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Feynman Rules for QED Feynman rules for QED The Feynman Rules for QED Setting up Amplitudes Casimir's Trick Trace Theorems Slides from Sobie and Blokland Physics 424 Lecture 16 Page 1 2 Page and ¢ ¥ ¨ © § ¨ © and © ¦ positrons £¥¤ equations § ¦ £¥¤ § 16 Dirac ¨ © e ¤ and ¢ the Lectur and ¦ satisfy £ satisfy § ¦ © § £ § spin) § = § © § Electrons (s § and ¦ ¦ £¥¤ £¥¤ and § and ¢ ¦ ¦ © £ £¥¤ and ¢ ¢ ¢ ¦ ¦ ¤ £¥¤ £ ¢ ¨ © spinors ¢ adjoints ¢ orthogonality ¢ normalization ¢ completeness ¡ ¡ ¡ ¡ ¡ 424 Physics 3 Page ¨ $ % "! # 16 e - * Lectur ¨ Photons ¨ * , ) +* ¦ & £¥¤ % ( ¨ % ' gauge ¦ condition ¦ and £ £¥¤ ¨ % entz % ¦ % & & ¤ ¥ £ Lor orthogonality normalization Coulomb Completeness % % % % ¡ ¡ ¡ ¡ ¡ 424 Physics The Feynman Rules for QED The Feynman rules provide the recipe for constructing an amplitude . from a Feynman diagram. ¡ Step 1: For a particular process of interest, draw a Feynman diagram with the minimum number of vertices. There may be more than one. 1 2/ 4 3 / © 1 0/ / Physics 424 Lecture 16 Page 4 The Feynman Rules for QED ¡ Step 2: For each Feynman diagram, label the four-momentum of each line, enforcing four-momentum conservation at every vertex. Note that arrows are only present on fermion lines and they represent particle flow, not momentum. 1 2/ 4 3 / © 1 2/ / Physics 424 Lecture 16 Page 5 Step 3: The amplitude depends on 1. Vertex factors 2. Propagators for internal lines 3. Wavefunctions for external lines Physics 424 Lecture 16 Page 6 Vertex Factors 5 Every QED vertex, 6 5 < 798;: contributes a factor of 6 . 8 : is a dimensionless coupling constant and is related to the ? fine-structure constant by 8 : = > @ A Physics 424 Lecture 16 Page 7 Propagators 798 < 6 Each internal photon connects two vertices of the form : 798 B 6 and : , so we should expect the photon propagator to contract the indices C and D . 7 8 E < B ? Photon propagator: F Internal fermions have a more complicated propagator, K G 7 HJI /F ? ? Fermion propagator: F I E The sign of F matters here — we take it to be in the same direction as the fermion arrow. Physics 424 Lecture 16 Page 8 External Lines Since both the vertex factor and the fermion propagators @ @ involve L matrices, but the amplitude must be a scalar, the external line factors must sit on the outside. Work backwards along every fermion line using: N N M M 6 6 5 in 5 out 5 in 5 out in out T P PRQ O Q O S S < < Physics 424 Lecture 16 Page 9 Matrix elements I G K G K < U 798 V follow fermion lines backward to give O 6 O 5 6 5 Physics 424 Lecture 16 Page 10 11 Page the in [ Q_^ K ents B 6 ? ^ curr \ \ 798 : G II N N ` ` ? P Q W 5 5 two ? the K B X < to \ 6 E 798 E 16 e Z elements G]\ Lectur [ OYZ X Z oportional \ \ K < pr Ma` Ma` 6 5 5 is Matrix 798 : G OYX P . W element below matrix The diagram 424 Physics And Finally... Step 4: The overall amplitude is the coherent sum of the individual amplitudes for each diagram: b b b H Hdc > Z ? c c ? ? f f f f b b b H H e > Z ? c c c Step 4a: Antisymmetrization. Include a minus sign between diagrams that differ only in the exchange of two identical fermions. Physics 424 Lecture 16 Page 12 Examples There are only a handful of ways to make tree-level diagrams in QED. Today, we will construct amplitudes for Bhabha scattering N N G K G K M M g g 5 6 5 6 5 5 5 5 and Compton scattering . Next week, we will undertake thorough calculations for Mott N G K G K h h M g g 5 5 6 6 scattering 5 5 , pair annihilation . You will P N G K i i M g examine fermion pair-production via 5 5 for your assignment. Physics 424 Lecture 16 Page 13 14 ? ^ Page \ \ [ [ N N ` ` Qn^ OnZ 5 5 K K B B 6 6 798 : 798 : 6 G G ? ? P Q PRQ W W ? ? K K B ? B X bml < < \ \ E H X Z 798 798 E E \ \ E Scattering Z Z ` ` M M G]\ G]\ 5 5 bkj [ [ O Z Q ^ 16 ? > e b ^ K K < < \ \ 6 6 N N ` ` Lectur Bhabha e 5 5 798 798 : : G G X X O O P P W W 7 7 6 > > Example: b j l b X Z \ \ ` ` M M 5 5 Antisymmetrization 424 Physics B < 15 ? ? S ^ S ? Page < B \ T T \ ` M S S X X 6 ` 5 p p OYZ OJZ K K B B 6 6 798 798 : : G G ? b ? ? K K H I I E E H