5Perturbative renormalization

III Advanced Quantum Field Theory


5.4 Renormalization of QED
That was pretty disappointing. How about the other theory we studied in QFT,
namely QED? Does it exist?
We again try to do dimensional regularization again. This will be slightly
subtle, because in QED, we have the universe and also a spinor space. In genuine
QED, both of these have dimension 4. If we were to do this properly, we would
have to change the dimensions of both of them to
d
, and then do computations.
In this case, it is okay to to just keep working with 4-dimensional spinors. We
can just think of this as picking as slightly different renormalization scheme than
MS.
In d dimensions, the classical action for QED in Euclidean signature is
S[A, ψ] =
Z
d
d
x
1
4e
2
F
µν
F
µν
+
¯
ψ
/
Dψ + m
¯
ψψ
,
where
/
Dψ = γ
µ
(∂
µ
+ iA
µ
)ψ.
Note that in the Euclidean signature, we have lost a factor of
i
, and also we have
{γ
µ
, γ
ν
} = 2δ
µν
.
To do perturbation theory, we’d like the photon kinetic term to be canonically
normalized. So we introduce
A
new
µ
=
1
e
A
old
µ
,
and then
S[A
new
, ψ] =
Z
d
d
x
1
4
F
µν
F
µν
+
¯
ψ(
/
∂ + ie
/
A)ψ + m
¯
ψψ
.
The original photon field necessarily has [
A
old
] = 1, as it goes together with the
derivative. So in d dimensions, we have
[e] =
4 − d
2
.
Thus, we find
[A
new
] = [A
old
] − [e] =
d − 2
2
.
From now on, unless otherwise specified, we will use A
new
, and just call it A.
As before, we introduce a dimensionless coupling constant in terms of an
arbitrary scale µ by
e
2
= µ
4−d
g
2
(µ).
Let’s consider the exact photon propagator in momentum space, i.e.
∆
µν
(q) =
Z
d
d
x e
iq·x
hA
µ
(x)A
ν
(0)i
in Lorenz gauge ∂
µ
A
µ
= 0.
We can expand this perturbatively as
+
1PI
+
1PI 1PI
+ ···
The first term is the classical propagator
∆
0
µν
(q) =
1
q
2
δ
µν
−
q
µ
q
ν
q
2
,
and then as before, we can write the whole thing as
∆
µν
(q) = ∆
0
µν
(q) + ∆
0ρ
µ
(q)Π
σ
ρ
(q)∆
0
σν
(q) + ∆
0ρ
µ
Π
σ
ρ
∆
0λ
σ
Π
κ
λ
∆
0
κν
+ ··· ,
where
−
Π
ρσ
(
q
) is the photon self-energy, given by the one-particle irreducible
graphs.
We will postpone the computation of the self-energy for the moment, and
just quote that the result is
Π
σ
ρ
(q) = q
2
δ
σ
ρ
−
q
ρ
q
σ
q
2
π(q
2
)
for some scalar function π(q
2
). This operator
P
σ
ρ
=
δ
σ
ρ
−
q
ρ
q
σ
q
2
is a projection operator onto transverse polarizations. In particular, like any
projection operator, it is idempotent:
P
σ
ρ
P
λ
σ
= P
λ
ρ
.
This allows us to simply the expression of the exact propagator, and write it as
∆
µν
(q) = ∆
0
µν
(1 + π(q
2
) + π
2
(q
2
) + ···) =
∆
0
µν
1 − π(q
2
)
.
Just as the classical propagator came from the kinetic term
S
kin
=
1
4
Z
F
µν
F
µν
dx =
1
2
Z
q
2
δ
µν
−
q
µ
q
ν
q
2
˜
A
µ
(−q)
˜
A
ν
(q) d
d
q,
so too our exact propagator is what we’d get from an action whose quadratic
term is
S
quant
=
1
2
Z
(1 − π(q
2
))q
2
δ
µν
−
q
µ
q
ν
q
2
˜
A
µ
(−q)
˜
A
ν
(q) d
d
q.
