5 QCD on the Lattice
227
N f = 0
N f = 0
300
250
200
150
f B s
]
V
e
M
[
N f = 2
N f = 2+1
t
x
e
,
1
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Fig. 5.20 Recent lattice results for f Bs . From left to right the results are taken from the following
papers. N f = 0 : [194–207]; N f = 2 : [203, 204, 207, 208], [209]; N f = 2 + 1 : [192, 210].
The labels indicate the method used to treat the b-quark in the simulation (“ext” and “int” stand
for extrapolations and interpolations to the mass of the b-quark, respectively. The horizontal lines
represent the (non-lattice) result from [130], with the quoted uncertainty
and with N f = 2 flavour of dynamical Wilson quarks, the CP-PACS Collaboration
find [203]
f
N f =2
B s
f
N f =0
B s
= 1.14 ± 0.05.
(5.238)
The “non-lattice” determination of f B s via fits using the experimental results for the
angles of the unitarity triangle as input [130] also point to a larger value compared
to the quenched theory, as can be seen from the horizontal band in the compilation
in Fig. 5.20.
In current unquenched simulations, systematic effects such as lattice artefacts
and the renormalization of local operators are not yet controlled at a similar
level compared to the quenched approximation. Thus, despite the fact that these
calculations are much more “realistic” in that they include sea quarks, the quoted
overall uncertainties are still relatively large.
B-Parameters ˆ
B B d and ˆ
B B s Following the recent experimental determination
of the mass difference s at the Tevatron [211, 212], lattice determinations of
the B-parameters ˆ
B B d and ˆ
B B s have received much attention. Although the first
calculations date back to the 1990s, relatively few results are available, due to
several specific technical difficulties. First, the complicated renormalization and
mixing patterns of four-quark operators which afflict lattice calculations of the kaon
B-parameter ˆ
B K are also encountered in the b-quark sector. Second, there is the
227
N f = 0
N f = 0
300
250
200
150
f B s
]
V
e
M
[
N f = 2
N f = 2+1
t
x
e
,
1
0
0
2
E
P
A t
x
e
,
1
0
0
2
2
-
D
C
Q
K
U t
x
e
,
1
0
0
2
1
-
D
C
Q
K
U S
S
F
,
3
0
0
2
I
I
-
e
m
o
R T
E
Q
H
+
t
n
i
,
7
0
0
2
1
-
A
H
P
L
A T
E
Q
H
+
S
S
F
,
7
0
0
2
2
-
A
H
P
L
A L
A
N
F
, 8
9
9
1
C
L
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A
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8
9
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A
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F L
A
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8
9
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1
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8
9
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9
9
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,
1
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2
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C
Q
R
N
,
3
0 0
2
D C
Q L
J
D
C
Q
R
N
,
3
0
0
2
D
C
Q
P
H D
C
Q R
N
,
5 0 0
2
D C
Q P
H
Fig. 5.20 Recent lattice results for f Bs . From left to right the results are taken from the following
papers. N f = 0 : [194–207]; N f = 2 : [203, 204, 207, 208], [209]; N f = 2 + 1 : [192, 210].
The labels indicate the method used to treat the b-quark in the simulation (“ext” and “int” stand
for extrapolations and interpolations to the mass of the b-quark, respectively. The horizontal lines
represent the (non-lattice) result from [130], with the quoted uncertainty
and with N f = 2 flavour of dynamical Wilson quarks, the CP-PACS Collaboration
find [203]
f
N f =2
B s
f
N f =0
B s
= 1.14 ± 0.05.
(5.238)
The “non-lattice” determination of f B s via fits using the experimental results for the
angles of the unitarity triangle as input [130] also point to a larger value compared
to the quenched theory, as can be seen from the horizontal band in the compilation
in Fig. 5.20.
In current unquenched simulations, systematic effects such as lattice artefacts
and the renormalization of local operators are not yet controlled at a similar
level compared to the quenched approximation. Thus, despite the fact that these
calculations are much more “realistic” in that they include sea quarks, the quoted
overall uncertainties are still relatively large.
B-Parameters ˆ
B B d and ˆ
B B s Following the recent experimental determination
of the mass difference s at the Tevatron [211, 212], lattice determinations of
the B-parameters ˆ
B B d and ˆ
B B s have received much attention. Although the first
calculations date back to the 1990s, relatively few results are available, due to
several specific technical difficulties. First, the complicated renormalization and
mixing patterns of four-quark operators which afflict lattice calculations of the kaon
B-parameter ˆ
B K are also encountered in the b-quark sector. Second, there is the