5 QCD on the Lattice
193
Instead of Parisi’s “boosted” coupling ˜
g other expansion parameters have been
suggested, which are expected to accelerate the convergence of the perturbative
series [83]. While mean-field improvement is a general procedure, which is easily
adapted to a wide range of actions and operators, it is difficult to estimate the
effectiveness of the resummation and, in turn, the size of higher-order corrections.
Also, a principal problem is the identification of the running scale with the cutoff,
since it is difficult to separate renormalization effects from lattice artefacts.
5.5.5 The Running Coupling from the Lattice
Having discussed the non-perturbative renormalization of QCD in detail, we shall
now present results for the running coupling constant, α s , from two different
approaches. This complements the discussion in Sect. 4.6, where determination of
α s from experimental data has been described in detail. Any lattice calculation of
α s proceeds along the following steps:
1. A non-perturbative definition of the coupling must be provided in terms of
some quantity which can be evaluated in lattice simulations with high precision.
This amounts to specifying the running coupling in a particular renormalization
scheme, α X (aμ 0 ), which can be related to the MS scheme of dimensional
regularization.
2. Scale setting: the matching to a hadronic scheme is performed via the calibration
of the lattice spacing, which yields the scale μ 0 at which α X is evaluated in units
of some physical quantity Q:
μ 0 [MeV] = (aμ 0 ) · a
−1
[MeV] = (aμ 0 ) ·
Q [MeV]
(aQ)
.
(5.141)
3. Running and matching: provided that the energy scale at which α X has been
determined is large enough, one can use perturbation theory to relate α X to the
coupling in the MS scheme, e.g.
α MS ( ¯
μ) = α X (μ) + c
(1)
X ( ¯
μ/μ)α X (μ)
2
+ . . . .
(5.142)
4. The -parameter can be determined from the asymptotic behaviour of α X via
Eq. (5.113).
The attentive reader has surely noticed that the above steps follow closely the
general strategy for non-perturbative renormalization via an intermediate renormalization scheme outlined in Sect. 5.5.1 and Fig. 5.12.
193
Instead of Parisi’s “boosted” coupling ˜
g other expansion parameters have been
suggested, which are expected to accelerate the convergence of the perturbative
series [83]. While mean-field improvement is a general procedure, which is easily
adapted to a wide range of actions and operators, it is difficult to estimate the
effectiveness of the resummation and, in turn, the size of higher-order corrections.
Also, a principal problem is the identification of the running scale with the cutoff,
since it is difficult to separate renormalization effects from lattice artefacts.
5.5.5 The Running Coupling from the Lattice
Having discussed the non-perturbative renormalization of QCD in detail, we shall
now present results for the running coupling constant, α s , from two different
approaches. This complements the discussion in Sect. 4.6, where determination of
α s from experimental data has been described in detail. Any lattice calculation of
α s proceeds along the following steps:
1. A non-perturbative definition of the coupling must be provided in terms of
some quantity which can be evaluated in lattice simulations with high precision.
This amounts to specifying the running coupling in a particular renormalization
scheme, α X (aμ 0 ), which can be related to the MS scheme of dimensional
regularization.
2. Scale setting: the matching to a hadronic scheme is performed via the calibration
of the lattice spacing, which yields the scale μ 0 at which α X is evaluated in units
of some physical quantity Q:
μ 0 [MeV] = (aμ 0 ) · a
−1
[MeV] = (aμ 0 ) ·
Q [MeV]
(aQ)
.
(5.141)
3. Running and matching: provided that the energy scale at which α X has been
determined is large enough, one can use perturbation theory to relate α X to the
coupling in the MS scheme, e.g.
α MS ( ¯
μ) = α X (μ) + c
(1)
X ( ¯
μ/μ)α X (μ)
2
+ . . . .
(5.142)
4. The -parameter can be determined from the asymptotic behaviour of α X via
Eq. (5.113).
The attentive reader has surely noticed that the above steps follow closely the
general strategy for non-perturbative renormalization via an intermediate renormalization scheme outlined in Sect. 5.5.1 and Fig. 5.12.