Expanding
π
(
q
2
) around
q
2
= 0, the leading term just corrects the kinetic term
of the photon. So we have
S
quant
∼
1
4
(1 − π(0))
Z
F
µν
F
µν
d
d
x + higher derivative terms.
One can check that the higher derivative terms will be irrelevant in d = 4.
Computing the photon self-energy
We now actually compute the self energy. This is mostly just doing some horrible
integrals.
To leading order, using the classical action (i.e. not including the countert-
erms), we have
Π
ρσ
1−loop
=
A
σ
q
A
ρ
p − q
p
= g
2
µ
4−d
Z
d
4
p
(2π)
d
Tr
(−i
/
p + m)γ
ρ
p
2
+ m
2
−i(
/
p −
/
q + m)γ
σ
(p − q)
2
+ m
2
,
where we take the trace to take into account of the fact that we are considering
all possible spins.
To compute this, we need a whole series of tricks. They are just tricks. We
first need the partial fraction identity
1
AB
=
1
B − A
1
A
−
1
B
=
Z
1
0
dx
((1 − x)A + xB)
2
.
Applying this to the two denominators in our propagators gives
1
(p
2
+ m
2
)((p − q)
2
+ m
2
)
=
Z
1
0
dx
((p
2
+ m
2
)(1 − x) + ((p − q)
2
+ m
2
)x)
2
=
Z
1
0
dx
(p
2
+ m
2
+ −2xpq + q
2
x)
2
=
Z
1
0
dx
((p − xq)
2
+ m
2
+ q
2
x(1 − x))
2
Letting p
0
= p − qx, and then dropping the prime, our loop integral becomes
g
2
µ
4−d
(2π)
d
Z
d
d
p
Z
1
0
dx
tr((−i(
/
p −
/
qx) + m)γ
ρ
(−i(
/
p −
/
q(1 − x)) + m)γ
σ
)
(p
2
+ ∆)
2
,
where ∆ = m
2
+ q
2
x(1 − x).
We next do the trace over Dirac spinor indices. As mentioned, if we are
working in
d
dimensions, then we should be honest and work with spinors in
d
dimensions, but in this case, we’ll get away with just pretending it is four
dimensions.
We have
tr(γ
ρ
γ
σ
) = 4δ
ρσ
, tr(γ
µ
γ
ρ
γ
ν
γ
σ
) = 4(δ
µν
δ
ρσ
− δ
µν
δ
ρσ
+ δ
µσ
δ
ρν
).
Then the huge trace expression we have just becomes
4
− (p + qx)
ρ
(p − q(1 − x))
σ
+ (p + qx) · (p − q(1 − x))δ
ρσ
− (p + qx)
σ
(p − q(1 − x))
ρ
+ m
2
δ
ρσ
.
Whenever
d ∈ N
, we certainly get zero if any component of
p
σ
appears an odd
number of times. Consequently, in the numerator, we can replace
p
ρ
p
σ
7→
p
2
d
δ
ρσ
.
Similarly, we have
p
µ
p
ρ
p
ν
p
σ
7→
(p
2
)
2
d(d + 2)
(δ
µν
δ
ρσ
+ δ
µρ
δ
νσ
+ δ
µσ
δ
ρν
)
The integrals are given in terms of Γ-functions, and we obtain
Π
ρσ
1−loop
(q) =
−4g
2
µ
4−d
(4π)
d/2
Γ
2 −
d
2
Z
1
0
dx
1
∆
2−d/2
δ
ρσ
(−m
2
+ x(1 − x)q
2
) + δ
ρσ
(m
2
+ x(1 − x)q
2
) − 2x(1 − x)q
ρ
q
σ
.
And, looking carefully, we find that the m
2
terms cancel, and the result is
Π
ρσ
1−loop
(q) = q
2
δ
ρσ
−
q
ρ
q
σ
q
2
π
1−loop
(q
2
),
where
π
1−loop
(q
2
) =
−8g
2
(µ)
(4π)
d/2
Γ
2 −
d
2
Z
1
0
dx x(1 − x)
µ
2
∆
2−d/2
.
