Improved determination of $B_K$ with staggered quarks

0.36 MB·

by  Taegil Bae

Improved determination of B with staggered quarks
K
Taegil Bae,1 Yong-Chull Jang,1 Hwancheol Jeong,1 Chulwoo Jung,2 Hyung-Jin Kim,2
Jangho Kim,1 Jongjeong Kim,1 Kwangwoo Kim,1 Sunghee Kim,1 Weonjong Lee,1 Jaehoon
Leem,1 Jeonghwan Pak,1 Sungwoo Park,1 Stephen R. Sharpe,3 and Boram Yoon4
(SWME Collaboration)
1Lattice Gauge Theory Research Center, FPRD, and CTP,
Department of Physics and Astronomy, Seoul National University, Seoul, 151-747, South Korea
2Physics Department, Brookhaven National Laboratory, Upton, NY11973, USA
3Physics Department, University of Washington, Seattle, WA 98195-1560, USA
4Los Alamos National Laboratory, Theoretical Division T-2, MS B283, Los Alamos, NM 87545, USA
(Dated: February 24, 2014)
WepresentresultsforthekaonmixingparameterB obtainedusingimprovedstaggeredfermions
K
onamuch enlarged setofMILCasqtad lattices. Compared toourpreviouspublication, which was
4 based largely on a single ensemble at each of the three lattice spacings a ≈ 0.09 fm, 0.06 fm and
1 0.045 fm, we have added seven new fine and four new superfine ensembles, with a range of values
0 of the light and strange sea-quark masses. We have also increased the number of measurements
2
on one of the original ensembles. This allows us to do controlled extrapolations in the light and

b strange sea-quark masses, which we do simultaneously with the continuum extrapolation. This
e reduces the extrapolation error and improves the reliability of our error estimates. Our final result
F is Bˆ =0.7379±0.0047(stat)±0.0365(sys).
K

1
PACSnumbers: 11.15.Ha,12.38.Gc,12.38.Aw
2
Keywords: latticeQCD,BK,CPviolation

]
t
a I. INTRODUCTION quadrupling the number of measurements on the previ-
l
- ouslyusedsuperfineensemble. (SeeTableIIbelow.) The
p
main impact of these improvements is that we can now
e The kaon B-parameter, BK, is one of the important
doacontrolledextrapolationinm andm ,whichwedo
h hadronic inputs into the unitary triangle analysis of fla- (cid:96) s
[ vor physics. Fully controlled, first-principles calculations simultaneously with the continuum extrapolation. This
are only available using lattice QCD, and consistent re- leads to better understood and, in most cases, numeri-
2
cally smaller systematic errors. It also removes the need
v sults utilizing several fermion discretizations have now
tousethecoarseensembles, withtheentireanalysisnow
8 been obtained [1–4]. For a recent review, see Ref. [5].
4 Among these results are those we have previously pre- carriedoutusingresultsfromthethreefinestlatticespac-
0 sented using improved staggered fermions for both va- ings.
0 lence and sea quarks [2, 6] Here we provide a significant Inthisreport, werefertoRefs.[6]and[2]forexplana-
2. update of these results, in which the control of several tions of many technical details. Since those papers were
0 sources of systematic error is markedly improved. published, several updates have appeared in conference
4 proceedings, the most recent being Ref. [7]. This report
Our previous result (Ref. [2]) was based largely on a
1 is based on our final data set, and the result supersedes
single gauge ensemble at each of three lattice spacings
v: (a ≈ 0.045, 0.06 and 0.09 fm—dubbed “ultrafine”, “su- earlier ones.
i perfine” and “fine” hereafter). These ensembles had ap-
X
proximately the same physical values for the light sea-
r
a quark mass (m(cid:96), the degenerate up and down mass) and II. DATA SAMPLE
the strange sea-quark mass (m ), allowing a continuum
s
extrapolation. Extrapolating m and m to their physi- The kaon B parameter B is defined as
(cid:96) s K
cal values was not, however, possible using these ensem-
bles. Instead, we used results from “coarse” ensembles (cid:104)K0|O (µ,R)|K¯0(cid:105)
B (µ,R)= ∆S=2 (1)
(a ≈ 0.12 fm), for which a range of values of m(cid:96) was K 8f2M2/3
K K
available, to argue that the extrapolations in m and m
(cid:96) s
would lead to only a small shift in B . These ensembles where R is the renormalization scheme in which the op-
K
(cid:80)
were too coarse, however, to be included in our contin- erator O = [s¯γ (1−γ )d][s¯γ (1−γ )d] is de-
∆S=2 ν ν 5 ν 5
uum extrapolation. fined, with µ is the corresponding renormalization scale.