H I I X Z Scattering ? ? Z \ \ b ? K K X 6 ` ` M ? / / \ \ X H E 5 \ \ H E > b Z Z / / \ \ G G Z Z 16 7 7 e \ \ G G ^ ? e \ \ ` M Lectur K K 6 < < ` 5 Compton 6 6 7 7 8 8 : : G G ^ ^ O O P P o o 7 7 > > Example: ? X Z Z b b antisymmetrization \ \ 6 ` ` M 5 No 424 Physics Polarized Particles A typical QED amplitude might look something like W [ < brq s P OnZ Q S ? X < The Feynman rules won't take us any further, but to get a number for b we will need to substitute explicit forms for the P O Q StX Z ? wavefunctions of the external particles: , , and < . If all external particles have a known polarization, this might be a reasonable way to calculate things. More often, though, we are interested in unpolarized particles. Physics 424 Lecture 16 Page 16 Spin-Averaged Amplitudes If we do not care about the polarizations of the particles then we need to 1. Average over the polarizations of the initial-state particles 2. Sum over the polarizations of the final-state particles ? f f in the squared amplitude b . We call this the spin-averaged amplitude and we denote it by u v ? f f b Note that the averaging over initial state polarizations involves summing over all polarizations and then dividing by the u v ? f f number of independent polarizations, so b involves a sum over the polarizations of all external particles. Physics 424 Lecture 16 Page 17 18 Page have we z that OYZ y z 6 x x OYZ z s ? x O T y y | [ s 6 6 ? OYZ suppose O x y y [ s 6 6 s ? P s O ? ? x x x ? O O O Z s O P O Z P W and w w w { O Z P [ [ [ [ [ W ? ? ? ? ? O O O O O s s s s s Sums 16 q b OYZ OYZ OYZ OnZ OYZ e P P P P P W W W W W further Lectur Spin even q q q q q ? f b things f simplify Let's Then 424 Physics 19 Page the over | OnZ s P K K ? | OYZ (summing s P H I H I ? O P ? [ { OJZ / \ G / \ G [ s ? above O s OnZ P > O Z P W { O P l } OYZ P elation W r 16 e O l } ? q q Lectur ? Z q ~ f b ? f ed-amplitude l } b f f 2), l squar completeness the the paticle in ? O P of ? O spins Applying to 424 Physics 20 that Page can the so we esent again, epr r indices, [ K K once [ we K Z O Z K if G [ OYZ P K over H I O Z P OYZ OYZ P but G OYZ Z elation OYZ , G / \ G r W K G W OYZ P W G number 16 summations e a > > > > q ? f [ b Lectur OYZ just f with completeness is OYZ P W l the side apply as we it multiplication , right-hand get ewrite we r matrix The Finally 424 Physics In total, we have W [ b s P OYZ O q ? V u v ? { | P G G K f f K b s s HJI H e \ \ I q Z Z ? ? U / / Z The factor of ? is from the averaging over initial spins, O O assuming exactly one of Z and ? corresponds to an initial-state particle. If they are both in the initial state (e.g., Z pair annihilation), the factor is ^ . If neither is in the initial state (e.g., pair production), the factor is V . Physics 424 Lecture 16 Page 21 22 | K Page use in HJI we / \ G ? sum, s P K amplitudes spin K I H the I rick. E / \ G T in Z s 's / rick G \ { elation T r esent > s ' spin-averaged PRQ l } pr > 16 T Casimir e e [ Q l } ar O as ? K ? Q Lectur s G Z ~ O P W l } completeness calculating Casimir [ known O of is Z s spinors e O; P W traces esponding ocedur of pr corr antiparticle This If terms the 424 Physics 23 Page K ourselves G K find G to > K H -matrices. 6 K K K going G G G G e = 16 we'r e races involving T traces: > > > > Lectur rick, K K K K T = H 's traces G about G of G G lot a Casimir of identities Because General calculating 424 Physics 24 Page similar a build In to . B der 8 K @ B or 6 B K < in > < 6 6 6 L < E 6 G B 6 B 6 < B B < need 6 < 6 @ e E E 8 < B 8 @ 6 U 6 ar G U B B will 6 6 < Blocks U E 6 U U > > we and 16 B e @ < B 6 > > > > that 8 B identities > Lectur < < < < ` 6 6 6 < 8 B 6 6 trace < Building 6 that identities that find major we show complicated can two e ou mor fashion, The Y 424 Physics 25 Page odd any U B K < of 8 6 < B 6 8 H H trace U K K B B B 6 the < G V < 8 6 is < B 8 @ G < B G U < 8 as E 8 > K @ 8 o, G V Identities zer 8 B > > > > < is K 16 B 8 is: e 6 race < @ 6 T Lectur G matrix > K 6 6 identity 6 B 6 single < trace 6 a -matrices.
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