182
H. Wittig
From the asymptotic scaling behaviour at high energies one can extract the
fundamental scale parameter of QCD via
= lim
μ→∞
μ(b 0 ¯
g
2 )
−b 1 /2b 2
0 e
−1/2b 0 ¯
g 2
,
¯
g ≡ ¯
g(μ).
(5.113)
Like the running coupling itself, the -parameter depends on the chosen renormalization scheme. 10 A related, but less commonly used variable is the renormalization
group invariant (RGI) quark mass
M f = lim
μ→∞
¯
m f (2b 0 ¯
g
2 )
−d 0 /2b 0
, f = u, d, s, . . . ,
¯
m ≡ ¯
m(μ).
(5.114)
Unlike , the RGI quark masses are scheme-independent quantities. Instead of
using the running coupling and quark masses of Eq. (5.110), one can parameterize
QCD in an entirely equivalent way through the set
, M u , M d , M s , M c , M b , M t .
(5.115)
At the non-perturbative level these quantities represent the most appropriate parameterization of QCD, since their values are defined without any truncation of
perturbation theory.
The perturbative renormalization of QCD is accomplished by replacing the bare
parameters with renormalized ones, whose values are fixed by considering the highenergy behaviour of Green’s functions, usually computed in the MS-scheme of
dimensional regularization. However, at low energies it is convenient to adopt a
hadronic renormalization scheme, in which the bare parameters are eliminated in
favour of quantities such as hadron masses and decay constants (see Sect. 5.2.4).
Since QCD is expected to describe both the low- and high-energy regimes of the
strong interaction, one should be able to express the quantities of Eq. (5.115), which
are determined from the high-energy behaviour, in terms of hadronic quantities.
In other words, by matching a hadronic renormalization scheme to a perturbative
scheme like MS one achieves the non-perturbative renormalization of QCD at all
scales. In particular, one can express the fundamental parameters of QCD (running
coupling and masses, or, equivalently, the -parameter and RGI quark masses) in
terms of low-energy, hadronic quantities. This amounts to predicting the values of
these fundamental parameters from first principles.
10 The expressions for b 0 and b 1 , as well as the -parameter have already been shown in Sect. 5.2.4.
H. Wittig
From the asymptotic scaling behaviour at high energies one can extract the
fundamental scale parameter of QCD via
= lim
μ→∞
μ(b 0 ¯
g
2 )
−b 1 /2b 2
0 e
−1/2b 0 ¯
g 2
,
¯
g ≡ ¯
g(μ).
(5.113)
Like the running coupling itself, the -parameter depends on the chosen renormalization scheme. 10 A related, but less commonly used variable is the renormalization
group invariant (RGI) quark mass
M f = lim
μ→∞
¯
m f (2b 0 ¯
g
2 )
−d 0 /2b 0
, f = u, d, s, . . . ,
¯
m ≡ ¯
m(μ).
(5.114)
Unlike , the RGI quark masses are scheme-independent quantities. Instead of
using the running coupling and quark masses of Eq. (5.110), one can parameterize
QCD in an entirely equivalent way through the set
, M u , M d , M s , M c , M b , M t .
(5.115)
At the non-perturbative level these quantities represent the most appropriate parameterization of QCD, since their values are defined without any truncation of
perturbation theory.
The perturbative renormalization of QCD is accomplished by replacing the bare
parameters with renormalized ones, whose values are fixed by considering the highenergy behaviour of Green’s functions, usually computed in the MS-scheme of
dimensional regularization. However, at low energies it is convenient to adopt a
hadronic renormalization scheme, in which the bare parameters are eliminated in
favour of quantities such as hadron masses and decay constants (see Sect. 5.2.4).
Since QCD is expected to describe both the low- and high-energy regimes of the
strong interaction, one should be able to express the quantities of Eq. (5.115), which
are determined from the high-energy behaviour, in terms of hadronic quantities.
In other words, by matching a hadronic renormalization scheme to a perturbative
scheme like MS one achieves the non-perturbative renormalization of QCD at all
scales. In particular, one can express the fundamental parameters of QCD (running
coupling and masses, or, equivalently, the -parameter and RGI quark masses) in
terms of low-energy, hadronic quantities. This amounts to predicting the values of
these fundamental parameters from first principles.
10 The expressions for b 0 and b 1 , as well as the -parameter have already been shown in Sect. 5.2.4.