5 QCD on the Lattice
181
At a certain separation r b one observes a crossing of energy levels and a continuing
flat behaviour of the ground state energy. Near the crossing point one actually
observes a repulsion of the energy levels, which is characteristic for the breaking
phenomenon. The diagonalization of the matrix correlator also yields information
on the composition of the states in the spectral decomposition. Indeed, for distances
r < r b the combination of operators describing the ground state is dominated by
Wilson loops, whereas for r > r b , two-meson operators are the most relevant.
5.5 Fundamental Parameters of QCD
We have noted already that QCD is parameterized in terms of the gauge coupling
and the masses of the quarks. In order to make predictions for cross sections, decay
rates and other observables, their values must be fixed from experiment. As was
discussed in detail in Sect. 4.3 , the renormalization of QCD leads to the concept of
a “running” coupling constant, which depends on some momentum (energy) scale
μ, and the same applies to the quark masses 9 :
α s (μ) ≡
¯
g 2 (μ)
4π
, ¯
m u (μ), ¯
m d (μ), ¯
m s (μ), ¯
m c (μ), ¯
m b (μ), ¯
m t (μ).
(5.110)
The property of asymptotic freedom implies that the coupling becomes weaker as
the energy scale μ is increased. This explains why the perturbative expansion of
cross sections in the high-energy domain allows for an accurate determination of α s
from experimental data.
The scale dependence of the coupling and the quark masses is encoded in the
renormalization group (RG) equations, which are formulated in terms of the βfunction and the anomalous dimension τ ,
μ
∂ ¯
g(μ)
∂μ
= β( ¯
g),
μ
∂ ¯
m(μ)
∂μ
= ¯
mτ ( ¯
g).
(5.111)
At high enough energy the RG functions β and τ admit perturbative expansions
according to
β( ¯
g) = −b 0 ¯
g
3
− b 1 ¯
g
5
+ . . . ,
τ ( ¯
g) = −d 0 ¯
g
2
− d 1 ¯
g
4
+ . . . .
(5.112)
Here, b 0 , b 1 and d 0 = 8/(4π) 2 are universal, while the higher coefficients depend
on the adopted renormalization scheme.
9 As usual we denote the running parameters by a bar across the symbol.
181
At a certain separation r b one observes a crossing of energy levels and a continuing
flat behaviour of the ground state energy. Near the crossing point one actually
observes a repulsion of the energy levels, which is characteristic for the breaking
phenomenon. The diagonalization of the matrix correlator also yields information
on the composition of the states in the spectral decomposition. Indeed, for distances
r < r b the combination of operators describing the ground state is dominated by
Wilson loops, whereas for r > r b , two-meson operators are the most relevant.
5.5 Fundamental Parameters of QCD
We have noted already that QCD is parameterized in terms of the gauge coupling
and the masses of the quarks. In order to make predictions for cross sections, decay
rates and other observables, their values must be fixed from experiment. As was
discussed in detail in Sect. 4.3 , the renormalization of QCD leads to the concept of
a “running” coupling constant, which depends on some momentum (energy) scale
μ, and the same applies to the quark masses 9 :
α s (μ) ≡
¯
g 2 (μ)
4π
, ¯
m u (μ), ¯
m d (μ), ¯
m s (μ), ¯
m c (μ), ¯
m b (μ), ¯
m t (μ).
(5.110)
The property of asymptotic freedom implies that the coupling becomes weaker as
the energy scale μ is increased. This explains why the perturbative expansion of
cross sections in the high-energy domain allows for an accurate determination of α s
from experimental data.
The scale dependence of the coupling and the quark masses is encoded in the
renormalization group (RG) equations, which are formulated in terms of the βfunction and the anomalous dimension τ ,
μ
∂ ¯
g(μ)
∂μ
= β( ¯
g),
μ
∂ ¯
m(μ)
∂μ
= ¯
mτ ( ¯
g).
(5.111)
At high enough energy the RG functions β and τ admit perturbative expansions
according to
β( ¯
g) = −b 0 ¯
g
3
− b 1 ¯
g
5
+ . . . ,
τ ( ¯
g) = −d 0 ¯
g
2
− d 1 ¯
g
4
+ . . . .
(5.112)
Here, b 0 , b 1 and d 0 = 8/(4π) 2 are universal, while the higher coefficients depend
on the adopted renormalization scheme.
9 As usual we denote the running parameters by a bar across the symbol.