The main improvements since Ref. [2] are the inclu- The standard scheme used in phenomenology is the MS
sion of many additional ensembles and an increase in scheme with naive dimensional regularization (NDR) for
the number of measurements on ensembles used previ- γ . We match our lattice-regulated operators to this
5
ously. Specifically, we have added results on seven more scheme, usually called NDR, using one-loop matching
fine and four more superfine ensembles, while more than factorsfromRef.[8]. Weusetheensemblesgeneratedby
2
√ √
TABLEI.MILCasqtadensemblesusedtocalculateBK. am(cid:96) TABLEII.Valuesfora2, LP and SP ontheMILCensem-
and ams are the masses, in lattice units, of the light and bles used for our chiral-continuum extrapolation. a is deter-
strange sea quarks, respectively. “ens” indicates the number mined from the mass-dependent values for r /a obtained by
1
of configurations on which “meas” measurements are made. theMILCcollaboration[9,11],togetherwithr =0.3117fm.
1
Ensembles added since Ref. [2] are denoted new while that √ √
with improved statistics is denoted update. Note that the ID a2(0.01fm2) LP(MeV) SP(MeV)
numbering of the ID tags for fine and superfine lattices does F6 0.673 350 598
not follow the ordering of am(cid:96). F4 0.706 485 765
F3 0.708 422 766
a (fm) am/am geometry ID ens × meas status
l s F1 0.710 346 764
0.12 0.03/0.05 203×64 C1 564×9 old
F5 0.710 294 751
0.12 0.02/0.05 203×64 C2 486×9 old
F2 0.710 243 755
0.12 0.01/0.05 203×64 C3 671×9 old
F7 0.710 248 596
0.12 0.01/0.05 283×64 C3-2 275×8 old
F9 0.710 174 759
0.12 0.007/0.05 203×64 C4 651×10 old
S3 0.348 440 694
0.12 0.005/0.05 243×64 C5 509×9 old
S4 0.348 383 694
0.09 0.0062/0.0186 283×96 F6 950×9 new S1 0.346 314 696
0.09 0.0124/0.031 283×96 F4 1995×9 new S2 0.347 262 695
0.09 0.0093/0.031 283×96 F3 949×9 new S5 0.348 222 691
0.09 0.0062/0.031 283×96 F1 995×9 old U1 0.192 316 703
0.09 0.00465/0.031 323×96 F5 651×9 new
0.09 0.0031/0.031 403×96 F2 959×9 new
0.09 0.0031/0.0186 403×96 F7 701×9 new
0.09 0.00155/0.031 643×96 F9 790×9 new ing rise to four degenerate tastes in the continuum limit.
0.06 0.0072/0.018 483×144 S3 593×9 new Thisunwanteddegeneracyisremovedbythefourth-root
0.06 0.0054/0.018 483×144 S4 582×9 new prescription, and we assume that this leads to the cor-
0.06 0.0036/0.018 483×144 S1 749×9 update rect continuum limit. The effect of this prescription, as
0.06 0.0025/0.018 563×144 S2 799×9 new well as that of using a mixed action, can be incorpo-
0.06 0.0018/0.018 643×144 S5 572×9 new rated into a chiral effective theory describing the stag-
0.045 0.0028/0.014 643×192 U1 747×1 old geredfermionformulation: staggeredchiralperturbation
theory (SChPT) [6, 13–15]. In particular, we use SU(2)
SChPT at next-to-leading order (NLO) to obtain the
functional form needed for extrapolations in light quark
theMILCcollaborationwithN =2+1flavorsofasqtad
f masses and a2.
