Baryogenesis in a First-Order Electroweak Phase Transition

Universiteit van Amsterdam

Faculty of Science

Bachelor Thesis

Baryogenesis in a First-Order Electroweak

Phase Transition

Supervisor:

dr. Marieke Postma

Second Assessor:

dr. Ben W. Freivogel

July 17, 2018

Abstract

In this thesis I study how an abundance of matter over antimatter could have arisen in the electroweak phase transition in the early universe. I start by discussing the conditions under which baryogenesis can take place and how these conditions can be met within the standard model. The Higgs mechanism is what breaks the electroweak symmetry and causes the electroweak phase transition; it will thus play a crucial role in electroweak baryogenesis. I will go in depth on the Higgs mechanism and the dynamics of the electroweak phase transition. Electroweak baryogenesis can only take place in a first-order electroweak phase transition. In a first-order electroweak phase transition, the Higgs field tunnels from a false vacuum to the true vacuum,leading to the formation of Higgs bubbles. These Higgs bubbles are the cause of the baryon to antibaryon asymmetry. A first-order electroweak phase transition is not possible within the standard model. In this thesis, I therefore extend the standard model by coupling a singlet scalar particle S to the Higgs. To see if this opens the door for electroweak baryogenesis, I perform a parameter scan over the parameters that come with coupling the singlet particle to the Higgs. This results in a region of parameter space where electroweak baryogenesis is possible.

Populaire Samenvatting

We leven in een wereld vol met materie; alles wat we kunnen zien en aanraken is opgebouwd uit deeltjes. Atomen zijn de bouwstukken van onze wereld. De term atoom komt van het griekse woord voor ondeelbaar. Echter bestaan atomen ook nog uit deeltjes; protonen, neutronen en elektronen. Het elektron is een fundamenteel deeltje, wat inhoudt dat het niet opgebouwd is uit andere deeltjes. Protonen en neutronen bestaan uit deeltjes die we quarks noemen. De up-quark heeft een elektrische lading van +3/2 en de down-quark een lading van −1/3. Zo bestaat een proton uit twee up-quarks en ´e´en down-quark en het neutron uit ´e´en up-quark en twee down-quarks.

In 1928 ontdekte Paul Dirac dat er ook iets moest zijn als antimaterie. Dit bleek uit zijn vergelijkingen voor de relativistische quantummechanica. Het idee gaat als volgt: elk fundamentele deeltje heeft een antideeltje. Voor het elektron is er bijvoorbeeld het anti-elektron. Deeltjes en hun antideeltjes hebben exact dezelfde eigenschappen maar ze hebben tegenovergestelde lading. Het elektron heeft bijvoorbeeld een negatieve lading en het anti-elektron heeft een positieve lading. Overige eigenschappen, zoals massa, zijn hetzelfde. De natuurwetten zien geen verschil tussen deeltjes en antideeltjes. Als atomen uit antineu-tronen, antiprotonen en anti-electronen zouden bestaan zou de wereld er hetzelfde uitzien. Het bestaan van antimaterie is in 1932 bevestigd, toen is er een deeltje gedetecteerd met de massa van het elektron maar met positieve lading. Dit is het antideeltje van de elektron, ook wel het positron genoemd. Positronen worden zelfs in het dagelijks leven gebruikt, zoals in de PET scan die je in elk ziekenhuis tegenkomt.

De wereld om ons heen bestaat vrijwel volledig uit materie en niet uit antimaterie. Deze waarneming komt ons logisch voor maar leidt echter tot een probleem. Immers, als de natuurwetten voor materie en antimaterie hetzelfde zijn, hoe kan het dan dat er meer materie is dan antimaterie? Een antwoord op deze vraag is te vinden in theori¨en die het ontstaan van materie - baryogenese - beschrijven. De naam baryogenese komt van de term baryon. Een baryon is een deeltje wat uit drie quarks bestaat. Protonen en neutronen zijn voorbeelden van baryonen. Een combinatie van drie antiquarks wordt een antibaryon genoemd.

Er zijn meerdere theorie¨en die baryogenese proberen te verklaren. Deze theori¨en zijn het eens dat het overschot aan materie is ontstaan in het vroege universum. E´´en van deze theorie¨en beweert dat het overschot aan materie ontstond tijdens de zogenaamde elektrozwakke faseovergang; elektrozwakke baryogenese. Om elektrozwakke baryogenese te begrijpen moeten we eerst kennis maken met het Higgs mechanisme, de elektrozwakke theorie en de basis van quantumveldentheorie. Quantumveldentheorie vertelt ons dat elk deeltje een eigen quantumveld heeft. Het elektron heeft het elektronveld, een up-quark heeft heeft up-quarkveld, de foton (een foton is een lichtdeeltje) heeft het elektromagnetische veld, enzovoort. Het deeltje zelf is een excitatie in zijn quantumveld, dat kun je zien als een golf in dat quantumveld.

Elk quantumveld, behalve het Higgsveld, heeft een vev gelijk aan 0. Hier staat vev voor ’vacuum expectation value’. In het Nederlands kan dit het best vertaald worden naar ’constante achtergrondwaarde’. Het Higgsveld heeft dus overal een bepaalde achtergrond-waarde. Dit betekent echter niet dat het universum gevuld is met allemaal Higgs deeltjes. Probeer het te vergelijken met de zee. De zee is het Higgsveld wat overal een waarde heeft, net als dat de zee overal een bepaalde diepte heeft. Een golf in de zee kan dan het Higgs deeltje voorstellen. Het feit dat het Higgsveld een achtergrondwaarde heeft zorgt ervoor dat deeltjes die interacties met het Higgsveld aangaan een massa krijgen. Ook is deze achtergrondwaarde van belang bij de elektrozwakke faseovergang.

Het Higgsveld had niet altijd een constante achtergrondwaarde. Toen het universum nog een fractie van een seconde bestond, bij extreem hoge temperaturen, had het Higgsveld net als de andere quantumvelden een vev van 0. Bij deze hoge temperaturen worden de elektromagnetische kracht en de zwakke kernkracht beschreven door dezelfde theorie: de elektrozwakke theorie. In het dagelijks leven hebben we veel te maken met elektromag-netisme, deze theorie vertelt ons alles over hoe elektriciteit en magnetisme werken. De zwakke interactie komt niet veel voor in ons dagelijks leven. Dit is de kernkracht die ver-antwoordelijk is voor radioactief beta-verval. Pas wanneer de temperatuur daalt krijgt het Higgsveld een vev en splitsen elektromagnetisme en de zwakke interactie. Dit wordt de elektrozwakke faseovergang genoemd.

Het is mogelijk dat tijdens deze faseovergang een overschot aan materie is ontstaan. Echter is daar wel een voorwaarde aan. De faseovergang moet namelijk van eerste orde zijn. Een voorbeeld van een eerste-orde faseovergang is het koken van water. In plaats van dat water geleidelijk van vloeibaar naar gas gaat, is het koken van water juist een wild proces waar bubbels van gas in het water ontstaan. Hetzelfde geldt voor de elektrozwakke faseovergang. Als deze eerste orde is ontstaan er bubbels waarin het Higgsveld een vev heeft. Wanneer de bubbels eenmaal ontstaan zijn, groeien ze met snelheden in de buurt van de lichtsnelheid. Deeltjes en antideeltjes botsen niet op dezelfde manier met de wanden van deze bubbels. Hierdoor ontstaat binnen de bubbels een overschot aan materie.

Het kan ook zijn dat de elektrozwakke faseovergang tweede orde is. Denk bijvoorbeeld aan het smelten van ijs.1 Bij een tweede orde faseovergang krijgt het Higgsveld geleidelijk aan overal een vev. Echter ontstaan er dan geen Higgsbubbels en is het dus niet mogelijk om een overschot aan materie te krijgen. Het doel van deze scriptie is om te bepalen of de elektrozwakke faseovergang eerste orde of tweede orde is, en dus ook of het ontstaan van materie mogelijk is in de elektrozwakke faseovergang.

