226
H. Wittig
larger than the uncertainty quoted above. Further discussions and compilations of
lattice data for f B s /f B can be found in [133, 193].
Estimates for absolute values of heavy-light decay constants are also highly
desirable, especially since f B is hard to determine experimentally, even at the Bfactories, since the B → τ ν τ decay rate is suppressed. For f B s the suppression
is even stronger, and thus the prospects for an experimental determination of this
quantity are extremely uncertain. The main issues facing lattice calculations are the
influence of lattice artefacts in conjunction with the renormalization of the axial
current, and the dependence of results on the number of dynamical quark flavours.
As an example for one of the most advanced quenched calculations for f B s
we shall briefly discuss the result by the ALPHA collaboration [198], which also
illustrates the interplay between various methods to treat the b-quark. In Ref. [198]
the results obtained in the static approximation were combined with data computed
around the mass of the charm quark. Provided that estimates for the decay constants
in both datasets have been extrapolated to the continuum limit, a subsequent
interpolation in the heavy quark mass yields the desired result for f B s . The ansatz
for the interpolation is based on the expression
f PS
√
m PS = C PS (M// MS ) γ
1 +
δ
m PS
,
(5.235)
where f PS is a generic heavy-light decay constant, γ , δ are real constants, and the
factor C PS arises from the matching between the static approximation and QCD
with fully relativistic quarks. Thus, using the static approximation as the limiting
case removes the systematic error due to the uncontrolled extrapolation to the mass
of the b-quark. The resulting estimate for f B s is [198]
f B s = 193 ± 6 MeV,
N f = 0.
(5.236)
Non-perturbative renormalization has been employed in both the static approximation and the relativistic formulation. Except for the unknown systematic error due
to quenching, the quoted error contains all uncertainties. The above result has been
confirmed by the approach based on the finite-size scaling method [199].
Turning now to unquenched simulations, we compare the above value to the
result by the HPQCD Collaboration [192], which was obtained using NRQCD for
the b-quark, while N f = 2 + 1 rooted staggered quarks were used as sea quarks.
Here, the estimate for f B s results from a combination of the value for f B and the
ratio f B s /f B already quoted in Eq. (5.234). In this way one obtains
f B s = 259 ± 32 MeV.
(5.237)
Thus, in spite of the large error, it appears that the inclusion of dynamical quark
effects increases the estimate for heavy-light decay constants. This is also supported
by other simulations. For instance, using their simulation results in quenched QCD
H. Wittig
larger than the uncertainty quoted above. Further discussions and compilations of
lattice data for f B s /f B can be found in [133, 193].
Estimates for absolute values of heavy-light decay constants are also highly
desirable, especially since f B is hard to determine experimentally, even at the Bfactories, since the B → τ ν τ decay rate is suppressed. For f B s the suppression
is even stronger, and thus the prospects for an experimental determination of this
quantity are extremely uncertain. The main issues facing lattice calculations are the
influence of lattice artefacts in conjunction with the renormalization of the axial
current, and the dependence of results on the number of dynamical quark flavours.
As an example for one of the most advanced quenched calculations for f B s
we shall briefly discuss the result by the ALPHA collaboration [198], which also
illustrates the interplay between various methods to treat the b-quark. In Ref. [198]
the results obtained in the static approximation were combined with data computed
around the mass of the charm quark. Provided that estimates for the decay constants
in both datasets have been extrapolated to the continuum limit, a subsequent
interpolation in the heavy quark mass yields the desired result for f B s . The ansatz
for the interpolation is based on the expression
f PS
√
m PS = C PS (M// MS ) γ
1 +
δ
m PS
,
(5.235)
where f PS is a generic heavy-light decay constant, γ , δ are real constants, and the
factor C PS arises from the matching between the static approximation and QCD
with fully relativistic quarks. Thus, using the static approximation as the limiting
case removes the systematic error due to the uncontrolled extrapolation to the mass
of the b-quark. The resulting estimate for f B s is [198]
f B s = 193 ± 6 MeV,
N f = 0.
(5.236)
Non-perturbative renormalization has been employed in both the static approximation and the relativistic formulation. Except for the unknown systematic error due
to quenching, the quoted error contains all uncertainties. The above result has been
confirmed by the approach based on the finite-size scaling method [199].
Turning now to unquenched simulations, we compare the above value to the
result by the HPQCD Collaboration [192], which was obtained using NRQCD for
the b-quark, while N f = 2 + 1 rooted staggered quarks were used as sea quarks.
Here, the estimate for f B s results from a combination of the value for f B and the
ratio f B s /f B already quoted in Eq. (5.234). In this way one obtains
f B s = 259 ± 32 MeV.
(5.237)
Thus, in spite of the large error, it appears that the inclusion of dynamical quark
effects increases the estimate for heavy-light decay constants. This is also supported
by other simulations. For instance, using their simulation results in quenched QCD