ j3 a43 j≠4 −1 X i ∂i i ∂i 3 i ∂i 4 0 i i a2 k þ a2 34 f4 þ a2 44 f4 ¼ : ð67Þ A5 A α a α ; 74 l fð Þ fð Þ ¼ i αi jð Þ j4 ð Þ a54 j≠5

One can then define the following changes in variables: which can be rewritten as a matrix equation

3 0 1 2 3 0 1 0 i i i i i i i i 1−10 i i 1 00 a21α a31α a41α a α a α f4 ¼ g34lk þ g34lk0þg34lk þ f5 ð68Þ A2 51 11 B C B i i i i i i i i C B i i C BA3 C B a22α a32α a42α a52α C B a12α C B C ¼ −B C B C: @ A @ i αi i αi i αi i αi A @ i αi A 4 0 1 2 00 4 A4 a23 a33 a43 a53 a13 f4 ¼ g44 k þ g44 k0þg44 k þ f5 ð69Þ l l l i i i i i i i i i i A5 a24α a34α a44α a54α a14α to cancel the second equation’s dependence on l (actually ð75Þ this gives us a couple too many parameters so we can set these to 1). This equation then can also be solved with the Inverting these equations and rescaling by multiplying 3 4 through by the denominators gives the very simple usual techniques to write f5 and f5 in terms of a new variable l. The remaining three equations are a function of structure 3 k, l and f5 only. We can use a similar trick to remove the X8 dependency on l in the third equation with the variables α ϵbcde i j k l δ αn A1ð iÞ¼ a2ba3ca4da5e iþjþkþl−n i n¼0 f3 ¼ g0 l þ g1 l0 þ g0 l00 þ f4 ð70Þ X8 5 53l 53l 53k 5 α − ϵbcde i j k l δ αn A2ð iÞ¼ a1ba3ca4da5e iþjþkþl−n i n¼0 1 X8 k ¼ gkll þ fk: ð71Þ α ϵbcde i j k l δ αn A3ð iÞ¼ a1ba2ca4da5e iþjþkþl−n i n¼0 The remaining equations are now in cascading form and X8 α − ϵbcde i j k l δ αn can be solved using the methods described above. The A4ð iÞ¼ a1ba2ca3da5e iþjþkþl−n i derivation of the solution to this overly pessimistic case n¼0 sketched above is very cumbersome. However, even in this X8 α ϵbcde i j k l δ αn case the solution is identical to what one would get if one A5ð iÞ¼ a1ba2ca3da4e iþjþkþl−n i : ð76Þ used a parametrized ansatz from the start. Using the n¼0 parametrized ansatz method we can write down the The analogous functions in the MSSM also have this form, solutions immediately without setting pen to paper, although using a permutation symbol in that case is probably overkill since there would only be two terms 10 Z X z α −α contracted with it and one of those terms is zero two out of α iz iyΔ − β Xj ¼ xiAjð iÞe e ðyÞ i ð72Þ three times. Nevertheless this seems to be the standard i¼0 form. The coefficients xi are determined as before by the equation with αi being the 10 roots to the equation αj−1 −1 ~ x~ ¼½ i d ð77Þ X5 k k 10 10 A ðαÞa 5α ð73Þ ~ 0 … 0 1 T j j i with d ¼½ ; ; ; =ða5mÞ with a5m being the coefficient j¼1 10 of α in the fifth equation once AjðαiÞ has been substituted in. The boundary conditions are determined by insisting α 1 ∞ 0 with A1ð Þ¼ and each field is well behaved at and Xj as well as Xj are

043504-8 GENERAL ANALYTIC METHODS FOR SOLVING COUPLED … PHYSICAL REVIEW D 93, 043504 (2016) 6 Z all continuous at the bubble wall, which means inverting X z α −α 0 δ iz iy ϵ − δ − δ β the equations 1l ¼ e xi e ½ l0 al3ðzÞ 1 i i¼0 6 Z ∀ γ ≤ 0 X z yi ¼ i α z −α y 0 Z δ2l ¼ e i x e i ½−δ1lδa ðzÞ − δ2β ð81Þ ∞ i l3 i 0 −αiy i¼ βi ¼ dye ΔðyÞ ∀ αi ≥ 0 X0 X 0 α β and so forth. Let us now turn our attention to the slightly ¼ Ajð iÞ i þ yn more complicated case where we no longer assume a is Xi;j nX small. The equation structure we will be left with is very 0 ¼ αiAjðαiÞβi þ γnyn: ð78Þ similar to the case when there are multiple CP-violating i;j n source terms that are not proportional to each other that we considered in Sec. IV. Let us define a series of error That such a cumbersome problem was reduced to an functions as before. The corrections to the densities are elementary one shows the power of the parametrized ansatz method. If we had multiple CP-violating source terms we H ¼ H0 þ δ1H þ δ2H þ … would then simply replace ΔðzÞ for f½Δ1ðzÞ; Δ2ðzÞand T ¼ T0 þ δ1T þ δ2T þ … solve as before. Q ¼ Q0 þ δ1Q þ δ2Q þ …; ð82Þ V. BEYOND AN ULTRATHIN where H0, Q0 and T0 solve the QTEs in the ultrathin wall WALL APPROXIMATION approximation. We can then look at terms that are first Consider the earlier example given in Sec. II. There are order only to set up a perturbative expansion. Using the fact VEV-dependent relaxation and source terms in the third that Q0;H0 and T0 solve the original set of QTEs we can transport equation that are assumed to switch off in the write QTEs purely in terms of their error functions, unbroken phase. For a thick-walled VEV profile this i ∂iδ i ∂iδ 0 approximation might become a poor one. We would like aT1 1T þ aQ1 1Q ¼ to derive an analytical method to go beyond step-function 0 0 0 δa ðzÞH0 þ δa ðzÞQ0 þ δa ðzÞT0 VEVs. The second transport equation (9) has VEV- H2 Q2 T2 i ∂iδ i ∂iδ i ∂iδ 0 dependent relaxation terms proportional to a. For now þaH2 1H þ aQ2 1Q þ aT2 1T ¼ let us simplify the situation by assuming that a is very small 0 0 0 δa ðzÞH0 þ δa ðzÞQ0 þ δa ðzÞT0 so we only need to seek corrections to the thin-wall H3 Q3 T3 i ∂iδ i ∂iδ i ∂iδ ϵ approximation in Eq. (10). We will return to the more þaH3 1H þ aQ3 1Q þ aT3 1T ¼ ðzÞ: ð83Þ general case later. To do so we define a series of error functions, This can be rewritten in the exact same form as the system we solved in Sec. IV with two CP-violating sources [i.e. ΔðzÞ¼ΘðzÞΔðzÞþð1 − ΘðzÞÞΔðzÞ ≡ Δ0ðzÞþϵðzÞ Eqs. (45)], 0 0 0 a 3ðzÞ ≡ a 3ðzÞΘðzÞþa 3ðzÞΘð−zÞ i ∂iδ i ∂iδ 0 l l l aT1 1T þ aQ1 1Q ¼ 0 − 0 − 0 0 Θ − ¼ al3ðzmaxÞ ½al3ðzmaxÞ al3ðzÞ þ al3ðzÞ ð zÞ i ∂iδ i ∂iδ i ∂iδ Δ aH2 1H þ aQ2 1Q þ aT2 1T ¼ 1ðzÞ 0 δ 0 ¼ al3 þ al3ðzÞ i ∂iδ i ∂iδ i ∂iδ Δ aH3 1H þ aQ3 1Q þ aT3 1T ¼ 2ðzÞ: ð84Þ lðzÞ¼l0ðzÞþδ1lðzÞþδ2lðzÞþ…: ð79Þ VI. A NUMERICAL COMPARISON Here l0ðzÞ solves the original transport equations in the In this section we look at how large an error in the ultrathin wall regime and δil is a correction of order i.We will take advantage of the fact that these error functions are baryogenesis is caused by taking the fast rate approxima- finite for all z. This allows the possibility of a perturbative tion and the ultrathin wall approximation. We also look to expansion. Our third transport equation can be written in reproduce a numerical result in the literature as a check of 1 Γ terms of the error functions, our method. Corrections of Oð = YÞ were discussed in [18] which found that the corrections to the baryon asymmetry − i i i could be of Oð1Þ in the case where Γ and Γ were similar a ∂ l0 þ a ∂iðδ1l þ δ2l þ …Þ Y M l3 l3 size in the broken phase. Since a numerical analysis on the δ 0 δ δ … Δ ϵ þ al3ðzÞðl0 þ 1l þ 2l þ Þ¼ ðzÞþ ðzÞ: ð80Þ effects of equilibrating top Yukawa and stop triscalar terms have already been studied in detail we mostly look at fast The corrections to m can be found order by order, rate corrections as a check. We are more interested in

043504-9 G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) TABLE I. The base set of parameters used for our numerical TABLE II. Table of thick wall corrections to the baryon study. The diffusion constants are taken from Ref. [27] and Δβ is asymmetry. taken from Ref. [28]. LW 250=T 100=T 25=T 10=T 2.5=T 6 6 100 110 DT =T DQ = DH =T δρE ρE B= B 0.18 0.15 0.14 0.14 0.12 Γ− 2 2 Γ 2 Γ− a 0.05 MðxÞ 0.5vðxÞ =T HðxÞ a M Γ 100 Y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vw 0.05 Lw =T 2 2 100 T 100 Δβ 0.015 v ¼ vu þ vd well known and can be found in Ref. [16], so we do not repeat it here. We find that our exact solution in the ultrathin ρE determining whether corrections that go beyond the ultra- wall approximation, B, differs from the approximate ρA thin wall approximation are large. In Table I we give the set solution, B by a factor of parameters we use for our numerical analysis. We also ρE − ρA need to define an appropriate space-time-dependent VEV B B ≈ 0.58 ð90Þ profile, ρE þ ρA B B 3x vðxÞ¼0.5v 1 − tanh which is consistent with the size of corrections found in L w [18]. The first-order corrections to the baryon asymmetry 3 arising from deviations from the ultrathin wall regime β 1 − 0 5Δβ 1 x ðxÞ¼ . þ tanh are typically moderate to small. Lw In Fig. 1 we show the correction to the density n for Δ 0 025β0 2 L ðxÞ¼ . ðxÞvðxÞ : ð85Þ several values for the bubble wall thickness. Even though We can then calculate numerical solutions to the the baryon asymmetry is weakly dependent on LW in the ultrathin wall regime, for the very large values of LW we densities Q, T and H which allow us to calculate nL ¼ consider this dependency becomes important. The typical Q þ Q1L þ Q2L ¼ 5Q þ 4T. This density, nLðzÞ, acts as a seed for the baryon density which satisfies the equation correction for our set of parameters is typically small but [20,21] nontrivial and grows monotonically with LW as expected, which is shown in Table II. Indeed recent work has shown n ρ00 − ρ0 − R ρ Γ F that bubble wall width for the electroweak phase transition Dq vw B ðzÞ B ¼ wsðzÞ nLðzÞð86Þ 2 in the NMSSM can have a large range of values [25].Asa with nF the number of fermion families, where the check on our calculations we find that the correction indeed relaxation term is given by goes to zero as Lw ↦ 0 although the absolute minimum is numerically very difficult to take as it involves evaluating 9 n −1 3 Γ 1 sq numerical integrals of very sharply peaked functions. RðzÞ¼ wsðzÞ 4 þ 6 þ 2 ð87Þ and nsq is the number of squark flavors. Not much is lost treating the weak sphaleron rate profile as a step function— that is compared to treating the VEV profile as a step function when calculating source and relaxation terms— although in principle one could use the same techniques as in Sec. V to derive corrections to this equation as well. Our sphaleron rates then have the form [22–24]

Γ 120 α5 Θ − ws ¼ T w ð zÞð88Þ Γ 128 3 α4 and ss ¼ð = ÞT S [23] respectively. The baryon asymmetry is then Z Γ 0 nF ws ρ − xR=vW B ¼ 2 nLðxÞe dx: ð89Þ FIG. 1. The density nL in the symmetric phase near the bubble vW −∞ wall as a function of z. The exact solution with first-order corrections to the ultrathin wall approximation is given by the As a point of comparison, we would like to compare our magenta line and the solution without such corrections is given in solutions with the analytic solution derived under the fast blue. The parameters used to calculate nLðZÞ are LW ¼ 250=T rate approximation in the ultrathin wall regime. The (top left), LW ¼ 100=T (top right), LW ¼ 25=T (bottom left) and 10 solution for nLðzÞ and ultimately the baryon number is LW ¼ =T (bottom right).

043504-10 GENERAL ANALYTIC METHODS FOR SOLVING COUPLED … PHYSICAL REVIEW D 93, 043504 (2016) discussions relating to this work. This work was supported through the J. L. Williams Scholarship, Australian Postgraduate Award Scholarship. In addition, part of this work was made possible through the Keith Murdoch Scholarship via the American Australian Association as well as the Center of Excellence for Particle Physics.

APPENDIX A: EXTRA DETAILS IN DERIVING THE SOLUTION In this appendix we give extra details for how to solve the FIG. 2. Reproducing Winslow and Tulin’s purely numerical calculation. Here the blue, magenta and golden lines correspond equation to the densities nQ3 , nu3 and nH respectively. i ∂i i ∂i 0 aQ1 Q þ aT1 T ¼ : ðA1Þ Finally for the sake of safety we would like a direct We can write T as a function of Q, using the usual methods comparison to a numerical calculation. We use the para- to write metrized ansatz approach to solve Eqs. (23) in Ref. [26] in order to produce their Fig. 3. This is shown in Fig. 2 where Z X z i 1 1 κ −κ ∂ Q we are able to reproduce almost an almost identical plot (we z y i − β T ¼ 2 e e aQ1 i dy i : κ∓ − κ ∂y remove the sum of densities to reduce clutter as little new aT1 information is given by this function). The small error can ðA2Þ probably be attributed to our use of an ultrathin wall approximation. The first change of variables we use is of course

VII. CONCLUSION −κ 0 h ¼ e Q ðA3Þ In this work we have sketched out a method to derive exact solutions to quantum transport equations in a variety which leads to the following identities: of cases including large sets of QTEs and situations where κ there are multiple CP-violating sources with nontrivially z 0 Q ¼ e h different space-time dependency. We have also produced a κ 0 κ 0 00 z very quick and powerful method using a parametrized Q ¼ð h þ hÞe 2 κ ansatz to derive the same solution. This method was found 00 κ 0 2κ 00 000 z Q ¼ð h þ h þ hÞe : ðA4Þ to turn cases which are cumbersome to the point of being almost intractable by standard methods to an elementary This allows us to write the integrand for T in terms of h problem. Furthermore a general structure of the solution and remove the integral, was made manifest when solving for large systems of κ Z equations. The third result of this paper was to sketch out a 1 X z z e −κ y way of going beyond the ultrathin wall approximation. Our T ¼ 2 e a κ∓ − κ numerical analysis demonstrated that the correction to the T1 0 κ 1 κ2 2 0 baryon asymmetry coming from the smallest correction to × ½ðaQ1 þ aQ1 þ aQ1Þh the ultrathin wall regime can be small to moderate but a1 2a2 κ h00 a2 h000 nonetheless nontrivial even for moderately thick walls. It þð Q1 þ Q1 Þ þ Q1 X κ would interesting to see if other models have larger 1 e z ¼ 2 corrections from thick bubble walls. Our numerical results κ∓ − κ aT1 analyzing corrections to the fast rate approximation were 0 κ 1 κ2 2 consistent with those found in [18] which found that these × ½ðaQ1 þ aQ1 þ aQ1Þh corrections can be very large. Our method ensures that 1 2 2 κ 0 2 00 þðaQ1 þ aQ1 Þh þ aQ1h : ðA5Þ these corrections are taken into account while still keeping to a largely analytical framework. To remove the exponents we use the change of variables

κ ACKNOWLEDGMENTS z j ¼ e h ðA6Þ I would like to acknowledge Csaba Balazs, Peter Winslow and Michael Ramsey Musolf for some useful from which we can derive the usual identities

043504-11 G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) −κ 2 1 0 z 00 0 h ¼ e j Q ¼ aT1k þ aT1k þ aT1k ðA16Þ −κ h0 ¼ðj0 − κ j Þe z as before. Finally, substituting in our rescaled expression 2 −κ 00 00 − 2κ 0 κ z h ¼ðj j þ jÞe : ðA7Þ for k we get X 1 We can also use the second line in the above equation to 2 000 1 − κ 2 00 T ¼ ½aQ1k þðaQ1 ∓aQ1Þk κ∓ − κ derive an identity that relates Q to both new variables j and its derivative, 0 − κ 1 0 − 0 κ × ðaQ1 ∓aQ1Þk aQ1 ∓k 0 − κ − 2 00 − 1 0 − 0 Q ¼ j j: ðA8Þ ¼ aQ1k aQ1k aQ1k ðA17Þ

Our expression for T is now free of any exponentials, as required.

1 X 1 0 1 0 2 00 APPENDIX B: A NOTE ON EXTRA T ¼ 2 ½aQ1j þ aQ1j þ aQ1j: ðA9Þ κ∓ − κ aT1 INTEGRATION CONSTANTS Throughout this work we have ignored extra integration We would like to write both T and Q in terms of a single constants that are produced in the process of reducing a set variable. This task is achieved by the choice of QTEs down to a single differential equation. The Z contexts in which we have ignored these constants vary z κ∓z −κ∓y but throughout we reference this appendix since the reason k ¼ e e jdy ðA10Þ is the same. To demonstrate this let us consider the simplest nontrivial example. Let us have a system which can be which can be inverted to write described by the following set of QTEs:

0 − κ i i i i j ¼ k ∓k: ðA11Þ a11∂ X1 þ a21∂ X2 ¼ 0 i i i i a ∂ X1 þ a ∂ X2 ¼ ΔðzÞ: ðB1Þ Both expressions for Q are now satisfied, 12 22

Solving the first equation for X1 after using the usual tricks 0 − κ Q ¼ jþ þjþ we have 00 − κ 0 − κ 0 − κ ¼ðk −k Þ þðk −kÞ κ 1 X z i i e −κ z 00 − κ κ 0 κ κ X1 −a ∂ k e β ¼ k ð þ þ −Þk þ þ −k ðA12Þ ¼ 21 þ 2 a11 κ∓ − κ X κ ↦ − i ∂i zβ and a21 k þ e 0 − κ i i Q ¼ j− þj− X2 ¼ a11∂ k: ðB2Þ 00 0 0 k − κ k − κ− k − κ k ¼ð þ Þ ð þ Þ β The extra integration coefficients should immediately 00 − κ κ 0 κ κ ¼ k ð þ þ −Þk þ þ −k: ðA13Þ rouse suspicion since the system we started with was a set of two coupled second-order differential equations and κ i We would like to write in terms of the coefficients aX1; such a system is completely specified by four boundary pffiffiffiffiffiffiffiffiffiffiffiffiffi conditions for every region it is solved in. Therefore −a1 ða1 Þ2 − 4a0 a2 solving the equations including the use of variable changes κ ¼ T1 T1 T1 T1 ðA14Þ 2a2 should not introduce the need for more initial conditions. T1 As an analogy we could make a strange change of variables which leads to the identities that shifts Q by a constant c. If we solve the system we should find that consistency demands that c ¼ 0 (rather 1 than another condition being produced that specifies c)or κ κ − aT1 þ þ − ¼ 2 that c is redundant. These are the same requirements that a 1 β T will be imposed on . Before we demonstrate that the 0 β ¼ 0 let us first show that they can be removed by κ κ aT1 þ − ¼ 2 : ðA15Þ another variable change. Specifically one can take aT1

Rescaling k we get k ¼ k2 þ f ðB3Þ

043504-12 GENERAL ANALYTIC METHODS FOR SOLVING COUPLED … PHYSICAL REVIEW D 93, 043504 (2016) where f is any function that satisfies the equations Δ0ðzÞ; Δ00ðzÞ and Δ00 are null while the coefficient of X ΔðzÞ is unity. This gives κ − i ∂ − zβ a21 if ¼ e X xi ¼ 0 i i a11∂ f ¼ 0: ðB4Þ Xi αixi ¼ 0 The solution is then Xi X 2 κ α x ¼ 0 i i f ¼ Ae ðB5Þ i X α3 2 2 − 2 2 −1 i xi ¼ða11a22 a12a21Þ ; ðB9Þ with i

β as in the usual case. However, we now have the additional A ¼ : ðB6Þ i i condition that the coefficient of exp κ−z β−B−; this gives a21κ ¼ the extra condition One can then just substitute the functions Q and T in terms X xi j j of k2 into the second equation and derive the usual solution. k − k κiþj 1 κ − α ða22a11 a12a21Þ − ¼ ðB10Þ For completeness we nonetheless consider the case where i − i β we keep in the set of equations. Substituting these solutions into the second equation gives which gives an overdetermined set of equations unless X β− ¼ 0. Alternatively we could make use of the last line in j j κ i − i ∂iþj i κi zβ Δ Eq. (B9) to write ða22a11 a12a21Þ kþ ða12 Þe ¼ ðzÞ X 4 Z j j κ X z ↦ i − i ∂iþj zβ Δ α −α κ ða22a11 a12a21Þ kþ Be ¼ ðzÞ: ðB7Þ iz iy Δ − β y − β k ¼ xie dye ½ ðyÞ AiB e i i¼0 κ Treating the term proportional to exp z as a inhomoge- ðB11Þ neity one could naively write with 4 Z X z α −α κ iz iy Δ − β y − β κ k ¼ xie dye ½ ðyÞ B e i : i j iþj A α κ− − α a a : i¼0 i ¼ ið iÞ 12 21 2 2 2 2 ðB12Þ a12a21 − a11a22 ðB8Þ Upon substituting this solution back into Eq. (B7) we κ 0 We can immediately see that þð−Þ ¼ in the broken indeed get that the equation is satisfied. However, X1 and κ−z (symmetric) phase in order to have a well-behaved function X2 are now independent of e . So, depending on the at infinity. As for the β− term let us try and derive the treatment of β−, it is either zero for self-consistency or coefficients xi; we will find an inconsistency unless β− ¼ 0 irrelevant in that it is a redundancy in the theory with now (we expect such an inconsistency due to the arguments physical consequences. For our purposes it is most con- given earlier). We demand that the coefficients of venient to set it to zero.

[1] A. D. Linde, Rep. Prog. Phys. 42, 389 (1979). [6] S. Eidelman et al. (Particle Data Group Collaboration), [2] M. Trodden, Rev. Mod. Phys. 71, 1463 (1999). Phys. Lett. B 592, 1 (2004); D. N. Spergel et al. (WMAP [3] R. Allahverdi and A. Mazumdar, New J. Phys. 14, 125013 Collaboration), Astrophys. J. Suppl. Ser. 148, 175 (2003). (2012). [7] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Annu. Rev. [4] S. Inoue, G. Ovanesyan, and M. J. Ramsey-Musolf, Nucl. Part. Sci. 43, 1 (1993). arXiv:1508.05404 [Phys. Rev. D (to be published)]. [8] M. Quiros, arXiv:hep-ph/9901312. [5] T. Liu, M. J. Ramsey-Musolf, and J. Shu, Phys. Rev. Lett. [9] J. Schwinger, J. Math. Phys. (N.Y.) 2, 407 (1961). 108, 221301 (2012). [10] K. T. Mahanthappa, Phys. Rev. 126, 329 (1962).

043504-13 G. A. WHITE PHYSICAL REVIEW D 93, 043504 (2016) [11] P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. (N.Y.) [21] J. M. Cline, M. Joyce, and K. Kainulainen, J. High Energy 4, 1 (1963). Phys. 07 (2000) 018. [12] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964). [22] D. Bodeker, G. D. Moore, and K. Rummukainen, Phys. Rev. [13] R. A. Craig, J. Math. Phys. (N.Y.) 9, 605 (1968). D 61, 056003 (2000). [14] K. C. Chou, Z. B. Su, B. L. Hao, and L. Yu, Phys. Rep. 118, [23] G. D. Moore, Phys. Rev. D 62, 085011 (2000). 1 (1985). [24] G. D. Moore and K. Rummukainen, Phys. Rev. D 61, [15] A. Riotto, Phys. Rev. D 58, 095009 (1998). 105008 (2000). [16] C. Lee, V. Cirigliano, and M. J. Ramsey-Musolf, Phys. Rev. [25] J. Kozaczuk, S. Profumo, L. Haskins, and C. L. Wainwright, D 71, 075010 (2005). J. High Energy Phys. 01 (2015) 144. [17] P. Huet and A. E. Nelson, Phys. Rev. D 53, 4578 (1996). [26] P. Winslow and S. Tulin, Phys. Rev. D 84, 034013 [18] V. Cirigliano, M. J. Ramsey-Musolf, S. Tulin, and C. Lee, (2011). Phys. Rev. D 73, 115009 (2006). [27] M. Joyce, T. Prokopec, and N. Turok, Phys. Rev. D 53, 2930 [19] D. J. H. Chung, B. Garbrecht, M. J. Ramsey-Musolf, and S. (1996). Tulin, J. High Energy Phys. 12 (2009) 067. [28] J. M. Moreno, M. Quiros, and M. Seco, Nucl. Phys. B526, [20] M. Carena, M. Quiros, M. Seco, and C. E. M. Wagner, Nucl. 489 (1998). Phys. B650, 24 (2003).