1_{Het smelten van ijs is eigenlijk een eerste orde faseovergang, maar geeft een idee van hoe een tweede}

Contents

1 Introduction 4

2 Basics of Baryogenesis 6

2.1 Requirements for Baryogenesis . . . 6

2.2 Standard Model CP-violation . . . 8

2.3 Current Theories on Baryogenesis . . . 9

3 Higgs Mechanism and Electroweak Theory 12 3.1 Lagrangian Formalism in Field Theory and Gauge Invariance . . . 12

3.2 Spontaneous U (1) Symmetry Breaking . . . 14

3.3 Higgs Mechanism: the general idea . . . 16

3.4 Electroweak Theory and Standard Model Higgs . . . 18

4 Electroweak Phase Transition and Baryogenesis 20 4.1 First- and Second-Order Phase Transitions . . . 20

4.2 Electroweak Baryogenesis . . . 21

4.3 Extension to Standard Model: extra scalar field S . . . 23

4.4 Corrections and Thermal Potential . . . 25

5 Tunneling in Field Theory 28 5.1 Bounce Formalism . . . 29

5.2 Barrier Penetration in Field Theory . . . 31

5.3 Tunneling at Non-Zero Temperatures . . . 32

6 Bubble Nucleation in Electroweak Phase Transition 33 6.1 Results of Parameter Scan . . . 34

1

Introduction

In this thesis I will be studying electroweak baryogenesis. Theories of baryogenesis de-scribe how an abundance of matter over antimatter could have arisen. Before we discuss baryogenesis, let us shortly review the discovery of antimatter and what antimatter actu-ally is. In 1928 Paul Dirac derived an equation describing spin-half particles - electrons for example - in a way that satisfied quantum mechanics as well as special relativity[1]. This equation always has two solutions2. At the time, only the solution describing the electron was recognized, while the other solution was neglected. Dirac first saw this second solution as a mathematical artifact. However, this second solution describes a particle with the exact same mass and spin as the electron, but with opposite charge: an anti-electron. This equation applies to all spin-half particles, not only the electron. Hence, the two solutions to Dirac’s equation describe a particle and its antiparticle. A particle and its antiparticle have the same mass, spin and lifetime. When a particle and its corresponding antiparti-cle collide, they annihilate. In 1932 the existence of antipartiantiparti-cles was confirmed when a particle with the exact same mass and spin as the electron but with opposite charge was detected[2].

In the same way that particles can combine to form larger structures - protons, atoms and molecules for example - antiparticles can form antiprotons, antiatoms and antimolecules. Since the laws of physics are the same for antiparticles, a world consisting of antimatter would look no different than from our own. A combination of three quarks is called a baryon. Accordingly, a combination of three antiquarks is called an antibaryon. The uni-verse is made out of baryons. After all, both protons and neutrons are baryons since they consist out of three quarks. The observation that the universe is made out of baryons forms a problem: if antimatter and matter have the exact same properties except opposite charge, how can it be that we live in a universe with only baryons and not antibaryons? The emergence of an abundance of baryons, or matter in general, is referred to as baryogenesis. Multiple theories attempt to explain baryogenesis, all of which state that the baryon-antibaryon asymmetry arose shortly after the big bang. In this thesis I will focus on one of the theories, namely that of electroweak baryogenesis. This theory proposes that the abundance of matter over antimatter arose during the electroweak phase transition in the early universe. For electroweak baryogenesis, this phase transition is required to be first-order. The theory is as follows. In a first-order electroweak phase transition the Higgs field tunnels from the false vacuum to the true vacuum3_{. In the true vacuum the Higgs has a}

vacuum expectation value (vev). At high temperatures, which is when the Higgs field has no vev, the universe is in the electroweak symmetric phase. After the phase transition the Higgs mechanism has broken the electroweak symmetry. We refer to this as the broken phase. Instead of smoothly gaining a vev, such as in a second-order phase transition, bubbles of the new true vacuum form and start to expand. This is where baryogenesis can happen. Due to CP violation, particles and antiparticles collide with the bubble wall in a different way. In front of the wall of the bubble, an excess of baryons is created due to sphaleron processes. These sphaleron processes almost only happen outside the bubble.

2_{Dirac’s equation actually has four solutions, these describe the different spin states of the electron.}

Two solutions are for the two spin states of the electron and two solutions for the spin states of the anti-electron.

3_{A potential can have multiple minimums. The global minimum of this potential is the groundstate.}

This is called the true vacuum. A local minimum in the potential which is not the groundstate is referred to as the false vacuum.

Figure 1: A visualization of the expanding Higgs bubbles. Inside the expanding bubbles the Higgs has a nonzero vacuum expectation value. Particles in the plasma collide with the expanding bubble wall. This is where CP violation takes place. Sphaleron processes outside the bubble convert the CP asymmetry into a baryon asymmetry.

When the bubble wall moves outward the baryons enter the bubble and are there to remain. In order to maintain this baryon asymmetry the phase transition must be strong. If the phase transition is not strong enough sphaleron processes inside the bubble would wash out the baryon asymmetry.

Given that baryogenesis can happen only in a first-order phase transition, we must now determine whether the electroweak phase transition in our universe was first-order or second-order. A first-order electroweak phase transition is not possible within the standard model due to the value of the Higgs mass[3]. We must look beyond the standard model to find a first order electroweak phase transition. There are multiple ways we can extend the standard model. One way it can be done is by introducing a new particle S. This particle is a singlet and scalar particle, hence its abbreviation S. By coupling this particle to the Higgs, the dynamics of the electroweak phase transition change. Because of this change in dynamics, a first-order electroweak phase transition could be possible after all, keeping the possibility of electroweak baryogenesis.

In this thesis I will first elaborate on the fundamentals of baryogenesis in chapter 2. These include the the fundamental requirements for baryogenesis and sources of CP viola-tion and baryon number violaviola-tion in the standard model. Here we discuss how sphaleron processes violate baryon number and we touch upon the main theories of baryogenesis. I then introduce the Higgs mechanism and electroweak theory in chapter 3, as they are needed to understand electroweak baryogenesis. In this chapter I also elaborate on gauge invariance and the Lagrangian formalism, which are needed to explain the Higgs mecha-nism. I will then discuss the dynamics of the electroweak phase transition, and explain how a strong first-order phase transition could lead to baryogenesis in chapter 4. This will also be where the singlet scalar particle S is introduced. In chapter 5 I proceed to develop the formalism needed to study tunneling in field theory, which is essential when looking at a first-order phase transition in field theory. By introducing the new particle S we also introduce some new free parameters, such as its mass and how strong it is coupled to the Higgs boson. All these parameters influence the phase transition. In chapter 6 I will determine for which region of parameter space a strong first-order electroweak phase transition - and with that baryogenesis - is possible.

2

Basics of Baryogenesis

Now, why is a theory of baryogenesis needed? The need for a theory on baryogenesis comes from the measured baryon-antibaryon asymmetry. In daily life it is obvious that there is an abundance of matter over antimatter. However, one could claim that there may be large regions of the universe in which there is more antimatter than matter, such that there is no net matter-antimatter asymmetry. In this case, there would be no baryon-antibaryon asymmetry and no need for a theory of baryogenesis. However, this cannot be the case. After all, where a region of antimatter meets a region of matter, particles start to annihilate. We should then see constant bursts of gamma rays coming from these border regions. Yet, this has never been observed and thus this hypothesis can be disregarded.

There is a way to measure the baryon-antibaryon asymmetry of the universe. It can be expressed in terms of the photon density of the universe

η = nB− nB¯ nγ

, (2.1)

where nB and n_B¯ are the baryon and antibaryon number density respectively. There

are multiple ways this quantity can be measured. The value of η has influence on the abundances of heavier elements created during nucleosynthesis. The baryon-antibaryon asymmetry also has influence on the relative sizes of the Doppler peaks in the cosmic microwave background. Both methods agree on a value of η ≈ 6 · 10−10 [4][5].

