Subatomic Physics: Particle Physics Lecture 4. Quantum Electro-Dynamics & Feynman Diagrams. Antimatter. Virtual Particles. Yukawa Potential for QED

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1 Ψ e (r, t) exp i/ (E)(t) (p) (r) Subatomic Physics: Particle Physics Lecture Quantum ElectroDynamics & Feynman Diagrams Antimatter Feynman Diagram and Feynman Rules Quantum description of electromagnetism Virtual Particles Yukawa Potential for QED Antimatter (Lecture ) KleinGordon equation predicts negative energy solutions. Dirac Interpretation: The vacuum is composed of negative energy levels with E<0. Each level is filled with two electrons of opposite spin: the Dirac Sea. A hole in the sea with charge!e and E<0, appears as a state with charge e and E>0. This idea lead Dirac to predict the positron, discovered in 9. FeynmanStueckelberg Interpretation: negative energy particles moving backwards in space and time correspond to... positive energy antiparticles moving forward in space and time x e (x, t) E>0 Ψ e ( r, t) exp i/ ( E)( t) ( p) ( r) e! (!x,!t) E<0 t

2 Feynman Diagrams A Feynman diagram is a pictorial representation of a particular process (decay or scattering) at a particular order in perturbation theory. Feynman diagrams can be used to represent and calculate the matrix elements, M, for scattering and decays. Feynman diagrams are very useful and powerful tools. We will use them a lot in this course. We use them a lot in our research! e µ Conventions Time flows from left to right (occasionally upwards) )/")&,# e Richard Feynman receiving the 967 Noble prize in physics for his invention of this technique. Fermions are solid lines with arrows Antifermion are solid lines with backward pointing arrows. Bosons are wavy (or dashed) lines Use Feynman Rules to calculate M at different orders in perturbation theory. µ time Quantum Electrodynamics (QED) QED is the quantum theory of electromagnetic interactions. Classical electromagnetism: Force between charged particle arise from the electric field Q ˆr E = π 0 r act instantaneously at a distance Quantum Picture: Force between charged particle described by exchange of photons. Strength of interaction is related to charge of particles interacting. "))/* e.g. electonproton scattering ep!ep propagated by the exchange of photons M(ep ep) e q m charge of electron photon propagator charge of proton Feynman rules: Vertex term: each photon!charged particle interaction gives a factor of fermion charge, Q. Propagator term: each photon gives a factor of /(q m )=/q where q is the photon fourmomentum. Matrix element is proportional to product of vertex and propagator terms. e

3 Electromagnetic Vertex Basic electromagnetic process: Initial state charged fermion (e, µ, " or quark antiparticles) Absorption or emission of a photon Final state charged fermion Examples: e! "e! # ; e! #"e! α Mathematically, EM interactions are described by a term for the interaction vertex and a term for the photon propagator QED Conservation Laws Momentum, energy and charge is conserved at each vertex Fermion flavour (e, µ,!, u, d...) is conserved: e.g. u"u# allowed, c"u# forbidden Coupling strength Matrix element is proportional to the fermion charge: M e Alternatively use the fine structure constant, # α = e π 0 c = e 7 strength of the coupling at the vertex is α π 5 Virtual Particles The force between two charged particles is propagated by virtual photons. A particle is virtual when its fourmomentum squared, does not equal its rest mass: m X = EX p X Allowed due to Heisenberg Uncertainty Principle: can borrow energy to create particle if energy ($E=mc ) repaid within time ($t), where!e!t " % Example: electronpositron scattering creating a muon pair: e e!!µ µ!. p p e µ p e q q = m µ p Four momentum conservation: Momentum transferred by the photon is: Squaring, p p = p p q = (p p ) = (p p ) q = (p ) (p ) p p = m e (E E p p ) > 0 In QED interactions mass of photon propagator is nonzero. Only intermediate photons may be virtual. Final state ones must be real! 6

4 Basic QED Processes All of these described by the same basic vertex term,! Q None of above processes is physical as they violate energymomentum conservation: p = (p e p e ) = m Join two together to get a real processes 7 Perturbation Theory QED is formulated from time dependent perturbation theory. Perturbation series: break up the problem into a piece we can solve exactly plus a small correction. e.g. for e e! "µ µ! scattering. Many more diagrams have to be considered for a accurate prediction of!(e e! "µ µ! ). As # is small the lowest order in the expansion dominates, and the series quickly converges! For most of the course, we will only consider lowest order contributions to processes. Lowest Order M α nd Order e e M α rd Order: e e µ µ M α 6 8 e e µ µ µ µ......

5 QED Coupling Constant Strength of interaction between electron and photon However, # s not really a constant... An electron is never alone: it emits virtual photons, these can convert to electron positron pairs... α = e π 0 7 Any test charge will feel the e e! pairs: true charge of the electron is screened. At higher energy (shorter distances) the test charge can see the bare charge of the electron. At large R test charge sees screened e charge Test Charge Test Charge At small R test charge sees bare e charge # varies as a function of energy and distance 9 Yukawa Potential The quantum and classical descriptions of electromagnetism should agree. Yukawa developed theory whereby exchange of bosons describes force / potential. KleinGordon equation: t Ψ(r, t) = c Ψ(r, t) m c Ψ(r, t) Nontime dependent solutions obey: Ψ(r) = m c Ψ(r) Spherically symmetric solutions of this are: Ψ( r ) = g πr exp mc r Interpret this as a potential, V, caused by a particle of mass, m. V (r) = g πr exp r R For electromagnetic force, m=0, g=e. V EM (r) = e πr with R = Potential felt by a charged particle due to the exchange of a photon. mc 0

6 QED Scattering Examples Elastic electronproton scattering:!" #$ e!"! p!e #! p σ M e q = 6π α q Momentum transferred to photon from e! : q = (p f p i ) = p f p i p f p i = m e (E f E i p f p i cos θ) M e q = πα q E f E i sin (θ/) Rutherford scattering: e! Au!e! Au, can neglect recoil of the gold atoms: E=Ei=Ef σ Z π α E sin (θ/) Inelastic e! e!µ µ! e e Momentum transferred by photon: For this situation need full density of states, &, (which we won t do...) σ = 6πE µ µ q = (p e p e ) = s M e s = πα s M = πα s Summary of Lecture Relativistic quantum mechanics predicts negative energy particles: antiparticles. Two interpretations: a negative energy particle travelling backwards in time. a hole in a vacuum filled with negative energy states. Quantum Electro Dynamics (QED) is the quantum mechanical description of the electromagnetic force. Electromagnetic force propagated by virtual photons: q = m Feynman diagrams can be used to illustrate QED processes. Use Feynman rules to calculate the matrix element, M. All QED interactions are described by a fermionfermionphoton vertex: Strength of the vertex is the charge of the fermion, Qf. Fermion flavour and energymomentum are conserved at vertex. The photon propagator ~ /q where q is the momentum transferred by the photon. M is proportional to product of vertex and propagator terms.