AN INTRODUCTION TO QED & QCD By Dr G Zanderighi University of Oxford Lecture notes by Prof N Evans University of Southampton Lecture delivered at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 51 - - 52 - Contents 1 Introduction .............................................................................................. 55 1.1 Relativity Review...................................................................................... 56 2 Relativistic Wave Equations .................................................................. 58 2.1 The Klein-Gordon Equation .................................................................... 59 2.2 The Dirac Equation................................................................................... 62 2.3 Solutions to the Dirac Equation .............................................................. 63 2.4 Orthogonality and Completeness........................................................... 65 2.5 Spin ............................................................................................................. 66 2.6 Lorentz Covariance................................................................................... 68 2.7 Parity, charge conjugation and time reversal ....................................... 71 2.8 Bilinear Covariants................................................................................... 74 2.9 Massless (Ultra-relativistic) Fermions ................................................... 75 3 Quantum Electrodynamics..................................................................... 77 3.1 Classical Electromagnetism..................................................................... 77 3.2 The Dirac Equation in an Electromagnetic Field.................................. 79 3.3 g – 2 of the Electron................................................................................... 80 3.4 Interactions in Perturbation Theory....................................................... 82 3.5 Internal Fermions and External Photons............................................... 84 3.6 Summary of Feynman Rules for QED ................................................... 86 4 Cross Sections and Decay Rates............................................................ 88 4.1 Transition Rate.......................................................................................... 88 4.2 Decay Rates................................................................................................ 90 4.3 Cross Sections............................................................................................ 91 4.4 Mandelstam Variables.............................................................................. 92 5 Processes in QED and QCD................................................................... 93 5.1 Electron-Muon Scattering........................................................................ 93 5.2 Electron-Electron Scattering.................................................................... 95 5.3 Electron-Positron Annihilation............................................................... 96 5.4 Compton Scattering.................................................................................. 98 5.5 QCD Processes.......................................................................................... 99 6 Introduction to Renormalization ........................................................ 101 6.1 Ultraviolet (UV) Singularities............................................................... 101 6.2 Infrared (IR) Singularities...................................................................... 102 6.3 Renormalization...................................................................................... 102 6.4 Regularization ......................................................................................... 105 7 QED as a Field Theory .......................................................................... 107 7.1 Quantizing the Dirac Field .................................................................... 107 7.2 Quantizing the Electromagnetic Field ................................................. 109 Acknowledgements ............................................................................................. 112 References.............................................................................................................. 112 Pre-School Problems ........................................................................................... 113 Rotations, Angular Momentum and the Pauli Matrices............................. 113 Four Vectors ...................................................................................................... 113 Probability Density and Current Density ..................................................... 114 - 53 - - 54 - 1 Introduction The aim of this course is to teach you how to calculate transition amplitudes, cross sections and decay rates, for elementary particles in the highly successful theories of Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD). Most of our work will be in understanding how to compute in QED. By the end of the course you should be able to go from a Feynman diagram, such as the one for e−e− → μ−μ− in figure 5, to a number for the cross section. To do this we will have to learn how to cope with relativistic, quantum, particles and anti-particles that carry spin.Infactall these properties of particles will emerge rather neatly from thinking about relativistic quantum mechanics. The rules for calculating in QCD are slightly more complicated that in QED, as we will briefly review, however, the basic techniques for the calculation are very similar. We have a lot to cover so will necessarily have to take some short cuts. Our main fudge will be to work in relativistic quantum mechanics rather than the full Quantum Field Theory (QFT) (sometimes referred to as ‘second quantization’). We will be in good company though since we will largely follow methods from Feynman’s papers and text books such as Halzen and Martin. In quantum mechanics a classical wave is used to describe a particle whose motion is subject to the Uncertainty Principle. In a full QFT the wave’s motion itself is subject to the Uncertainty Principle too - the quanta of that field are what we then refer to as particles. Luckily at lowest order in a perturbation theory calculation one neglects the quantum nature of the field and the two theories give the same answer. At higher orders the quantum nature of the field gives rise to virtual pair creation of particles - in the quantum mechanics version of the story these are included in a more ad hoc fashion as we will see. Luckily the simultaneous QFT course will give you a good grounding in more precise methodologies. Thus our starting point will be ordinary Quantum Mechanics and our first goal (section 2) will be to write down a ‘relativistic version’ of Quantum Mechanics. This will lead us to look at relativistic wave equations, in particular the Dirac equation, which describes particles with spin 1/2. We will also develop a wave equation for photons and look at how they couple to our fermions (section 3) - this is the core of QED. A perturbation theory analysis will result in quantum mechanical probability amplitudes for particular processes. After this, we will work out how to go from the probability amplitudes to cross sections and decay rates (section 4). We will look at some examples of tree level QED processes. Here you will get hands-on experience of calculating transition amplitudes and getting from them to cross sections (section 5). We will restrict ourselves to calculations at tree level but, at the end of the course (section 6), we will also take a first look at higher order loop effects, which, amongst other things, are responsible for the running of the couplings. For QCD, this running means that the coupling appears weaker when measured at higher energy scales and is the reason why we can sometimes do perturbative QCD calculations. However, in higher order calculations divergences appear and we have to understand — at least in principle — how these divergences can be removed. In reference [1] you will find a list of textbooks that may be useful. - 55 - 1.1 Relativity Review An event in a reference frame S is described by the four coordinates of a four-vector (in units where c =1) xμ =(t, x), (1.1) where the Greek index μ ∈ {0, 1, 2, 3}. These coordinates are reference frame dependent. The coordinates in another frame S are given by xμ, related to those in S by a Lorentz Transformation (LT) μ → μ μ ν x x =Λνx , (1.2) where summation over repeated indices is understood. This transformation identifies xμ as a contravariant 4-vector (often referred to simply as a vector). A familiar example of aLT is a boost along the z-axis, for which ⎛ ⎞ γ 00−βγ ⎜ ⎟ ⎜ 0100⎟ Λμ = ⎜ ⎟ , (1.3) ν ⎝ 0010⎠ −βγ 00 γ with, as usual, β = v and γ =(1− β2)−1/2.LT’s can be thought of as generalized rotations. The “length” of the 4-vector (t2 −|x|2) is invariant to LTs. In general we define the Minkowski scalar product of two 4-vectors x and y as μ ν μ x · y ≡ x y gμν ≡ x yμ, (1.4) where the metric 1ifμ = ν gμν = g = diag(1, −1, −1, −1),gμλg = gμ = δμ = , (1.5) μν λν ν ν 0ifμ = ν has been introduced. The last step in eq. (1.4) is the definition of a covariant 4-vector (sometimes referred to as a co-vector), ν xμ ≡ gμν x . (1.6) This transforms under a LT according to → ν xμ xμ =Λμ xν . (1.7) Note that the invariance of the scalar