3Interacting fields
III Quantum Field Theory
3.4 Feynman diagrams
This is better, but still rather tedious. How can we make this better? In
the previous example, we could imagine that the term that contributed to the
integral above represented the interaction where the two
ψ
-particles “exchanged”
a
φ
-particle. In this process, we destroy the old
ψ
-particles and construct new
ones with new momenta, and the
φ
-particle is created and then swiftly destroyed.
The fact that the
φ
-particles were just “intermediate” corresponds to the fact
that we included their contraction in the computation.
We can pictorially represent the interaction as follows:
ψ
ψ
ψ
ψ
The magical insight is that every term given by Wick’s theorem can be interpreted
as a diagram of this sort. Moreover, the term contributes to the process if and
only if it has the right “incoming” and “outgoing” particles. So we can figure
out what terms contribute by drawing the right diagrams.
Moreover, not only can we count the diagrams like this. More importantly,
we can read out how much each term contributes from the diagram directly!
This simplifies the computation a lot.
We will not provide a proof that Feynman diagrams do indeed work, as it
would be purely technical and would also involve the difficult work of what it
actually means to be a diagram. However, based on the computations we’ve
done above, one should be confident that at least something of this sort would
work, maybe up to some sign errors.
We begin by specifying what diagrams are “allowed”, and then specify how
we assign numbers to diagrams.
Given an initial state and final state, the possible Feynman diagrams are
specified as follows:
Definition
(Feynman diagram)
.
In the scalar Yukawa theory, given an initial
state |ii and final state |f i, a Feynman diagram consists of:
–
An external line for all particles in the initial and final states. A dashed
line is used for a φ-particle, and solid lines are used for ψ/
¯
ψ-particles.
φ
ψ
¯
ψ
–
Each
ψ
-particle comes with an arrow. An initial
ψ
-particle has an incoming
arrow, and a final
ψ
-particle has an outgoing arrow. The reverse is done
for
¯
ψ-particles.
φ
ψ
¯
ψ
–
We join the lines together with more lines and vertices so that the only loose
ends are the initial and final states. The possible vertices correspond to
the interaction terms in the Lagrangian. For example, the only interaction
term in the Lagrangian here is
ψ
†
ψφ
, so the only possible vertex is one
that joins a φ line with two ψ lines that point in opposite directions:
φ
ψ
¯
ψ
Each such vertex represents an interaction, and the fact that the arrows
match up in all possible interactions ensures that charge is conserved.
–
Assign a directed momentum
p
to each line, i.e. an arrow into or out of
the diagram to each line.
φ
ψ
¯
ψ
p
p
2
p
1
The initial and final particles already have momentum specified in the
initial and final state, and the internal lines are given “dummy” momenta
k
i
(which we will later integrate over).
Note that there are infinitely many possible diagrams! However, when we lay
down the Feynman rules later, we will see that the more vertices the diagram has,
the less it contributes to the sum. In fact, the
n
-vertices diagrams correspond
to the nth order term in the expansion of the S-matrix. So most of the time it
suffices to consider “simple” diagrams.
Example
(Nucleon scattering)
.
If we look at
ψ
+
ψ → ψ
+
ψ
, the simplest
diagram is
ψ
ψ
ψ
ψ
On the other hand, we can also swap the two particles to obtain a diagram of
the form.
ψ
ψ
ψ
ψ
These are the ones that correspond to second-order terms.
There are also more complicated ones such as
ψ
ψ
ψ
ψ
This is a 1-loop diagram. We can also have a 2-loop diagram:
ψ
ψ
ψ
ψ
If we ignore the loops, we say we are looking at the tree level.
To each such diagram, we associate a number using the Feynman rules.
Definition
(Feynman rules)
.
To each Feynman diagram in the interaction, we
write down a term like this:
(i) To each vertex in the Feynman diagram, we write a factor of
(−ig)(2π)
4
δ
4
X
i
k
i
!
,
where the sum goes through all lines going into the vertex (and put a
negative sign for those going out).
(ii)
For each internal line with momentum
k
, we integrate the product of all
factors above by
Z
d
4
k
(2π)
4
D(k
2
),
where
D(k
2
) =
i
k
2
− m
2
+ iε
for φ
D(k
2
) =
i
k
2
− µ
2
+ iε
for ψ
Example.
We consider the case where
g
is small. Then only the simple diagrams
with very few vertices matter.
ψ
ψ
ψ
ψ
p
1
p
0
1
p
2
p
0
2
k
ψ
ψ
ψ
ψ
p
1
p
0
2
p
2
p
0
1
k
As promised by the Feynman rules, the two diagrams give us
(−ig)
2
(2π)
8
(δ
4
(p
1
− p
0
1
− k)δ
4
(p
2
+ k −p
0
2
) + δ
4
(p
1
− p
0
2
− k)δ
4
(p
2
+ k − p
0
1
)).
Now integrating gives us the formula
(−ig)
2
Z
d
4
k
(2π)
4
i(2π)
8
k
2
− m
2
+ iε
(δ
4
(p
1
− p
0
1
− k)δ
4
(p
2
+ k − p
0
2
) + δ
4
(p
1
− p
0
2
− k)δ
4
(p
2
+ k − p
0
1
)).
Doing the integral gives us
(−ig)
2
(2π)
4
i
(p
1
− p
0
1
)
2
− m
2
+
i
(p
1
− p
0
2
)
2
− m
2
δ
4
(p
1
+ p
2
− p
0
1
− p
0
2
),
which is what we found before.
There is a nice physical interpretation of the diagrams. We can interpret
the first diagram as saying that the nucleons exchange a meson of momentum
k
=
p
1
− p
0
1
=
p
2
− p
0
2
. This meson doesn’t necessarily satisfy the relativistic
dispersion relation
k
2
=
m
2
(note that
k
is the 4-momentum). When this
happens, we say it is off-shell, or that it is a virtual meson. Heuristically, it
can’t live long enough for its energy to be measured accurately. In contrast, the
external legs are on-shell, so they satisfy p
2
i
= µ
2
.