043504-14

Chapter 5

Solving bubble wall proﬁles

5.1 Introductory remarks

To calculate the rate a false vacuum decays, one is required to calculate a solution to the classical equations of motion known as the bounce. The bounce is the solution where the field starts near the false vacuum and asymptotes to the true vacuum. Finding the bounce is necessary if one wishes to examine both phase transitions and vacuum stability [96, 97, 98, 99, 100, 101, 102, 103, 104]. Unfortunately, the bounce solution is notoriously difficult to calculate. Although solving a set of coupled non linear inhomogeneous equations numerically is typically a moderate but nonetheless tractable challenge, the situation is complicated by the fact that there is an easily found trivial solution where the field stays in one vacuum or the other. For a lot of numerical methods this means that the trivial solution is a powerful basin of attraction making the bounce solution difficult to find [105]. I provide a new method for performing such a calculation by first curve fitting the bounce solution in the entire parameter space of the most general renormalizable tree level potential of a single field. I then propose, similarly to another algorithm [106], that the bounce solution of a multifield problem can be approximated by the solution to a single field problem which I have in analytic form. Specifically the solution to the multifield problem is the one dimensional ansatz plus some arbitrary function which I refer to as a perturbation. The perturbation is then given by the solution to a set of linearized differential equations due to similar methods given in the previous chapter. This sets up a fast converging perturbative series. I test our method against a known result.

5.2 Declaration for thesis chapter 5

Declaration by candidate In the case of the paper present contained in chapter 5, the nature and extent of my contribution was as follows:

119 Publication Nature of contribution Extent of contribution Came up with the idea and derivation, wrote first draft of paper and implemented numerical treat- 6 70% ment of multifield case. Involved in discussions throughout. The following coauthors contributed to the work. If the coauthor is a student at Monash, their percentage contribution is given: Author Nature of contribution Extent of contribution Set up numeric code for single field case and Csaba Balazs involved in curve fitting. Involved in discussions throughout. Edited draft and worked on figure presentation. Involved with initial discussion and ideas and Sujeet Akula worked heavily on making the ideas in the paper more transparent. Involved in discussions throughout. The undersigned hereby certify that the above declaration correctly reflects the nature and extent of the candidate and co-authors’ contributions to this work.

120

5.3 Published material for chapter 5: Semi-analytic techniques for calculating bubble wall proﬁles

122 Eur. Phys. J. C (2016) 76:681 DOI 10.1140/epjc/s10052-016-4519-5

Regular Article - Theoretical Physics

Semi-analytic techniques for calculating bubble wall proﬁles

Sujeet Akula1,2,a, Csaba Balázs1,2,3,b, Graham A. White1,2,3,c 1 School of Physics and Astronomy, Monash University, Melbourne, VIC 3800, Australia 2 ARC Centre of Excellence for Particle Physics at the Terascale, Monash University, Melbourne, VIC 3800, Australia 3 Monash Centre for Astrophysics, Monash University, Melbourne, VIC 3800, Australia

Received: 22 September 2016 / Accepted: 15 November 2016 / Published online: 9 December 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We present semi-analytic techniques for find- solution to an ansatz for an arbitrary potential. Since the ing bubble wall profiles during first order phase transitions ansatz is only an approximate solution, in the next step we with multiple scalar fields. Our method involves reducing the derive a perturbative expansion that converges to the exact problem to an equation with a single field, finding an approx- solution. For each term in the perturbative series we provide imate analytic solution and perturbing around it. The pertur- a semi-analytic solution. Since our starting potential is arbi- bations can be written in a semi-analytic form. We assert that trary, and because the convergence of the perturbative expan- our technique lacks convergence problems and demonstrate sion is independent of the potential, our method is completely the speed of convergence on an example potential. general. To derive the ansatz we take advantage of the fact that the multi-field problem can be approximated by finding the solu- 1 Introduction tion to a single-field potential. The approximate tunneling solution can be found along the field direction that connects The decay of a false vacuum is a complex problem with the true and false vacua. This single-field potential can be numerous applications in cosmology [1–6] and is particularly solved in terms of a single parameter. To improve the initial important in the study of baryogenesis [7–26] (although there ansatz we compute correction functions to the ansatz in a are mechanisms for producing the baryon asymmetry that do manner analogous to Newton’s method of finding roots. The not require calculating the decay of the false vacuum [27– result is a perturbative series of corrections that are expected 33]). Calculating tunneling rates is also an important problem to converge quadratically. in the study of vacuum stability [34–40]. The differential equations that define these corrections While the physics of the tunneling process is qualitatively can be solved analytically in terms of eigenvalues of the well understood [41], quantitatively it is a complicated prob- mass matrix and a function of the initial ansatz. In doing lem that involves solving a set of highly nonlinear coupled so we use techniques that were recently employed to ana- differential equations usually requiring a numerical solution. lytically solve number densities across a bubble wall [49]. The two techniques that are most commonly used to solving Although the technique has elements in common with New- the tunneling problem are path deformation [42,43] and min- ton’s method it does not share its trouble with null derivatives imizing the integral of the squared equations of motion for a giving divergent corrections or oscillations around the solu- set of parametrized functions [44,45], although other meth- tion. We also argue that the other problems with Newton’s ods also exist [46]. Exact solutions only exist for specialized method are generically not relevant to our method. cases [47,48]. The layout of this paper is as follows. In Sect. 2 we give In this paper we offer a new approach by solving the tun- a brief overview of the false vacuum problem. In Sect. 3 we neling problem semi-analytically. First, we give an analytic develop an ansatz form that approximately solves a general variety of multi-field potentials with a false vacuum, where the potential is specified by a single parameter. In Sect. 4 we derive the perturbative corrections to the ansatz forms and discuss the convergence. In Sect. 5 we use this method to a e-mail: [email protected] solve a problem which can be directly compared with the b e-mail: [email protected] literature. Concluding remarks are given in Sect. 6. c e-mail: [email protected] 123 681 Page 2 of 9 Eur. Phys. J. C (2016) 76 :681

2 Fate of the false vacuum 3 Approximate solution to the multi-ﬁeld potential

Consider a potential of multiple scalar fields V (φi ) with at 3.1 Reducing to a single-field potential least two minima. The trivial solution to the classical equations of motion is stationary extremizing the potential. This The bounce solution can be approximated by the bounce solution typically gives the field a non-zero vacuum expecta- solution of a single differential equation as follows [42,43]. tion value and is responsible for giving standard model par- First make a shift of fields such that the false vacuum is ticles their mass via electroweak symmetry breaking. The at the origin in field space. The true vacuum is then at ˆ ˆ other, less obvious solution is one where the fields continu- vφ1 where φ1 is a unit vector that points in the direction ously vary from one minimum to another. In this case, the of the true vacuum. Then define a complete set of unit ˆ false vacuum decays into the true vacuum via tunneling pro- vectors orthogonal to φ1 and rewrite the potential in the cesses, and is termed the ‘bounce solution’ [41]. If this is rotated basis V (ϕ1,ϕ2,...) → V (φ1,φ2,...). Then con- ˆ achieved within a first order phase transition, regions of the sider the potential only in the φ1 direction between the min- new vacuum appear and expand as bubbles in space. In this ima, V (φ1, 0,...). One can then solve the single equation of paper we are interested in calculating the profile of the bub- motion ble, that is, the spacetime dependence of the bubble nucle- ∂2φ ( − ) ∂φ ∂ (φ , ,...) ation. 1 + d 1 1 − V 1 0 = 2 0(4) The spatial bubble profile is obtained by extremizing the ∂ρ ρ ∂ρ ∂φ1 Euclidean action to derive an initial ansatz that approximately solves the full classical equations of motion. Let us therefore turn our atten- d 1 2 SE = d x ∂μϕi + V (ϕi ) (1) tion to the most general renormalizable tree level potential 2 with a single field, where d = 4 for zero temperature tunneling and d = 3for V (ϕ) = M2ϕ2 + bϕ3 + λϕ4. (5) finite temperature tunneling relevant to cosmological phase transitions. The nucleation rate per unit volume is An approximate expression for the effective action of a similar potential was found in Ref. [50]. The above potential can φ = ϕ ϕ ϕ − SE be rescaled min where min is the global minimum of = ( ) T A T e (2) the above potential. Then the rescaled potential has a global minimum at φ = 1. To ensure that the effective action is where A(T ) is a temperature dependent prefactor propor- unaffected by this rescaling, we also make the replacement 2 tional to the fluctuation determinant, T is the temperature and ρ → ϕminρ. We paramatrize the rescaled potential as SE is the euclidean action for the bounce solution which sat- (4α − 3) isfies the classical equations of motion. In the case of a spher- V (φ) = Eφ2 + Eφ3 − αEφ4. (6) ically symmetric bubble the classical equations of motion are 2 Tunneling between two vacua requires the existence of a ∂2ϕ ∂ϕ ∂ i + 2 i − V = potential barrier or “bump” separating the two minima. As 2 0(3) ∂ρ ρ ∂ρ ∂ϕi parametrized in Eq. (6), this type of barrier can only exist if E < 0 and α ∈ (0.5, 0.75).3 To illustrate this point, we present in Fig. 1 the potential in φ given in Eq. (6) for both and the bounce solution satisfies the conditions ϕ (0) ≈ vtrue, i i the edge choices of α and the mean allowed choice, using ϕ (∞) = vfalse and ϕ(0) = 0.1 Here ρ is the ordinary 3D i i i several choices of E. One can see that, for α = 1 ,wehave spherical coordinate, as we are considering finite tempera- 2 exactly the Mexican hat potential (albeit shifted to φ = 0.5) ture, and vtrue and vfalse are the vacuum expectation values i i with degenerate minima, and for α = 3 , there is no potential of the field ϕ in the true and false vacua, respectively. The 4 i barrier between false and true minima. equations of motion resemble the classical solution of a ball rolling in a landscape of shape-V with ρ playing the role of time, but including a ρ-dependent friction term. 2 This definition of α differsfromthatof[50] but the physical principles are the same. 3 This is assuming the three turning points are in the positive φ direction with the local minimum at the origin. The rest of potentials with three 1 The first is not a boundary condition unlike the other two. It is instead turning points are covered by this analysis simply by making combina- the condition that differentiates the bounce from a trivial solution. tions of the transformations φ → φ + a and φ →−φ. 123 Eur. Phys. J. C (2016) 76 :681 Page 3 of 9 681

| | Thus, one can factor E √out of the equations of motion by further rescaling ρ → ρ/ |E|. Then the equations of motion only depend on α. Under the scaling we have introduced, ⎧ ⎨ϕ → φ = ϕ ϕ ϕ min ⎩ρ → √min ρ (7) |E| the Euclidean action becomes φ3 ∂φ 2 m 2 SE = 4π √ dρρ − V (φ) (8) |E| ∂ρ

where4 V ≡ V/|E|, and we have integrated over the angular variables assuming isotropy. The integral in Eq. (8) must only depend on α,asin φ3 m SE = 4π √ f (α). (9) |E| Meanwhile, we will approximate the rescaled ﬁeld itself with the well-known “kink” solution [44,45] 1 ρ − δ(α) φ ≈ 1 − tanh (10) 2 Lw(α)

parametrized by the offset δ and the bubble wall width Lw. Thus it remains to determine the α-dependent functions δ(α), Lw(α) from the kink solution, and f (α) in the Euclidean action. We first evenly sample values of α within (0.5, 0.75), then numerically solve the full bubble profile using conventional techniques. Next, for each value of α, we fit the kink solution given in Eq. (10) to the full solution, extracting Lw and δ. Lastly, we numerically integrate to find f in the Euclidean action. This results in a tabulation of values for Lw, δ, and f , for each value of α. Using the apparent α dependence and intuition from our parametrization of the potential, we find ansatz functional forms in terms of α for each of these parameters. The offset δ should diverge at the boundaries α = 0.5 and α = 0.75, and is found to be quite small otherwise. It also appears to have approximate odd parity about the mean allowed value of α = 0.625. We modeled this with odd powers of non-removable poles at the boundaries of α, along with an offset and a linear correction about the mean: Fig. 1 We present our rescaled scalar potential parametrized by E and ⎡ ⎤ α given in Eq. (6). Each panel displays −E = 1, 2, 5, 10, 20. The top (2n−1) 2 α α = 1 5 α − 5 and bottom panels are the edge choices of , with 2 on top and ⎣ 8 ⎦ δ(α) ≈ δ0 +k α − + an . α = 3 on the bottom. The middle panel has the mean value of α = 5 8 α − 1 α − 3 4 8 n=1 2 4 (11) 3.2 Developing ansatz solutions We then fit these parameters using the tabulated values. In deriving an approximate solution to the potential in Eq. (6), we first note that the effective potential is proportional to E. 4 V does not have any dependence on |E|. 123 681 Page 4 of 9 Eur. Phys. J. C (2016) 76 :681

Table 1 The ﬁtted values for the parameters that deﬁne the approximate ansatz functions for f which is used in the Euclidean action, the bubble wall width Lw, and the offset δ from the kink solution f (α) Lw(α) δ(α) Parameter Value Parameter Value Parameter Value f0 0.0871 0 1.4833 δ0 2.2807 p 1.8335 c 0.4653 k −4.6187 q 3.1416 r 18.0000 a1 0.5211 5 s 0.7035 a2 × 10 7.8756

The bubble wall width Lw is dominated by two asymp- α = 3 α → 1 totes. It diverges at 4 , and become small as 2 .We used this form to model the asymptotic behavior, 1 r c Lw(α) ≈ α − + . (12) 0 2 α − 3 s 4

As before, the normalization 0, the two exponents r and s, and the coefficient c are fit using the tabulated values from the full numerical calculation. Interestingly, we find almost exactly that r = 18. (The full fitted parameters are given in Table 1.) The Euclidean action determined by f (α) diverges at α = 1 α = 3 2 and is zero at 4 . This is modeled by α − 3 p f (α) = f 4 (13) 0 α − 1 q 2 where only a normalization parameter and exponents need to be fitted. In Table 1 we present all the fitted values that go into these ansatz approximate forms. The numerical values computed for these functions as well as the resulting fits are given in Fig. 2. We did not estimate uncertainties in the full numerical calculations nor in the fitted parameters, though in principle this could be done. Thus we are not able to compute rigorous measures of the goodness of fits. As these fits are only used to Fig. 2 We present the fits to the offset δ (top panel) and the bubble form a base ansatz solution which then receives perturbative wall width Lw (middle panel) of the kink solution, and the integral f (bottom panel) appearing in the Euclidean action. The numerically corrections, such an undertaking lies outside the scope of computed values are presented along with the fitted curves, as well as this work. We do, however, provide in Fig. 2 the residuals the residuals between the fitted curves and tabulated values. 3 We note that |E| scales as |bφ | so SE /T scales as vergent perturbative corrections. The process is largely anal- φm φm T b , where b is the cubic coupling of the unscaled field, ogous to Newton’s method for finding roots of functions. as in Eq. (5). Also b controls the height of the barrier sepa- Here, we iteratively determine functional corrections to the rating the two minima. ansatz form.

4.1 Perturbative corrections to the ansatz 4 Perturbative solution We first note that along the trajectory in field space from the In the previous section, we developed fitted curves to esti- false vacuum to the true vacuum, the magnitude of any of mate the parameters of the well known kink solution. In this the fields in φ ={φi (ρ)} generically does not exceed the section, we will take advantage of rescaling to compute con- distance between the two minima. That is, 123 Eur. Phys. J. C (2016) 76 :681 Page 5 of 9 681

|φi (ρ)| |vtrue − vfalse|. (14) where M is the mass matrix ∂2V (φ) If we rescale our fields as described in Sect. 3, the distance M = ij ∂φ ∂φ . (21) between the ansatz and the actual solution is bounded by 1, i j A but is usually much smaller than 1. (This is illustrated with In this way, zik is the ith element of the eigenvector of M that the concrete example presented in Sect. 5.) Let us call the λ2 has eigenvalue k. It is necessary, however, to use both posi- ansatz to φi , Ai with correction εi , so that φi = Ai + εi . tive and negative roots, ±λk. Thus we must further introduce Applying this to Eq. (3) yields ˜ λ j and z˜ij with ∂2 A ∂2ε 2 ∂ A 2 ∂ε i + i + i + i λ˜ = λ , λ˜ =−λ , λ˜ = λ , ... (22) ∂ρ2 ∂ρ2 ρ ∂ρ ρ ∂ρ 1 1 2 1 3 2 ˜ = ˜ = ˜ = ˜ = ... 2 zi1 zi1, zi2 zi1, zi3 zi2, zi4 zi2, (23) ∂V (φ) ∂ V (φ) = + ε j (15) ˜ ∂φ ∂φ ∂φ so that in λ j and z˜ij, the index j = 1, 2,...,2n. Finally, we i A j i j A use the techniques described in [49] to arrive at the particular where A ≡ {Ai }. We can then rearrange the above to separate solution, terms that depend only on the ansatz forms from those involv- 2n n λ˜ ρ ing the unknown correction functions, ε . This leaves us with ≶ e j ρ −λ˜ ≶ ≶ i ε = z˜ te j t h B (t) dt − β . ε i ij ρ jk k j a set of coupled inhomogeneous differential equations for i : = = 0 j 1 k 1 ∂2ε 2 ∂ε ∂2V (φ) (24) i + i − ε = B (ρ). 2 j i (16) ∂ρ ρ ∂ρ ∂φi ∂φj A In the above, the functions Bk are defined in Eq. (17), and ≶ Here the functions B (ρ) are the inhomogeneous part of the β i the constants j are determined by boundary and matching ε differential equations for i , and they are given by conditions. The constants h j k that appear in the integrand are determined by the 2n2 constraint equations ∂V (φ) ∂2 A 2 ∂ A B (ρ) ≡ − i − i . (17) i ∂φ ∂ρ2 ρ ∂ρ 2n i A ˜ z˜ijh jkλ j = δik, (25) One can see that the value of the functions B represents how i j=1 well the ansatz forms solve the equations of motion. This 2n can be seen not only as the definition of Bi are the equa- z˜ijhik = 0 . (26) tions of motion where the fields are taken to be A , but also i j=1 because if Bi were zero, then the differential equations for × the corrections to the ansatz εi would become homogeneous. Each of the above equations are n n matrix equations, giving We can linearize and approximately solve these differen- 2n2 total constraints. tial equations analytically by approximating the mass matrix To account for the corrections to the mass matrix we rela- ε0 by a series of step functions with a correction which we will bel the solution we found i , substitute into the differential ε = ε0 +δε +... η (ρ) use to form the convergent series of perturbations. Consider- equations i i i and restore ij in the differ- ing only the homogeneous case (Bi = 0), Eq. (16)issolved ential equations. Keeping only terms to first order we write by introducing ε of the form λρ ze ∂2ε 2 ∂ε ∂2V (φ) ε ∼ . (18) i + i − ε = B (ρ), ρ ∂ρ2 ρ ∂ρ ∂φ ∂φ j i i j A 2 0 0 This will reduce the equation to an eigenvalue problem in the ∂ δεi + ε 2 ∂δεi + ε i + i − δε + ε0 M¯ + η = B (ρ). mass matrix. It is therefore convenient to define the homo- ∂ρ2 ρ ∂ρ j j ij ij i geneous solutions with the notation (27) λ ρ z e k 0 εh = ik (19) Since ε solve the initial differential equations in terms of ik ρ ¯ i Mij by definition we can make an immediate simplification. where the index i refers to field φi , and the index k will Keeping only terms up to first order in our expansion we then run over the n eigenvalues of the mass matrix, and k is not write εh ε = summed over. Substituting ik for i in Eq. (16) with Bi 0 ∂2δε 2 ∂δε i + i − δε M¯ − η ε0 = 0, yields ∂ρ2 ρ ∂ρ j ij ij j = λ2 ∂2δε ∂δε Mijz jk k zik (20) i 2 i ¯ 0 + − δεj Mij = ηijε . (28) j ∂ρ2 ρ ∂ρ j 123 681 Page 6 of 9 Eur. Phys. J. C (2016) 76 :681

This once again is a set of coupled linear differential equations which has the same form as the original set of differential equations. So the solution is the same as before with η ε0 δε ij j replacing Bi . One then finds the series of k up to a desired tolerance to find each value of εi . This process con- tinues until the equations of motion are satisfied up to the desired tolerance.

4.2 Observations on convergence

Newton’s method is well known to have four major issues. In our analogous form, these issues would be:

1. If the initial ansatz function is too far from the true function, convergence will be slow. Fig. 3 The tunneling trajectory in the space of the x and y fields at the 2. Oscillating solutions where ε(n)(ρ) ≈−ε(n+1)(ρ). level of the base ansatz (solid straight line), and including the first three 3. Divergent corrections that arise in Newton’s method if iterative corrections (solid curved lines). Also included is the numerical result, independent of our semi-analytic method, from Ref. [43](dashed the function’s derivative becomes undefined or zero. The curve) equivalent issue will be discussed in detail below. 4. Being in the wrong basin of attraction and converging to ∂2 (n) ∂ (n) ∂ the wrong function. Ai 2 Ai V = + + = 0. ∂ρ2 ρ ∂ρ ∂φ ( + ) ( ) ( + ) i A n 2 ,...,A n ,A n 2 ,... 1 i i1 We will demonstrate that our method as applied here does (30) not suffer from these problems with the exception of issue 4, where in principle a local minimum could be closer to Thus, in the case of a single field, an oscillating solution the initial ansatz than the closest bounce-like extrema. This, means that corrected field at the order where oscillation however, is a limitation of all other known algorithms for begins has solved the equations of motion exactly. In the φ finding bubble wall profiles. Meanwhile, we have already multi-field case, as the fields j =i converge without oscillat- demonstrated in Sect. 4.1, our that our algorithm is free from ing corrections, the changes to the derivative of the poten- the first problem as the guess of the initial ansatz ensures that tial energy will diminish and thus will resemble the single- the error functions are generically bounded by 1 (but should field case. In the case that more than one field has begun to be much less than 1). For the remaining two issues, a little receive oscillating corrections, this could prevent a rapid con- more care is needed. vergence but does not necessarily preclude it as the equations Let us examine the issue of oscillating solutions. Let the for the fields are still coupled. updated function be In Newton’s method of finding roots, a major issue is when the derivative of the function becomes zero or undefined. The n−1 closest analogy to our method is the case where the mass (n) (0) (k) 2 A = A + ε , (29) ∂ V (φ) i i i matrix ∂φ φ ( ) becomes zero or singular. In fact this is = i j A n k 1 not an issue, as we can demonstrate. In the case that the mass (0) ε(k) matrix is zero, the differential equations become where Ai is the initial ansatz and i are the correction 2 functions. Suppose that, for field φi , the successive cor- ∂ ε 2 ∂ε i + i = B (ρ). (31) rection functions begin oscillating at iteration n, so that ∂ρ2 ρ ∂ρ i ( ) ( + ) ε n =−ε n 1 . But this means that the equations of motion i i This is easily solved as (n+2) = (n) + ε(n) + ε(n+1) for Ai Ai i i can be written before ρ β− dy y Taylor expanding as ε = β + 1 + x2 B(x) x i 0 2 d (32) ρ 0 y 0 (n) (n) (n+ ) (n) (n) (n+ ) ∂2[A + ε + ε 1 ] 2 ∂[A + ε + ε 1 ] β β i i i + i i i where 0 and −1 are both zero if the mass is zero every- ∂ρ2 ρ ∂ρ where. The case to consider is when the mass matrix is sin- ∂V (φ) gular. This in fact does arise quite typically at some spatial + ∂φ (n) (n) (n+1) points, but this is not a problem because the matrix inverse i A +ε +ε k k k is not needed, and zero eigenvalues can be treated easily by 123 Eur. Phys. J. C (2016) 76 :681 Page 7 of 9 681

Fig. 4 The base ansatz form, first three iterations of corrections and to the ansatz form of the x and y fields are given in the bottom left and the full numerical result of the x and y fields are presented in the top left bottom right panels, respectively and top right panels, respectively. The first three iterative corrections using singular value decomposition or strategic placement of that the minimum is at u = 1 and then we divide by |E| to the step functions. Also, this issue is avoided if the full inho- get mogeneous differential equation is directly solved numeri- V (u, 0) cally. = 0.36u2 − u3 + 0.57u4. (34) |E|