In this section we develop the ingredients that are needed to have a viable theory of baryogenesis. The three basic requirements for a viable theory of baryogenesis will be dis-cussed in the following subsection. Then, in section 2.2 we look at how these requirements can be met in the standard model. In the last subsection of this chapter we will discuss some current theories on baryogenesis.

2.1

Requirements for Baryogenesis

In order to create and maintain a baryon asymmetry there are three conditions that must be met. They are commonly referred to as Sakharov’s requirements. First of all, baryon number must be violated. Secondly we must have C and CP violation. Finally, there must be a loss of thermal equilibrium[6].

As stated above, a source of baryon number violation is required. There are processes in the standard model that allow baryon number to be violated; it was shown that baryon number is violated in a nonperturbative process called a sphaleron process[7]. Nonpertur-bative implies that it cannot be represented by an infinite series of Feynman diagrams. Even though it is a nonperturbative process, we can visualize it as is done in figure 2. In this process both lepton number and baryon number change with 3 units. The quantity B − L is conserved while the quantity B + L is violated. Here B is baryon number and L is lepton number. Sphaleron processes play an important role in all major theories on baryogenesis, as it is the main source of B violation at high temperatures. It is important to note that these processes are highly suppressed when electroweak symmetry is broken.

As previously stated, a source of C and CP violation is required for there to be more baryons than antibaryons. To see why this is the case we must first look at what the C, P and CP operations do to particles. The C operator flips the charge of a given particle, meaning that a particle becomes its antiparticle. The P operator mirrors the universe; the x direction becomes the negative x direction, the y direction becomes the negative y

Figure 2: A schematic overview of a sphaleron process. The shaded circle at the vertex implies that this process is nonperturbative. In the above diagram three leptons, one for each flavor, are converted into nine antiquarks. This happens in such a way that charge is conserved. Note that B − L is conserved but B + L is violated.

direction, and so forth. When we let the P operator act on a certain particle it flips the helicity of the particle. The CP operator is the product of the C and P operators. It flips both the charge and the helicity of a particle. In the case of a left-handed quark the operations would look as follows

C : qL → ¯qL , P : qL→ qR and CP : qL→ ¯qR. (2.2)

It is necessary that we have both C and CP violation. To illustrate this we can look at the process where a particle X decays into two quarks with the same helicity. The conservation of CP implies

Γ(X → qLqL) = Γ( ¯X → ¯qRq¯R) and Γ(X → qRqR) = Γ( ¯X → ¯qLq¯L), (2.3)

which, when taking taking the sum, implies that we must have

Γ(X → qLqL) + Γ(X → qRqR) = Γ( ¯X → ¯qRq¯R) + Γ( ¯X → ¯qLq¯L)

Γ(X → qq) = Γ( ¯X → ¯q ¯q). (2.4)

Baryon number is thus conserved if CP is conserved, even though C violation states

Γ(X → qLqL) 6= Γ( ¯X → ¯qLq¯L). (2.5)

Processes where C and CP are violated can create a baryon asymmetry. For the baryon asymmetry to be maintained, the system must be out of thermal equilibrium - one of Sakharov’s requirements. If not, the process that inverses baryon production will happen at the same rate as baryon production itself. For example, consider the process where particle X decays into a particle Y and a baryon B. In thermal equilibrium we would have

Γ(X → Y + B) = Γ(Y + B → X) (2.6)

2.2

Standard Model CP-violation

Now that we have reviewed the three requirements for baryogenesis, we can consider op-tions for C and CP violation within the standard model. In addition to B violation from sphaleron processes, C and CP violation are also present in the standard model. C viola-tion in the standard model is not easy to find. Only left-handed particles and right-handed antiparticle interact with the weak force. Right-handed particles and left-handed antipar-ticles don’t participate in the weak interaction. Looking at equation 2.2 we can see that the weak interaction violates C and P. The CP violation in the standard model originates from the complex phases in the CKM matrix. The elements of the CKM matrix describe the coupling of the different quarks to the W bosons of the weak force. In the Wolfenstein parameterization the CKM matrix is given by[8]

VCKM=   Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb  =   1 −1₂λ2 _{λ Aλ}3_{(ρ − iη)} −λ 1 − 1₂λ2 _Aλ2

Aλ3(1 − ρ − iη) −Aλ2 ₁



 (2.7)

The CP transformation of an element in the CKM matrix is given by its complex conjugate

CP : Vij → Vij∗. (2.8)

The elements of the CKM matrix appear in calculating the matrix element for a Feyn-man diagram involving the weak interaction. For example, when a top-quark decays into a bottom-quark and a W+ _{boson, we get a factor V}∗

tb at the vertex. For processes in

particle physics, the amplitude is calculated by adding up the matrix elements of all the Feynman diagrams and then taking the absolute value. Thus, in order to have measurable CP violation multiple diagrams must contribute4_.

This form of the Wolfenstein parameterization is accurate up to O(λ3). The diagonal elements are all of order O(1). This implies that the top-quark, mostly decays into the bottom-quark, the charm-quark into the strange-quark and the up-quark into the down-quark. The off-diagonal elements are all on the order of different powers of λ. For example, processes where the top-quark decays into a down-quark are thus less likely. There are four free parameters in the CKM matrix. The Wolfenstein parameterization is not the only way complex phases in the CKM matrix can be expressed, here only the elements Vub and Vtd

have complex phases. In different parameterizations, other elements carry complex phases, this is done by redefining the fields. Redefining the fields does not have an effect on observables. There is only one free parameter that determines the amount of CP violation. The amount of CP violation in any parameterization can be expressed in terms of the Jarlskog invariant

J = |Im(VijVklVkj∗V ∗

il)|. (2.9)

The free parameters of the CKM matrix can be determined by measuring the amount of CP violation in the decay of neutral mesons. There are multiple ways CP violation can manifest itself[8]. There is direct CP violation, in which a neutral meson P0 _{and its CP}

conjugated partner do not have the same decay rates

4_{If we had just one diagram with a CP dependent complex phase, the absolute value would be the}

Figure 3: One of the two lowest-order Feynman diagrams describing mixing in the Bs system.

The mixing happens via the weak interaction. There are two internal top-quark lines, any of the other up-type quarks could have been chosen as well. At the vertices we see the CP dependent elements of the CKM matrix.

Γ P0 → f ) 6= Γ ¯P0 → ¯f
, (2.10)

where f is the final product of the decay process. Then we have CP violation in mixing. Mixing is when neutral mesons transform into their own antiparticle via the weak inter-action. This process is referred to as neutral meson oscillation. In figure 3 we have an example of a mixing diagram in the Bs system. For CP violation in mixing we have

Γ P0 → ¯P0
6= Γ ¯P0→ P0
_{. (2.11)}

The third way in which CP violation can manifest is due to interference between decays with and without mixing. Before a neutral meson decays, it is allowed to oscillate to its aniparticle. Because multiple diagrams contribute, their CP dependent and CP indepen-dent complex phases interfere, giving us a measurable source of CP violation. In this type of CP violation we have

Γ P0_{(→ ¯P}0₎→ f
6= Γ ¯P0_(→P0₎→ f
. (2.12)

where the brackets indicate that it does not have to oscillate. It could also oscillate multiple times. In fact, it cannot be determined how many times the meson oscillated.

The measured value of the Jarlskog invariant is J = 3.1(2) · 10−5 [9]. Even though the CKM matrix of the standard model provides a source for CP violation, it is too small to explain baryogenesis. Therefore we need to look at mechanisms beyond the CKM matrix or standard model.

2.3

Current Theories on Baryogenesis

Multiple theories try to explain the baryon asymmetry we see in the universe. All of these theories state that the matter- antimatter asymmetry arose in the very early universe. The three main theories are leptogenesis, electroweak baryogenesis and GUT baryogenesis. In this thesis, the focus lies on electroweak baryogenesis. A short summary of leptogenesis and GUT baryogenesis will be given in this subsection, the latter of which is no longer a viable theory for baryogenesis.

Leptogenesis is the most appealing theory to explain the abundance of matter to antimatter[10]. The theory of leptogenesis is based on an asymmetry in the neutrino

sector. Right-handed neutrinos and left-handed antineutrinos are absent in the standard model. They can be included by introducing the seesaw mechanism.

