5Quantizing the Dirac field
III Quantum Field Theory
5.2 Yukawa theory
The interactions between a Dirac fermion and a real scalar field are governed by
the Yukawa interaction. The Lagrangian is given by
L =
1
2
∂
µ
φ∂
µ
φ +
1
2
µ
2
φ
2
+
¯
ψ(iγ
µ
∂
µ
− m)ψ −λφ
¯
ψψ.
where
µ
is the mass of the scalar and
m
is the mass of the Lagrangian. This is
the full version of the Yukawa theory. Note that the kinetic term implies that
[ψ] = [
¯
ψ] =
3
2
.
Since [φ] = 1 and [L] = 4, we know that
[λ] = 0.
So this is a dimensionless coupling constant, which is good.
Note that we could have written the interaction term of the Lagrangian as
L
int
= −λφ
¯
ψγ
5
ψ.
We then get a pseudoscalar Yuakwa theory.
We again do painful computations directly to get a feel of how things work,
and then state the Feynman rules for the theory.
Example. Consider a fermion scattering ψψ → ψψ.
ψ
ψ
ψ
ψ
p
q
p
0
q
0
We have initial and final states
|ii =
p
4E
p
E
q
b
s†
p
b
r†
q
|0i
|fi =
p
4E
p
0
E
q
0
b
s
0
†
p
0
b
r
0
†
q
0
|0i.
Here we have to be careful about ordering the creation operators, because they
anti-commute, not commute. We then have
hf| =
p
4E
p
0
E
q
h0|b
r
0
q
0
b
s
0
p
0
.
We can then look at the O(λ
2
) term in hf|(S − 1) |ii. We have
(−iλ)
2
2
Z
d
4
x
1
d
4
x
2
T
¯
ψ(x
1
)ψ(x
1
)φ(x
1
)
¯
ψ(x
2
)ψ(x
2
)φ(x
2
)
The contribution to scattering comes from the contraction
:
¯
ψ(x
1
)ψ(x
1
)
¯
ψ(x
2
)ψ(x
2
): φ(x
1
)φ(x
2
).
The two
ψ
’s annihilate the initial state, whereas the two
¯
ψ
create the final state.
This is just like the bosonic case, but we have to be careful with minus signs
and spinor indices.
Putting in |ii and ignoring c operators as they don’t contribute, we have
:
¯
ψ(x
1
)ψ(x
1
)
¯
ψ(x
2
)ψ(x
2
): b
s†
p
b
r
†
q
|0i
= −
Z
d
3
k
1
d
3
k
2
(2π)
6
2
p
E
k
1
E
k
2
[
¯
ψ(x
1
)u
m
k
1
][
¯
ψ(x
2
)u
n
k
2
]e
−ik
1
·x
1
−ik
2
·x
2
b
m
k
1
b
n
k
2
b
s
†
p
b
r†
q
|0i
where the square brackets show contraction of spinor indices
= −
1
2
p
E
p
E
q
[
¯
ψ(x
1
)u
r
q
][
¯
ψ(x
2
)u
s
p
]e
−iq·x
1
−ip·x
2
−[
¯
ψ(x
1
)u
s
p
][
¯
ψ(x
2
)u
r
q
]e
−ip·x−iq·x
2
|0i.
The negative sign, which arose from anti-commuting the
b
’s, is crucial. We put
in the left hand side to get
h0|b
r
0
q
0
b
s
0
p
0
[
¯
ψ(x
1
)u
r
q
][
¯
ψ(x
2
)u
s
p
]
=
1
2
p
E
p
0
E
q
0
[¯u
s
p
0
u
r
q
][¯u
r
0
q
0
u
s
p
]e
ip
0
·x
1
+iq
0
·x
2
− [¯u
r
0
q
0
u
r
q
][¯u
s
0
p
0
u
s
p
]e
ip
0
·x
2
+iq
0
·x
1
.
Putting all of this together, including the initial relativistic normalization of the
initial state and the propagator, we have
hf|S − 1 |ii
= (−iλ)
2
Z
d
4
x
1
d
4
x
2
(2π)
4
d
3
k
(2π)
4
ie
ik·(x
1
−x
2
)
k
2
− µ
2
+ iε
[¯u
s
0
p
0
· u
s
p
][¯u
t
0
q
0
· u
r
q
]e
ix
1
·(q
0
−q)+ ix
2
·(p
0
−p)
− [¯u
s
0
p
0
u
r
q
][u
r
0
q
0
u
s
p
]e
ix
1
·(p
0
−q)+ ix
2
·(q
0
−p)
= i(−iλ)
2
Z
d
4
k(2π)
4
k
2
− µ
2
+ iε
[¯u
s
0
p
0
· u
s
p
][u
r
0
q
0
· u
r
q
]δ
4
(q
0
− q + k)δ
4
(p
0
− p + k)
−[¯u
s
0
p
0
· u
y
q
][¯u
r
0
q
0
· u
s
p
]δ
4
(p
0
− q + k)δ
4
(q
0
− p + k)
.
So we get
hf|S − 1 |ii = iA(2π)
4
δ
4
(p + q −p
0
− q
0
),
where
A = (−iλ)
2
[¯u
s
0
p
0
· u
s
p
][¯u
r
0
q
0
· u
r
q
]
(p
0
− p)
2
− µ
2
+ iε
−
[¯u
s
0
p
0
· u
r
q
][¯u
r
0
q
0
· u
s
p
]
(q
0
− p)
2
− µ
2
+ iε
!
.