Sphaleron Solutions and Their Phenomenology in the Electroweak Theory

University of Groningen

Master thesis

Physics

Sphaleron solutions and their phenomenology in the electroweak theory

Supervisor: Author: Prof. Dr. Dani¨el Boer Tomas Heldeweg Second Reader: Prof. Dr. Rob Timmermans

July 30, 2018

Abstract The absence of anti-matter in the universe necessitates a process that violates the baryon number in the early universe. Such a process is found through sphaleron transitions in the electroweak theory. The sphaleron is an unstable static finite-energy solution to the classical field equations. It is a saddle point of the energy functional and its energy is associated with the height of the potential barrier between vacuum states with different winding numbers. Transition between topologically distinct vacua realize a violation of the baryon number through the ABJ anomaly. In the early universe temperatures are high enough so that transitions over the barrier could occur via thermal excitation. It is possible that such a transition could also occur in high energy collisions if the center of mass energy is comparable to the energy of the sphaleron. Here an overview of the construction of the SU(2) sphaleron is given. Furthermore, the probability of detecting a sphaleron transition in a high energy collision experiment is discussed. Contents

1 Introduction 2

2 Introduction to Gauge theory 3 2.1 Gauge theory ...... 3 2.2 Scalar ﬁeld theory ...... 5 2.3 Electroweak theory ...... 7 2.4 Outlook ...... 10

3 Homotopy classes and its applications 12 3.1 Homotopy classes ...... 12 3.2 Kinks and anti-kinks ...... 13 3.3 Topological conservation laws ...... 16 3.4 SSB and application of homotopy classes ...... 16 3.5 Application Homotopy classes ...... 17

4 Instantons 19 4.1 Instantons in quantum mechanics ...... 19 4.2 Instanton in quantum ﬁeld theory ...... 20 4.3 Winding vacuum number ...... 23 4.4 Relation between tunneling of vacua and instanton ...... 25 4.5 Theta vacuum ...... 26 4.6 The ABJ anomaly and instantons ...... 27 4.7 Strong CP problem and small θ ...... 29 4.8 B+L Anomaly in electroweak theory ...... 30

5 Sphalerons 32 5.1 Sphaleron solution in scalar ﬁeld theory ...... 32 5.2 Higgs vacuum manifold ...... 33 5.3 Construction of the SU(2) Sphaleron ...... 34

6 Phenomenology 39 6.1 Baryogenesis ...... 39 6.2 Baryon number violation and Sphaleron transition rate ...... 39 6.3 C and CP violation ...... 40 6.4 Departure from thermal equilibrium ...... 40 6.5 Concluding remarks ...... 42

7 Detection of ∆(B + L) processes 43 7.1 Probability of detecting a (B + L) violating process ...... 43 7.2 Tye & Wong Construction ...... 44 7.3 Validity Tye & Wong approach ...... 46 7.4 Detection of ∆(B + L) processes in the Tye & Wong approach ...... 46

8 Discussion 50 8.1 Energy of the sphaleron ...... 50 8.2 Baryogenesis ...... 50 8.3 Tye & Wong approach ...... 50 8.4 Higher order sphalerons ...... 50

9 Conclusion 51

1 1 Introduction

Quantum Field Theory has proved to be an excellent tool for describing fundamental interactions between particles in nature. It is used as a mathematical framework for the Standard Model, which describes 3 of the 4 (known) fundamental forces in the universe: the weak, strong and electromagnetic force. The Standard Model is arguably the greatest creation in physics. Although it has its flaws, it has been able to accurately predict the existence of various particles, most notably the Higgs boson in recent years. In the Standard Model particles are described by fields in spacetime. The Lagrangian of the fields describe the interactions and dynamics of the particles. The construction of the Lagrangian is based on the symmetries of the forces. The total gauge symmetry group of the Standard Model is given by SU(3) × SU(2) × U(1). Each particle may behave differently under a transformation of a symmetry group. One subsector of the Standard Model is the electroweak theory. As the name suggests, the electroweak theory describes the electromagnetic and weak force. In the electroweak theory there exist various non-trivial solutions. One of these non-trivial solutions is called the sphaleron. It is not unique to the electroweak theory but its existence there has very interesting implications; it could play an important role in generating the baryon asymmetry of the universe. The mystery of the observed discrepancy between matter and anti-matter in the universe has been a topic of active research for many years. The importance of understanding the discrepancy is quite trivial: if it were not there we would not exist. Moreover, a model that would accurately predict the observed asymmetry of matter and anti-matter could lead us one step closer to obtaining a beloved theory of everything. Unfortunately, the Standard Model has, so far, been unsuccessful in explaining the observed asymmetry. In order to realize successful baryogenesis, i.e. the process that produced the baryon asymmetry, Sakharov[44] postulated three conditions that have to be met. One of these conditions is that there is a process which violates the baryon number. The sphaleron process in the electroweak theory supplies this violation. This process can only take place in high energy/temperature environments. In the early universe the temperature would have been high enough so that these processes could occur readily. Although we cannot recreate conditions as they were in the early universe we might be able to detect sphaleron processes in high energy collision experiments. Detecting such a process would be a beautiful dis- covery and solidifies the electroweak theory as an essential part of models describing the fundamental forces. The bottleneck of detecting such a process is the center of mass energy in the collider experiments. This raises the question whether it is feasible to detect such processes in state-of-the-art particle accelerators such as the LHC, or as a further matter whether it is even possible to detect a sphaleron process in the near future.

The outline of this master thesis is as follows. In chapter 2 a brief introduction to gauge theories and the construction of the electroweak theory will be given. In chapter 3 the application of homotopy classes and topological laws will be discussed. In chapter 4 the ﬁrst non trivial structures will be constructed, such as the winding number vacuum and instanton conﬁgurations. Thereafter it will be shown how they can lead to anomalous conservation laws. In chapter 5 the construction of the sphaleron in the electroweak theory will be given. The role of the sphaleron in the baryon asymmetry will be discussed in chapter 6. In chapter 7 the possibility of detecting processes that violate baryon number in collider experiments will be discussed. In chapter 8 several aspects of the sphaleron will be discussed. Finally, in chapter 9 some concluding remarks will be made. Chapter 2-4 of this thesis will largely be based on the books Gauge Theory of Elementary Particle Physics by Cheng & Li [13] and Coleman’s Erice Lectures [16]. Chapter 5 will be based on the paper by Manton [36]. Finally, chapter 7 will be based on a recent paper by Tye and Wong [50] and the papers by Ellis et al. [19][20]

2 2 Introduction to Gauge theory

Gauge theory is a field theory in which the Lagrangian is invariant under a group of local transformations. This is done by promoting global symmetries to local symmetries. Doing this requires the addition of gauge fields to the theory and changing the derivatives into covariant derivatives. In this chapter we will first look at a simple fermionic field theory and promote the global symmetries to local ones to sketch the method of obtaining the gauge theory. After that we will give a short introduction to the Electroweak theory, which is the subject theory of this thesis.

2.1 Gauge theory Consider the Lagrangian for a free massive fermion ﬁeld given by:

µ L0 = ψ(x)(iγ ∂µ − m)ψ(x), (2.1)

where ψ is a doublet

ψ ψ = 1 . (2.2) ψ2

The Lagrangian is invariant under global transformations of the Lie group SU(2). The group SU(2) consists out of all 2 × 2 unitary matrices with a unit determinant. Since SU(2) is a Lie group it has an associated Lie algebra su(2). A Lie algebra consist out of a basis T a which adheres to the Lie bracket

a b ab c [T ,T ] = fc T (2.3)

ab a where fc are called the structure constants. The basis T generates all elements of the Lie algebra and are often referred to as generators. Elements of the Lie group can be obtained by exponentiating the elements of the Lie algebra and we can therefore use the generators of the Lie algebra to generate the associated Lie group. We can ﬁnd elements of the SU(2) group through

a U(θ) = exp {iθ Ta}, (2.4)

where the θa’s are the (real) group parameters. In the fundamental representation of SU(2) the generators Ta are related to the Pauli spin matrices τa as 1 T = τ . (2.5) a 2 a The doublet of equation (2.2) transforms under the fundamental representation. The transformation law of an element U ∈ SU(2) acting on ψ and ψ can be written as

ψ(x) → ψ0(x) = Uψ(x), (2.6) 0 ψ(x) → ψ (x) = ψ(x)U †. (2.7)

If we plug in the transformed ﬁelds into the Lagrangian the exponential terms cancel leaving the Lagrangian invariant. We can then conclude that the Lagrangian is invariant under SU(2). If we now change our global SU(2) symmetry into a local one by changing θa into θa(x),

θa(x)τ ψ(x) → ψ0(x) = exp i a ψ(x), (2.8) 2 the Lagrangian will no longer be invariant under this transformation since our derivative term transforms as

0 0 −1 ψ(x)∂µψ(x) → ψ (x)∂µψ (x) = ψ(x)∂µψ(x) + ψ(x)U (θ)[∂µU(θ)] ψ(x), (2.9)

3 To recover the invariance one can explicitly construct the so-called gauge-covariant derivative Dµ that re- places ∂µ. We demand that this covariant derivative acting on ψ transforms the same way as ψ:

0 Dµψ(x) → [Dµψ(x)] = U(θ)Dµψ(x). (2.10)

If this condition is satisfied the Lagrangian will be invariant under local SU(2) transformations. This can a only be done by adding fields to our theory, which are the gauge fields Aµ(x). For every group generator there is a corresponding gauge field, since we have 3 generators, the Pauli matrices, we have 3 gauge fields (a = 1, 2, 3). Then we construct the gauge-covariant derivative:

a τaA D = ∂ − ig µ . (2.11) µ µ 2

Plugging this in (2.10) we can ﬁnd the transformation law for the gauge ﬁelds,

0a a τaA τaA ∂ − ig µ (U(θ)ψ) = U(θ) ∂ − ig µ ψ. (2.12) µ 2 µ 2

0a Solving for τaAµ yields:

0a a τaA τaA i µ = U(θ) µ U −1(θ) + U(θ)∂ U −1(θ), (2.13) 2 2 g µ which defines the transformation law of the gauge fields. At this point the Lagrangian reads, g L = ψ(x)iγµ∂ ψ(x) + ψ(x)τ Aa ψ(x) − mψ(x)ψ(x). (2.14) 0 µ 2 a µ To make the gauge fields dynamical variables there should be a term in the Lagrangian containing their derivative. This term should meet several requirements. It, of course, must be gauge/Lorentz invariant to keep the Lagrangian gauge/Lorentz invariant and should also be of 4 dimensions or less to keep the theory renormalizable. One way to construct such a term is to look at the commutation relation of covariant derivatives acting upon a field vector,

[Dµ,Dν ]ψ(x) = (DµDν − Dν Dµ)ψ(x). (2.15)

From the deﬁnition of the covariant derivative (2.11) we ﬁnd that the commutation relation of covariant derivatives is a rank-2 tensor, τ [D ,D ] = ig a F a . (2.16) µ ν 2 µν so that

a a a a a τaF τaA τ A τaA τ A µν = ∂ µ − ∂ a ν − ig µ , a ν . (2.17) 2 µ 2 ν 2 2 2

The new term contains derivatives of the gauge fields, as was desired. To see how it transforms we first use (2.10) to find,

0 [(DµDν − Dν Dµ)ψ(x)] = U(θ)(DµDν − Dν Dµ)ψ(x). (2.18)

and then by substituting (2.16) we obtain

0a a τaFµν U(θ)ψ = U(θ)τaFµν ψ, 0a a −1 τaFµν = U(θ)τaFµν U (θ). (2.19)

4 From this point out we will often use the notation τ A ≡ a Aa (2.20) µ 2 µ τa a Fµν ≡ F (2.21) 2 µν and refer to Fµν as the field strength tensor. Continuing, to find the gauge invariant term in the Lagrangian one can consider the trace of one or the product of multiple Fµν matrices. To make the term Lorentz invariant the space indices of Fµν have to be contracted so a suitable term with conventional normalization is: 1 1 L = − Tr[F F µν ] = − F a F aµν . (2.22) dyn 2 µν 4 µν In the last line it was used that 1 Tr[F F µν ] = Tr[F a τ aF bµν τ b], µν 4 µν 1 = Tr[τ aτ b]F a F bµν , 4 µν 1 = δabF a F bµν , 2 µν 1 = F a F aµν . (2.23) 2 µν where the trace property Tr[τ aτ b] = 2δab of the Pauli matrices was used. So the complete gauge invariant Lagrangian is given by, 1 L = − Tr[F F µν ] + ψ (iγµD − m) ψ. (2.24) 2 µν µ Now that we have a feeling for how gauge fields are introduced in the Lagrangian we will move on to a simple complex scalar field gauge theory in which spontaneous symmetry breaking occurs. After that we can move on to the electroweak theory, which is an extension of the simple theory and see how the Higgs field gives rise to massive bosons through the so called Higgs Mechanism.

2.2 Scalar field theory Consider a Lagrangian for a free scalar field φ given by 1 L = − f f µν + (D φ)†Dµφ − V (φ), (2.25) 2 µν µ where φ is a complex scalar field. The Lagrangian is invariant under local U(1) transformations eiα(x) and consequently we have a covariant derivative given by

Dµ = ∂µ + igaµ. (2.26)

The transformation law of the gauge ﬁeld is given by 1 a → a0 = a − ∂ α(x). (2.27) µ µ µ g µ The dynamical gauge ﬁeld is given by

fµν = ∂µaν − ∂ν aµ. (2.28)

5 In these equations aµ is the gauge field that appears to keep the Lagrangian invariant under local U(1) transformations. The potential term is given by λ V (φ) = −µ2φ∗φ + (φ∗φ)2, (2.29) 2 where the first term can be thought of as the mass term and the latter term as the self interaction term. The potential term has multiple minima when µ2 > 0 given by µ2 φ∗φ = |φ|2 = . (2.30) λ The minima describe a circle in the complex plane as seen in figure 1 where the parameterization φ + iφ φ = 1 √ 2 , (2.31) 2 is used.

Figure 1: The mexican hat potential [45]

The so-called vacuum expectation value is therefore given by r µ2 hφi = . (2.32) λ

Any one of these minima can be chosen to be the true vacuum state φ0 of the theory and consequently the U(1) symmetry will be broken. Goldstone’s theorem states that a massless Goldstone boson should appear for every generator of a broken symmetry (we will give a proof of Goldstone’s theorem in section 3.4). U(1) has only one generator and so there should be 1 massless Goldstone boson. If we expand the Lagrangian around the new vacuum φ0 through

φ1 + iφ2 φ = φ0 + √ , (2.33) 2 and insert it into the potential term (2.29) we find 1 1 V (φ) = − µ4 + (2µ2)φ2 + interaction terms. (2.34) 2λ 2 1 √ The φ1 field attains a mass of m = 2µ. Since φ2 has no mass term it is the field associated with the massless Goldstone boson. The derivative terms in the Lagrangian are also influenced by the shifted scalar field (2.33) 1 1 √ (D φ)†Dµφ = (∂ φ )2 + (∂ φ )2 + g2φ2a aµ + 2gφ a ∂µφ + interaction terms. (2.35) µ 2 µ 1 2 µ 2 0 µ 0 µ 2

6 From the kinematic terms we can see that the gauge ﬁeld aµ has obtained a mass term given by

2 2 µ Lm = g φ0aµa . (2.36) We can exploit the U(1) gauge invariance of the Lagrangian to rotate away all the terms containing the Goldstone boson field φ2. By collecting the φ2 terms and the gauge mass term we can rewrite them as follows √ 2 1 2 2 2 µ µ 2 2 1 (∂µφ2) + g φ0aµa + 2gφ0aµ∂ φ2 = g φ0 aµ + √ ∂µφ2 , (2.37) 2 2gφ0 and if we now compare this to the transformation law of the gauge field (2.27) then we can rotate the term away with a gauge transformation α = − √φ2 . Doing this removes the Goldstone boson from the 2φ0 Lagrangian. Often it is said that in this process the Goldstone boson has been eaten by the gauge field to give it its mass. This process is called the Higgs Mechanism. From this point onwards we will use a slightly different notation for potential terms that have a nonzero vacuum expectation value, in this case

λ 2 V (φ) = φ∗φ − v2 , (2.38) 2

q µ2 where v is the vacuum expectation value. If we fill in v = λ then we find that this potential term is shifted µ4 by 2λ compared to the potential term given in equation (2.29). The term is a constant in the Lagrangian and we can therefore ignore it without a loss of generality. It also neatly cancels with the first term on the right hand side of equation . We will now explore the Higgs Mechanism in the electroweak theory.

