4Wilsonian renormalization
III Advanced Quantum Field Theory
4.6 Calculating RG evolution
We now want to actually compute the RG evolution of a theory. To do so, we of
course need to make some simplifying assumptions, and we will also leave out a
lot of details. We note that (in
d >
2), the only marginal or relevant operators
that involves derivatives is the kinetic term (
∂ϕ
)
2
. This suggests we can find a
simple truncation of the RG evolution by restricting to actions of the form
S[ϕ] =
Z
d
d
x
1
2
(∂ϕ)
2
+ V (ϕ)
,
and write the potential as
V (ϕ) =
X
k
Λ
d−k(d−2)
g
2k
(2k)!
ϕ
2k
.
In other words, we leave out higher terms that involve derivatives. This is known
as the local potential approximation (LPA). As before, we split our field into low
and high energy modes,
ϕ = φ + χ,
and we want to compute the effective action for φ:
S
eff
Λ
[φ] = −~ log
Z
C
∞
(M)
[Λ,Λ
0
]
Dχ e
−S[φ+χ]
.
This is still a very complicated path integral to do. To make progress, we assume
we lower the cutoff just infinitesimally, Λ = Λ
0
− δ
Λ. The action at scale Λ now
becomes
S[φ + χ] = S[φ] +
Z
d
d
x
1
2
(∂χ)
2
+
1
2
χ
2
V
00
(φ) +
1
3!
χ
3
V
000
(φ) + ···
,
where it can be argued that we can leave out the terms linear in χ.
Since we’re just doing path integral over modes with energies in (Λ
− δ
Λ
,
Λ],
each loop integral takes the form
Z
Λ−δΛ≤|p|≤Λ
d
d
p ··· = Λ
d−1
δΛ
Z
S
d−1
dΩ ··· ,
where dΩ denotes an integral over the unit (
d −
1) sphere. Since each loop
integral comes with a factor of
δ
Λ, to leading order, we need to consider only
1-loop diagrams.
A connected graph with
E
edges and
L
loops and
V
i
vertices of
χ
-valency
i
(and arbitrarily many valency in φ) obeys
L − 1 = E −
∞
X
i=2
V
i
.
Note that by assumption, there are no single-χ vertices.
Also, every edge contributes to two vertices, as there are no
χ
loose ends.
On the other hand, each vertex of type i has i many χ lines. So we have
2E =
X
i
V
i
.
Combining these two formulae, we have
L = 1 +
∞
X
i=2
(i − 2)
2
V
i
.
The number on the right is non-negative with equality iff
V
i
= 0 for all
i ≥
3.
Hence, for 1-loop diagrams, we only need to consider vertices with precisely two
χ-lines attached. Thus, all the contributions look like
, , , . . .
We can thus truncate the action as
S[φ + χ] − S[φ] =
Z
d
d
x
1
2
(∂χ)
2
+
1
2
χ
2
V
00
(φ)
.
This is still not very feasible to compute. We are only going to do this integral
in a very specific case, where φ is chosen to be constant.
We use the fact that the Fourier modes of
χ
only live between Λ
−δ
Λ
< |p| <
Λ.
Then taking the Fourier transform and doing the integral in momentum space,
we have
S[φ + χ] − S[φ] =
Z
Λ−δΛ<|p|≤Λ
d
d
p
2(2π)
d
˜χ(−p)(p
2
+ V
00
(φ))˜χ(p)
=
Λ
d−1
δΛ
2(2π)
d
[Λ
2
+ V
00
(φ)]
Z
S
d−1
dΩ ˜χ(−Λˆp) ˜χ(Λˆp).
The
χ
path integral is finite if we work on a compact space, say
T
d
with side
length
L
, in which case there are only finitely many Fourier modes. Then the
momenta are
p
µ
=
2π
L
n
µ
, and the path integral is just a product of Gaussians
integrals. Going through the computations, we find
e
−δ
Λ
S
=
Z
Dχ e
−(S[φ+χ]−S[φ])
= C
π
Λ
2
+ V
00
(φ)
N/2
,
where
N
is the number of
χ
modes in our shell of radius Λ and thickness
δ
Λ,
and C is some constant. From our previous formula, we see that it is just
N = vol(S
d−1
)Λ
d−1
δΛ ·
L
2π
d
≡ 2aΛ
d−1
δΛ,
where
a =
vol(S
d−1
)
2(2π
d
)
=
1
(4π)
d/2
Γ(d/2)
L
d
.
