6Weak decays
III The Standard Model
6.1 Effective Lagrangians
We’ll only consider processes where energies and momentum are much less than
m
W
, m
Z
. In this case, we can use an effective field theory. We will discuss
more formally what effective field theories are, but we first see how it works in
practice.
In our case, what we’ll get is the Fermi weak Lagrangian. This Lagrangian
in fact predates the Standard Model, and it was only later on that we discovered
the Fermi weak Lagrangian is an effective Lagrangian of what we now know of
as electroweak theory.
Recall that the weak interaction part of the Lagrangian is
L
W
= −
g
2
√
2
(J
µ
W
+
µ
+ J
µ†
W
−
µ
) −
g
2 cos θ
W
J
µ
n
Z
µ
.
Our general goal is to compute the S-matrix
S = T exp
i
Z
d
4
x L
W
(x)
.
As before,
T
denotes time-ordering. The strategy is to Taylor expand this in
g
.
Ultimately, we will be interested in computing
hf|S |ii
for some initial and final states
|ii
and
|fi
. Since we are at the low energy
regime, we will only attempt to compute these quantities when
|ii
and
|fi
do not
contain
W
±
or
Z
bosons. This allows us to drop terms in the Taylor expansion
having free W
±
or Z components.
We can explicitly Taylor expand this, keeping the previous sentence in mind.
How can we possibly get rid of the
W
±
and
Z
terms in the Taylor expansion?
If we think about Wick’s theorem, we know that when we take the time-ordered
product of several operators, we sum over all possible contractions of the fields,
and contraction practically means we replace two operators by the Feynman
propagator of that field.
Thus, if we want to end up with no
W
±
or
Z
term, we need to contract all
the
W
±
and
Z
fields together. This in particular requires an even number of
W
±
and
Z
terms. So we know that there is no
O
(
g
) term left, and the first
non-trivial term is O(g
2
). We write the propagators as
D
W
µν
(x − x
0
) = hT W
−
µ
(x)W
+
ν
(w
0
)i
D
Z
µν
(x − x
0
) = hT Z
µ
(x)Z
ν
(x
0
)i.
Thus, the first interesting term is
g
2
one. For initial and final states
|ii
and
|fi
,
we have
hf|S |ii = hf|T
1 −
g
2
8
Z
d
4
x d
4
x
0
J
µ†
(x)D
W
µν
(x − x
0
)J
ν
(x
0
)
+
1
cos
2
θ
W
J
µ†
n
D
Z
µν
(x − x
0
)J
ν
n
(x
0
)
+ O(g
4
)
|ii
As always, we work in momentum space. We define the Fourier transformed
propagator
˜
D
Z,W
µν
(p) by
D
Z,W
µν
(x − y) =
Z
d
4
p
(2π)
4
e
−ip·(x−y)
˜
D
Z,W
µν
(p),
and we will later compute to find that
˜
D
µν
is
˜
D
Z,W
µν
(p) =
i
p
2
− m
2
Z,W
+ iε
−g
µν
+
p
µ
p
ν
m
2
Z,W
!
.
Here
g
µν
is the metric of the Minkowski space. We will put aside the computation
of the propagator for the moment, and discuss consequences.
At low energies, e.g. the case of quarks and leptons (except for the top quark),
the momentum scales involved are much less than
m
2
Z,W
. In this case, we can
approximate the propagators by ignoring all the terms involving
p
. So we have
˜
D
Z,W
µν
(p) ≈
ig
µν
m
2
Z,W
.
Plugging this into the Fourier transform, we have
D
Z,W
µν
(x − y) ≈
ig
µν
m
2
Z,W
δ
4
(x − y).
What we see is that we can describe this interaction by a contact interaction,
i.e. a four-fermion interaction. Note that if we did not make the approximation
p ≈
0, then our propagator will not have the
δ
(4)
(
x − y
), hence the effective
action is non-local.