staggered sea quarks and a Symmanzik-improved gauge
action[9]. Forvalencequarks,weuseHYP-smearedstag-
geredfermions[10]. Theadvantagesofthismixedaction
set-up are explained in Ref. [6]. III. DATA ANALYSIS AND FITTING
We use the MILC asqtad lattices listed in Table I. As
already noted, the eleven extra ensembles compared to We calculate B (1/a,NDR) on each ensemble fol-
K
Ref. [2] allow us to control the continuum and sea-quark lowing the method explained in Ref. [6], using multi-
mass extrapolations with much greater confidence. To ple measurements on each configuration. For the va-
give a sense of the range of these parameters, we present lence d quarks we use the four masses (in lattice units)
in√Table II values for a2, the light sea-quark pion mass amx =amnsom×{0.1,0.2,0.3,0.4},wheremnsomisanomi-
( LP) and the mass of the unphy√sical flavor-non-singlet nalstrangequarkmasswhichliesfairlyclosetothephys-
s¯s state composed of sea quarks ( SP). We will extrap- icalvalue. Forcoarse,fine,superfineandultrafineensem-
olate/interpolate to physical sea-quark masses using LP bleswetakeamnsom =0.05,0.03,0.018and0.014,respec-
and SP, while simultaneously extrapolating a2 to zero. tively. For the valence s quarks we use the three masses
We exclude the coarse ensembles from Table II as they am =amnom×{0.8,0.9,1.0}. With these values we are
y s
are not used in the final extrapolation. extrapolating to the “physical” d¯d mass (158 MeV—the
We see from Table II that both fine and superfine lat- massofaflavor-non-singletd¯dstate)usinglatticevalence
tices have a substantial range of pion masses, in the for- pions with masses in the range ∼ 200−400 MeV. For
mercasereachingdownalmosttothephysicalvalue. The the valence s quark we are extrapolating from the range
√
fine lattices also have two values for S , allowing in- ∼550−620MeVtothe“physical”value686MeV.Thus
P
terpolationtothe“physical”value,0.6858GeV[12]. For bothvalenceextrapolationsarerelativelyshort. Inaddi-
the continuum extrapolation, the relevant quantity for tion,sincem /m ≤1/2andm ∼mphys,theyaredone
x y y s
our action is a2, and we see that this varies by almost a in a regime where SU(2) ChPT should be valid.
factor of four. We perform the chiral and continuum extrapolations
Staggeredfermionsintroduceanunwanted“taste”de- in three stages. First, the valence d quark is extrapo-
gree of freedom, with each lattice staggered flavor giv- lated to mphys, using a functional form based on NLO
d
3
TABLE III. Fit functions and quality. TABLE IV. Parameters of representative fits.
fit type fit function Constraints χ2/d.o.f fit type c c c c
1 2 3 4
B1 1+a2+L +S 2, 3, 4 1.48 B1/N1 0.542(7) 0.7(3) -0.17(1) 0.00(1)
P P
B2 B1+a2L +a2S 2, 3, 4, 5, 6 1.47 B4 0.54(1) 0.3(4) -0.17(2) -0.00(2)
P P
B3 B1+αs2+a2αs+a4 2, 3, 4, 7, 8, 9 1.47 fit type c5 c6 c7 c8 c9
B4 B2+α2+a2α +a4 2, 3,..., 9 1.47 B4 -0.2(8) 0.2(1) 0.1(2) 0.1(2) 0.01(3)
s s
N1 same as B1 none 1.91
0.6
SChPT, although including higher-order analytic terms. 0.58
Details are as in Ref. [2]. This “X-fit” is done separately coarse
on each ensemble, using a correlated fit with Bayesian 0.56
priors. We correct for non-analytic contributions from
taste breaking in valence and sea pions and from the un- K 0.54
B
physicalvalueofm usingtheone-loopchirallogarithms
(cid:96)
(which are predicted without unknown constants). This 0.52
correction depends on the ensemble, and ranges in size fine
from 3−5%. The X-fits on all the new ensembles are 0.5 superfine
ultrafine
very similar to those displayed in Refs. [2, 6, 7].