There are multiple ways to implement the seesaw mechanism. Here we discuss the simplest type I seesaw. Consider a matrix describing the mass eigenstates of the neutrinos

¯ νL ν¯R
 0 m m M  νL νR  , (2.13)

where we have M  m. Here m is the Dirac mass and M is the Majorana mass. We will not go into depth on how these mass terms come to be. In order to find the masses of the neutrinos we need to find the eigenvalues of the mass matrix. In the case where M is much larger than m the eigenvalues are given by

mh ≈ M and ml ≈

m2

M (2.14)

where mh stands for the heavy mass eigenvalue and ml for the light mass eigenvalue. Note

that the chiral eigenstates νL and νR are not mass eigenstates. The mass eigenstates are

now approximated by νh ≈ − m MνL+ νR and νl ≈ νL− m MνR. (2.15)

Note that the heavy neutrino state is mostly the right-handed neutrino and that the light neutrino state consists mostly of the left-handed neutrino. The value of M may be orders of magnitude larger than the electroweak scale. This type I seesaw mechanism explains why observed left-handed neutrinos are orders of magnitude lighter than other standard model particles and why we have not yet observed right-handed neutrinos.

Because the right-handed neutrino is much heavier than the standard model particles it is allowed to decay, unlike the left-handed neutrino. However, since this is also the case for left-handed antineutrinos, we must introduce a source of CP violation in order to have leptogenesis. Similar to CP violation in neutral meson decay, we can have CP violation in heavy neutrino decay by introducing CP dependent complex phases. The sum of multiple decay diagrams then leads to different decay rates for the right-handed neutrino and left-handed anti-neutrino[10], see figure 4. The CP violation in heavy neutrino decay is not enough to maintain a lepton-antilepton asymmetry, as we still need a process that violates baryon number and the system needs to be out of equilibrium. Since the decaying neutrinos are superheavy, the system is automatically out of equilibrium. Sphaleron processes convert the lepton asymmetry into a baryon asymmetry. The combination of sphaleron processes and CP violation in heavy neutrino decay leads to leptogenesis.

Figure 4: The three main diagrams involving CP violation in heavy neutrino decay[11]. Ni

denotes a heavy mass eigenstate neutrino, and thus not a flavor eigenstate. The vertices in these diagrams carry CP dependent complex phases. The sum of these diagrams results in CP violation.

There are only few restrictions to leptogenesis. The restriction on neutrino masses, which must be 10−3eV < mν < 0.1eV , has already been met[10]. This was verified

by measurements on neutrino oscillations. However, because of the superheavy mass of the right-handed neutrino, it is very hard to get experimental evidence for leptogenesis. Currently, some experiments exist trying to determine whether CP violation is present in the neutrino sector, however, current experiments are not sensitive enough.

In addition to electroweak baryogenesis and leptogenesis, we have baryogenesis in grand unification - GUT baryogenesis. In grand unified theories (GUT), leptons and quarks are coupled via X and Y bosons. Similar to standard model CP violation, the coupling elements between the leptons and the quarks can have CP dependent complex phases. Interference of multiple diagrams leads to measurable CP violation in processes involving these X and Y bosons. Together with sphaleron processes, this can lead to a net baryon-antibaryon asymmetry.

There are however strong restrictions to GUT baryogenesis and grand unification in general. For one, measurements on proton decay have significantly reduced the amount of viable GUT models[11]. Another drawback is that theories involving grand unification cannot be tested in the near future because of the high energies involved. Still, the most im-portant imim-portant reason why GUT baryogenesis is no longer relevant is inflation. During the GUT phase transition, magnatic monopoles were created, yet they have never actually been observed. Inflation offers an explanation for this: after the GUT phase transition, the universe expanded extremely fast. This would reduce the number density of magnetic monopoles to essentially zero. At the same time, inflation also creates an impossibility for GUT baryogenesis. Indeed, if any baryons had been created during the GUT phase tran-sition, their number density would also have been reduced to zero, which simply cannot be the case.

Regardless of their theoretical viability, both leptogenesis and GUT baryogenesis share the same inconvenience: they cannot be experimentally verified in the near future . Elec-troweak baryogenesis however, might be, making it the most interesting theory to consider. Electroweak baryogenesis can be tested in particle colliders that are being planned for the near future, such as the International Linear Collider[12]. Another possible source of in-formation on electroweak baryogenesis are measurements on the electric dipole moment of the electron. If the electroweak phase transition was strong enough and of first-order, the planned LISA detector would be able to detect gravitational waves originating from this strong phase transition.

3

Higgs Mechanism and Electroweak Theory

The Higgs mechanism plays a crucial role in the electroweak phase transition. It is what causes the electroweak phase transition to happen. For that reason, we will go in depth on the Higgs mechanism and electroweak theory in this chapter. In order to come to a proper understanding of the Higgs mechanism and electroweak theory, we must first discuss the fundamentals of quantum field theory, the theory that describes forces between particles. Central to quantum field theory are the Lagrangian formalism and the principle of gauge invariance. A theory must be gauge invariant in order to be a viable quantum field theory. This can be accomplished by introducing a new field with a corresponding gauge boson into the theory. This gauge boson is required to be massless. I will elaborate upon gauge invariance and the Lagrangian formalism in quantum field theory in section 3.1.

The gauge boson in a theory is required to be massless, this is because adding a mass term to the Lagrangian of the theory would break gauge invariance, making the theory inviable. This confronts us with a contradiction, as the gauge bosons of the weak interaction - the W and Z bosons - are known to be massive. In order to explain the masses of the W and Z bosons, the Higgs mechanism was introduced. The Higgs mechanism gives masses to the W and Z bosons by breaking the electroweak symmetry, without breaking gauge invariance. These concepts - symmetry breaking and the Higgs mechanism - will be discussed in section 3.2 and 3.3 respectively. Finally, I will discuss electroweak theory in section 3.4.

3.1

Lagrangian Formalism in Field Theory and Gauge Invariance

In quantum mechanics we describe particles with wavefunctions. These wavefunctions satisfy a certain wave equation that depends on the properties of the system in question. In contrast, quantum field theory describes particles as excitations in a quantum field. The dynamics of a quantum field can be described with Lagrangian mechanics. Lagrangian mechanics in quantum field theory works the same as how it does in classical mechanics. In quantum field theory, instead of using the total Lagrangian, we use the Lagrangian density. Integrating over the Lagrangian density gives the total Lagrangian

L = Z

L(φi, ∂µφi) d3~x, (3.1)

where the Lagrangian density is a function of a set of fields φi and the corresponding

derivatives ∂µφi. Similar to classical mechanics, the dynamics are described by the

Euler-Lagrange equations ∂µ  ∂L ∂(∂µφi)  − ∂L ∂φi = 0. (3.2)

The Higgs field is a scalar field, which means that the Higgs boson has zero spin. The Lagrangian density that describes a scalar field goes as

L = 1

2(∂µφ)(∂

µ_{φ) − V (φ). (3.3)}

Here V (φ) represents the interaction potential of the field. The first term in this Lagrangian can be thought of as a kinetic term. In the case of a massive relativistic spin-half field satisfying Dirac’s equation, the Lagrangian is given by

LD = i ¯ψγµ∂µψ − m ¯ψψ (3.4)

where ψ is a four-component spinor field. One example of a massive relativistic spin-half field is the field corresponding to the electron. For a relativistic abelian5 vector field Aµ, such as the electromagnetic field, the Lagrangian goes as

LA = −

1 4F

µν_F

µν where Fµν = ∂µAν − ∂νAµ. (3.5)

where Fµν _{is the field strength tensor. For electromagnetism the electric and magnetic}

fields are contained in this field strength tensor.

A viable quantum field theory is required to be locally gauge invariant. The theory is said to be locally gauge invariant if the Lagrangian does not change under a local gauge transformation. In the case of quantum electrodynamics, the Lagrangian is invariant under a U (1) local phase transformation. This corresponds to transforming the field as follows

ψ(x) → eiqχ(x)ψ(x). (3.6)

It gives the field a position and time dependent complex phase. Note that we distinguish between local and global gauge invariance. In a global phase transformation the complex phase would not depend on position and time.

