134
15. QCD II: Asymptotic Freedom, the Renormalization Group
We must now point out to the reader an error in the foregoing analysis, in
the case of a gauge theory. The quantity Z 2 is not gauge invariant in QED
(or QCD), and hence γ 2 depends on the choice of gauge. This is really no
surprise, because the full fermion propagator itself is not gauge invariant (the
free-field propagator is gauge invariant, of course). What ultimately matters
is that the complete physical amplitude for any process, at a given order of
α, be gauge invariant. Thus the analysis given above really only applies – in
this simple form – to non-gauge theories, such as the ABC model of chapter
6, or to gauge-invariant quantities.
This is an appropriate point at which to consider the treatment of quark
masses in the RGE-based approach. Up to now we have simply assumed
that the relevant |q
2
| is very much greater than all quark masses, the latter
therefore being neglected. While this may be adequate for the light quarks u,
d, s, it seems surely a progressively worse assumption for c, b and t. However,
in thinking about how to re-introduce the quark masses into our formalism,
we are at once faced with a difficulty: how are they to be defined? For an
unconfined particle, such as a lepton, it seems natural to define ‘the’ mass as
2
the position of the pole of the propagator (i.e. the ‘on-shell’ value p = m
2 ), a
definition we followed in chapters 10 and 11. Significantly, renormalization is
required (in the shape of a mass counter-term) to achieve a pole at the ‘right’
physical mass m, in this sense. But this prescription is inherently perturbative,
and cannot be used for a confined particle, which never ‘escapes’ beyond the
range of the non-perturbative confining forces, and whose propagator can
therefore never approach the form ∼ (/ p − m)
−1 of a free particle.
Our present perspective on renormalization suggests an obvious way forward. Just as there was, in principle, no necessity to define the QED coupling
parameter e via an on-shell prescription, so here a mass parameter in the Lagrangian can be defined in any way we find convenient; all that is necessary
is that it should be possible to determine its value from some measurable
quantity (for example, quark masses from lattice QCD predictions of hadron
¯
masses). Effectively, we are regarding the ‘m’ in a term such as −mψ ˆ (x)ψ ˆ (x)
as a ‘coupling constant’ having mass dimension 1 (and, after all, the ABC
¯ ˆ
coupling itself had mass dimension 1). Incidentally, the operator ψ(x)ψ ˆ (x) is
gauge invariant, as is any such local operator. Taking this point of view, it is
clear that a renormalization scale will be involved in such a general definition
of mass, and we must expect to see our mass parameters ‘evolve’ with this
scale, just as the gauge (or other) couplings do. In turn, this will get translated into a |q
2
|-dependence of the mass parameters, just as for α(|q
2
|) and
α s (|q
2
|).
The RGE in such a scheme now takes the form
[
]
∑
∂
∂
∂
μ
2
+ β(α s )
+
γ i (α s ) + γ m (α s )m
R(|q
2
|/μ
2 , α s , m/|q|) = 0
∂μ 2
∂α s
∂m
i
(15.79)
where the partial derivatives are taken at fixed values of the other two vari
15. QCD II: Asymptotic Freedom, the Renormalization Group
We must now point out to the reader an error in the foregoing analysis, in
the case of a gauge theory. The quantity Z 2 is not gauge invariant in QED
(or QCD), and hence γ 2 depends on the choice of gauge. This is really no
surprise, because the full fermion propagator itself is not gauge invariant (the
free-field propagator is gauge invariant, of course). What ultimately matters
is that the complete physical amplitude for any process, at a given order of
α, be gauge invariant. Thus the analysis given above really only applies – in
this simple form – to non-gauge theories, such as the ABC model of chapter
6, or to gauge-invariant quantities.
This is an appropriate point at which to consider the treatment of quark
masses in the RGE-based approach. Up to now we have simply assumed
that the relevant |q
2
| is very much greater than all quark masses, the latter
therefore being neglected. While this may be adequate for the light quarks u,
d, s, it seems surely a progressively worse assumption for c, b and t. However,
in thinking about how to re-introduce the quark masses into our formalism,
we are at once faced with a difficulty: how are they to be defined? For an
unconfined particle, such as a lepton, it seems natural to define ‘the’ mass as
2
the position of the pole of the propagator (i.e. the ‘on-shell’ value p = m
2 ), a
definition we followed in chapters 10 and 11. Significantly, renormalization is
required (in the shape of a mass counter-term) to achieve a pole at the ‘right’
physical mass m, in this sense. But this prescription is inherently perturbative,
and cannot be used for a confined particle, which never ‘escapes’ beyond the
range of the non-perturbative confining forces, and whose propagator can
therefore never approach the form ∼ (/ p − m)
−1 of a free particle.
Our present perspective on renormalization suggests an obvious way forward. Just as there was, in principle, no necessity to define the QED coupling
parameter e via an on-shell prescription, so here a mass parameter in the Lagrangian can be defined in any way we find convenient; all that is necessary
is that it should be possible to determine its value from some measurable
quantity (for example, quark masses from lattice QCD predictions of hadron
¯
masses). Effectively, we are regarding the ‘m’ in a term such as −mψ ˆ (x)ψ ˆ (x)
as a ‘coupling constant’ having mass dimension 1 (and, after all, the ABC
¯ ˆ
coupling itself had mass dimension 1). Incidentally, the operator ψ(x)ψ ˆ (x) is
gauge invariant, as is any such local operator. Taking this point of view, it is
clear that a renormalization scale will be involved in such a general definition
of mass, and we must expect to see our mass parameters ‘evolve’ with this
scale, just as the gauge (or other) couplings do. In turn, this will get translated into a |q
2
|-dependence of the mass parameters, just as for α(|q
2
|) and
α s (|q
2
|).
The RGE in such a scheme now takes the form
[
]
∑
∂
∂
∂
μ
2
+ β(α s )
+
γ i (α s ) + γ m (α s )m
R(|q
2
|/μ
2 , α s , m/|q|) = 0
∂μ 2
∂α s
∂m
i
(15.79)
where the partial derivatives are taken at fixed values of the other two vari