Thesecondstage,or“Y-fit”,istheextrapolationfrom 0.48
0 0.1 0.2 0.3 0.4 0.5
our three values of m to mphys. The dependence on m
y s y L
is expected to be analytic, and we find that a linear fit P
works well. Examples of such fits are shown in Ref. [6];
those on the new ensembles are very similar. FIG. 1. BK(2GeV,NDR) vs. LP (GeV2), with the B1 fit.
The third and final extrapolation is that in m , m The N1 fit is indistinguishable. The result of the extrapo-
(cid:96) s
and a2. Here we improve on Refs. [2, 6] both by having lation is shown by the (black) triangle. The fit function for
thesuperfineensemblesisplottedusingtheaveragevaluesof
a larger range of lattice parameters and by doing these
a2 and S , while for the fine ensembles the average of the
extrapolations simultaneously. As in these earlier works, P
values on ensembles F1, F2 and F4 is used. Coarse lattice
we find that we cannot obtain good fits if we include
results are shown for comparison; they are not included in
results from the coarse lattices and so exclude them. the fit. Results from ensembles F6 and F7 are not shown (as
Wenowdescribethefitfunctionsandfittingapproach explained in the text), but are included in the fit.
usedinthesimultaneouschiral-continuumextrapolation.
Our first fit function assumes that the lattice operator is
perfectlymatchedtothatinthecontinuum. ThenSU(2) ±2 to be a fairly conservative choice for the constraints.
SChPT at NLO predicts a linear residual dependence on These details, along with the resulting quality of fit, are
m and a2 (after taste-breaking in the chiral logarithms collected in Table III. The parameters of both fits turn
(cid:96)
has been removed by hand). The only constraint on the outtobealmostidenticalandaregiveninTableIV.Both
m dependence is that it must be analytic, but for our fits are reasonable, with the sizes of the constrained co-
s
short interpolation or extrapolation it is likely to be well efficients c −c well within the expected range of ±2.
2 4
described by a linear dependence. The appropriate fit Thus the constraints are not important for this fit.1
function is thus The quality of the resulting fit is illustrated by Fig. 1.
Acomplicationindisplayingthefitisthatwithinthefine
L S
f1(a2,LP,SP)=c1+c2(aΛQ)2+c3Λ2P +c4Λ2P , (2) ensembles there is a range of values of a2 and SP (and
X X similarly for the superfine ensembles) and this feature
cannot be displayed in a two-dimensional plot. The fit
where we are using L and S as stand-ins for m and
P P (cid:96) lines shown are for average values of a2 and S , and,
m , respectively. Taking Λ = 0.3 GeV and Λ = 1.0 P
s Q X even with a perfect fit, would not pass exactly through
GeV (i.e. respectively a typical QCD scale and chiral
the corresponding points. As can be seen from Table II,
expansion scale) we expect c −c = O(1). We stress
2 4 this is a small effect except for ensembles F6 and F7,
that Eq. (2) is only valid for a small range of S around
P whichhavesignificantlysmallervaluesofS . Thusthese
the physical value. In particular, the linear dependence P
two ensembles are not included in the figure (although
on S is not assumed or expected to remain valid down
P they are included in the fit itself). The results on the
to S = 0, and there is no a priori expectation that
P
c ≈ c . The latter relation would only hold were we in
4 3
the regime where SU(3) ChPT at NLO was valid.
Whenusingtheformf1 weeitherapplyBayesiancon- 1 Despitehavingalmostidenticalfitparameters,theχ2/d.o.f.val-
straints, c = 0±2, to c , c and c (fit B1) or leave uesforthetwofitsdifferbecauseinaBayesianfitoneaugments
i 2 3 4
all four coefficients free (fit N1). We consider the range bothχ2 andtheeffectivenumberofdatapoints[16].
4
0.02 c α2term,aswellastermsc (aΛ )2αandc (aΛ )4aris-
7 8 Q 9 Q
ultrafine ingfromhigher-orderdiscretizationerrors. Whenadding
0.015
superfine these terms we find that Bayesian constraints (for which
fine
0.01 weusec −c =0±2)areneededforstablefits. Wehave
7 9
considered both fits in which only these three terms are
0.005
addedtof (fitB3)andinwhichallthetermsdescribed
K 1
B 0 above are included (fit B4). The full list of fits we use is
∆
shown in Table III.
