Feynman rules for QED

Transcription

1 Feynman rules for QD The Feynman Rules for QD Setting up Amplitudes Casimir s Trick Trace Theorems Slides from Sobie and lokland Physics 424 Lecture 1 Page 1

2 lectrons and positrons and (s = spin) satisfy the Dirac equations spinors and adjoints and satisfy and orthogonality and normalization and completeness and Physics 424 Lecture 1 Page 2

3 % % ' % % Photons $% "!# Lorentz condition % orthogonality & normalization & % ) Coulomb gauge and %( Completeness - *, * +* & % % Physics 424 Lecture 1 Page 3

4 . The Feynman Rules for QD The Feynman rules provide the recipe for constructing an amplitude Feynman diagram. from a Step 1: For a particular process of interest, draw a Feynman diagram with the minimum number of vertices. There may be more than one Physics 424 Lecture 1 Page 4

5 The Feynman Rules for QD 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 Physics 424 Lecture 1 Page

6 Step 3: The amplitude depends on 1. Vertex factors 2. Propagators for internal lines 3. Wavefunctions for external lines Physics 424 Lecture 1 Page

7 8 = Vertex Factors very QD vertex, contributes a factor of. 798;: is a dimensionless coupling constant and is related to the fine-structure constant by 8 : Physics 424 Lecture 1 Page 7

8 D C F F 7 I F F Propagators ach internal photon connects two vertices of the form and, so we should expect the photon propagator to contract the indices and. 798 : 798 : Photon propagator: 7 8 Internal fermions have a more complicated propagator, Fermion propagator: HJI The sign of matters here we take it to be in the same direction as the fermion arrow. Physics 424 Lecture 1 Page 8

9 @ N N M M Q O S S xternal Lines Since both the vertex factor and the fermion propagators involve matrices, but the amplitude must be a scalar, the external line factors must sit on the outside. Work backwards along every fermion line using: in out in out in out T PRQ P O Physics 424 Lecture 1 Page 9

10 O Matrix elements I follow fermion lines backward to give V O 798 U Physics 424 Lecture 1 Page 10

11 X ^ X Matrix elements II The matrix element is proportional to the two currents in the diagram below : W P Q Q_^ OY 798 : W P OYX ] N ` Ma` N ` Ma` Physics 424 Lecture 1 Page 11

12 b f b f e And Finally... Step 4: The overall amplitude is the coherent sum of the individual amplitudes for each diagram: Hdc c c b H b f H c c c b H b f Step 4a: Antisymmetrization. Include a minus sign between diagrams that differ only in the exchange of two identical fermions. Physics 424 Lecture 1 Page 12

13 g M N Mg N Mg N h i Mg N xamples There are only a handful of ways to make tree-level diagrams in QD. Today, we will construct amplitudes for habha scattering and Compton scattering. Next week, we will undertake thorough calculations for Mott scattering, pair annihilation. You will examine fermion pair-production via for your assignment. h g P i Physics 424 Lecture 1 Page 13

14 ^ X ^ X e X xample: habha Scattering N ` M ` N ` M ` N ` M ` N ` M ` bml bkj b Antisymmetrization : W P Q Qn^ O 798 : W P OX 7 b j ] 798 On 798 : W PRQ Q^ 798 : W P OX 7 b l H ] Physics 424 Lecture 1 Page 14

15 ^ ^ e p o 7 S I X o p H b 7 S I xample: Compton Scattering M ` X ` M ` X ` ` ` M ` M ` b H b b No antisymmetrization HI X 7 S T X 798 : OY 7 8 : P O^ b 7 HI S T X 798 : OJ 7 8 : P O^ H Physics 424 Lecture 1 Page 1

16 b Polarized Particles A typical QD amplitude might look something like SX Q s W P On brq The Feynman rules won t take us any further, but to get a number for we will need to substitute explicit forms for the wavefunctions of the external particles:,, and. Q P O StX 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 1 Page 1

17 fb f 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 in the squared amplitude We call this the spin-averaged amplitude and we denote it by. v fb u f Note that the averaging over initial state polarizations involves summing over all polarizations and then dividing by the number of independent polarizations, so v fb u f sum over the polarizations of all external particles. involves a Physics 424 Lecture 1 Page 17