Thus, the second term in the S-matrix expansion becomes
−
Z
d
4
x
ig
2
8m
2
W
J
µ†
(x)J
µ
(x) +
m
2
W
m
2
Z
cos
2
θ
W
J
µ†
n
(x)J
nµ
(x)
.
We want to define the effective Lagrangian
L
eff
W
to be the Lagrangian not involving
W
±
, Z such that for “low energy states”, we have
hf|S |ii = hf|T exp
i
Z
d
4
x L
eff
W
|ii
= hf|T
1 + i
Z
d
4
x L
eff
W
+ ···
|ii
Based on our previous computations, we find that up to tree level, we can write
iL
eff
W
(x) ≡ −
iG
F
√
2
J
µ†
J
µ
(x) + ρJ
µ†
n
J
nµ
(x)
,
where, again up to tree level,
G
F
√
2
=
g
2
8m
2
W
, ρ =
m
2
W
m
2
Z
cos
2
θ
W
,
Recall that when we first studied electroweak theory, we found a relation
m
W
=
m
Z
cos θ
W
. So, up to tree level, we have
ρ
= 1. When we look at higher levels,
we get quantum corrections, and we can write
ρ = 1 + ∆ρ.
This value is sensitive to physics “beyond the Standard Model”, as the other
stuff can contribute to the loops. Experimentally, we find
∆ρ ≈ 0.008.
We can now do our usual computations, but with the effective Lagrangian rather
than the usual Lagrangian. This is the Fermi theory of weak interaction. This
predates the idea of the standard model and the weak interaction. The
1
m
2
W
in
G
F
in some sense indicates that Fermi theory breaks down at energy scales near
m
W
, as we would expect.
It is interesting to note that the mass dimension of
G
F
is
−
2. This is to
compensate for the dimension 6 operator
J
µ†
J
µ
. This means our theory is
non-renormalizable. This is, of course, not a problem, because we do not think
this is a theory that should be valid up to arbitrarily high energy scales. To
derive this Lagrangian, we’ve assumed our energy scales are m
W
, m
Z
.
Computation of propagators
We previously just wrote down the values of the
W
±
and
Z
-propagators. We
will now explicitly do the computations of the
Z
propagator. The computation
for
W
±
is similar. We will gloss over subtleties involving non-abelian gauge
theory here, ignoring problems involving ghosts etc. We’ll work in the so-called
R
ε
-gauge.
In this case, we explicitly write the free Z-Lagrangian as
L
Free
Z
= −
1
4
(∂
µ
Z
ν
− ∂
ν
Z
µ
)(∂
µ
Z
ν
− ∂
ν
Z
µ
) +
1
2
m
2
Z
Z
µ
Z
µ
.
To find the propagator, we introduce an external current
j
µ
coupled to
Z
µ
. So
the new Lagrangian is
L = L
free
Z
+ j
µ
(x)Z
µ
(x).
Through some routine computations, we see that the Euler–Lagrange equations
give us
∂
2
Z
ρ
− ∂
ρ
∂ · Z + m
2
Z
Z
ρ
= −j
ρ
. (∗)
We need to solve this. We take the ∂
ρ
of this, which gives
∂
2
∂ · Z − ∂
2
∂ · Z + m
2
Z
∂ · Z = −∂ · j.
So we obtain
m
2
Z
∂ · Z = −∂ · j.
Putting this back into (∗) and rearranging, we get
(∂
2
+ m
2
Z
)Z
µ
= −
g
µν
+
∂
µ
∂
ν
m
2
Z
j
ν
.
We can write the solution as an integral over this current. We write the solution
as
Z
µ
(x) = i
Z
d
4
y D
Z
µν
(x − y)j
ν
(y), (†)
and further write
D
2
µν
(x − y) =
Z
d
4
p
(2π)
4
e
−ip·(x−y)
˜
D
2
µν
(p).
Then by applying (∂
2
+ m
2
Z
) to (†), we find that we must have
˜
D
2
µν
(p) =
−i
p
2
− m
2
Z
+ iω
g
µν
−
p
µ
p
ν
m
2
Z
.