Compton Scattering in Scalar QED
Christian Johnson
December 1, 2015
1
Calculating Matrix Elements
There are three Feynman diagrams to consider when calculating the matrix
elements for φγ → φγ in Scalar QED:
1.1
S-Channel
p1
p4
k
p2
p3
Figure 1: S-channel diagram
The matrix element can be found by following the Feynman rules:
iM = (−ie)(pµ2 + k µ )1µ [
k2
i
](−ie)(k ν + pν3 )4?
ν
− m2
(1)
Kinematics requires that k µ = pµ1 + pµ2 = pµ3 + pµ4 , therefore we can replace
the ks above:
iM =
−ie2
(2pµ2 + pµ1 )1µ (2pν2 + pν4 )4ν?
k 2 − m2
(2)
But pµi iµ = 0 so:
M=
−e2
−e2
(2p
·
+p
·
)(2p
·
+p
·
)
=
(2p2 ·1 )(2p3 ·4 ) (3)
2 1
1 1
3 4
4 4
k 2 − m2
k 2 − m2
However, p~1 = −p~2 and p~3 = −p~4 , and the polarization vectors i are purely
transverse. So p2 · 1 = p1 · 1 = 0 and p3 · 4 = p4 · 4 = 0. Therefore the
S-channel does not contribute to the scattering amplitude:
M =0
1
(4)
1.2
T-Channel
p3
p1
k
p4
p2
Figure 2: T-channel diagram
Again, we calculate the amplitude from the Feynman rules:
iM = iµ (−ie)(k µ + pµ3 )[
k2
i
](−ie)(pν2 + k ν )4?
ν
− m2
(5)
Using kinematics to rewrite k µ = pµ3 − pµ1 = pµ2 − pµ4 :
M=
e2
(2pµ3 − pµ1 )(2pν2 − pν4 )1µ 4?
ν
2p2 · p4
(6)
Simplifying with the fact that pµi iµ = 0 yields:
M=
1.3
e2
(2p3 · 1 )(2p2 · ?4 )
2p2 · p4
(7)
Seagull Channel
Figure 3: Seagull-channel diagram. For some reason the legs weren’t labeled
by LATEX
2
The seagull diagram contributes a factor of:
2
?
M = 2e2 gµν µ1 ν?
4 = 2e 1 · 4
(8)
Therefore the total amplitude at tree level is:
M = 2e2 1 · ?4 +
2
1
(p3 · 1 )(p2 · ?4 )
p2 · p4
(9)
Calculating Cross Section
The differential cross section is given by:
dσ
1
=
|M |2
2
2
dΩ
64π ECM
(10)
dσ
dσ
Since we want d cos
θ instead of dΩ , we multiply this by 2π. Then it’s simply
a matter of plugging everything in and simplifying:
dσ
e4
(p3 · 1 )(p2 · ?4 )
= 2 2
1 · ?4 +
d cos θ
p2 · p4
8π ECM
3
2
(11)