6Quantum electrodynamics
III Quantum Field Theory
6.5 Computations and diagrams
We now do some examples. Here we will not explain where the positive/negative
signs come from, since it involves some tedious work involving going through
Wick’s theorem, if you are not smart enough to just “see” them.
Example. Consider the process
e
−
e
−
→ e
−
e
−
.
We again can consider the two diagrams
p, s
q, r
p
0
, s
0
q
0
, r
0
µ
ν
p, s
q, r
p
0
, s
0
q
0
, r
0
µ
ν
We will do the first diagram slowly. The top vertex gives us a factor of
−ie[¯u
s
0
p
0
γ
µ
u
s
p
].
The bottom vertex gives us
−ie[¯u
r
0
q
0
γ
ν
u
r
q
].
The middle squiggly line gives us
−
iη
µν
(p
0
− p)
2
.
So putting all these together, the first diagram gives us a term of
−i(−ie)
2
[¯u
s
0
p
0
γ
µ
u
s
p
][¯u
r
0
q
0
γ
µ
u
r
q
]
(p
0
− p)
2
!
.
Similarly, the second diagram gives
−i(−ie)
2
[¯u
s
0
p
0
γ
µ
u
s
q
][¯u
r
0
q
0
γ
µ
u
r
p
]
(p − q)
2
!
,
Example. Consider the process
e
+
e
−
→ γγ.
We have diagrams of the form
p, s
q, r
E
µ
, p
0
E
ν
, q
0
This diagram gives us
i(−ie)
2
[¯v
r
q
γ
ν
(
/
p −
/
p
0
+ m)γ
µ
u
s
p
]
(p − p
0
)
2
− m
2
E
µ
(p
0
)E
ν
(q
0
).
Usually, since the mass of an electron is so tiny relative to our energy scales, we
simply ignore it.
Example
(Bhabha scattering)
.
This is definitely not named after an elephant.
We want to consider the scattering process
e
+
e
−
→ e
+
e
−
with diagrams
p, s
q, r
p
0
, s
0
q
0
, r
0
p, s
q, r
p
0
, s
0
q
0
, r
0
These contribute
−i(−ie)
2
−
[¯u
s
0
p
0
γ
µ
u
s
p
][¯v
r
q
γ
µ
v
r
0
q
0
]
(p − p
0
)
2
+
[¯v
r
q
γ
µ
u
s
p
][¯u
s
0
p
0
γ
µ
v
r
0
q
0
]
(p + q)
2
!
.
Example (Compton scattering). Consider scattering of the form
γe
−
→ γe
−
.
We have diagrams
u
q
¯u
q
0
E
µ
(p)
E
ν
(p
0
)
p + q
u
q
¯u
q
0
E
µ
(p)
E
ν
(p
0
)
q −p
0
Example. Consider the
γγ → γγ
scattering process. We have a loop diagram
Naively, we might expect the contribution to be proportional to
Z
d
4
k
k
4
,
This integral itself diverges, but it turns out that if we do this explicitly, the
gauge symmetry causes things to cancel out and renders the diagram finite.
Example
(Muon scattering)
.
We could also put muons into the picture. These
behave like electrons, but are actually different things. We can consider scattering
of the form
e
−
µ
−
→ e
−
µ
−
This has a diagram
e
−
e
−
µ
−
µ
−
We don’t get the diagram obtained by swapping two particles because electrons
are not muons.
We can also get interactions of the form
e
+
e
−
→ µ
+
µ
−
by
e
−
e
+
µ
−
µ
+