194
H. Wittig
First we discuss the determination of α s from the Schrödinger functional. The
definition of the running coupling is somewhat technical in this case. The starting
point is the effective action of Eq. (5.123); the classical field configurations at the
boundaries at x 0 = 0, T can be parameterized in terms of a real variable η:
C = C(η),
C
= C
(η).
(5.143)
For explicit expressions we refer the reader to the original article [84]. The
associated effective action is defined by
(η) = − ln Z[C
(η), 0, 0; C(η), 0, 0]
(5.144)
and admits a perturbative expansion in terms of the bare coupling g 0 , viz.
=
1
g 2
0
0 + 1 + g
2
0 2 + . . . .
(5.145)
A renormalized coupling can then be defined in terms of the effective action via
1
¯
g 2
SF (L)
=
∂
∂η
∂
∂η
0 (η)
η=0, m=0
.
(5.146)
This definition is imposed at vanishing quark mass, m = 0, and provided that the
aspect ratio T /L has been fixed, the spatial dimension is the only scale in the theory,
such that ¯
g SF (L) runs with the box size L. From the perturbative expansion of (η)
one easily infers that ¯
g 2
SF (L) = g 2
0 at tree level. The quantity on the right-hand side
is given in terms of plaquettes attached to the SF boundaries and can be computed
with good statistical precision.
If L max denotes the largest box size for which ¯
g SF is computed, then the scale
is set by expressing L max in terms of some known dimensionful quantity, for
instance, by computing the combination L max /r 0 in the continuum limit and using
r 0 = 0.5 fm.
The finite-size scaling procedure described earlier in Sect. 5.5.1 allows to
compute the scale evolution of ¯
g SF over several orders of magnitude. In particular,
each of the horizontal steps in Fig. 5.14 can be repeated for several values of the
lattice spacing, so that the continuum limit is reached by taking a/L → 0 for
fixed physical box size L. The resulting scale evolution of α SF ≡ ¯
g 2
SF /4π is
shown in Fig. 5.15 and compared to the perturbative evolution. Although the nonperturbatively determined points are described very well by perturbation theory,
using the three-loop expression for the RG function, one should realize that this
behaviour may be specific to the SF scheme and should not be generalized to other
schemes.
H. Wittig
First we discuss the determination of α s from the Schrödinger functional. The
definition of the running coupling is somewhat technical in this case. The starting
point is the effective action of Eq. (5.123); the classical field configurations at the
boundaries at x 0 = 0, T can be parameterized in terms of a real variable η:
C = C(η),
C
= C
(η).
(5.143)
For explicit expressions we refer the reader to the original article [84]. The
associated effective action is defined by
(η) = − ln Z[C
(η), 0, 0; C(η), 0, 0]
(5.144)
and admits a perturbative expansion in terms of the bare coupling g 0 , viz.
=
1
g 2
0
0 + 1 + g
2
0 2 + . . . .
(5.145)
A renormalized coupling can then be defined in terms of the effective action via
1
¯
g 2
SF (L)
=
∂
∂η
∂
∂η
0 (η)
η=0, m=0
.
(5.146)
This definition is imposed at vanishing quark mass, m = 0, and provided that the
aspect ratio T /L has been fixed, the spatial dimension is the only scale in the theory,
such that ¯
g SF (L) runs with the box size L. From the perturbative expansion of (η)
one easily infers that ¯
g 2
SF (L) = g 2
0 at tree level. The quantity on the right-hand side
is given in terms of plaquettes attached to the SF boundaries and can be computed
with good statistical precision.
If L max denotes the largest box size for which ¯
g SF is computed, then the scale
is set by expressing L max in terms of some known dimensionful quantity, for
instance, by computing the combination L max /r 0 in the continuum limit and using
r 0 = 0.5 fm.
The finite-size scaling procedure described earlier in Sect. 5.5.1 allows to
compute the scale evolution of ¯
g SF over several orders of magnitude. In particular,
each of the horizontal steps in Fig. 5.14 can be repeated for several values of the
lattice spacing, so that the continuum limit is reached by taking a/L → 0 for
fixed physical box size L. The resulting scale evolution of α SF ≡ ¯
g 2
SF /4π is
shown in Fig. 5.15 and compared to the perturbative evolution. Although the nonperturbatively determined points are described very well by perturbation theory,
using the three-loop expression for the RG function, one should realize that this
behaviour may be specific to the SF scheme and should not be generalized to other
schemes.