4 QCD: The Theory of Strong Interactions
99
coupling vanishes asymptotically at large Q 2 is called (ultraviolet) “asymptotically
free”. An important result that has been proven is that in four spacetime dimensions
all and only non-abelian gauge theories are asymptotically free.
Going back to Eq. (4.28) we replace β(α) ∼ ±bα 2 , do the integral and perform
a simple algebra. We find
QED :
α(t) ∼
α
1 − bαt
(4.46)
and
QCD :
α(t) ∼
α
1 + bαt
(4.47)
A slightly different form is often used in QCD. Defining 1/α = b log μ 2 // 2
QCD
we can write:
α(t) ∼
1
1
α + bt
=
1
b log
μ 2
2
QCD
+ b log
Q 2
μ 2
=
1
b log
Q 2
2
QCD
(4.48)
We see that α(t) decreases logarithmically with Q 2 and that one can introduce a
dimensional parameter QCD that replaces μ. Often in the following we will simply
write for QCD . Note that it is clear that depends on the particular definition of
α, not only on the defining scale μ but also on the renormalisation scheme (see, for
example, the discussion in the next session). Through the parameter b, and in general
through the β function, it also depends on the number n f of coupled flavours. It is
very important to note that QED and QCD are theories with “decoupling”: up to the
scale Q only quarks with masses m << Q contribute to the running of α. This is
clearly very important, given that all applications of perturbative QCD so far apply
to energies below the top quark mass m t . For the validity of the decoupling theorem
[11] it is necessary that the theory where all the heavy particle internal lines are
eliminated is still renormalisable and that the coupling constants do not vary with
the mass. These requirements are true for the mass of heavy quarks in QED and
QCD, but are not true in the electroweak theory where the elimination of the top
would violate SU (2) symmetry (because the t and b left quarks are in a doublet) and
the quark couplings to the Higgs multiplet (hence to the longitudinal gauge bosons)
are proportional to the mass. In conclusion, in QED and QCD, quarks with m >> Q
do not contribute to n f in the coefficients of the relevant β function. The effects of
heavy quarks are power suppressed and can be taken separately into account. For
example, in e + e − annihilation for 2m c < Q < 2m b the relevant asymptotics is for
n f = 4, while for 2m b < Q < 2m t n f = 5. Going accross the b threshold the β
function coefficients change, so the α(t) slope changes. But α(t) is continuous, so
that changes so as to keep constant α(t) at the matching point at Q ∼ o(2m b ).
The effect on is large: approximately 5 ∼ 0.65 4 .
99
coupling vanishes asymptotically at large Q 2 is called (ultraviolet) “asymptotically
free”. An important result that has been proven is that in four spacetime dimensions
all and only non-abelian gauge theories are asymptotically free.
Going back to Eq. (4.28) we replace β(α) ∼ ±bα 2 , do the integral and perform
a simple algebra. We find
QED :
α(t) ∼
α
1 − bαt
(4.46)
and
QCD :
α(t) ∼
α
1 + bαt
(4.47)
A slightly different form is often used in QCD. Defining 1/α = b log μ 2 // 2
QCD
we can write:
α(t) ∼
1
1
α + bt
=
1
b log
μ 2
2
QCD
+ b log
Q 2
μ 2
=
1
b log
Q 2
2
QCD
(4.48)
We see that α(t) decreases logarithmically with Q 2 and that one can introduce a
dimensional parameter QCD that replaces μ. Often in the following we will simply
write for QCD . Note that it is clear that depends on the particular definition of
α, not only on the defining scale μ but also on the renormalisation scheme (see, for
example, the discussion in the next session). Through the parameter b, and in general
through the β function, it also depends on the number n f of coupled flavours. It is
very important to note that QED and QCD are theories with “decoupling”: up to the
scale Q only quarks with masses m << Q contribute to the running of α. This is
clearly very important, given that all applications of perturbative QCD so far apply
to energies below the top quark mass m t . For the validity of the decoupling theorem
[11] it is necessary that the theory where all the heavy particle internal lines are
eliminated is still renormalisable and that the coupling constants do not vary with
the mass. These requirements are true for the mass of heavy quarks in QED and
QCD, but are not true in the electroweak theory where the elimination of the top
would violate SU (2) symmetry (because the t and b left quarks are in a doublet) and
the quark couplings to the Higgs multiplet (hence to the longitudinal gauge bosons)
are proportional to the mass. In conclusion, in QED and QCD, quarks with m >> Q
do not contribute to n f in the coefficients of the relevant β function. The effects of
heavy quarks are power suppressed and can be taken separately into account. For
example, in e + e − annihilation for 2m c < Q < 2m b the relevant asymptotics is for
n f = 4, while for 2m b < Q < 2m t n f = 5. Going accross the b threshold the β
function coefficients change, so the α(t) slope changes. But α(t) is continuous, so
that changes so as to keep constant α(t) at the matching point at Q ∼ o(2m b ).
The effect on is large: approximately 5 ∼ 0.65 4 .