Quantum Electrodynamics

2012 Matthew Schwartz II-6: Quantum electrodynamics 1 Introduction Now we are ready to do calculations in QED. We have found that the Lagrangian for QED is 1 = F2 + i ψ¯ D ψ m ψ¯ ψ (1) L − 4 µν − with D µ ψ = ∂µψ + ieA µ ψ. We have also introduced quantized Dirac fields d3 p 1 ψ( x) = ( as us e − ipx + bs † vs eipx) (2) (2 π) 3 p p p p s 2 ωp X Z d3 p p1 ψ¯ ( x) = ( bs v¯ s e − ipx + as † u¯ s eipx ) (3) (2 π) 3 p p p p s 2 ωp X Z The spinor creation and annihilation operatorsp for spinors must anticommute by the spin-statistics theorem: s † s ′ † s s ′ s † s ′ † s s ′ a p , a q = a p, a q = b p , bq = b p, b q = 0 (4) n o′ ′ n o s s † s s † 3 3 a , a = b , b = δss ′ (2 π) δ ( p q) (5) p q p q − n o n o A basis of spinors for each momentum pµ can be written as √p σ ξs √p σ ηs us ( p) = · , vs ( p) = · (6) √p σ¯ ξs √p σ¯ ηs · ! − · ! T T with ξ1 = η1 = (1 , 0) and ξ2 = η2 = (0, 1) . These spinors satisfy 2 us ( p) u¯s ( p) = p + m s =1 X2 (7) vs ( p) v¯s ( p) = p m − s =1 X We also calculated the Feynman propagator for a Dirac spinor: d4 p i( p + m) 0 T ψ(0) ψ¯ ( x) 0 = eipx (8) h | { }| i (2 π) 4 p2 m2 + iε Z − In this lecture we will derive the Feynman rules for QED and then perform some important calculations. 2 QED Feynman rules The Feynman rules for QED can be read right off of the Lagrangian just as in scalar QED. The only subtlety is possible extra minus signs coming from anti-commuting spinors within the time- ordering. First we write down the Feynman rules, then derive the supplementary minus sign rules. A photon propagator is represented with a squiggly line i pµ pν = − gµν (1 ξ) (9) p2 + iε − − p2 ¡ 1 2 Section 2 Unless we are explicitly checking gauge invariance, we will usually work in Feynman gauge, ξ = 1 , where the propagator is igµν = − ( Feynman gauge) p2 + iε A spinor propagator is a solid line with an arrow i( p + m) ¡ = p2 m2 + iε − The arrow points to the right forparticles and to the left for anti-particles. For internal lines, the arrow points with momentum¡ flow. External photon lines get polarization vectors b =ǫ µ( p) ( incoming) (10) ⊗ ⋆ b = ǫ ( p) ( outgoing) (11) ⊗ µ Here the blob means the rest of the diagram. External fermion lines get spinors, with u spinors for electrons and v spinors for positrons. =us ( p) ⊗ =u¯ s ( p) ⊗¡ = v¯ s ( p) ¡⊗ =vs ( p) ¡⊗ External spinors are on-shell (they are forced to be on-shell by LSZ). So for external spinors, we can use the equations of motion ¡ ( p m) us ( p) = u¯ s ( p)( p m) = 0 (12) − − ( p + m) vs ( p) = v¯ s ( p)( p + m) = 0 (13) which will simplify a number of calculations. Expanding the Lagrangian 1 2 µ µ = F + ψ¯ ( iγ ∂µ m) ψ e ψ¯ γ ψ A µ (14) L − 4 µν − − µ we see the interaction is int = e ψ¯ γ ψ A µ. Since there is no factor of momentum, the Feynman rule is the sameL for any− combination of incoming or outgoing fields (unlike in scalar QED) e+ e− e− + e = = = e− = ieγµ (15) e− + + − e e ¡ ¡ ¡ ¡ µ The µ on the γ will get contracted with the µ of the photon which will either be in the gµν of the photon propagator (if the photon is internal) or the ǫ µ of a polarization vector (if the photon is external). µ µ The γ = γαβ as a matrix will always get sandwiched between spinors, as in µ µ u¯ γ u = u¯α γαβ uβ (16) for e − e − scattering, or v¯ γµ u for e+ e − annihilation, etc. The barred spinor always goes on the µ left, since the interaction is ψ¯ A µ γ ψ. If there is an internal fermion line between the ends, the fermion propagator goes between the end spinors: γν γµ p p p i( p2 + m) 1 2 3 = ie 2 u p γµ γν u p ǫ2 q ǫ1 q ( ) ¯( 3) 2 2 ( 1 ) µ( 2 ) ν( 1 ) (17) − p2 m + iε ¡ − ν µ ǫ1 ǫ2 QED Feynman rules 3 µ µ µ µ µ µ where the photon momenta are q1 = p2 p1 and q2 = p3 p2 . In this example, the 3 γ-matrices get multiplied and then sandwiched between− the spinors.