PHY489 Lecture 15 Quantum Electrodynamics • Follow Griffiths sections 7.5-7.7 + example 7.7 from section 7.8, using the e-µ- e-µ- scattering as an example. Feynman rules for Quantum Electrodynamics e-µ- e-µ- scattering Spin-averaging of amplitudes Scattering of an electron from a heavy spin-1/2 particle Feynman Rules for Quantum Electrodynamics 1. Notation: label incoming and outgoing four-momenta & corresponding spins; label the internal four momenta; assign arrows to lines as follows: • Arrows on external lines indicate whether they represent particles or anti- particles (the latter flow backwards in time). • Arrows on external photon lines point forwards. • Arrows on internal lines are in the direction that preserves the “direction of flow”: each vertex must have at least one arrow entering and one arrow leaving. - - - - EXAMPLE: e µ e µ scattering (in QED ! No weak interaction contribution) so at low energies, where the weak µ µ µ µ interaction contribution is negligible p2, s2 p4, s4 (suppressed by the high mass of the weak gauge bosons) e.g. 1 1 γ q γ vs. q2 q2 − M 2c2 p1, s1 p3, s3 e e e e for the propagator. Think of the µ here as a heavy charged particle with M >> me. Feynman Rules for QED cont’d 2. External Lines: contribute factors to M as follows: • electrons incoming u e.g. fermion outgoing ✝ 0 u ≡ u γ • positrons incoming v ✝ 0 e.g. anti-fermion outgoing v ≡ v γ µ • photons incoming ε ⎫ ⎬ polarization vectors: see § 7.4 µ* ⎭ outgoing ε µ 3. Vertices: each vertex contributes a factor of i g γ (photon is spin-1). (here my g is Griffiths g ≡ e 4 π / c = 4 π α ). e 3. Propagators: each internal line contributes a factor of: i γ µq + mc • electrons and positrons ( µ ) [ e.g. internal fermion line ] q2 − m2c2 gµν • photons −i 2 q Feynman Rules for QED cont’s 5. Conservation of energy and momentum: for each vertex write a factor 4 2π δ 4 k + k + k ( ) ( 1 2 3 ) enforcing overall energy and momentum conservation at that vertex. (here each k represents a 4 momentum; incoming four momenta are positive and outgoing are negative). 6. Integrate over all internal momenta: for each internal momentum q write −4 a factor of 2 π d 4 q and integrate. ( ) 7. Cancelling the remaining δ-function (expressing overall energy and momentum conservation) leaves you with -iM. 8. Anti-symmetrization: include a relative minus sign between diagrams differing only by the exchange of two incoming (or outgoing) electrons (or positrons) or of and incoming electron with an outgoing positron (or vice versa) [ see next slide ]. Anti-symmetrization of QED diagrams Recall the ABC model scattering process AABB. There are two diagrams that contribute at lowest order: We summed the amplitudes for these two diagrams to get the total amplitude (so, with a relative positive sign). p2, s2 p3, s3 p2, s2 p3, s3 For fermions the relative sign between _ p , s such diagrams is negative. 