5 Comparison with a solved example We then use our analytic formulas to write the ansatz and make the appropriate rescalings to u(ρ) and ρ such that the ( , ) We apply our method with the sample potential given in [43] ansatz is the solution to the original 1D potential. In the x y basis the ansatz is 2 2 2 2 V (x, y) = (x + y ) 1.8(x − 1) + 0.2(y − 1) − δ . ρ − 0.437 x(ρ) = 1.046 1 − tanh , (35) (33) 1 ρ − 0.437 For δ = 0.4 the potential deforms quite dramatically from y(ρ) = 1.663 1 − tanh . (36) 1 the initial Ansatz so the convergence will be slower than for a typical case. Wemake a rotation in ﬁeld basis (x, y) → (u,v) Note that the wall width is only equal to 1 due to the rescaling. such that u traces a straight line path from the origin to the We have to sanitize our initial ansatz to set the derivative to global minimum and v is of course orthogonal to u. Our one zero as ρ → 0 or the correction diverges due to the φ/t term dimensional potential is then given writing the potential in in the differential equations. To achieve this we subtract from the rotated basis and setting v to zero. We then rescale such our initial ansatz 123 681 Page 8 of 9 Eur. Phys. J. C (2016) 76 :681

Fig. 5 The error function, B, of the ansatz solution x0 when applied to the ﬁeld equations, and the same for the ansatz solution including the ﬁrst three perturbative corrections denoted by x(n) = x(n−1) + ε(n)

( ) ( − ) ( ) ∂x ∂x φ n (ρ) = φ n 1 (ρ) + ε n (ρ), with φ = x, y. (39) δx(ρ) = − ρ , φ ∂ρ exp ∂ρ (37) ρ=0 ρ=0 (n) Figure 4 also includes the error functions εφ (ρ) to illustrate ∂y ∂y δy(ρ) = − ρ . the overall and diminishing magnitude of corrections to the ∂ρ exp ∂ρ (38) ρ=0 ρ=0 fields in successive perturbations. In Fig. 5 we show the error function to the ansatz for the (ρ) If one uses a small amount of step functions to approximate x field, Bx , which arises from the inhomogeneous part of the spacetime dependent mass matrix one can find that the Eq. (16). The error function is given for the bare ansatz solu- (ρ) corrections δεi are slowly converging. In particular it is use- tion of x and the first three perturbative corrections. We ful to have step functions for regions where m12(ρ) = 0 and point out that the magnitude of the error is reduced by roughly m2 < 0 as the functional form of the solutions changes in a factor of 5–10 from each perturbative correction, and that i (3) these regions. In the former the differential equations decou- the error function for x (ρ) has reduced in magnitude by a (0)(ρ) ple for a region, for the latter, some of the exponents αi are factor of 300 compared to that of x . imaginary (but the εi (ρ) remains real). In Fig. 3 we show each iteration of the trajectory in the 6 Conclusion (x(ρ), y(ρ)) field space, along with the numerical trajectory as derived in [43]. The algorithm essentially converges after In this work we presented a new method to calculate the 3 perturbations. In Fig. 4 we show the x and y fields starting bubble profile in a bounce solution for a multi-field poten- with the base ansatz forms x(0) and y0, and then including tial with a false vacuum. The method uses fitted functions the first three perturbative corrections, denoted by to estimate the parameters of the single-field kink solution 123 Eur. Phys. J. C (2016) 76 :681 Page 9 of 9 681 which is used as an ansatz form. It then applies this form 19. X. Chang, R. Huo, Phys. Rev. D 89(3), 036005 (2014). doi:10. to the full multi-field potential, which receive perturbative 1103/PhysRevD.89.036005 correction functions that are reduced to elementary numeri- 20. M. Lewicki, T. Rindler-Daller, J.D. Wells, JHEP 06, 055 (2016). doi:10.1007/JHEP06(2016)055 cal integrals. We have argued that the perturbative series of 21. M. Dhuria, C. Hati, U. Sarkar, Phys. Lett. B 756, 376 (2016). doi:10. corrections should converge quadratically, and is immune to 1016/j.physletb.2016.03.018 the issues of the analogous Newton’s method. This method is 22. C. Lee, V. Cirigliano, M.J. Ramsey-Musolf, Phys. Rev. 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123

Chapter 6

Particle cosmology in the NMSSM

6.1 Introductory remarks

The discovery of a Higgs boson of 125 GeV [107, 108] was the completion of the Standard Model and one of the most remarkable discoveries of the last century. It also makes electroweak baryogenesis difficult in the MSSM [47] and impossible in the Standard Model [109]. For the Standard Model the issue is that the electroweak phase transition is not strongly first order for a Higgs mass heavier than 40 GeV. Finally the Standard Model has no dark matter candidate and it is difficult∼ for the Higgs to catalyze inflation. For the MSSM such a heavy Higgs requires some amount of fine tuning – this is known as the little hierarchy problem [110]. Furthermore the MSSM requires light stops to catalyze a strongly first order electroweak phase transition which is virtually incompatible with collider searches. The NMSSM can catalyse a strongly first order phase transition through a weak scale singlet. Furthermore, the LSP provides a good dark matter candidate and there are many flat directions which can be exploited to catalyze inflation. In this work I explore the parameter space to verify the compatibility of the NMSSM these three pillars of particle cosmology – electroweak baryogenesis, dark matter and inflation.

6.2 Declaration for thesis chapter 6

Declaration by candidate In the case of the paper present contained in chapter 6, the nature and extent of my contribution was as follows:

Publication Nature of contribution Extent of contribution Worked on analytic derivations and numerical codes. Worked to understand the background 6 needed and explained this to collaborators. Heav- 50% ily involved in writeup. Contributed to discussions throughout The following coauthors contributed to the work. If the coauthor is a student at Monash, their percentage contribution is given:

133 Author Nature of contribution Extent of contribution Along with Anupam Mazumdar came up with Csaba Balazs the idea for the paper. Wrote a section and contributed to discussions throughout. Heavily involved in implementation of numerical code. Co-wrote the paper with me apart from two Ernestas Pukartas sections written by Csaba Balazs and Anupam. Involved with discussions throughout Along with Csaba Balazs came up with the con- Anupam Mazumdar cept. Wrote a section and contributed to ideas throughout. The undersigned hereby certify that the above declaration correctly reﬂects the nature and extent of the candidate and co-authors’ contributions to this work.

134

6.3 Published material for chapter 6: Baryogenesis, dark matter and inﬂation in the Next-to-Minimal Supersymmetric Standard Model

136 JHEP01(2014)073 a,b,c Springer October 9, 2013 January 16, 2014 : December 12, 2013 December 26, 2013 : : : , Received 10.1007/JHEP01(2014)073 Revised Published and Graham White Accepted e doi: Published for SISSA by Ernestas Pukartas e [email protected] , [email protected] , . 3 Anupam Mazumdar, 1309.5091 The Authors. a,b,c,d Beyond Standard Model, Supersymmetric Standard Model, CP violation, c

Explaining baryon asymmetry, dark matter and inflation are important ele- , [email protected] E-mail: [email protected] ARC Centre of Excellence forVictoria Particle 3800 Physics, Australia Monash University, Australian Collaboration for Accelerator Science,Victoria Monash 3800 University, Australia Consortium for Fundamental Physics,Lancaster, Lancaster LA1 University, 4YB, U.K. School of Physics, MonashVictoria University, 3800, Australia Monash Centre for Astrophysics,Victoria Monash 3800 University, Australia b c e d a Open Access Article funded by SCOAP the XENON1T and LUX experiments. Keywords: Renormalization Group ArXiv ePrint: microwave background (CMB) radiationdark through matter, inflation, and the successful observedthe baryogenesis. relic recent density Higgs Constrains for mass byinvisible bound), collider Z branching decay measurements ratios width (such of arexenon as also rare, nuclei imposed. flavour would violating We fall explore decays, within where and current dark bounds the matter of interactions XENON100 with and the projected limits for In this paper weModel explore (NMSSM) these by issues studyingtransition, within the the Next-to-Minimal conditions for Supersymmetric abundancedriven a Standard of by strongly a the first gauge invariant lightest order flatpresent direction electroweak supersymmetric the of phase particle regions MSSM - of (LSP), made parameter up and of space right inflation which handed can squarks. yield We successful predictions for cosmic Abstract: ments of a successful theory that extends beyond the Standard Model of particle physics. Csaba Balázs, Baryogenesis, dark matter and inflationnext-to-minimal in supersymmetric the standard model JHEP01(2014)073 ]. 6 ]. 10 5 ]. 9 , 8 ]. Since inflation 1 16 ]. The thermal WIMP 7 3 ]. For a review on inflation, see [ 4 – 5 2 – 1 – 14 14 18 8 4 1 18 partners are sufficient toexpansion keep of them the in Universe thermalscenario and bath is their until attractive abundance they due freezes decouple todark out due its matter [ to predictive and power the potential inplays link estimating this to the role the abundance of weak of dark scale. the matter Within ideally. the For reviews MSSM on the dark neutralino matter, see [ minimally by embedding inflation within aSupersymmetric visible Standard sector of Model BSM, (MSSM) such [ as the Minimal The lightest supersymmetric particle (LSP)is is a an ideal weakly interacting dark massive matter particle candidate (WIMP), since and it its interactions with the super- background (CMB) radiation measureddilutes matter, by the the end Planck ofdegrees inflation satellite must of [ excite all freedom thewith without relevant the Standard any seed Model initial (SM) excess perturbations of for the dark structure matter formation. This and can dark be achieved radiation, along Dark matter is required to form the observed large scale structures of the Universe [ Inflation can explain the current temperature anisotropy in the cosmic microwave 5.1 Brief review5.2 of inflation Parameter space for inflation, dark matter and baryogenesis 3.1 The scalar potential at high temperature 4.1 Scanning the parameter space for dark matter and first order phase transition 2. 1. 1 Introduction Beyond the Standard Model (BSM)three particle cosmological physics cornerstones. should aspire to include the following A Solving the toy model 5 Gauge invariant inflaton 6 Conclusions 3 Baryon asymmetry 4 Constraints applied Contents 1 Introduction 2 The next-to-minimal supersymmetric standard model JHEP01(2014)073 ] 16 ) and/or ]. So we ]. C ] could be 13 23 21 , 22 we show model , is necessary to 10 4.1 ] and TEXONO [ ].) The NMSSM has 15 14 ] and CDMS [ ) violation, charge ( 20 B ]. Here we supersede that study by a new 24 ], CRESST-II [ – 2 – ]. This asymmetry can be achieved by satisfying 19 10 ]: baryon number ( 11 ]. 12 and finally conclude. ], CoGeNT [ 5 18 , ) violation, and the out-of-equilibrium condition. These ingredients 17 ]. CP 26 , 25 we summarize the constraints applied to the theory, and in we examine the baryon washout condition and derive an algorithm to find regions . Some of the dark matter particles, in particular those with high singlino fraction, 3 4 σ on baryon asymmetry, see [ realize Big Bang nucleosynthesisthe [ three Sakharov conditions [ charge-parity ( are present within thestrong first SM, order but phase transition with to a keep the 126 baryon GeV asymmetry Higgs intact. the For a model review lacks sufficiently Baryon asymmetry, at the observed level of one part in about 10 In the next section we briefly summarize the relevant features of the NMSSM. In The joint parameter space for satisfactory neutralino dark matter and first order elec- In this work, we present a highly efficient algorithm to delineate regions of the NMSSM The MSSM provides all three ingredients, however the electroweak baryogenesis in its 3. section of its parameter spacesection satisfying a stronglypoints first jointly satisfying order inflation, electroweakimplications baryogenesis, phase on and transition. inflation dark in matter In abundance. We discuss algorithm for finding regions whereering electroweak the baryogenesis parameter can space be forMSSM successful, neutralino squarks, by dark and consid- updating mater all andfrom experimental successful the constraints, inflation especially LHC driven the [ by vital the Higgs limits potentially explained by dedicatedalso affect search other of properties lightneutralino of can DM. be dark The very matter. NMSSM light even singlet With in sector a the presence can sizable of singlinotroweak heavy phase fraction super-partners transition the [ has lightest been studied in [ where the neutralino relic abundancethan 3 exceeds the upper limittend to imposed escape by the Planck most bydirect stringent more bounds detection given experiments, by but thesignals XENON100 they [ from fall close DAMA to [ the regions where tantalizing positive a strongly first order electroweakmined phase by transition, the singlet since sector the andHiggs order becomes parameter mass. essentially independent is of now the deter- Standard Model-like with strongly first order phase transition. After finding these regions we reject model points electroweak baryogenesis scenario (where thetry CP-violating reside phases in catalyzing the theasymmetry asymme- gaugino with sector) a it is 126turn GeV getting our standard-like attention hard, Higgs to if the bosonric not electroweak Standard in impossible, baryogenesis Model a to in (NMSSM). the generate naturalmore (For Next enough flexibility a manner to as review [ it Minimal on introduces Supersymmet- the a NMSSM, new see Standard [ Model gauge singlet which helps achieving context is becoming more and more constrained by the LHC data. In the simplest MSSM JHEP01(2014)073 κ ]) = 29 d (2.4) (2.5) (2.6) (2.7) (2.8) (2.1) (2.2) (2.3) b H , T ) 0 1 higher. The b H ], and when , λ + . 1 34 c b . H 4 | . + h . S = ( c | ∗ d . 2 tri u κ H V b ∗ u , H + h . + + 3 term (as defined in ref. [ 3 H 2 ˆ , 2 S | S ] 2 µ d | κ κ d 1 3 29 , H NMSSM 2 H κλS H u κA ) gaugino | + V 2 d 1 3 d V H + d | NMSSM 2 H + ] and H soft 2 b + + H V λ m u d 29 − ] which is defined by the superpotential + b 2 u + H H ) + u b 2 S scalar H | 2 28 =0 | ( , NMSSM λ V u H soft d B 2 2 | V S 27 H g + + | H λ | – 3 – u + 8 + =0 2 H + 2 1 λA µ MSSM | set to zero [ g 2 soft m denotes the electric charge of the component. The | MSSM − softH V u 0 B 2 R-symmetry on the non-renormalizable sector of the NMSSM. MSSM V = = , softH | H = 2 ± i | V conserving, scale invariant, version of the the Next-to- S MSSM D = ( | Z b H 2 2 s V 3 + | W soft Z m S =0 MSSM D V | softH = B 2 The MSSM terms above are [ V | V = λ 1 = W ]) breaks the global U(1) Peccei-Quinn symmetry [ = V 14 MSSM κ soft V NMSSM soft is the MSSM superpotential without the V denote the scalar components of the neutral Higgs superfields. During NMSSM H =0 S µ V | ) are SU(2) doublet (singlet) Higgs superfields: b S and . The superscript in ( ]. It has been shown by Panagiotakopoulos and Tamvakis that the cosmological domain wall T is a free parameter, perturbativity up to the Grand Unification Theory scale MSSM ) 33 u,d + λ W 2 u,d – H b b H 30 H , In this version of the NMSSM the formation of cosmological domain walls during electroweak symmetry 0 2 1 b H restricts it below 0.7.cubic singlet In coupling this work we respect this limitbreaking and is do a concern. notrefs. [ take The solutionproblem of is the eliminated domain after wall imposing a problem has been discussed, amongst others, in While and the NMSSM It is important for electroweak baryogenesis that the Higgs potential receives contributions both from the MSSM Here electroweak symmetry breaking thevacuum neutral expectation components value. of these will acquire a non-zero contains the MSSM soft terms with Above and ( corresponding soft supersymmetry breaking scalar potential In this work weMinimal consider Supersymmetric the Standard Model(see, [ for instance, ref. [ 2 The next-to-minimal supersymmetric standard model JHEP01(2014)073 B the (2.9) 2 | gauge (2.11) (2.10) 15 Y N entry can | violation in 1 M CP e ] S, , . ) 15 14 a , the neutral components             ˜ 3 N G i i denote vacuum expectation the higgsino and a d u i c W + i ¯ ˜ G 2 S H H 0 | u 3 0 0 h h h X e h κ 14 λ H λ M whose entries provide the mixing are individually conserved with a N − 14 | + ij i i N i L i − u + u . The mass of the lightest neutralino N 2 ˜ 2 S + H W b H 2 S h h i √ | h √ 0 0 2 d 1 λ ¯ ˜ g and g 13 W e gauge boson H − 2 − N | B L 13 M i i is the smallest entry of the mass matrix, the d N d 2 2 + H H i – 4 – h h √ + √ 0 1 ˜ 2 S B g 0 g h , is an exactly conserved quantity in the SM (and ¯ ˜ κ B − f W L 1 2 gauge couplings, and 12 the wino, ]. Unfortunately, there is not enough M − 0 ( M L N 2 , and the singlet | 1 2 B d 1 35 + , 12 b H = M e N 12 B |             11 and N = violation, and departure from thermal equilibrium. Remarkably, u and SU(2) e χ gaugino = b ) basis, is given by the symmetric matrix [ H V 0 1 Y ˜ S M e CP χ , u ˜ H , d ˜ gives the bino, H and/or , 2 | 3 C ˜ are the U(1) 11 W , the neutral component of the SU(2) i 2 N , b | B 1 − g In this work we the lightest neutralino is the dark matter candidate. Neutralinos in the ˜ B, i is small, that is the Peccei-Quinn limit. This limit is also motivated by the desire to − the baryon and leptonwithin numbers, NMSSM). While atgood low approximation, temperatures at very highthrough temperatures sphaleron baryon processes number [ violationthe in standard unsuppressed CKM matrix to generate the observed baryon asymmetry. In the NMSSM, 3 Baryon asymmetry As mentioned above, threeviolation, conditions have to bethese met conditions to can generate be baryon met asymmetry: in the Standard Model of particle physics. The difference of where singlino fraction. When,lightest for neutralino example, tends torender be the lightest singlino neutralino dominated. to acquire Alternatively, mostly a bino small admixture. Here values. To obtain the massThis eigenstates, can we have be to done diagonalizeamongst with the neutralino the gauginos, mass help higgsinos matrix. ofgiven and a by: unitary the matrix singlino. The lightest neutralino, for example, is ture of the lightest( neutralino is controlled by the neutralino mass matrix which, in the of each Higgs doublet originates from soft supersymmetry breaking where the fields above are the fermionic components of the vector superfields. The admix- κ obtain a light dark matter candidate, as indicated inNMSSM the are next admixtures paragraph. ofboson the fermionic components of five superfields: the U(1) vanishes this symmetry is restored. For simplicity, in this work we consider the limit where JHEP01(2014)073 ]. 53 (3.1) ]. This 52 – denotes the 50 ϕ ] violation. Here 47 – ]. As the Universe 37 CP 36 , where } c , ϕ ) only approximately quan- 0 { 3.1 violating processes near the bubble ]. As noted before the baryon washout , . considered a toy model which included 54 CP 1 . ]. The Standard Model cannot satisfy the et al and γ ]. To this end we require [ 49 , C – 5 – ≡ 12 c c 48 T ϕ ]. Seeing as we also wish to satisfy other, in some cases rather violation can be added to the gaugino and singlino sectors. Pro- 54 CP ] and consider higher order temperature corrections, loop corrections 54 violating phases in the NMSSM to a later work. is the temperature at which the effective potential obtains degenerate minima. CP c T At very high temperatures electroweak symmetry is restored [ the tree level effective potentialtemperature of corrections. the Our NMSSM strategy atmodel will the be from Peccei-Quinn to limit ref. derive with [ aand the semi-analytic small largest solution to deviations the frommodel toy solution. the If so we only called scan Peccei-Quinn regions of limit parameter as space where perturbations our approximations to hold, the toy Previous analysis of the electroweakQuinn phase limit transition found within that theis NMSSM the heavily near parameter constrained the space [ Peccei- strict, that experimental satisfies and the cosmological baryonefficient constraints, washout way we condition are of motivated findingphase to transition. regions find In a of a numerically parameter prior analysis, space Wagner that allow for a strongly first order with electroweak baryogenesis, therewe is focus also on the therelevant baryon issue washout of condition insufficient and postpone the detailed3.1 investigation of the The scalar potential at high temperature baryon washout condition forexperimental a constraint Higgs also like all particle butRecent work rules of however out has mass suggested electroweak about that baryogenesisder the 125 in phase GeV NMSSM transition the is [ and MSSM compatible the with [ observedcondition a Higgs strongly is mass first [ not or- the only hurdle that prevents the Standard Model from being consistent where This is known as thetifies baryon the washout baryon condition. washout. A Equationformalism more ( of precise condition electroweak can baryogenesis be obtained [ in the gauge invariant dampened within the brokenwalls phase. can create The athe large baryon baryon asymmetry asymmetry. will If be preserved the [ phase transition is strongly first order sition is first order, theoccurs electroweak symmetry when is the broken effective throughHiggs tunneling potential vacuum processes. has expectation This degenerate valuethe (VEV). minima otherwise In such symmetric a vacuum,is case broken until bubbles everywhere the of in phaseand broken the therefore transition phase Universe. baryon grow is violating in Within complete processes the are and symmetric unsuppressed symmetry whereas phase they sphaleron are processes, exponentially vided that the thirdasymmetry in Sakharov the condition NMSSM. isparameter In space met, a this it strong paper is first we order possible focus electroweak to phase on transition generate thecools, can the latter: the be electroweak observed achieved? where symmetry is in broken the during NMSSM the phase transition. If the phase tran- sufficient amount of JHEP01(2014)073 (3.6) (3.4) (3.5) (3.2) (3.3) 2 2 ) 9 for bosons . , φ T ( b 4 a , ϕ m is the toy model ˜ 2 λ ! T 2 . It should be noted + ) 2 2 and 1 φ T V log 2 s . ( T 2 ) V 2 4 ϕ m T φ ) . 2 ( + f 2 φ . 2 a , T ( π m x aϕ V T 2˜ m r 64 V /T as the terms that violate the + − ) − g e φ + ∆ 2 S log ( V ± ϕ − + ∆ 4 2 s m ) 1 2 loop ϕ φ T T . (We also use the short hand that , 3). The term

( ) ) 2 . We have also identified the log term π V S / 2 φ φ λ T 3 s m ) ( ( ϕ 64 V 2 2 log E ϕ − + ] / / γ + ∆ g κ 2 3 3 2 s ] ] − π π 55 S 2 2 ϕ + + ∆ ) ) V κA 2 s 12 12 πe dxx 2 2 φ φ m ( ( T T – 6 – ∞ ) ) = ( 0 m m + ∆ + [ [ φ φ Z f 3 ( ( T + + for bosons. The high temperature expansion is up a 2 2 2 48 48 g g 4 V π − m m T 2 − − − − ET ϕ and g g 2 2 ) = ∓ 2 − T T ) ± ) ) ∼ ≡ ,T 2 ], which is given by E g as the loop corrections, ∆ s φ φ γ ϕ ( ( 24 2 − 1) 2 2 24 24 − ϕ, ϕ m m cT loop πe 1) = ( + + V + g g V ± 2 = (4 T, m, ≡ ∼ ϕ ( b 2 5. The temperature dependent effective potential is a function of the a V . T, m M ( 1 .) Within one loop accuracy under the high temperature expansion, it is +1) V 2 d = . H T V + /T T, m, is the number of fermionic or bosonic degrees of freedom respectively. Similarly ) ( 2 u φ to highlight that these terms will be treated as a perturbation. To keep notation V ± ( H g T m q V The one loop temperature corrections to the effective potential are: ≡ Here we have definedPeccei-Quinn ∆ limit (which iseffective potential approximately from ref. [ Higgs field and theϕ singlet field which wegiven denote by ones that we treat asthat a the perturbation) high and temperature their approximation sumbe for we assumed the denote one to as ∆ loop beand temperature valid. fermions corrections cannot respectively. Indeedhave it Our is numerical only scans valid stay when away from this limit, we generally for fermions. Here as ∆ compact, all small temperature corrections that are not included in our toy model (i.e. the for bosons, and where the argument is +to for an fermions overall and temperature dependent constant [ our cosmological and experimental constraints. it is numerically efficient to produce a very large volume of points that easily satisfy all of JHEP01(2014)073 (3.9) (3.7) (3.8) ) = 0, (3.10) (3.11) . Note c δγ T ) = 0. Let + , away from 6= x ,T , γ T 0 ϕ s . , . Similarly, the c δγ . ) δϕ 2 (0 for . ,γ T 2 ϕ δϕ s ˜ + λ ) 2 2 V 2 δ ,ϕ + s λ c ˜ ϕ c ϕ ϕ T/ϕ 2 +