-0.005
Wefindagainthataddinghigher-ordertermsdoesnot
-0.01 improve the quality of the fits (see Table III), and also
has little impact on the resulting fit parameters. For ex-
-0.015
ample,asshowninTableIV,thetermscommonbetween
-0.02 fits B1/N1 and B4 are almost identical, while the extra
0 5 10 15
terms in B4 are all small and consistent with zero.
ens ID
In summary, our data can be described well by the
simpleformf ,butisalsoconsistentwiththeextraterms
FIG. 2. Residuals ∆B for fit B1. The superfine lattices are 1
K aslongastheyhavesmallcoefficients. Inlightofthis,we
ordered by decreasing L (S3, S4, S1, S2 and S5), while the
P have not extended the fits to include the other possible
fine lattices are ordered F6, F4, F3, F1, F5, F2, F7 and F9.
NNLO terms, e.g. that proportional to L2. We also
These are the orders used in Tables I and II. P
concludethatitisreasonabletousefitB1forourcentral
value,whilequotingthemaximumdifferencebetweenthe
coarse ensembles are also shown, although they are not results from B1 and {B2, B3, B4} fits (which turns out
included in the fit. It is clear that the slope versus LP is to be for fit B4) as an extrapolation systematic.
significantly different on the coarse ensembles, and also
that there are large discretization errors. These are the
features that make fits including the coarse ensembles IV. ERROR BUDGET
unstable,ashighordertermsareneededtoincludethem.
The complication of having different values of a2 and We present the error budget in Table V. The largest
SP canbeavoidedbyconsideringtheresiduals∆BK(i)= errorisfromusingone-loopmatching,whichweestimate
BK(i)−f1(i), where i labels the ensembles. These are to be ∆BK/BK = α2, with α evaluated at scale 1/a
shown in Fig. 2. for the ultrafine lattice. This error is unchanged from
As already noted, the values of the parameters c2−c4 Ref. [2]. In principle, one can determine the size of the
lie in the expected range. In particular, if one writes the α2 contribution from the chiral-continuum fit, given a
c1andc2termsintheformc1[1+(aΛ)2]thenwefindΛ≈ sufficientlyextensivesetoflatticeensembles. Indeed,this
350MeV, which is a reasonable scale for a discretization term is included in fit B4 (with coefficient c ). However,
7
error. We also note that the SU(3) symmetry relation itisclearfromtheresultsoftheprevioussectionthatwe
c3 = c4 does not hold. Indeed, we find no significant do not have enough ensembles to pin down c7, especially
dependence on SP in the vicinity of the strange quark given the large number of parameters in the fit. Thus,
mass. although fit B4 finds a small value, c = 0.1(2), we do
7
We now turn to our other fits. SU(2) SChPT is a notthinkthisissufficientlyreliabletotakeatfacevalue,
joint expansion in a2 and LP (ignoring possible factors and prefer the conservative approach of taking c7 = ±1
ofαmultiplyinga2),soatNNLOweexpectatermofthe to estimate the matching error.
formc5(aΛQ)2(LP/Λ2X)withcoefficientc5 ∼O(1). Infit The errors from continuum and sea-quark mass ex-
B2weincludethisterm, aswellasitsSU(3)counterpart trapolations have been significantly reduced compared
c6(aΛQ)2(SP/Λ2X), with coefficients constrained as for to Ref. [2]. Previously, these errors were separate, and
c2 − c4. In this case we find that the constraints are were estimated to be 1.9% from the continuum extrapo-
neededtoobtainsensiblefits. Theresultingfit,however, lation, 1.5% from the am extrapolation, and 1.3% from
(cid:96)
liesveryclosetoB1,and,asshowninTableIII,doesnot the am extrapolation [2]. The combined error was thus
s
have an improved χ2/d.o.f.. This simply reflects the fact 2.8%. The addition of the new ensembles and the use of
that our data has similar slopes versus LP on the fine acombinedextrapolationhasreducedthiserrorto0.9%.
and superfine lattices.
