4Wilsonian renormalization
III Advanced Quantum Field Theory
4.4 Renormalization group flow
We now study the renormalization group flow. In other words, we want to
understand how the coupling constants actually change as we move to the
infrared, i.e. take Λ
→
0. The actual computations are difficult, so in this section,
we are going to understand the scenario rather qualitatively and geometrically.
We can imagine that there is a configuration space whose points are the
possible combinations of the
g
i
, and as we take Λ
→
0, we trace out a trajectory
in this configuration space. We want to understand how these trajectories look
like.
As in most of physics, we start at an equilibrium point.
Definition
(Critical point)
.
A critical point is a point in the configuration
space, i.e. a choice of couplings g
i
= g
∗
i
such that β
i
(g
∗
i
) = 0.
One such example of a critical point is the Gaussian theory, with all couplings,
including the mass term, vanishing. Since there are no interactions at all, nothing
happens when we integrate out modes. It is certainly imaginable that there are
other critical points. We might have a theory where the classical dimensions
of all couplings are zero, and also by a miracle, all quantum corrections vanish.
This happens, for example, in some supersymmetric theories. Alternatively, the
classical dimensions are non-zero, but the quantum corrections happen to exactly
compensate the effect of the classical dimension.
In either case, we have some couplings
g
∗
i
that are independent of scale, and
thus the anomalous dimension
γ
φ
(
g
∗
i
) =
γ
∗
φ
would also be independent of scale.
This has important consequences.
Example.
At a critical point, the renormalization group equation for a two-point
function becomes
0 =
Λ
∂
∂Λ
+ β
i
(g
∗
i
)
∂
∂g
i
+ 2γ
φ
(g
∗
i
)
Γ
(2)
Λ
(x, y).
But the β-function is zero, and γ
φ
is independent of scale. So
Λ
∂
∂Λ
Γ
(2)
Λ
(x, y) = −2γ
∗
φ
Γ
(2)
Λ
(x, y).
On the other hand, on dimensional grounds, Γ must be of the form
Γ
(2)
Λ
(x, y, g
∗
i
) = f(Λ|x − y|, g
∗
i
)Λ
d−2
for some function
f
. Feeding this into the RG equation, we find that Γ must be
of the form
Γ
(2)
Λ
(x, y, g
∗
i
) =
Λ
d−2
c(g
∗
i
)
Λ
2∆
φ
|x − y|
2∆
φ
∝
c(g
∗
i
)
|x − y|
2∆
φ
,
where
c
(
g
∗
i
) are some constants independent of the points. This is an example of
what we were saying before. This scales as
|x − y|
−2∆φ
, instead of
|x − y|
2−d
,
and the anomalous dimension is the necessary correction.
Now a Gaussian universe is pretty boring. What happens when we start
close to a critical point? As in, say, IA Differential Equations, we can try to
Taylor expand, and look at the second derivatives to understand the behaviour
of the system. This corresponds to Taylor-expanding the
β
-function, which is by
itself the first derivative.
We set our couplings to be
g
i
= g
∗
i
+ δg
i
.
Then we can write
Λ
∂g
i
∂Λ
g
∗
i
+δg
i
= B
ij
({g
k
})δg
j
+ O(δg
2
),
where
B
ij
is (sort of) the Hessian matrix, which is an infinite dimensional matrix.
As in IA Differential Equations, we consider the eigenvectors of
B
ij
. Suppose
we have an “eigencoupling” σ
j
. Classically, we expect
g
i
(Λ) =
Λ
Λ
0
d
i
−d
g
i
(Λ
0
),
and so
δg
j
=
δ
ij
gives an eigenvector with eigenvalue
d
i
−d
. In the fully quantum
case, we will write the eigenvalue as ∆
i
− d, and we define
γ
i
= ∆
i
− d
i
to be the anomalous dimension of the operator. Since
σ
j
was an eigenvector, we
find that
Λ
∂σ
i
∂Λ
= (∆
i
− d)σ
i
.
Consequently, we find
σ
i
(Λ) =
Λ
Λ
0
∆
i
−d
σ
i
(Λ
0
)
to this order.
Suppose ∆
i
> d
. Then as we lower the cutoff from Λ
0
to 0, we find that
σ
i
(Λ)
→
0 exponentially. So we flow back to the theory at
g
∗
i
as we move to
lower energies. These operators are called irrelevant.
Assuming that quantum corrections do not play a very large role near the
critical point, we know that there must be infinitely many such operators, as we
can always increase
d
i
, hence ∆
i
by adding more derivatives or fields (for
d >
2).
So we know the critical surface is infinite dimensional.
On the other hand, if ∆
i
< d
, then
σ
(Λ) increases as we go to the infrared.
These operators hence become more significant. These are called relevant
operators. There are only finitely many such relevant operators, at least for
d > 2. Any RG trajectory emanating from g
∗
i
is called a critical trajectory.
We can draw a picture. The critical surface
C
consisting of (the span of) all
irrelevant modes, and is typically infinite dimensional with finite codimension:
A generic QFT will start at scale Λ
0
with both relevant and irrelevant operators
turned on. As we flow along the RG trajectory, we focus towards the critical
trajectory. This focusing is called universality.
This is, in fact, the reason we can do physics! We don’t know about the
detailed microscopic information about the universe. Further, there are infinitely
many coupling constants that can be non-zero. But at low energies, we don’t
need to know them! Most of them are irrelevant, and at low energies, only the
relevant operators matter, and there is only finitely many of them.
One thing we left out in our discussion is marginal operators, i.e. those
with ∆
i
=
d
. To lowest order, these are unchanged under RG flow, but we
have to examine higher order corrections to decide whether these operators are
marginally relevant or marginally irrelevant, or perhaps exactly marginal.
Marginally relevant or marginally irrelevant may stay roughly constant for
long periods of RG evolution. Because of this, these operators are often important
phenomenologically. We’ll see that most of the couplings we see in QED and
QCD, apart from mass terms, are marginal operators, at least to lowest order.
If we use the classical dimension in place of ∆
i
, it is straightforward to figure
out what the relevant and marginal operators are. We know that in
d
dimensions,
the mass dimension [
φ
] =
d−2
2
for a scalar field, and [
∂
] = 1. Focusing on the
even operators only, we find that the only ones are
Dimension d Relevant operators Marginal operators
2 φ
2k
for all k > 0 (∂φ)
2
, φ
2k
(∂φ)
2
for all k > 0
3 φ
2k
for k = 1, 2 (∂φ)
2
, φ
6
4 φ
2
(∂φ)
2
, φ
4
> 4 φ
2
(∂φ)
2
Of course, there are infinitely many irrelevant operators, and we do not attempt
to write them out.
Thus, with the exception of
d
= 2, we see that there is a short, finite list of
relevant and marginal operators, at least if we just use the classical dimension.
Note that here we ignored all quantum corrections, and that sort-of defeats
the purpose of doing renormalization. In general, the eigen-operators will not be
simple monomials, and can in fact look very complicated!