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 Electrons and positrons
¡ spinors ¦ £¥¤ ¦ £¥¤ ¨ ©
¢ ¢ § and (s = spin) satisfy the Dirac equations and ¨ ©
§
¡ adjoints ¨ © ¨ © ¥ § ¢ ¢ © § § © ¢
and satisfy and
¡ orthogonality £ ¦ £ ¦ £ ¦ £ ¦ ¢ ¢ § § and
¡ normalization ¢ ¢ § § and
¡ completeness ¦ £¥¤ ¦ £¥¤ ¦ £¥¤ ¦ £¥¤ ¢ ¢ © § § ©
¤ and ¤
Physics 424 Lecture 16 Page 2 Photons "! ¨ ¨ $ % #
¡ Lorentz condition ¥ %
¡ orthogonality &
% % £ ¦ £ ¦
¡ normalization &
' % %
¡ Coulomb gauge ) % % and (
¡ Completeness +* ¨ ¨ ¨ - , &
% % * * ¦ £¥¤ ¦ £¥¤ ¤
Physics 424 Lecture 16 Page 3 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 6 Page on lines 16 lines e Lectur depends external internal for for amplitude factors The opagators avefunctions 3: ertex W V Pr 3. 1. 2.
Step
424 Physics 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 Matrix elements II
The matrix element is proportional to the two currents in the diagram below. 798 E B < W G K [ W G K [ B < 798 798 P P Q 6 Q_^ OYX 6 OYZ ? : : ? G]\ K \ Z X E N Ma` 5 \ 5 \ ^ X ` 6 N Ma` 5 \ 5 \ Z ? `
Physics 424 Lecture 16 Page 11 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 6 5 6 5 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 Example: Bhabha Scattering N N M M 5 \ 5 \ 5 \ 5 \ ^ ^ X X ` ` ` ` 6 6 N N M M 5 \ 5 \ 5 \ 5 \ Z ? Z ? ` ` ` ` b bkj bml e
> Antisymmetrization E 798 E B < W G K [ W G K [ B < b 798 7 798 P P Q 6 Qn^ O 6 O > ? X Z j : : ? G]\ K \ Z X E 798 E B < G K [ W G K W [ B < b 798 798 7 PRQ P 6 OnZ 6 O Q > ^ ? X : : l ? G]\ K H \ Z ?
Physics 424 Lecture 16 Page 14 Example: Compton Scattering M M 6 \ 5 \ 6 \ 5 \ ^ ^ X X ` ` ` ` 6 \ 6 \ ? ? M M ` ` 5 \ 5 \ Z Z ` ` b b b H e
> No antisymmetrization Z ? p o G K 7 H \ \ I Z X E G K G K T B < b 798 7 7 P 6 OYZ 8 6 O / / S S > ^ ? Z X : < : B ? ? G K \ \ I Z X E E p o K G 7 H H \ \ I Z ? G K G K T < B b 7 798 7 P 8 6 6 O OJZ / / S S > ^ ? ? X : : B G ? K ? < H \ \ I Z ? E
Physics 424 Lecture 16 Page 15 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 Spin Sums
Let’s simplify things even further and suppose that we have W [ b s P O O q Z ? ? T f f W W [ [ b s s P P OnZ O O O q ? Then Z ? x w z x y W [ s s P OYZ O O 6 O q ? ? Z w z x x x y [ W s s P O 6 OYZ OYZ O q ? ? z w x x y y y [ W s s P O 6 6 6 OYZ OYZ O q ? ? | { P [ W s s P P OYZ OYZ O O q ? ?
Physics 424 Lecture 16 Page 18 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 21 state
Page |
(e.g.,
K Z initial
an HJI state spins, the
to Z in
/
\
G
s P is
initial K
initial ? the
esponds
I H over in
. ?
neither V
corr ?
/
G \
If O
is
s [
.
?
{ O
both
^
Z s
e is 16
ar OYZ
and
P W e factor
averaging
U V
Z O Lectur the the they factor
of
q q
If
b v om
the ? one
fr
f b
is
?
f
Z u oduction), have particle.
of e exactly pr we pair annihilation), factor total,
initial-state In The pair assuming (e.g.,
424 Physics Casimir’s Trick
This procedure of calculating spin-averaged amplitudes in terms of traces is known as Casimir’s Trick. { | P T G W W [ [ G K K s s s s P P H HJI O; O O O \ I \ > ? Z ? Z / / G K
If antiparticle spinors Q are present in the spin sum, we use the corresponding completeness relation l l G K } } PRQ Q \ I > E / Z ? l ~ }
Physics 424 Lecture 16 Page 22 Traces
Because of Casimir’s Trick, we’re going to find ourselves calculating a lot of traces involving 6 -matrices.