18 s s s q x z s y O w s q z xy x s O w s q z y x s y y O w s q s q Spin Sums Let s simplify things even further and suppose that we have O W P O b q T W P O O W P On fb f O Then O x O W P OY OY x O W P OY OY x O W P OY OY P s { P O O W P OY Physics 424 Lecture 1 Page 18

19 s q O { q q OY P s { P O O fb f W P OY Applying the completeness relation HI l} P O l} ~ l} to in the squared-amplitude above (summing over the spins of paticle 2), P O O P s s fb f HI On P On lƒ OJ W P O Physics 424 Lecture 1 Page 19

20 fb f l The right-hand side is just a number, but if we represent the matrix multiplication with summations over indices, we can rewrite it as O P OY P O OY P OY OY W P OY OY W OY W P OY Finally, we apply the completeness relation once again, so that we get q W HI Physics 424 Lecture 1 Page 20

21 b q s { e q V In total, we have v fb u f V U O s W P OY HI P s HJI The factor of assuming exactly one of and corresponds to an initial-state particle. If they are both in the initial state (e.g., pair annihilation), the factor is. If neither is in the initial state (e.g., pair production), the factor is. is from the averaging over initial spins, O ^ O Physics 424 Lecture 1 Page 21

22 { Š Q I Q Casimir s Trick This procedure of calculating spin-averaged amplitudes in terms of traces is known as Casimir s Trick. T O s W P O O s W P O; s HI P s HJI Œ ŒŽ If antiparticle spinors are present in the spin sum, we use the corresponding completeness relation ~ l} l} PRQ l} Physics 424 Lecture 1 Page 22

23 = Traces ecause of Casimir s Trick, we re going to find ourselves calculating a lot of traces involving -matrices. eneral identities about traces: H H = Physics 424 Lecture 1 Page 23

24 uilding locks The two major identities that we will need in order to build more complicated trace identities are 8 ` U 8 L You can show and 8 š. In a similar fashion, we find that U 8 U U Physics 424 Lecture 1 Page 24

25 U H U 8 8 œ 8 œ œ š Simple Trace V The simplest trace identity is: matrix is zero, as is the trace of any odd The trace of a single number of -matrices. -matrices, For 2 U V šž œ 8 H8 š 8 š 8 Physics 424 Lecture 1 Page 2

26 Ÿ X y 7 Traces With. The vertex factor for weak interactions involves. y inspection, -matrices), (an even number of Since š Also, Physics 424 Lecture 1 Page 2

27 Ÿ V š S The Non-Trivial Trace Only with 4 (or more) other nonzero trace involving : -matrices can we obtain a š œ 7 œ š where the totally antisymmetric tensor is defined as œ ]ª VH for even permutations of 0123 for odd permutations of 0123 if any 2 indices are the same Physics 424 Lecture 1 Page 27

28 Contractions of the «Tensor Since S š œ is completely antisymmetric, we will get zero when we contract this with any tensor that is symmetric in 2 indices, such as 8 or H. Only contractions with another antisymmetric tensor survive: S š œ S š œ S š œ S š œ S š œ S U š œ š œ ž... Physics 424 Lecture 1 Page 28

29 X I X ± X ² X D C xample 1 One of the traces involved in habha scattering is ± W HI HJI We can expand this out to create 4 terms, but 2 of these terms (the ones linear in ) will involve 3 -matrices, and are therefore zero. Thus, HI H I 8 H This result will be contracted with another trace that is covariant (i.e., ) in and as opposed to contravariant. Physics 424 Lecture 1 Page 29

30 ² L ² UH Uµ I xample 2 It isn t always a joyous task to contract 2 traces together. ³ Consider valuating the traces, 8 H W 8 ] H W H ] { U V ] ² H { I Physics 424 Lecture 1 Page 30

31 Summary The Feynman rules for QD provide the recipe for translating Feynman diagrams into mathematical expressions for the amplitude. If we are interested in the spin-averaged amplitude v fb u f then we need not ever use explicit fermion spinors and photon polarization vectors. Instead, Casimir s Trick allows us to calculated spin-averaged amplitudes in terms of traces of -matrices. With practice, -matrix traces can be taken quite quickly. Physics 424 Lecture 1 Page 31