The Dirac Lagrangian from equation 3.4 is not invariant under a local U (1) gauge transformation. This is due to the derivatives that appear in the Lagrangian. When the derivatives act on the field, they will also act on the position- and time-dependent complex phase. This can be fixed by introducing a new gauge field Aµ. We make the following

transformations

∂µ→ ∂µ+ iqAµ and Aµ→ Aµ− ∂µχ. (3.7)

Substituting the transformations of 3.7 into the Dirac Lagrangian 3.4 makes it gauge invariant. Including the Lagrangian for the vector field Aµ gives us

LQED = ¯ψ(iγµ∂µ− me)ψ + e ¯ψγµψAµ−

1 4FµνF

µν

. (3.8)

This is the Lagrangian describing the interaction between electrons and photons, or in other words, quantum electrodynamics. The second term in this Lagrangian represents the interaction between photons and electrons. Such an interaction term can be represented by a Feynamn diagram, as shown in figure 5. The Lagrangians for quantum chromodynamics (QCD) and the weak interaction can be derived in a similar way. Instead of imposing U (1) symmetry, we require the Lagrangians to be invariant under SU (3) and SU (2)Llocal phase

transformations for QCD and the weak interaction respectively.

If the photon happened to be massive, there would be a mass term in 3.5. This mass term would then break gauge invariance. In fact, adding a mass term for the gauge boson in any gauge theory will break gauge invariance. This forms a problem as we already know that the gauge bosons of the weak interaction, the W and Z bosons, are massive. Therefore, there must be a mechanism that gives masses to these bosons without breaking gauge invariance. The Higgs mechanism is such a mechanism.

5_{Abelian is a term from group theory. In an abelian group the elements of the group are commutative,}

meaning that it does not matter if we first do operation A and then operation B or vice versa. The U (1) group is an abelian group.

Figure 5: The interaction vertex of quantum electrodynamics. This particular diagram shows a photon forming an e+e− pair. If rotated, this diagram would represent an electron emitting a photon, or an e+e− pair annihilating and producing a photon.

3.2

Spontaneous U (1) Symmetry Breaking

At the core of the Higgs mechanism lies something referred to as spontaneous symmetry breaking. In the standard model, the Higgs mechanism breaks the electroweak U (1)Y ×

SU (2)L symmetry6[13]. For simplicity, we will only look at U (1) symmetry breaking.

Consider the Higgs field to be a complex scalar field of the form

φ(x) = √1

2(φ1(x) + iφ2(x)). (3.9)

Since the field is complex the corresponding Lagrangian is given by

L = (∂µφ)∗(∂µφ) − V (φ). (3.10)

Now consider a potential of the following form

V (φ) = µ2(φ∗φ) + λ(φ∗φ)2. (3.11) Note that the Lagrangian possesses a global U (1) symmetry as it is invariant under the transformation φ → eiθ_{φ. In order for our potential to have a finite minimum we must have}

λ > 0. When µ2 > 0 this potential has an absolute minimum at φ = 0. In the case where µ2 _{< 0 the potential has a local maximum at φ = 0, this specific potential is referred to}

as the Mexican hat potential, see figure 6. The potential has an infinite set of degenerate minima at |φ| = r −µ2 2λ = v √ 2, (3.12)

which forms a circle in the complex plane. The ground state of the field will be on a particular point on this circle, breaking the U (1) symmetry. Since this value of the field is the ground state it will be the vacuum expectation value of the field. Spontaneous symmetry breaking thus gives the Higgs field a vacuum expectation value.

Next to generating a vacuum expectation value, spontaneous U (1) symmetry breaking gives rise to one massive Higgs boson and one massless particle. To see this we must expand the Higgs field around the ground state. Because of the gauge invariance of our Lagrangian

6_{The U (1) × SU (2) group is a product of the U (1) group and the SU (2) group. The subscripts Y and L}

imply that the corresponding groups only act on weak hypercharge and left-handed particles respectively. Weak hypercharge comes from electroweak theory and will be explained in section 3.4.

Figure 6: left : Visualization of equation 3.13. The circle represents the set of minima described by equation 3.12. This circle forms the ground state of the field. Excitations in η and ξ are represented by a yellow and green arrow respectively. right : The Mexican hat potential from equation 3.11.

we can choose the ground state to be on the real axis so that we have φ = v/√2 at the ground state. From here we can expand the field as

φ(x) = √1 2 v + η(x)
e iξ(x)/v _{≈ √}1 2 v + η(x) + iξ(x)
(3.13)

where two new fields η and ξ are introduced. Note that excitations in η are in the direction where V (φ) changes whereas excitations in ξ are in the direction where the potential does not change. Substituting this expression for φ into our original Lagrangian 3.10 we obtain a Lagrangian describing the interactions between the two new fields

L = 1 2(∂µη)(∂ µ_{η) +} 1 2(∂µξ)(∂ µ_{ξ) − λv}2_η2_{− V} int, (3.14)

where the interaction potential Vint is given by

Vint = 1 4λη 4₊ 1 4λξ 4₊ 1 2λη 2_ξ2_{+ λvη}3_{+ λvηξ}2_{. (3.15)}

From the η2 term in this Lagrangian we can see that the field η describes a massive particle with a corresponding mass of m2

η = 2λv2. This particle is the Higgs boson. The field ξ

has no mass, the corresponding particle is known as a Goldstone boson. Goldstone bosons naturally arise when a certain gauge symmetry is broken7. They play an important role

7_{There will be one Goldstone boson for each generator of the symmetry that is broken. This is known}

as the Goldstone theorem. The U (1) group has only one generator and thus we only have one Goldstone boson.

Figure 7: The vertices corresponding to the last three terms. They appear in the same order as in equation 3.15.

in the Higgs mechanism as we will discover later on. Each term in 3.15 corresponds to a certain interaction that can be represented by a Feynman diagram. The last three terms corresponds to the diagrams shown in figure 7.

3.3

Higgs Mechanism: the general idea

Now that we are familiar with spontaneous symmetry breaking, we can continue to discuss the Higgs mechanism. We will now apply the principle of local gauge invariance to the Higgs field. The Lagrangian from 3.10 is invariant under a global U (1) phase transformation. However it is not invariant under a local phase transformation. This is because of the derivatives appearing in the first term of the Lagrangian. Consider the local U (1) phase transformation

φ → eigχ(x)φ. (3.16)

In order to make 3.10 invariant under this transformation, we introduce a gauge field Bµ

by making the following two transformations

∂µ→ ∂µ+ igBµ and Bµ → Bµ− ∂µχ. (3.17)

Using the potential from equation 3.11 we can write the resulting Lagrangian as follows

L = − 1 4F

µν

Fµν + (∂µφ)∗(∂µφ) − µ2|φ|2− λ|φ|4

− igBµφ∗(∂µφ) + ig(∂µφ)∗Bµφ + g2BµBµ|φ|2. (3.18)

where Fµν _{is the field strength tensor of the field B}µ_{. Since our ground state is at φ = v/}√₂

we can expand φ around this point as done in 3.13. Doing this results in the following Lagrangian L =1 2(∂µη)(∂ µ_{η) − λv}2_η2₊1 2(∂µξ)(∂ µ_{ξ) −} 1 4F µν_F µν+ 1₂g2v2BµBµ − V (η, ξ, B) + gvBµ(∂µξ). (3.19)

Again we have our massive scalar η and one massless Goldstone boson ξ. The BµBµ

term is a mass term for our gauge field. Three- and four-point interactions between the different fields are included in the potential V (η, ξ, B). The last term is a direct coupling between the gauge boson and the Goldstone boson. It appears as if the gauge field and the Goldstone field can mix, see the corresponding Feynman diagram in figure 8.

