RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-6. Particle Physics

Update on lattice QCD calculation of neutral B meson mixing
in static limit of b quark
T. Ishikawa,∗1 Y. Aoki,∗2,∗1 T. Izubuchi,∗3,∗1 C. Lehner,∗3 and A. Soni,∗3 for RBC Collaboration
The Cabibbo–Kobayashi–Maskawa (CKM) matrix
is a key part of elementary particle physics, and constraints on the elements Vts and Vtd can be obtained
from B 0 − B 0 mixing, where highly nonperturbative
hadronic weak matrix elements play an essential role.
We perform simulations on this subject using static
heavy quark as the treatment of b quark.1)2) The final
values of B meson decay constants and mixing matrix
elements and ξ parameter in the static b quark limit
are summarized as
fB = 218.8(6.5)stat (16.1)sys [MeV],
fBs = 263.5(4.8)stat (18.7)sys [MeV],
fBs /fB = 1.193(20)stat (35)sys ,fB B̂B = 240(15)stat (17)sys [MeV],
fBs B̂Bs = 290(09)stat (20)sys [MeV],ξ = 1.208(41)stat (44)sys ,∗2∗3

fBf Bs240(4)

210(5)270260250240230220(6)

RIKEN Nishina Center
Nagoya University, KMI
Brookhaven National Laboratory






fBs /fBfB



B B f Bs

B BsξFig. 1. Error budget for final quantities. The height of thebars indicates total error, whereas the relative size ofthe colors is determined by squared errors.1)

(1)
(2)
(3)

where statistical (stat) and systematic (sys) errors aregiven. In Fig. 1, the error budget is presented. Amongthe uncertainties presented in Fig. 1, a large portion of
the error consists of statistical error, chiral extrapolation uncertainty, and renormalization uncertainty; it isthus important to reduce them to obtain more reliableresults.
The first step in improving the current results is
reducing the statistical errors using the All-ModeAveraging (AMA) technique.3) This involves locating many source points in the measurement but using an approximation in obtaining quark propagators
to greatly reduce computational cost. We show thepreliminary results of the AMA calculation in Fig. 2and compare them with the current results (withoutAMA). Although the AMA calculation is still ongoing,
the presented error reduction is quite encouraging.
The systematic uncertainties involve chiral extrapolation and renormalization error. For the chiral extrapolation, we are currently focusing our effort on physicalpion simulation, where we use RBC/UKQCD’s 2+1
flavor 483 × 96 domain-wall fermion ensemble. By
this calculation, most of the chiral extrapolation uncertainty is removed. Currently, the renormalization
error is large for non-ratio quantities because only oneloop perturbation is employed. We are, however, investigating a possibility of nonperturbative method forthe matching using RI/MOM scheme and coordinate
space method,4) where removing power divergence is∗1

280

fB [MeV]

230

270

220

260

1.3

fBs [MeV]

fBs /fB

1.2

250
current

AMA

1/2
fB B̂B [MeV]

current

AMA

current
320
310
300
290
280
270

current

AMA

1/2
fBs B̂Bs [MeV] 1.3

AMA

ξ

1.2
current

AMA

current

AMA

Fig. 2. Comparison of chiral and continuum extrapolated
physical quantities between current and AMA results.
The error denotes only statistical one.

essential to the existence of the static heavy quark.
Although the static limit is a good approximation
for b quark, it is known to have O(ΛQCD /mb ) ∼ 10%
uncertainty. To obtain precise results, we need to remove or reduce this uncertainty systematically. Theheavy quark expansion of a heavy-light quantity Φhl ,which has a finite asymptotic limit as mQ → ∞, iswritten as∞  ΛQCD pγp
Φhl (1/mQ ) = Φhl (0) exp, (7)mQ
p=1
where mQ is the heavy quark mass, which is heavier
than the QCD scale ΛQCD . Our project finally includes simulations in light quark mass region (typicallyc quark mass region) for obtaining expansion parameters γp s. Combining the static result with the lighter
quark mass enables us to obtain results with much lessuncertainty, where our static limit results play a crucialrole as a valuable “anchor point”.
References
1) Y. Aoki et al.: arXiv:1406.6192 [hep-lat].
2) T. Ishikawa et al.: PoS LATTICE 2013, 410 (2014).
3) T. Blum et al.: Phys. Rev. D 88, 094503 (2013).
4) K. Cichy et al.: Nucl. Phys. B 865, 268 (2012).

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