Consequently, up to field-independent numerical factors, integrating out
χ
leads
to a change in the effective action
δ
Λ
S
eff
δΛ
= a log(Λ
2
+ V
00
(φ))Λ
d−1
L
d
.
This diverges as
L → ∞
! This is infrared divergence, and it can be traced to our
simplifying assumption that
φ
is everywhere constant. More generally, it is not
unreasonable to believe that we have
δ
Λ
S
eff
δΛ
= aΛ
d−1
Z
d
d
x log(Λ
2
+ V
00
(φ)).
This isn’t actually quite the result, but up to the local approximation, it is.
Now we can write down the
β
-function. As expected, integrating out some
modes has lead to new terms. We have
Λ
dg
2k
dΛ
= [k(d − 2) − d]g
2k
− aΛ
k(d−2)
∂
2k
∂φ
2k
log(Λ
2
+ V
00
(φ))
φ=0
.
As before, the first term does not relate to quantum corrections. It is just due to
us rescaling our normalization factors. On the other hand, the 2
k
th derivative is
just a fancy way to extract the factor of φ
2k
in term.
We can actually compute these things!
Example. Then we find that
Λ
dg
2
dΛ
= −2g −
ag
4
1 + g
2
Λ
dg
4
dΛ
= (d − 4)g
4
−
ag
6
(1 + g
2
)
+
3ag
2
4
(1 + g
2
)
2
Λ
dg
6
dΛ
= (2d − 6)g
6
−
ag
8
(1 + g
2
)
+
15ag
4
g
6
(1 + g
2
)
2
−
30ag
3
4
(1 + g
2
)
3
.
Note that the first term on the right hand side is just the classical behaviour
of the dimensionless couplings. It has nothing to do with the
χ
field. The
remaining terms are quantum corrections (
∼ ~
), and each comes from specific
Feynman diagrams. For example,
ag
4
1 + g
2
involves one quartic vertex, and this comes from the diagram
There is one
χ
propagator, and this gives rise to the one 1 +
g
2
factor in the
denominator.
The first term in β
4
comes from
The second term comes from
Note that g
2
is just the dimensionless mass coupling of χ,
g
2
=
m
2
Λ
2
.
At least perturbatively, we expect this to be a relevant coupling. It increases
as Λ
→
0. Consequently, at scales Λ
m
, the quantum corrections to these
β-functions are strongly suppressed! This makes sense!
The Gaussian fixed point
From the formulae we derived, we saw that there is only one critical point, with
g
∗
2k
= 0 for all
k ≥
2. This is free since there are no vertices at all, and hence no
corrections can happen. This is just the free theory.
In a neighbourhood of this critical point, we can expand the
β
-functions in
lowest order in δg
i
= g
i
− g
∗
i
. We have
β
2k
= Λ
∂g
2k
∂Λ
= [k(d − 2) − d]g
2k
− ag
2k+2
.
Writing this linearized β-function as
β
2i
= B
ij
g
2j
,
we see that
B
ij
is upper triangular, and hence its eigenvalues are just the diagonal
entries, which are
k(d − 2) − d = 2k − 4,
in four dimensions.
So vertices
φ
2k
with
k ≥
3 are irrelevant. If we turn them on at some scale,
they become negligible as we fall towards the infrared. The mass term
g
2
is
relevant, as we said before, so even a small mass becomes increasingly significant
in the infrared. Of course, we are making these conclusions based on a rather
perturbative way of computing the
β
function, and our predictions about what
happens at the far infrared should be taken with a grain of salt. However, we
can go back to check our formulation, and see that our conclusion still holds.