, ϕ c λ ) = + 2 0. These points however T β = ˜ γ 2 s 2 + δϕ ,T ∂ < γ m 2 s ∂V 0 ,γ 2 s + , s cos (4 ˜ m c ϕ + m 4 ( , (0 β c . ϕ ˜ γ s ϕ 2 ( ϕ V 2 | ) ˜ ,γ a V δϕ sin s S is continuous in its three arguments 2 ,γ ,ϕ V + s c λ 4 ϕ V ,ϕ

. c + . We will denote the ﬁelds, + ∆ ϕ T 2 9 Eϕ

γ ˜ . γ β, 1 s T ∂ β ˜ ∂V 2 ϕ loop + 3 2 & ∂ and ˜ V ∂V 4 – 7 – sin c s S sin d ˜ γ + ˜ ϕ λϕ V ) 2 H γϕ λϕ + ∆ − c 2 , acquires large loop corrections from the stop mass. ∆ m ϕ T δγ β β, ˜ λ − − δϕ V 2 δ + + 4 ,γ = s cos β cos , γ ˜ γϕ 2 ,ϕ β is also zero by deﬁnition when the derivative is evaluated s c c δγ = 0. It is easy to see that ˜ 2 2 2 ϕ ϕ c ) + (∆

g cos δ . Let us assume that ϕ sin ϕ , we can write − T x u 8 , γ ˜ λ + ˜ γ ϕ + s ϕ 2 2 H 2 or ∂ 1 s ˜ ϕ ∂V ˜ , we can then write: g ϕ/ m λA ˜ , γ ϕ ϕ c Gϕ , − , respectively. It is also useful to define ˜ + c 2 = = = ϕ c loop ( = ϕ 2 ˜ a δϕ T ˜ 2 λ = ˜ γ V M + + ∆ ˜ γ c and S ∂ ≈ ∂V ϕ c ( ∂ϕ/∂ T V + ∆ T can be written as a function of , respectively. From the small change formula in three variables, we can write: ∆ V ≡ We recall that at the critical temperature the effective potential obtains degenerate , δγ s V Noting that The first term onreasons discussed the above. right Furthermore, the handwith derivative side of respect our of to toy model the either at effective above Lagrangian its equation minimum. is∆ identical Setting to the zero left for hand the side of the above equation to zero and defining where the nontemperature trivial is VEV a small of perturbationsinglet the to VEV the full at tree the temperature level criticalδϕ critical and dependent VEV, the potential inverse order at parameter the both obtain critical small corrections, the critical temperature anddenoted the by non zero VEVthat at this temperature forthe our respective toy minima model as benear ˜ the critical temperature. It is then apparent that In the region where temperatureis corrections possible to for the a bareare first singlet relatively order mass rare phase become and transition important we to it lose occur little when in ignoringminima them. with one minima at In the last equationNote the that parameter we have not includedTherefore any our temperature corrections scheme to is the valid bare in mass the of the region singlet. where where JHEP01(2014)073 ]. < 1. 61 . , this (3.13) (3.12) γ 1118 . δγ + γ ] coupled to . 58 ]. We find that ) ] and CRESST- – 1 CL i.e. 0 ] and LUX [ γ ( 15 56 σ 60 [ F 128 [ = 220 km/s and the . ≡ 0 v 2 s . We therefore calculate < } m γE 2 are small compared to the 2 β − G. a h ˜ λ 2˜ 0 1 V e ≡ χ G tan q 2 , λ λ + ) 2 ) is significantly smaller than any 2 2 s ) v micrOMEGAs2.4 2 2 m δγ v , λ, A λ − 2 s ]. + λ + m + γ 62 2 s { ( + ) m 2 s F (2 m γE , the circular velocity s 2 ( 3 – 8 – ˜ a − m 2 ˜ λ − ( √ 2 ˜ λ ˜ a such that = 544 km/s [ 2 q : we illustrate the bounds on a neutralino-nucleon inter- v } 3 GeV/cm . + β esc = 2 v 2 4. The baryon washout condition is satisfied for . = 0 cγ tan 0 , M 0 1 λ − e χ . − 5 GeV. ρ . . Most importantly, we require the dark matter relic abundance to = 174 GeV. This gives us the relation: ] are highlighted. γE δγ : we require that model points satisfy an upper limit on dark matter ). Additionally, we impose current limits from various experiments, 2 1 v , λ, A 129 s + m ˜ 2 λ < 3.13 ] to calculate observables and performed a scan over NMSSM parameter : we impose the LHC bound on the Higgs boson mass by taking the with respect to gamma evaluated at the vev at the critical temperature. { 128 [ h . − 59 T 0 V < 0 = < m ]. We also consider the projected bounds of XENON1T [ 2 5 h . 20 1 0 e χ action cross-section as measuredII[ by the XENON100 experimentThese [ bounds are derivedthe using Galactic standard halo assumptions,Galactic i.e. escape dark velocity matter density in Higgs mass combined theoretical and experimental121 uncertainties within the following range, i.e. Direct dark matter detection Relic abundance relic abundance observed by thein Planck satellite, large i.e. part Ω of thefor parameter the space total the lightest Planckconsidered. neutralino measured The is value, points not enough which andΩ pass to multi-component account the dark Planck matter constraint within should 3 be • • • for values of ranges shown in table be consistent with Planck along withtransition the constraint of of eq. strongly first ( orderas electroweak phase we enumerate them below. 4 Constraints applied To find regions consistentNMSSMTools [ with experiment we used equation is a functionδγ of only four parameters: of its five components.derivative We of ensure that allFinally components we of insist that ∆ derive the following equation The details of this calculation are given in the appendix. Note that, apart from Using the condition of degenerate minima occurring at a critical temperature, it is easy to temperature VEV is Finally we solve our toy model. We begin this calculation by insisting that the zero JHEP01(2014)073 ]. > 1, . 2, / ) 64 + 1 [ j Z (4.1) e χ e χ j > 4 1 m − e χ < M 10 with 0 1 → e × χ j ]. ]. This limits − e χ m e 67 1 68 26) + e . χ e 0 ]. ( , → σ ). This relationship 2 ± 65 / [ 3 − ] ] 9 e 55 . ] ] ] ] ] 2.11 − ] + 0 1 26 68 1 e e 2 , , 209 GeV [ 2 χ [ Z 63 64 65 69 66 10 [ [ [ [ [ m 25 14 M > Source × [ [ 4 ) ) = (3 5 0 2 . − e χ 4 ]: 4 sγ 9 1 − m 9 − 14 < → − 2 10 + 10 ) b 10 0 1 ( × 2 14 e × χ B × 4 GeV SUSY µ N 128 m 5 GeV 5) . . 22) . . ± 5 2 0 + 3MeV . . 0 1 Value 5 209 GeV . ]. 2 . < 13 < δa 103 +1 − < ] and ± 4 : 4 1 2 > 24 9 . N . > – 9 – ( 63 2 − 125 3 [ : we require the supersymmetric contribution to 43 . F π − 9 10 ( 2 G (3 − 3 √ while electroweak baryogenesis in the NMSSM has a × Z 10 values [ 4 : in the light neutralino regions where ) β 12 . M ) 2 − × β 0 2 ) µ = − e χ 0 1 5 2 + e 0 1 χ . . 2 sγ m 0 1 e 1 χ µ + 1 h µ h e 0 1 χ e +1 − χ 0 1 e χ + → e 2 χ → m δa → . m 0 1 → b Z are Higgsino fractions coming from eq. ( Ω 3 e s χ ( Z Γ Quantity B B 2 Γ m 14 ( A null result in LEP searches on a process ( : we enforce limits on the branching ratios of flavour violating decays, N B : we take the lower LEP bound on a mass of chargino to be 2 ) = ]. − boson decay width µ 66 and ], which can be translated into ( + Z µ 2 14 13 . List of the experimental constraints which we imposed in our NMSSM scan. N → 2 to be in the range: pb [ s 5 GeV [ 2 . − B − ( µ lightest neutralino must mostly beor either no a admixture from bino, higgsinos. a wino, or a singlino with minimal where is derived assuming three massless neutrinos. In order to satisfy this constraint the Invisible one has to take intoAnalytical account expression of for the this invisible decay process width is of given Z by boson [ into neutralinos. Chargino mass 103 sets an upper bound10 on the neutralino productionthe cross-section mass of the lightest neutralino from below. hanced for large values ofpreference tan for moderate tan Muon anomalous magnetic moment g Flavour physics B Baryogenesis does not conflict with these constraints, since these processes are en- Table 1 Our constrains are summarized in table We used a general limit on the chargino mass, however for possible caveats, see ref. [ • • • • 2 JHEP01(2014)073 ], 71 comes µ inflaton. e d e d e u ]. The masses of left and 70 ] GeV β 7] . sin β 01] GeV . [1:65] Range [0 : 0 cos 6000 GeV 6000 GeV 6000 GeV 3300 GeV 6000 GeV 4000 GeV 6000 GeV -5000 GeV -2500 GeV -2500 GeV λ [0 : 0 [0:10000] GeV [10:3000] GeV [10:4000] GeV A [800:6000] GeV 128 which is the upper value on the dark . 0 [0 : < – 10 – 2 R L R e confidence level. As mentioned above, neutralinos τ e τ b h e L m m m σ 2 R e c e Q = = = CDM m κ m L β L R R 3 R 1 2 3 t · b τ λ e t s e e e µ µ e λ µ Q = = κ A A A A M M M m m m m tan m R L A 1 e u to be positive. The first four parameters are constrained by the = = e = Q Parameter m 2 s R L ] bounds on squarks and sleptons. R m e e e d e e m 73 m m m shows the bino (green dots), wino (red stars), higgsino (blue squares), and ] and CMS [ 1 72 . Scan ranges and fixed values of the NMSSM parameters. The upper bound on shows the parameter ranges of our scan. We fixed the first and second genera- sition 2 Figure ATLAS [ singlino (pink diamonds) fractionsof of the the lightest lightest neutralino. neutralinomatter We below abundance restrict Ω set the by relic Planck density at 3 tion sfermionic masses tocontributions to high electron values and in nuclearright order electric handed dipole stops to moments are avoid [ adjustedvaried large the to electroweak potential yield gaugino the suspersymmetric masses measured inogy. a value wide Varying of range the the to gluino Higgs exploreThe boson mass dark selected mass. matter is ranges phenomenol- important We also guarantee to that determine our the spectrum running does of not the conflict with the LEP [ 4.1 Scanning the parameter space for dark matterTable and first order phase tran- Table 2 from our requirementstrongly for first ordermatter phase phenomenology transition. and inflation, We which vary we shall the discuss gaugino below. masses in order to explore the dark JHEP01(2014)073 , ], β 20 sin ). The β 63 GeV ≈ cos 3.13 128 set by 0 1 . λ e χ 0 ]. m . We only show < 74 0 1 e χ 2 µ < A of the Planck central ], CRESST-II [ h m 0 1 have a large singlino σ e χ 18 0 1 -ray line observed from , Ω e χ ]. γ 1 ) = 0, see eq. ( 17 m < γ ( F 128 [ . 0 ]. 1118 . ]. The peculiar clustering of the < 23 1 and 2 h 1 0 1 e χ Ω < 1118 . – 11 – confidence level [ )) and the mass of the lightest chargino. Also, light σ 4.1 boson. As this, and the previous, figure shows our model sets an upper bound on the parameter Z 2 s m ]. All the points with the smallest 128 at 3 . section. 0 21 4 2 have to have small higgsino fraction due to strict limits on the < / 2 Z h 0 1 e χ < m 0 1 e χ , we show the dark matter relic density dependence on , we show how spin independent dark matter-nucleon scattering experiments m 2 ] and CDMS [ 3 19 . Bino (green dots), wino (red stars), higgsino (blue squares), and singlino (pink diamonds) . On the right hand panel we only show the points that fall within 3 4 In figure In figure The heavier neutralino can have a larger higgsino fraction. This is also connected to already mentioned in the relatively light and dedicatedexcess searches interactions for in light backgroundCoGeNT neutralino detected [ DM by potentially DAMA/LIBRA couldfraction, [ and explain are ruled outfrom by XENON100 the experiment. XENON1T We and also LUX show the experiments. projected These bounds bounds are based on the assumptions points also have the potential tothe explain Galactic the Centre origin in of terms the of 130 GeV the annihilation ofprobe a the 130 scenarios GeV with neutralino thepoints [ constraints listed that in are table Planck. circled in It is black interesting fall to within note the that range quite a 0 few points lie in the regions where LSP is points in this plot isis understood due as to follows. neutralino Thedepletes first, annihilation the smaller through group neutralino the around abundance 126second, GeV making larger Higgs. it group possible This ofhiggsino resonant to component points annihilation coupling satisfy to originate the from Planck the bound. lightest The neutralinos with an enhanced which allows larger higgsino fraction in theparagraph lightest this neutralino. opens As up we shall more argue annihilation in channels the next to satisfypoints the for relic which density the constraint. that dark matter is relic to density satisfy to Ω falls below the upper limit from Planck, invisible Z boson decay (fromdark eq. matter ( regions aresophisticated very scanning fine techniques, see tuned for and instance, require ref. separate [ detailedthe analysis fact and that the more positivity of condition of the stronglysection first order electroweak phasevalue transition for and the pass relic abundance all of constraints dark listed matter: in 0 with masses Figure 1 components of the lightest neutralino for scanned model points. All points shown satisfy the JHEP01(2014)073 128. . 0 < 2 h 0 1 e χ Ω < 1118 . . 4 signal region from CRESST-II are shown. We σ – 12 – 128, are highlighted in black circles. . 0 < 2 h 0 1 e χ Ω < 1118 . . Spin independent direct detection cross section vs. mass of the neutralino in our scan. , i.e. 0 . Relic abundance versus the mass of the dark matter particle for our scan. The blue lines ). They also pass all the constraints listed in section σ 3.13 also show projected boundsrelic for abundance XENON1T accounts and forwithin LUX 3 the experiments. full dark The mattter points content where of neutralino the Universe measured by Planck Figure 3 The current bounds from XENON100 and the 2 All the points satisfyeq. the ( condition for a first order electroweak phase transition, as indicated by Figure 2 indicate the bounds on dark matter density implied by the Planck satellite: 0 JHEP01(2014)073 can 1199. β . . Lower β = 0 being positive. ) are distributed 2 s observed m region is the MSSM 3.13 λ , for a particular tan so we avoided scanning those. and low β β κ dependent and for large tan tend to cluster around lower values. sin β β β being small is violated. Similarly the cos V λ A – 13 – 5000 GeV fixed to be able to satisfy the Higgs that are tan − are mainly within a 100-200 GeV range because, as V = µ t is inconsistent with the baryon washout condition, just A β ). Here we take the Planck central value Ω is a relic of our approximations rather than any real difficulty λ observed Ω ]. We kept and / 0 1 13 λ e χ A , we show how the relevant parameters that enter eq. ( 4 . Distribution of the parameters which are relevant for baryogenesis in our scan. too large so that our assumption of ∆ V by a factor (Ω In figure Since in most of the cases that we have found the relic abundance is significantly lower Figure 4 values are constrained because of the invisible Z decay and the chargino mass, which SI , however, originates from baryogenesis since the low mass of 125 GeVmass [ bound more easily.mentioned Values above, of large valuesThis are translates constrained into upper byµ bound, the lower requirement than of The breakdown is duemake to ∆ terms in ∆ upper bound on in satisfying the baryonλ washout condition in thatlimit parameter range. and it The is lower difficult bound to on satisfy the baryon washout condition in the MSSM for a Higgs in the scans. In theThis top is left not panel we because can athat see higher that our tan tan approximations break down for the large tan than the set limits byσ the direct detection experiments,As we we need see, to most lower ofand the the XENON1T cross points experiments. section that fall below XENON100 will be tested very soon by LUX JHEP01(2014)073 . µ (5.3) (5.1) (5.2) as an e d e d could still e u e e e L e L ], where inflation ), we can consider ] iθ ) = 0 there needs to 78 , 4 γ -flat directions, for a , ( φe 2 77 D ]. While F 5 (Φ = , - and 6 θ p 10 F φ M + 6 p . 6 M φ e , d ]. 6 3 p 6 y + 3 Φ e M 80 A d , √ 6 y − + 79 2 e u ⊃ – 14 – φ 2 φ = W m φ 1 2 in their respective potentials [ , in order to satisfy condition a e ] could also be good inflaton candidates. All the inflaton ) = 76 φ [ ( d correspond to the right handed squarks. In fact within the V H e d u is much easier met at low values. ) through H ]. In this paper, we will consider one further step and we wish to e u, potential which would potentially ruin the flatness and therefore the A.8 A.8 83 as an inflaton for this study. inflection point – d e , for which the scalar potential is then given by [ d ] and 81 e H d φ u e 3 u , ]. Out of these flat directions, we will be interested in studying 2 [ ], where (1), and the scalar component of the superfield Φ is given by: SH 75 4 e e – e flat direction is lifted by a higher order superpotential term of the following L enters eq. ( 2 ∼ O e ], L e λ d e y d 75 A e u Previous studies on MSSM inflation only considered the overlap between the parameter Within the NMSSM with the introduction of a singlet it becomes necessary to include , and so condition λ After minimizing the potentialthe along real the part angular of direction where 5.1 Brief review ofThe inflation form [ space for a successfulrelic inflation abundance and [ the neutralinoconstrain primordial as inflation the along with dark neutralinobaryogenesis. matter, dark matter which and satisfies condition the for sufficient tions to the success of inflation driven solelybe by a the good gauge invariant inflatonare inflaton not candidate [ constrained in by our theconcentrate dark current on matter scenario, and the but baryogenesis in constraints, our therefore we analysis only slepton masses can be driven forand sufficiently explain large the seed e-foldings perturbationsbeen of for confirmed inflation the by to temperature the anisotropy explain recent in the Planck the data current CMB, [ Universe which has the dynamics of a singlet field. Since the singlet here is not gauged there will be contribu- Within the MSSM therereview see are [ nearly 300inflaton gauge-invariant [ MSSM, candidates provide Since be some tuning between fourthA and fifth terms. This fine tuning increases with increasing 5 Gauge invariant inflaton set bounds on the higgsino component of the neutralino which is directly related to low JHEP01(2014)073 ' (5.8) (5.9) (5.4) (5.5) (5.6) (5.7) (5.12) (5.13) (5.10) (5.11) (5.14) , while COBE 0 φ N is a positive y A , ] 2 , , ∆ ] ] ) ), where 2 during the process). The mass 2 , √ . 4 φ , , ) α ∆ ( 4 θ ). The inflationary perturbations 2 ( 4 ) φ V √ -term respectively ( 2 . α . ( O y . p ( ) 0 2 α e COBE , 2 d 2 ( 0 A φ + O M 2 m p N α φ m ) in O 4 [ ( α COBE 0 φ / + 2 M 1 + φ + O N m 0 e 2 d 2 0 sin φ 2 + 3 3 p φ 2 φ m 1 + 4 45 − 2 1 COBE cot[ 2 φ 2 φ 10 0 1 M m √ + ∆ N 2 ≡ φ m φ 2 m √ – 15 – 2 p e 2 u λ ∆ α 2 φ ' 4 m α 15 , 2 y 2 0 m M √ m φ φ 4 A inf 900 = 40 m = − ) = 32 ) = ) = 4 H 2 φ ≡ 0 0 0 π ) = 0 0 0 φ φ φ 5 m 2 8 φ ( ( ( φ 0 = 1 √ ∆ ( 000 V s 00 V V n = V H δ is the number of e-foldings between the time when the observation- are the soft breaking mass and the COBE A N 1, there exists a point of inflection ( and ] 4 φ 2 is given by: α m e d e d The amplitude of the initial perturbations and the spectral tilt are given by: For [ e u respectively, where In the above, ally relevant perturbations are generated till the end of inflation and follows: and The Hubble expansion rate during inflation is given by [ at which where Note that the masses arewill now be VEV dependent, able i.e. LHC to will constrain be the ablequantities inflaton to are mass constrain related only the toshall at masses each discuss at the below. other the scale via LHC of renormalization scale. inflation, group However i.e. equations both (RGEs), the as physical we quantity since its phasefor is absorbed by a redefinition of where JHEP01(2014)073 ]: ], and 4 (5.15) 1 = 3 and i 128 [ . 0 < 2 , where h k 0 1 e e χ d j Ω e variance of the spectral d i < σ e u 1 -term are given by [ we can determine the masses ± y , , 1118 A . 2 1 2 1 g g 1 2 1 a, b, c, d and the M M 9 8 5 5 2 − + 5 is the relativistic degrees of freedom 10 . + 2 3 2 3 g × ]. The reheat temperature at which all the g 3 2 3 84 M 196 = 231 M . 3 , the estimation of the reheat temperature can flat direction, in order to relate the low energy 4 16 ∗ φ g e – 16 – d = 2 e m 2 d 2 ζ 1 e π u 1 π . P . The running of the inflaton mass is mainly determined 6 4 0 2 − φ − , where = = 0 φ 2 φ y ˆ φ µ ˆ µ d 50. Since inflaton is made up of squarks, reheating and ther- d m dm dA ˆ ˆ µ µ ∼ p ) 4 4 / p 1 ∗ /M 0073. The brown lines show the running of the inflaton mass, where they g ) . are in thermal equilibrium (kinetic and chemical equilibrium) is given 2 0 0 π φ ( ± 15 V / , we can assign the flat direction combination to be: 3 e u 9606 . 120 4)ln( . / . Blue region depicts the parameter space for inflation where it yields the right amplitude k = 0 = s 6= rh n We can then use the RGEs for the j T 9 + (1 . 6= by the Planck data. The RGEs for the inflaton mass and the Since the requirement for asquark, successful i.e. baryogenesis implicitly constraintsi the right handed physics that we can probe at the LHC with the high energy inflation, which is constrained present within NMSSM. Since alldetermine the the soft physical SUSY parameters breaking arebe mass, fixed made in rather this accurately. model once we 5.2 Parameter space for inflation, dark matter and baryogenesis 66 malisation happens instantly as showed indegrees ref. of [ freedom by of density perturbations intilt, the CMB, i.e. intersect with the bluestrongly region first depict order the phase correct transition.of relic From the these abundance, intersections, inflaton 0 atby the the inflationary bino scale and gluino masses, see table Figure 5 JHEP01(2014)073 , we (5.16) 3 ). Once j . 6= 0 i φ 6= . The parameters 3 M are U(1) and SU(3) 128. In table . . The gaugino masses 3 Inflaton RGE Inflaton RGE 0 Constrained by and 5 M order phase transition order phase transition order phase transition order phase transition . The blue region shows 1 < 5 st st st st M Dark Matter abundance Dark Matter abundance 2 1 1 1 1 , which satisfy the condition and ), we know the mass of the . The brown lines show the h , CMB temperature anisotropy CMB temperature anisotropy s 0 1 1 e χ n 5.4 = 0 M Ω 3 . ]: inflaton candidate (3 3 7 4 . < 963 i j . 2 i d 123 112 − e d . . 796 861 g 02906 i M a, b, c, d 1084 5349 6 . 2 0 126 e . d 0 137 3 6 6= 3. It includes central value of density 1118 e u β . k 7 . 5 6= 8 515 . . c 276 124 j − . . 772 10418 59 1375 4281 3279 – 17 – 3 . 2 0 . 0 176 . Since from eq. ( 5 3 2 i g π ), and also accommodate neutralino as a dark matter i 6 . α 16 2 4 . / variation in spectral tilt 985 3.13 . 3 b 042 112 − . . 524 06520 ) = σ 2006 4986 2120 2 − . 2 0 119 are the associated couplings. i . 1 0 142 g 4 3 ( ± = g β 3 8 . α as an inflaton for 3 and 214 17 k . . a 659 119 − e 1 d . . 06405 1127 2151 5269 1425 j g 5 . 2 0 61 and . e d 0 165 is the VEV at which inflation occurs, 3 3 2 0 e u π φ 16 are constrained from baryogenesis point of view, and this in turn uniquely determine , we show the four benchmark points, = / 5 eff 0 2 µ β ](GeV) h 0 1 (GeV) β, µ λ = 11 14 = ˆ e χ 0 1 (GeV) (GeV) (GeV) (GeV) . We show the benchmark points that are depicted in figure (GeV) , where inflation can happen. This can be seen in figure e χ tan 1 3 φ λ 1 0 Ω 10 µ eff tan φ m α A m M M µ × , Parameter [ In figure To solve these equations, we need to take into account of the running of the gaugino λ 0 φ for a successful baryogenesis, eq. ( which satisfies the relicsummarise abundance the relevant constraint parameters 0 the for Standard NMSSM Model. required to explain the Universe beyond scale the parameter of perturbations together with mass of the inflatondetermined at by a the RGEs, particular and scale it and is its mostly sensitive running to from bino high and scale gluino to masses. low scale is with inflaton at the electroweak scale, by using the RGEs we are able to evolve it to the high gaugino masses, and masses and coupling constants which are given by, see [ the mass of theagain lightest we stop reiterate which that setsbe without the significantly our mass less approximation for strict. scheme, the The mass constraints of on the baryogenesis inflaton would is given at thewhere inflationary scale ˆ Table 3 which enter in the RGλ, equations A are mainly sensitive due to different JHEP01(2014)073 (A.4) (A.1) (A.2) (A.3) 2 ) 2 c . EP is supported by 4 c ϕ . 2 ϕ 4 c 2 λ ˜ a 2 + ˜ λϕ = 0 λ 2 s . 2 c + . m ) 3 ( ) to get 4 c 2 λϕ 2 c ϕ ) ϕ ϕ 2 c 2 + 4 c 2 + ) 2 E A.2 ϕ ˜ a ϕ 2 c 2 s λ 2 2 2 s T ϕ 2 λ m 2 3 ) + m 2 c λ 2 2 s + − − ϕ a 2 s 2 s 2 + m 2˜ 4 c 2 c m λ 2 s m ϕ ϕ (2 2 ( ) to get a second equation ) 4 ˜ 2 λ a m + ( T 2˜ ϕ − 2 s ( 2 2 times eq. ( + A.2 – 18 – 2 4 c ˜ a / m 3 c -flat direction and it is made up of right handed ( M − is ˜ D λϕ − Eϕ 2 c = c = ˜ ϕ λ T c 3 c ,T ϕ − c 2 c c 2 ϕ c 2 ϕ

