5 QCD on the Lattice
199
In this expression, the subscript “exp” denotes the experimental values for the
respective quantities, while the matrix element G bare
PS is given by
G
bare
PS
m PS =m K
≡ G
bare
K =
0
( ¯
5 s) lat
K
.
(5.157)
The pseudoscalar decay constant f bare
PS parameterizes the matrix element of the
unrenormalized axial current, i.e.
f
bare
PS m PS
m PS =m K
≡ f
bare
K m K =
0
( ¯
0 γ 5 s) lat
K
.
(5.158)
The renormalization factor Z M relates the bare current quark mass to the RGinvariant mass. Thus, the task for lattice calculations is to compute the ratio
f bare
PS Q/G bare
PS for a generic pseudoscalar state and tune the bare quark mass such
that m PS = m K . By combining the result with the renormalization factor Z M
and the experimental value of m 2
K /Q 2 , the RGI quark masses in units of Q
are obtained up to lattice artefacts of order a p , where p is characteristic of the
details of the discretization. Since the RGI quark masses are scale- and schemeindependent quantities, the factor Z M depends only on the bare coupling g 0 .
Using the Schrödinger functional as the intermediate renormalization scheme, nonperturbative estimates of Z M computed for O(a) improved Wilson fermions within
a wide range of bare couplings, have been published in Refs. [75] and [78]. In this
case, Z M is given by
Z M (g 0 ) =
M
¯
m SF (μ 0 )
Z A (g 0 )
Z P (g 0 , aμ 0 )
,
(5.159)
where the ratio M/ ¯
m SF (μ 0 ) is computed via the finite-size scaling procedure. The
transition between lattice regularization and the SF-scheme is accomplished by
determining Z P and the renormalization factor Z A of the axial current. 14 Note that
the dependence on the intermediate matching scale μ 0 drops out completely in this
expression. Finally, the conversion to the MS-scheme is performed by considering
Z m (g 0 , aμ) ≡
¯
m MS (μ)
M
Z M (g 0 ),
(5.160)
where the ratio ¯
m MS (μ)/M can be computed through the numerical integration of
the perturbative approximation of the anomalous dimension τ and the β-function at
four loops. This yields [35, 78]
¯
m MS (2 GeV)
M
=
0.7208, N f = 0
0.7013, N f = 2 .
(5.161)
14 If the fermionic discretization preserves chiral symmetry Z A = 1, while for Wilson fermions Z A
should be computed non-perturbatively. For the SF this was performed in Refs. [91, 92].
199
In this expression, the subscript “exp” denotes the experimental values for the
respective quantities, while the matrix element G bare
PS is given by
G
bare
PS
m PS =m K
≡ G
bare
K =
0
( ¯
5 s) lat
K
.
(5.157)
The pseudoscalar decay constant f bare
PS parameterizes the matrix element of the
unrenormalized axial current, i.e.
f
bare
PS m PS
m PS =m K
≡ f
bare
K m K =
0
( ¯
0 γ 5 s) lat
K
.
(5.158)
The renormalization factor Z M relates the bare current quark mass to the RGinvariant mass. Thus, the task for lattice calculations is to compute the ratio
f bare
PS Q/G bare
PS for a generic pseudoscalar state and tune the bare quark mass such
that m PS = m K . By combining the result with the renormalization factor Z M
and the experimental value of m 2
K /Q 2 , the RGI quark masses in units of Q
are obtained up to lattice artefacts of order a p , where p is characteristic of the
details of the discretization. Since the RGI quark masses are scale- and schemeindependent quantities, the factor Z M depends only on the bare coupling g 0 .
Using the Schrödinger functional as the intermediate renormalization scheme, nonperturbative estimates of Z M computed for O(a) improved Wilson fermions within
a wide range of bare couplings, have been published in Refs. [75] and [78]. In this
case, Z M is given by
Z M (g 0 ) =
M
¯
m SF (μ 0 )
Z A (g 0 )
Z P (g 0 , aμ 0 )
,
(5.159)
where the ratio M/ ¯
m SF (μ 0 ) is computed via the finite-size scaling procedure. The
transition between lattice regularization and the SF-scheme is accomplished by
determining Z P and the renormalization factor Z A of the axial current. 14 Note that
the dependence on the intermediate matching scale μ 0 drops out completely in this
expression. Finally, the conversion to the MS-scheme is performed by considering
Z m (g 0 , aμ) ≡
¯
m MS (μ)
M
Z M (g 0 ),
(5.160)
where the ratio ¯
m MS (μ)/M can be computed through the numerical integration of
the perturbative approximation of the anomalous dimension τ and the β-function at
four loops. This yields [35, 78]
¯
m MS (2 GeV)
M
=
0.7208, N f = 0
0.7013, N f = 2 .
(5.161)
14 If the fermionic discretization preserves chiral symmetry Z A = 1, while for Wilson fermions Z A
should be computed non-perturbatively. For the SF this was performed in Refs. [91, 92].