2.3 Electroweak theory The electroweak theory, also referred to as the Weinberg-Salam theory, combines two of the four fundamental interactions, namely the electromagnetic and weak interaction. The Lagrangian of the electroweak theory consists of many parts but we only consider the bosonic part for now. The Lagrangian is then given by 1 1 L = − Tr[F F µν ] − f f µν + (D Φ)†(DµΦ) − V (Φ), (2.39) 2 µν 4 µν µ where

a a a abc b c Fµν = ∂µAν − ∂ν Aµ + g AµAν , (2.40)

fµν = ∂µaν − ∂ν aµ, (2.41) 1 1 D Φ = ∂ Φ − igτ aAa Φ − ig0a Φ, (2.42) µ µ 2 µ 2 µ and

1 2 V (Φ) = λ Φ†Φ − v2 . (2.43) 2

Φ is called the Higgs ﬁeld which is a two-component complex vector Φ1 † ∗ ∗ Φ = , Φ = Φ1 Φ2 , (2.44) Φ2

a The gauge symmetries in the theory are SU(2)L × U(1)Y . In the Lagrangian the Aµ are the gauge ﬁelds related to the SU(2)L symmetry with coupling constant g and aµ are the gauge ﬁelds related to the U(1)Y with coupling constant g0.

7 In nature weak interactions only aﬀect left-handed particles (or right-handed anti-particles). This has led to the construction of the weak isospin T . Left-handed particles are assigned the quantum number T = 1/2 and right handed particles T = 0. Left handed particles are then grouped into doublets of T 3 = ±1/2 that behave similarly under weak interactions and right handed particles are singlets. An element of UL ∈ SU(2)L can be found by exponentiating the Lie algebra just as in section 2.1

a UL = exp {iθa(x)T }, (2.45)

a where T are the generators and θa are the group parameters. The fundamental representation is obtained for the generator 1 T a = τ a, (2.46) 2 where τ a are again the Pauli matrices. Left handed doublets transform under the fundamental representation a pf SU(2)L and right handed singlets transform under the trivial representation of SU(2)L (T = 0). The U(1)Y symmetry is based on the quantum number called the weak hypercharge Y . This quantum number relates the electromagnetic charge to the third component of the weak isospin so that

Q = T 3 + Y. (2.47)

As an example, a left handed electron has T 3 = −1/2,Q = −1 and and a left handed electron neutrino has 3 T = 1/2,Q = 0 so therefore the weak hypercharge of left handed leptons is given by Y (lL) = −1/2. An element of uY ∈ U(1)Y can be generated by n o uY = exp iθ(x)Yˆ . (2.48)

The Higgs ﬁeld transforms as a doublet under SU(2)L and has a hypercharge Y = 1/2. The complete transformation law under SU(2)L × U(1)Y of the Higgs ﬁeld can therefore be written as

τ 1 Φ → Φ0 = exp i a θa(x) + iα(x) .Φ (2.49) 2 2

Prior to the addition of the Higgs field the gauge theories implied that the force-carrying particles were massless, but empirically they had been found to have mass. Addition of the Higgs field dealt with this problem through spontaneous symmetry breaking. Just as in the previous chapter we will see that because of the nonzero expectation value of the Higgs field the gauge fields obtain mass. The potential (2.43) has a † v2 2 local maximum at Φ = 0 and its minima, Φ Φ = 2 , form a spherical shell in C -space. Since all the minima in the shell are physically equivalent, i.e. they can be rotated into each other through gauge transformations, we can choose the theory to have a vacuum expectation value of

1 0 hΦi = √ . (2.50) 2 v If take this form to be the true ground state of the theory then 3 out of the 4 symmetries will be broken. The remaining symmetry can be found by considering what transformation leaves the vacuum expectation value invariant. If we take the transformation law (2.49) with θ1 = θ2 = 0 and θ3 = α we ﬁnd

τ 1 1 0 eiα(x) 0 1 0 exp i( 3 α(x) + α(x)) √ = √ , 2 2 2 v 0 1 2 v 1 0 = √ . (2.51) 2 v

8 which indeed leaves the vacuum expectation invariant. There should thus be one massless gauge boson corresponding to this unbroken symmetry. We can parametrize the Higgs field as 1 0 Φ(x) = √ exp {iτ aξa(x)} , (2.52) 2 H(x) + v where H(x) and ξa(x) are real fields. The H(x) fields correspond to the Higgs boson and the ξa(x) fields correspond Goldstone bosons. We have 3 Goldstone bosons corresponding to the number of generators of the 3 broken symmetries. The Goldstone bosons correspond to excitations in angular directions on the spherical shell, while the Higgs bosons correspond to excitations in the radial direction. We can exploit the invariance of the Lagrangian (2.39)under gauge transformations of SU(2)L to rotate the Goldstone bosons away, analogous to what we did in the previous section. By applying the SU(2)L gauge transformation with θa = −2ξa

a a UL(ξ) = exp {−iτ ξ (x)}, (2.53) we ﬁnd

Φ0(x) = U(ξ)Φ(x), (2.54) 1 0 Φ0(x) = √ , (2.55) 2 H(x) + v The kinematical terms give the mass terms of the gauge fields. Substituting the shifted field (2.55) in the kinetic terms of the Lagrangian yields g2 1 (D Φ0)†(DµΦ0) = ∂ H∂µH + (v + H)2[(A01)2 + (A02)2] + (v + H)2[g0a − gA03]2. (2.56) µ µ 8 µ µ 8 µ µ and we find several mass terms. We can define the W ± bosons as A01 ∓ A02 W ± ≡ µ √ µ . (2.57) µ 2 From this the mass term g2v2 1 W †W µ, (2.58) 4 2 µ is obtained and consqequently the mass of the W bosons is given by gv M = . (2.59) W 2 There are 2 more gauge bosons that can be extracted from the last term in (2.56). They can be found by writing the quadratic term as a matrix

v2 v2 g2 −gg0 A03µ [g0a − gA03]2 = A03 a , (2.60) 8 µ µ 8 µ µ −gg0 g02 aµ and diagonalizing it via the orthogonal transformation cos θ − sin θ O = W W , (2.61) sin θW cos θW with g0 g sin θW = , cos θW = , (2.62) pg2 + g02 pg2 + g02

9 accordingly we ﬁnd

v2 g2 −gg0 A03µ v2 g2 + g02 0 A03µ A03 a O†O O†O = A03 a O† O . (2.63) 8 µ µ −gg0 g02 aµ 8 µ µ 0 0 aµ

Furthermore we deﬁne Zµ A03µ ≡ O , (2.64) Aµ aµ

to ﬁnd that v2 g2 + g02 0 A03µ v2 A03 a O† O = (g2 + g02)Z Zµ + 0 · A Aµ. (2.65) 8 µ µ 0 0 aµ 8 µ µ and so we ﬁnd the massless A boson (the photon), corresponding to the unbroken symmetry, and the Z boson with mass v p M = g2 + g02. (2.66) Z 2 Furthermore, the mass of the Higgs particle can be obtained trivially from the potential term(2.43) √ MH = 2λv. (2.67)

Concluding, through spontaneous symmetry breaking the gauge ﬁelds have become massive and we have obtained 3 massive gauge bosons corresponding to the 3 Goldstone bosons.

2.4 Outlook Now that we have a general idea of how gauge fields are constructed and how they can become massive through spontaneous symmetry breaking we will explore the concept of homotopy classes and topological conservation laws in field theories. In field theories we are often interested in finding finite energy or finite action field configurations. To obtain such configurations we need to put constraints on the fields. For example, given the action of a pure static (no time dependence) SU(2) gauge theory, Z 3 µν S = d xFµν F , (2.68) where the integral is over the whole space R3 and

Fµν = ∂µAµ − ∂ν Aν − ig [Aµ,Aν ] . (2.69)

In order to obtain ﬁnite action solutions we need to have the constraint

Fµν −−−−→ 0, (2.70) |x|→∞

10 in other words Fµν must vanish at inﬁnity or else the action becomes inﬁnite. This implies from equation (2.69) that

Aµ −−−−→ 0, (2.71) |x|→∞ or Aµ can be a gauge transformation of Aµ = 0, called a pure gauge, at inﬁnity

i −1 Aµ −−−−→ U∂µU . (2.72) |x|→∞ g

So smooth configurations of Aµ(x) that satisfy the constraint (2.72) are finite action configurations. These solutions are clearly not unique; there are an infinite number of configurations for Aµ(x) possible, which can already be seen from (2.72) as the group SU(2) is continuous and thus there are an infinite number of pure gauge configurations. Topologically, we can think of all the configurations that satisfy (2.72) as a space of configurations, which will be referred to as the field configuration space in the remainder of the thesis. The functions U 0s of (2.72) can be seen as maps from the points at infinity to the elements of the gauge group SU(2), i.e. maps from space into the gauge group. The points at infinity in R3 are clearly a sphere denoted by S2 and so in mathematical notation we have U : S2 → SU(2). Maps like this can be placed in so-called homotopy classes, which we will talk about in more detail in the next chapter.

11 3 Homotopy classes and its applications

Homotopy classes play a big role in both the instanton and sphaleron solutions. In the ﬁrst part of this chapter a formal deﬁnition of homotopy classes will be given, accompanied by a simple illustrative example. The homotopy classes are characterized by index, which has many names and here we will refer to it as the winding number. Finally we will explore the application of homotopy classes and winding numbers in gauge theories.

3.1 Homotopy classes Mappings from one topological space to another are said to be homotopic when they are continuously deformable into each other. Formally, consider the set of mappings fs : X → Y between manifolds X and Y and let fs(x) be a continuous map for all s ∈ [0, 1] and x ∈ X. If the mappings fs(x) are continuous in s for all x then two mappings fs1(x) and fs2(x) with s1, s2 ∈ [0, 1] can be continuously deformed into one another and are thus homotopic. The full set of functions that are homotopic belong in the same homotopy class. A simple illustrative example can be made by considering the mapping from S1 → S1, i.e. a mapping from a circle to a circle. Let the ﬁrst circle be parameterized by an angle θ ranging from 0 to 2π and the second by 1 1 the unimodular complex numbers, so that we have fs(θ): Sθ → Suni. So some standard mappings would be the trivial mapping

f (0)(θ) = 1, (3.1)

and the identity mapping

f (1)(θ) = eiθ. (3.2)

By taking powers of the identity mapping we can create other mappings

f (ν)(θ) = [f (1)(θ)]ν = eiνθ, (3.3)

1 where ν is an integer called the winding number, since it indicates how many times we wind around Suni 1 1 1 when going around Sθ . It turns out that every mapping from Sθ → Suni can be put in one of the homotopy 1 classes given by (3.3) characterized by ν ∈ Z. Formally we can use the notation Π1(S ) = Z to indicate n the characterization. More generally the notation Πn(G) = S is used for the mappings f : S → G and the associate winding number belongs in the set S. A nice analogy to this example is when you imagine wrapping a rubber band around your ﬁnger. You can stretch and deform the rubber band into all kind of shapes, but you can always deform it so that it goes back to its original shape. The amount of times you wrap the rubber band around your ﬁnger is analogous to the winding number. You can never unwrap the rubber band without going out of the plane of the rubber band. The winding number can be directly expressed by the following integral formula

i Z 2π 1 df(θ) ν = − dθ . (3.4) 2π 0 f(θ) dθ One can see that direct substitution of (3.3) into the integral yields

i Z 2π 1 Z 2π − dθe−iνθiνeiνθdθ = dθνdθ, (3.5) 2π 0 2π 0 = ν. (3.6)

The same can be done for the mappings from S3 → S3 where each homotopy class is also deﬁned by an 3 j integer winding number, i.e. Π3(S ) = Z (in general we have Πj(S ) = Z). Unfortunately, there is no nice

12 analogy to this (unless you are able to imagine wrapping 4 dimensional spheres each other). The winding number in this case can be expressed by the integral formula 1 Z ν = − dθ dθ dθ ijkTr f∂ f −1f∂ f −1f∂ f −1 , (3.7) 24π2 1 2 3 i j k where the θ’s parameterize S3 and f is a mapping from S3 → S3. A general rule for homotopy classes, which will turn out to be useful, comes as a consequence of Bott’s periodicity theorem [10]: The homotopy classes for unitary groups are given by ( 0, if k is even Πk(U(N)) = Πk(SU(N)) = , (3.8) Z, if k is odd

k+1 for k > 1 and N ≥ 2 .

3.2 Kinks and anti-kinks The following sections will use the same line of reasoning as Coleman’s book [16]. Understanding how we can use topology, and subsequently homotopy classes, to find finite energy or finite action solution in field theories is most easily done by exploring such solutions in a relatively easy field theory. An excellent start would be to consider the a scalar field theory in one space and one time dimension. The Lagrangian density of this theory is given by 1 L = ∂ φ∂µφ − V (φ), (3.9) 2 µ with 1 V (φ) = λ(φ2 − v2)2. (3.10) 2

The theory has a non zero vacuum expectation value v and thus it has multiple minima located at φ = ± √1 v. 2 As we will see, this theory there exist static stable non-trivial finite energy solutions to the equations of motion because of the multiple minima in the potential term. In order to obtain finite energy solutions we require φ to go to a zero of V (φ) when x → ±∞. This can be seen from the time-independent Hamiltonian of the theory Z ∞ 1 x H = ∂xφ∂ φ + V (φ) dx. (3.11) −∞ 2 If the potential term does not go to a minimum of V (φ) at x → ±∞ then the integral diverges. If the vacuum expectation value would be zero then a trivial finite-energy solution would be one where V (φ) = 0 for all x. However, having 2 discrete minima implies that there might be solutions for φ that attain one minimum at x = −∞ and the other minimum at x = ∞ or vice versa. To obtain such solution we can use the equations of motion (e.o.m.). The e.o.m. are found through the variational principle

Z 1 −δL = δ ∂ φ∂µφ + V (φ) dx = 0, (3.12) 2 µ so that the e.o.m. are given by

∂V (φ) ∂ ∂µφ + = 0. (3.13) µ ∂φ

13 One intuitive way of ﬁnding solution for the e.o.m. is by comparing the theory to a problem of a particle moving in a potential −V (x). The least action principle for such a problem is given by " # Z Z 1 dx2 δS = δ dtL = δ + V (x) dt = 0. (3.14) 2 dt

The equations of motion obtained through the Euler-Lagrange equations are mathematically equivalent to the those of the scalar field theory. The variables have been changed to φ → x and x → t. The potential −V (x) is sketched in figure 2. Obtaining finite energy solutions similarly requires the particle to go to zeros

Figure 2: Quartic potential with maxima ± √1 v 2 of −V (x) as t → ±∞. Here the trivial solutions are those where the particle remains at the top of the hills of the potential for all t. Non trivial solutions would be where the particle starts at one of the tops at t = −∞ and ends at the other at t = −∞. The potential energy of the particle is exactly converted in enough kinetic energy so that the particle is at rest at the top of the hill at t = ∞. Then from the energy conservation law we know that

1 dx2 − V (x) = 0. (3.15) 2 dt

Changing back to the scalar ﬁeld theory by x → φ and t → x we ﬁnd

1 dφ2 = V (φ). (3.16) 2 dx

The formal solution to this equation is

φ Z − 1 x = ± dφ0 [2V (φ0)] 2 . (3.17) φ0

In this integral the φ0 is an arbitrary parameter coming from the translational invariance of the e.o.m., i.e. there is no explicit position dependence x in equation (3.13). The value of φ0 is the value where φ crosses from negative to positive. The translational invariance implies that it doesn’t matter when this crossing happens, as long as φ begins and ends at φ = ± √1 v. The integral yields two solutions for φ 2 1 φ+ = √ v tanh(µx), 2 1 φ− = −√ v tanh(µx), (3.18) 2

