Section 9: Quantum Electrodynamics

Physics 8.33 Section 9: Quantum Electrodynamics May c W. Taylor 8.33 Section 9: QED / 6

9. Feynman rules for QED Field content: A µ(x) gauge field, ψ(x) Dirac spinor Action Z» S = d x ψ(iγ µ D µ m)ψ Z F =» d x ψ(i/ m)ψ ea µ ψγ µ ψ FµνFµν Feynman rules: Fermion propagator Photon propagator EM coupling vertex p k µ ν i a b µ i /p m + iε k + iε ieγ µ ab (gµν ( ξ) kµkν k ) (Feyn. gauge ξ = ) Incoming photon: ɛ µ, outgoing photon: (ɛ µ ) Incoming fermion/antifermion: u s (p)/ v s (p) Outgoing fermion/antifermion: ū s (p)/v s (p) c W. Taylor 8.33 Section 9: QED / 6

Comments on QED Feynman rules (momentum) arrows on external edges: incoming/outgoing p conserved at vertices R over undetermined momenta signs: must fix order of in/out fermions loops contribute (-) factor! Example: a ( ψψ)( ψψ)a + (note: ( ψψ) commutes) Fermion loop: ( ψψ)( ψψ)( ψψ) Can now compute general diagrams, e.g. e e e e : e- e- e- + e- e + e e + e, e + e γγ : homework e- e- e- e- c W. Taylor 8.33 Section 9: QED 3 / 6

Example: e + e µ + µ Basic reaction in e + e colliders used to calibrate. m µ 5.7MeV m e e, µ, τ 3 generations of leptons. Z S = [ ψ e(i /D m e)ψ e + ψ µ(i /D m µ)ψ µ FµνFµν ] same Feynman rules, fermions are e ± or µ ±. µ - k, r p, s e - µ + k, r q = p + p p, s e + im = [ v s (p )( ieγ µ )u s (p)][ igµν ][ū r (k)( ieγ ν )v r (k )] q Using ( vγ µ u) = u γ µ+ γ v = u γ γ µ v = ūγ µ v M = e ` v(p )γ µ u(p)ū(p)γ ν v(p ) `ū(k)γ µv(k ) v(k )γ νu(k) q average over initial s, s, sum over final r, r X M(s, s r, r ) = e q Tr[( /p m e)γ µ (/p + m e)γ ν ]Tr[(/k + m µ)γ µ(/k m µ)γ ν] s,s,r,r c W. Taylor 8.33 Section 9: QED / 6

X Me + e µ + µ = e q Tr[( /p m e)γ µ (/p + m e)γ ν ]Tr[(/k + m µ)γ µ(/k m µ)γ ν] Need Tr γ µ γ ν, Tr γ µ γ ν γ λ,... Tr(γ µ... γ µ k+ ) = Tr(γ 5 γ 5 γ µ... γ µ k+ ) = Tr(γ µ γ ν ) = Tr(γµ γ ν + γ ν γ µ ) = Tr (a)(g µν ) = g µν Tr(γ µ γ ν γ λ γ σ ) = (g µν g λσ g µλ g νσ + g µσ g νλ ) Tr[(/p m)γ µ (/p + m)γ ν ] = [ g µν (p p + m ) + p µ p ν + p ν p µ ] So (k k )(p p ) terms cancel and X h M = e (k p)(k p ) + (k p )(k i p) + m µ(p p ) + m e(k k ) + m em q µ spins = e s [(m e + m µ) + (m e + m µ)(s t u) + (t + u )] (s = (p + p ), t = (p k), u = (p k ) ; k k = s m µ, p k = t m e m µ,...) c W. Taylor 8.33 Section 9: QED 5 / 6