ϕ 2 ET ) = c ∂V ) equal to eq. ( T 2 ∂ϕ ( 2 ) and set it to 3 2 M ϕ A.1 M + A.1 2 c cT Using the condition that the potential at the critical VEV is equal to the potential at ARC Centre of Excellence for Particle Physics at the Tera-scale = 0 we have a second equation Finally we can set eq. ( research environment during part of this project. A Solving the toyThe model first derivative with respect to ϕ We can then use eq. ( Da Silva and Lingfeithe Wang for valuable discussions.STFC This ST/J501074. work was AM supported isfor in supported Fundamental part Physics by by the undertute Lancaster-Manchester-Sheffield STFC of Consortium grant Theoretical ST/J000418/1. Physics, China CB for providing thanks partial the financial Kavli support Insti- and an excellent the CMB, but also all the relevant matter requiredAcknowledgments for baryogeneis and dark matter. We thank Michael Morgan forport invaluable throughout help, this continuous project. discussions, feedback and EP sup- also would like to thank Sofiane Boucenna, Jonathan lightest neutralinos can bedensity limits. generated Part thermally of whichdetection these constraints. model satisfies Finally, points the we also have present shownwith pass that dark inflation, the the presented most where matter scenario stringent the issquarks. dark fully inflaton matter consistent The is direct visible a sector inflation would explain not only the temperature anisotropy of In this work we examinedto-Minimal inflation, Supersymmetric baryogenesis Standard and Model. darkcosmological matter We requirements in have can found the be context thatticular, of these simultaneously we Next- three have accommodated shown important byeven that the a with theory. strongly recent first LHC In order constrains par- phase applied. transition can Then be we easily demonstrated achieved that an abundance of 6 Conclusions JHEP01(2014)073 , , (A.5) (A.6) (A.7) Nucl. , . Using γ (2007) 019 MSSM flat 06 2 s ]. m γE 2 − a ˜ λ 2˜ JCAP G q SPIRE , 2 Gauge invariant MSSM λ IN + . 2 s ][ . 2 m ) ! ). It is proportional to ]. 2 c − . 2 c ]. ϕ 2 s 2 2 + ϕ γE ) A.3 2 m λ 2 c ) 2 ˜ a − ϕ SPIRE 2 s ˜ 2 a + 2 ˜ λ SPIRE λ IN γE 2 s m λ ) and it is straight forward to show that s 2 2 IN δ ][ , Princeton University Press, Princeton m − a A-term inflation and the smallness of neutrino m + ( s ][ 2˜ 2 + ˜ λ 2 s hep-ph/0605035 ( 2 s + + [ √ 2 m 2 s ( ˜ a m – 19 – 2 m − γE q − ( Planck 2013 results. XVI. Cosmological parameters − 2 ˜ λ 1 + + λ

2 2 c = ]. 2 we have G = 1 ϕ λ cγ 2 c 2 c hep-ph/0608138 Particle physics models of inflation and curvaton scenarios Eγ ), which permits any use, distribution and reproduction in = ϕ ϕ − = (2006) 191304 [ Cosmic abundances of stable particles: Improved analysis 2 arXiv:1001.0993 ]. 2 c ]. SPIRE [ ϕ 97 IN cγ γE [ (A.8) + ) γ SPIRE SPIRE ˜ 2 λ ( IN IN [ (2007) 018 − F ) to get the following equation CC-BY 4.0 ][ (2011) 85 ≡ Principles of physical cosmology 07 This article is distributed under the terms of the Creative Commons A.6 (1991) 145 0 = 497 collaboration, P. Ade et al., Phys. Rev. Lett. JCAP , , B 360 hep-ph/0610134 Phys. Rept. U.S.A. (1993). Phys. masses direction inflation: Slow roll,[ stability, fine tunning and reheating Planck arXiv:1303.5076 inflaton P.J.E. Peebles, P. Gondolo and G. Gelmini, R. Allahverdi, K. Enqvist, J. Garc´ıa-Bellido,A. Jokinen and A. Mazumdar, A. Mazumdar and J. Rocher, R. Allahverdi, K. Enqvist, J. Garc´ıa-Bellidoand A. Mazumdar, R. Allahverdi, A. 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Rev. ]. ]. ]. , arXiv:1211.2674 (2001) 151 06 [ 380 arXiv:1110.3726 b collaboration, D 82 [ (2003) 421 SPIRE SPIRE SPIRE IN IN arXiv:1103.5758 IN inflation and the neutralino[ dark matter Rev. [ history in light of Planck data waves and constraints from Planck and a solution to the fine-tuning problem 1847 021801 Heavy Flavor Averaging Group C-hadron and tau-lepton properties as of early 2012 spectra: the pMSSM parameter space with neutralino dark matter [ JCAP bremsstrahlung Rept. DELPHI collisions at [ NMSSM The LEP Electroweak Working Group B 606 DELPHI collisions up to 31 LHC R. Allahverdi, B. Dutta and A. Mazumdar, R. Allahverdi, B. Dutta and Y. Santoso, L. Wang, E. Pukartas and A. Mazumdar, S. Choudhury, A. Mazumdar and S. Pal, http://lepsusy.web.cern.ch/lepsusy/www/inos A. Arbey, M. Battaglia and F. Mahmoudi, K. Enqvist, A. Mazumdar and P. Stephens, S. Hotchkiss, A. Mazumdar and S. Nadathur, G. Tomar, S. Mohanty and S. Rao, K. Enqvist and A. Mazumdar, A. Chatterjee and A. Mazumdar, https://twiki.cern.ch/twiki/bin/view/AtlasPublic/SupersymmetryPublicResults https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsSUS A. Menon, D. Morrissey and C. Wagner, S. Abel, S. Khalil and O. Lebedev, [81] [82] [79] [80] [66] [64] [65] [77] [78] [74] [75] [76] [71] [72] [73] [68] [69] [70] [67] [63] JHEP01(2014)073 (2011) D 87 (2013) D 83 D 88 Phys. Rev. , Phys. Rev. , Non-perturbative Phys. Rev. , Probing the supersymmetric inflaton ]. – 24 – Xenophobic dark matter SPIRE ]. ]. IN ][ SPIRE SPIRE IN IN ][ ][ arXiv:1205.2815 [ arXiv:1103.2123 arXiv:1306.2315 [ [ 015021 and dark matter link(2013) via 023529 the CMB, LHC and XENON1T experiments production of matter and rapid123507 thermalization after MSSM inflation R. Allahverdi, A. Ferrantelli, J. Garc´ıa-Bellidoand A. Mazumdar, J.L. Feng, J. Kumar and D. Sanford, C. Boehm, J. Da Silva, A. Mazumdar and E. Pukartas, [84] [85] [83]

Chapter 7

Gravitational waves as a probe of vacuum stability via high scale phase transitions

7.1 Introductory remarks

The recent discovery of gravitational waves [111] has led to a surge in interest in this new way of gaining information about the Universe [112, 113, 114, 115, 116, 117, 118, 119]. Of particular relevance to particle cosmology are gravitational wave signatures left by cosmic phase transitions [120, 121, 122]. There are proposed gravitational wave detectors that are sensitive to cosmic phase transitions, LISA and aLIGO. If the electroweak phase transition was strongly first order this will leave a stochastic gravitational background at a frequency and amplitude that can be detected by LISA. In contrast, aLIGO is insensitive to phase transitions at the weakscale and is instead sensitive to phase transitions occurring at a much higher scale, T between about 106 and 108 GeV [12]. Unfortunately the time-frame of LISA is several decades in the future in contrast to aLIGO. It is therefore of more pressing interest whether there are physical motivations for their to be a phase transition occuring at a temperature that could leave relic gravitational wave backgrounds visible to aLIGO. In this paper I consider the problem of vacuum stability in the Standard Model and note the coincidence of scales between the sensitivity of aLIGO and the instability scale in the Standard Model. Adding a gauge single to the Standard Model is arguably the simplest method to improve its stability. This can occur through improved running or through a threshold correction to the Higgs self coupling. Such a gauge singlet can also acquire a vacuum expectation value at some time during the history of the Universe. If it does so through a strongly first order phase transition then it will contribute to the relic gravitational wave background. I present three bench mark points where the Higgs self coupling remains positive up to the GUT scale, (1016) GeV, as well as having a strongly first order phase transition, fast enoughO bubble nucleation for the phase transition to proceed, peak frequency and amplitude that is potentially visible to aLGIO and acceptable zero temperature phenomenology. I also present a scan with slightly weaker criteria where there is a moderate to large threshold correction to the Higgs self coupling

163 and a strongly ﬁrst order singlet phase transition occurs at the right temperature. To our knowledge this is the ﬁrst paper that gives a physical motivation for a phase transition that leaves a relic visible to aLIGO.

7.2 Declaration for thesis chapter 7

Declaration by candidate In the case of the paper present contained in chapter 7, the nature and extent of my contribution was as follows:

Publication Nature of contribution Extent of contribution Found the hole in the literature about physical motivations for phase transitions potentially visible to aLIGO and suggested this direction. Calcu- lated the high scale phase transition and its thermal properties. Analysed bench marks in detail 6 calculating the gravitational wave amplitude and 60% frequency, the nucleation temperature. Calculated the running of each of the bench marks. Worked on producing multiple plots. Worked substantially on the draft writing many sections and editing all others. Contributed to discussions throughout. The following coauthors contributed to the work. If the coauthor is a student at Monash, their percentage contribution is given: Author Nature of contribution Extent of contribution Digitized the plot for aLIGO sensitivity and super- imposed our data on top. Wrote the introduction Csaba Balazs and was heavily involved in editing. Contributed to discussions throughout. Set up scan to verify zero temperature phenomenology is correct and implemented the com- munication with my code. Wrote several sections Andrew Fowlie and edited all others. Substantially expanded understanding of vacuum stability. Contributed to discussions throughout. Wrote a section. Contributed to discussions Anupam Mazumdar throughout. The undersigned hereby certify that the above declaration correctly reﬂects the nature and extent of the candidate and co-authors’ contributions to this work.

164

7.3 Published material for chapter 7: Gravitational waves at aLIGO and vacuum stability with a scalar singlet extension of the Standard Model

Begins overleaf

166 PHYSICAL REVIEW D 95, 043505 (2017) Gravitational waves at aLIGO and vacuum stability with a scalar singlet extension of the standard model

† ‡ Csaba Balázs,1,2,* Andrew Fowlie,1, Anupam Mazumdar,3,4, and Graham A. White1,§ 1ARC Centre of Excellence for Particle Physics at the Tera-Scale, School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800, Australia 2Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800, Australia 3Consortium for Fundamental Physics, Physics Department, Lancaster University, LA1 4YB Lancaster, United Kingdom 4Kapteyn Astronomical Institute, University of Groningen, 9700 AV Groningen, The Netherlands (Received 30 November 2016; published 8 February 2017) A new gauge singlet scalar field can undergo a strongly first-order phase transition (PT) leading to gravitational waves (GW) potentially observable at aLIGO and stabilizes the electroweak vacuum at the same time by ensuring that the Higgs quartic coupling remains positive up to at least the grand unification (GUT) scale. aLIGO (O5) is potentially sensitive to cosmological PTs at 107–108 GeV, which coincides with the requirement that the singlet scale is less than the standard model (SM) vacuum instability scale, which is between 108 GeV and 1014 GeV. After sampling its parameter space, we identify three benchmark points with a PT at about T ≈ 107 GeV in a gauge singlet extension of the SM. We calculate the nucleation temperature, order parameter, characteristic time scale, and peak amplitude and frequency of GW from bubble collisions during the PT for the benchmarks and find that, in an optimistic scenario, GW from such a PT may be in reach of aLIGO (O5). We confirm that the singlet stabilizes the electroweak vacuum while remaining consistent with zero-temperature phenomenology as well. Thus, this scenario presents an intriguing possibility that aLIGO may detect traces of fundamental physics motivated by vacuum stability at an energy scale that is well above the reach of any other experiment.

DOI: 10.1103/PhysRevD.95.043505

I. INTRODUCTION [7–11], fragmentation of the inflaton or any scalar condensate [12–14], cosmic strings [15,16], or a cosmological The recent detection of gravitational waves (GW) by the phase transition (PT) accompanying either the breakdown of LIGO Collaboration opened a new observational window a fundamental symmetry or a scalar field acquiring a vacuum for the early Universe [1]. Among the most exciting expectation value (VEV). If this PT is first order, then GW prospects is the observation of GW from cosmological are created by violent collisions between expanding bubble events that happened well before the first observable walls of the new vacuum (see, e.g., Refs. [3,17–46]), which photons were created [2]. Not limited by recombination, can be potentially constrained by the current and future GW GW can be used to directly probe fundamental physics, observatories, such as the future space mission eLISA reaching to considerably higher energies than any other [47,48], and also possibly by aLIGO within the next 5 years existing experiments. There are potentially several known [49]. Recently, it has been shown that these GW are sources of observable GW, which can be split into three detectable by BBO or DECIGO [50–54]. categories [3]: (i) binary black hole mergers, mergers of In the present work, we explore the detectability of GW binary neutron stars or a neutron star and a black hole, or originating from fundamental physics at the upgraded supernova core collapse, with a duration between a milli- LIGO detector, aLIGO, in the near future (2020–2022) second and several hours; (ii) long duration signals, i.e., [55–58]. It is known that the frequency of GW from the from spinning neutron stars; and (iii) stochastic background electroweak PT is too low to be detected at aLIGO [21,22]. arising from the superposition of unresolved astrophysical Therefore, our main emphasis here is to seek GW accom- sources. The latter can be a stochastic background of GW panying an earlier PT with physics beyond the Standard which can also arise from cosmological events, such Model (BSM). In search of detectable primordial GW at as during primordial inflation [4–6], resonant preheating aLIGO (LIGO run phase O5), we provide a simple but concrete particle physics model which can yield the observed amplitude and peak frequency for GW which *[email protected] † [email protected] have been recently proposed in Ref. [49]. In the current ‡ [email protected] paper we analyze a framework which is an extension of the §[email protected] standard model (SM) of elementary particles with a gauge

2470-0010=2017=95(4)=043505(10) 043505-1 © 2017 American Physical Society BALÁZS, FOWLIE, MAZUMDAR, and WHITE PHYSICAL REVIEW D 95, 043505 (2017) 1 1 1 1 singlet scalar (SSM) (see, e.g., Refs. [59,60]). Indeed, this 2 2 4 2 2 3 4 V0ðH; SÞ¼μ jHj þ λjHj þ M S þ κS þ λ S is the simplest example of BSM physics that could enhance 2 2 S 3 2 S electroweak vacuum stability [61–63]. Besides this, such a 2 1 2 2 þ κ1SjHj þ κ2S jHj ; ð1Þ simple choice for physics beyond the SM could also help 2 us understand primordial inflation [64–66] (for a review where MS is the mass parameter of the singlet, κ is a see Ref. [67]). dimensionful coupling, λ is the singlet quartic coupling, As noticed before in Ref. [49], aLIGO is potentially S and κ1 2 are singlet-Higgs couplings. The above potential sensitive to cosmological PTs occurring at scales 107 GeV ; 108 is the most general gauge invariant, renormalizable scalar to GeV, which raises the question of whether such a potential with the considered particle content. The linear new scale emerges in BSM physics. It is a well-established 3 operator m S is removed by a shift in the singlet field result that the observed values of the top-quark mass, Higgs without loss of generality. mass and strong coupling drive the SM Higgs quartic To account for changing field properties during cosmo- coupling, via renormalization evolution, to negative values Λ ∼ 1010 logical PTs, we consider a one-loop effective potential with at about I GeV. The latter is known as the Higgs or – finite-temperature corrections (i.e., a free energy). As the vacuum instability scale [68 72]. The SM scalar potential Universe cools the free energy develops a deeper minimum is believed to be metastable; although we live in a false in the singlet direction, there is a PT to a new ground state vacuum, the probability of tunneling to the true vacuum is ≳ 130 and the singlet acquires a VEV, although no symmetries are negligible, and for a heavy Higgs boson, mh GeV, broken. If there is a discontinuity in the order parameter the Higgs potential would be stable [70]. In this paper we show two important results, which we γ ≡ hSi=T; can summarize below: (i) It is possible to realize a successful strong first-order i.e., the PT is first order, bubbles spontaneously emerge in PT in the singlet direction with the nucleation temper- the Universe in which the singlet VEV is nonvanishing 7 8 h i ≠ 0 ature within the range of 10 –10 GeV, which would S . We will scan over the Lagrangian parameters at the give rise to a GW signal within the frequency range high scale; guarantee that a strongly first-order PToccurs at a 7 8 of aLIGO, i.e., 10–100 Hz. We will establish this critical temperature in the range ð10 ; 10 Þ GeV by solving by taking into account finite-temperature corrections, for Lagrangian parameters; and impose the constraints on first incorporated in Ref. [73] in the context of the weak-scale parameters by requiring that the Higgs mass be next-to-minimal supersymmetric SM. 125 1 GeV and that the VEV be 246 GeV. This typically Oð Þ (ii) We carefully compute the running of the couplings requires dimensionful parameters to be TC and dimen- in the SSM at two loops, and conclude that for the sionless parameters to be Oð1Þ at the high scale. GW from range of parameters we have scanned, parameters high-energy PTs were considered in Ref. [74]. that yield a strong first-order PT could also amelio- A fraction of the latent heat from the PT could ultimately rate the SM Higgs metastability. In this paper we be released in collisions between bubbles, which result shall provide three benchmark points, where the in striking GW signatures. This occurs at the bubble scale of BSM physics would leave an undeniable nucleation temperature, TN, which is typically similar to ≲ footprint in the GW signal, potentially within the the critical temperature, TN TC, i.e., the temperature at range of aLIGO (O5). which the original ground state and emerging ground state Our paper is organized as follows. In Sec. II, we first are degenerate. We will calculate the nucleation temper- explain the SSM model. In Sec. III, we discuss what range ature in order to calculate the peak frequency and the of parameters of the singlet can yield strong first-order amplitude of the GW resulting from the singlet PT. PT, and what are the conditions to be fulfilled. In Sec. IV, we briefly discuss GW amplitude and frequency from the III. PHASE TRANSITIONS IN A TEMPERATURE first-order PT. In Sec. V, we discuss the Higgs vacuum IMPROVED POTENTIAL stability in the presence of a singlet-Higgs interaction, In this section we investigate whether the SM extended and in Sec. VI we discuss our numerical results. In Sec. VII, we conclude with our results and discuss briefly future with a singlet can produce GW at a strongly first-order PT directions. which could be detected by aLIGO. Acceptable low-energy phenomenology, including standard Higgs properties and vacuum stability, is imposed. To achieve such a scenario we II. SINGLET EXTENSION OF THE require the following cosmological history. STANDARD MODEL (1) Higgs and singlet fields are in true, stable vacuum at We consider the SM plus a real scalar (see, e.g., the origin at high temperature. 7 8 Refs. [59,60]) that is a singlet under the SM gauge groups (2) At T ≈ TN ≈ TC ∈ ð10 ; 10 Þ GeV, the singlet and carries no, e.g., discrete charges. Thus, our model is acquires a VEV in a strongly first-order PT gen- described by the tree-level scalar potential erating GW, potentially in reach of aLIGO. [The

043505-2 GRAVITATIONAL WAVES AT ALIGO AND VACUUM … PHYSICAL REVIEW D 95, 043505 (2017) temperature was chosen to coincide with the peak frequency sensitivity in aLIGO (O5).] (3) At low temperature, the Higgs acquires a VEV, hHi ≈ 246 GeV, resulting in the correct weak scale, Higgs mass, and satisfying constraints on Higgs- singlet mixing. We will calculate the critical and nucleation temperatures numerically as functions of the Lagrangian parameters. This is needed to calculate the frequency and amplitude of GW originating from bubble collisions. The first step is to include finite-temperature corrections to the effective potential. The one-loop finite-temperature corrections to the scalar potential have the form [75,76] 2 T4 X m2 X m ΔV ¼ J b þ J f ; ð Þ FIG. 1. The effective potential (i.e., free energy) for benchmark T 2π2 B 2 F 2 2 b T f T SSM II, shown above, below and at the critical temperature, TC, at which the minima are degenerate, and at the nucleation where JB and JF are thermal bosonic and fermionic functions, temperature, TN. respectively, and the sums are over field-dependent boson and fermion mass eigenvalues. We also add zero-temperature (2) There exists a critical temperature, TC, at which the one-loop Coleman-Weinberg corrections [75,76], two minima are degenerate, X g m2 m2 Δ ¼ i i i − ð Þ Vj ¼ Vj : ð6Þ VCW 64π2 log μ2 ni ; 3 F T i This is illustrated in Fig. 1 for a benchmark point μ summed over massive particles, where is the renormaliza- tabulated in Table I by SSM I. tion scale, chosen to minimize large logarithms; mi is a field- (3) The order parameter at the critical temperature dependent mass eigenvalue; gi is the number of degrees of ¼ 3 2 hSi freedom associated with the massive particle; and ni = γ ≡ ; ð7Þ for scalars and fermions and 5=6 for massive gauge bosons TC (up to an overall sign for fermions). must be substantial [i.e., Oð1Þ] in order to yield a Note that when one considers a PT in the singlet strong first-order PT. The fact that S is a gauge singlet direction the only relevant masses are field-dependent mass means that we do not need to concern ourselves with eigenvalues of both the CP even and CP odd scalar mass subtleties involving gauge invariance [75]. matrices as well as the charged Higgs. Also, there are no (4) Bubbles form, expand, dominate the Universe and issues with gauge dependence. The final corrections to the violently collide. finite-temperature effective potential are the Debye masses For the first-order PT generating GW, we fix the critical Δ VD which result in the Lagrangian bare mass terms temperature and order parameter, and solve for Lagrangian Δ 2 ∝ 2 obtaining corrections of the form mT T [77]. Thus, parameters at the high scale such that the conditions hold. we consider the one-loop finite-temperature potential The peak frequency and peak amplitude of the resulting GW are controlled by the nucleation temperature, TN, V ¼ V0 þ ΔV þ ΔV þ ΔV : ð4Þ D T CW which is the temperature at which a 1=e volume fraction The conditions for a strongly first-order PT generating (given by the Guth-Tye formula [78]) of the Universe is in GW are that the true vacua. By dimensional analysis, this approximately (1) There are at least two minima, occurs once ∂ ∂ pðtÞt4 ≈ 1; ð8Þ V ¼ V ¼ 0 ð Þ ∂ ∂ : 5 S F S T where pðtÞ is the probability per unit time per unit volume The calligraphic subscripts indicate the expression that a critical bubble forms. As a function of temperature, T should be evaluated in the true ( ) and false −S ð ; ð ; ÞÞ 1 4 E T Sb r T (F) vacua. pðTÞ ≈ T e T ; ð9Þ

S ð ; ð ; ÞÞ 1The vacua are degenerate at the critical temperature. We, where E T Sb r T is the Euclidean action evaluated however, always refer to the deepest minimum at zero temper- along a so-called bounce solution. The Euclidean action is ature as the true minimum. defined as