14 often called respectively the kink and the anti-kink solution in literature. As a check, if we plug in these solutions into the Hamiltonian (3.11) we obtain an energy of 2 E = v2, (3.19) 3 and thus the kink solutions yield a ﬁnite energy as required. The next step is to check the stability of the time-independent ﬁnite energy solutions. Take the solution

φ(x, t) = f(x) + δ(x, t), (3.20)

where f(x) is the time-independent solution (one of the kink solutions (3.18) and δ(x, t) is a small perturbation. Substituting this into the equation of motion (3.13) yields 1 ∂ ∂µ(f + δ) + 2λ(f + δ)(f 2 + 2δf + δ2 − v2) = 0. (3.21) µ 2 Collecting only ﬁrst order perturbation terms we ﬁnd 1 ∂ ∂µδ + 2λ(3f 2 − v2)δ = 0, µ 2 ∂2V (f) ∂ ∂µδ + δ = 0. (3.22) µ ∂f 2 Since the time dependence only appears through derivatives the equation is invariant under time translations, i.e. if g(x, t) is a solution for δ(x, t) then so is g(x, t + c). This fact allows us to write the time component of the small perturbation as a superposition of normal modes

X iωnt δ(x, t) = Re ane Ψn(x), (3.23) n

where Ψn and ωn obey the equation

∂2V (f) −∂ ∂xΨ + Ψ = ω2 Ψ . (3.24) x n ∂f 2 n n n

The equation is a 1-dimensional Schrödingerequation. The solutions for φ(x, t) are stable if and only if no eigenvalues of the Schrödingerequation are negative, i.e. if the eigenfrequencies ωn are real. Recall that solutions of the e.o.m. (3.13) are invariant under spatial translations. From the invariance it follows that if f(x) is a solution to the equations of motion then f(x + c) is also a solution. The solution f(x) itself is not invariant under spatial translation; f(x) is different from f(x + c). We can remove this spatial translation ∂ non-invariance of f(x) by applying the generator of spatial translations ∂x to the e.o.m. to obtain ∂3 ∂ ∂V (f) − f(x) + = 0, (3.25) ∂x3 ∂x ∂f which can be rewritten to ∂2 ∂2V (f) ∂f − + = 0. (3.26) ∂x2 ∂f 2 ∂x

Comparing this equation to the Schrodinger (3.24) we see it is the schrodinger equation with eigenfunction ∂f ψ0 = ∂x and eigenvalue 0. If we look at the shape of the derivative of our kink solution in figure 3 we find that it has no nodes. This is a property of monotonic functions (always increasing/decreasing functions). It is well known that in the Schrödingerequation the eigenfunction with no nodes has the lowest eigenvalue. Therefore there are no eigenvalues lower than 0 and the solution is stable under small perturbations.

15 Figure 3: Derivative of kink solution

3.3 Topological conservation laws We have shown that there exist non-trivial finite-energy solutions in a (1 + 1) dimensional scalar field theory with muliple minima which are stable under small perturbations. We will now show how the solutions are stable in a topological sense. We have mentioned earlier that we can consider all static finite energy solutions as a field configuration space. We can further divide the space of static finite energy solutions in subspaces, where each type of solution is its own subspace. In total we have 4 subspaces corresponding to the 2 trivial solutions and the kink and anti-kink solution. Recall that finite energy solution are obtained only if φ(±∞, t) is a zero of V (φ). These solutions are a constant, φ = ± √1 v at x → ±∞ and so 2

∂0φ(±∞, t) = 0, (3.27) i.e. there is no way to go from one finite energy solution to another at spatial infinity, hence the subspaces are disconnected. The disconnected subspaces give us a topological conservation law. This line of argumentation holds independently of the equations of motions, it is just a consequence the disconnectedness of the subspaces of static finite energy solutions. One final thing to note is that the static finite energy fields φ are mappings from the set of infinities {−∞, ∞} in R1 to the set of minima of the potential {−v, v}. In higher dimensional theories we will encounter field configurations that can be classified by mappings as seen in section 3.1. Then we can classify the disconnected subspaces, if the theory has any, of the configuration by the winding numbers we discussed in section 3.1. At the end of this chapter we will give some examples of such theories. Before we give some examples we will generalize spontaneous symmetry breaking in such a way that we can easily identify the set of minima of the potential by just knowing the (broken) symmetries of the theory.

3.4 SSB and application of homotopy classes Consider a scalar ﬁeld gauge theory with gauge group G. A general form of the Lagrangian would be 1 L = − F a F aµν + D φ†Dµφ(x) − V (φ). (3.28) 4 µν µ

The ground state of the theory would be one where φ = φ0 is a constant so that the covariant derivatives disappear in the Hamiltonian and where φ0 is a minimum of V (φ), precisely as we have done before. We denote the set of minima φ0 of the potential by M and so ( ) ∂V (φ) M = φ0 : = 0 . (3.29) ∂φ φ=φ0

Since G is the gauge group of the theory, V (φ) is invariant when acted upon by an element g ∈ G. Now let H be a subgroup of G, where h ∈ H if and only if hφ0 = φ0, i.e. an element h ∈ H leaves the vacuum value φ0 invariant. Every φ0 thus has its own subgroup H that leaves the ground state invariant. Since an 0 element g leaves the potential V invariant then an element g acting on φ0 must return another φ0 ∈ M, so

16 0 gφ0 = φ0. Therefore we can identify the set M as the left coset of G denoted by G/H = gh : h ∈ H. This implies that if we know the gauge group G and the subgroup H that leaves the vacuum expectation value invariant then we can describe the minima of the theory by the left coset G/H. We can further use this result to see how Goldstone bosons appear. Recall that transformations U of a gauge group G that is a Lie a group can be put in exponential form by U = exp {iθaT }. Inﬁnitesimally this transforms the ﬁeld φ as

0 a 2 φ → φ = φ + iθaT φ + O(α ), (3.30)

a so that δφ = iθaT φ. The variation in the potential due to the transformation is given by ∂V δV = δφ, ∂φ ∂V = iθ T aφ = 0, (3.31) ∂φ a

and must be zero since U ∈ G leaves V invariant. If we take the derivative (3.31) at φ = φ0 we ﬁnd

2 ∂(δV ) ∂V a ∂ V a = iθaT + iθaT φ0 = 0. (3.32) ∂φ ∂φ ∂φ ∂φ φ=φ0 i j φ=φ0

The ﬁrst term on the right hand side is clearly zero since V (φ0) is a minimum of the potential. Then we are left with

2 ∂ V a T φ0 = 0. (3.33) ∂φ ∂φ i j φ=φ0 The second derivative term can be thought of as a mass matrix, this becomes evident when we take the taylor expansion of the potential around φ0

2 ∂V 1 ∂ V 2 V (φ) = V (φ0) + (φ − φ0) + (φ − φ0) + ... (3.34) ∂φ 2 ∂φ ∂φ φ=φ0 i j φ=φ0

1 2 2 Comparing the second derivative term to a mass term 2 m φ then it can interpreted as a mass matrix

2 ∂ V = Mij. (3.35) ∂φ ∂φ i j φ=φ0

i Generators T that leave the vacuum value φ0 invariant, and thus belong to the group H, satisfy equation i j i (3.33) since T φ0 = 0. But generators T that don’t leave the vacuum value φ0 invariant T φ0 6= 0 must have a zero value mass matrix. Concluding, we can identify a massless Goldstone boson for every generator T j of the broken gauge group G. The number of Goldstones bosons is equal to the number of generators of G minus the number of generators of H1. In the next section we will look at some examples of gauge theories where we can see the application of the theory discussed in this chapter.

3.5 Application Homotopy classes Consider the U(1) gauge theory discussed in section 2.2 with the potential term term given by

λ V (φ) = (φ∗φ − v2)2, (3.36) 2 1In Lorentz-noninvariant gauge theories this can be diﬀerent. In this thesis we will not encounter such gauge theories, for further reading on theories with an anomalous number of Goldstone Bosons one can read [39]

17 in (2 + 0) dimensions. Now if we want to find the finite energy solutions then we know from section 3.2 that we have to find mappings from the spatial infinities into the set of zeros of the theory. The set of infinities is clearly given by S1 since we are working in two space dimensions. The gauge group G is given by U(1) and after the SSB the unbroken symmetry group H is given by the trivial group. Therefore the left coset G/H is just U(1) and describes the set of zeros. The group U(1) describes a circle in the complex plane and can therefore topologically be seen as S1. Thus the finite energy solutions are given by the mappings from S1 → S1. As we have discussed in section 3.1 such mappings can be categorized by integer winding 1 numbers Π1(S ) = Z. In this case the winding number tracks how many times you wind around the set of zeros S1 (field configuration space) when you go around the circle of infinities once. Therefore the field configuration space can be split up in an infinite amount of disconnected subspaces, each categorized by an integer winding number. The trivial solutions are in the subspace with winding number 0 and non trivial solutions have non-zero integer numbers. Colemans book Aspects of Symmetry [16] offers more pedagogical examples: Consider the following theory that only has trivial solutions. Let G be SO(3) and the potential is given by λ V (φ) = (|φ|2 − v2)2, (3.37) 2 in (2 + 0) dimensions. The scalar fields are now 3-vectors and the set of zeros is given by |φ|2 = v2. If we take one of the solutions to be the true ground state we fix one axis of SO(3), so for example 0 φ = 0 , (3.38) v would be an eligible solution. The theory will remain invariant under a SO(2) subgroup of SO(3), which will be the unbroken gauge group H. Therefore the set of zeros G/H is given by is SO(3)/SO(2) =∼ S2. Thus the finite energy solutions will be mappings from the space of infinities which is S1, as in the last example, into the set of zeros S2. This mapping is a trivial one since you can continuously deform a circle on a sphere 2 to a point, therefore : Π1(S ) = I. So there is only one homotopy class: the trivial mapping. If we take the same example but in (3 + 0) dimensions then the set of infinities is a sphere S2. The 2 2 2 mapping would be S → S and is given by an integer winding number, Π2(S ) = Z. Hence the finite energy solutions can again be categorized in an infinite amount of disconnected subspaces categorized by a winding number.

Although we did not find the explicit form of the solutions in the examples above we can still learn a lot about a theory through only the topological properties. To further find the explicit form of the solutions for non-trivial solutions we can use the integral form of the winding number found in section 3.1. We have so far only considered the finite energy solutions of field theories. The way we obtained the classification through homotopy classes of the finite energy solutions was done in a similar manner for all the examples. The requirement was that the set of infinities can be seen as a n-sphere Sn so that we can create mappings Πn(G/H). It is therefore not possible for a (1 + 0) dimensional to use homotopy classes as a way to classify the solutions. In the next chapter we will discuss finite-action solutions called instantons. We can also use homotopy classes in a similar manner to classify these solutions. As we will see in the next chapter, the vacuum of the discussed theory can also be described using homotopy classes. In order to use homotopy classes for the vacuum structure we will use the process called compactification. Compactification, as the name suggests, makes a topological space a compact space. The general way to do this is by identification of the points at infinity. The simplest example to see how this works is to consider the real line, which contains all the points in R1. The space is not compact since the points go off to infinity in both the positive and negative direction. Now by identification of the points at infinity, i.e. by calling the points at infinity at both the negative and the positive end ∞ we compactify the space. This is because we can connect the end points of both the negative and positive end of the lines since they are both ∞. Topologically, if we attach the ends of a line to each other we obtain a circle S1. The same line of reasoning can be used for higher dimensions so n n n that through compactification we go from R to S . This S is then used to find the classification Πn(G).

18 4 Instantons

Instantons are finite-action field configurations in classical field theories in Euclidean space. They are also local minima of the Euclidean action, which means that they are solutions to the classical Euclidean equations of motion. The reasons we are interested in these kind of solutions is because in the semiclassical approximation, or small ~ approximation, the Euclidean path integral is dominated by these minima of the action. Semiclassical approximations are great in describing quantum mechanical effects. One prime example is that the tunneling amplitude of a particle through a barrier can be found using the semiclassical approximation. To see why we work in Euclidean space and how these configurations dominate the Euclidean path integral we first investigate the instanton solution that comes from tunneling in quantum mechanics, which is basically (0 + 1)D dimensional quantum field theory.

4.1 Instantons in quantum mechanics The Lagrangian of a particle with no spin and unit mass moving in a potential V (x) is given by

1 dx2 L = − V (x). (4.1) 2 dt From the path integral in quantum mechanics we know that the quantum mechanical amplitude of a particle starting at xi and ending at xf is given by Z iHt/ −iS/ hxf |e ~|xii = N [dx]e ~. (4.2)

iHt/ The left side contains the position eigenstates |xii and |xf i and the time evolution operator e ~. On the right hand side we have N as the normalization factor and S the action given by " # Z T/2 1 dx2 S = dt − V (x) . (4.3) −T/2 2 dt

The integral integrates over all possible paths that obey the boundary conditions xi = x(−T/2) and xf = x(T/2). If we now take the semiclassical approximation ~ → 0 then the integral will be dominated by the stationary points of the action. This is because the approximation causes the exponent on the right hand side to oscillate very rapidly. Integrals that exhibit this behaviour can be solved using the steepest descent method. As we know from classical mechanics, paths that yield a local minimum of the action, i.e. stationary paths are solution to the equations of motions and thus classical paths. In the steepest descent method we ﬁrst expand around the classical paths by X x(t) = xcl + cnxn(t), (4.4) n where the xn functions obey the boundary conditions xn(±T/2) = 0. Now imagine that we have a quartic potential as in ﬁgure 4.

Figure 4: Quartic potential with minima ±a

19 The potential has two minima at x = ±a and a local maximum at x = 0. If we want to calculate the quantum mechanical amplitude of the particle at x = −a, tunneling through the barrier and ending at x = a then we need to solve Z ha|eiHt/~| − ai = N [dx]e−iS/~. (4.5)

Since there are no classical paths that describe the tunneling we have no classical paths to expand around in equation (4.4). So how can we continue with the steepest descent method? The trick is to consider the coordinate transformation called the Wick rotation

t → −iτ. (4.6)

The Wick rotation transforms Minkowski space into Euclidean space since

2 2 2 2 2 ds = −(dt) + dxi → dτ + dxi (4.7)

The quantum mechanical amplitude is now given by Z ha|e−Hτ/~| − ai = N [dx]e−SE /~, (4.8)

2 where SE is the Euclidean action given by

0 " # Z T /2 1 dx2 SE = dτ m − − V (x) . (4.9) −T 0/2 2 dτ

The equation of motion in Euclidean space is

d2x = −V 0(x), (4.10) dτ 2 which is the equation of motion of a particle moving through a potential −V (x) with a total energy of

1 dx E = − V (x). (4.11) 2 dτ This should ring a bell, we’ve seen this before in the kink solutions of chapter 2. After the Wick rotation there are classical paths we can take in imaginary time, paths that move from the hill top x = −a to the hill top x = a. These paths are called instantons and diﬀer from the kink solutions in that they are structures in time. Concluding, by using a Wick rotation we went from Minkowski space to Euclidean space and we we’re able to identify paths that obey the Euclidean equations of motion, which in turn dominate the exponential term since we’re using a semiclassical approximation. Now that we’ve got a feel for how instanton solutions come by we can move on to the (3 + 1) dimensional case.

4.2 Instanton in quantum ﬁeld theory In order to calculate quantum mechanical amplitudes in QFT we need to solve n-point correlations functions. These functions can be solved by ﬁnding the generating functional of a theory. Given the pure gauge Lagrangian 1 L = − Tr[F F µν ], (4.12) 2 µν 2One should always check for poles in the integrand when doing a Wick rotations since their contribution can interfere with the validity of the Wick rotation and whether the contour integrals contribute. In this case there is no pole in the rotation.