X spins M eē µ µ = e q h (k p)(k p ) + (k p )(k i p) + m µ(p p ) + m e(k k ) + m em µ Assume E m e, set m e ; OK since me M µ ; write m = mµ Kinematics in center of mass (COM) frame p = (E, Eẑ) k = (E, k) p = (E, Eẑ) k = (E, k) where k + m = E, k z = k cos θ. q = (p + p ) = E (= s) p p = E p k = p k = E E k cos θ p k = p k = E + E k cos θ X spins M = e E [m E + (E E k cos θ) + (E + E k cos θ) ] = e [( + m m ) + ( E E ) cos θ] c W. Taylor 8.33 Section 9: QED 6 / 6

For E m e, e + e µ + µ matrix element X M = e [( + m m ) + ( E E ) cos θ] spins Thus ( dσ dω )cm = E (v) k π (E ( X M ) cm) = r α»( m + m m ) + ( 6E E E E ) cos θ [v, E cm E] Using R dω = R π sin θdθ R π, R cos θ π/3, r σ tot = πα m m ( + 3Ecm E E ) For E m, dσ dω cm α E cm ( + cos θ) σ tot πα 3E cm [Note: Couplings + dim. analysis σ α E ; just need π 3 ] c W. Taylor 8.33 Section 9: QED 7 / 6

q σ tot = πα m ( + m ) p (E m)/m for E = m + ɛ 3Ecm E E Compare with tau particle production: σ(e + e τ + τ )/σ(e + e µ + µ ) Experimentally measured τ + τ production, fit gives m τ 78 ± 7 MeV [P & S] c W. Taylor 8.33 Section 9: QED 8 / 6

Helicity structure Take ultrarelativistic limit m e, m µ (E m e, m µ) ξl Helicities separate. Compute, e.g. σ(e R e+ L µ R µ+ L ) ξ R + γ 5 ξ η Write (in general spinor representation) γ 5 = γ 5 = η ξ η = ξ P ± = ± γ5 P ± = P ± P ±P = projection ops. P ±u(p) R L spinor R L fermion P±v(p) L R antifermion To select e R/L, insert v(p )γ µ ± γ 5 u(p) = v (p ) e R/L annihilates only with e+ L/R (photon is spin ) ± γ 5 γ γ µ u(p) c W. Taylor 8.33 Section 9: QED 9 / 6

Example: e R e+ L M = X µ R µ+ L e s,s,r,r q ` v(p )γ µ P +u(p)ū(p)γ ν P +v(p ) `ū(k)γ µp +v(k ) v(k )γ νp +u(k) Tr[/p γ µ P ±/pγ ν P ±] = Tr[/p γ µ /pγ ν ± γ 5 So M = e q Similarly, So as promised i ] = h g µν (p p ) + p µ p ν + p ν p µ iε αµβν p αp β h (p k)(p k ) + (p k )(p k) + ε αµβν ε ρµσνp αp β k ρ k σi = 6e q (p k )(p k) = e ( + cos θ) dσ dω (e R e + L µ R µ + L ) = α ( + cos θ) Ecm e L e + R µ L µ + R : e R e + L µ L µ + R, e L e + R µ R µ + L : α Ecm α Ecm X α ( + cos θ) Ecm ( + cos θ) ( cos θ) c W. Taylor 8.33 Section 9: QED / 6

NR limit E m µ (E still m e) Choose e R e+ L, p = Eẑ u(p) = EB @ v(p )γ µ u(p) = E ( ) C A σ µ σ µ v(p ) = E B @ B Photon has J z =, ε + = (ˆx + iŷ); how about ū(k)γ µ v(k )? u(k) = ξ m ξ v(k ) = ξ m ξ M(e R e + L µ µ + ) = e q ( 8mEξ» M = e Tr ξξ θ indep., need J z = ξ = ξ ξ, ξ = @ C A C A = E(,, i, ) j ū(k)γ µ v(k, µ = ) = mξ σ i ξ, µ = i ξ ) = e ξ = e dσ dω Spin avg.: ξ α (erē L µµ) = k Ecm E dσ dω = α E cm c W. Taylor 8.33 Section 9: QED / 6 k E