043505-3 BALÁZS, FOWLIE, MAZUMDAR, and WHITE PHYSICAL REVIEW D 95, 043505 (2017) TABLE I. Benchmark points, at the scale Q ¼ 250 GeV, that exhibit GW potentially in reach of aLIGO (O5), vacuum stability, and acceptable low-energy phenomenology. The peak amplitudes were calculated numerically for β=HN from Eq. (14). 2ð 2Þ λ κð Þ κ ð Þ κ λ ð Þ γ ð Þ β Ω Point MS GeV S GeV 1 GeV 2 mS GeV TC GeV TN=TC =HN GW SSM I 4.2 × 1014 0.064 2.1 × 107 −4.9 × 105 0.14 0.53 4.5 × 107 2.8 3.7 × 107 0.44 118 1.3 × 10−9 SSM II 6.9 × 1014 0.073 2.8 × 107 −7.3 × 105 0.15 0.51 5.5 × 107 2.9 4.2 × 107 0.45 110 1.3 × 10−9 SSM III 1.3 × 1015 0.13 7.4 × 107 −1.4 × 106 0.09 0.40 1.3 × 108 2.3 8.2 × 107 0.35 45 6 × 10−9

Z ∞ ð Þ 2 β S ð Þ S ¼ 4π 2 dS r þ ð Þ ð Þ ≈ E TN ð Þ E r dr V S; T ; 10 ; 15 0 dr HN TN and is a functional of the singlet field, SðrÞ. A bounce up to an Oð1Þ factor. We solved the right-hand side in solution is a solution to the classical equation of motion for Eq. (13). We attempt to calculate β by numerical differ- the singlet [79]. That is, we must solve entiation of the action with respect to temperature in Eq. (14); however, to reflect uncertainties in our calcu- ∂2S 2 ∂S ∂VðS; TÞ þ ¼ ; lation, we furthermore present results from varying the time ∂ 2 ∂ ∂ r r r S scale of the PT in the range 1 ≤ β=HN ≤ 200. The lower S0ð0Þ¼0;Sð∞Þ¼0; ð11Þ bound is from causality [83]—the characteristic size of a bubble cannot exceed a horizon—and the upper bound is ð ; Þ for Sb r T , where the effective potential is defined in slightly greater than the approximation in Eq. (15). Eq. (4). In a radiation dominated Universe, temperature and The latent heat is parametrized by time are related by Δρ π2 4 rffiffiffiffiffiffiffiffiffiffi α ≡ ρ ≡ g⋆TN ð Þ 45 M ρ where N 30 : 16 2 ¼ P ð Þ N T t 3 pffiffiffiffiffi ; 12 16π g⋆ The denominator ρN is the energy density of the false where g⋆ ≈ 100 is the number of relativistic degrees of vacuum and g⋆ ¼ 107.75 is the number of relativistic freedom and MP is the Planck mass. Combining Eqs. (8), degrees of freedom at the nucleation temperature TN. (9) and (12) results in the condition that the Euclidean The numerator, Δρ, is the latent heat in the transition action satisfies between the true and false vacuum, S ð ; ð ; ÞÞ E TN Sb r TN TN dV dV ≈ 170 − 4 ln − 2 ln g⋆: ð13Þ Δρ ¼ − − − ð Þ 1 V TN V TN ; 17 TN GeV dT F dT T

We solve for the nucleation temperature TN in Eq. (13) by evaluated at the nucleation temperature, where V is the bisection, finding the bounce solution and the resulting temperature improved scalar potential (i.e., free energy) Euclidean action for every trial temperature. To find a and subscripts indicate true (T ) and false (F) vacua. bounce solution, we approximate the bounce solution by The bubble wall velocity—a factor that influences the perturbing about an approximate kink solution [80]. amplitude of GW—is slowed by friction terms arising from interactions with particles in the plasma. In the high-scale IV. GRAVITATIONAL WAVES PT that we are considering, because there are fewer friction terms than in the electroweak phase transition (EWPT) in The amplitude of GW from a first-order PT depends on 2 the SM, we expect that vw ≈ 1 in general. The efficiency of the wall velocity of a bubble, vw; the latent heat released in converting latent heat into GW—the final factor affecting Δρ the transition between the true and false vacuum, ; the GW—is denoted by ϵ. Because in our scenario γ ≳ 1.75 efficiency of the conversion of latent heat to GW; and the (i.e., we consider a very strongly first-order PT), one finds duration of the transition. The latter is parametrized by that ϵ ≈ 1. We take ϵ ¼ 1 throughout. S S S Combining all the factors, from numerical simulations d 4 dln E=T E β ≡ − ¼ H ð14Þ using the so-called envelope approximation (see, e.g., dt N dlnT T tN TN S ¼ S 2 where 4 E=T is the four-dimensional Euclidean action In supersymmetric models, the wall velocity of bubbles in an for a bounce solution to the equations of motion, tN is the EWPT tends to be heavily suppressed by strongly interacting ¼ − _ scalars [84]. In the SM, the wall velocity in an EWPT is nucleation time and H T=T. The characteristic time significantly higher without these friction terms. Thus, for a scale of the PT is 1=β. We can approximate the time scale high-scale PT in the SSM, with even fewer friction terms, we by [81,82] expect vw ≃ 1.

043505-4 GRAVITATIONAL WAVES AT ALIGO AND VACUUM … PHYSICAL REVIEW D 95, 043505 (2017) Ref. [85] for an analytic calculation), the peak amplitude modifying the beta function for the quartic coupling (at one of the GW strength, defined as the energy density per loop by a fish diagram) or by negative corrections to the logarithmic frequency interval in units of the critical energy Higgs mass. The latter implies that a Higgs mass of about density of the Universe, due to bubble collisions measured 125 GeV,as required by experiments, could be achieved with today, is given by a quartic coupling larger than that in the SM, and could be realized by tree-level mixing which should result in a 31 6 2 α 2 4 3 Ω ≃ 10−9 . HN ϵ2 vw negative correction, as eigenvalues are repelled by mixing GW × β α þ 1 2 0.43 þ vw [62,63]. A quartic coupling sufficiently greater than that 1 100 3 in the SM could ensure that the quartic coupling remains × ; ð18Þ positive until the Planck scale, though it should remain g⋆ perturbative until that scale. where g⋆ ¼ 107.75 in our model. The factors are Oð1Þ for a There are, however, additional stability conditions in the PT at a nucleation temperature 107 GeV ≲ T ≲ 108 GeV. SSM, such as N pffiffiffiffiffiffiffi The peak amplitude is Oð10−9Þ for α ≃ 1 and γ ≃ 2. The λ ≥ 0; λ ≥ 0; and κ2 ≥−2 λ λ; ð22Þ aLIGO experiment, LIGO running phase O5, should be S S Ω ≳ 5 10−10 sensitive to amplitudes greater than about GW × that result from considering large-field behavior in the Oð10Þ–Oð100Þ 4 4 at about Hz [58,86]. H ¼ 0, S ¼ 0 and λH ¼ λSS directions in field space. The peak amplitude observable today occurs at the peak Note that if κ2 is negative, the latter condition is equivalent to λ ≥ 0 frequency SM , that is, the SM vacuum stability condition. In this case, stability cannot be improved by a threshold correction, 1=6 ≃ 16 5 fN TN g⋆ ð Þ though it could be improved by modified RGEs (see the f0 . Hz × 8 19 HN 10 GeV 100 Appendix). Thus, we consider κ2 > 0. To ensure perturbative unitarity, we followed Ref. [87]. Because in our solutions the where fN is the peak frequency at the nucleation time, Higgs and singlet are approximately decoupled, it resulted λ ≲ 4 2 0.62β in a constraint that S . below the GUT scale. fN ¼ : ð20Þ We ensure that the mixing angle between the doublet 1.8 − 0.1v þ v2 w w and singlet is negligible, such that our model agrees with The peak frequency of GW from a PT coincides with experimental measurements indicating that the Higgs is aLIGO’s maximum sensitivity at about 20 Hz if the nucle- SM-like. There is, however, a residual threshold correction 7 8 to the SM quartic. After eliminating the mass squared terms ation temperature is about 10 GeV≲TN ≲10 GeV [49]. by tadpole conditions, the tree-level mass-squared matrix ð Þ V. VACUUM STABILITY in the basis h; s reads 2 After the discovery of the Higgs boson, and subsequent λv κ1 þ κ2v 2 ¼ S ð Þ determinations of its mass, the stability of the SM vacuum M 1 v : 23 κ1v þ κ2vSv ð4λSvS þ κÞvS − 2 κ1v was reexamined [68–72]. At large-field values, the SM vS effective potential is approximately The off-diagonal elements lead to mixing between mass 1 and interaction eigenstates, described by a mixing angle V ðhÞ¼ λðμ ≈ hÞh4; ð21Þ eff 2 κ þ κ 3 θ ≈− 1 2vS v þ O v ð Þ tan 4λ þ κ 3 : 24 and for stability it is sufficient to ensure that, given an initial SvS vS vS value of the quartic coupling at low energy, the renorm- As the mixing is small, we use the same notation for mass alization group (RG) evolution is such that the quartic and interaction eigenstates. The mass eigenvalues are coupling is positive at least until the Planck scale. approximately The result is sensitive to low-energy data—notably the top-quark mass, Higgs mass and strong coupling—in the ðκ þ κ Þ2 2 ≈ λ − 1 2vS 2 ð Þ quartic coupling’s renormalization group equation (RGE). mh v ; 25 vSð4λSvS þ κÞ With present experimental data, however, it is believed that Λ ≃ 1010 1 2 2ðκ þ κ Þ2 the quartic coupling turns negative at about I GeV, 2 ≈ ð4λ þ κÞ − v κ − 1 2vS ð Þ mS vS SvS 1 ; 26 referred to as the SM Higgs instability scale. The SM Higgs 2 vS κ þ 4λSvS potential is believed to be metastable; although we live in a Oð 4 2Þ false vacuum, the probability of tunneling to the true vacuum neglecting terms v =vS . As stressed in Refs. [62,63],in is negligible [70]. the limit v=vS → 0, the singlet only partially decouples. This instability can be remedied in simple extensions of While the mixing vanishes (tan θ → 0), a negative tree- the SM, including the SSM, which could alleviate it by level contribution to the Higgs mass survives:

043505-5 BALÁZS, FOWLIE, MAZUMDAR, and WHITE PHYSICAL REVIEW D 95, 043505 (2017) ðκ þ κ Þ2 2 ¼ λ − 1 2vS 2 ≤ λ 2 ð Þ Higgs mass parameter in the tree-level tadpole equations. mh v v : 27 vSð4λSvS þ κÞ To approximately satisfy limits on Higgs-singlet mixing from hadron colliders (see, e.g., Ref. [89]), we required Thus, the quartic coupling in the SM plus a singlet that a tiny mixing angle between Higgs and singlet scalars, ≈ 125 −6 achieves mh GeV is greater than that in the SM (or tan θ ≤ 10 . We tuned the Higgs mass by bisection in equivalently, there is a threshold correction to the quartic the Higgs quartic such that mh ¼ 125 1 GeV. We found coupling in an effective theory in which the singlet is simultaneous solutions to the low-energy constraints and integrated out from the SM plus singlet), which improves GW requirements by iterating between the weak scale and the stability of the Higgs potential. That is, the critical temperature. 2 In Table I we present three benchmark points with GW ðκ1 þ κ2v Þ Δλ ¼ S ≥ 0 ð Þ amplitudes potentially within the reach of aLIGO (O5), ð4λ þ κÞ : 28 vS SvS acceptable zero-temperature phenomenology and a sub- 2 stantial threshold correction to the tree-level Higgs quartic If κ → 0 and κ1 → 0, Δλ → κ =4λ , reproducing the 2 S for improved vacuum stability. The running of the Higgs expression in Refs. [62,63]. Substantial κ1 in the numerator or cancellations involving κ in the denominator could, however, help generate a sizable threshold correction. There are, however, subtleties: the conditions in Eq. (22) were necessary, but insufficient for stability. For example, in Ref. [62] it was shown that for a Z2 symmetric potential and renormalization scales μ ≲ MS,ifκ2 > 0, the SM vacuum stability condition, λ ≡ λ − Δλ ≥ 0 ð Þ SM ; 29 is required to avoid deeper minima in the S ¼ 0 direction. μ ≲ ≲ Λ We thus require MS I, that is, that the singlet scale is less than the SM instability scale. This ensures that although there is an instability scale at which the SM vacuum stability condition is broken, λ ðμ ¼ Λ ≳ Þ 0 ð Þ SM I MS < ; 30 FIG. 2. Running of the Higgs quartic λ in the SM and for our the vacuum may in fact be stable, as we may violate the SM solutions in the SSM. All lines correspond to mh ≃ 125 GeV. vacuum stability condition at scales μ ≳ MS. We trust that lessons from the Z2 symmetric case are applicable to our general potential in Eq. (1). Thus, in this paper, we describe our model as stable if the couplings satisfy the large-field conditions on vacuum stability in Eq. (22) and the SM μ ≲ ≲ Λ vacuum stability condition in Eq. (29) for MS I. We leave a detailed analysis to a future work.

VI. NUMERICAL RESULTS As well as generating GW potentially within reach of aLIGO and improving vacuum stability, our models must satisfy low-energy experimental constraints on the weak scale (i.e., the Z-boson mass), the Higgs mass and Higgs- singlet mixing, and be free from Landau poles below the GUT scale. We fixed an order parameter, 1.75 ≲ γ ≲ 5, and 7 8 a critical temperature of 10 GeV ≲ TC ≲ 10 GeV. We included low-energy constraints by building two-loop FIG. 3. Peak amplitudes and frequencies of GW for our SSM RGEs in SARAH-4.8.2 [88] by modifying the SSM model benchmark points from our approximate numerical calculation of and constructing a tree-level spectrum generator by finding β=HN (squares), with uncertainty represented by varying between consistent solutions to the tree-level tadpole equations and β=HN ¼ 1 and β=HN ¼ 200 (lines). The shaded regions indicate diagonalizing the weak-scale mass matrix. Our spectrum LIGO sensitivities during various phases of running [58,86]. All generator guaranteed the correct weak scale by tuning the lines intersect the sensitivity of aLIGO (LIGO running phase O5).

043505-6 GRAVITATIONAL WAVES AT ALIGO AND VACUUM … PHYSICAL REVIEW D 95, 043505 (2017) −8 8 quartic for our three benchmarks and in the SM is shown in 10 GeV ≤ jκ1j ≤ 10 GeV Fig. 2, demonstrating that for our benchmarks, the quartic −8 10 ≤ κ2 ≤ 2 coupling remains positive below the Planck scale, unlike in 1012 2 ≤ 2 ≤ 1018 2 the SM. Note that the running of the Higgs quartic coupling GeV MS GeV is sensitive to the precise values of the top Yukawa, yt, and 7 8 10 GeV ≤ TC ≤ 10 GeV the strong coupling, g3. The experimental measurements for 7 2.3 ≤ γ ≤ 3: ð31Þ yt and g3 were boundary conditions at Q ≈ 10 GeV; this introduced an error of up to about 3% in their weak-scale κ λ γ values for our benchmarks. As such the running for SSM III We traded the Lagrangian parameters and S for TC and is pessimistic; its quartic running is probably steeper. For by solving Eqs. (5) and (6), and λ and μ2 by requiring benchmark SSM I, the quartic coupling hits a Landau pole correct Higgs and Z-boson masses. A substantial fraction of above the GUT scale. We illustrate that our benchmark our MC solutions could exhibit GW in reach of aLIGO; points result in peak amplitudes and frequencies of GW however, calculating the amplitude of GW accurately potentially within reach of aLIGO (O5) in Fig. 3. However, requires a thorough lattice simulation. note that here we have varied 1 ≤ β=HN ≤ 200. When selecting our benchmarks, however, we found that We selected our benchmarks from thousands of solutions if γ ≳ 3, the rate of tunneling is sometimes too slow for a found by Monte Carlo (MC) sampling SSM parameters at PT to dominate the Universe, with this being the case more the GW scale, Q ¼ TC, from the intervals often as γ approaches 5. That is, it is impossible to satisfy

7 8 FIG. 4. Scatter plots of solutions in the SSM that exhibit strongly first-order PT at TC ∈ ð10 ; 10 Þ GeV, acceptable weak-scale phenomenology, and no Landau poles below the GUT scale. For the benchmark points shown, in addition, we checked that the PT results in GW signatures are potentially within reach of aLIGO (O5).

043505-7 BALÁZS, FOWLIE, MAZUMDAR, and WHITE PHYSICAL REVIEW D 95, 043505 (2017) condition Eq. (13) for any temperature. This is consistent instability scale. Indeed, the original analysis that proposed with Ref. [90], in which no solutions with γ > 5 were the existence of a heavy singlet leading to a tree-level boost found. Since we desire a completed PT, we discarded in the Higgs quartic coupling promoted the case where the solutions with an order parameter γ ≳ 5. This may, in fact, mass of the singlet was 107–108 GeV [62]. This is precisely be optimistic, as Ref. [90] indicates that completed PTs in the region where the stochastic background is visible at γ ≈ 5 S with are rare and as we require a lower value of E=T aLIGO. It should be stressed, though, that it is also possible since the nucleation temperature is 5 orders of magnitude to boost the stability of the vacuum with a lighter singlet. higher than the EW scale [see Eq. (13)]. On the other hand, With planned LIGO running phases sensitive to GW if the order parameter γ ≲ 2.3, the amplitude of GW may be amplitudes below 10−9, it is interesting to consider moti- too far below aLIGO (O5) sensitivity for all but the most vations for a PT at 107–108 GeV, which, on a logarithmic optimistic estimate of the peak amplitude. There is there- scale, lies about halfway between the EW and the grand fore a “Goldilocks region” for the strength of the PT, unification scales. One exotic possibility is EW baryo- 2.3 ≲ γ ≲ 3, for which GW could be observed at aLIGO. genesis through a multistep PT with the first transition Thus, to roughly select GW amplitudes in reach of aLIGO, at around 107–108 GeV as proposed in Ref. [91]. This we sampled from 2.3 ≲ γ ≲ 3. presents another intriguing possibility about physically We scatter our MC solutions in Fig. 4. We find that motivated PTs occurring at such a high scale. This and moderate Higgs quartics of λ ∼ 0.35 are common, although other scenarios we leave to future work. there are outliers at λ ≳ 0.4. We see in Fig. 4(a) that the dimensionless singlet-Higgs coupling is moderate, κ2 ≲ 0.1. ACKNOWLEDGMENTS We find, unsurprisingly, in Figs. 4(b) and 4(c) that dimensionful parameters are similar to the critical temperature, We thank Bhupal Dev, Eric Thrane and Peter Athron 7 mS ∼ κ1 ∼ TC ∼ 10 GeV. The Higgs-singlet couplings for helpful discussions. This work in part was supported by appear correlated in Fig. 4(d). This is likely due to the the Australian Research Council Centre of Excellence for fact that the Higgs-singlet mixing angle is reduced for Particle Physics at the Terascale. A. M. is supported by κ1 ∼−2κ2vs. The sizes of the Higgs-singlet couplings are Science and Technology Facilities Council Grant No. ST/ related to the threshold correction in Eq. (28), which we J000418/1. require to be moderate. There exist points with a Higgs quartic larger than in the benchmark SSM I that may suffer APPENDIX: SSM β FUNCTIONS from Landau poles in the Higgs quartic below the GUT scale. We generated beta functions from our modified SSM VII. DISCUSSION AND CONCLUSIONS model in SARAH-4.8.2 [88]. The beta functions for λS and κ2 were such that the quartics remained positive. The GW detectors, such as LIGO, are a novel way of probing former is positive at one loop, new physics. In this work, we studied the detectability of primordial GW in the context of the SM augmented with a 2 1L 2 2 16π βλ ¼ κ2 þ 36λ ; ðA1Þ single real scalar field that is a singlet under all SM gauge S S groups. The scale of the scalar singlet (its mass and VEV) though there are negative terms at two loops, and the latter 107–108 was motivated by vacuum stability to be GeV. We is proportional to κ2 at one loop, have shown that, with this scale, the singlet dynamics leads 1 to a strongly first-order PT that generates GW potentially 2 1L 2 2 16π βκ ¼ κ2ð−9g − 45g þ 60λ within reach of aLIGO (LIGO run phase O5). Selected 2 10 1 2 from a wide sample over the parameter space, we presented 2 þ 60yt þ 40κ2 þ 120λSÞ; ðA2Þ three benchmark points with detailed calculations of the peak GW frequency and amplitude, demonstrating that for and at two loops. Thus at two loops it cannot change sign. an optimistic estimate of the peak frequency and amplitude, There is, furthermore, an additional contribution to the beta they lie within aLIGO sensitivity. The most optimistic function of the SM quartic, scenario, of course, arises for β=H ∼ Oð1Þ. N 27 9 9 9 While it is known that eLISA is able to probe PTs at or 16π2β1L ¼ g4 þ g2g2 þ g4 − g2λ near the EW scale, to our knowledge this work is the first to λ 100 1 10 1 2 4 2 5 1 2 2 2 4 2 discuss a physical motivation for a PT to leave a relic − 9g2λ þ 12λ þ 12λyt − 12yt þ κ2; ðA3Þ background potentially detectable by aLIGO. Our result is due to the coincidence of aLIGO sensitivity with the EW which could improve vacuum stability.

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043505-10 Chapter 8

Effective ﬁeld theory, electric dipole moments and electroweak baryogenesis

8.1 Introductory remarks

Effective field theory allows to build physics models from bottom-up by including higher and higher dimensional interactions in the Lagrangian approximating better and better an ultraviolet complete theory. This technique is also used to make the scope of experimental constraints more model independent [123, 124, 125]. They have been used for example in applying dark matter direct and indirect detection constraints [126]. Unfortunately, some of these effective operators were used outside the range of validity of effective field theory [127]. This motivated the use of simplified models which minimally extend the Standard Model particle content [128, 129, 130, 131, 132, 133]. (Simplified models can also be viewed as a lowest order effective field theory that, by adding more effective operators, begins to asymptote the complete model.) There has been little done on effective field theory in the study of electroweak baryogenesis. As such one needs to explore the basic framework before discussing questions of the range of validity and ultraviolet completion. The strength of the experimental constraints on CP violating effective operators due mostly to negative searches for permanent electric dipole moments is impressive – constraining some operators up to the multi TeV level [134, 135, 136, 137]. This motivates an effective field theory approach that attempts to broaden the scope on these limits. However, successful electroweak baryogenesis requires not only new sources of CP violation, but new weak scale physics to catalyse a phase transition. In principle, these two conditions could have a unified explanation – say one extra particle. I consider the more general case where they are unrelated. In this case, the many possibilities that could catalyse a strongly first order electroweak phase transition is a peripheral concern and it is more practical to sweep such details under the rug and instead parametrize the details of the phase transition. When I calculate the baryogenesis produced by some test operators I find that surprisingly, the scale of CP violation can be quite high if the conditions of the phase transition are ideal (generally a very strongly first order phase transition, a thin bubble wall propagating at a slow speed). A second surprising finding is that

177 when one sweeps details of the physics that leads to a strongly ﬁrst order phase transition under a rug, the degeneracy of certain dimension 6 operators is lifted. For such claims to be veriﬁed one will eventually need to look at the UV completion of such operators to see if one can avoid contributions from the heavy physics from becoming Boltzmann suppressed. With these issues in mind I nonetheless present a rigorous step toward linking EDM constraints to baryogenesis calculations in a model independent way.

8.2 Declaration for thesis chapter 8

Declaration by candidate In the case of the paper present contained in chapter 8, the nature and extent of my contribution was as follows:

Publication Nature of contribution Extent of contribution Came up with idea, numerical implementation, analytic demonstration of key concepts in the pa- 8 50% per, wrote large parts of the paper and contributed to discussions throughout. The following coauthors contributed to the work. If the coauthor is a student at Monash, their percentage contribution is given: Author Nature of contribution Extent of contribution Invovled with writeup. Contributed to discussions Csaba Balazs throughout Worked heavily on EDM calculations, write up Jason Yue and contributed to discussions throughout. The undersigned hereby certify that the above declaration correctly reﬂects the nature and extent of the candidate and co-authors’ contributions to this work.