In this process the amount of degrees of freedom has been changed. Let’s look at our original Lagrangian, where the Higgs was described by equation 3.9. Here we have two degrees of freedom; one for φ1 and one for φ2. We also have a massless gauge boson

con-tributing degrees of freedom. A gauge field has a certain polarization. If the gauge boson

Figure 8: The last term in the Lagrangian from equation 3.19. This term is a direct coupling between the gauge boson and the Goldstone boson. As this diagram shows, the direct coupling implies that the Goldstone boson and the gauge boson can mix.

is massless it can only have transversal polarization. The massless gauge boson thus con-tributes two degrees of freedom giving us a total of four. Now we look at the Lagrangian from 3.19. Our gauge boson now has a mass which allows it to have a longitudinal polar-ization, we thus get three degrees of freedom from the Gauge boson. We also have degrees of freedom from η and the Goldstone boson, each contributing one, giving us a total of five degrees of freedom.

We can solve this by letting the gauge boson eat the Goldstone boson. Note that we can combine the Goldstone and the gauge field terms from the Lagrangian as follows

1 2(∂µξ)(∂ µ ξ) + gvBµ(∂µξ) + 1₂g2v2BµBµ= 1₂g2v2  Bµ+ 1 gv(∂µξ) 2 . (3.20)

We can now get rid of the Goldstone field by making the following gauge transformation

Bµ+

1

gv∂µξ → Bµ, (3.21)

and now 3.19 becomes

L = 1 2(∂µη)(∂ µ_{η) − λv}2_η2₋1 4F µν_F µν+ 1₂g2v2BµBµ− V (η, B). (3.22)

Now we want to rewrite the expression for the Higgs. Note that in equation 3.13 the Goldstone field is in the complex phase of the Higgs field

φ(x) = √1

2 v + η(x)
e

iξ(x)/v_{. (3.23)}

But since our Lagrangian is invariant under a local U (1) gauge transformation, we can rotate the complex phase away and rewrite the field as

φ(x) = √1

2 v + η(x)
≡ 1 √

2 v + h(x)
, (3.24)

where we now introduced the Higgs field h. The Goldstone field has disappeared from our expression for φ. It has been eaten by the gauge boson Bµ. Working in this gauge we can

rewrite the Lagrangian one last time as

L =1 2(∂µh)(∂ µ_{h) − λv}2_h2₋ 1 4FµνF µν₊ 1 2g 2_v2_B µBµ + g2vBµBµh + 1₂g2BµBµh2− λvh3− 1₄λh4. (3.25)

This Lagrangian describes a massive Higgs scalar, a massive gauge field, interactions between the gauge field and the Higgs, and self-interactions of the Higgs. The masses of the gauge boson and the Higgs boson are given by

mB = gv and mH =

√

2λv2_{. (3.26)}

Other particles that interact with the Higgs will also obtain a mass. This is how particles like electrons and quarks obtain their mass. Note that this is different from how gauge bosons get massive.

3.4

Electroweak Theory and Standard Model Higgs

The Higgs mechanism in the standard model is achieved by imposing a local U (1)Y ×

SU (2)Lsymmetry. Even though the Lagrangian from 3.25 was obtained using a local U (1)

phase transformation, the method for the standard model Higgs remains the same. In the standard model the Higgs is a weak isospin doublet given by

φ =φ + φ0  = √1 2  χ1(x) + iχ2(x) v + h(x) + iχ3(x)  , (3.27)

where the upper component of the doublet is charged. In a weak isospin doublet the upper component and lower component always differ by one charge. The complex conjugate of φ+ will give the negatively charged φ−. There appear three Goldstone bosons in this doublet. The gauge symmetry of a certain theory reveals a lot about the force it describes. Therefore, let’s look further into the U (1)Y×SU (2)Lsymmetry of the electroweak theory in

order to understand the electroweak force. Similar to the local U (1) phase transformation, with a U (1)Y × SU (2)L symmetry we require the Lagrangian to be invariant under the

following transformation

φ → expigW~α(x) · ~T + 1₂ig0Y ζ(x)φ, (3.28)

where gW and g0 are the gauge couplings and Y is weak hypercharge. ~T are the three

generators of the SU (2) group. These generators can be written in terms of the well-known Pauli matrices

~

T = 1₂~σ. (3.29)

Now in order for our system to be invariant under this gauge transformation we substitute the derivatives in our Lagrangian from 3.10 by

∂µ→ ∂µ+1₂igW~σ · ~Wµ+1₂ig0Y Bµ. (3.30)

Instead of introducing one new gauge field as in 3.17, we have introduced four new gauge fields. They are Wµ(1), Wµ(2), Wµ(3) and Bµ.

To find the masses of the gauge bosons after symmetry breaking, we follow the same steps as in section 3.3. As it turns out, the previously introduced gauge fields Wµ(3) and

Bµ are not mass eigenstates. Instead the mass eigenstates are the gauge fields belonging

to the photon and the Z boson

Aµ = Wµ(3)sin(θW) + Bµcos(θW)

Zµ = Wµ(3)cos(θW) − Bµsin(θW). (3.31)

At zero temperature it might seem as if electromagnetism and the weak interaction are two completely separate forces. This equation however shows that the electromagnetic field and the gauge field of the Z are mixes of the electroweak gauge fields. The corresponding masses are given by

mA= 0 and mZ = 1₂v

q g2

W + g02. (3.32)

In equation 3.31 we introduced the weak mixing angle θW, also known as the Weinberg

tan(θW) =

g0 gW

(3.33)

The W(1)_{and W}(2) _{gauge bosons do turn out to be mass eigenstates. However, the physical}

W+ and W− bosons of the weak interaction are linear combinations of these two bosons W_µ±= √1 2 W (1) µ ∓ iW (2) µ
. (3.34)

The mass of the W boson is then given by

mW = 1₂gWv. (3.35)

After the electroweak symmetry is broken, the W and Z bosons have obtained masses. The three Goldstone bosons from 3.27 are eaten by these W and Z bosons, similar to how the Goldstone boson from section 3.3 was eaten by the Bµ field. The degrees of freedom

belonging to these three Goldstone bosons have become the degrees of freedom of the longitudinal polarization states of the W and Z bosons. Now that the Goldstone bosons have been ”gauged away”, the Higgs doublet can be written in the unitary gauge

φ = √1 2  0 v + h(x)  . (3.36)

When the weak interaction and electromagnetism are separated, we talk about weak isospin and electric charge as good quantum numbers. In the unified electroweak theory, we instead consider weak isospin and weak hypercharge as good quantum numbers. Weak hypercharge Y was introduced in 3.30. Weak hypercharge is a linear combination of electric charge and weak isospin described by

Y = 2 Q − I_W(3)
. (3.37)

Only after electroweak symmetry is broken, electric charge becomes a good quantum num-ber. However, even though electromagnetism and the weak interaction are now separate forces, the couplings are closely related

4

Electroweak Phase Transition and Baryogenesis

Now that the basics of electroweak theory and of the Higgs mechanism have been discussed, we can go into depth on the electroweak phase transition. Shortly after the big bang, at high temperatures, the weak interaction and electromagnetism were unified under the elec-troweak force. At around 10−12 seconds after the big bang, the universe had cooled down enough for the weak interaction and electromagnetism to separate from the electroweak force. The W(i) and B bosons of the electroweak force were replaced with the photon and the W± and Z bosons. These W and Z bosons obtained their mass due to spontaneous symmetry breaking.

In the upcoming sections I discuss how decreasing temperatures caused the Higgs mech-anism to break the electroweak symmetry. The potential of the Higgs changes with tem-perature. Below a certain temperature, electroweak symmetry is spontaneously broken and thus the electroweak phase transition happens. Since distinguishing between first- and second-order phase transitions is central to my thesis, I discuss the difference between first-and second-order phase transitions in section 4.1. In the section that follows I will explain the core of this thesis, namely how baryogenesis could have taken place in a first-order electroweak phase transition. A first-order electroweak phase transition is not possible within the standard model. Therefore, in section 4.3, I extend the standard model by coupling a singlet scalar field S to the Higgs. Coupling this particle to the Higgs influences the dynamics of the electroweak phase transition. In section 4.4, I will discuss how the Higgs potential changes with temperature and how coupling a new particle to the Higgs influences the electroweak phase transition.

