3.14.1
Vacuum Polarization and the
β
Function
In QCD, there are four types of diagrams, a quark loop, two gauge boson loops, and a ghost loop.
The book also draws a diagram for the sum of contributions from counterterms, which are treated
in perturbation theory.
In renormalized perturbation theory we can write the coupling of a fermion to the gauge field as
μ
4

d
2
g
R
Z
1
A
a
μ
¯
ψ
i
γ
μ
T
a
ij
ψ
j
=
μ
4

d
2
g
R
Z
1
Z
2
√
Z
3
A
μ
(0)
¯
ψ
(0)
i
γ
μ
T
a
ij
ψ
(0)
j
(3.14.3)
where we are explicit about the scale dependence from dim reg. Thus the bare coupling, which must
be independent of the renormalization scale, is
g
0
=
μ
4

d
2
g
R
Z
1
Z
2
√
Z
3
(3.14.4)
Differentiating with respect to
μ
and expanding in perturbation theory tells us that
β
(
g
R
) =
μ
d
dμ
g
R
=
g
R

2

μ
d
dμ
δ
1

δ
2

1
2
δ
3
(3.14.5)
with
=
d

4. But the
δ
i
only depend on
μ
through their
g
R
dependence. So we can solve this
relation perturbatively giving
β
(
g
R
) =

2
g
R
+
2
g
2
R
∂
∂g
R
δ
1

δ
2

1
2
δ
3
(3.14.6)
Using the 1loop counterterm values obtained from modified minimal subtraction, we find that
β
(
g
R
) =

2
g
R

g
3
R
16
π
2
11
3
C
A

4
3
n
f
T
F
(3.14.7)
Note that the
ξ
dependence cancels, giving a gauge invariant result.
226
We could have also obtained the same result from
β
(
g
R
) =

2
g
R
+
2
g
2
R
∂
∂g
R
δ
A
3

3
2
δ
3
(3.14.8)
but this gives exactly the same result, due to gauge invariance. There is only one charge setting the
strength of the strong interactions.
Specializing to QCD in the standard model, we have
N
= 3, so
C
A
= 3, and we write
α
s
=
g
2
s
4
π
.
We also ahve
T
F
= 1
/
2, so at one loop we have
μ
d
dμ
α
s
=

α
2
2
2
π
11

2
n
f
3
(3.14.9)
In nature we have
n
f
= 6, but this varies with scale, because they drop out of the RG below the
scale of their masses. However, as long as
n
f
<
17, note that the
β
function for
α
s
is negative, which
means that
the strong couplings gets stronger at low energies, and weaker at high energies
, unlike
electromagnetism.
In particular, since
α
s
→
0 at very high energies, QCD exhibits
asymptotic freedom
, which just
means that the quarks and gluons are free at high energies. Conversely, QCD gets very strong at a
fixed scale Λ
QCD
, where we can solve the RG to write
α
s
=
2
π
11

2
n
f
3
1
log
μ
Λ
QCD
(3.14.10)
Measuring
α
s
at any scale
μ
determines the QCD scale, which in reality is very rougly near 1 GeV.
The QCD scale sets the characteristic energy scale of strongforce bound states, such as the proton.
3.15
Higgs Mechanism
We can combine our knowledge of Goldstone bosons and nonAbelian gauge theories to understand
how to describe massive interacting spin 1 bosons.
Let us try to imagine a theory of massive, charged (interacting) spin 1 particles. In such a theory,
one would naively guess that at very high energies
E
m
A
, the masses of the spin 1 particles
should be unimportant for the physics. This is certainly true for fermions and scalar particles – at
high energies their masses have a negligible effect on physics.
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 Fall '12
 Halperin
 Physics, mechanics, Quantum Field Theory