178

8.3 Published material for chapter 8: Effective ﬁeld theory, electric dipole moments and electroweak baryogenesis

Begins overleaf

180 JHEP03(2017)030 Springer March 7, 2017 : February 21, 2017 December 16, 2016 : : , Published 10.1007/JHEP03(2017)030 Accepted Received doi: Published for SISSA by b,c [email protected] , and Jason Yue a . 3 1612.01270 Graham White The Authors. a c Cosmology of Theories beyond the SM, CP violation, Eﬀective ﬁeld theories,

Negative searches for permanent electric dipole moments (EDMs) heavily con- , [email protected] ARC Centre of Excellence forThe Particle University Physics of at Sydney, NSW the 2006, Terascale, School Australia of Physics, E-mail: [email protected] ARC Centre of Excellence forMonash Particle University, Physics at theVictoria Terascale School 3800, of Physics Australia and Astronomy, Department of Physics, NationalTaipei Taiwan 116, Normal Taiwan University, b c a Open Access Article funded by SCOAP Keywords: Thermal Field Theory ArXiv ePrint: Consequently, derivative operators cannot betors eliminated (via in the terms equations ofphase non-derivative of transition. opera- motion) Thus, if we one re-classifyourselves the is to independent agnostic third dimension to generation sixthe the operators, quarks, BAU new restricting gauge (as physics bosons that anon-derivative and leads function operator, to the and of the relate Higgs. the it bubble to Finally, wall the we width EDM calculate constraints. and the cutoff) for a derivative and a (BAU) produced during aaspects strongly of first order the electroweak highusual phase kink scale solution transition. for physics the The drivingof bubble thermal wall the the profile. Higgs phase field We find transition yieldvacuum that CPV are expectation operators contributions involving paramaterised value to derivatives (vev), by the BAU while the containing non-derivative derivatives operators of the lack Higgs such contributions. Abstract: strain models of baryogenesisolating utilising (CPV) various operators. higherconnection dimensional between Using charge these effective and EDM field parity constraints vi- theory, and we the create baryon a asymmetry model of independent the universe Csaba Balazs, Effective field theory, electric dipoleelectroweak baryogenesis moments and JHEP03(2017)030 3 5 5 ] closely resembles 2 , 1 ] via electroweak baryogenesis. 8 to simultaneously generate enough 1 9 – 1 – 8 12 7 16 vertices vertices 1 t DD 5 ]. ]. The SM also falls short in the amount of charge (C) and 9 5 O O 10 ] the electroweak phase transition (EWPT) does not provide a sufficient 1 4 17 ]. These two facts alone are enough to motivate the existence of new physics 7 , 6 ] and references therein for a pedagogical review) within the SM because with a Higgs 3 Physics models entailing new particles or interactions can introduce charge-parity vio- For an approach where EWBG is achieved without adding particle content to the SM nor invoking 6.1 Space-time dependent cutoff 4.2 Contributions from 4.3 Contributions from 4.4 Calculating the baryon asymmetry Higgs 4.1 Constructing new CPV sources with higher dimensional operators 1 experimental constraints. Under this motivation,Model we by consider effective an extension dimension of sixically the operators. utilises Standard To two achieve such electroweak baryogenesis, higher one dimensional typ- operators higher dimensional operators, see [ responsible for baryon asymmetry. lating (CPV) phases to assist explainingThe the observed use BAU of [ effective fieldadhering theories to (EFTs) allows a one specific to test model a or large class framework. of models This without greatly facilitates the connection with that of the Standard Model (SM).(cf. This [ rules out the mechanism ofmass electroweak baryogenesis of 125 GeV [ departure from equilibrium [ charge-parity (CP) violation to(BAU) generate [ the observed baryon asymmetry of the universe 1 Introduction The Higgs boson discovered at the Large Hadron Collider (LHC) [ 6 Numerical results and discussion 7 Conclusions 5 EDM constraints 3 Operators classification 4 Electroweak baryogenesis with higher dimensional operators Contents 1 Introduction 2 Removing redundancies of operators with derivative coupling to the JHEP03(2017)030 – – 6 31 26 violating sources CP , with their respective we demonstrate that the 4 ] (particularly applied to 2 . The ], whilst evading ever tighter 3 ]. This bridge can be used to 18 , 16 25 – 17 – 13 22 violating operator basis (cf. e.g. [ ]. . We present resulting BAU in section 21 CP 5 800 GeV on the scale of new physics that . . 7 – 2 – violating interactions with a space-time varying bubble violating dimension six operators that are candidates for CP CP ]) place a bound of Λ 20 , 19 . Finally we conclude with section ] and references therein). Considerable amount of literature have been devoted ] for an approach of connecting EFTs and UV complete models in the context of Higgs flavour 12 30 – 6.1 While building the above bridge, we point out that the degeneracy between certain 10 2 violation and a strongly first order phase transition (SFOPT) at the electroweak scale The structure of this paper is outlined as follows. In section The most promising operators for BAU generation are those that contain at least one In this work, we argue that it possible to build a relatively direct bridge between the See [ 2 ]) that is capable of generating the baryon asymmetry. ]). before briefly discussing thesection possibility of space-time varying masses of heavy particles in violation. redundancy between various operators isthen lost catalogue during the the full electroweak setproducing phase of the transition. BAU We via theare electroweak mechanism calculated in section usingEDM the constraints closed derived subsequently time in section path formalism in section due to the equationsresult is of an increased motion. sensitivity tobaryon the asymmetries It width are of involves calculated the show a electroweak that bubbleconstrain current derivative wall. EDM the coupling measurements The can available respective to meaningfully parameter space. the Higgs and the wall as well as a stronglyenhancement of coupled such SM interaction field, during i.e. theefficient a electroweak top phase mode quark transition or for becomes a thechosen gauge baryogensis. most within boson. the The Consequently, new resonant the catalogue two aforementioned of qualitatively bridge. the different One operators of operators presented the in are operators this chosen paper is normally to considered demonstrate redundant different dependencies onnecessary the then assumed to extend profile the33 of higher the dimensional space-time varying vacuum. It is Higgs field to accommodate then classify the UV completion(s)29 corresponding to the EFThigher (for dimensional some examples operators see isderivative [ ones lifted. via the Usually,broken classical derivative in equations operators BAU of calculations are since motion. traded the to However, CPV such sources non- corresponding degeneracy to may these be operators have could boost the strength of the phase transition [ EDM constraints on aproduced higher by such dimensional an operator operatorparametrised and by by assuming the the a maximal strongly bubble first baryon wall order width, asymmetry phase velocity, transition etc.) (which [ is (cf. [ to generate sufficient CPV viaconstraints dimension from six searches operators for [ permanenties electric dipole of moments a (EDMs).top-Higgs SFOPT Similarly, sector stud- catalysed [ by dimension six operators [ CP JHEP03(2017)030 ]: 34 CPV , (2.2) (2.3) (2.4) (2.1) =6 D O CPV , . =6 and violating phase ! D n as O ··· , CP =4+ n + DD ) n O V,D m =4+ ( ∆ ) =4+ O V,D m ( ∆ ) V,D m O . ( ∆ O n m n,m 1 Λ Λ c m n n,m N Λ H. c ∈ µ O X D the operators + n,m n,m X µ † † D + SM operator may contain derivatives, and there R ∂ both t L ∂H ∂H L ∂ – 3 – CPV , Q + − CPV , =6 = † H = D =6 SM µ operators, each possibly with a different cutoff scales. D L O H ∂H D DD n ∂ O contributing to both the BAU as well as EDMs, and µ µ ensure that the EWPT is strongly first order (see for O

D D 3 2 CPV =4+ n CPV µ R R c t t Λ D ) L L V,D =4+ m + Q Q ( ∆ 2 2 ) O 1 1 V,D SM Λ Λ m ( ∆ L O = → = L is an operator DD ]). In general, the O 2 1 CPV 11 , Λ , =6 10 D O departure from thermal equilibrium. baryon number violation, charge and charge-parity violation, and The second and third conditions can be satisfied within the SMEFT framework by Usually, the classical equations of motion are used to eliminate derivative operators as Higgs We note that one can in principle have many such operators. The approach we adopt here is to inspect 3 (3) (1) (2) each one separately as a sole source of CPV (in addition to the CKM phase). use of the field equations Explicitly, one can use the classical equations of motion to rewrite cause the Sakharov conditions are metexist. only if For a concrete example consider an example operator of the class The derivatives on the Higgs in the above operator can be typically eliminated by making in the Cabibbo-Kobayashi-Maskawa (CKM) matrix,baryon but asymmetry. it The is third too conditionto feeble also catalyse fails to a in provide the strongly enough SM first as order the electroweak Higgs phase mass transition. isadding too higher heavy dimensional operators to the SM Lagrangian can be a single or several redundant. However, we will show insection) this that section one qualitatively should (numerically exercise in cautionfor a when subsequent baryon eliminating asymmetry derivative operators calculations with within EOMs the EFT framework. The reason for this is be- In electroweak baryogenesis, thecondition as SU(2) the sphalerons anomaloushigh are baryon temperature. responsible number In for violating the meeting processes SM, the the become first second unsuppressed condition at is met through a where the set ofexample operators [ A successful explanation of the BAU necessarily fulfils the three Sakharov conditions [ 2 Removing redundancies of operators with derivative coupling to the JHEP03(2017)030 , ) ). n 4 DD − 2.4 have (2.7) (2.5) (2.6) O =4+ (Λ O ) . V,D m ( . ∆ ∂V/∂H ¨ CP 4 O O ¨ 1 S 2 Λ 1 Λ violating sources O and O ) does not yield the + CP ) as free parameters, takes the form: DD + ¨ 2.6 T CP O ¨ j ( !# rather than the ). However, by ignoring S ) v x n T R ( ( d v j , v

† d and =4+ . Y R ))] i SM e ) in eq. ( w ) x e Q and V,D ( ∂H L m ∂V v ( ) is introduced to describe bubble ∆ 2.7 parametrises the distance perpen- w LY µ w x ij z O

L ( z ∂ L , has a cubic sensitivity to the bub- can be safely neglected in eq. ( ) v µ i bounce solution ij x ∂ n ( DD i R − v )( t ), one can derive the t O x L R ∂ ( t 2.4 Q v ) determined by the operators L − t T ∂ Q ( ! 1 + tanh ). Using eq. ( − violating sources typically depend on the deriva- – 4 – ) v − j x ( − ) 2.7 v ))

). Speciﬁcally, the CPV source resulting from the CP T Q x 2 ( and † T ( u H and instead leaving v ( 4 v SM † Y v w µ n ∂H R (3) symmetry, the ≈ ∂ L ∂V u µ λH ) O ∂ z

=4+ and ( ( − t ij t v ∂ ) 2 w ) for the explicit form of the sources. ∂ V,D ) i µ ) m L x ( ∆ x 4.8 ( R ( O v t R , and the cutoff Λ. , one could in principle consider every single possibility for the set v [ " L v 2 2 Ht Q 1 1 = Λ Λ CPV L , i Q ) − =6 ∼ ∼ x ( = D O H DD ¨ h and their Wilson coefficients to calculate CP O ¨ DD ∂V/∂H S ¨ n measures the width of the bubble wall and O CP O ¨ S w ). The strength of the CPV source due to the CPV operators then become very =4+ L T For each In BAU calculations, however, the ) ( The reader is referred to ( V,D v 4 m ( ∆ O the precise structure of we can then draw as direct a bridge as possible between BAU calculations and EDM limit. operator with derivative couplingble to wall the width Higgs, whereasof the non-derivative operator hassensitive a to the quartic exact sensitivity structuresensitivity to of that the the occurs value set in of EDM operators calculations. Here, dicular to the wall.there are no With free both parameterssame left result, in not eq. even ( approximately.different This dependencies is on due to the fact that figuration in the finite temperaturethe effective true action vacuum. which Assuming interpolates an the false vacuum to One can immediately seeexplicit that when the a two specific expressionsformation form do for during not the the agree Higgs electroweak in profile phase general. of transition. This is The made profile is a stationary field con- reads tive of the space-timesupposedly varying degenerate vacuum operators shown during in the eq.to ( phase lowest transition. order in For the example, inverse for cutoff: the After substituting in the SM Lagrangian and evaluating the derivatives, the operator In EDM calculations, operatorsThe use suppressed of by the Λ equationssince of a motion hierarchy to of relatevalue scale (vev), different is operators established is justified between in the this constant context Higgs vacuum expectation JHEP03(2017)030 – F as , 37 2 (3.2) (3.1) D ] (see D , µ ), there 32 ψ x D ( v HF, 2 HF. 2 , ψ 2 and not violating operator , ψ 2 being much sharper 2 HD D n CP 2 µ HD D 2 =4+ ) D, ψ , and one with no derivative V,D 2 m ( ∆ H DD D, ψ O 2 2 O -even analogue is given in [ H 2 , ψ 2 CP H 2 , ψ 2 H 2 ,F 3 – 5 – combination to cancel the SU(2) charge. There- H 2 2 ], while the ,F ψ 2 31 violating sources to D , ψ 2 2 D CP 2 the operators satisfying the above constraints. We find 34 . 3 1 ,FH 3 should not appear. Under these constraints the possible classes violation. We use the closed time path (CTP) formalism [ H ,FH 2 4 H sensitivity. This behaviour is attributed to the derivative structure 2 D CP DF,F 4 2 violating source terms for two operators which facilitate resonantly 2 − , ψ FD 4 ,H D CP -odd operators from [ 2 2 violating interactions with the bubble wall. cannot appear without a D . , ψ 4 CP 1 4 ). t H ].) We list in table ,H T O CP ( 2 v 35 , ψ applied to fermions, derivative operators, (dual) field strength tensors and Higgs , D 2 4 ,H 33 D H 6 H 2 and F H ] to calculate w When the Higgs field developsare a operators space-time varying which vacuum interfere expectationexotic with value, new the sources standard of 41 top quark vev insertionenhanced diagram to give couplings 4 Electroweak baryogenesis with4.1 higher dimensional operators Constructing new CPV sources with higher dimensional operators also [ in total that fulfil allare of usually our considered constraints, redundant. includingthe 19 We latter with considered can higher operators be derivative with couplings formedthe that by field taking the strength sum tensor.tensive of study the We — first will operator one and with select an a two operator second qualitatively involving derivative different coupling, operators for an ex- of operators are We take the In order for the contributions tomust the involve BAU at be resonantly least enhanced,with one a single space-time varying Higgsfore, terms operator such and as one other field. Terms the Higgs and gaugeand bosons need be considered.operators With respectively, the one should symbolic obtain meanings 12 of operator classes, 8 of which involve the Higgs 3 Operators classification In this section, we classify andwhich count the can operators involving no derivative couplings longer withwhose be Higgs contribution to considered the BAU redundant. areThis suppressed In means by small that electroweak Yukawa only couplings barogenesis, the can be the left neglected. handed SM third fields generation quark doublet, the right-handed top, This is due tothan the the sensitivity expected of Λ of the operators and theL changing the profile of the vev during the EWPT as controlled by The result is that redundancy between derivative and non-derivative operators is lifted. JHEP03(2017)030 ) ) ˜ and H ˜ ) ) ) ) H ν ˜ H H ) ) H ν ˜ ν ν H H H H H H µ D H H ν ν µ D ) D D ν ν i,ν i,ν µ D µ ( ( D D D ↔ ↔ ↔ ↔ i i ← D D R µ D D D D τ τ )( )( † † t † † ) ) † † R µ D † † t H H − R H H ( H ( H ↔ µ t D ( ( 2 H H 2 µ µ µ R ) µ µ ↔ t D µν µν i i µν µν µν µν D D D D ˜ L L σ σ D D B B ( H 2 H ( ( ˜ HD W W ( ( † µ µ L L Q µ Q µ 2 H µ µ µν µν i i 2 Q Q ψ µν µν D D ↔ D D ˜ iH FH D D B B † † ˜ D W W † H H := µ H )( ) ˜ ˜ D W W 2 H ) H ˜ ˜ BH BH BH BH H HW HW H H DD µ µ 2 2 2 2 σDD σDD † 2 2 2 2 D DtDH H ↔ ↔ O 4 (1) (2) (1) (2) D D D D (1) (2) (1) (2) D D D D D O O † D H O i † O O O O ( † O O O O H H ( H H ( L i L a µν µν ] involving at least one Higgs and one Q R 2 2 2 µν i t Q 36 τ ) ) D D D µ µ ) ˜ ˜ HG b a a 4 4 4 µ γ γ HW ˜ HB γ (2 (2 (1 H H H L R t L R R Q t O O O R t Q a i t τ T µν D ) H 2 – 6 – HF µν σ µν H µ 2 H H σ µ L σ H i ψ ) ↔ ν µ D H L 2 L Q ↔ i D ↔ µ i † D ψ Q D Q H † i 2 µ H D † H H D D H † † )( )( H † H † tG tB tW 4 µ 4 . O O H H O D D 2 ν D H Ht (3) (1) 2 Hq Hq 2 ) D D i O O O ( D µ τ H µν µν µ iµν iµν µν µν D aµν aµν B B ← ˜ ( D ˜ ˜ B B W W G G i i µν µν R − µν µν i i ˜ µν µν a µν µν a W W 4 4 4 4 µ ˜ B B Ht G G W W ) D D D D ) D L b a 2 2 2 2 i H H 2 i i H H τ Q (3 (3 (2) (1) H H H H H H H H ( † † τ τ H † † 3 † † † † † O O O O 2 H H H H H H H F H H 2 iH H List of dimension six operators based on [ † ψ := H ˜ ˜ ˜ H G B W ˜ i WB µ HG H HB H HW H 1 ↔ HWB H D t O O O O O O i O O † O field or fermion field.H Here we follow the definition that Table 1. other field thatdue is to either the ainvolving equations Standard (i) of two Model motion derivatives are gauge acting no boson a long or Higgs applicable, a field one top and has (ii) quark. to one be derivative Since cautious acting with redundancies on the either classes gauge JHEP03(2017)030 (4.6) (4.3) (4.4) (4.5) (4.1) (4.2) ]. As a 43 , . . # 42 ) , ) ) , x, y ) ∗ R ( ) E x, y m R R ( ( t E L F R S + ( t h F S k R ]. The effects of mixing taking the lowest order ) h E − E )( y 46 x ) + 3 ∗ ( R – + ) , k L v E ) + = 0 y L E µ 44 ( L k ( E y ∂ ( v E F µ ρ ( 2 ∗ i ), the self energy is h ) . The first can be treated in the ∂ . c F Λ x 2 t ∗ i h ( c Γ Λ v DD L/R for both operators is just the usual , t 2 ) ) + O 1 k 2 , x y → 2 , µ ) ) ( ( − ) + 0 k ) ) just produces the following correction − x x v y F k 2 ( x ( ( ∗ t ( and ∗ n R ( + v − v y F λ F E

∗ t n R − H g 11 L 2 y i (Λ ) t E c E − y Λ L O O 3

− – 7 – E ) x ) ( − x · ) = 1 ) = ( x " x x ik 1 + ( v ( ( v − 2 i Im > < e F F µ c Λ g g ) is then expanded near 4 ∂ R 7→ ) y k µ vertices ω dk ( t 4 π ∂ 2 L v ) + d 1 Γ 2 i k ) t (2 ω x c x Λ ( ( O v ∞ Z t v 0 y Z ) + . 2 x 1 ) = ) ( − y v x t ( ) = y − v x + 1) 2 T | term has a derivative coupling. For simplicity we will ignore interactions x, y ( 2 t ( λ y π βx | ) = S e 2 tot DD conserving term to lowest order in Λ C ) is the density of states and Σ O x, y N ( , k ) = ( 0 conserving relaxation term up to CP = x k tot ( ( t Σ ρ F Γ n CP We will consider two exotic operators, The We also use the vev-insertion approximation (VIA) where the BAU production is 4.2 Contributions from The to the Standard Model with usual way by defining the self energy as resonant relaxation termvarying arising vacuum. from The interactions term term. between In the this top case and the lowest the order space-time is the zeroth order and we find where whereas the with gauge bosons. Making the replacements perform a resummation tosimplification, all we orders ignore in the the hole vevswith modes following multiparticle in the the states quark techniques in plasma in thelater, [ [ thermal more precise bath numerical as study. well Under as these resummation assumptions, will the also quark be propagator left reads to a dominated by physics in frontsmall of compared the to advancing both bubble theproduce nucleation wall. the temperature THis resonant and CPV is the sources.The valid effective mass degreesin when splitting of of the freedom the particles are vev then mass that is those eigenbasis belonging space-time of varying vevs the are symmetric treated (unbroken) perturbatively phase. under such approximation. Their interactions One could with the JHEP03(2017)030 # ) (4.9) (4.7) (4.8) R (4.10) ) ]. This ∗ R w 2 −E E ) 2 ( ( ) R ,L . F F . L E n n ) x Λ] + − ( − − E , v L ) ) L ∗ t R µ E L L ( E ∂ Γ E E ( 3 , ( ( ) f R f t x n n ( Γ . The result is v , µ x R ) t 2 2 x = k k , , m z − + − L . t z Λ] t ∗ ( R R , Γ m E E [ L conserving relaxation term comes t L L ∗ t I 2 Γ E ) E y i , Λ x ) c ( R ]. Solving the contour integrals we find CP " x t 00 + ( 0 Γ v , is given by 47 , Im v

] for notational convenience. x · ) Im ) R [ R t before going to zero. We therefore linearise x x I , or two Higgs doublet models which all have = ω ( dk ( i 1 7→ 2 00 w t L v c z , m v k 2 ω L L O – 8 – 3 t Λ ) −

