5 QCD on the Lattice
197
in terms of α V [87]:
α
(3)
MS
(e
−5/6 q) = α
(3)
V (q) +
2
π
α
(3)
V (q)
2 − (0.3111 . . .)
α
(3)
V (q)
3
,
(5.154)
which yields α
(3)
MS
(3.26 GeV). This coupling is then translated to α
(5)
MS
(M Z ) via the
numerical integration of the four-loop RG β-function, including the effects from
quark mass thresholds at m c and m b , which finally yields
α
(5)
MS
(M Z ) = 0.1170 ± 0.0012.
(5.155)
This result is included in the world average of α
(5)
MS
(M Z ) = 0.1176 ± 0.002 in
Ref. [61]. It is also in very good agreement with the non-lattice global estimate of
α
(5)
MS
(M Z ) = 0.1182 ± 0.0027 [88].
The running and matching in this approach is done perturbatively, involving
energy scales from M Z down to m c . In this sense the method may be regarded
as similar in spirit to, say, the determination of α s from the semi-leptonic branching
ratio of τ decays, as in both cases the coupling is extracted from the perturbative
expansion of a particular observable. While for τ -lepton decays an experimentally
measured quantity is considered, it is the non-perturbatively computed data for the
Wilson loops in the lattice approach which are expressed in terms of the running
coupling. This contrasts with the Schrödinger functional approach, where also the
running is computed non-perturbatively, albeit with considerable numerical effort.
The error on the result in Eq. (5.155) is rather small. It is left for future studies
to confirm this level of precision, which must entail further investigations into the
influence of lattice artefacts, as well as the validity of the fourth root trick.
5.5.6 Light Quark Masses
We shall now apply the general framework of non-perturbative renormalization
to the determination of quark masses. Typically one distinguishes the “light”
u, d, s quarks from the “heavy” c, b, t quarks. At first, this distinction may seem
rather arbitrary. It is actually based on the relative magnitude of the quark masses
compared with the chiral symmetry breaking scale χ , which separates “soft”
from “hard” momentum scales. Masses and momenta well below χ break chiral
symmetry only softly, so that spontaneous chiral symmetry breaking still dominates
over the explicit breaking generated by non-zero values of the quark masses. Gasser
and Leutwyler [89, 90] have demonstrated that QCD with u, d, s flavours can be
studied via an “effective” theory of Goldstone boson fields. This approach, called
Chiral Perturbation Theory (ChPT), has an SU(3) L ⊗ SU (3) R chiral symmetry,
which is spontaneously broken to the SU(3) vector subgroup. The associated
Goldstone bosons are then identified with the pions, kaons and η-mesons, whose
197
in terms of α V [87]:
α
(3)
MS
(e
−5/6 q) = α
(3)
V (q) +
2
π
α
(3)
V (q)
2 − (0.3111 . . .)
α
(3)
V (q)
3
,
(5.154)
which yields α
(3)
MS
(3.26 GeV). This coupling is then translated to α
(5)
MS
(M Z ) via the
numerical integration of the four-loop RG β-function, including the effects from
quark mass thresholds at m c and m b , which finally yields
α
(5)
MS
(M Z ) = 0.1170 ± 0.0012.
(5.155)
This result is included in the world average of α
(5)
MS
(M Z ) = 0.1176 ± 0.002 in
Ref. [61]. It is also in very good agreement with the non-lattice global estimate of
α
(5)
MS
(M Z ) = 0.1182 ± 0.0027 [88].
The running and matching in this approach is done perturbatively, involving
energy scales from M Z down to m c . In this sense the method may be regarded
as similar in spirit to, say, the determination of α s from the semi-leptonic branching
ratio of τ decays, as in both cases the coupling is extracted from the perturbative
expansion of a particular observable. While for τ -lepton decays an experimentally
measured quantity is considered, it is the non-perturbatively computed data for the
Wilson loops in the lattice approach which are expressed in terms of the running
coupling. This contrasts with the Schrödinger functional approach, where also the
running is computed non-perturbatively, albeit with considerable numerical effort.
The error on the result in Eq. (5.155) is rather small. It is left for future studies
to confirm this level of precision, which must entail further investigations into the
influence of lattice artefacts, as well as the validity of the fourth root trick.
5.5.6 Light Quark Masses
We shall now apply the general framework of non-perturbative renormalization
to the determination of quark masses. Typically one distinguishes the “light”
u, d, s quarks from the “heavy” c, b, t quarks. At first, this distinction may seem
rather arbitrary. It is actually based on the relative magnitude of the quark masses
compared with the chiral symmetry breaking scale χ , which separates “soft”
from “hard” momentum scales. Masses and momenta well below χ break chiral
symmetry only softly, so that spontaneous chiral symmetry breaking still dominates
over the explicit breaking generated by non-zero values of the quark masses. Gasser
and Leutwyler [89, 90] have demonstrated that QCD with u, d, s flavours can be
studied via an “effective” theory of Goldstone boson fields. This approach, called
Chiral Perturbation Theory (ChPT), has an SU(3) L ⊗ SU (3) R chiral symmetry,
which is spontaneously broken to the SU(3) vector subgroup. The associated
Goldstone bosons are then identified with the pions, kaons and η-mesons, whose