4 4 p1, s1 p4, s4 p1, s1 Here for e−e− → e−e− More on anti-symmetrization of QED diagrams + − + − Consider electron positron scattering: e e → e e (Bhabha scattering) p2, s2 p3, s3 p2, s2 p3, s3 _ p , s p , s p1, s1 p4, s4 1 1 4 4 Relative negative sign since diagrams differ by exchange of incoming positron and outgoing electron: = Electron-Muon Scattering in QED − − − − Now back to e µ → e µ scattering in QED: µ µ p2, s2 p4, s4 We had: q γ Now apply Feynman rules to obtain M. p1, s1 p3, s3 e e Procedure is to write down terms working backwards in time along each fermion line: s s 4 electron line: u ( 3 ) p ig µ u( 1 ) p 2 4 p p q ( 3 ) γ ( 1 ) ( π ) δ ( 1 − 3 − ) outgoing vertex incoming δ-function for conservation of energy electron coupling electron and momentum at electron vertex. spinor spinor s s 4 muon line: u ( 4 ) p igγ ν u( 2 ) p 2π δ 4 p + q − p ( 4 ) ( 2 )( ) ( 2 4 ) Electron-Muon Scattering cont’d µ µ p2, s2 p4, s4 −ig q γ Propogator is µ ν . Applying the Feynman rules we obtain q2 p1, s1 p3, s3 e e [ spin indices have been suppressed ] 4 4 −ig 4 d q ⎡u p igγ µ u p 2π δ 4 p − p − q ⎤ µν ⎡u p igγ ν u p 2π δ 4 p + q − p ⎤ ⎢ ( 3 ) ( 1 )( ) ( 1 3 )⎥ 2 ⎢ ( 4 ) ( 2 )( ) ( 2 4 )⎥ 4 ⎣ ⎦ q ⎣ ⎦ 2π ∫ ( ) Use first δ-function for integration over d4q. This leaves: 2 −ig 4 ⎡u p γ µu p ⎤ ⎡u p γ u p ⎤ 2π δ 4 p + p − p − p 2 ⎣ ( 3 ) ( 1 )⎦ ⎣ ( 4 ) µ ( 2 )⎦( ) ( 1 2 3 4 ) p − p ( 1 3 ) Canceling the overall δ-function leaves us with -iM: g 2 M = ⎡u p γ µu p ⎤ ⎡u p γ u p ⎤ 2 ⎣ ( 3 ) ( 1 )⎦ ⎣ ( 4 ) µ ( 2 )⎦ p − p ( 1 3 ) Electron-Muon Scattering cont’d µ µ p2, s2 p4, s4 g 2 M = ⎡u p γ µu p ⎤ ⎡u p γ u p ⎤ 2 ⎣ ( 3 ) ( 1 )⎦ ⎣ ( 4 ) µ ( 2 )⎦ q γ p p ( 1 − 3 ) p1, s1 p3, s3 e e Each of component of each of these factors has the form: ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜4 ⎟ This is just a number, so the above expression is just a ( 1 x 4 )⎜ 4 x 4 ⎟ ⎜x ⎟ number (e.g. a scalar quantity), which we will learn to ⎜ ⎟ ⎜1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ calculate. But first, we need to learn how to deal with spin. Dealing with Spin If we know the spins of the incoming and outgoing particles, we can write down the appropriate spinors and do the matrix multiplication. In practice, we often have (experimentally) beams of particles with random − − − − spins, and in doing an experiment ( for example e µ → e µ ) we might measure only the number of particles scattered at a particular angle. This means we effectively average over the initial spins and sum over the final spins. Can do this too when we calculate the amplitude. Could compute the amplitude Mi for each possible configuration, and then sum 2 and average, to get the so-called spin-average amplitude M . However, we can also calculate this directly. 