General identities about traces: G K G K G K H H > G K G K = = > K G G K > K G K G G K > >
Physics 424 Lecture 16 Page 23 Building Blocks
The two major identities that we will need in order to build more complicated trace identities are B < @ 8 8 > B < K G < B B < U L 6 6 8 > ` @ @ < B < B
6 6 6 6 6 6 8 > > You can show that < and < . In a similar fashion, we find that K G B < B < < B U 6 6 6 6 8 6 6 > E < < B < B U 6 6 6 6 > E < B B @ U 6 6 > E B U 6 > E
Physics 424 Lecture 16 Page 24 Simple Trace Identities G K @ V
The simplest trace identity is: >
The trace of a single 6 matrix is zero, as is the trace of any odd number of 6 -matrices. G K G K < B < B B < U H
6 6 6 6 6 6 6 For 2 -matrices, > G K B < U U 8 > G K B < V 8 > B < @ 8 >
G K < B B B < < B < @ H 6 6 6 6 8 8 8 8 8 8 > E
Physics 424 Lecture 16 Page 25
Traces With ¡
The vertex factor for weak interactions involves 6 . ¡ G K ¢
6 By inspection, > . ¡ y Z ? X 7
6 6 6 6 6 6 Since > (an even number of -matrices), ¡ G K < ¢ 6 6 > ¡ G K < B ¢ 6 6 6 6 >
Also, ¡ G K < B ¢ 6 6 6 >
Physics 424 Lecture 16 Page 26
The Non-Trivial Trace
Only with 4 (or more) other 6 -matrices can we obtain a ¡ nonzero trace involving 6 : ¡ G K < B B < @ 7 6 6 6 6 6 S >
where the totally antisymmetric tensor is defined as ¥ V E ¦§¦©¨ for even permutations of 0123 B < ¤£ V H S for odd permutations of 0123 ¦§¦]ª ¢ if any 2 indices are the same
Physics 424 Lecture 16 Page 27 Contractions of the « Tensor B <
Since S is completely antisymmetric, we will get zero when we contract this with any tensor that is symmetric in 2 < < G K B < B B H 8 \ \ \ \ ? Z indices, such as or Z ? .
Only contractions with another antisymmetric tensor survive: B < @ U S S > E B < B < ¯ ® S S > ¬ E B < ¬ B < ¯ ¯ ¯ ¯ U S S > ° E E ° ° B < ¬ ¬ ¬ . .
Physics 424 Lecture 16 Page 28 Example 1
One of the traces involved in Bhabha scattering is W G K G [ K < B ± H HJI 6 \ \ I 6 > /Z /X
We can expand this out to create 4 terms, but 2 of these terms 6 (the ones linear inI ) will involve 3 -matrices, and are therefore zero. Thus, ? K G K G B < B < ± H \ 6 \ I 6 6 6 > /Z /X ? < < G G K K B B B < B < @ @ H H \ \ \ \ \ \ 8 I 8 > Z² X E X Z Z X
This result will be contracted with another trace that is B < C D covariant (i.e., B < as opposed to contravariant ) in and .
Physics 424 Lecture 16 Page 29 Example 2
It isn’t always a joyous task to contract 2 traces together.
G K G K ³ < B 6 \ 6 \ 6 \ 6 \ > Z ? Z ? Consider / / < / B / Evaluating the traces, < < W G K [ ³ B B B < @ H \ \ \ \ \ \ 8 > Z ? ² E ? Z Z ? W G]\ K [ @ H L \ \ \ \ \ 8 Z² ? Z ? Z ? E < B B < B < ?´? ? ? ? ? { | G]\ K G K G K @ @ U ® U V H H \ \ \ \ \ \ \ > Z² ? ² Z ? Z² ? E Z ? ? ?¶? | { K G]\ U µ H \ I I > ? Z ² Z
Physics 424 Lecture 16 Page 30 Summary
The Feynman rules for QED provide the recipe for translating Feynman diagrams into mathematical expressions for the amplitude. u v ? f f b If we are interested in the spin-averaged amplitude 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 6 -matrices.
With practice, 6 -matrix traces can be taken quite quickly.
Physics 424 Lecture 16 Page 31