The interesting term is
φ
4
, which is marginal in
d
= 4 to lowest order. This
means we have to go to higher order. To next non-trivial order, we have
Λ
dg
4
dΛ
= 3ag
2
4
+ O(g
2
4
g
2
),
where we neglected
g
6
as it is irrelevant. Using the specific value of
a
in
d
= 4,
we find that, to this order,
1
g
4
(Λ)
= C −
3
16π
2
log Λ.
Equivalently, we have
g
4
(Λ) =
16π
2
3
log
µ
Λ
−1
for some scale
µ
. If we have no higher order terms, then for the theory to make
sense, we must have g
4
> 0. This implies that we must pick µ > Λ.
How does this coefficient run? Our coupling
g
4
is marginal to leading order,
and consequently it doesn’t run as some power of Λ. It runs only logarithmically
in Λ.
This coupling is irrelevant. In the infrared limit, as we take Λ
→
0, we find
g
4
→
0, and so we move towards the Gaussian fixed point. This is rather boring.
On the other hand, if we take Λ
→ ∞
, then eventually
µ
Λ
hits 1, and we
divide by zero. So our perturbation theory breaks! Notice that we are not saying
that our theory goes out of control as Λ
→ ∞
. This perturbation theory breaks
at some finite energy scale!
Recall that last term, we were studying
φ
4
theory. We didn’t really run into
trouble, because we only worked at tree level (and hence wasn’t doing quantum
field theory). But if we actually try to do higher loop integrals, then everything
breaks down. The φ
4
theory doesn’t actually exist.
The Wilson–Fisher critical point
Last time, we ignored all derivatives terms, and we found, disappointedly, that
the only fixed point we can find is the free theory. This was bad.
Wilson–Fisher, motivated by condensed matter physics rather than funda-
mental physics, found another non-trivial fixed point. What they did was rather
peculiar. They set the dimension to
d
= 4
−ε
, for some small values of
ε
. While
this might seem rather absurd, because non-integral dimensions do not exist
(unless we want to talk about fractals, but doing physics on fractals is hard),
but we can still do manipulations formally, and see if we get anything sensible.
They proved that there exists a new fixed point with
g
∗
2
= −
1
6
ε + O(ε
2
)
g
∗
4
=
ε
3a
+ O(ε
2
)
g
∗
2k
∼ O(ε
k
).
for k ≥ 3. To study the behaviour near this critical point, we again expand
g
i
= g
∗
i
+ δg
i
in the
β
-function we found earlier to lowest non-trivial order, this time expanding
around the Wilson–Fisher fixed point.
If we do this, then in the (g
2
, g
4
) subspace, we find that
Λ
∂
∂Λ
δg
2
δg
4
=
ε
3
− 2 −a
1 +
ε
6
0 ε
δg
2
δg
4
.
The eigenvalues and eigenvectors are
ε
3
− 2 and ε, with eigenvectors
σ
1
=
1
0
, σ
2
=
−a
3 +
ε
2
2(3 + ε)
Notice that while the mass term itself is an eigenvector, the quartic coupling is
not! Using the asymptotic expansion
Γ
−
ε
2
∼ −
2
ε
− γ + O(ε),
where
γ ≈
0
.
577 is the Euler–Mascheroni constant, plus fact that Γ(
x
+ 1) =
xΓ(x), we find that in d = 4 − ε, we have
a =
1
(4π)
d/2
1
γ(d/2)
d=4−ε
∼
1
16π
2
+
ε
32π
2
(1 − γ + log 4π) + O(ε
2
),
Since
ε
is small, we know that the eigenvalue of
σ
1
is negative. This means it is
a relevant operator. On the other hand
σ
4
, is an irrelevant operator. We thus
have the following picture of the RG flow:
g
4
g
2
I
II
III
IV
We see that theories in region I are massless and free in the deep UV, but flows
to become massive and interacting in the IR. Theories in region II behaves
similarly, but now the mass coefficient is negative. Consequently,
φ
= 0 is a local
maximum of the effective potential. These theories tend to exhibit spontaneous
symmetry breaking.
Finally, theories in III and IV do not have a sensible continuum limit as
both couplings increase without bound. So at least within perturbation theory,
these theories don’t exist. They can only manifest themselves as effective field
theories.