The key point is that this diverges when
d
= 4, because Γ(0) is a pole, and so
we need to introduce counterterms.
We set d = 4 −ε, and ε → 0
+
. We introduce a counterterm
S
CT
[A, ψ, ε] =
Z
1
4
δZ
3
F
µν
F
µν
+ δZ
2
¯
ψ
/
Dψ + δM
¯
ψψ
d
d
x.
Note that the counterterms multiply gauge invariant contributions. We do not
have separate counterterms for
¯
ψ
/
∂ψ
and
¯
ψ
/
Aψ
. We can argue that this must be
the case because our theory is gauge invariant, or if we want to do it properly,
we can justify this using the Ward identities. This is important, because if we
did this by imposing cutoffs, then this property doesn’t hold.
For Π
1−loop
µν
, the appropriate counterterm is δZ
3
. As ε → 0
+
, we have
π
1−loop
(q
2
) ∼ −
g
2
(µ)
2π
2
Z
1
0
dx x(1 − x)
2
ε
− γ + log
4πµ
2
∆
+ O(ε)

.
The counterterm
×
= −
δZ
3
4
Z
F
µν
F
µν
d
4
x,
and must be chosen to remove the
1
ε
singularity. In the
MS
scheme, we also
remove the −γ + log 4π piece, because we hate them. So what is left is
π
MS
(q
2
) = +
g
2
(µ)
2π
2
Z
1
0
dx x(1 − x) log
m
2
+ q
2
x(1 − x)
µ
2
.
Then this is finite in
d
= 4. As we previously described, this 1-loop correction
contains the term ∼ log
m
2
+q
2
µ
2
. So it is small when m
2
+ q
2
∼ µ
2
.
Notice that the log term has a branch point when
m
2
+ q
2
x(1 − x) = 0.
For x ∈ [0, 1], we have
x(1 − x) ∈
0,
1
4
.
So the branch cut is inaccessible with real Euclidean momenta. But in Lorentzian
signature, we have
q
2
= q
2
− E
2
,
so the branch cut occurs when
(E
2
− q
2
)x(1 − x) ≥ m
2
,
which can be reached whenever
E
2
≥
(2
m
)
2
. This is exactly the threshold energy
for creating a real e
+
e
−
pair.
The QED β-function
To relate this “one-loop” exact photon propagator to the
β
-function for the
electromagnetic coupling, we rescale back to A
old
µ
= eA
new
µ
, where we have
S
(2)
eff
[A
old
] =
1
4g
2
(1 − π(0))
Z
F
µν
F
µν
d
4
z
= −
1
4
1
g
2
(µ)
−
1
2π
2
Z
1
0
dx x(1 − x) log
m
2
µ
2

Z
F
µν
F
µν
d
4
z.
We call the spacetime parameter
z
instead of
x
to avoid confusing it with the
Feynman parameter x.
Since nothing physical can depend on the arbitrary scale µ, we must have
0 = µ
∂
∂µ
1
g
2
(µ)
−
1
2π
2
Z
1
0
dx x(1 − x) log
m
2
µ
2

.
Solving this, we find that to lowest order, we have
β(g) =
g
2
12π
2
.
This integral is easy to do because we are evaluating at
π
(0), and the then
log
term does not depend on x.
This tells us that the running couplings are given by
1
g
2
(µ)
=
1
g
2
(µ
0
)
+
1
6π
2
log
µ
0
µ
.
Now suppose µ ∼ m
e
, where we measure
g
2
(m
e
)
4π
≈
1
137
,
the fine structure constant. Then there exists a scale µ
0
given by
m
0
= m
e
e
6π
2
/g
2
(m
e
)
∼ 10
286
GeV,
where g
2
(µ
0
) diverges! This is known as a Landau pole.
Yet again, the belief is that pure QED does not exist as a continuum quantum
field theory. Of course, what we have shown is that our perturbation theory
breaks down, but it turns out more sophisticated regularizations also break
down.
Physics of vacuum polarization
Despite QED not existing, we can still say something interesting about it.
