4Wilsonian renormalization
III Advanced Quantum Field Theory
4.3 Correlation functions and anomalous dimensions
Let’s say we now want to compute correlation functions. We will write
S
Λ
[φ, g
i
] =
Z
M
d
d
x
"
1
2
(∂φ)
2
+
m
2
2
φ
2
+
X
i
g
i
Λ
d−d
i
0
O
i
(φ, ∂φ)
#
,
where, as before, we will assume
m
2
is one the of the
g
i
. Note that the action
of the
φ
we produced by integrating out modes is not
S
Λ
[
φ, g
i
(Λ)], because
we had the factor of
Z
Λ
sticking out in the action. Instead, it is given by
S
Λ
[Z
1/2
φ, g
i
(Λ)].
Now we can write a general n-point correlation function as
hφ(x
1
) ···φ(x
n
)i =
1
Z
Z
≤Λ
Dφ e
−S
Λ
[Z
1/2
Λ
φ,g
i
(Λ)]
φ(x
1
) ···φ(x
n
).
We can invent a canonically normalized field
ϕ(x) =
p
Z
Λ
φ,
so that the kinetic term looks right. Then defining
hϕ(x
1
) ···ϕ(x
n
)i =
1
Z
Z
≤Λ
Dφ e
−S
Λ
[ϕ,g
i
(Λ)]
ϕ(x
1
) ···ϕ(x
n
),
we find
hφ(x
1
) ···φ(x
n
)i = Z
−n/2
Λ
hϕ(x
1
) ···ϕ(x
n
)i.
Note that we don’t have to worry about the factor of
Z
1/2
Λ
coming from the
scaling of the path integral measure, as the partition function
Z
is scaled by the
same amount.
Definition (Γ
(n)
Λ
). We write
Γ
(n)
Λ
({x
i
}, g
i
) =
1
Z
Z
≤Λ
Dφ e
−S
Λ
[φ,g
i
]
φ(x
1
) ···φ(x
n
) = hϕ(x
1
) ···ϕ(x
n
)i.
Now suppose 0
< s <
1, and that we’ve chosen to insert fields only with
energies
< s
Λ. Then we should equally be able to compute the correlator using
the low energy theory S
sΛ
. We’re then going to find
Z
−n/2
sΛ
Γ
(n)
sΛ
(x
1
, ··· , x
n
, g
i
(sΛ)) = Z
−n/2
Λ
Γ
(n)
sΛ
(x
1
, ··· , x
n
, g
i
(Λ)).
Differentiating this with respect to s, we find
Λ
d
dΛ
Γ
(n)
Λ
(x
1
, ··· , x
n
, g
i
(Λ)) =
Λ
∂
∂Λ
+ β
i
∂
∂g
i
+ nγ
φ
Γ
(n)
Λ
({x
i
}, g
i
(Λ)) = 0.
This is the Callan-Symanzik equation for the correlation functions.
There is an alternative way of thinking about what happens when change Λ.
We will assume we work over
R
n
, so that it makes sense to scale our universe
(on a general Riemannian manifold, we could achieve the same effect by scaling
the metric). The coordinates change by
x 7→ sx
. How does Γ
Λ
(
x
1
, . . . , x
n
, g
i
)
relate to Γ
Λ
(sx
1
, . . . , sx
n
, g
i
)?
We unwrap the definitions
Γ
(n)
Λ
({sx
i
}, g
i
) =
1
Z
Z
≤Λ
Dφ e
−S
Λ
[φ,g
i
]
φ(sx
1
) ···φ(sx
n
)
We make the substitution
ϕ
(
x
) =
aφ
(
sx
), with a constant
a
to be chosen later so
that things work out. Again, we don’t have to worry about how D
φ
transforms.
However, this change of variables does scale the Fourier modes, so the new cutoff
of ϕ is in fact sΛ. How the S
Λ
[φ, g
i
] transform? Using the chain rule, we have
S
sΛ
[ϕ, g
i
] =
Z
M
d
d
x
"
1
2
(∂ϕ)
2
+
m
2
2
ϕ
2
+
X
i
g
i
(sΛ)
d−d
i
0
O
i
(ϕ, ∂ϕ)
#
Putting in the definition of ϕ, and substituting y = sx, we have
= s
−d
Z
M
d
d
y
"
1
2
a
2
s
2
(∂φ)
2
+
m
2
2
ϕ
2
+
X
i
g
i
(sΛ)
d−d
i
0
O
i
(aφ, as∂φ)
#
,
where all fields are evaluated at
y
. We want this to be equal to
S
Λ
[
φ, g
i
]. By
looking at the kinetic term, we know that we need
a = s
(d−2)/2
.
By a careful analysis, we see that the other terms also work out (or we know
they must be, by dimensional analysis). So we have
Γ
(n)
Λ
({sx
i
}, g
i
) =
1
Z
Z
≤Λ
Dφ e
−S
Λ
[φ,g
i
]
φ(sx
1
) ···φ(sx
n
)
=
1
Z
Z
≤sΛ
Dϕ e
−S
sΛ
[ϕ,g
i
]
s
(d−2)n/2
ϕ(x
1
) ···ϕ(x
n
)
= s
(d−2)n/2
Γ
(n)
sΛ
({x
i
}, g
i
)
Thus, we can write
Γ
n
Λ
(x
1
, ··· , x
n
, g
i
(Λ)) =
Z
Λ
Z
sΛ
n/2
Γ
n
sΛ
(x
1
, ··· , x
n
, g
i
(sΛ))
=
Z
Λ
s
2−d
Z
sΛ
n/2
Γ
n
Λ
(sx
1
, ··· , sx
n
, g
i
(sΛ)).
Note that in the second step, we don’t change the values of the
g
i
! We are just
changing units for measuring things. We are not integrating out modes.
Equivalently, if y
i
= sx
i
, then what we found is that
Γ
n
Λ
y
1
s
, ··· ,
y
n
s
, g
i
(Λ)
=
Z
Λ
s
d−2
Z
sΛ
n/2
Γ
m
Λ
(y
1
, ··· , y
n
, g
i
(sΛ)).
What does this equation say? Here we are cutting off the Γ at the same energy
level. As we reduce
s
, the right hand side has the positions fixed, while on the
left hand side, the points get further and further apart. So on the left hand side,
as
s →
0, we are probing the theory at longer and longer distances. Thus, what
we have found is that “zooming out” in our theory is the same as flowing down
the couplings g
i
(sΛ) to a scale appropriate for the low energy theory.
Infinitesimally, let s = 1 − δs, with 0 < δs 1. Then we have
Z
Λ
(1 − δs)
2−d
Z
(1−δs)Λ
1/2
≈ 1 +
d − 2
2
+ γ
φ
δs,
where γ
s
is the anomalous dimension of φ we defined before.
Classically, we’d expect this correlation function
hφ
(
sx
1
)
···φ
(
sx
n
)
i
to scale
with s as
d − s
2
n
,
since that’s what dimensional analysis would tell us. But quantum mechanically,
we see that there is a correction given by γ
φ
, and what we really have is
∆
n
φ
=
d − 2
2
+ γ
φ
n
.
So the dependence of the correlation on the distance is not just what we expect
from dimensional analysis, but it gains a quantum correction factor. Thus, we
say γ
φ
is the “anomalous dimension” of the field.