ELECTROWEAK PHYSICS Theory for the Experimentalist

ELECTROWEAK PHYSICS Theory for the experimentalist Chris Hays Oxford University CONTENTS Introduction vii 1 The geometry of forces 1 1.1 The fiber bundle of the universe 2 1.2 Spacetime metric 3 1.3 Connections 5 1.4 Curvature 6 1.5 Principle of least action 7 1.6 Conservation laws 9 2 Path integrals and fields 11 2.1 Non-relativistic path integral 12 2.2 Perturbation theory 13 2.2.1 Green’s functions 14 2.3 Path integral of a scalar field 15 2.3.1 Free-field transition amplitude 16 2.3.2 Interacting-field transition amplitude 17 2.4 Path integral of a fermion field 18 2.5 Path integral of a gauge field 19 2.5.1 Free-field generating functional 21 3 The Higgs mechanism 23 iii iv CONTENTS 3.1 Self-interacting scalar field theory 23 3.1.1 Real scalar field 23 3.1.2 Complex scalar field 24 3.2 Gauged scalar field theory 25 3.2.1 U(1)-charged scalar field 25 3.2.2 SU(2)-charged scalar field 26 3.2.3 Propagators after symmetry breaking 27 4 The Electroweak theory 29 4.1 The Electroweak Lagrangian 29 4.2 Electroweak symmetry breaking 30 4.2.1 Scalar field Lagrangian 31 4.2.2 Fermion field Lagrangian 33 4.3 Electroweak propagators 34 5 Cross sections and Feynman diagrams 37 5.1 Scattering matrix 37 5.2 Cross sections and lifetimes 38 5.3 Feynman rules 40 6 Scalar renormalization 47 6.1 Renormalized Lagrangian 47 6.2 Regularization 49 6.2.1 Propagator loop corrections 49 6.2.2 Vertex loop correction 50 6.3 The renormalization group 51 7 QED renormalization 55 7.1 QED divergences 55 7.2 Fermion self energy 56 7.3 Vacuum polarization 57 7.4 Vertex correction 61 8 Electron magnetic moment 65 8.1 Leading order 65 8.2 Next-to-leading order 66 8.3 Status 68 9 Muon decay 69 9.1 Tree-level prediction 69 9.2 Lifetime corrections 73 9.3 Experimental measurements 75 CONTENTS v 10 Z boson production 77 10.1 Z boson measurements 77 10.1.1 Cross section for Z/γ∗ production 78 10.1.2 Longitudinal asymmetry 81 10.1.3 Forward-backward asymmetry 83 11 W boson mass 85 11.1 Tree-level relations 85 11.2 Renormalization 86 11.3 One-loop results 88 11.3.1 Top-bottom loop 89 12 The Higgs boson 93 12.1 Higgs boson mass 93 12.2 Higgs boson production at hadron colliders 95 12.2.1 Gluon fusion [27] 96 12.2.2 Vector-boson fusion 97 12.3 Higgs boson production at lepton colliders 101 12.4 Higgs boson decay 103 13 Meson mixing 109 13.1 CP Violation 109 13.2 Mixing overview 110 13.3 Mixing at leading order 111 13.4 CP violation in mixing 115 14 Neutrino masses 117 References 119 INTRODUCTION The Standard Model of particle physics is naturally divided into the electroweak and strong interactions: each has a unique complication that affects both phenomenology and practical calculations. In the case of the strong interaction, it is the non-perturbative coupling at low momentum transfer that causes rich phenomena, which at present can only be described by heuristic models or intensive numerical calculations. In the case of the electroweak interaction, it is the breaking of the underlying symmetry that leads to a wide range of phenomena and complicates calculations. The electroweak interaction is the focus of this text. The underlying symmetries of both sets of interactions arise from their descriptions as quantum gauge field theories. To gain a fundamental understanding of such theories, we proceed in the following steps: first, we discuss the geometrical construction of a classical gauge field theory (Chapter 1); second, we describe the quantization of the field and its physical interpretation (Chapter 2); third, we present the mechanism by which the gauge symmetry of the theory can be broken (Chapter 3); and finally, we provide the realization of these concepts in nature, the electroweak theory (Chapter 4). After this introduction of the underlying principles of the theory, we turn to practical calculations. This begins with a discussion of path integrals and Feynman diagrams (Chapter 5), which are ubiquitous calculational tools that are indispensible in perturbative calculations. We follow with the application of these tools to calculations of scattering cross sections and particle widths (Chapter 6). At this point we can begin performing leading-order calculations of specific processes; however, the precision of electroweak measurements de- mands the inclusion of higher order calculations. Here we run into a fly in the ointment of quantum field theory: as soon as we go beyond leading order, we are faced with quantum contributions that produce unphysical results. This is resolved with a program of renormal- vii viii INTRODUCTION ization, which we describe in the context of both scalar and gauge field theories (Chapter 7). The discussion of renormalization leads naturally into the specific measurements that fix the three parameters required to describe