216
H. Wittig
Lattice calculations of weak hadronic matrix elements is a major activity within
the lattice community, and a thorough coverage of all aspects would easily fill
an entire chapter. We shall therefore concentrate on some of the most important
quantities, and point out the main conceptual issues. It is strongly recommended that
the reader consult the regular reviews of the topic at the annual lattice conferences,
e.g. [131–135].
5.7.1 Weak Matrix Elements in the Kaon Sector
In the kaon sector, K 0 − ¯
K 0 mixing is one of the most important processes. The
B-parameter B K parameterizes the non-perturbative contribution to indirect CP
violation. It is defined by the ratio of the relevant operator matrix element to its
value in the so-called “vacuum saturation approximation”:
B K (μ) =
¯
K 0
Q (μ)
K 0
8
3 f 2
K m 2
K
.
(5.212)
Here, μ denotes the renormalization scale at which the = 2 four-quark operator
Q , defined by
Q
=
¯
sγ μ (1 − γ 5 )d
¯
sγ μ (1 − γ 5 )d
≡ O VV+AA − O VA+AV ,
(5.213)
is considered. The relation between K and the CKM matrix elements is provided
by the RG-invariant B-parameter ˆ
B K . In NLO perturbation theory ˆ
B K is related to
B K (μ) via
ˆ
B K =
¯
g(μ) 2
4π
γ 0 /2b 0
1 + ¯
g(μ)
2
b 0 γ 1 − b 1 γ 0
2b 2
0
B K (μ),
(5.214)
where γ 0 , γ 1 denote the coefficients in the perturbative expansion of the anomalous
dimension of Q . Since QCD is parity-conserving, the physically relevant
operator in the above expression is the parity-even combination O VV+AA . The
typical left-handed chiral structure of this operator, which is characteristic for
weak transitions, poses a problem for lattice calculations if Wilson fermions are
employed. In this case the discretization breaks chiral symmetry explicitly, and
thus O VV+AA mixes under renormalization with operators involving the opposite
chirality. Therefore, the general renormalization pattern is
O
R
VV+AA (μ) = Z(g 0 , aμ)
O
bare
VV+AA +
4
i=1
i (g 0 )O
bare
i
(5.215)
H. Wittig
Lattice calculations of weak hadronic matrix elements is a major activity within
the lattice community, and a thorough coverage of all aspects would easily fill
an entire chapter. We shall therefore concentrate on some of the most important
quantities, and point out the main conceptual issues. It is strongly recommended that
the reader consult the regular reviews of the topic at the annual lattice conferences,
e.g. [131–135].
5.7.1 Weak Matrix Elements in the Kaon Sector
In the kaon sector, K 0 − ¯
K 0 mixing is one of the most important processes. The
B-parameter B K parameterizes the non-perturbative contribution to indirect CP
violation. It is defined by the ratio of the relevant operator matrix element to its
value in the so-called “vacuum saturation approximation”:
B K (μ) =
¯
K 0
Q (μ)
K 0
8
3 f 2
K m 2
K
.
(5.212)
Here, μ denotes the renormalization scale at which the = 2 four-quark operator
Q , defined by
Q
=
¯
sγ μ (1 − γ 5 )d
¯
sγ μ (1 − γ 5 )d
≡ O VV+AA − O VA+AV ,
(5.213)
is considered. The relation between K and the CKM matrix elements is provided
by the RG-invariant B-parameter ˆ
B K . In NLO perturbation theory ˆ
B K is related to
B K (μ) via
ˆ
B K =
¯
g(μ) 2
4π
γ 0 /2b 0
1 + ¯
g(μ)
2
b 0 γ 1 − b 1 γ 0
2b 2
0
B K (μ),
(5.214)
where γ 0 , γ 1 denote the coefficients in the perturbative expansion of the anomalous
dimension of Q . Since QCD is parity-conserving, the physically relevant
operator in the above expression is the parity-even combination O VV+AA . The
typical left-handed chiral structure of this operator, which is characteristic for
weak transitions, poses a problem for lattice calculations if Wilson fermions are
employed. In this case the discretization breaks chiral symmetry explicitly, and
thus O VV+AA mixes under renormalization with operators involving the opposite
chirality. Therefore, the general renormalization pattern is
O
R
VV+AA (μ) = Z(g 0 , aμ)
O
bare
VV+AA +
4
i=1
i (g 0 )O
bare
i
(5.215)