20 then the Euclidean action is given by 1 Z S (A ) = d4xTr[F F µν ], (4.13) E µ 2 µν and the generating functional has the shape Z −SE (Aµ)/ Z = D[Aµ]e ~. (4.14)

The generating functional is very reminiscent of the partition function in statistical mechanics in the sense that integration is taken over all possible paths and the probability of each path occurring is weighted by the exponent of the Euclidean action of that path. We are interested in doing a semiclassical approximation and so, just as before, we want to find the minima of the action. However, the Euclidean action has a tendency to diverge when the Euclidean volume goes to infinity. Therefore it is more accurate to say that the finite action configurations dominate the path integral, since the minima of the action are definitely finite action configurations. To obtain finite action solutions we need Fµν to vanish at infinity,

Fµν −−−−→ 0. (4.15) |x|→∞ Naively we could say that this condition is met when the gauge fields are 0 at infinity. But they can also be a gauge transformation of 0 at infinity,

i −1 Aµ −−−−→ U∂µU , (4.16) |x|→∞ g also known as a pure gauge field. The set of infinities in Euclidean R4 space form a three-sphere S3 and thus the gauge transformations U are mappings from S3 into the gauge group. Here we only consider the gauge group to be SU(2), which will be sufficient for the purposes of this thesis.3 The SU(2)-manifold is equivalent to S3. This can be seen from the definition of the group

α −β SU(2) = : α, β ∈ , |α|2 + |β|2 = 1 . (4.17) β α C

If we write out the complex numbers as α = x1 + iy1 we ﬁnd that

2 2 2 2 x1 + y1 + x2 + y2 = 1, (4.18)

which is the equation for a three-sphere in Euclidean R4-space and so the U’s are a mapping from S3 → S3. As we have seen in chapter 2 these U’s can be put in homotopy classes categorized by a winding number ν. In this case the winding number is given by 1 Z ν = − dθ dθ dθ ijkTr U∂ U −1U∂ U −1U∂ U −1 . (4.19) 24π2 1 2 3 i j k

There is a nice mathematical trick which allows us express the winding number in terms of Fµν . Firstly we deﬁne the gauge-dependent current 2 Kµ = 4µνρσTr[A ∂ A − igA A A ], (4.20) ν ρ σ 3 ν ρ σ often referred to in literature as a Chern-Simons current. It will become clear shortly why we introduce this term. Then one can check that

µ µν ∂µK = 2Tr[Fµν F˜ ], (4.21)

3For gauge theories with SU(N) with N > 2 the homotopy classes are still given by an integer winding number due to Bott’s peridiodicity theorem (3.8). The general approach in this section will therefore be applicable to SU(N), diﬀering only via the structure constants of the theory.

21 where 1 F˜µν = µνρσF , (4.22) 2 ρσ

4 is the dual of Fµν . If we take the integral of our gauge-dependent current over the whole Euclidean R -space we can, according to Gauss’s theorem, write it as a surface integral Z Z 4 µ µ d x∂µK = dσµK . (4.23) S The surface integral is then over the S3 at infinity. At infinity we know that our gauge potential should be a pure gauge field (4.16). Substituting this in the current we find that 4 Kµ = µνρσTr U∂ U −1U∂ U −1U∂ U −1 . (4.24) 3g2 ν ρ σ This term should look familiar. Plugging this into the integral at equation (4.23) we will find it to be very similar to the expression for the winding number ν (4.23). Z 4 Z dσ Kµ = dσ µνρσTr U∂ U −1U∂ U −1U∂ U −1 , (4.25) µ 3g2 µ ν ρ σ S S 4 Z = dθ dθ dθ ijkTr U∂ U −1U∂ U −1U∂ U −1 , (4.26) 3g2 1 2 3 i j k 32π2ν = − , (4.27) g2 and so with (4.21) we can deduce that

Z 16π2ν Tr[F F˜µν ]d4x = − . (4.28) µν g2 To ﬁnd instanton solutions we make use of the positivity condition in Euclidean space

Z 2 4 Tr Fµν ± F˜µν d x ≥ 0. (4.29)

We can rewrite the integrand in (4.29) as

2 Fµν ± F˜µν = 2 Fµν Fµν ± Fµν F˜µν , (4.30) so that we have the inequality

Z Z 2 µν 4 ˜µν 4 16π |ν| Tr Fµν F d x ≥ Tr Fµν F d x = . (4.31) g2 Thus we can conclude that the action is bound by

1 Z 8π2|ν| S = F F µν d4x ≥ . (4.32) E 2 µν g2 This bound is in the literature often referred to as a Bogomol’nyi bound. The bound implies from (4.29) that the action is minimized when

Fµν = ±F˜µν . (4.33)

22 And now remains the final task of finding gauge fields that satisfy this relation. Belavin et al. [7] were the first to construct such a non trivial solution for ν = 1 which is given by 2 1 Aa = ηa x , (4.34) µ g |x|2 + λ2 µν ν

a where λ is an arbitrary parameter often called the instanton size and ηµν is the ’t Hooft symbol with the following properties

a aij a ai a aj a ηij = , ηi4 = δ , η4j = −δ , η44 = 0, (4.35) where we use the convention that the 4th index is the time component When contracted with the Pauli matrices we can more explicitly write the instanton solution as

1 (−τ x + x × τ ) 1 τ · x A(x) = 4 ,A (x) = − . (4.36) g |x|2 + λ2 4 g |x|2 + λ2

Now that we’ve constructed an instanton solution we will look into the vacuum structure of a pure gauge theory. As we will see the vacuum structure of such a theory can be described by winding numbers as well.

4.3 Winding vacuum number ’t Hooft first came with the idea that the physical interpretation of the instanton solution was the tunneling between vacuum states with different topological/winding numbers [48]. From the path integral formulation we know that the vacuum to vacuum transition amplitude is given by the sum over all possible paths between the initial and final vacuum. The amplitude of each individual path is weighted by the exponent of the action. Given the Euclidean action 1 Z S = d4xTr[F F µν ] (4.37) E 2 µν we will find out that the vacuum structure of a non-Abelian gauge theory is highly non-trivial. We want the vacuum state to have the lowest energy. The Hamiltonian of this gauge theory at any time x4 is given by 1 Z H = d3x[F (x)]2 (4.38) 2 µν we can easily see that the lowest energy value is obtained when

Fµν (x) = 0 (4.39)

This means, as we have seen before, that all pure gauge ﬁelds produce a zero ﬁeld, i A (x) = U(x)∂ U −1(x) (4.40) µ g µ

where the functions U(x) are continuous functions from R4 to the gauge group G. In the path integral formulation we must sum over all physically distinct configurations Aµ(x) that connect the initial vacuum state with the final. Here the emphasis lies on physically distinct configurations since a gauge transformation of a physical state yields a physically equivalent state. To remove this gauge redundancy to prevent overcounting physically equivalent configurations we impose a gauge fixing condition. In principle it is arbitrary which gauge fixing condition we impose as long as we remove the gauge redundancy. In this case we will work in the temporal gauge

A4(x) = 0 (4.41)

23 The temporal gauge is a partial gauge fixing condition, we still have a remaining gauge freedom since time- independent gauge transformations yield 0 for the temporal gauge i A (x) = U(x)∂ U −1(x) = 0 (4.42) 4 g 4 Therefore the vacuum will be described by the time independent configuration i A (x) = A (x) = U(x)∂ U −1(x) (4.43) i i g i Imposing the temporal gauge has now made the functions U(x) mappings from R3 into the gauge group G. To classify these functions we make use of the remaining gauge freedom. We impose that at spatial infinities

Ai(|x|) |x|→∞ = 0 (4.44) by choosing

U(|x|) |x|→∞ = 1 (4.45) at spatial infinity (more generally U(x) can be chosen to be a constant at infinity). The Euclidean R3 space with infinity identified is equivalent to S3, so we have compactified R3. By imposing this final gauge fixing condition the functions U(x) have become mappings from S3 to the gauge group G. For the purposes of this thesis it will be sufficient if we take the gauge group to be SU(2). As we know these functions can be put in homotopy classes categorized by winding numbers. To prevent confusion we will label the winding numbers of the vacua with roman letters and the instanton configurations with greek letters. The general configuration of the vacuum states can be found using the following functions for U(x) in equation (4.43) " # iπx · σ Un(x) ≡ exp n (4.46) p|x|2 + λ2

The functions Un(x) that are in the same homotopy class are all related through so called small gauge transformations; every function U 0(x) can be obtained from another function U(x), given that it is in the same homotopy class as U(x), by a small gauge transformation. A small gauge transformation V is defined to be continuously obtainable from the identity gauge transformation V = 1. Thus all small gauge transformation are in the trivial homotopy class. Such a transformation transforms the vacuum as i A0 (x) = V (x)A (x)V −1(x) + V (x)∂ V −1(x) (4.47) µ µ g µ i i = V (x)U(x)∂ U −1(x)V −1(x) + V (x)∂ V −1(x) (4.48) g µ g µ i = V (x)U(x)∂ [V (x)U(x)]−1 (4.49) g µ or alternatively U 0(x) = V (x)U(x) (4.50) Applying a small gauge transformation on a vacuum state will yield another vacuum state with the same winding number n. Since there are an infinite number of small gauge transformations there are an infinite amount of vacuum configurations with winding number n that are physically equivalent. We call gauge transformations that change the winding number of vacuum state into a vacuum state with a different winding number large gauge transformations. A large gauge transformation is therefore not continuously obtainable from the identity gauge transformation. Large transformations transform between physically distinct configurations. We can conclude that the vacuum has a very rich non-trivial structure. It consists of a countably infinite amount of vacuum configurations in different homotopy classes characterized by a winding number that cannot be transformed into each other through small gauge transformations.

24 4.4 Relation between tunneling of vacua and instanton The following two sections will lean on the explanation given in the book by Cheng & Li[13]. Both the vacua and the instanton solution are in homotopy classes characterized by a winding number. At the beginning of the previous section it was mentioned that the instanton solution connects vacuum states with a diﬀerent winding number such that:

inst Ai (x, x4 = −∞) = Ai(x) of vacuum state |ni (4.51) inst Ai (x, x4 = ∞) = Ai(x) of vacuum state |n + νi (4.52) The n is the winding number of the vacuum and the ν is the winding winding number of the instanton. Recall that the instanton solution with winding number ν = 1 is given by 1 τ · x 1 (−τ x + x × τ ) A (x) = − , A(x) = 4 (4.53) 4 g |x|2 + λ2 g |x|2 + λ2

In the previous section we were working in the temporal gauge A4(x) = 0. To transform to this gauge we use a gauge transformation U(x), i A0 (x) = U(x)A (x)U −1(x) + U(x)∂ U −1(x) µ µ g µ

0 where now we want A4(x) = 0. This implies that i U(x)∂ U −1(x) = −U(x)A (x)U −1(x) g 4 4 g ∂ U −1(x) = − A (x)U −1(x) 4 i 4 −1 iτ · x −1 ∂4U (x) = − 2 2 2 U (x) (4.54) x4 + x + λ Solving for U −1(x) yields −1 −iτ · x x4 U (x) = exp √ arctan √ + θ0 (4.55) x2 + λ2 x2 + λ2 where θ0 is the integration constant. The integration constant can be found by considering the boundary conditions at x4 → ∞. At this boundary we should have i A0 (x) = U(x)∂ U −1(x),A0 (x) = 0 (4.56) i g i 4 By direct calculation we ﬁnd that 1 θ = (n + )π (4.57) 0 2 is the right answer, because then we ﬁnd

iπτ · x U(x4 = −∞) = exp √ n (4.58) x2 + λ2 iπτ · x U(x4 = ∞) = exp √ (n + 1) (4.59) x2 + λ2

and so the instanton solution of (4.53) does indeed connect two vacuum states: at x4 = −∞ the conﬁguration attains the form of a vacuum state with winding number n and at x4 = ∞ the conﬁguration attains the

25 form of a vacuum state with winding number n + 1. The fact that the instanton mediates tunneling implies that in the calculation of the quantum mechanical amplitude of vacuum-vacuum transition should include an instanton term so that (in Minkowski space) we have Z −iHt −i R d4x(L+JI(x)) hm|e |niJ = [dA]ν=n−me (4.60) where J is the source term for the instanton I(x).

4.5 Theta vacuum In the chapter 3.3 we concluded that there are inﬁnitely degenerate vacuum states |ni categorized by a winding number n. This degeneracy of vacuum states prevents us from picking a suitable true vacuum state. A large gauge transformation Tm with winding number m can transform a vacuum state

X −inθ X −i(n0−m)θ 0 imθ Tm |θi = e |n + mi = e |n i = e |θi (4.63) n n0

Thus the superposition of winding number vacuum states is an eigenstate of large gauge transformation Tµ as is required for a true vacuum state. Furthermore the |θi states should be orthogonal so that 0 −iHt 0 hθ |e |θiJ = δ(θ − θ )IJ (θ) (4.64)

IJ (θ) can be determined by ﬁrst writing out the left hand side as

0 −iHtu X imθ0 −inθ −iHt hθ |e |θiJ = e e hm|e |niJ (4.65) m,n Z X −i(n−m)θ im(θ0−θ) −i R d4x(L+JI(x)) = e e [dA]ν=n−me (4.66) m,n

Now we can solve for IJ (θ) and making the substitution ν = n − m we find Z X −iνθ −i R d4x(L+JI(x)) IJ (θ) = e [dA]ν=n−me (4.67) ν We can now substitute the integral expression for the winding number ν (4.28) into this equation to find X Z Z θg2 I (θ) = e−iνθ [dA] exp −i d4x(L − Tr[F F˜µν ] + JI(x)) (4.68) J ν=n−m 16π2 µν ν We may then combine the Lagrangian and the theta term into an effective Lagrangian given by θg2 L = L − Tr[F F˜µν ] (4.69) eff 16π2 µν Now one might wonder why we did not include this term in the pure gauge Lagrangian from the beginning since it has the right dimension and it is gauge-invariant. The reason is that the term can be expressed as a divergence of a current (4.23) and it can therefore be expressed as a surface term in the action integral and since variations are assumed to vanish at the boundary we can leave this term out of the action, but with this space and gauge group we can construct an instanton configuration that does not vanish at the boundary and therefore the term survives. The term does not conserve parity P but conserves charge conjugation C and thus it violates CP conservation.

26 4.6 The ABJ anomaly and instantons So far we have been discussing the instanton solutions in a pure gauge Lagrangian. Eventually we would like to investigate what happens to fermions in an instanton background, but to do this we need to learn a little bit about the Adler-Bell-Jackiw anomaly in quantum field theory. The anomaly comes from the non-conservation of the current of the chiral symmetry. There are different ways to encounter this anomaly and here we choose to find it through the functional integral. Given the Lagrangian comprising of a single fermion field that is gauge invariant under a gauge group SU(N) 1 L = ψ(iγµD − m)ψ − Tr[F F µν ] (4.70) µ 2 µν Under a (global) chiral transformation the fermionic field transforms as

5 ψ → eiαγ ψ (4.71)

5 † 5 ψ → eiαγ ψ γ0 = ψeiαγ (4.72) where in the last line the anti-commutation relation of gamma matrices {γ5, γµ} = 0 was used. If we fill in these fields into the Lagrangian and using the anti-commutation relation again we find that the Lagrangian transforms as

5 1 L → ψ(iγµD − e2iαγ m)ψ − Tr[F F µν ] (4.73) µ 2 µν Thus we conclude that the Lagrangian is symmetric under the chiral transformations if the fermions are massless. Then, in the massless case, Noether’s theorem states that for every symmetry in the Lagrangian there is a conserved current jµ. We can then construct the Noether current for this chiral symmetry as follows ∂L ∂L αjµ = δψ + δψ (4.74) ∂(∂µψ) ∂(∂µψ)

5 The inﬁnitesimal transformation of (4.72) is eiαγ ψ ∼ (1 + iαγ5)ψ and thus δψ ∼ iγ5αψ and so ﬁlling in jµ yields the conserved current

µ µ 5 j5 = −ψγ γ ψ (4.75) Now the chiral right- and left-handed components of a spinor are found through the chiral projection operator PR,L so that 1 ± γ5 ψ = P ψ = ψ (4.76) R,L R,L 2 Using the chiral notation the zeroth component of the current can be written as

0 † † j5 = ψLψL − ψRψR (4.77) Thus the zeroth component is the diﬀerence in number density between chiral right-handed and chiral left- handed particles. The corresponding charge can be found through Z 5 3 0 Q ≡ d xj5 (4.78)

Since we integrate over space the charge gives the diﬀerence between the number of chiral right-handed and chiral left-handed particles. Now we are going to explore the same chiral transformation in quantum ﬁeld theory. In the path integral formalism the functional integral for this theory is given by Z Z 4 Z = DψDψDAµ exp i d xL (4.79)