Classically, when we want to do electromagnetism in a dielectric material, the
electromagnetic field causes the dielectric material to become slightly polarized,
which results in a shift in the effective electromagnetic potential. We shall see
that in QED, vacuum itself will act as such a dielectric material, due to effects
of virtual electron-positron pairs.
Consider the scattering two (distinguishable) Dirac spinors of charges
e
1
and
e
2
. The S matrix for the process is given by
S(1 2 → 1
0
2
0
) = −
e
1
e
2
4π
δ
4
(p
1
+ p
2
− p
1
0
− p
2
0
)¯u
1
0
γ
µ
u
1
∆
µν
(q)¯u
2
0
γ
ν
u
2
,
where q = p
1
− p
0
1
is the momentum of the photon propagator.
1
0
1
2
0
2
p
1
p
0
1
p
2
p
0
2
Here we are including the exact photon propagator in the diagram. It is given by
∆
0
µν
(q)
1 − π(q
2
)
,
and so we can write
≈
−e
1
e
2
4π
δ
(4)
(p
1
+ p
2
− p
1
0
− p
2
0
)¯u
1
0
γ
µ
u
1
∆
0
µν
¯u
2
0
γ
ν
u
2
(1 + π(q
2
) + ···).
So the quantum corrections modify the classical one by a factor of (1+
π
(
q
2
)+
···
).
To evaluate this better, we note that
¯u
i
γ
µ
u
i
∆
0
µν
¯u
2
0
γ
ν
u
2
= ¯u
1
0
γ
µ
u
1
¯u
2
0
γ
µ
u
2
1
q
2
.
In the non-relativistic limit, we expect |q
2
| |q| and
¯u
1
0
γ
µ
u
1
≈
giδ
m
1
m
0
1
0
,
where
m
1
, m
1
0
are the
σ
3
(spin) angular momentum quantum numbers. This
tells us it is very likely that the angular momentum is conserved.
Consequently, in this non-relativistic limit, we have
S(1 2 → 1
0
2
0
) ≈
e
1
e
2
4π|q|
2
δ
(4)
(p
1
+ p
2
− p
1
0
− p
2
0
)(1 + π(|q|
2
))δ
m
1
m
0
1
δ
m
2
m
0
2
.
This is what we would get using the Born approximation if we had a potential of
V (r) = e
1
e
2
Z
d
3
q
(2π)
3
1 + π(|q|
2
)
|q|
2
e
iq·r
.
In particular, if we cover off the
π
(
|q|
2
) piece, then this is just the Coulomb
potential.
In the regime |q|
2
m
2
e
, picking µ = m
e
, we obtain
π(|q|
2
) = π(0) +
g
2
(µ)
2π
2
Z
1
0
dx x(1 − x) log
1 +
x(1 − x)|q|
2
m
2
≈ π(0) +
g
2
(µ)
60π
2
|q|
2
m
2
.
Then we find
V (r) ≈ e
1
e
2
Z
d
3
q
(2π)
3
1 + π(0)
q
2
+
g
2
60π
2
m
2
+ ···
e
iq·r
= e
1
e
2
1 + π(0)
4πr
+
g
2
60π
2
m
2
δ
3
(r)
.
So we got a short-range modification to the Coulomb potential. We obtained
a
δ
-function precisely because we made the assumption
|q|
2
m
2
e
. If we did
more accurate calculations, then we don’t get a
δ
function, but we still get a
contribution that is exponentially suppressed as we go away form 0.
This modification of the Coulomb force is attributed to “screening”. It leads
to a measured shift in the energy levels of
`
= 0 bound states of hydrogen. We
can interpret this as saying that the vacuum itself is some sort of dielectric
medium, and this is the effect of loop diagrams that look like
1
0
1
2
0
2
p
1
p
0
1
p
2
p
0
2
The idea is that the (genuine) charges
e
1
and
e
2
polarizes the vacuum, which
causes virtual particles to pop in and out to repel or attract other particles.
So far, the conclusions of this course is pretty distressing. We looked at
φ
4
theory, and we discovered it doesn’t exist. We then looked at QED, and it
doesn’t exist. Now, we are going to look at Yang–Mills theory, which is believed
to exist.