P509 Standard Model Phenomenology
Office: N/A
Email: ajainphysics@gmail.com
This is a one semester graduate course on symmetries and the Standard Model of particle physics, covering the following broad areas:
- Group theory, Lie algebras and representations
- Poincaré representations and particle multiplets
- Local symmetry - abelian and nonabelian gauge theory
- Spontaneous symmetry breaking - Goldstone and Higgs mechanisms
- Symmetry structure of the Standard Model, and beyond...
See the course syllabus for further details.
This is a one-semester graduate course on symmetries in particle physics, covering the following topics:
(*) covered in more detail if time permits
- Introduction
- Symmetries, kinematics and constraints
- Group Theory and Lie Algebras
- Basic group theory
- Lie algebras and representations
- Applications - global symmetries in field theory
- Spacetime Symmetries, Representations and Particles
- Poincaré group
- Scalar, vector and spinor representations
- Particles and fields, Coleman-Mandula theorem
- Local Symmetry and Gauge Theory
- Abelian gauge theory
- Nonabelian gauge theory
- Spontaneous Symmetry Breaking
- Global symmetry breaking - Goldstone mechanism
- Local gauge symmetry breaking - Higgs mechanism
- Standard Model Structure
- Chirality and the electroweak gauge group
- Higgs sector and symmetry breaking
- Flavour symmetry and interactions
- Standard Model Lagrangian
- Beyond the Standard Model*
- Symmetry constraints on new physics
- running couplings, grand unification, ...
(*) covered in more detail if time permits
This course covers the symmetries of the Standard Model of particle physics, within the framework of classical field theory. As such, I'm not aware of a text that covers the course content that closely. However, there are many that cover this material as part of a more extensive study of either group theory and Lie algebras, or the Standard Model as a quantum field theory. The following list should provide useful references for various topics:
Group Theory in Physics:
Gauge Theory and the Standard Model:
Group Theory in Physics:
- Lie Algebras in Particle Physics, H. Georgi
- Groups, Representations and Physics, H. Jones
- Group Theory in Physics: An Introduction, J. Cornwell
Gauge Theory and the Standard Model:
- Gauge Theories in Particle Physics (Vol 2), I. Aitchison and A. Hey
- The Standard Model: A Primer, C. Burgess and G. Moore
- Gauge Theory of Elementary Particle Physics, T.-P. Cheng and L.-F. Li
Further online material for the course will be provided, including:
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The course will assessed according to the following three components:
The final assessment component will be an individual project, requiring some background research and a written report of approximately 15 pages. Possible projects will be outlined on this website as the semester progresses.
If the application of this scheme would result in grades deemed by the instructor to be inconsistent with the University's grading descriptions (to be found on p.38 of the current Undergraduate Calendar), percentages will be assigned which are consistent with them. NB: Use of calculators in exams On all examinations the only acceptable calculator is the Sharp EL-510R. This calculator can be bought in the Bookstore for about $10. DO NOT bring any other calculator to the examinations. |
The final assignment for the course will be an individual project focussed on a particular aspect of group theory as it applies to the Standard Model of Particle Physics, or possibly some extensions. Assessment will be via a written report. You should try to prepare your report carefully so that it covers the basic physical picture, some of the relevant calculational details, and that you focus on what you found to be the most important or interesting aspects.
The precise choice of topic is open, provided it relates directly to one or more aspects of symmetry and group theory applied to particle physics and/or other field theories, but should be reasonably well-defined. Some possible topics are outlined below, but you are free to choose something else - just come and talk to me to get the topic approved.
NB: The (*)'d topics at the end are a bit more technical.
The underlying quark model arose through a recognition that the hadrons filled out representations of SU(2) and SU(3). This project should discuss part of the history, the modern identification of light hadrons within the SU(3) framework, and also more recent spectroscopy of possible exotics.
Some references:
The Standard Model generation structure admits a large global flavour symmetry. The individual quark and lepton generations can mix due to a mismatch of mass and electroweak eigenstates. This project should discuss the structure of the Standard Model flavour symmetry, some of the phenomenology of flavour-changing transitions, and the Kobayashi-Maskawa mechanism of CP violation.
Some references:
At low energies, QCD is best described in terms of the dynamical breaking of global chiral symmetry, and the associated pseudo-Goldstone bosons (pions and kaons). A systematic field theory describing the interactions of these Goldstone bosons can be written down on symmetry grounds, known as chiral perturbation theory. This project should discuss the structure of the chiral lagrangian, and a couple of applications.
Some references:
An interesting model scenario for incorporating baryons into the low energy chiral Lagrangian describing pions, is via a soliton solution (a skyrmion). While not expected to be quantitatively correct for the specific number of colours and flavours in QCD, it describes a certain theoretical limit with a large number of colours. This project should explain how the skyrmion solution can arise in the chiral Lagrangian, and some of its properties.
Some references:
The Standard Model gauge group, and also its field content, suggest that the entire structure may unify into a larger (grand unified) gauge group at very high energy scales. We will discuss this briefly at the end of the course. This project should discuss how the Standard Model may fit into a unified gauge theory, and some of the possible tests.
Some references:
The mechanism of electroweak symmetry breaking is the part of the Standard Model which is still essentially untested. Thus there are several variants of the Higgs sector that should soon be tested at the LHC. One simple generalization of the Standard Model Higgs is the 2-Higgs doublet model (which also forms part of supersymmetric scenarios). This project should explain the structure of the 2-Higgs doublet model, its spectrum, and some possible experimental tests.
Some references:
Classic examples of global and local spontaneous symmetry breaking that occur in condensed matter physics include the formation of superfluids and superconductors respectively. This project should discuss how these phases arise through the breaking of symmetries and the relevant degrees of freedom.