27 The functional integral should be stationary under a coordinate transformation. By applying the same transformation (4.72) on the functional integral Fujikawa showed that under the chiral transformation the measure of the functional integral transforms as [23]

Z α(x)g2 DψDψ → exp −i d4x Tr[F F˜µν ] DψDψ (4.80) 8π2 µν

If we now combine the result of the transformation of the measure and the transformation of the Lagrangian density in (4.73) we ﬁnd that the functional integral transforms as

Z Z α(x)g2 Z → DψDψDA exp i d4x L + α(x)∂ jµ − Tr[F F˜µν ] (4.81) µ µ 5 8π2 µν

The next step is to vary the functional integral with respect to α(x) and we obtain

Z 2 δZ µ g ˜µν i R d4xL = DψDψDAµi(∂µj5 − 2 Tr[Fµν F ])e = 0 (4.82) δα(x) α=0 8π This should be equal to zero since the functional integral should be stationary under a coordinate transformation. This condition is met when g2 ∂ jµ = Tr[F F˜µν ] (4.83) µ 5 8π2 µν This result is known as the ABJ anomaly. If we integrate both sides over the whole space we have

Z Z g2 d4x∂ jµ = d4x Tr[F F˜µν ] (4.84) µ 5 8π2 µν Now if we rewrite the left hand side of the equation to Z Z Z 4 µ 3 0 i d x∂µj5 = dx0 d x(∂0j5 − ∂ij5) (4.85) Z Z Z 5 3 i = dx0∂0Q − dx0 d x∂ij5 (4.86)

Here we have used the fact that we work in Minkowski space and (4.78). The last terms in (4.85) vanishes because we can use Gauss theorem to write the volume integral of the divergence of a current as a surface integral of the flux of the current through the surface of the volume. Under the assumption that at spatial infinities this flux is zero the term disappears so we are left with

Z Z g2 dx ∂ Q5 = d4x Tr[F F˜µν ] (4.87) 0 0 8π2 µν Z g2 Q5(t = ∞) − Q5(t = −∞) = ∆Q5 = d4x Tr[F F˜µν ] (4.88) 8π2 µν The left hand side is now the change in the final and initial difference in right- and left-handed particles. The right hand side at this point must be very familiar and we can use (4.19) to find that under a background instanton configuration with winding number ν the difference is given by

∆Q5 = 2ν (4.89)

This result implies that in the case of massless fermions instantons convert left handed particles into right handed particles or vice versa for anti-instantons. This result can be promoted to multiple fermion ﬂavours

28 by multiplying by Nf on the right hand side. In 1976 ’t Hooft [48] computed that after integrating out the gauge ﬁelds the vacuum transition amplitude with fermions in an instanton background is proportional to Z ¯ ¯ ¯ κ Dψ Dψ exp S0,ψ + Jψψ det(ψRψL) (4.90)

Then one can ﬁnd an eﬀective Lagrangian term that mimics the resulting amplitude

2 Leff = Cg−8e−8π/g Lf + h.c. (4.91)

f where h.c. stands for hermitian conjugate (for anti instantons) and L is a 2Nf fermionic interaction term which couples left handed to right handed particles. The resulting vertex is shown in ﬁgure 5

Figure 5: ’t Hooft vertex [18]

and is often referred to as the ’t Hooft vertex. So the quantum mechanical amplitude scales with the 2 exponent of the classical action of the instanton solution e−8π/g .

4.7 Strong CP problem and small θ In the previous section we have considered the instanton eﬀect on massless fermions. Going back to the result of considering the theta vacuum, we had to add a term to the Lagrangian (4.69), which violates CP symmetry. This term is very similar to the term (4.80) and we can therefore rotate the theta vacuum term θ away if we apply the chiral rotation α = − 2 . This implies that the theta term is unphysical since it can be rotated away by a gauge transformation. But if we turn on the quark masses then the application of the chiral rotation results in a complex mass term which has a phase of θ, which violates violates CP-symmetry

mψψ¯ → meiθψψ¯ (4.92) or using the chiral notation

¯ ¯ ∗ ¯ mψψ → mψLψR + m ψRψL (4.93) with, in this case, m = |m|e−iθ/2. Thus we can no longer remove the θ term from the Lagrangian. In QCD, where the gauge group is SU(3), this is a problem since the strong interaction seemingly conserves CP-symmetry, although there is no theoretical proof that it should. The CP violating term also introduces

29 an electric dipole moment for the neutron [5]. This allows us to set an upper limit for the θ term inferred from the upper limit of the neutron’s electric dipole moment[4]. The resulting upper limit is |θ| < 10−10. A proposed solution to deal with the CP violating term was that in the case of multiple fermion flavours one can introduce flavoured chiral rotations. This allows one to choose only one of the quark flavours to have the CP violating term. If the mass of this quark vanishes then so would the CP violating term and CP symmetry will be restored. The most natural choice for the quark mass that would vanish is the up quark since it is the lightest quark. However, using the ’t hooft vertex one can see that using multiple flavours generate an effective mass for the up quark which is dependent on the renormalization scheme

Figure 6: Joined internal lines of heavier quarks yield an eﬀective mass of the upquark [17]

By joining the lines of heavier quark masses, as shown in fig 6, these internal lines create an effective mass term for the up quark through their propagators given by [17] mdms µeff ∼ (4.94) Λqcd where Λqcd is an integration constant coming from the renormalization scheme and sets the overall energy scale. This solution therefore does not work since the quark could be zero at one energy scale but attains a value in another scheme. The solution to the problem should be scheme independent. Another proposed solution is the existence of so called axions in the Peccei-Quinn Mechanism. The idea of this mechanism is that the θ parameter is promoted to a dynamical field, where excitations of the field represent a new particle called the axion. The axion is also a good candidate for dark matter and therefore a subject of active research. The Peccei-Quinn Mechanism is beyond the scope of this thesis, but interested readers are referred to [40].

4.8 B+L Anomaly in electroweak theory So far we have investigated the global chiral anomaly in SU(N) gauge theories and how instantons play a µν role in ﬂipping the handedness of fermions by having a nonzero contribution in the Tr[Fµν F˜ term. In the Electroweak theory the global U(1) symmetry has an anomalous current given by µ X ¯i µ i jl = ψl γ ψl (4.95) i µ X ¯i µ i jB = ψBγ ψB (4.96) i

30 where the summation goes over the number of generations of fermions (i = 1, 2, 3). The ABJ anomaly relates to the currents as N g2 jµ = jµ = f Tr[F F˜µν ] (4.97) l B 16π2 µν Then, using the same algebra as in section 4.6, we can ﬁnd the relation

∆B = ∆l = Nf ν (4.98)

Therefore when a vacuum transition occurs ∆(B −L) is conserved while ∆(B + L) = 2νNf . The implication of this anomaly is that a transition between vacuum states can lead to an asymmetry in the baryon number. This process could possibly explain baryogenesis in the early universe, i.e. the observed discrepancy between matter and antimatter which we will discuss in more detail in the chapter 6. The calculation of the quantum 2 2 mechanical amplitude of this process to occur based on ’t Hooft’s paper [49] leads to a prefactor of e−16π /g . The electroweak theory has a coupling factor g ∼ 0.65 and thus the prefactor is of the order ∼ 10−163. Therefore the probability of such a reaction to occur is heavily suppressed. Consequently it seems that the instanton process would be unobservable in experiments which will be discussed in more detail in chapter 5. There is also another process which causes a transition between vacuum states, which is called the sphaleron process. The sphaleron process is a process where instead of tunneling through the barrier the system goes over the barrier. The sphaleron itself is the field configuration of the Higgs and gauge fields transitioning between two topologically distinct vacuum configurations. The energy of this configuration is the energy of the potential barrier between topologically distinct vacuum states. In the next chapter we will discuss the construction of the sphaleron.

31 5 Sphalerons

The sphaleron solution is a static unstable finite-energy solution of the field equations. In the electroweak theory these solutions can be found by considering non-contractible loops in field configuration space. The sphaleron solution then belongs to the point at which the non-contractible loop attains maximal energy. The energy of the sphaleron configuration can then be regarded as the height of the potential barrier between two physically equivalent vacuum states with different winding numbers. A transition between such vacuum states leads to breaking of conservation laws as we have seen with the instantons, but they are suppressed in the electroweak theory because of the weak coupling constant of the theory. However transitions between the vacuum states might be possible by overcoming the barrier through high energy collision experiments. Therefore the determination of the sphaleron solution and consequently the height of the potential barrier gives us an indication at which energies we can expect these transitions to happen. The sphaleron could also have played a bigger role in the early universe and is therefore a candidate to explain the baryon asymmetry of the universe. Before we construct the sphaleron solution in the electroweak theory it is convenient to first construct such a solution in a simple theory.

5.1 Sphaleron solution in scalar ﬁeld theory In this section we will evaluate a simple gauge theory which has a sphaleron solution. This example will follow along the same lines as was given in Manton’s book Topological Solitons[37]. Consider the gauge theory with gauge group U(1) and the Lagrangian

µν † µ L = fµν f + (Dµφ) D φ − V (φ) (5.1) with complex scalar ﬁeld φ and

fµν = ∂µaν − ∂ν aµ

Dµφ = ∂µφ − iaµφ 1 V (φ) = λ(φ†φ − v2)2 (5.2) 2 The equations of motion are given by 1 D Dµφ + 2λ(φ†φ − v2)φ = 0 µ 2 µν † ν ν † 2∂µf − i(φ D φ − (D φ) φ) = 0 (5.3)

We are interested in ﬁnding static ﬁnite energy solutions in this theory. The static potential energy is given by Z † 1 E = ((D1φ) D φ + V (φ)) dx (5.4)

In the gauge a1 = 0 we can find the static field equation 1 ∂ ∂1φ + λ(φ†φ − v2)φ = 0 (5.5) 1 2 This equation is very similar to the equation of motion in section 3.2 and the static finite energy solutions are therefore given by the kink equations (3.18). Last time we discussed the stability of the solution by perturbing the system and we found that it was stable. This is not necessarily the case here since we are now working with a complex scalar field. The kink solution is actually a sphaleron solution in this theory and thus unstable. An intuitive way to see that the kink solution can be interpreted as a sphaleron solution is as follows: Consider the trivial static finite energy solution given by φ = √1 v, a = 0 for all values of x. 2 1

32 In the complex scalar gauge theory we also have a non trivial solution φ(x) = √1 veiα(x), a = ∂α where 2 1 α attains the value of 0 at x = −∞ and 2π at x = ∞. The trivial and non-trivial vacuum configurations are related by a gauge transformation with winding number 1 and both have zero energy. Consider now the path that connects these two configurations through a gauge transformation eiα(x). This rotation is smooth and thus after a rotation of eiπ the path will come across the kink solution. After another rotation by eiπ the configuration will become the non-trivial vacuum configuration. The total path starts and ends at the vacuum starts passing through the kink solution at the midpoint between the two vacuum configurations where it attains maximal energy. From the kink solution we can wind back around the complex plane in two directions to a configuration which has the same endpoints at both the spatial infinities as depicted in figure 7. Since the kink solution is the midpoint between the vacuum state with winding number ν = 0 and 1 ν = 1 we can assign it the fractional winding number ν = 2 .

Figure 7: The path that connects the trivial vacuum configuration and the non-trivial vacuum configuration passes the kink configuration (left) at the midpoint. We can unwind from the kink solution back to a configuration that has the same value at spatial infinities (right).

A helpful analogy is a rotating pendulum. If we rotate a pendulum from θ = 0 to θ = 2π (which are the analogons of the vacuum conﬁgurations) then it will certainly cross the point θ = π which is clearly unstable. The beginning and the ending point of the pendulum are physically equivalent, just like the trivial and unit winding number vacuum state are physically equivalent, i.e. they are related through a gauge transformation and both yield zero energy. In this way we can imagine the path connecting the two physically equivalent vacuum solution as a non-contractible loop, which it has a winding number associated with it and passes the unstable sphaleron conﬁguration.

5.2 Higgs vacuum manifold In chapter 1.2 we went over the structure of the electroweak theory and mentioned that spontaneous symmetry breaking break 3 out of the 4 symmetries. Recall that the subgroup that left the vacuum value invariant after SSB was a linear combination of a U(1) subgroup of SU(2)L and U(1)Y , namely

τ 1 exp i( 3 α(x) + α(x)) hΦi = hΦi (5.6) 2 2

and the quantum numbers associated with the symmetry group satisfy

Q = T 3 + Y (5.7)

where Q is the electromagnetic charge, T 3 is the weak isospin and Y is the hypercharge. Therefore the transformation that leaves the vacuum expectation value invariant (5.6) in conjunction with the relation (5.7) imply that the unbroken gauge group is given by U(1)Q, which is the gauge group of electromagnetism, often written as U(1)EM. Elements of this symmetry group can be generated with exp iαQ. Now from chapter 2.4 we learned that we can describe the vacuum manifold of a theory by the left coset G/H of the symmetry group of the Lagrangian. Here we have the symmetry group G is SU(2)L ×U(1) and the unbroken ∼ ∼ 3 gauge group H is U(1)EM. Therefore the vacuum manifold is described by SU(2)×U(1)/U(1) = SU(2) = S .

33 5.3 Construction of the SU(2) Sphaleron In this section we will construct the SU(2) sphaleron of the electroweak theory based on the original paper by Manton [36]. To find static (unstable) finite-energy solutions in the electroweak theory we first need to find the constraints for which the energy is finite. We then construct a suitable ansatz for the field configurations of Aµ and Φ which adhere to these constraints. It is convenient to first work out the solution for the sphaleron with only the SU(2) gauge field. This is realized by working in the limit that the weak mixing angle vanishes so that the U(1) gauge fields aµ may be set to 0. The influence of the U(1) gauge fields will be discussed later on. The Lagrangian of the electroweak theory is then given by 1 L = − Tr[F F µν ] + (D Φ)†(DµΦ) − V (Φ), (5.8) 2 µν µ where

a a a abc b c Fµν = ∂µAν − ∂ν Aµ + g AµAν , (5.9) 1 D Φ = ∂ Φ − igτ aAa Φ, (5.10) µ µ 2 µ and

1 2 V (Φ) = λ Φ†Φ − v2 . (5.11) 2

The (static) energy functional of the electroweak theory is given by

Z 1 E = d3x Tr[F F ij] + (D Φ)†(DiΦ) + V (Φ) (5.12) 2 ij i

For the energy to be finite the Higgs field must asymptotically go to a minimum of V (Φ). In spherical coordinates this means that at r → ∞ we can construct a limiting field

Φ∞(θ, φ) = lim Φ(r, θ, φ) (5.13) r→∞ which is a smooth function of θ and φ and satisﬁes the condition

|Φ∞|2(θ, φ) = (5.14)

Since the limiting ﬁeld is spherically symmetric we can choose the ﬁxing condition

1 0 Φ∞(θ = 0) = √ v (5.15) 2 1 The limiting field is a spherical shell at spatial infinity descried by two angles and is therefore topologically S2. If we compare the limiting field to the vacuum configuration of the Higgs field which was given by

v 0 Φvac(x) = √ (5.16) 2 1 then it becomes clear that the limiting field can be regarded as a map from the spatial infinities S2 in to the vacuum manifold of the Higgs field, which is topologically equivalent to S3, thus Φ∞ : S2 → S3. 3 Since Π2(S ) = I we can contract this map to a single point so there are no non-trivial static finite energy configurations. The next step is to construct non-contractible loops in the field configuration space. The construction of the non-contractible loops is done by the introduction of a new parameter µ ∈ [0, π]. The parameter µ attains a different value for every map Φ∞ starting and ending at the vacuum configuration. The Manton construction uses the parameterization as sketched in figure 8. In this parameterization the

34 Figure 8: The parameterization of the Manton construction[36]

asymptotic Higgs ﬁeld is given by

v sin µ sin θeiφ Φ∞(µ, θ, φ) = √ (5.17) 2 e−iµ(cos µ + i sin µ cos θ)