4.1

First- and Second-Order Phase Transitions

When temperature decreases the Higgs potential changes. The way the potential changes with temperature determines the order of the phase transition. Without discussing the definition of first- and second-order phase transitions we will apply it to the electroweak phase transition. In a second-order phase transition the Higgs field smoothly goes from 0 to the vacuum expectation value. As the temperature lowers, the original minimum of the potential slowly becomes a local maximum, see figure 9. The critical temperature is reached when V00(φ = 0) = 0. After that, the field ”rolls” into the new groundstate. If the electroweak phase transition was second-order, then the Higgs field smoothly developed a non-zero vacuum expectation value everywhere.

In a first-order electroweak phase transition, the Higgs field does not gradually go from 0 to its vacuum expectation value. Consider the Higgs potential at high temperatures, where there is only one global minimum which is at φ = 0. When the temperature decreases, a new local minimum appears, see figure 9. As long as the temperature is above the critical temperature, the minimum at φ = 0 is the global minimum. When the temperature reaches its critical point, the minima are degenerate. When the temperature goes below the critical temperature, the minimum at φ 6= 0 is lower than the minimum at φ = 0. The minimum at φ 6= 0 is therefore the groundstate, we now call this minimum the true vacuum. The local minimum at φ = 0 is now referred to as the false vacuum. Classically, the false vacuum would be stable since there is a potential energy barrier between the two minima. However, as will be elaborated on in chapter 5, the field is allowed to tunnel from the false vacuum to the true vacuum. Instead of gradually going from 0 to its vacuum expectation value, such as in a second-order transition, the Higgs field abruptly goes from its initial

Figure 9: A visual representation of first- and second-order transitions on the left and right respectively. Here the potential is sketched for different values of temperature. In a first-order phase transition, the critical temperature is when the two minima are degenerate. In a second-order phase transition, the critical temperature is when V00(0) = 0. Note that in the first-order case there is a potential energy barrier between the two minima while in the second-order case the field can roll into the groundstate.

value to the vacuum expectation value.

In field theory, such a first-order phase transition leads to formation of Higgs bubbles. The process is as follows: the Higgs field tunnels to the true vacuum at a certain point in space and time due to quantum and thermal fluctuations. A small bubble where the field is in the true vacuum appears. Outside this bubble the field is still in the false vacuum. The bubble will then expand because it is energetically favorable to do so8_{. This expansion}

continues until the entire universe is in the true vacuum. The exact dynamics of bubble formation will be discussed in chapter 5, but the information we have now is enough to have a basic understanding of electroweak baryogenesis.

4.2

Electroweak Baryogenesis

In this section I will explain how baryogenesis could have taken place in the electroweak phase transition. For the purpose of this section, assume that the electroweak phase transition is first order. After all, baryogenesis can only take place in a first-order phase transition. Shortly after the big bang, when temperatures were high, the Higgs potential had a global minimum at φ = 0. When the temperature dropped below the critical temperature, a new true vacuum formed and φ = 0 became the false vacuum. The Higgs field, depending on the circumstances, tunneled from the false vacuum to the true vacuum at a certain point in space and time.

After the Higgs field has tunneled to the new true vacuum, a bubble of the true vacuum is formed and starts to expand. Inside the bubble, the Higgs field has a nonzero vacuum expectation value (vev). The value of the Higgs field as a function of position is sketched in figure 10. Note that it is not simply a step function. When the particles enter the bubble, they become massive due to the Higgs vev. Particles that are outside of the bubble will be collide with the expanding bubble wall. In these collisions CP is violated, resulting in a position dependent CP asymmetry as sketched i n figure 11. Just outside of the bubble,

8_{As we will see in chapter 5, expanding is not always energetically favorable due to surface tension of}

Figure 10: The value of the Higgs field sketched as a function of position. In this figure z = 0 is the position of the bubble wall. Points on the left of the y axis are inside the bubble and on the right of the axis is outside the bubble. The dynamics of the instanton will be described in chapter 5. Note that it is not a step function but that it does have a high gradient at z = 0

there is an abundance of qL+ ¯qR compared to qR+ ¯qL. This is compensated by an excess

of qR+ ¯qL inside the bubble. There is no net difference in the total amount of qL+ ¯qR

compared to qR+ ¯qL.

Sphaleron processes outside the bubble will convert the CP asymmetry into a baryon asymmetry. Sphalerons only interact with left-handed quarks and leptons but not with right-handed quarks and leptons. These sphaleron processes allow left-handed quarks and left-handed antiquarks to be converted into antileptons and leptons respectively. Note however that in front of the bubble wall there is more ¯qL than qL. This means more

antiquarks are converted into leptons than quarks being converted into antileptons. The CP asymmetry that in front of the wall is thus being transformed into a baryon asymmetry in front of the wall. Inside the bubble, the sphaleron processes are highly suppressed. The strength of the phase transition is an indicator of how strongly sphaleron processes are suppressed inside the bubble. I will discuss what the strength of the phase transition is later on. Because sphalerons are suppressed inside the bubble, no significant baryon asymmetry develops inside the bubble. The baryon asymmetry is sketched as a function of position in figure 11.

If the bubble would not be expanding, the baryon asymmetry in front of the bubble wall would eventually be washed out. This is due to other processes in the plasma which allow CP to be violated, compromising the initial CP asymmetry. Because the bubble wall moves outward, the baryons that were created in front of the wall are now inside the bubble. Since we assume the phase transition to be strong enough, the sphalerons are

Figure 11: left : Sketch of the CP asymmetry as a function of position. Again, the z = 0 is the bubble wall and z > 0 corresponds to points outside the wall. The CP has an antisymmetric shape, the total CP asymmetry vanishes. right : Baryon asymmetry versus position. Outside of the bubble wall the CP asymmetry is converted into a baryon asymmetry. Inside the bubble there is no baryon asymmetry.

highly suppressed inside the bubble and the baryons are there to remain. When the bubble has expanded until the entire universe is in the true vacuum, there is a surplus of baryons in the universe. It is this way that the electroweak phase transition – provided that the transition is strong enough – might constitute an explanation for baryogenesis.

There are restrictions on electroweak baryogenesis however. The most important one was already alluded into the last paragraph: a first-order electroweak phase transition can only cause a surplus of baryons if the phase transition is strong enough. Sphaleron pro-cesses happen mainly in the electroweak symmetric phase. They can still happen when electroweak symmetry is broken but are suppressed. Inside the bubble, electroweak symme-try is broken and thus sphalerons are suppressed. How much the sphalerons are suppressed is described by the ratio between the vacuum expectation value of the Higgs and the nu-cleation temperature. For baryogenesis we will need

v(TN)

TN

> 1. (4.1)

Note that the vev of the Higgs is a function of temperature. The nucleation temperature is the temperature at which the bubble forms and starts to expand.

A second restriction on electroweak baryogenesis has to do with the expansion of the bubble wall [12]. In electroweak baryogenesis, the CP asymmetry that arose in front of the wall is converted into a baryon asymmetry. However, if the velocity of the bubble wall is too large, there is not enough time for sphaleron processes to convert the CP asymmetry into a baryon asymmetry. Thus for baryogenesis to occur, the velocity of the bubble wall must not become highly relativistic. How to determine whether the bubble wall motion becomes highly relativistic?. Two factors should be considered in this regard: the difference in energy density inside the bubble and outside the bubble, and the pressure of the plasma outside of the bubble.

The first factor is the difference in energy density inside and outside the bubble. The true vacuum inside the bubble has a lower energy than the false vacuum outside the bubble. This difference in energy density inside the bubble and outside the bubble causes the bubble wall to accelerate. The second factor is the pressure from the plasma. Particles in the plasma outside the bubble are massless but inside the bubble the particles become massive due to the vev of the Higgs. This gives rise to a difference in pressure, slowing down the bubble wall. For the bubble wall not to reach relativistic speeds, the pressure and energy difference must be balanced

∆V < ∆P, (4.2)

where ∆V and ∆P are the difference in energy density and pressure respectively. This criterion states that the pressure difference caused by the plasma must be greater than the gain in energy density in order to keep the bubble wall motion nonrelativistic. This equation is referred to as the Bodeker-Moore criterion.