interactions of the electroweak gauge bosons with fermions (in the massless fermion approximation). The first parameter is taken to be the electromagnetic coupling, which describes the strength of the fermion-fermion-photon interaction and is measured using the magnetic moment of the electron (Chapter 8); this interaction is one of the elements of the renormalization program described in the prior chapter. The second parameter is the weak coupling, which describes the fermion-fermion- W boson interaction, and is indirectly determined using precise measurements of the muon lifetime (Chapter 9). The final parameter is the expectation value of the vacuum energy of the scalar field, which is indirectly determined using precise measurements of the Z boson mass (Chapter 10). With the input gauge boson parameters in hand, we can now make a first-order prediction of the W boson mass, and demonstrate its sensitivity to other electroweak parameters through higher order corrections (Chapter 11). We then discuss the determination of the remaining parameters: the Higgs boson mass and flavour-diagonal Higgs-fermion- fermion Yukawa couplings (Chapter 12), measured through direct production of the Higgs boson; and off-diagonal Yukawa couplings manifested through the CKM matrix, measured through flavour-changing charged current processes such as Bs mixing (Chapter 13). A completely open question is the nature of neutrinos and the corresponding parameters to describe their interactions. Their mixing can be parameterized with the PMNS matrix, akin to the CKM matrix, but we do not know the neutrino mass terms. These issues are touched on in Chapter 14. CHAPTER 1 THE GEOMETRY OF FORCES The development of the general theory of relativity was a conceptual breakthrough in understanding the force of gravity. Instead of the mysterious “action at a distance”, gravity could be explained by the simple premise that all energy warps space and time. The curved trajectory of a particle in a gravitational field can be described as an unaltered path through a curved spacetime1; in mathematical terms, the path is a geodesic, the shortest distance from point a to point b. It turns out that this concept can be extended to the strong and electroweak gauge forces, though here the curved space is internal. In this description, charged matter warps the internal space, causing the trajectories of other charged particles to curve in the projected spacetime. To describe this phenomenon, an additional layer of mathematics is required that combines spacetime with the internal space into a fiber bundle. This structure has more information than is physically accessible, however, and the internal space must be factored out in the end to account for the gauge independence of physical processes. The detailed mathematical description of gauge forces is not generally provided in particle physics textbooks. It is included here for several reasons: first, it provides a valuable perspective in understanding the misnomer of “symmetry breaking”; second, it gives useful context for defining physically measurable quantities; and third, it describes the non-local 1One can visualize the effect as a bunching of space (or a stretching of time) near an object; space (time) is effectively more (less) dense in the vicinity of matter. A particle approaching matter will take less time to travel a given distance, accelerating towards the object. This effect is uniform for all matter, a crucial requirement for this intepretation. Electroweak Physics. 1 By Chris Hays 2 THE GEOMETRY OF FORCES aspects of the theory, which have applications in Dirac monopoles and the Aharanov-Bohm effect. A broader understanding of this geometrical construction could lead to more applications, in particular those that extend into the weak sector of the theory. This chapter provides a heuristic overview of the mathematics required to understand both the curvature of spacetime and of internal spaces. The spacetime manifold is intro- duced and combined with internal spaces into a fiber bundle. To describe particle trajectories, several additional structures are required: the spacetime metric tensor, the spacetime and fiber connections, and their resulting curvature. Armed with this structure, a trajectory is then described as an integrated path length over a curved space; this is the familiar action of physics. Choosing the shortest path leads to the gravitational and gauge force field equations. This minimization is modified in the quantum theory: the minimum path length corresponds to the peak of a probability amplitude, rather than the exclusive path of a particle. This will be discussed in more detail in the next chapter. The above description applies to a single particle travelling through a background space.

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