In this parameterization the family of maps Φ∞(µ), i.e. S2 × S1 → S3, is topologically equivalent to a single 3 3 ∞ 3 map Ψ : Sdom → SHiggs and as such Π1(Φ ) = Π3(S ) = Z. Manton’s proof [36] that they are equivalent is as follows: Consider the function

p(µ, θ, φ) = (sin µ sin θ cos φ, sin µ sin θ sin φ, sin2 µ cos θ + cos2 µ, sin µ cos µ(cos θ − 1)). (5.18)

3 This function assigns every combination of (µ, θ, φ) to a point on p(µ, θ, φ) Sdom. The function p(µ, θ, φ) has the following properties: (i) It is continuous. (ii) The angles θ and φ can be regarded as spherical polar coordinates since p(µ, θ, φ) = (µ, θ, φ + 2π) and p is independent of φ when θ = 0, π. (iii) For all µ ∈ [0, π] p(µ, θ = 0, φ) = (0, 0, 1, 0). (iv) For all θ ∈ [0, 2π] p(µ = 0, π, θ = 0, φ) = (0, 0, 1, 0). (v) Each point p on 3 Sdom is obtained atleast once for a combination of (µ, θ, φ) and if p is not (0, 0, 1, 0) then for every value of µ = (0, π) we obtain a unique point (θ, φ) of S2. Through these properties we can deﬁne the single map Ψ as

∞ Ψ(p) = ΦRe(µ(p), θ(p), φ(p)), (5.19) where   ReΦ1 ∞ ImΦ1 ΦRe =   . (5.20) ReΦ2 ImΦ2 and therefore, through this parameterization, the single map is equivalent to the loop of maps Φ∞. Thus we have a non-contractible loop in the field configuration space. The non-contractible loop passes at the 1 midpoint µ = 2 the sphaleron configuration analogous to the kink in section 4.1. Now that we have an expression for the Higgs field at infinity we move on to find an expression for the gauge fields. Before we find an expression for the gauge fields we first remove the gauge redundancy by imposing the polar gauge fixing condition Ar = 0. With this gauge fixing condition the energy functional of (5.12) will become Z h 1 1 1 E = Tr(∂rAθ∂rAθ) + Tr(∂rAφ∂rAφ) + Tr(FθφFθφ) r2 r2 sin2 θ r4 sin2 θ † † 1 † 1 † i 2 + (∂rΦ) ∂rΦ + (DθΦ) DθΦ + (DφΦ) DφΦ + V (φ) r sin θ dr dθ dφ (5.21) r2 r2 sin2 θ

35 in spherical coordinates. From (5.21) it is clear that the derivative terms should disappear at infinity in order to obtain finite energy solutions. Therefore we need to find gauge fields at infinity that satisfy

∞ ∞ DθΦ = 0,DφΦ = 0 (5.22)

Suitable expressions for the gauge ﬁelds at inﬁnity are i i A∞ = − ∂ U ∞(U ∞)−1,A∞ = − ∂ U ∞(U ∞)−1 (5.23) θ g θ φ g φ where √ ∞∗ ∞ ∞ 2 Φ2 Φ1 U = ∞∗ ∞ v −Φ1 Φ2 eiµ(cos µ − i sin µ cos θ) sin µ sin θeiφ = (5.24) − sin µ sin θe−iφ e−iµ(cos µ + i sin µ cos θ) so that v 0 Φ∞ = √ U ∞ (5.25) 2 1

This property ensures that (5.22) is satisﬁed. To see this we use that the covariant derivative of the electroweak theory is given by

DµΦ = ∂µΦ − igAµΦ (5.26)

So that if we fill this in in equation (5.23) and substitute the expression for the gauge fields (5.24) we obtain ∞ ∞ ∞ −1 ∞ ∞ v ∞ 0 ∂θ,φΦ − ∂θ,φU (U ) Φ = ∂θ,φΦ − √ ∂θ,φU 2 1 ∞ ∞ = ∂θ,φΦ − ∂θ,φΦ = 0 (5.27) where the property (5.25) was used twice. We now have a proper construction for the Higgs field and the gauge fields at infinity to obtain finite energy solutions. Consider now the following ansatz for the fields throughout R3 v 0 v Φ(r, θ, φ, µ) = (1 − h(r))√ + √ h(r)Φ∞(θ, φ, µ) 2 e−iµ cos µ 2 ∞ Aθ(r, θ, φ, µ) = f(r)Aθ (θ, φ, µ) ∞ Aφ(r, θ, φ, µ) = f(r)Aφ (θ, φ, µ)

Ar(r, θ, φ, µ) = 0 (5.28)

The functions f(r) and h(r) have the following behavior at spatial inﬁnity

lim h(r) = 1, lim f(r) = 1 (5.29) r→∞ r→∞ This way we obtain our finite-energy field configurations at spatial infinity. Also to ensure smoothness at the origin the functions must obey 1 lim h(r) = 0, lim f(r) = 0 (5.30) r→0 r→0 r

36 Of course, the functions should also be chosen such that the field configurations result in finite-energy. If we insert the ansatz (5.28) of the fields into the energy functional (5.21) then after integration over the angles φ and θ we obtain " Z 4 h df 2 2 i E = sin2 µ + [f(1 − f)]2 sin4 µ g2r2 dr r2 v2 hdh2 2 i + sin2 µ + ([h(1 − f)]2 sin2 µ − 2fh(1 − f)(1 − h) cos2 µ sin2 µ + [f(1 − h)]2 cos2 µ sin2 µ) 2 dr r2 # λv4 + (h2 − 1)2 sin4 µ 4πr2 dr (5.31) 4

1 The energy is thus dependent on the parameter µ and attains a maximal value at µ = 2 π for most choices of f and h[32]. Then we want to find the functions f and h which minimize this maximal energy, since this will be yield the true sphaleron configuration. To find the functions f and h we use the variational principle 1 on (5.31) with µ = 2 π to find the equations d2f g2r2v2 r2 = 2f(1 − f)(1 − 2f) − h2(1 − f) dr2 4 d dh r2 = 2h(1 − f)2 + λv2r2(h2 − 1) (5.32) dr dr Solutions for f and h that satisfy these equations and the boundary equations yield the minimum maximal energy. Unfortunately equations (5.32) cannot be solved analytically, however Burzlaff [12] has shown that a smooth solution does exist. The mass of the Higgs particle is 125 GeV [3] and the vacuum expectation value of the electroweak theory is 246 GeV. Using these values one can compute a numerical solution for f and h to the equations (5.32) that minimizes the potential. In the paper by Tye & Wong they computed the functions f and h and they can be approximated by f(r) ≈ 1 − sech(1.154mwr) and h(r) = tanh(1.056mwr) [50]. The shape of the functions is given in figure 9.

Figure 9: Radial proﬁle functions f and h

Using these functions in equation (5.31) the energy of the sphaleron was computed to be Esphaleron = 9.11 TeV. This value can be thought of as the barrier height between vacua with diﬀerent winding numbers. To

37 see this intuitively one can consider the parameter µ to be a rescaled version of the winding number so that at µ = 0 → n = 0 and µ = π → ν = 1. Then we can regard the sphaleron solution to have a half integer winding number just as in section 5.1 sitting at the mid-point on a path, the non-contractible loop, between topologically distinct vacuum states. A transitions between vacuum states through the instanton process would then occur in the following manner: In energetic situations such as in the early universe or high energy collisions the field configurations can form to become a sphaleron configuration which would subsequantly decay into a vacuum state with a different winding number. In the next chapter we discuss how the existence of the sphaleron solution and the interpretation of it being the barrier between vacuum states could play a role in the baryon asymmetry of the universe.

38 6 Phenomenology

In this chapter we will qualitatively investigate how the sphaleron possibly could have played a role in generating the observed baryon asymmetry in the early universe. We will discuss the probability of a sphaleron process transition occurring at early times in the universe and how leptogenesis might lead to baryogenesis.

6.1 Baryogenesis The current cosmological model theorizes that the Big Bang should have created equal amounts of matter and antimatter. However the observed imbalance between matter and antimatter in the universe is about

n − n ¯ η = B B = 6 × 10−10 (6.1) γ

where nB and nB¯ are the baryon and antibaryon density and γ is the photon density. This result is obtained from the anisotropies in the Cosmic Microwave Background, observed by space telescopes such as Planck and WMAP.[46]. Since there was no asymmetry directly after the big bang there must have been a process which has caused the discrepancy in matter and antimatter. Such a process must adhere to three properties according to Sakharov[44]:

• There must be a fundamental process causing the violation of baryon number • This process must violate C and CP since the conservation of the C and CP symmetry would lead to the creation of the same amount of matter and antimatter particles. • Departure from thermal equilibrium since in thermal equilibrium the process that creates excess baryons is cancelled by the inverse process. In the next sections it will be discussed whether these conditions are met within the standard model.

6.2 Baryon number violation and Sphaleron transition rate In section 4.8 we discussed that instanton processes are not seemingly possible in the electroweak theory due to the size of the coupling constant. Here we will evaluate the probability of going over the sphaleron barrier through thermal excitations and in that way achieve baryon number violation. It is useful to ﬁrst go back to the pendulum analogy as sketched in section 5.2. Classically, the pendulum obtains a potential energy of V (π) = 2mgh, where h is the length of the pendulum at the critical point θ = π. At least this amount of energy is required to overcome the critical point. Now imagine that we place the pendulum in a thermal heat bath, where thermal excitations could cause the pendulum to go over the unstable point. From − 2mgh statistical mechanics we know that this probability scales with the Boltzmann factor as e T . Thus the transition rate is unsuppressed when T 2mgh. The same argument can be made for transitions over the potential barrier between vacuum states. Therefore the sphaleron transition rate Γ is proportional to

−ESph Γ ∝ e T (6.2) and these transition will be unsuppressed when T ESph and baryon number violating processes could have occurred readily. The ﬁrst estimation of this rate for T < TW , where TW is the temperature of the electroweak phase transition, was done by Arnold and McLerran [2]. Based on their methodology a more accurate calculation was done in [28] and the transition rate per volume is given by

3 4 −E Γ ESph mw(T ) 4 Sph ∼ T e T (6.3) V T T

39 At very high temperatures in the early universe the electroweak symmetry was not broken. The transitional phase between the broken and unbroken symmetry is called the electroweak phase transition and takes place at temperature TW ∼ 100 GeV. In the unbroken electroweak theory the potential barrier between vacuum states vanishes and thus the VEV vanishes as well. Since there was no barrier between vacuum states the sphaleron transitions were not suppressed by the Boltzmann factor. Therefore transitions between vacuum states happened readily. Lattice computations have found that the sphaleron transition rate per volume is [9] Γ ∼ 25α5 T 4 (6.4) V W Thus ∆(B + L) processes could have happened readily at high temperatures in the early universe and the ﬁrst Sakharov condition is met.

6.3 C and CP violation Within the standard model there are sources for CP violating processes. We already have seen one in section 4.6/4.7, but because θ has such a small value this source plays a negligible role in the observed baryon asymmetry in the universe. The prime example for the CP violation in the standard model is found within the CKM matrix. The CKM matrix describes the strength of ﬂavour changes in weak decays. The standard parameterization of the CKM matrix is given by

     iδ13    Vud Vus Vub 1 0 0 c13 0 s13e c12 s12 0 Vcd Vcs Vcb  = 0 c23 s23  0 1 0  −s12 c12 0 (6.5) iδ13 Vtd VtsVtb 0 −s23 c23 −s13e 0 c13 0 0 1

iδ13 In this parametrization the elements Vcd,Vtd,Vcs,Vts and Vub pick up the complex phase term e which violates CP. However in other parameterizations of the CKM matrix the phase term can be found in diﬀerent elements of Vij. Quantifying the amount CP violation requires an invariant measure of the CP violation, independent of the parameterization. Such an invariant was ﬁrst constructed by Jarlskog [26] as

2 2 2 2 2 2 I =2J(mt − mc )(mt − mu)(mc − mu) (6.6) 2 2 2 2 × (mb − md)(ms − md) (6.7) where J is the Jarlskog invariant given by

2 J = c12c23c13s12s23s13 sin δ (6.8) Thus in order to obtain nonzero CP violation we need J 6= 0 and degenerate quark masses. The parametrization independent measure I of CP violation has thereafter been used to quantify the amount of CP violation in the standard model. Unfortunately Gavela et al. [24] concluded that the amount of CP violation in the CKM matrix could not explain the observed baryon asymmetry in the universe.

6.4 Departure from thermal equilibrium A type of particle is in thermal equilibrium when its interaction rate is larger than the expansion rate of the universe. If this condition is met then the products of the interaction have the possibility to undergo the inverse reaction. If the expansion rate of the universe is larger than this will not happen and the type of particle becomes frozen out. The expansion rate (or Hubble rate) H in the early universe is given by √ T H ∼ g∗ T (6.9) Mp

18 where Mp ∼ 10 GeV is the reduced Planck mass and g∗ ∼ 100 is the eﬀective massless degrees of freedom in the early universe. In the unbroken symmetry phase the sphaleron transition rate per unit volume is given

40 Figure 10: Difference between first (left) and second (right) order phase transition where Tc is the temperature at which the electroweak phase transition occurs[14]. by equation (6.4). To compare the rates we need to pick an appropriate volume for the sphaleron transition. 1 The appropriate choice is T 3 , since this is the average space that a particle in a thermal bath occupies. 5 Therefore the sphaleron rate scales with ∼ αW T . So at very high temperatures the sphaleron is out of equilibrium. Around the temperatures of the order ∼ 1010GeV the sphaleron transition rate becomes larger than the expansion of the universe and thus the transitions will be in thermal equilibrium. But after the electroweak phase transition the transition rate becomes exponentially suppressed (6.3) and the transition rate eventually decreases below the expansion rate of the universe. Although the sphalerons go out of thermal equilibrium this does not necessarily mean that a net baryon number is created. It is important that the electroweak phase transition is a first order transition and not a second order. The difference between a first and second order phase transition is sketched in figure 10. In a first order transition a local minimum appears. The barrier between the central minimum and the local minimum separates the broken and unbroken phase. Eventually the local minimum becomes the true vacuum state. The first order phase transition then proceeds via bubble nucleation. Inside the bubble the symmetry is broken and the sphaleron transitions eventually go out of thermal equilibrium while outside the symmetry is unbroken and the sphaleron transitions are in thermal equilibrium. These bubbles appear randomly throughout the universe, grow and coalesce to eventually fill the entire universe. Illustratively one can see in figure 11 how the first order phase transition causes baryon production in order to realize the baryon symmetry.

Figure 11: Baryon production in ﬁrst order electroweak phase transition.[38]

Due to CP violating interactions in the bubble wall incident fermions on the wall will be reﬂected or transmitted in diﬀerent amounts based on their chirality. Subsequently there will be a chiral asymmetry

41 around the bubble wall with more left handed fermions being reflected. The electroweak sphaleron only couples to left handed fermions and therefore outside of the bubble wall a net amount of baryons will be created. These baryons will thereafter be passed by the bubble wall and if the sphaleron transitions are sufficiently suppressed in the broken phase then the net baryon number survives. This is only the case when the phase transition is strongly first order. [15] On the other hand a second order phase transition happens uniformly and continuously across the universe resulting in a negligible asymmetry of baryons. [35] Whether the electroweak phase transition is first order or not depends on several parameters, one of them being the Higgs mass. An upper limit was set on the Higgs mass in order to realize a first order transition. It was determined to be mH < 72 GeV[27]. In order for the sphaleron processes to be suppressed sufficiently in the unbroken phase the upper limit is even lower mh < 35 GeV. Therefore it seems that within the standard model it is not possible to have a first order phase transition since the mass of the Higgs boson is mh ∼ 1250GeV .

6.5 Concluding remarks As we have seen in the previous sections the three Sakharov conditions are met in the standard model. Unfortunately, the amount of CP violation is not enough to account for the observed baryon asymmetry and the electroweak phase transition is not a ﬁrst order phase transition. Therefore it seems the baryon asymmetry cannot be realized within the standard model and thus we require extensions of the standard model. An example of a solution is that the baryogenesis is realized through leptogenesis. The sphaleron process plays a key role as it would convert the excess lepton number from leptogenesis into a baryon number. A good candidate process for the leptogenesis is predicted by the see-saw mechanism. The see-saw mechanism requires neutrinos to be their own anti-particle, i.e. they are majorana particles. This mechanism cannot be included in the standard model as it that violates gauge invariance, but could be possible in GUT models. Other solutions can be found within the Minimal Supersymmetric Standard Model where there are new sources of CP violation. These solutions are not in the scope of this thesis, interested readers are referred to respectively [11] for baryogenesis through leptogenesis and [43] for baryogenesis in supersymmetric extensions. Although the second and third Sakharov condition require extensions of the standard model it is clear that the electroweak sphaleron plays an essential role in the observed baryon asymmetry in the universe. In the next chapter we will discuss the possibility of detection a sphaleron process in collider experiments.

