5 QCD on the Lattice
217
Thus, in order to determine the physical matrix element, one must not only
determine the overall renormalization factor Z, but also the mixing coefficients
i . Several techniques have been developed [136–138] to address this problem,
which is merely an inconvenience rather than a serious obstacle. In a formulation
based on staggered fermions the problem is absent, since the remnant U(1) ⊗ U(1)
symmetry protects the operator from mixing with other chiralities. However, a
drawback of the staggered formulation is the broken flavour (“taste”) symmetry,
which may lead to significant complications [139]. Fermionic discretizations based
on the Ginsparg-Wilson relation, such as domain wall or overlap fermions do not
suffer from the mixing problem, whilst preserving all flavour symmetries. Finally,
the mixing problem can also be circumvented for Wilson-like discretizations in the
context of twisted-mass QCD [140, 141]. With the help of a suitably chosen flavour
rotation (see Eq. (5.51)), the matrix element of O VV+AA in QCD can be mapped
exactly onto that of the parity-odd operator O VA+AV in the chirally twisted theory,
viz.
¯
K
0
O
bare
VA+AV
K
0
tmQCD
= i
¯
K
0
O
bare
VV+AA
K
0
QCD
.
(5.216)
It has been shown that O VA+AV renormalizes purely multiplicatively [142], i.e.
all mixing coefficients vanish. The overall multiplicative, scale-dependent renormalization factor of O VA+AV which yields the physical matrix element has been
determined non-perturbatively [143], using the finite-size scaling procedure based
on the Schrödinger functional formalism described in Sect. 5.5.2.
We now give a summary of the current status of B K . Here, the calculation
by the JLQCD Collaboration [154], based on staggered quarks in the quenched
approximation, has served as a benchmark result for a long time. Their result,
for which the perturbatively renormalized matrix element was extrapolated to the
continuum limit, has since been confirmed by many other calculations employing
different fermionic discretizations and different renormalization techniques. These
include domain wall [148, 149] and overlap quarks [150, 151], as well as the
Wilson formulation [153, 155]. Moreover, a calculation employing twisted mass
QCD has been completed [152], which includes non-perturbative renormalization
and a thorough investigation of the continuum limit.
Recently, results for B K from simulations with dynamical quarks have become
available, both for N f = 2 [146, 147] and N f = 2 + 1 flavours [144, 145]. A
compilation of quenched and unquenched results is shown in Fig. 5.19. Although
the figure suggests a trend in the data which points to slightly lower estimates for
ˆ
B K if dynamical quarks are switched on (see Fig. 5.19), the quoted uncertainties are
still too large to point to a significant deviation. In particular, a systematic study
of the continuum limit in the unquenched case is not yet available. It is interesting
to compare the results for ˆ
B K to the non-lattice determination in Ref. [130]. Here,
the determinations of the angles of the unitarity triangle from experimental data
in conjunction with direct measurements of M d , ,M s and K allow to fit the
values of several of the quantities in Eq. (5.211), which incorporate the hadronic
217
Thus, in order to determine the physical matrix element, one must not only
determine the overall renormalization factor Z, but also the mixing coefficients
i . Several techniques have been developed [136–138] to address this problem,
which is merely an inconvenience rather than a serious obstacle. In a formulation
based on staggered fermions the problem is absent, since the remnant U(1) ⊗ U(1)
symmetry protects the operator from mixing with other chiralities. However, a
drawback of the staggered formulation is the broken flavour (“taste”) symmetry,
which may lead to significant complications [139]. Fermionic discretizations based
on the Ginsparg-Wilson relation, such as domain wall or overlap fermions do not
suffer from the mixing problem, whilst preserving all flavour symmetries. Finally,
the mixing problem can also be circumvented for Wilson-like discretizations in the
context of twisted-mass QCD [140, 141]. With the help of a suitably chosen flavour
rotation (see Eq. (5.51)), the matrix element of O VV+AA in QCD can be mapped
exactly onto that of the parity-odd operator O VA+AV in the chirally twisted theory,
viz.
¯
K
0
O
bare
VA+AV
K
0
tmQCD
= i
¯
K
0
O
bare
VV+AA
K
0
QCD
.
(5.216)
It has been shown that O VA+AV renormalizes purely multiplicatively [142], i.e.
all mixing coefficients vanish. The overall multiplicative, scale-dependent renormalization factor of O VA+AV which yields the physical matrix element has been
determined non-perturbatively [143], using the finite-size scaling procedure based
on the Schrödinger functional formalism described in Sect. 5.5.2.
We now give a summary of the current status of B K . Here, the calculation
by the JLQCD Collaboration [154], based on staggered quarks in the quenched
approximation, has served as a benchmark result for a long time. Their result,
for which the perturbatively renormalized matrix element was extrapolated to the
continuum limit, has since been confirmed by many other calculations employing
different fermionic discretizations and different renormalization techniques. These
include domain wall [148, 149] and overlap quarks [150, 151], as well as the
Wilson formulation [153, 155]. Moreover, a calculation employing twisted mass
QCD has been completed [152], which includes non-perturbative renormalization
and a thorough investigation of the continuum limit.
Recently, results for B K from simulations with dynamical quarks have become
available, both for N f = 2 [146, 147] and N f = 2 + 1 flavours [144, 145]. A
compilation of quenched and unquenched results is shown in Fig. 5.19. Although
the figure suggests a trend in the data which points to slightly lower estimates for
ˆ
B K if dynamical quarks are switched on (see Fig. 5.19), the quoted uncertainties are
still too large to point to a significant deviation. In particular, a systematic study
of the continuum limit in the unquenched case is not yet available. It is interesting
to compare the results for ˆ
B K to the non-lattice determination in Ref. [130]. Here,
the determinations of the angles of the unitarity triangle from experimental data
in conjunction with direct measurements of M d , ,M s and K allow to fit the
values of several of the quantities in Eq. (5.211), which incorporate the hadronic