5 QCD on the Lattice
237
K 0 and ¯
K 0 states. Obviously, such a calculation must be performed in the nonperturbative regime of QCD since it involves typical hadronic scales.
Other CKM matrix elements, such as V us , V ub and V cb are related to weak
processes involving kaons, D- and B-mesons, which are described by a variety
of leptonic decay constants (and their ratios), form factors of semi-leptonic meson
and baryon decays, as well as the B-parameters that encode strong interaction
contributions to B 0 − ¯
B 0 and B 0
s − ¯
B 0
s mixing. All these quantities have been studied
in lattice QCD for many years, and increasingly precise estimates with controlled
systematic errors have appeared over the past decade. They have been instrumental
for recent analyses of the unitarity of the CKM matrix [324–327].
Similar considerations apply to SM parameters such as the strong coupling
constant α s and the masses of the quarks. While the asymptotic scaling behaviour
of α s gives rise to the dimensionful -parameter that encodes the intrinsic scale
of QCD, the quark masses are external parameters. Providing the link between
experimentally accessible quantities and quark masses, as well as expressing the
-parameter in units of some measurable low-energy quantity has been a primary
task for lattice QCD. Lattice calculations have also be instrumental for determining
the coupling constants of effective descriptions of QCD, such as the low-energy
constants of Chiral Perturbation Theory.
The importance of accurate, model-independent determinations of SM parameters and input quantities for flavour physics has led to the foundation of the Flavour
Lattive Averaging Group (FLAG). Updates of the FLAG report have appeared at
regular intervals since the publication of its first edition in 2010 [245]. As part of its
mission, FLAG issues global estimates and averages of lattice results, provided that
they satisfy a set of defined quality criteria. FLAG estimates are quoted separately
according to the sea quark content of the calculations that enter the global analyses,
i.e. whether they have been obtained with a degenerate doublet of u, d quarks
(N f = 2) or with an additional dynamical strange (N f = 2 + 1) and charm quark
(N f = 2+1+1). The current status of lattice QCD calculations of quark masses, the
strong coupling, decay constants, form factors, mixing parameters and low-energy
constants is summarized in Tables 1 and 2 of the 2016 FLAG report [247]. The
FLAG webpage 22 contains additional updates. Below we comment on the current
status of a few selected quantities.
Quark Masses According to FLAG, the strange quark mass is known to 1%
precision, while the accuracy in the determination of the average u and d quark
mass, ˆ
m ≡
1
2 (m u + m d ), varies between 1–5 %, depending on the sea quark
content [328–332, 332–340]. Thanks to the recent progress in including the effects
of isospin breaking in lattice QCD calculations, estimates for the masses of the
individual u and d quarks could also be obtained, typically with 2 − 5 % precision
[261, 262, 264, 330]. Furthermore, the masses of the heavy quarks have been
determined with excellent precision [328, 330–332, 335, 337, 341–348].
22 http://flag.unibe.ch/.
237
K 0 and ¯
K 0 states. Obviously, such a calculation must be performed in the nonperturbative regime of QCD since it involves typical hadronic scales.
Other CKM matrix elements, such as V us , V ub and V cb are related to weak
processes involving kaons, D- and B-mesons, which are described by a variety
of leptonic decay constants (and their ratios), form factors of semi-leptonic meson
and baryon decays, as well as the B-parameters that encode strong interaction
contributions to B 0 − ¯
B 0 and B 0
s − ¯
B 0
s mixing. All these quantities have been studied
in lattice QCD for many years, and increasingly precise estimates with controlled
systematic errors have appeared over the past decade. They have been instrumental
for recent analyses of the unitarity of the CKM matrix [324–327].
Similar considerations apply to SM parameters such as the strong coupling
constant α s and the masses of the quarks. While the asymptotic scaling behaviour
of α s gives rise to the dimensionful -parameter that encodes the intrinsic scale
of QCD, the quark masses are external parameters. Providing the link between
experimentally accessible quantities and quark masses, as well as expressing the
-parameter in units of some measurable low-energy quantity has been a primary
task for lattice QCD. Lattice calculations have also be instrumental for determining
the coupling constants of effective descriptions of QCD, such as the low-energy
constants of Chiral Perturbation Theory.
The importance of accurate, model-independent determinations of SM parameters and input quantities for flavour physics has led to the foundation of the Flavour
Lattive Averaging Group (FLAG). Updates of the FLAG report have appeared at
regular intervals since the publication of its first edition in 2010 [245]. As part of its
mission, FLAG issues global estimates and averages of lattice results, provided that
they satisfy a set of defined quality criteria. FLAG estimates are quoted separately
according to the sea quark content of the calculations that enter the global analyses,
i.e. whether they have been obtained with a degenerate doublet of u, d quarks
(N f = 2) or with an additional dynamical strange (N f = 2 + 1) and charm quark
(N f = 2+1+1). The current status of lattice QCD calculations of quark masses, the
strong coupling, decay constants, form factors, mixing parameters and low-energy
constants is summarized in Tables 1 and 2 of the 2016 FLAG report [247]. The
FLAG webpage 22 contains additional updates. Below we comment on the current
status of a few selected quantities.
Quark Masses According to FLAG, the strange quark mass is known to 1%
precision, while the accuracy in the determination of the average u and d quark
mass, ˆ
m ≡
1
2 (m u + m d ), varies between 1–5 %, depending on the sea quark
content [328–332, 332–340]. Thanks to the recent progress in including the effects
of isospin breaking in lattice QCD calculations, estimates for the masses of the
individual u and d quarks could also be obtained, typically with 2 − 5 % precision
[261, 262, 264, 330]. Furthermore, the masses of the heavy quarks have been
determined with excellent precision [328, 330–332, 335, 337, 341–348].
22 http://flag.unibe.ch/.