42 7 Detection of ∆(B + L) processes

In this chapter we will discuss the probability of detecting ∆(B + L) violating processes in collider experiments. The fact that at high temperatures transitions accross the sphaleron barrier occur suggest that they could also occur in scattering reactions if the energy is comparable to the sphaleron energy. We will also discuss a recent controversial paper written by Tye & Wong [50] which suggests that these process could occur at lower energies than previously thought by exploiting the periodic nature of the vacuum.

7.1 Probability of detecting a (B + L) violating process

Recall that a transition between adjacent vacuum states ∆ν = 1 results in a change ∆(B + L) = 2νNf . An example of such a reaction with 3 quark ﬂavors would be

q + q → 3¯l + 7¯q (7.1)

In collider experiments the temperature is near zero. Earlier calculations suggested that the reaction could therefore only take place through tunneling. Accordingly, the calculated quantum mechanical amplitude semiclassically was done using instanton configurations. They concluded that these reactions were highly 2 2 suppressed by the factor e16π /g [25] just as we had discussed in section 4.8. But one would expect that these reaction could take place classically if the center of mass energy is comparable to the sphaleron energy. A way of connecting the two results was proposed by Arnold and McLerran [2]. They suggested that when the reaction takes place classically then the two incoming particles fields must configure into a physical sphaleron which subsequently decays in the 3¯l + 7¯q and also into many W, Z and Higgs bosons: ¯ q + q → 3l + 7¯q + nwW + nzZ + nhΦ (7.2) where nw, nz and nh are the number of W, Z and Higgs bosons respectively. They proved that the instan- tonic calculations break down in the limit where there are many other particles and how one can obtain unsuppressed results in this limit. Subsequently Ringwald [42] and Espinosa [22] were the first to give a quantitative calculation of the cross section of (7.2) using semiclassical techniques and found that it is en- hanced by an exponential power of the center of mass energy. The resulting cross section for such reactions (7.2) has the following shape

−16π2 √ σ2→any = f(s) exp F [ s/E ] (7.3) B+L g2 0 √ where E0 ∼ 18 TeV, is of the order of twice the sphaleron energy, and F [ s/E0] is called the ’holy grail’ function is of the form √ √ √ √ 9 s 4/3 9 s 2 s8/3 F [ s/E0] = 1 − + + O (7.4) 8 E0 16 E0 E0 √ s √ It is a series of fractional powers of . Unfortunately this means that at center of mass energies s ≥ E0 E0 the holy grail function becomes unreliable. At low center of mass energies this term becomes unity and the transition is suppressed as required. The√ function f(s) is a prefactor dependent on the center of mass energy, which for center of mass energies mw s ESph attains a value of [29]

7 1 8π2 2 f(s) = 2 2 (7.5) mW g This value is not expected to change a lot and we are more interested in the holy grail function since together with the suppression factor it dominates the cross section. As mentioned, the holy grail function (7.3) is unreliable at high energies, but limits on the cross section have been set. Numerical results of Bezrukov et

43 √ al. [8] found that for center of mass energies s ≥ 20 TeV√ the transitional probability is still suppressed by a factor 10−20. At even higher center of mass energies s ∼ 30 it has been estimated that the cross section becomes of the order σ = 10−3 fb [41] which is already a lot higher and the reactions (7.2) could potentially be detected. Reactions of the type (7.2) are easily detectable since they create a huge amount of particles. This is because beside 3¯l + 7¯q the bosons will further decay in many other particles. Unfortunately the LHC is currently running at peak energies of 13 TeV and therefore the observation of ∆(B +L) violating reactions does not seem feasible in the near future. However, the aforementioned paper by Tye & Wong suggest that these reactions could possibly occur readily at energies around the sphaleron energy.

7.2 Tye & Wong Construction The result from Tye & Wong is based on the construction of Bloch waves based on the periodic nature of the sphaleron potential. In the paper they modify the original constructions of the sphaleron by Manton, which we discussed in section 5.3, and refer to it as the constant mass construction. The difference between the Manton and constant mass construction is the definition of the parameter µ. In figure 12 the difference is sketched

Figure 12: The diﬀerence between the Manton construction (a) and the constant mass construction (b)[50]

The µ parameter in the Manton construction represents the angle relative to a tangent plane of the sphere while the µ parameter in the constant mass construction represents the latitude at which the loops are constructed. The essential step is to make the µ parameter time dependent. They do this by making the static ﬁeld conﬁgurations time dependent, i.e. ∂A ∂A ∂µ ∂Φ ∂Φ ∂µ i = i , = (7.6) ∂t ∂µ ∂t ∂t ∂µ ∂t This way they can construct a Lagrangian which now consists out of a kinetic mass term

1 µ˙ 2 L = m 2 − V (µ) (7.7) 2 mW The mass term in the Lagrangian therefore consists out of the derivatives of the field configurations with 1 ˙ respect to µ. Then one introduces Q = µ/mW so that L = 2 mQ−V (Q). Using this parametrization ensures that the kinetic term has the right dimension. From this Lagrangian one can define the canonical conjugate momentum and Hamiltonian for Q

2 ∂L πQ πQ = = mQ,˙ H = πQ˙ − L = + V (Q) (7.8) ∂Q˙ 2m

44 ∂ Then by imposing canonical quantization on the variable Q so that [Q, πQ] = i and H → −i ∂Q one can obtain the time-independent Schr¨odingerequation

1 ∂2 − + V (Q) Ψ(Q) = EΨ(Q) (7.9) 2m ∂Q2

The mass in the kinetic term 7.7 cannot be dependent on µ in order to construct the time independent Schr¨odingerequation. By applying the gauge transformation (U ∞)−1 and thereafter replacing U ∞ by U which is given by

(cos µ − i sin µ cos θ) sin µ sin θeiφ U = (7.10) − sin µ sin θe−iφ (cos µ + i sin µ cos θ)

they remove the dependence of µ in the mass term of (7.7). The complete constant mass ansatz is then given by

v 0 v 0 Φ(r, θ, φ, µ) = (1 − h(r))√ U −1 + √ h(r) 2 cos µ 2 1 ∞ Aθ(r, θ, φ, µ) = f(r)Aθ (θ, φ, µ) ∞ Aφ(r, θ, φ, µ) = f(r)Aφ (θ, φ, µ)

Ar(r, θ, φ, µ) = 0 (7.11)

with i A = − (1 − f(r))∂ U(U)−1 (7.12) θ,φ g φ

Using this construction yields for the mass term in the Lagrangian (7.7)

8πm Z m = W d(m r)[4(f − 1)2 + 2(m r)2(h − 1)2] (7.13) g2 W W

which yields a value of m = 17.1 TeV when the radial proﬁle function f(r) and h(r) that minimize the potential are inserted (see section 5.3). They then solve the Schr¨odingerequation (7.9) using Bloch waves owing to the periodic nature of the potential V (Q). These solutions can be put in bands and bandgaps, where bands represent regionsm where there is a solution and gaps regions where there are none, using the semiclassical WKB quantization for periodic potentials [47]. The resulting probability of overcoming a potential barrier using this method can be summarised to a cross section of

16π2 √ σ ∝ exp(c S( s)) (7.14) g2 √ for reactions of the type (7.1). Here c ∼ 2 and the function S( s) is called the suppression factor and its behaviour is given by the solid line in ﬁgure 13

45 √ √ Figure 13: The behaviour of S( s) (solid line) and F ( s) to only ﬁrst order (dot-dash line) and second order (dashed line). The suppression factors are normalised to -1 at zero center of mass energy.[50]

√ The ﬁgure also gives a comparison to the ’holy grail’ function (7.3). As we can see S( s) approaches 0 faster and remains at 0 once the center of mass energy is higher than the sphaleron energy. Consequently these processes could be observed at LHC, since their peak center of mass energy is around 13 TeV. The validity of the approach in the Tye & Wong paper needs scrutiny and we will give some points of criticism in the next section.

7.3 Validity Tye & Wong approach The Lagrangian in (7.7) is seemingly gauge non-invariant. According to their paper the mass term in the original Manton construction is dependent on µ and therefore they cannot use the WKB approximation, but after applying the gauge transformation (U ∞)−1 and thereafter replacing U ∞ by U, which is equivalent to a gauge transformation, the mass becomes finite. In this gauge their methodology works while in the gauge of the Manton construction it doesn’t. Therefore their methodology is also gauge non-invariant. They also compute the mass term in the Lagrangian (7.7) following a different construction of the sphaleron called the AKY construction [1]. Using this construction they found a mass term of m = 22.5 TeV which is 1.3 times larger than that of the Manton construction. The sphaleron solution of both constructions are equivalent and the difference in mass stems from the different shape of the periodic potential of the two constructions. Albeit it small, the difference in the mass parameter yields a difference in the cross sections. This is perhaps a more general point of criticism, because an accurate calculation of the mass term requires a better analysis of the shape of the periodic sphaleron potential. The validity of the Tye & Wong approach requires detailed analysis. The Tye & Wong approach could be favoured by the detection of sphaleron transitions at center of mass energies around 9 TeV. After their publication several papers [19] [20] have come out discussing the probability of detecting a sphaleron process using their approach. In the next section we will give an overview of the results of these papers.

7.4 Detection of ∆(B + L) processes in the Tye & Wong approach First we will discuss the probability of observing a ∆(B + L) process is in proton-proton collisions at the LHC. In the proton-proton collisions the violating occurs via the scattering of quarks in the protons. These type of reactions are then given by

q + q → 3¯l + 7¯q + X (7.15)

46 for ∆ν = −1 and

q + q → 3l + 11q + X (7.16)

for∆ν = 1. The X consists out of bosons which will decay further into particle anti-particle pairs as mentioned in section 7.1. Note that at very high center of mass energies one could theoretically have a transition over multiple barriers leading to reactions of the type

q + q → n(3l + 11q) + X (7.17)

for ∆ν = n. Here we will only consider unit sphaleron transitions as they are the most likely to occur. The overall cross section of the Tye & Wong approach is unknown (7.14). In [19] they propose that the scale is 1 p of the order ∼ 2 . The cross section then picks up a prefactor ∼ 2 . The constant p will determine the mw mw true scale of the cross section. One also needs to take in to account that the sphaleron process takes place via scattering of quarks inside proton. The quarks inside the protons will carry a fraction of the energy of dLab 4 the proton. This eﬀect can be described by the so-called parton luminosity function dE . . The total cross section is then given by

Z 2 p X dLab 16π √ σ(∆ν = ±1) = dE exp(c S( s)) (7.18) m2 dE g2 W ab √ The contribution of the parton luminosity function as a function of the center of mass of the quarks sˆ for collisions between quarks belonging to the lightest two generations is given in ﬁgure 14

Figure 14: Contribution from parton luminosity function as a function of center of mass energy of quarks.[19]

√ The calculations of ﬁgure 14 were based on the nominal values c = 2, s = 13 TeV, Esph = 9 TeV and p = 1. As we can see the largest contibution comes from uu quark collisions. Using these nominal values the total cross section as a function of the center of mass energy of the incoming protons was calculated in [19]. The results are plotted in ﬁgure 15

4For the derivation of this term and the basics of collider physics the reader is referred to the book [21]

47 Figure 15: Total sphaleron transition cross section as a function of center of mass energy of the protons.[19]

From ﬁgure 15 we read that at a center of mass energy of 13 TeV the cross section of a reaction is around ∼ 10 fb. In the second run of the LHC they expect to obtain atleast 100 fb−1 of data at center of mass energies of 13 TeV. Therefore it could be possible that if p = 1 a sphaleron process could be detected, however p has a very high uncertainty and is likely to be lower than 1. The non-detection of a sphaleron can also be used to set an upper limit p.

There is also a probability of detecting sphaleron transitions in the IceCube detector. In the IceCube detector neutrinos could interact with quarks in the ice and the following sphaleron induced reaction

q + ν → 8¯q + 2¯l + X (7.19) for a ∆ν = −1 transition. The leptons produced by this interaction could cause cherenkov radiation given that they are energetic enough. The subsequent decay of the heavy quark products of this reaction could also produce leptons that are energetic enough to create cherenkov radiation. Given that both leptons are energetic enough they would then cause a double bang signature to occur. Other than the double-bang signature simulations of the above reaction done in [20] there are no distinctive signatures. The non detection of sphaleron induced processes in the LHC could also be used to set an upper limit on the prefactor p. The result of the analysis done in [20] is given in ﬁgure 16

Figure 16: Comparison of the upper limit on the prefactor p that can be set in the LHC via the reaction (7.15) or by 4 years of IceCube data.[20]

Since the run 2 of the LHC is expected to obtain 100 fb−1 of data at a center of mass energy of 13 TeV

48 the LHC will set a lower upper limit for p for Esph = 9. The LHC is therefore favoured in ﬁnding a sphaleron induced transition.

49 8 Discussion

Here we will summarise and give some additional points of discussion regarding the sphaleron.

8.1 Energy of the sphaleron In section 5.3 we worked in the limit where the weak mixing angle vanishes for simplicity’s sake. The influence of the U(1) gauge fields on the energy of the sphaleron was evaluated by Klinkhamer & Manton in [32]. They concluded that turning the weak mixing angle on results in a decrease of the sphaleron energy. Subsequently, in the papers by Kunz et al.[34] and Klinkhamer et al. [31] it was calculated that turning the weak angle on to a value of θw = .5 results in a decrease of the sphaleron energy by only a percent and the resulting difference in the cross sectional analysis will be negligible. Moreover the resulting cross sections given in section 7.4 were calculated using Esph = 9 TeV, which is already ∼ 1% lower than Esph = 9.11 TeV .

8.2 Baryogenesis In chapter 6 we discussed the inﬂuence of the sphaleron transitions on the observed asymmetry in universe. Although the Standard Model itself is not enough to explained the observed asymmetry, the (B + L) mechanism itself is necessary to fulﬁll the Sakharov conditions. Therefore the sphaleron transition process could be the baryon number violating mechanism in other theories such as GUT or supersymmetry.

8.3 Tye & Wong approach In the previous chapter we discussed the probability of finding a sphaleron induced process in the LHC and in the IceCube detector. The non-detection sets an upper limit on the prefactor p. The uncertainty in this prefactor is very large and it could be that p 1. If p is in this region then the center of mass energy needs to be very large in order to overcome the suppression by p favouring the original calculation using the holy grail function. Moreover, detection of sphaleron induced transitions at lower energies than what is predicted with the holy grail function do not necessarily confirm that the Tye & Wong approach is valid. This is because one can the arbitrarily choose the prefactor p to attain the correct value. It is possible that the methodology of finding the true cross section might be entirely different from the cross sections obtained by both the Tye & Wong approach and the holy grail approach.

8.4 Higher order sphalerons The electroweak sphaleron that was constructed in section 5.3 was found using non-contractible loops in the ﬁeld conﬁguration space. It turns out that there are also other sphalerons: the electroweak sphaleron S∗ and the SU(3) sphaleron Sˆ. The sphaleron S∗ can be constructed by considering non-contractible 2-spheres in the electroweak theory [30] and the sphaleron Sˆ can be constructed by considering non-contractible 3-spheres in SU(3) theories[33]. The sphaleron S∗ plays a role in the consistency and dynamics of the electroweak theory and the and the sphaleron Sˆ is related to the Bardeen anomaly [6]. Interested readers are referred to the aforecited papers.

