Quantum ElectroDynamics III Feynman diagram Dr.Farida Tahir Physics department CIIT, Islamabad
Human Instinct What? Why? Feynman diagrams Feynman diagrams Feynman diagrams How?
What? Graphic way to represent exchange forces Describing a variety of particle interactions Developed by Richard Feynman when working on the development of QED 1942
Why? To calculate Decay rates and Scattering cross section
Calculation of Decay rates and Scattering cross sections Need two information Dynamical Evaluation of relevant Feynman diagram to determine the amplitude M for the process. Kinematical The phase space factor, it depends on the masses, energies, and momenta of the participants.
How? Ingredients (Rules) Recipe (Structure)
Feynman rules: Electron (e - ) & Positron (e + ) (Dirac eq.) Photons (Maxwell eq.) Dirac Equation in electromagnetic field (concept of gauge symmetry)
The Feynman rules for Electron (e - ) Positron (e + ) i ( ) X. P h s ψ ( X ) = ae u ( P) ψ ( X ) Free = ae i h ( X. P) v ( P) u ( λ γ mc) u = 0 ( λ Dirac γ + mc) v = 0 P λ ( λ γ P mc) = 0 ( λ Adjoint v γ P + mc) = 0 λ P λ λ s= 1,2 u s u s = u 1 u 2 = 0 uu 2mc Orthogonal = Normalized ( λ γ p + mc) λ Completeness v 1 v 2 = 0 vv = 2mc s= 1,2 v s v s = λ ( γ p mc) λ
The Feynman rules for (γ) Photon (γ) A μ ( X ) = ae i h ( X. P) ε μ () s Free μ ε Pμ ε μ ε = 0 * ( 1) μ (2) = * ε μ ε μ = 1 0 Lorentz condition Orthogonal Normalized s= 1,2 ( ε ) ( ) s ε s = δij pˆ i pˆ j ( ) i ( ) j Completeness
Complete Lagrangian for f & γ Lagrangian density describing the fermionic field in the presence of an electromagnetic field is L = Ψ( X ) [ ( ) ] λ λν λ γ i λ qaλ m Ψ( X ) F Fλν J Aλ 1 L = Ψ( X ) Ψ 4 [ ] λ λν λ λ iγ λ m Ψ( X ) F Fλν ( J + qψ( X ) γ ( X )) A λ Current coupled to A λ, describe the interaction vertex. 1 4 Feynman diagram Current produce by Dirac particle
Structure of Feynman Diagrams
Structure of Feynman Diagrams Fermions represented by straight lines with arrows pointing in direction of time flow Forward-facing arrows represent particles Backward-facing arrows represent antiparticles Photons and weak bosons, W - and W + and Z 0 are squiggly lines Gluons are curly lines
Line types External lines Enter and leave diagram Represent real observable particles real particles and must have E 2 = p 2 + m 2 Referred to as particle being on mass shell
Line types Internal lines Connect vertices, called propagators Represent virtual particles that cannot be observed Do not have to obey relativistic mass, energy, momentum relationship: (mc 2 ) 2 = E 2 -(pc) 2 Referred to as particle being off mass shell
Virtual At the point of emission (or absorption) the vertices gives a contradiction with Einstein relation between energy and mass. Vertices Conserve: energy, momentum, & charge for all types of interactions Determine order of perturbation contributes to the particular calculation Same number of arrows enter as leave
How to write currents by using F.D For each QED vertex, write factor μ i eγ For each internal photon line having momentum k write factor α g αβ β id F αβ ( k ) = i 2 k + iε For each internal lepton line having momentum p write factor p m is p = 1 + F ( ) i = 2 p m + iε p m 2 + iε
For each external (initial) fermions u r (p) For each external (final) fermions u r (p) For each external (initial) anti-fermions v r (p) For each external (final) anti - fermions v r (p)
For each initial photon ε rα (k) * For each final photon ε rα ( k) k α α k
Examples Basic vertices Electromagnetic Charge particle enters, emits (or absorbs) a photon and exits. 2 e α = c = 1 h 137 -ieγ μ
Current ' p μ p L = J μ J μ + ν K J μ = u id r μν ( p) i eγ u ( p') ( k) μ μν g i k + iε = 2 r J ν = u r ( K + p2) ieγνur ( p2) p K + p2 2
) ( ) ( ') ( ) ( 2 2 2 p u ie p K u i K g i p u ie p u M r r r r fi ν μν μ γ ε γ + = ) ( ) ( ') ( ) ( 2 2 2 2 p u p K u p u p u i K e i r r r r μ γ μ γ ε + =
Lowest Order Fundamental Processes Scattering e Mott μ eμ e Bhabha e + e Elastic e + Møller ee ee Inelastic Pair production + γ γ e e Compton eγ eγ Pair annihilation e e + γ γ Muon production + + e e μ μ
Scattering e + + + e e + e e + e + γ +γ γ + + γ e + e e + γ e + γ
Møller scattering e e ee Electron-electron scattering (Moller scattering) Two electron enters, a photon passes between them
Bahabha scattering e e + e e + Twist into any topological configuration Rule: Particle line running backward in time is interpreted as the corresponding antiparticle going forward e - e + annihilate to form a photon which produces a new e - e +
Bhabha & Moller scattering are related by cross symmetry + + + + 2 3 4 e e e e 1 A A C + B C + D + C B + D + D A + B
Compton scattering
Remember Dominant contribution comes from tree level 2 e α = h c = 1 137 Ignore higher order contribution
QED Interactions Higher order diagrams Charge screening
Recall Forces coupling Strength Range Particles Strong α s 1 10-15 Gluons; m=0 Electromagnetic α 1/137 Photon; m=0 Weak α 10-6 w 10-18 W &Z boson Gravity α g 10-39 graviton; m=0