. ∞ y m ) 0 ( [ x x Z v I ( ) ) ) 1 + v x x x ) vertices ( ( ( x 0 0 v ( v v when 7→ 3 3 000 − DD ) ) t x v ) x x Γ y O ( ( ( v v v ∗ t 2 3 y ) i ∗ ∗ t t Λ 2 2 c x y y . The correction to the SM i i ( Λ Λ x c c v = Im ∗ t z C Im Im 2 violating source we expand to first order in y 2 i N Λ C C c π 2 2 w N N v violating sources arising from violating source term, involving the third derivative of the Higgs coming from . The time derivative of the vev profile is then a spatial derivative times the π π CP | w w v v x = Im operator requires some care since it involves a derivative coupling to the Higgs. CP CP − violating source controlled by the first derivative of the vev. We note that there is t = 2 = 2 DD ¨ DD w CP O ¨ v O S | is sufficiently gentle near the phase boundary [ The CP DD ¨ z CP O ¨ = S The derivative coupling causes the operatorthan the to be much morethe sensitive to the bubble width the transport equations by setting thiswill correction be to a its somewhat average a valuedensities between conservative at [0 assumption all as far this from correction the will bubble not wall. relax thethe number next to leading order expansion around Note that the correctionThe to usual the practice relaxation in term solvingtions involves these which the transport means second equations assuming is the derivativeThere to relaxation of are terms linearise the are the some vev. a differential ambiguity constant equa- interm value this in varies procedure the quite broken in rapidly phase. that with the correction to the above relaxation the vacuum near from the zeroth order term in the expansion where we have implicitly defined the function 4.3 Contributions from The Once again, we replace the Higgs field with a space-time varying vacuum and expanding wall velocity. In line with theto VIA, we assume that the variation of bubble wall with respect Only the zerothus component also contributes ignore under thez the bubble assumption wall of curvature and spatial work isotropy. in the Let rest frame of the bubble wall For the new JHEP03(2017)030 (4.11) (4.12) (4.13) (4.14) , , X ¨ X ¨ CP O ¨ CP O ¨ S S , + − bosons vanishes as in 5 ) 5 U y ]. Neglecting the bubble ± U ( SS 48 X SS W ¨ CP O ¨ 2Γ S + Γ , y i − α ) − H H T k H H k y, e . B + − 0 d (100). While such an analysis has not k , − H z Q L Q O 0 b Q n k Q 9( . We then use the diffusion approximation Z n Q | k + − x + + , z + − i T − L R T T t t H α k t T – 9 – T k n n n − T w k e v − ) | = = = , Y i Γ α , are derived in reference [ Q Y = Q T Q ( H k 2 Y z − H X H k + Γ these are A 1 = } − Q x Q 1 k 5 Q Q Q U 6 k Q =1 k − i X − T − T DD, t , we consider it worth solving the transport analytically using the k T ) = T T T k X z k thus reducing the problem to a set of coupled differential equations ∈ { ( M n X 2 X Γ M Y ∇ − ]. In the broken phase the solution is 49 = Γ = Γ = = 6 we can then relate the chemical potentials to the number densities. For with µ µ µ / J 2 X T Q H µ T µ µ O i ∇ · ∂ ∂ ∂ µ i k = i in an example modelbeen (the done MSSM) in by the a SM+ techniques factor in of [ to the rest frame ofto the bubble write wall in a single space-timesphaleron variable. and three We bodybeen do Yukawa shown not rates that use are this the fast assumption usual compared can simplification to cause that a an the diffusion underestimate strong time of as the it baryon has asymmetry Systematically calculating the sources for eachspecies self energy leads term to involving the an above particle network of coupledoperator transport equations. Using the usual relationship where and the three body Yukawawall rates, curvature Γ we can reduce the problem to a one dimensional one by changing variables When calculating the BAU,very we fast make and the thatthe in usual symmetric the phase. assumption VIA that Wesuppressed the ignore gauge by interactions chemical small interactions with coupling potential are constants. particlethe for species Specifically, following the the linear whose number combinations interactions densities are we consider are when the bubble wall becomes verythe thin. source has Nonetheless, an we expect increasedfunctional the sensitivity methods, qualitative to result as the that it wall comes widthitself, from to rather the be than derivative true coupling our even to if approximation the one scheme. space-time uses varying Wigner 4.4 vev Calculating the baryon asymmetry a danger that the VIA approximation becomes cruder for derivative couplings particularly JHEP03(2017)030 ] ) = 51 , z (5.1) ( (4.20) (4.17) (4.18) (4.19) (4.15) (4.16) ]. We ]. The 50 L n 55 23 ) is given , i α 22 ( is then given X B A Y , ) z ( and L i n is identified as the coef- , y i ws ψ Γ , x i x, F 2 n , satisfies the equation [ , β ) d ) i B x z . ρ , α ( − L , µν R z n i Q γ x ψF e D − , showing that the electron EDM gives = Θ( . 1 κ µν , 3 y − 2 σ B + 4 T Q e 5 ws ρ ∗ X,s Γ 2 w D g R 0 v A 2 ψγ 2 ) 4 −∞ 15 f z π Z 45 p 6 2 =1 id − – 10 – i X = S ± − = Q Θ( ws w R s = Γ v D ) = − z F + ]). ) ( = n κ z 2 ( X ± EDM 57 , 0 B − κ L ρ ]. The baryon asymmetry of the universe, w = 56 , v corresponding to a charged fermion 54 . The baryon number density, B – − Y T ψ 20 , ) d 52 [ z +4 ( 19 . The procedure for how to derive , 00 B T Q } ρ 5 W 16 -violation in the Higgs sector are necessary to realise electroweak baryo- Q α = 5 D CP L 3 120 Q, T, H Q ≈ + is the number of fermion families. The relaxation parameter is given by ∈ { L ws 2 F X n Q ]. From these solutions one can then define the left handed number density The dipole moment + 49 L 1 As argued before, one is led to focus on the top-Higgs sector in electroweak baryogenesis ficient of the five dimensional operator in the effective Lagrangian the most stringent bound sinceis it is obtained weakly from sensitive to measurementstherefore hadronic using uncertainties. focus polar This on bound moleculehensive thorium contributions and monoxide to systematic (ThO) electron treatment [ operators to EDMs (cf. a (eEDM) e.g. future [ and study delay that will a include more other compre- dimension six electric dipole moments (EDMs) ofnection electron, between EDMs neutron, and molecules electroweak andsensitivity baryogenesis atoms. of have these been A low suggested direct energy inthreshold con- observables [ owes corrections to as contributions from highexperimental operator scale constraints mixing physics are and summarised is in run table down and integrated out. The present 5 EDM constraints New sources of genesis. These sources, however, are severely constrained via their contributions to the where and the entropy density is where Γ by where Q where in [ and in the symmetric phase we have JHEP03(2017)030 , T its ) h ξ (5.2) (5.3) + , v ), mercury ] n violation in (0 2 61 1 ] , ] ] √ , CP 59 60 58 55 [ [ = ]. 2 t 2 h 69 H – m m cm cm [ cm ), neutron ( cm [ 66 e e e e e , 2 = 173 GeV is the physical f 27 t 30 26 29 S ξ t via two-loop Barr-Zee type − − − − y 5 m violation interactions of this e P e 10 d sin 10 10 10 y violating top-Higgs coupling of = 0. If there is 5 × × × × ξ CP + iγ 6 2 0 h.c., 7 . . . . CP 1 + 6 3 8 + 2 t 2 h and ξ < < < R < m m t ]. We add that other degrees of freedom γ γ | /v | L | | cos t t e n Hg 1 Tl 65 h d m f d d , d | | | | 2 iξ P t e tth f 16 y √ 2 2 t t S e – 11 – y y y √ √ := t operator around its vev. With 2 h − − v SM t R R operator leads to H t t Tl y Hg e n e 1 L L SM e t t t = y 205 t t m 199 O t m m y . Such contribution is given by SM t are defined in [ − y 1 2 , 3 = 6= 0) it induces contributions to 1 ) parametrises the magnitude of the top-Higgs coupling, and f π α ξ L ⊃ − t Tl) atoms at 90% C.L. y (4 ]) for a more pedagogical discussion of the derivation. Two-loop Barr-Zee diagrams contributing to the electron EDM. 205 3 16 64 = e e are assumed to be in their mass eigenstate and Type Molecule/Atom Bounds d (1) coupling. At the non-derivative level, ) by expansion of the Neutron , h ] as shown figure O Diamagnetic Current limits on electric dipole moments of the electron ( Figure 1. R Paramagnetic 5.2 62 Electron (ThO) ] (based on [ , t 63 L t phase. In the SM one has Firstly we discuss how the See [ Hg) and thallium ( 5 199 give sizable contribution to thebeen EDM studied via in the detail same in the the Barr-Zee context of type two diagram.the Higgs form This doublet ( have models [ where the loop functions (not present in our analysis), e.g. charged Higgs boson, may interact with the top quark to where top mass. In addition, CP the top-Higgs coupling ( diagram [ due to the sort are encoded in Table 2. ( JHEP03(2017)030 2 2, λv CP . In π/ (5.4) (5.5) R , and h.c.. t = = 1 m + 2 iξ µ R − CP t e φ L ), since four- t and 2 7→ h 2.5 2 h √ R t h.c.. } m for the dimension- 1 TeV for both + = 2 -violation comes only C R & t 2 λv L µ ∈ 3 2 t CP operators to the electron 2 . α ] and references therein for h − m √ t 2 ξ DD 70 iξ } µ e {z O ]. Following the previous steps, t − 2 2 y t v 20 Λ ξ is chosen to absorb the effects of 2 1 t t and DD iξ Λ c α c e {z 1 t t y + + 3 O α α in terms of derivative free operators as in | 4. The electron EDM bound on the latter | h.c. − DD − violating sources resulting from the operators π/ + R h.c. O t R R – 12 – = t L + t L CP come from from the first term of ( ˜ t Ht R } v } CP v respectively. We set the nucleation temperature to L . The value of ˜ φ Ht Q DD 4 L 2 2 CP 2 ) with the momentum dependent top-Higgs vertex. Dif- O . Making the assumption that = 173 GeV. Currently, we take v Λ Q iφ R t H 1 e λv i t † m 5.3 ) c 1 2 ˜ m iξ Ht H e = through the same two-loop diagram, shown in figure {z H t L + 1 2 phase can be identified with − † t e 1 m Q Λ c 2 α d m and figure µ iξ λH operator, the top-Higgs interaction contains a derivative. In prin- + e CP {z 3 DD,t t 3000 GeV for 2 − c α 1 m 2 √ 2 | DD & DD Λ µ O c shows the contributions of the − ( 2 2 + = DD Λ in figure α c L ⊃ − ). 1 = 6 operators and that the scale of the operator is set by the cutoff, one t and to reproduce phase of the higher dimensional operator is observed. Particularly, a cutoff of + 2 O 2.5 ) = 0 and 1 d 1 α α √ h | . Figure CP 3 − DD,t and 3600 GeV is required to remain consistent with the current constraints In both of these cases, one assumes a generic coefficient The dominant constraints on In case of the O H = † & DD L ⊃ − H discussions on experimental constraints of such coupling). 6 Numerical results andWe discussion plot the BAU producedO by the new Λ but is relaxed tooperator Λ is weaker, withphases. the cutoff One scale should roughlypure scalar required keep electron to in Yukawa coupling be mind with Λ its that SM when value (cf. interpreting [ these results, one assumes a the but we note that this relation can beEDM modified as by a pure functionon Higgs of the effective the operators cutoff such scale as Λ. For the former, operator a strong dependence four top Yukawa coupling from the sets Im ( one obtains fering from the discussion ofcorresponds the to baryon the production one duringvalid well the then after EWPT, to the the use EWPT Higgs classical and vevequation here EOMs is ( to hence recast not space-time dependent. It is fermion operators to do not lead to sensitive observables [ this leads to These operators are brought intosuch their case, mass the basis physical by a field redefinition ciple, this contributes to one can derive an analogue of ( JHEP03(2017)030 �� =π/� =π/� �� = 2 since this is  ϕ ϕ  . The first value γ /T �� ) T of the Wilson coefficient ( (d) (b) v Λ[��] Λ[��] CP iφ �� := e γ = 1 �� DD,t 2 such that new coupling constants c π/

� �

� � � � � � � � � � � � � � �� ] [ | / | ] [ | / |

�� - - CP φ – 13 – �� during the electroweak phase transition for a critical denotes the phase γ CP �� =π/� =π/� φ ��  ϕ ϕ  ). We then set the value of the vev deep within the broken violating phase 5 �� ]. Generically, a smaller value of the wall velocity produces a larger CP 71 (c) (a) Λ[��] Λ[��] 100 [ �� operators. Here ≥ (cf. section c i DD T O Two loop contribution to the electron electric dipole moment via a top quark due to = �� 1 and 1 t DD,t

� �

c � � � � � � � � � � � � � �

�� ] [ | / | |[ | / | ] O

= 100 GeV and the

�� - - n is the minimal value oforder unity phases — transition since necessary this tobroken is sufficiently phase suppress the thereby sphaleron approximate preserving interactions condition the deep for baryonan in number. a approximate the strongly maximum The value higher first for value is temperature T are phase to obtain two different values of the order parameter Figure 2. the appearing in front of the operator. The horizontal line corresponds to the experimental limit. JHEP03(2017)030 obs B 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 /η B η 1. As . 7000 7000 1 t 6000 6000 05 and 0 =0.1 =0.1 .  γ=1 γ=2 W W violating sources 5 TeV, due to its v v . 5000 5000 CP is very sensitive to the in the plane of the bubble is about a TeV whereas 1 t DD 4000 4000 O DD Λ[GeV] O O 3000 3000 2000 2000 is relatively gentle. w L violating operator 1000 1000 5 0 0 5 20 15 10 25 20 15 10 25 CP – 14 – 7000 7000 which is expected given that the 6000 6000 =0.05 =0.05 γ=2 γ=1 operator is significantly higher, about 3 γ W W v v 1 t O 5000 5000 to some power. One should note that the Standard Model with Λ). The dependence on γ , 4000 4000 w , the minimum cutoff for the operator Λ[GeV] L 5 3000 3000 The baryon asymmetry due to the 2000 2000 As expected, the baryon asymmetry due to the operator

1000 1000 5 0 5 0

15 10 25 20 10 20 15 25

× T × T w w L L explained in section the minimum cutoff foreffect the on the top quark Yukawa. bubble wall width. In both cases, a thin bubble wall is favoured with a large proportion of are all proportional to a light Higgs has aparticles in larger the wall plasma velocity. which The mightapproach. also wall be velocity Therefore, heavy can we enough be to canby suppressed justify keeping once an by the again effective additional field wall parametrise theory velocity our as ignorance a of free such parameter particles and just setting it to values 0 Figure 3. wall width vs. cutoff ( BAU as does a larger value of JHEP03(2017)030 obs B 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 /η B η 7000 7000 is large. This 6000 6000 DD γ =0.1 =0.1 γ=1 γ=2 W W  v v in the plane of the 5000 5000 DD so the suppression due O 2 Λ 4000 4000 / Λ[GeV] 4 ) T ( 3000 3000 v is quite steep. ). However, explaining the baryon w operator with extremely large values L T 1 ( t 2000 2000 v O operator. This also means that the BAU . It would be very interesting to see how violating operator DD DD 1000 1000 CP 5 0 0 5 O 20 15 10 25 20 15 10 25 O – 15 – 7000 7000 6000 6000 =0.05 =0.05 γ=2 γ=1 ). This means that this operator may be completely ruled | W W violating source scales as i v v Λ). The dependence on c , | w / 5000 5000 for the operator L CP w operator is not viable with about a 6-fold increase in the minimal L 1 t 4000 4000 Λ[GeV] O is more sensitive to the value of 3000 3000 1 t O The baryon asymmetry due to the 2000 2000 ]. 43 , Remarkably the BAU can be produced by the

1000 1000 5 0 5 0 42

15 10 25 20 10 20 15 25

× T × T w w L L is due to theto fact the that cutoff the is notfor as operator severe as itasymmetry is with for the the value of Λ (or equivalently Λ for very small values of strongly this effect persistsin when [ one goes beyond the VIA by usingof techniques the described cutoff if the wall width and velocity are small but the order parameter Figure 4. bubble wall width vs. cutoff ( the parameter space already ruled out. However, the baryon asymmetry diverges quickly JHEP03(2017)030 (6.3) (6.2) (6.1) ) is large enough as T ( 0 . ) x . ( . So for an order parameter 2 c v ) to parametrise our ignorance of x /T ( ) w x c ) + κ L T . T ( ( v 2 0 ) ansatz Λ x ( 2 := ) ∆Λ 1 + tanh x ( 2 – 16 – ∂ κ ∂x has to be high enough to justify the new physics 2 ) ∆ 0 x 1 ( 1 + 4 0 ) + ∆Λ Λ κ T ( )) we can make an 2 0 x 0. Using intuition about the generic behaviour of space-time Λ ) = ( x β ( → → κ ) 2 1 T Λ ( violating source will be of the order 0 CP 75 one might need to produce as much as 10 times of the observed baryon . ]. Including the effects of washout, with detailed calculations of the sphaleron 0 72 ≈ c ) is a space-time dependent function absorbing the effects of heavy physics which /T x ) ( c κ T Finally, we briefly consider the case where we definitely do not expect effective field In this section, we argue that a space-time dependent cutoff is not necessarily fatal for Not all baryon number produced during the electroweak phase transition is preserved ( v theory to work when Λ varying functions during the electroweakvariation of phase the transition ratio (such of asthe vevs CPV new phases, physics vevs and corrections to the with the fact that temperatureeffective is field also theory defined within to ait remain particular represents valid reference to Λ frame). be Forderivative heavy the of enough. Λ, Since one CPV couldbaryon source asymmetry. ask terms The if generically answer it will is is depend typically in no on principle in the possible the that case such where a Λ term can boost the vacuum expectation value of the Higgs. an effective field theory, although weexample, do not we claim do that not our discuss treatmentvarying any is cutoff subtleties comprehensive. is that For defined may within arise a from particular the frame fact of that reference the (although space-time comfort ourselves Here the EFT is ignorantsymmetry of. that An have example a for soft such mass situation but is acquire an some EFT contribution for to sparticles their in masses super- from the Within the approach ofparticles effective by the field inverse theory,symmetry of we breaking, their approximate mass the the squared. massthe propagators of If vev of the of a heavy the heavy particle other particle acquires field inherits some such a of that space-time its dependence mass via via dependence on the strengthof of the order parameter asymmetry [ energy, we leave to an interesting future project. 6.1 Space-time dependent cutoff in sensitivity by aboutcoupling an become order moderately of more magnitude accurate.asymmetry or has There measurements some is of moderate of the course dependence top the on caveat quark the that Yukawa nucleation the temperature. until baryon the phase transition is finished. The fraction that is preserved has a double exponential out as a sole explanation to the BAU in the foreseeable future if EDM searches improve JHEP03(2017)030 the (6.4) (6.5) 6 to zero ) is small. 0 T ( 0 which is acquired . 1 . t ) )) O i R t x Γ , Λ( , L t R t Γ , Γ , R t L t Γ , m , L R t t is not unique and the EFT framework m 1 ( t , m I L ] O t 2 ∗ t m y ( 1 t I y ] – 17 – ∗ i c = 6 operator that we test is t y )Im[ D x ( ˙ β )Im[ ) x x ( ( κ 2 ) ˙ v x ( = 2 v = 1 ¨ HDM t ¨ CP 2 ¨ CP O ¨ S S ) is the cutoff evaluated at a single space-time point for simplicity. This is, of i x violating source we get is Examining the connection between dimension six effective operators and the BAU, we One should not, however, take the comparisons between the above two CPV sources This is physically unrealistic as there is always a thermal mass mass but done for illustrative purposes. 6 traded to non-derivative ones, can be safelyscale. neglected since When they calculating are suppressed the byfiled BAU, the yield however, cutoff a operators CPV containing contribution a toIf derivative baryogenesis one of that trades involves the these the operators Higgs derivative to of non-derivative the ones Higgs then vev. one completely changes the nature work, we studied such a connection using the frameworkfound of that an the effective conventional degeneracy fieldof is theory. the broken Higgs between field operators and containing CPV theirna¨ıve counterparts derivatives analysis, related higher by the order equation contributions of which motion. arises According when to the derivative operators are the parameter spaceelectroweak of baryogenesis baryogenesis models models. willThe be number Consequently, either of in confirmed baryogenesis models, or thebounds however, ruled is near (including out rendering future EDM the by applicationindependent, various EDM limits) of direct searches. on experimental connection each between model EDM constraints impractical. and BAU This calculations. necessitates In a this model be coincidental, it would befield theory interesting works for in future calculating work the to BAU ascertain for how a well7 variety the of effective models where Conclusions Λ The growing sensitivity of electric dipole moment searches is increasingly constraining is necessarily somewhat ignorantthe of EFT the framework UV produces completion. thethe correct top What dependence quark, is on including the remarkable masses resonancetop here and effects, is Yukawa thermal coupling the that widths correct as of dependencephysics well all on as in the the a vev scenario variation profiles, where of the the EFT the framework space-time is dependence expected to of be the crude. heavy While this may UV complete theory in a case where we hadtoo no literally. right If to we expect replacedacquire this. the coefficients cutoff with with the complicatedexpected mass dependence since of the on the heavy second parameters physics Higgs beyond that doublet produces the we would SM. This is course, the most ambiguous partHiggs of doublet this model discussion. where We one can gets compare the above to the two Remarkably, the effective field theory framework reproduces much of the structure of the by integrating out a heavyCP Higgs in a two Higgs doublet model. If we set Λ Here Λ( Suppose we have the case where our JHEP03(2017)030 , 114 pp ]. The SPIRE IN ][ Phys. Rev. Lett. , (2012) 1 due to power-counting IOP Concise Physics n , GeV with the CMS (1998) 283 =4+ B 716 125 ) V,D ]). Also, more work needs to m ( ∆ 73 O B 532 arXiv:1207.7235 [ Phys. Lett. with SM operators is problematic as a , CPV ]. , Nucl. Phys. (2012) 30 , =6 – 18 – D O SPIRE ]. IN B 716 Combined measurement of the Higgs boson mass in ][ SPIRE ), which permits any use, distribution and reproduction in IN ]. ]. TeV with the ATLAS and CMS experiments 8 Phys. Lett. Observation of a new particle in the search for the Standard Model , SPIRE SPIRE IN IN and Observation of a new boson at a mass of collaborations, ][ ][ CC-BY 4.0 = 7 arXiv:1503.07589 This article is distributed under the terms of the Creative Commons s [ CMS A pedagogical introduction to electroweak baryogenesis ]. Nonetheless, we have made a step toward a more general test of the √ 74 and collaboration, collaboration, hep-lat/9805013 arXiv:1207.7214 ATLAS collisions at (2015) 191803 universality class of the[ electroweak theory CMS experiment at the LHC Morgan & Claypool, (2016) [ ATLAS Higgs boson with the[ ATLAS detector at the LHC We stress that the baryon asymmetry calculated from the normally neglected dimen- After re-classifying dimension six effective operators, we selected two simple dimension K. Rummukainen, M. Tsypin, K. Kajantie, M. Laine and M.E. Shaposhnikov, G.A. White, [4] [5] [2] [3] [1] References Research Council. JY will likework to during particularly his thank transitional Archil period. Kobakhidze for supportingOpen this Access. Attribution License ( any medium, provided the original author(s) and source are credited. Acknowledgments We like to thank W. Denkens,Ren J. de Chen Vries, Nodoka for Yamanaka, Michael Schmidt useful and Chuan- discussions. This work was partially supported by the Australian dynamics of the EWPT.scenarios The such approach as webe multistep suggest phase done does transitions analysing notin (cf. these apply references e.g. operators to [ [ using moreelectroweak complicated the baryogenesis full paradigm. Wigner functional approach presented connecting the effect ofoperators the can yield. EDM constraints Finally, wetime to discussed dependent the the possibility and amount of showed of thethe that effective baryon correct cutoff the asymmetry physics being effective these even space- field when theory we expect approach it captures to thesion be six bulk operators a of involving crude derivative approximation. coupling to the Higgs is more sensitive to the bubble arguments in the EOMsrapidly when varying Higgs relating wall profile destabilises the hierarchy between the vev and cutoffsix scale. operators (one containing abaryon derivative asymmetry. and We the also other subjected not) these and operators calculated to the EDM constrains, respective thereby directly of the CPV contribution to baryogenesis. The removal of JHEP03(2017)030 , ] , C , ]. ]. ]. ] (2007) 049 (2016) 011 SPIRE SPIRE SPIRE Phys. Rev. ]. 03 02 IN IN , IN ][ hep-ph/0309291 ][ ]. ][ [ ]. (2013) 21 SPIRE JHEP JHEP ]. 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Chapter 9

Conclusions and future work

Electroweak baryogenesis is a field where there is a large amount of work to be done. There is only one numerical toolkit available publicly which only covers a few of the calculations need [106]. There has been limited work on multistep phase transitions [11]. There is also much work to be done on examining the baryogenesis compatible parameter space of popular BSM models. Very little work has been done in the context of effective field theory to directly use experimental limits to confine higher dimensional CP violating operators. This in particular is a sub-field in its infancy as the validity range and scope of EFT requires study. The scale of new physics in preliminary studies was found to be surprisingly high. Nonetheless if these theoretical questions can be answered it presents an exciting opportunity to directly link EDM constraints to baryogenesis calculations. Finally, gravitational waves provide an exciting new direction to test physics at energy not otherwise accessible on Earth through colliders. In this thesis I have focused energy on each of these directions with an eye to the bigger picture of particle cosmology. Having said that, there is much work to be done in each of these directions. On the effective field theory front, one needs to thoroughly examine the validity limits of effective field theory as has been done for dark matter. Furthermore the fate of operator degeneracies needs to be understood on a deeper level. Finally there needs to be a systematic study of all CP violating higher dimensional operators in the Standard Model and in the case of extended scalar sectors. In the case of testing baryogenesis within existing particle physics models, in some soon to be published work, I examine the charged transport dynamics during the electroweak phase transition in the NMSSM. In this work I find that the baryon asymmetry can be very different if the singlet acquires a vacuum expectation value before or during the electroweak phase transition. Furthermore, others [97] have found that the parameter ∆β, which the baryon asymmetry is proportional to, can be an order of magnitude higher in the NMSSM compared to its MSSM value. There is ample motivation to perform a scan where the transport dynamics and the properties of the phase transition are examined simultaneously. Such a work naturally ties into the need for better numerical tools. I am in the process with several collaberators of developing a numerical toolkit that has enough flexibility to perform mutiple baryogenesis calculations: from phase transitions, to EDMs, to deriving and solving coupled transport equations.

205 In the case of extending the vanilla electroweak baryogenesis scenario to multistep phase transitions, again, very little has been done before this thesis. I have some soon to be published work where I look at the case where SU(3)C is spontaneously broken and then restored through the addition of leptoquark scalars. I demonstrate that this is still a testable framework and secondly that the baryon asymmetry can be produced during the colour break phase transition. A nice extension to the work done here would be to explore the experimental signatures more deeply. For instance, two step phase transitions can leave relic charge asymmetries. In the case I test the charge asymmetry is many orders of magnitude lower than current observational bounds. Perhaps there is a scenario where this asymmetry is large enough to be testable in the near future. Secondly anomalous violation of lepton flavour violation was recently explained through the leptoquark representation I used [138]. Understanding the impact on flavour physics of the colour breaking paradigm would be a rich extension to our project and there is already evidence that such a leptoquark might exist. Finally gravitational wave detectors provide a useful avenue to test the whole multistep paradigm. This is particularly important as the key phase transition that catalyzes the production of a baryon asymmetry might involve scalar particles far too heavy to be seen by either this or the next generations of particle colliders but might be detectable through gravitational wave detectors. Ligo is sensitive to cosmic phase transitions at an unusual energy scale but it is expected to be sensitive to relic gravitational backgrounds from cosmic phase transitions many decades earlier. I have written the first work outlining a physical motivation for a cosmic phase transition at such a scale but if there are other physical motivations that would be worth exploring.

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