50 9 Conclusion

In this thesis we have investigated the existence of a static unstable finite-energy solution in the electroweak theory called the sphaleron. We have seen that non-abelian gauge theories have a non-trivial vacuum with physically equivalent but topologically distinct winding number vacuum states. Anomalous processes can occur via transitions between these vacuum states. The anomalous process is dependent on the underlying gauge theory. In the electroweak theory a transition between vacuum states results in the violation of (B+L). Transitions can occur by tunneling through the potential barrier, which can be described by the instanton configurations, or via excitation over the barrier. The height of the potential barrier can be determined by the sphaleron configuration. Since these transitions violate (B + L) they could have played a pivotal role in explaining the observed baryon asymmetry of the universe by fulfilling one of the Sakharov conditions for successful baryogenesis. As the temperature in the early universe was very high these processes could have occurred readily through thermal excitations over the potential barrier. Transitions over the sphaleron might also be realised in collider experiments such as the LHC. Calculations√ of the cross section implies that these reactions might be observable at center of mass energies of s ∼ 30 TeV. Since the LHC is currently running at peak energies of 13 TeV (and later on 14 TeV) the observation of such a process is not feasible in the near future. A recent paper by Tye & Wong suggests that these reactions might occur at much lower center of mass energies. Their work needs critical examination but if their method is valid then we might be able to detect sphaleron processes in the near future. Detection of such a process would be a beautiful affirmation of the power of quantum field theory as a tool in describing particle physics and the universe around us.

51 References

[1] T. Akiba, H. Kikuchi, and T. Yanagida. “Static minimum-energy path from a vacuum to a sphaleron in the Weinberg-Salam model”. In: Phys. Rev. D 38 (6 Sept. 1988), pp. 1937–1941. doi: 10.1103/ PhysRevD.38.1937. url: https://link.aps.org/doi/10.1103/PhysRevD.38.1937. [2] Peter Arnold and Larry McLerran. “Sphalerons, small ﬂuctuations, and baryon-number violation in electroweak theory”. In: Phys. Rev. D 36 (2 July 1987), pp. 581–595. doi: 10.1103/PhysRevD.36.581. url: https://link.aps.org/doi/10.1103/PhysRevD.36.581.

[3] ATLAS√ and CMS Collaborations. “Combined Measurement of the Higgs Boson Mass in pp Collisions at s = 7 and 8 TeV with the ATLAS and CMS Experiments”. In: ArXiv e-prints (Mar. 2015). arXiv: 1503.07589 [hep-ex]. [4] C. A. Baker et al. “Improved Experimental Limit on the Electric Dipole Moment of the Neutron”. In: Physical Review Letters 97.13, 131801 (Sept. 2006), p. 131801. doi: 10.1103/PhysRevLett.97.131801. eprint: hep-ex/0602020. [5] Varouzhan Baluni. “CP-nonconserving effects in quantum chromodynamics”. In: Phys. Rev. D 19 (7 Apr. 1979), pp. 2227–2230. doi: 10.1103/PhysRevD.19.2227. url: https://link.aps.org/doi/10. 1103/PhysRevD.19.2227. [6] William A. Bardeen. “Anomalous Ward Identities in Spinor Field Theories”. In: Phys. Rev. 184 (5 Aug. 1969), pp. 1848–1859. doi: 10.1103/PhysRev.184.1848. url: https://link.aps.org/doi/10. 1103/PhysRev.184.1848. [7] A. A. Belavin et al. “Pseudoparticle solutions of the Yang-Mills equations”. In: Physics Letters B 59 (Oct. 1975), pp. 85–87. doi: 10.1016/0370-2693(75)90163-X. [8] F. Bezrukov et al. “Suppression of baryon number violation in electroweak collisions: numerical results”. In: Physics Letters B 574 (Nov. 2003), pp. 75–81. doi: 10.1016/j.physletb.2003.09.015. eprint: hep-ph/0305300. [9] D. Bödeker, G.D. Moore, and K. Rummukainen. “Hard thermal loops and the sphaleron rate on the lattice”. In: Nuclear Physics B - Proceedings Supplements 83-84 (2000). Proceedings of the XVIIth International Symposium on Lattice Field Theory, pp. 583–585. issn: 0920-5632. doi: https://doi. org/10.1016/S0920-5632(00)91745-6. url: http://www.sciencedirect.com/science/article/ pii/S0920563200917456. [10] Raoul Bott. “THE STABLE HOMOTOPY OF THE CLASSICAL GROUPS”. In: Proceedings of the National Academy of Sciences 43.10 (1957), pp. 933–935. issn: 0027-8424. doi: 10.1073/pnas.43. 10.933. eprint: http://www.pnas.org/content/43/10/933.full.pdf. url: http://www.pnas.org/ content/43/10/933. [11] W. Buchmüller,P. Di Bari, and M. Plümacher. “Some aspects of thermal leptogenesis”. In: New Journal of Physics 6 (Aug. 2004), p. 105. doi: 10.1088/1367-2630/6/1/105. eprint: hep-ph/0406014. [12] J. Burzlaff. “A classical lump in SU(2) gauge theory with a Higgs doublet”. In: Nuclear Physics B 233 (Mar. 1984), pp. 262–268. doi: 10.1016/0550-3213(84)90415-2. [13] T. P. Cheng and L. F. Li. GAUGE THEORY OF ELEMENTARY PARTICLE PHYSICS. 1984. isbn: 9780198519614. [14] J. M. Cline. “Baryogenesis”. In: ArXiv High Energy Physics - Phenomenology e-prints (Sept. 2006). eprint: hep-ph/0609145. [15] A. G. Cohen, D. B. Kaplan, and A. E. Nelson. “Progress in electroweak baryogenesis”. In: Annual Review of Nuclear and Particle Science 43 (1993), pp. 27–70. doi: 10.1146/annurev.ns.43.120193. 000331. eprint: hep-ph/9302210. [16] Sidney Coleman. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985. doi: 10.1017/CBO9780511565045.

52 [17] M. Creutz. “Ambiguities in the Up-Quark Mass”. In: Physical Review Letters 92.16, 162003 (Apr. 2004), p. 162003. doi: 10.1103/PhysRevLett.92.162003. eprint: hep-ph/0312225. [18] M. Creutz. “The t Hooft vertex revisited”. In: Annals of Physics 323 (Sept. 2008), pp. 2349–2365. doi: 10.1016/j.aop.2007.12.008. arXiv: 0711.2640 [hep-ph]. [19] J. Ellis and K. Sakurai. “Search for sphalerons in proton-proton collisions”. In: Journal of High Energy Physics 4, 86 (Apr. 2016), p. 86. doi: 10.1007/JHEP04(2016)086. arXiv: 1601.03654 [hep-ph]. [20] J. Ellis, K. Sakurai, and M. Spannowsky. “Search for Sphalerons: IceCube vs. LHC”. In: ArXiv e-prints (Mar. 2016). arXiv: 1603.06573 [hep-ph]. [21] R. K. Ellis, W. J. Stirling, and B. R. Webber. QCD and Collider Physics. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press, 1996. doi: 10.1017/ CBO9780511628788. [22] Olivier Espinosa. “High-energy behavior of baryon- and lepton-number violating scattering amplitudes in the standard model”. In: Nuclear Physics B 343.2 (1990), pp. 310–340. issn: 0550-3213. doi: https: //doi.org/10.1016/0550-3213(90)90473-Q. url: http://www.sciencedirect.com/science/ article/pii/055032139090473Q. [23] Kazuo Fujikawa. “Path-Integral Measure for Gauge-Invariant Fermion Theories”. In: Phys. Rev. Lett. 42 (18 Apr. 1979), pp. 1195–1198. doi: 10.1103/PhysRevLett.42.1195. url: https://link.aps. org/doi/10.1103/PhysRevLett.42.1195. [24] M. B. Gavela et al. “Standard Model Cp-Violation and Baryon Asymmetry”. In: Modern Physics Letters A 9 (1994), pp. 795–809. doi: 10.1142/S0217732394000629. eprint: hep-ph/9312215. [25] David J. Gross, Robert D. Pisarski, and Laurence G. Yaﬀe. “QCD and instantons at ﬁnite temperature”. In: Rev. Mod. Phys. 53 (1 Jan. 1981), pp. 43–80. doi: 10.1103/RevModPhys.53.43. url: https://link.aps.org/doi/10.1103/RevModPhys.53.43. [26] C. Jarlskog. “Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation”. In: Phys. Rev. Lett. 55 (10 Sept. 1985), pp. 1039–1042. doi: 10.1103/PhysRevLett.55.1039. url: https://link.aps.org/doi/10.1103/PhysRevLett.55. 1039.

[27] K. Kajantie et al. “Is There a Hot Electroweak Phase Transition at mH & mW ?” In: Phys. Rev. Lett. 77 (14 Sept. 1996), pp. 2887–2890. doi: 10.1103/PhysRevLett.77.2887. url: https://link.aps. org/doi/10.1103/PhysRevLett.77.2887. [28] S.Yu. Khlebnikov and M.E. Shaposhnikov. “The statistical theory of anomalous fermion number non- conservation”. In: Nuclear Physics B 308.4 (1988), pp. 885–912. issn: 0550-3213. doi: https://doi. org/10.1016/0550-3213(88)90133-2. url: http://www.sciencedirect.com/science/article/ pii/0550321388901332. [29] V.V. Khoze and A. Ringwald. “Total cross section for anomalous fermion-number violation via dis- persion relation”. In: Nuclear Physics B 355.2 (1991), pp. 351–368. issn: 0550-3213. doi: https : //doi.org/10.1016/0550-3213(91)90118-H. url: http://www.sciencedirect.com/science/ article/pii/055032139190118H. [30] F. R. Klinkhamer. “Construction of a new electroweak sphaleron”. In: Nuclear Physics B 410 (Dec. 1993), pp. 343–354. doi: 10.1016/0550-3213(93)90437-T. eprint: hep-ph/9306295. [31] F. R. Klinkhamer and R. Laterveer. “The sphaleron at ﬁnite mixing angle”. In: Zeitschrift f¨urPhysik C Particles and Fields 53.2 (June 1992), pp. 247–251. issn: 1431-5858. doi: 10.1007/BF01597560. url: https://doi.org/10.1007/BF01597560. [32] F. R. Klinkhamer and N. S. Manton. “A saddle-point solution in the Weinberg-Salam theory”. In: Phys. Rev. D 30 (10 Nov. 1984), pp. 2212–2220. doi: 10.1103/PhysRevD.30.2212. url: https: //link.aps.org/doi/10.1103/PhysRevD.30.2212.

53 [33] F. R. Klinkhamer and C. Rupp. “A sphaleron for the non-Abelian anomaly”. In: Nuclear Physics B 709 (Mar. 2005), pp. 171–191. doi: 10.1016/j.nuclphysb.2004.12.027. eprint: hep-th/0410195. [34] Jutta Kunz, Burkhard Kleihaus, and Yves Brihaye. “Sphalerons at finite mixing angle”. In: Phys. Rev. D 46 (8 Oct. 1992), pp. 3587–3600. doi: 10.1103/PhysRevD.46.3587. url: https://link.aps.org/ doi/10.1103/PhysRevD.46.3587. [35] V.A. Kuzmin, V.A. Rubakov, and M.E. Shaposhnikov. “On anomalous electroweak baryon-number non-conservation in the early universe”. In: Physics Letters B 155.1 (1985), pp. 36–42. issn: 0370- 2693. doi: https://doi.org/10.1016/0370-2693(85)91028-7. url: http://www.sciencedirect. com/science/article/pii/0370269385910287. [36] N. S. Manton. “Topology in the Weinberg-Salam theory”. In: Phys. Rev. D 28 (8 Oct. 1983), pp. 2019– 2026. doi: 10.1103/PhysRevD.28.2019. url: https://link.aps.org/doi/10.1103/PhysRevD.28. 2019. [37] Nicholas Manton and Paul Sutcliffe. Topological Solitons. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004. doi: 10.1017/CBO9780511617034. [38] David E Morrissey and Michael J Ramsey-Musolf. “Electroweak baryogenesis”. In: New Journal of Physics 14.12 (2012), p. 125003. url: http://stacks.iop.org/1367-2630/14/i=12/a=125003. [39] H. B. Nielsen and S. Chadha. “On how to count Goldstone bosons”. In: Nuclear Physics B 105 (Mar. 1976), pp. 445–453. doi: 10.1016/0550-3213(76)90025-0. [40] R. D. Peccei and Helen R. Quinn. “CP Conservation in the Presence of Pseudoparticles”. In: Phys. Rev. Lett. 38 (25 June 1977), pp. 1440–1443. doi: 10.1103/PhysRevLett.38.1440. url: https: //link.aps.org/doi/10.1103/PhysRevLett.38.1440. [41] A. Ringwald. “An upper bound on the total cross-section for electroweak baryon number violation”. In: Journal of High Energy Physics 10, 008 (Oct. 2003), p. 008. doi: 10.1088/1126-6708/2003/10/008. eprint: hep-ph/0307034. [42] A. Ringwald. “High energy breakdown of perturbation theory in the electroweak instanton sector”. In: Nuclear Physics B 330.1 (1990), pp. 1–18. issn: 0550-3213. doi: https://doi.org/10.1016/0550- 3213(90)90300-3. url: http://www.sciencedirect.com/science/article/pii/0550321390903003. [43] A. Riotto and M. Trodden. “Recent Progress in Baryogenesis”. In: Annual Review of Nuclear and Particle Science 49 (1999), pp. 35–75. doi: 10.1146/annurev.nucl.49.1.35. eprint: hep-ph/9901362. [44] A. D. Sakharov. “Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe”. In: Pisma Zh. Eksp. Teor. Fiz. 5 (1967). [Usp. Fiz. Nauk161,no.5,61(1991)], pp. 32–35. doi: 10.1070/ PU1991v034n05ABEH002497. [45] L. Senatore. “Lectures on Inflation”. In: New Frontiers in Fields and Strings (TASI 2015) - Pro- ceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics. Edited by POLCHINSKI JOSEPH ET AL. Published by World Scientific Publishing Co. Pte. Ltd., 2017. ISBN #9789813149441, pp. 447-543. Ed. by J. Polchinski and et al. 2017, pp. 447–543. doi: 10.1142/ 9789813149441_0008. arXiv: 1609.00716 [hep-th]. [46] D. N. Spergel et al. “First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: De- termination of Cosmological Parameters”. In: Astrophysical Journal, Supplement 148 (Sept. 2003), pp. 175–194. doi: 10.1086/377226. eprint: astro-ph/0302209. [47] U. P. Sukhatme and M. N. Sergeenko. “Semiclassical Approximation for Periodic Potentials”. In: eprint arXiv:quant-ph/9911026 (Nov. 1999). eprint: quant-ph/9911026. [48] G. ’t Hooft. “Computation of the quantum effects due to a four-dimensional pseudoparticle”. In: Phys. Rev. D 14 (12 Dec. 1976), pp. 3432–3450. doi: 10.1103/PhysRevD.14.3432. url: https: //link.aps.org/doi/10.1103/PhysRevD.14.3432.

54 [49] G. ’t Hooft. “Symmetry Breaking through Bell-Jackiw Anomalies”. In: Phys. Rev. Lett. 37 (1 July 1976), pp. 8–11. doi: 10.1103/PhysRevLett.37.8. url: https://link.aps.org/doi/10.1103/ PhysRevLett.37.8. [50] S.-H. H. Tye and S. S. C. Wong. “Bloch wave function for the periodic sphaleron potential and unsuppressed baryon and lepton number violating processes”. In: Physical Review D 92.4, 045005 (Aug. 2015), p. 045005. doi: 10.1103/PhysRevD.92.045005. arXiv: 1505.03690 [hep-th].

55 Acknowledgments

First and foremost, I would like to thank my supervisor DaniëlBoer. I thoroughly enjoyed working under his supervision. Whenever I got stuck on something he always knew how to nudge me in the right direction. It was always a pleasure to go to our meetings where he would patiently help me to wrap my head around conceptually difficult problems. Secondly, I would like to thank Rob Timmermans for being the second reader of thesis. I’m grateful that he could do this on such a short notice. I would also like to thank my ’roommates’ in the master students office Bas, Nick, Marenthe and Nikki. Although the subject of our theses were very different, our conversations and lunch breaks made the work place a fun place to be. (Admittedly, some days we might have had a little too much fun at the cost of getting work done!) Last but certainly not least, I would like to thank my girlfriend Nienke Bijen and my parents for their love and encouragement.