2 g µ s M = ⎡u 3 γ u 1 ⎤ ⎡u 4 γ u 2 ⎤ [with, e.g. u 3 ≡ u ( 3 ) p ] We had 2 ⎣ ( ) ( )⎦ ⎣ ( ) µ ( )⎦ ( ) ( 3 ) p p ( 1 − 3 ) 4 2 g ∗ * M = ⎡u 3 γ µu 1 ⎤ ⎡u 4 γ u 2 ⎤ ⎡u 3 γ νu 1 ⎤ ⎡u 4 γ u 2 ⎤ 4 ⎣ ( ) ( )⎦ ⎣ ( ) µ ( )⎦ ⎣ ( ) ( )⎦ ⎣ ( ) ν ( )⎦ p − p ( 1 3 ) * We therefore need to compute quantities of the form G ≡ ⎡ u a Γ 1 u b ⎤ ⎡ u a Γ 2 u b ⎤ ⎣ ( ) ( )⎦ ⎣ ( ) ( )⎦ where a, b label the appropriate spins and momenta and Γ1, Γ2 are two 4x4 matrices. Spin-averaging cont’d * We need to evaluate factors like G ≡ ⎡u a Γ u b ⎤ ⎡u a Γ u b ⎤ ⎣ ( ) 1 ( )⎦ ⎣ ( ) 2 ( )⎦ ⎡u a u b ⎤ Note that ( ) Γ 2 ( ) is just a number (or a 1x1 matrix), so ⎣ ⎦ * ✝ ✝ ✝ ✝ ⎡u a Γ u b ⎤ = ⎡u a Γ u b ⎤ ⎡ 0 ⎤ u b ✝ 0✝u a ( ) 2 ( ) ( ) 2 ( ) = u a γ Γ2u b = Γ2γ ⎣ ⎦ ⎣ ⎦ ⎢ ( ) ( )⎥ ( ) ( ) ⎣ ⎦ ✝ 0 0 ✝ 0 0 0 0✝ 0 = u b γ γ Γ2γ u a [since γ γ = 1 , γ = γ ] ( ) ( ) ( ) 0 ✝ 0 0 ✝ 0 = u b γ Γ2γ u a = u b Γ2u a where Γ ≡ γ Γ γ ( ) ( ) ( ) ( ) 2 2 * G ⎡u a u b ⎤ ⎡u a u b ⎤ ⎡u a u b ⎤ ⎡u b u a ⎤ So ≡ ( )Γ1 ( ) ( )Γ2 ( ) = ( )Γ1 ( ) ( )Γ2 ( ) ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Remember,we have to sum this quantity over all possible spins. Write it in this form to take advantage of the completeness relation for the spinors, when performing this spin sum: u(s)u (s) µ p mc ∑ = γ µ + s=1,2 ⎫ ⎪ See eqn 7.99 and ⎬ problem 7.24 (s) (s) µ p mc ∑ v v = γ µ − ⎪ s=1,2 ⎭ Summing G over the spin orientations of particle b we obtain ⎪⎧ s s ⎪⎫ G u a u( b ) p u ( b ) p u a ∑ = ( )Γ1 ⎨ ∑ ( b ) ( b )⎬Γ2 ( ) b spins s 1,2 − ⎩⎪ b = ⎭⎪ = u a Γ γ µ p + m c Γ u a ( ) 1 ( bµ b ) 2 ( ) = u a Γ p + m c Γ u a [ where p ≡ γ µ p : in general a/ ≡ γ µ a ] ( ) 1 ( / b b ) 2 ( ) / b bµ µ = u a Qu a [with Q = Γ p + m c Γ ] ( ) ( ) 1 ( / b b ) 2 [ Note that Q is still just a 4x4 matrix ] Need to also sum over possible spin states of particle a: ⎛ ⎞ s s G u ( a ) p Qu( a ) p ∑ ⎜ ∑ ⎟ = ∑ ( a ) ( a ) a−spins ⎝ b−spins ⎠ s =1,2 a ⎛ ⎞ s s G u ( a ) p Qu( a ) p Need to evaluate ∑ ⎜ ∑ ⎟ = ∑ ( a ) ( a ) a−spins ⎝ b−spins ⎠ s =1,2 a Write out the matrix multiplication explicitly: s s ⎪⎧ s s ⎪⎫ u ( a ) p Q u( a ) p = Q u( a ) p u ( a ) p ∑ ∑ ( a )i ij ( a ) j ∑ ij ⎨ ∑ ( a ) ( a )⎬ i, j=1,4 s =1,2 i, j=1,4 s =1,2 a ⎩⎪ a ⎭⎪ ji each of these 3 quantities is just a number u (s)Q u(s) Q u (s) u(s) [ see footnote on pg 251 of text ] ∑∑ ∑ i ij j ⇒ ∑∑ ij ∑ i j i j s=1,2 i j s=1,2 ✝ 0 ⎪⎧ s s ⎪⎫ ⎪⎧ s s ⎪⎫ using u u = u γ u = uu Q u ( ) u( ) = Q u( ) u ( ) i j ( )i j ( ) ji ∑ ij ⎨ ∑ i j ⎬ ∑ ij ⎨ ∑ ⎬ i, j ⎩⎪s=1,2 ⎭⎪ i, j ⎩⎪s=1,2 ⎭⎪ ji ⎛ ⎞ ⎪⎧ s s ⎪⎫ G = Q u( a ) p u ( a ) p = Q p + m c = Tr Q p + m c so ∑ ⎜ ∑ ⎟ ∑ ij ⎨ ∑ ( a ) ( a )⎬ ∑ ij ( / a a ) ji ( ( / a a )) i, j=1,4 a−spins ⎝ b−spins ⎠ i, j=1,4 ⎪sa =1,2 ⎪ ⎩ ⎭ ji u u u✝ 0 u uu Show that i j = γ j = ( ) (which we used on the previous slide).