Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Improved estimate of neutral B meson mixing
in static limit of b quark with AMA technique
T. Ishikawa,∗1 Y. Aoki,∗2,∗1 T. Izubuchi,∗3,∗1 C. Lehner∗3 and A. Soni∗3 (for RBC Collaboration)

∗1
∗2
∗3

before AMA

290

χ /d.o.f. = 1.5

250

fB [MeV]fB [MeV]

260

250
240
24c, HYP124c, HYP232c, HYP132c, HYP2
cont, phys

220
210
200
190

0

χ /d.o.f. = 2.6p-val = 0.02

270

260

230

2280

p-val = 0.17

270

AMA

290

228024023024c, HYP124c, HYP232c, HYP132c, HYP2cont, phys

220
2102001901020ml+mres [MeV]

01020ml+mres [MeV]

before AMA
χ /d.o.f. = 0.3

5p-val = 0.92

444AMA

23210

MB [(GeV) ]5MB [(GeV) ]The Cabibbo–Kobayashi–Maskawa (CKM) matrix
plays a key role in elementary particle physics, andconstraints on the elements Vts and Vtd can be obtained from B 0 − B 0 mixing. Treatments of b quark on
the lattice QCD are, however, challenging because ofa multi-scale problem where a large hierarchy in mass
exists between light quarks (u and d) and b quark. Asolution to this problem is using Heavy Quark Eﬀective
Theory (HQET), in which the theory is expanded by
1/mb , where mb is b quark mass. The leading order ofthe HQET is a static approximation of the b quark andis the formulation used in this work. While the static
approximation is known to have O(ΛQCD /mb ) ∼ 10%
uncertainty, it is useful for an interpolation strategy,
where the physical b quark mass point (∼ 4.2 GeV) isreached by the interpolation between the static limit
and simulations in the lower quark mass region (cquark mass region). For this purpose, high precisioncalculations in the static limit are required.
RBC/UKQCD Collaboration has worked on suchcalculations for several years 1)2)3) . While we havesuccessfully improved the estimate, the statistical error remains a major part of the total uncertainty. Recently, an eﬃcient method called All-Mode-Averaging(AMA)4) has been proposed for signiﬁcantly reducing
the statistical error. The AMA technique is an operator improvement using symmetries on the action andapproximations. The most useful symmetry is translational invariance of space-time. The idea involves
locating many source points in the measurement, butusing an approximation in obtaining quark propagators to reduce computational cost. As an approximation, we use a sloppy CG, where its stopping condition
is relaxed. A crucial point here is the existence ofcorrelation between the original and the approximatedoperators; thus, we choose the stopping condition sothat the good correlation is kept, but the computational cost is reduced signiﬁcantly. We also note thata low-mode deﬂation technique helps the cost reduction in addition to enhancement of the correlation inthe AMA procedure.
In Fig. 1, we show chiral and continuum extrapolation of the B meson decay constant fB and the neutral
B meson mixing matrix element MB without and withAMA. We use gluon ensembles with two lattice spacings, which are depicted as “24c” (coarse) and “32c”(ﬁne). “HYP1” and “HYP2” represent link smearingsin the static b action, whose results should coincide24c, HYP124c, HYP232c, HYP132c, HYP2cont, phys

1020ml+mres [MeV]2χ /d.o.f. = 1.0p-val = 0.414321024c, HYP124c, HYP232c, HYP132c, HYP2cont, phys

1020ml+mres [MeV]Fig. 1. Chiral and continuum extrapolation of fB and MB .

each other in the continuum limit. To obtain the AMAresults approximately 30% − 40% cost increase is required from the original one (“before AMA” in theﬁgures). A remarkable improvement in statistics is obtained compared with the small increase in the cost.
At the physical point of light quark mass and in thecontinuum limit, the central values only slightly moveby using the AMA; the statistical errors are, however,signiﬁcantly reduced. In addition, the improvement
makes a qualitative fact clear: the results with HYP2
smearing have very small scaling violation. We conclude that the AMA technique has potential to enable
us to reach statistically decisive results.
References
1) V. Gadiyak and O. Loktik, Phys. Rev. D 72, 114504(2005).
2) C. Albertus et al., Phys. Rev. D 82, 014505 (2010).
3) T. Ishikawa et al., LATTICE2013:410, 2013.
4) T. Blum et al., Phys. Rev. D 88, 094503 (2013).

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Nagoya University, KMI
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