4.3

Extension to Standard Model: extra scalar field S

Baryogenesis can only take place in the electroweak phase transition if this phase tran-sition is first-order. However, according to ref [3] there is no possibility for a first-order electroweak phase transition within the standard model. It was shown that a first-order phase transition is only possible with a Higgs mass up to 75 GeV. This is substantially

Figure 12: The Feynman diagrams corresponding to the last two terms in the interaction potential. Note that the external lines of the diagrams can be connected so that the diagrams become loop diagrams. As we will see later, these loop diagrams play a role in quantum corrections.

lower than the measured mH = 125 GeV. In order to have a model for electroweak

baryo-genesis we will therefore need to look beyond the standard model. The impossibilities that the standard model poses could be resolved by extending the Higgs sector such that a first-order phase transition is possible. There are multiple models that try to achieve this. In the minimal supersymmetric standard model (MSSM) there are two Higgs doublets. This model predicts that the standard model Higgs is a linear combination of these two doublets. Due to this extension, the dynamics of the phase transition change. Still, a first-order electroweak phase transition is ruled out in this model. Other supersymmetric models do allow for a first-order phase transition.

In this thesis, I extend the Higgs sector by coupling a singlet scalar field S to the Higgs. Singlet implies that it does not carry weak isospin. This singlet particle does not carry any of the standard model charges, i.e. color, weak hypercharge, electric charge. It only interacts with the Higgs field via a Higgs-portal interaction. The tree-level interaction potential of the Higgs and the singlet can be written as

V (h, S) = −1₂µ2h2+1₄λh4+1₂ν2S2+ 1₄ηS4+1₂κh2S2. (4.3) The first two terms in this potential are the same as in the original Higgs potential. The third term is an S2 term. This term strongly influences the mass of the singlet. The fourth term is a quartic self-interaction of the singlet. The last term in the interaction potential describes the coupling between the Higgs and the singlet via a Higgs portal interaction. With these last three terms come three free parameters: ν, η and κ. Terms in the interaction potential can be represented by vertices in a Feynman diagram. The Feynman diagrams belonging to the last two terms in the potential are shown in figure 12. This interaction potential is the tree-level potential. Tree-level implies that quantum corrections have not yet been included.

Here, the expression for the Higgs field slightly differs from equation 3.27. In the interaction potential 4.3, the Higgs field h is written to include the vacuum expectation value v. The Higgs doublet takes the following form

φ(x) = √1 2  0 h(x)  (4.4)

Extending the standard model comes with restrictions. The Higgs boson is prompted to decay into two singlet particles9_{. Measurements on the Higgs branching ratio do not}

show the Higgs decaying to this singlet particle. Since this decay has not been observed, it must be kinematically forbidden. We therefore have the restriction mS > 2mH. From

this interaction potential we can extract the mass of the singlet. When the electroweak symmetry is broken, the tree-level squared mass of the singlet is given by

m2_S = ∂ 2 ∂S2V (h, S)     (h,S)=(v,0) (4.5)

where v is the vacuum expectation value of the Higgs. Because of the significant mass of the singlet, and the fact that it does not have any interactions with other standard model particles, makes it an excellent dark matter candidate. This thesis will not focus on the dark matter aspect of this newly introduced particle.

Upon taking a second look at the interaction potential, it appears that the potential has a local minimum at (h, S) = (0, w) when ν2 _{< 0. As long as this minimum is not the}

true vacuum at zero temperature, no problems arise. It is possible that during the phase transition this state briefly was the ground state. Before the Higgs field develops a vev, the singlet field would briefly have a vev. This is referred to as a two-step transition. For ν2 _{> 0 there will be no local minima in the potential for S 6= 0. In this case we just have}

a one-step transition. To recap:

one-step : (0, 0) → (v, 0)

two-step : (0, 0) → (0, w) → (v, 0) (4.6)

4.4

Corrections and Thermal Potential

The electroweak phase transition is driven by the Higgs mechanism. The Higgs potential changes with temperature. At high temperatures, The Higgs potential preserves elec-troweak symmetry. As temperature decreases, the Higgs potential spontaneously breaks the electroweak symmetry. The potential of the Higgs changes with temperature due to quantum effects that will be discussed in this subsection.

The classical interaction potential from 4.3 does not give us the whole picture. There will be quantum corrections from loop diagrams contributing to the potential. With per-turbation theory we sum over all of these loop diagrams resulting in the Coleman-Weinberg potential [14] VCW(h, S) = 1 64π2 X i ni " m4_i(h, S)  log m 2 i(h, S) m2 i0  − 3 2  + 2m2_i(h, S)m2_i0 # , (4.7)

where i runs over all particles coupling to the Higgs including S and the Higgs boson itself. Here ni is the amount of degrees of freedom for a given particle and m0i is the mass of the

particle at (h, S) = (v, 0), it serves as a renormalization scale. Note that the masses of the particles depend on the values of h and S.

In quantum field theory, the mass of a scalar field is derived from the interaction potential. When only one field contributes to the interaction potential, the squared mass is equal to the second derivative of the potential with respect to the field. When multiple fields contribute to a potential we need to find the eigenvalues of the squared-mass matrix. In the case of the tree-level potential for the Higgs and the singlet from 4.3 it is given that

M_sqHS =  ∂2 h ∂h∂S ∂h∂S ∂S2  V =−µ 2 _{+ 3λh}2_{+ 2κS}2 _κhS κhS ν2_{+ 3ηS}2_{+ 2κh}2  (4.8)

At zero temperature, when electroweak symmetry is broken, the tree-level squared masses of the Higgs and the singlet are given by

m2_H(v, 0) = −µ2+ 3λv2

m2_S(v, 0) = ν2+ κv2 (4.9)

In the Coleman-Weinberg potential the sum is over all particles coupled to the Higgs. However, because the corrections are proportional to squared mass, I only include the top-quark and the W and Z bosons. The masses of the included particles are given by

m2_W = 1 4g 2_h2_{, m}2 Z = 1 4(g 2_{+ g}02 )h2 and m2_t = 1 2t 2_h2_{. (4.10)}

For the Coleman-Weinberg potential we need to know the degrees of freedom for each particle. The top-quark gets a minus sign because it is a fermion. The amount of degrees of freedom is then given by

n{h,S,W,Z,t} = {1, 1, 6, 3, −12}. (4.11)

As indicated earlier, the thermal corrections will determine the dynamics of the phase transition. The temperature corrections are described by the following potential

VT(h, S, T ) = X i niT4 2π2 Jb/f  m2 i(h, S) T2  , (4.12)

where the Jb/f function depends on whether the particle is a boson or a fermion. It is

calculated as follows Jb/f  m2 T2  = Z ∞ 0 log " 1 ∓ exp  − r k2_{+ m}2 T2 # k2dk, (4.13)

where a minus sign stands for the bosons and a plus sign for the fermions. For a higher degree of accuracy I include thermal corrections to the masses. This can be done by making the following substitution

m2_i → m2_i + Πi, (4.14)

where Πi is the thermal correction to the squared mass. These corrections can be obtained

by expanding 4.12 to leading order in m2/T2. This gives [17] Πh =  g02 16 + 3g 16+ λ 2 + t2 4 + η 12  T2, ΠS =  κ 3 + η 4  T2 and ΠW = 11g2 6 T 2 . (4.15)

The thermal masses of the Goldstone bosons and the photon are ignored. The thermal mass of the Z boson can be approximated by

ΠZ =

11 6 g

2_{+ g}02
_T2 _(4.16)

The Coleman-Weinberg potential from 4.7 and the thermal corrections from 4.12 deter-mine the effective shape of the potential at any given temperature. The effective potential is given by the sum of the tree-level potential and the two correction potentials