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 asym- metry 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 = (x)(iγ @µ − m) (x); (2.1) where is a doublet = 1 : (2.2) 2 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 fiθ Tag; (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 2 SU(2) acting on and can be written as 0 (x) ! (x) = U (x); (2.6) 0 (x) ! (x) = (x)U y: (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 (x) ! 0(x) = exp i a (x); (2.8) 2 the Lagrangian will no longer be invariant under this transformation since our derivative term transforms as 0 0 −1 (x)@µ (x) ! (x)@µ (x) = (x)@µ (x) + (x)U (θ)[@µU(θ)] (x); (2.9) 3 To recover the invariance one can explicitly construct the so-called gauge-covariant derivative Dµ that re- places @µ.