5Perturbative renormalization
III Advanced Quantum Field Theory
5.1 Cutoff regularization
Our discussion of renormalization has been theoretical so far. Historically, this
was not what people were studying when doing quantum field theory. Instead,
what they did was that they had to evaluate integrals in Feynman diagrams,
and the results happened to be infinite!
For example, consider the scalar
φ
4
theory in
d
= 4 dimensions, with action
given by
S[φ] =
Z
1
2
(∂φ)
2
+
1
2
m
2
φ
2
+
λ
4!
φ
4
d
4
x.
We want to compute the two point function
hφφi
. This has, of course, the tree
level diagram given by just the propagator. There is also a 1-loop diagram given
by
φ φ
k k
p
We will ignore the propagators coming from the legs, and just look at the loop
integral. The diagram has a symmetry factor of 2, and thus loop integral is
given by
−
λ
2(2π)
4
Z
d
4
p
p
2
+ m
2
.
This integral diverges. Indeed, we can integrate out the angular components,
and this becomes
−
λ
2(2π)
4
vol(S
3
)
Z
|p|
3
d|p|
|p|
2
+ m
2
.
The integrand tends to infinity as we take p → ∞, so this clearly diverges.
Well, this is bad, isn’t it. In light of what we have been discussing so far,
what we should do is to not view
S
as the Lagrangian in “continuum theory”,
but instead just as a Lagrangian under some cutoff
k
2
≤
Λ
0
. Then when doing
the loop integral, instead of integrating over all
p
, we should integrate over all
p
such that p
2
≤ Λ
0
. And this certainly gives a finite answer.
But we want to take the continuum limit, and so we want to take Λ
0
→ ∞
.
Of course the loop integral will diverge if we fix our coupling constants. So
we might think, based on what we learnt previously, that we should tune the
coupling constants as we go.
This is actually very hard, because we have no idea what is the “correct”
way to tune them. Historically, and practically, what people did was just to
introduce some random terms to cancel off the infinity.
The idea is to introduce a counterterm action
S
CT
[φ, Λ] = ~
Z
δZ
2
(∂φ)
2
+
δm
2
2
φ
2
+
δλ
4!
φ
4
d
4
x,
where
δz
,
δm
and
δφ
are some functions of Λ to be determined. We then set the
full action at scale Λ to be
S
Λ
[φ] = S[φ] + S
CT
[φ, Λ].
This action depends on Λ. Then for any physical quantity
hOi
we are interested
in, we take it to be
hOi = lim
Λ→∞
hOi computed with cutoff Λ and action S
Λ
.
Note that in the counterterm action, we included a factor of
~
in front of
everything. This means in perturbation theory, the tree-level contributions from
S
CT
would be of the same order as the 1-loop diagrams in S.
For example, in the above 1-loop diagram, we obtain further contributions
to the quadratic terms, given by
φ φ×
k
2
δZ
φ φ×
δm
2
We first evaluate the original loop integral properly. We use the mathematical
fact that vol(S
3
) = 2π
2
. Then the integral is
p
= −
λ
16π
2
Z
Λ
0
0
p
3
dp
p
2
+ m
2
= −
λm
2
32π
2
Z
Λ
2
0
/m
2
0
x dx
1 + x
=
λ
32π
2
Λ
2
0
− m
2
log
1 +
Λ
2
0
m
2
,
where we substituted x = p
2
/m
2
in the middle.
Including these counter-terms, the 1-loop contribution to hφφi is
λ
32π
2
Λ
2
0
− m
2
log
1 +
Λ
2
0
m
2
+ k
2
δZ + δm
2
.
The objective is, of course, to pick
δz
,
δm
,
δφ
so that we always get finite answers
in the limit. There are many ways we can pick these quantities, and of course,
different ways will give different answers. However, what we can do is that
we can fix some prescriptions for how to pick these quantities, and this gives
as a well-defined theory. Any such prescription is known as a renormalization
scheme.
It is important that we describe it this way. Each individual loop integral
still diverges as we take Λ
→ ∞
, as we didn’t change it. Instead, for each fixed
Λ, we have to add up, say, all the 1-loop contributions, and then after adding
up, we take the limit Λ → ∞. Then we do get a finite answer.
On-shell renormalization scheme
We will study one renormalization scheme, known as the on-shell renormalization
scheme. Consider the exact momentum space propagator
Z
d
4
x e
ik·x
hφ(x)φ(0)i.
Classically, this is just given by
1
k
2
+ m
2
,
where m
2
is the original mass term in S[φ].
In the on-shell renormalization scheme, we pick our counterterms such that
the exact momentum space propagator satisfies the following two properties:
–
It has a simple pole when
k
2
=
−m
2
phys
, where
m
2
phys
is the physical mass
of the particle; and
– The residue at this pole is 1.
Note that we are viewing k
2
as a single variable when considering poles.
To find the right values of
δm
and
δZ
, we recall that we had the one-particle
irreducible graphs, which we write as
Π(k
2
) =
k
1PI
,
where the dashed line indicates that we do not include the propagator contribu-
tions. For example, this 1PI includes graphs of the form
k k
k k
as well as counterterm contributions
×
k
2
δZ
×
δm
2
Then the exact momentum propagator is
∆(k
2
)
=
φ φ
k
+
φ φ
1PI
+
φ φ
1PI 1PI
+ ···
=
1
k
2
+ m
2
−
1
k
2
+ m
2
Π(k
2
)
1
k
2
+ m
2
+
1
k
2
+ m
2
Π(k
2
)
1
k
2
+ m
2
Π(k
2
)
1
k
2
+ m
2
+ ···
=
1
k
2
+ m
2
+ Π(k
2
)
.
The negative sign arises because we are working in Euclidean signature with
path integrals weighted by e
−S
.
Thus, if we choose our original parameter
m
2
to be the measured
m
2
phys
, then
in the on-shell scheme, we want
Π(−m
2
phys
) = 0,
and also
∂
∂k
2
Π(k
2
)
k
2
=−m
2
phys
= 0.
To 1-loop, the computations at the beginning of the chapter tells us
Π(k
2
) = δm
2
+ k
2
δZ +
λ
32π
2
Λ
2
0
− m
2
log
1 +
Λ
2
0
m
2
.
We see that no 1-loop contributions involve
k
, which we can see in our unique
1-loop diagram, because the loop integral doesn’t really involve k in any way.
We see that the second condition forces δZ = 0, and then we must have
δZ = O(λ
2
)
δm
2
= −
λ
32π
2
Λ
2
0
− m
2
log
1 +
Λ
2
0
m
2
+ O(λ
2
).
Here to 1-loop, we don’t need wavefunction renormalization, but this is merely a
coincidence, not a general phenomenon.
Of course, if we consider higher loop diagrams, then we have further correc-
tions to the counterterms.