238
H. Wittig
Running Coupling A milestone was achieved by the ALPHA collaboration, who
published [349] an estimate for α s (M 2
Z ) obtained by tracing the scale evolution
of the strong coupling non-perturbatively over several orders of magnitude into an
energy range where the application of perturbation theory can be considered safe (at
least as far as the quoted precision is concerned). Their main result is the determination of the -parameter in three-flavour QCD, i.e.
(3)
MS
= 341(12) MeV, which can
be matched to the -parameter in the five-flavour theory using perturbation theory,
giving
(5)
MS
= 215(10)(3) MeV. Finally, this is translated into the result for the
strong coupling [349]:
α
MS
s (M
2
Z ) = 0.11852(84).
(5.253)
The quoted error is 30% smaller than that of the 2016 PDG estimate of α s =
0.1181(11) [244]. The latter includes lattice results from Refs. [331, 335, 350–354].
Kaon Weak Matrix Elements The kaon B-parameter B K is now known with
an overall accuracy of 1.3% [336, 355–359]. Moreover, the calculations of matrix
elements relevant for K 0 − ¯
K 0 mixing have been extended to include operators that
arise in extensions of the Standard Model [355, 358–363].
Lattice QCD results for kaon leptonic decay constants (more precisely: the
ratio f K + /f π + ) and the form factor f + (0) describing semi-leptonic K → ππν
decays have now reached a level of precision that enables a competitive and modelindependent determination of V us (see Sect. 5.7.1 of the original review article).
Moreover, it is possible to test the unitarity of the first row in the CKM matrix, i.e.
the relation
|V ud |
2
+ |V us |
2
+ |V ub |
2
= 1,
(5.254)
by combining experimental information with lattice results for f + (0) and f K + /f π + .
Neglecting the contribution from |V ub | 2 ≈ 1.7 · 10 −5 , one finds that |V ud | 2 + |V us | 2
can be determined with a total precision at the percent level, by combining the
FLAG estimates 23 for f + (0) and f K + /f π + with the experimentally accessible
combinations |V us |f + (0) = 0.2165(4) and |V us /V ud |f K + /f π + = 0.2760(4)
[244, 364]. In QCD with dynamical light, strange and charm quarks (N f = 2+1+1)
the result is |V ud | 2 + |V us | 2 = 0.9797(74), which signals a slight tension of
2.7 standard deviations with the Standard Model. The precision of the unitarity test
can be sharpened considerably by replacing |V ud | with the value extracted from
neutron β-decay, i.e. |V ud | = 0.97417(21) [365]. It is then sufficient to provide
one additional constraint from lattice QCD, either in the form of f + (0) or the
ratio f K + /f π + . Inserting the lattice result for f + (0) yields |V ud | 2 + |V us | 2 =
0.99884(53), which again differs from unitarity by about 2σ . Using instead the
23 See the web update at http://flag.unibe.ch/.
H. Wittig
Running Coupling A milestone was achieved by the ALPHA collaboration, who
published [349] an estimate for α s (M 2
Z ) obtained by tracing the scale evolution
of the strong coupling non-perturbatively over several orders of magnitude into an
energy range where the application of perturbation theory can be considered safe (at
least as far as the quoted precision is concerned). Their main result is the determination of the -parameter in three-flavour QCD, i.e.
(3)
MS
= 341(12) MeV, which can
be matched to the -parameter in the five-flavour theory using perturbation theory,
giving
(5)
MS
= 215(10)(3) MeV. Finally, this is translated into the result for the
strong coupling [349]:
α
MS
s (M
2
Z ) = 0.11852(84).
(5.253)
The quoted error is 30% smaller than that of the 2016 PDG estimate of α s =
0.1181(11) [244]. The latter includes lattice results from Refs. [331, 335, 350–354].
Kaon Weak Matrix Elements The kaon B-parameter B K is now known with
an overall accuracy of 1.3% [336, 355–359]. Moreover, the calculations of matrix
elements relevant for K 0 − ¯
K 0 mixing have been extended to include operators that
arise in extensions of the Standard Model [355, 358–363].
Lattice QCD results for kaon leptonic decay constants (more precisely: the
ratio f K + /f π + ) and the form factor f + (0) describing semi-leptonic K → ππν
decays have now reached a level of precision that enables a competitive and modelindependent determination of V us (see Sect. 5.7.1 of the original review article).
Moreover, it is possible to test the unitarity of the first row in the CKM matrix, i.e.
the relation
|V ud |
2
+ |V us |
2
+ |V ub |
2
= 1,
(5.254)
by combining experimental information with lattice results for f + (0) and f K + /f π + .
Neglecting the contribution from |V ub | 2 ≈ 1.7 · 10 −5 , one finds that |V ud | 2 + |V us | 2
can be determined with a total precision at the percent level, by combining the
FLAG estimates 23 for f + (0) and f K + /f π + with the experimentally accessible
combinations |V us |f + (0) = 0.2165(4) and |V us /V ud |f K + /f π + = 0.2760(4)
[244, 364]. In QCD with dynamical light, strange and charm quarks (N f = 2+1+1)
the result is |V ud | 2 + |V us | 2 = 0.9797(74), which signals a slight tension of
2.7 standard deviations with the Standard Model. The precision of the unitarity test
can be sharpened considerably by replacing |V ud | with the value extracted from
neutron β-decay, i.e. |V ud | = 0.97417(21) [365]. It is then sufficient to provide
one additional constraint from lattice QCD, either in the form of f + (0) or the
ratio f K + /f π + . Inserting the lattice result for f + (0) yields |V ud | 2 + |V us | 2 =
0.99884(53), which again differs from unitarity by about 2σ . Using instead the
23 See the web update at http://flag.unibe.ch/.