Some references:
In cases where symmetries are spontaneously broken, the possibility arises for defects to form where a localized region remains in the unbroken phase. Important examples are vortices in condensed matter systems. There are also speculations that cosmic strings could form during phase transitions in the early universe. This project should discuss how these defects form through symmetry breaking, their topological stability, and the differences that arise when the symmetry is either global or local (i.e. with gauge fields).
Some references:
Theories with supersymmetry can also be spontaneously broken by the scalar field vacuum. Analogously to the emergence of a massless field representing a Goldtone boson, for each broken Lie group generator, the breaking of supersymmetry leads to a massless spinor field - the goldstino. The project should look at the supersymmetry transformations in the Wess-Zumino model with an appropriate symmetrybreaking potential, and observe how the goldstino emerges on expanding the Lagrangian about the symmetry breaking vacuum.
Some references:
This is a more mathematical project, exploring more detail of the classification of Lie algebras, due primarily to Cartan. The project should cover some aspects of the classification program, roots and weights, and Dynkin diagrams
Some references:
The precise choice of topic is open, provided it relates directly to one or more aspects of symmetry and group theory applied to particle physics and/or other field theories, but should be reasonably well-defined. Some possible topics are outlined below, but you are free to choose something else - just come and talk to me to get the topic approved.
NB: The (*)'d topics at the end are a bit more technical.
- SU(3) and light hadrons
The underlying quark model arose through a recognition that the hadrons filled out representations of SU(2) and SU(3). This project should discuss part of the history, the modern identification of light hadrons within the SU(3) framework, and also more recent spectroscopy of possible exotics.
Some references:
- PDG Review
- The Eightfold Way, J. Rosner (historical overview)
- Flavour Symmetry in the Standard Model, and CP Violation
The Standard Model generation structure admits a large global flavour symmetry. The individual quark and lepton generations can mix due to a mismatch of mass and electroweak eigenstates. This project should discuss the structure of the Standard Model flavour symmetry, some of the phenomenology of flavour-changing transitions, and the Kobayashi-Maskawa mechanism of CP violation.
Some references:
- Introduction to Flavor Physics, Y. Grossman
- Chiral symmetry breaking, and chiral perturbation theory
At low energies, QCD is best described in terms of the dynamical breaking of global chiral symmetry, and the associated pseudo-Goldstone bosons (pions and kaons). A systematic field theory describing the interactions of these Goldstone bosons can be written down on symmetry grounds, known as chiral perturbation theory. This project should discuss the structure of the chiral lagrangian, and a couple of applications.
Some references:
- Chiral perturbation theory, G. Ecker
- Skyrme Model
An interesting model scenario for incorporating baryons into the low energy chiral Lagrangian describing pions, is via a soliton solution (a skyrmion). While not expected to be quantitatively correct for the specific number of colours and flavours in QCD, it describes a certain theoretical limit with a large number of colours. This project should explain how the skyrmion solution can arise in the chiral Lagrangian, and some of its properties.
Some references:
- Nucleons as skyrmions, M. Oka and A. Hosaka
- Grand unified theories
The Standard Model gauge group, and also its field content, suggest that the entire structure may unify into a larger (grand unified) gauge group at very high energy scales. We will discuss this briefly at the end of the course. This project should discuss how the Standard Model may fit into a unified gauge theory, and some of the possible tests.
Some references:
- PDG Review
- Grand unified theories and proton decay, P. Langacker
- Two-Higgs Doublet Model (example of a non-SM Higgs sector)
The mechanism of electroweak symmetry breaking is the part of the Standard Model which is still essentially untested. Thus there are several variants of the Higgs sector that should soon be tested at the LHC. One simple generalization of the Standard Model Higgs is the 2-Higgs doublet model (which also forms part of supersymmetric scenarios). This project should explain the structure of the 2-Higgs doublet model, its spectrum, and some possible experimental tests.
Some references:
- The CP-conserving two-Higgs-doublet model, J. Gunion and H. Haber
- Symmetry breaking in superfluids and superconductors
Classic examples of global and local spontaneous symmetry breaking that occur in condensed matter physics include the formation of superfluids and superconductors respectively. This project should discuss how these phases arise through the breaking of symmetries and the relevant degrees of freedom.
Some references:
- Superconductivity for Particular Theorists, S. Weinberg
- Vortices in theories with symmetry breaking (global and local)
In cases where symmetries are spontaneously broken, the possibility arises for defects to form where a localized region remains in the unbroken phase. Important examples are vortices in condensed matter systems. There are also speculations that cosmic strings could form during phase transitions in the early universe. This project should discuss how these defects form through symmetry breaking, their topological stability, and the differences that arise when the symmetry is either global or local (i.e. with gauge fields).
Some references:
- Cosmic Strings (see section 2), M. Hindmarsh and T. Kibble
- Goldstinos and spontaneous (super)symmetry breaking(*)
Theories with supersymmetry can also be spontaneously broken by the scalar field vacuum. Analogously to the emergence of a massless field representing a Goldtone boson, for each broken Lie group generator, the breaking of supersymmetry leads to a massless spinor field - the goldstino. The project should look at the supersymmetry transformations in the Wess-Zumino model with an appropriate symmetrybreaking potential, and observe how the goldstino emerges on expanding the Lagrangian about the symmetry breaking vacuum.
Some references:
- Cartan classification of Lie algebras, roots and weights(*)
This is a more mathematical project, exploring more detail of the classification of Lie algebras, due primarily to Cartan. The project should cover some aspects of the classification program, roots and weights, and Dynkin diagrams
Some references:
- Lie Algebras in Particle Physics, H. Georgi