9.3 Recall scalar QED: p p µ ie(p + p ) µ B -p B Consider A + φ B, A φ + B vs. p φ A A im(a + φ(p) B) = im(a B + φ( p)) [as function of p] Only of p >, p > only one physical process. Related by analytic continuation. Same for fermions, if choose phases cleverly. With PS conventions, P u(p)ū(p) = /p + m = ( /p m) = P v( p) v( p) so ( ) for each crossing : = c W. Taylor 8.33 Section 9: QED / 6

Example: e µ e µ µ X spins - A k, r p, s e - µ + k, r q = p + p p, s e + M B = e q Tr `(/p + m e)γ µ (/p + m e)γ ν Tr = ( ) X MA(p = p, k = k ) e - B p e - p q = p - p µ - k k µ - (/k + m µ)γ µ(/k + m µ)γ ν For E m µ, X MA = e s (t + u ) crossing: s t, t u, u s X MB = e t (s + u ) Note: kinematics very different. dσ α = dω Ecm( cos θ) ( + ( + cos θ) ) as θ θ cm (cp. Rutherford cross-section) c W. Taylor 8.33 Section 9: QED 3 / 6

9.3 : e γ e γ p p + k k k p + p - k p p k k im = ( ie )ε µ ε ν `ū(p )γ µ (/p + /k + m)γ ν u(p) + ( ie )ε s m u m Note: (/p + m)γ µ u(p) = p µ u(p) γ µ (/p m)u(p) µ ε ν» im = ie ε µ ε νū(p γ µ /kγ ν + γ µ p ν ) + γν /k γ µ + γ ν p µ u(p) p k p k Gauge invariance (Ward Id.) [prove in general later] ff ε µ ε µ + ak µ ε µ ε µ + ak µ im = (as for scalar Compton) Need to sum Σε µε ν; Assume k µ = (k,,, k) Σε µε ν = B @ `ū(p )γ ν (/p /k + m)γ µ u(p) C A c W. Taylor 8.33 Section 9: QED / 6

Write M = M µ ε µ M = Σε v M v M µ ε µ Ward Identity M + M 3 = M = M 3 M = Σg µνm µ M ν = M + M + M + M 3 Valid for all k so Σε µε ν g µν Matrix element for e γ e γ» im = ie ε µ ε νū(p γ µ /kγ ν + γ µ p ν ) + γν /k γ µ + γ ν p µ u(p) p k p k Σ spins M G µνρσ AA = = e High energy: E m e» Tr + G µνρσ BA (/p + m) γµ /kγ ν + γ µ p ν gµρgνσ [Gµνρσ AA p k + G µνρσ AB + G µνρσ BB ] (/p + m) γσ /kγ ρ + γ σ p ρ p k Σ = e s Tr ˆγ µ /p γ µ(/p + /k)γ ν /pγ ν(/p + /k) + AB + BA + BB,... c W. Taylor 8.33 Section 9: QED 5 / 6

Compute AA contribution with γ µ /pγ µ = /p: Σ M = e s Tr ˆγ µ /p γ µ(/p + /k)γ ν /pγ ν(/p + /k) = e s Tr `/p (/p + /k)/p(/p + /k) = e s Tr `/p /k/p/k = 8e s (p k)(p k) = e u s AB = BA = BB e s/u exchange s and u (Recall: in COM frame, Σ M = e u s + s u dσ d cos θ = πα s u s s u dσ dσ = π = πα d cos θ dω s Full cross-section in lab frame (p = ): Klein-Nishina M e ) (COM frame) dσ d cos θ = πα ω» ω m ω ω + ω ω sin θ (notation: /ω /ω = ( cos θ)/m, k = (ω,,, ω), k = (ω, ω sin θ,, ω cos θ)) Observe sin θ correction to ultrarelativistic calculation; Cp. problem 5 3: For ω, [ ] ( + cos θ) (same for φ) c W. Taylor 8.33 Section 9: QED 6 / 6