# PX430 Gauge Theories for Particle Physics

## Lecturer: Tim Gershon

## Weighting: 7.5 CATS

The electromagnetic field is a gauge field. Gauge changes to the vector potential (A^{μ} → A^{μ} - ∂^{μ} Φ with Φ an arbitrary function of postion and time), combined with multiplication of the wavefunction of particles with (dimensionless) charge *q* by the phase factor, *e ^{i q Φ }*, leave all physical properties unchanged. This is called a gauge symmetry. In particle physics, this idea is generalized to (space- and time-dependent) unitary matrix-valued fields multiplying spinor wavefunctions and fields. This generalization of the theory of an electron in an electromagnetic field is the basis for current theories of elementary particles. The module starts with the theory of the electron in the electromagnetic field making the gauge symmetry explicit. It then discusses the gauge symmetries appropriate for the various theories and approximate theories used to describe other elementary particles and their interactions with their corresponding gauge fields.

**Aims:**

To follow from *Relativistic Quantum Mechanics* (which is a pre-requisite), to develop ideas of gauge theories and apply these to the field of particle physics. To study, in particular, the theory underpinning the Standard Model of Particle Physics and to highlight the symmetry properties of the theory. Quantum electrodynamics (QED) should be considered in some detail, and its success illustrated by comparison with experiments.

**Objectives:**

At the end of this module you should:

- have an appreciation of the theoretical framework of the Standard Model
- understand the symmetry properties associated with gauge invariance
- be able to calculate amplitudes for simple QED processes
- be able to discuss qualitatively properties of the strong and weak interactions

**Syllabus:**

- Introduction and revision: relativistic quantum mechanics and notation; the Klein Gordon equation; the Dirac equation and interpretation of negative energy solutions; quantum numbers and spin; revision of matrices, Hermitan, unitary, determinants
- Group theory: definition of a group, examples of discrete groups; continuous groups, Lie groups, examples: U(1), SU(2), SU(3)
- Gauge invariance: symmetries and conservation laws; current conservation; Noether's theorem; the gauge principle; examples: Maxwell's equations, quantum electrodynamics
- Quantum field theories: brief outline of the deeper theory; Feynman rules and diagrams
- Non Abelian gauge theories: SU(2) and the electroweak interaction; SU(3) and QCD; local nonAbelian gauge theory; gauge fields; self-interaction
- Quantum electrodynamics: perturbation theory; scattering and cross sections

**Commitment:** 13 Lectures and 2 problem classes

**Assessment:** 1.5 hour examination

The module has a website.

**Recommended Texts:**

IJR Aitchison and AJG Hey *Gauge Theories in Particle Physics*, IoPP

**Leads from**: PX395 The Standard Model and PX408 Relativistic Quantum Mechanics