5Perturbative renormalization
III Advanced Quantum Field Theory
5.2 Dimensional regularization
People soon realized this was a terrible way to get rid of the infinities. Doing
integrals from 0 to Λ
0
is usually much harder than integrals from 0 to
∞
.
Moreover, in gauge theory, it is (at least naively) incompatible with gauge
invariance. Indeed, say in U(1) gauge theory, a transformation
ψ(x) → e
iλ(x)
ψ(x)
can potentially introduce a lot of high energy modes. So our theory will not be
gauge invariant.
For these reasons, people invented a different way to get rid of infinite
integrals. This is known as dimensional regularization. This method of getting
rid of infinities doesn’t fit into the ideas we’ve previously discussing. It is just
magic. Moreover, this method only works perturbatively — it tells us how to
get rid of infinities in loops. It doesn’t give any definition of a regularized path
integral measure, or describe any full, coherent non-perturbative theory that
predicts the results.
Yet, this method avoids all the problems we mentioned above, and is rather
easy to use. Hence, we will mostly used dimensional regularization in the rest of
the course.
To do dimensional regularization, we will study our theory in an arbitrary
dimension
d
, and do the integrals of loop calculations. For certain dimensions,
the integral will converge, and give us a sensible answer. For others, it won’t. In
particular, for d = 4, it probably won’t (or else we have nothing to do!).
After obtaining the results for some functions, we attempt to analytically
continue it as a function of
d
. Of course, the analytic continuation is non-unique
(e.g. we can multiply the result by
sin d
and still get the same result for integer
d
), but there is often an “obvious” choice. This does not solve our problem yet
— this analytic continuation tends to still have a pole at
d
= 4. However, after
doing this analytic continuation, it becomes more clear how we are supposed to
get rid of the pole.
Note that we are not in any way suggesting the universe has a non-integer
dimension, or that non-integer dimensions even makes any sense at all. This is
just a mathematical tool to get rid of infinities.
Let’s actually do it. Consider the same theory as before, but in arbitrary
dimensions:
S[φ] =
Z
d
d
x
1
2
(∂φ)
2
+
1
2
m
2
φ
2
+
λ
4!
φ
4
.
In d dimensions, this λ is no longer dimensionless. We have
[φ] =
d − 2
2
.
So for the action to be dimensionless, we must have
[λ] = 4 − d.
Thus, we write
λ = µ
4−d
g(µ)
for some arbitrary mass scale
µ
. This
µ
is not a cutoff, since we are not
going to impose one. It is just some arbitrary mass scale, so that
g
(
µ
) is now
dimensionless.
We can then compute the loop integral
p
=
1
2
gµ
4−d
Z
d
d
p
(2π)
d
1
p
2
+ m
2
=
gµ
4−d
2(2π)
d
vol(S
d−1
)
Z
∞
0
p
d−1
dp
p
2
+ m
2
.
We note the mathematical fact that
vol(S
d−1
) =
2π
d/2
Γ(d/2)
.
While
S
d−1
does not make sense when
d
is not an integer, the right-hand
expression does. So replacing the volume with this expression, we can analytically
continue this to all d, and obtain
p
= µ
4−d
Z
∞
0
p
d−1
dp
p
2
+ m
2
=
1
2
µ
4−d
Z
∞
0
(p
2
)
d/2−1
dp
2
p
2
+ m
2
=
m
2
2
µ
m
4−d
Γ
d
2
Γ
1 −
d
2
.
The detailed computations are entirely uninteresting, but if one were to do this
manually, it is helpful to note that
Z
1
0
u
s−1
(1 − u)
t−1
du =
Γ(s)Γ(t)
Γ(s + t)
.
The appearance of Γ-functions is typical in dimensional regularization.
Combining all factors, we find that
p
=
gm
2
2(4π)
d/2
µ
m
4−d
Γ
1 −
d
2
.
This formula makes sense for any value of
d
in the complex plane. Let’s see what
happens when we try to approach
d
= 4. We set
d
= 4
− ε
, and use the Laurent
series
Γ(ε) =
1
ε
− γ + O(ε)
x
ε
= 1 +
ε
2
log x + O(ε
2
),
plus the following usual property of the Γ function:
Γ(x + 1) = xΓ(x).
Then, as d → 4, we can asymptotically expand
p
= −
gm
2
32π
2
2
ε
− γ + log
4πµ
2
m
2
+ O(ε)
.
Unsurprisingly, this diverges as
ε →
0, as Γ has a (simple) pole at
−
1. The
pole in
1
ε
reflects the divergence of this loop integral as Λ
0
→ ∞
in the cutoff
regularization.
We need to obtain a finite limit as
d →
4 by adding counterterms. This time,
the counterterms are not dependent on a cutoff, because there isn’t one. There
are also not (explicitly) dependent on the mass scale
µ
, because
µ
was arbitrary.
Instead, it is now a function of ε.
So again, we introduce a new term
φ φ×
δm
2
Again, we need to make a choice of this. We need to choose a renormalization
scheme. We can again use the on-shell renormalization. However, we could have
done on-shell renormalization without doing all this weird dimension thing. Once
we have done it, there is a more convenient way of doing this in dimensional
regularization:
(i) Minimal subtraction (MS): we choose
δm
2
= −
gm
2
16π
2
ε
so as to cancel just the pole.
(ii) Modified minimal subtraction (MS): We set
δm
2
= −
gm
2
32π
2
2
ε
− γ + log 4π
to get rid of some pesky constants, because no one likes the Euler–
Mascheroni constant.
In practice, we are mostly going to use the
MS
scheme, because we really, really,
hate the Euler–Mascheroni constant.
Note that at the end, after subtracting off these counter-terms and taking
ε → 0, there is still an explicit µ dependence! In this case, we are left with
−
gm
2
32π
2
log
µ
2
m
2
Of course, the actual physical predictions of the theory must not depend on
µ
, because
µ
was an arbitrary mass scale. This means
g
must genuinely be a
function of µ, just like when we did renormalization!
What is the physical interpretation of this? We might think that since
µ
is arbitrary, there is no significance. However, it turns out when we do actual
computations with perturbation theory, the quantum corrections tend to look like
log
Λ
2
µ
2
, where Λ is the relevant “energy scale”. Thus, if we want perturbation
theory to work well (or the quantum corrections to be small), we must pick
µ
to
be close to the “energy scale” of the scenario we are interested in. Thus, we can
still think of g(µ
2
) as the coupling constant of the theory “at scale µ
2
”.