− To see explicitly what is a matrix and what is a vector, we can add in the spinor indices µ i( p2 + m) ν µ i( p2 + m) βγ ν u¯( p3 ) γ γ u ( p1 ) = u¯α ( p3 ) γ γ uδ( p1 ) (18) p2 m2 + iε αβ p2 m2 + iε γδ 2 − 2 − If we tied the ends of the diagram above together we would get a loop p1 γν γµ µ ǫν ǫ (19) 1 p p 2 → → p2 For fermion loops we use the same convention as for scalar loops that the loop momentum goes in the direction of the particle-flow arrow. In the loop, since any possible intermediate states are allowed, we must integrate over the momenta of the virtual spinors as well as sum over their possible spins. The uδ u¯α¡then in Eq. (18) gets replaced by a propagator which sums over all possible spins. This is done automatically since the numerator of the propagator is ( p2 + m) δα = s s s uδ u¯α . We also must integrate over all possible momenta constrained by momentum conser- vation at each vertex. So the loop in Eq. (19) evaluates to P d4 p d4 p i = ( i e) 2 1 2 (2 π) 4 δ4( p + p M − − (2 π) 4 (2 π) 4 2 − Z i( p1 + m) βγ i( p2 + m) δα p ) ǫ2 ⋆( p) ǫ1 ( p) γµ γν 1 µ ν αβ p2 m2 + iε γδ p2 m2 + iε 1 − 2 − The extra minus sign is due to spin-statistics as will be explained shortly. Contracting all the µ µ µ µ µ spinor indices and replacing p1 by p + k and p2 by k p + k d4 k i( p + k + m) i( k + m) ǫν µ 2 2 ⋆ 1 µ ν i = 1 ǫ2 = e ǫ ǫ Tr γ γ (20) M p p µ ν (2 π) 4 ( p + k) 2 m2 + iε k2 m2 + iε k Z " − − # where the¡ trace is a trace of spinor indices. Computing Feynman diagrams in QED will often involve taking the trace of products of γ-matrices. A useful general rule is that the spinor matrices always get multiplied together in the direction opposite to the particle-flow arrow, which allows us to read off Eqs. (17) and (20) easily from the corresponding diagrams. 2.1 Signs Recall that spinors anti-commute within a time-ordered product: T ψ( x) ψ( y) = T ψ( y) ψ( x) (21) { } − { } Minus signs coming from such anti-commutations appear in the Feynman rules. It is easiest to see when they should appear by example. Consider Moller Scattering ( e − e − e − e − ) at tree-level. There are two Feynman diagrams, → for the t-channel (in Feynman gauge) p1 p3 µ igµν ν i t = = ( ie) u¯( p3) γ u( p1 ) − ( ie) u¯( p4) γ u( p2 ) (22) M ± − ( p p ) 2 − 1 − 3 p2 p4 ¡ 4 Section 2 and u-channel p1 p 3 µ igµν ν i u = = ( ie) u¯( p3 ) γ u( p2 ) − 2 ( ie) u¯( p4) γ u( p1 ) (23) M p4 ± − ( p p ) − 1 − 4 p2 ¡ The question is what sign should each diagram have? To find out, recall that these Feynman diagrams represent S-matrix elements. By the LSZ reduction theorem, they represent contributions to the Fourier transform of the Green’s function G ( x , x , x , x ) = Ω T ψ( x ) ψ¯ ( x ) ψ( x ) ψ¯ ( x ) Ω (24) 4 1 2 3 4 h | { 1 3 2 4 }| i with external propagators removed and external spinors added. The first non-zero contribution to this Green’s function in perturbation theory comes at order e2 in an expansion of free fields G = ( ie) 2 d4 x d4 y (25) 4 − Z Z 0 T ψ( x ) ψ¯ ( x ) ψ( x ) ψ¯ ( x ) ψ¯ ( x) A( x) ψ( x) ψ¯ ( y) A( y) ψ( y) 0 (26) × 1 3 2 4 D n o E where the big () indicate that the spinors inside are contracted. More explicitly, we can write G ( x , x , x , x ) = ( ie) 2 γµ γν d4 x d4 y 4 1 2 3 4 − β1 β2 β3 β4 Z Z ¯ ¯ ¯ µ ¯ ν 0 T ψα 1 ( x ) ψ ( x ) ψα 2 ( x ) ψ ( x ) ψ ( x) A ( x) ψβ2 ( x) ψ ( y) A ( y) ψβ4 ( y) 0 (27) × 1 α 3 3 2 α 4 4 β1 β3 In this form, we can anti-commute the spinors within the time-ordering before performing any contractions. To get Feynman diagrams out of this Green’s function, we have to perform contractions, which means creating fields from the vacuum and then annihilating them. To be absolutely cer- tain about the sign coming from the contraction, it is easiest to anti-commute the fields so that the fields which annihilate spinors are right next to the fields which create them.

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