An Introduction to QED And

An Intro duction to QED and QCD Je Forshaw Department of Physics and Astronomy University of Manchester Manchester M PL Lectures presented at the Scho ol for Young High Energy Physicists Rutherford Appleton Lab oratory September Contents Intro duction Units and Conventions Relativistic Wave Equations Wavefunctions and Fields The KleinGordon Equation The Dirac Equation Free Particle Solutions I Interpretation Free Particle Solutions I I Spin Normalisation and Gamma Matrices Lorentz Covariance Parity Bilinear Covariants Charge Conjugation Neutrinos Cross Sections and Decay Rates The Number of Final States Lorentz Invariant Phase Space LIPS Cross Sections Twobody Scattering Decay Rates Quantum Electro dynamics Quantising the Dirac Field Quantising the Electromagnetic Field Feynman Rules of QED ElectronMuon Scattering ElectronElectron Scattering ElectronPositron Annihilation e e e e e e and e e hadrons Compton Scattering Intro duction to Renormalisation Renormalisation of QED Renormalisation of Quantum Chromo dynamics A Pre Scho ol Problems B Answers to the Exercises Intro duction The traditional aim of this course is to teach you how to calculate amplitudes cross sections and decay rates particularly for quantum electro dynamics qed but in principle also for quantum chromo dynamics qcd By the end of the course you should be able to go from a Feynman diagram such as the one for e e in Figure a to a number for the cross section We will restrict ourselves to calculations at tree level but at the end of the course h amongst other things we will also lo ok qualitatively at higher order loop eects whic are resp onsible for the running of the qcd coupling This running means that the qcd coupling app ears weaker when measured at higher energy scales and is the reason why we can sometimes do p erturbative qcd calculations As you might guess the sort of diagrams which are imp ortant here have closed lo ops of particle lines in them in Fig ure b is one example contributing to the running of the qcd coupling the curly lines denote gluons In order to do our calculations we will need a certain amount of technology In particular we will need to describ e particles with spin esp ecially the spin leptons and quarks We will therefore sp end some time lo oking at the Dirac equation After this well work out how to go from quantum mechanical probability amplitudes to cross sections and decay rates Then with these to ols in hand we will lo ok at some examples of tree level qed pro cesses Here you will get handson exp erience of calculating transition amplitudes and getting from them to cross sections We then moveontoqcd This will entail a brief intro duction to renormalisation in both qed and qcd We will intro duce the idea of the running coupling and lo ok at asymptotic freedom in qcd In reference you will nd a list of textb o oks which may b e useful Units and Conventions I will use natural units c h so mass energy inverse length and inverse time all have the same dimensions vector a a a a scalar pro duct ab a b ab g a b From the scalar pro duct you see that the metric is if g diag g g if For c g and g are numerically the same e e a b Figure Examples of Feynman diagrams contributing to a e e and b the running of the strong coupling constant From the ab ove you would think it natural to write the space comp onents of a vector as i a for i However for vectors I usually write the comp onents as a This ought i not to cause confusion since for vectors the metric is just the unit matrix Generally sp eaking a little care must b e taken in getting the sign right for the comp onents of a vector Note that isacovector x x i i so r and r My convention for the totally antisymmetric LeviCivita tensor is if f g an even p ermutation of f g if an odd p ermutation otherwise Note that and p q r s changes sign under a parity transformation which is obvious b ecause it contains an odd number of spatial comp onents Relativistic Wave Equations The starting p oint for this course is the Schrodinger equation which can b e written quite generally as H i t where H is the Hamiltonian ie the energy op erator In this equation is the wave function describing the single particle probability amplitude I shall usually reserve the Greek symbol for spin fermions and for spin b osons So for pions and the like I shall write H i t In this course wewant to extend the nonrelativistic quantum mechanics familiar to undergraduates into the relativistic domain For example in nonrelativistic quantum mechanics you are used to writing H T V where T is the kinetic energy and V is the potential energy A particle of mass m and momentum p has nonrelativistic kinetic energy P T m where capital P is the op erator corresp onding to momentum p For a slow moving particle v c eg an electron in a Hydrogen atom this is adequate but for relativistic systems v c the Hamiltonian ab ove breaks down For a free relativistic particle the total energy E is given by the Einstein equation E p m Thus the square of the relativistic Hamiltonian H is simply given by promoting the momentum to op erator status H P m So far so go o d but now the question arises of how to implement the Schrodinger equation which is expressed in terms of H rather than H Naively the relativistic Schrodinger equation lo oks like p t P m ti t but this is dicult to interpret b ecause of the square ro ot There are two ways forward Work with H By iterating the Schrodinger equation we have t H t t This is known as the KleinGordon KG equation In this case the wavefunction describ es spinless b osons Invent a new Hamiltonian H which is linear in momentum and whose square is D equal to H given ab ove H P m In this case we have D t H t i D t which is known as the Dirac equation with H being the Dirac Hamiltonian In D this case the wavefunction describ es spin fermions as we shall see Wavefunctions and Fields You may be wondering why I am talking ab out wavefunctions while in your eld theory course Dave Dunbar is telling you ab out elds Some of you may even be wondering what is the dierence between awavefunction and a eld The bottom line is that single particle wavefunctions work just ne if you want to describ e systems where the particle number is conserved Problems come when you wanttoallow for relativistic eects In particular antiparticles and hence the p ossibility that particles can annihilate or be pair pro duced In fact these new concepts can be accommo dated within the wavefunction approach but inaway which is not really very satisfactory Well take a lo ok at the problems encountered in trying to cling to the wavefunction way of thinking shortly Rather than patch things up its much more app ealing simply to ditch the usual interpretation of as a wavefunction and to identify it as a eld This eld is then sub jected to the usual laws of quantum mechanics This means elevating the eld and its canonical momentum to the status of op erators which are then deemed to satisfy the usual commutation relations This is what Dave has b een doing in his course In the limit that particle number is conserved the theory is equivalent to the single particle w avefunction approach It is imp ortant to emphasise that a eld is very dierentfromawavefunction Think rst of a classical eld I nd it easier to picture a eld by rst dividing space into innitesimally small b oxes In each box is a ctitious particle and the amplitude of the particles displacement from equilibrium is the value of the eld as the point where the small b ox is If the eld satises some wave equation then you can think of the motions of the individual particles as being inuenced by whats going on in the neighb ourho o d eg consider a vibrating membrane the particles can b e thought of as little harmonic oscillators all coupled together The eld is dened in the limit of vanishingly small boxes and so describ es a system with an innite number of degrees of freedom The Maxwell eld of electromagnetism is a famous classical eld We can go ahead an demand that it also be consisten t with the laws of quantum mechanics This means we must quantise the motion of the innite number of little harmonic oscillators To do this we simply imp ose the commutation relation x p i where x is the displacement i i i th of the i oscillator and p is its momentum Note that x is simply the value of the eld i i th at the point where the i box is lo cated After doing this we have a theory which is consistent with b oth relativity and quantum mechanics Photons emerge as the quanta of the electromagnetic eld and so the theory naturally describ es systems with anynumber of particles Maxwells wave equations are not the only equations we can write down which are consistent with relativity We can also write down the KleinGordon and Dirac equations to o there are others but these are the most relevant ones for particle physics Quantis ing the corresp onding elds leads to quanta whichhave spin and spin resp ectively the Maxwell eld leads to spin quanta Letting the spin quanta carry electric charge means that they can interact with the spin quanta and the formalism has no problem dealing with varying numb ers of quanta Of course Im skipping the details so as to giveyou an overview DaveDunbars course aims to provide a fairly comprehensive intro duction to this whole area A nal word of caution Notation

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