5 QCD on the Lattice
195
Starting from μ 0 = 1/L max one obtains the coupling at μ = 2 9 /L max after nine
steps in the scaling procedure. At that point one can extract the -parameter by
evaluating the exact expression
SF = μ
b 0 ¯
g 2 (μ)
−b 1 /(2b 2
0 )
e −1/(2b 0 ¯
g 2 (μ)) exp
−
¯
g(μ)
0
dx
1
β(x)
+
1
b 0 x 3 −
b 1
b 2
0 x
,
(5.147)
where μ = 2 9 /L max . The integral can be computed using the three-loop approximation to the RG β-function in the SF scheme. Equation (5.147) yields the combination
SF L max , and knowledge of L max in physical units allows to express the
parameter in MeV. Conversion to the MS scheme is easily achieved, since the ratio
of -parameters in two different schemes is computable via a one-loop calculation
in which ¯
g 2
MS
is expanded in powers of ¯
g 2
SF . This gives
MS = SF · c .
(5.148)
The entire procedure of determining the -parameter via the Schrödinger functional
has so far been carried out for the pure SU(3) gauge theory (N f = 0) and for
QCD with two flavours of dynamical quarks. The values of the coefficient c are
2.04872(4) for N f = 0 [84] and c = 2.382035(3) for N f = 2 [85], and the
resulting values for MS are [75, 77]
(0)
MS
r 0 = 0.602 ± 0.048
⇔
(0)
MS
= 238 ± 19 MeV
(2)
MS
r 0 = 0.62 ± 0.04 ± 0.04 ⇔
(2)
MS
= 245 ± 16 ± 16 MeV,
(5.149)
where r 0 = 0.5 fm is used to convert into physical units. There is room for
improvement in several respects: for N f = 2 the extrapolation to the continuum
limit can be made more reliable by including simulations at smaller lattice spacings,
which should reduce the first of the two quoted errors. Also, the conversion into
physical units should be performed in terms of a quantity such as f π , which is
directly accessible in experiment. Finally, the calculation must be repeated with
more dynamical quark flavours, in order to allow for a direct comparison with
phenomenology, since all experimental determinations yield the -parameter for
N f = 4 or 5 quark flavours.
The determination of α s and MS via the Schrödinger functional is quite
involved. However, it is the only method so far, which allows to map out the running
of α s in a completely non-perturbative manner, including the systematic elimination
of lattice artefacts. In particular, perturbation theory is used only for energy scales
well above 50 GeV.
The second method that we will discussed here in some detail is the determination of α s via heavy quarkonia. Below we present an account of the calculation
published in [86]. Here, the dynamical quark effects of the light (u, d, s) quarks have
been accounted for in simulations with improved staggered quarks employing the
195
Starting from μ 0 = 1/L max one obtains the coupling at μ = 2 9 /L max after nine
steps in the scaling procedure. At that point one can extract the -parameter by
evaluating the exact expression
SF = μ
b 0 ¯
g 2 (μ)
−b 1 /(2b 2
0 )
e −1/(2b 0 ¯
g 2 (μ)) exp
−
¯
g(μ)
0
dx
1
β(x)
+
1
b 0 x 3 −
b 1
b 2
0 x
,
(5.147)
where μ = 2 9 /L max . The integral can be computed using the three-loop approximation to the RG β-function in the SF scheme. Equation (5.147) yields the combination
SF L max , and knowledge of L max in physical units allows to express the
parameter in MeV. Conversion to the MS scheme is easily achieved, since the ratio
of -parameters in two different schemes is computable via a one-loop calculation
in which ¯
g 2
MS
is expanded in powers of ¯
g 2
SF . This gives
MS = SF · c .
(5.148)
The entire procedure of determining the -parameter via the Schrödinger functional
has so far been carried out for the pure SU(3) gauge theory (N f = 0) and for
QCD with two flavours of dynamical quarks. The values of the coefficient c are
2.04872(4) for N f = 0 [84] and c = 2.382035(3) for N f = 2 [85], and the
resulting values for MS are [75, 77]
(0)
MS
r 0 = 0.602 ± 0.048
⇔
(0)
MS
= 238 ± 19 MeV
(2)
MS
r 0 = 0.62 ± 0.04 ± 0.04 ⇔
(2)
MS
= 245 ± 16 ± 16 MeV,
(5.149)
where r 0 = 0.5 fm is used to convert into physical units. There is room for
improvement in several respects: for N f = 2 the extrapolation to the continuum
limit can be made more reliable by including simulations at smaller lattice spacings,
which should reduce the first of the two quoted errors. Also, the conversion into
physical units should be performed in terms of a quantity such as f π , which is
directly accessible in experiment. Finally, the calculation must be repeated with
more dynamical quark flavours, in order to allow for a direct comparison with
phenomenology, since all experimental determinations yield the -parameter for
N f = 4 or 5 quark flavours.
The determination of α s and MS via the Schrödinger functional is quite
involved. However, it is the only method so far, which allows to map out the running
of α s in a completely non-perturbative manner, including the systematic elimination
of lattice artefacts. In particular, perturbation theory is used only for energy scales
well above 50 GeV.
The second method that we will discussed here in some detail is the determination of α s via heavy quarkonia. Below we present an account of the calculation
published in [86]. Here, the dynamical quark effects of the light (u, d, s) quarks have
been accounted for in simulations with improved staggered quarks employing the