We next consider the impact of operator matching er-
rors on the fit function. Since we use one-loop match-
only those lattice four-fermion operators composed of bilinears
ing,2 these errors are proportional to α2 (with α evalu-
having the same taste as the external pions [6], Thus there are
ated as the scale 1/a). Thus we also consider fits with a matchingcorrectionsproportionaltoα. These,however,appear
onlyatNNLOinSU(2)SChPT[6], andareoftheformαa2 or
αLP. Furthermore,thenumericalcoefficientsofthesetermsare
small[8]. ThuswechoosetotreatthemaseffectivelyofNNNLO,
2 Infact,wedonotdoacompleteone-loopmatching,sincewekeep anddonotincludetheminthefits.
5
The statistical error is essentially unchanged from
TABLE V. Error budget for B using SU(2) SChPT fitting.
Ref. [2]. As in that work, we have not accounted for the K
impact of auto-correlations. We do observe about a 20%
increase in the statistical error due to auto-correlations, cause error(%) memo
as reported in Ref. [17]. However, since there is signif- STATISTICS 0.64 jackknife
icant uncertainty in the size of this increase, and since matching factor 4.4 see text
 
discretization
this effect is much smaller than our current systematic  
am extrap 0.9 diff. of B1 and B4 fits
errors, we have decided to neglect it in this paper. (cid:96)
am extrap 
The error from the X-fits is estimated similarly to the s
X-fits 0.1 see text
approach used in Ref. [2], namely by considering the ef-
Y-fits 2.0 diff. of linear and quad.
fects of doubling the Bayesian priors and of switching to
finite volume 0.4 diff. of V =∞ and FV fit
the eigenvalue-shift method of fitting [18]. We find that r 0.3 r error propagation
1 1
neither of these changes lead to statistically significant f 0.1 132MeV vs. 124.4MeV
π
shifts in BK on any ensemble. If we combine the two TOTAL SYSTEMATIC 4.9
fractional shifts in quadrature we find an ≈ 0.1±0.1%
shift on essentially all fine and superfine ensembles.3 We
thus take this as our estimate of this (very small) effect.
130.41 MeV) for determining the central value of B ,
The error in Y-fits arises from the uncertainty in K
and then repeat the analysis using the decay constant in
the functional form used to extrapolate in the valence
the SU(2) chiral limit, f(0) =124.2MeV [9]. This is the
strange-quark mass. m . We have used linear fits for π
y
sameprocedureasinRefs.[2,6]. Inthiscasetheshiftin
our central value, but cannot rule out a small quadratic
B does vary significantly between ensembles, so we re-
component. Thus we have repeated the entire analysis K
peattheentireanalysisusingbothvaluesoff ,andtake
using quadratic Y-fits, finding a statistically significant π
the difference in final values as our estimate of the error.
2.0% downward shift in the final value of B . This we
K
As the Table shows, the resulting error is very small.
quote as the corresponding systematic error. This error
is much larger than that quoted in Ref. [2], but we think
the present estimate is both more conservative and more
reliable. V. CONCLUSION
Thefinite-volumeerrorisestimatedasinRefs.[2,19].
For our central value we do the X-fits with SChPT Our final results are
expressions including the (analytically known) finite-
volume corrections to the chiral logarithms. We then B (2GeV,NDR) = 0.5388±0.0034±0.0266 (3)
K
repeattheentireanalysisusinginfinite-volumechirallog- Bˆ = 0.7379±0.0047±0.0365 (4)
arithmsinX-fits, andtakethedifferencebetweenthere- K
sulting values of B as an error estimate. The rationale
K where the first errors are statistical and the second sys-
forthischoiceisthatone-loopchirallogarithmstypically tematic. Bˆ = B (RGI) is the renormalization group
provide only a semi-quantitative estimate of the size of K K
invariant value of B . This result supersedes our previ-
finite-volume effects. The resulting error is numerically ous result, Bˆ =0.7K27±0.004±0.038 [2], with which it
small. K
is completely consistent. Although the changes are nu-
The final two errors are those due to uncertainty in
merically small, they are significant. By adding many
the choices of scale and of the appropriate value of the
new ensembles we now can properly extrapolate in sea
pion decay constant to use in the chiral logarithms en-
quark masses (rather than estimate the effect of such an
tering X-fits. We follow the MILC collaboration and set
extrapolation and include it as an error). This is the
the scale using r = 0.3117(22) fm [20]. To obtain our
1 main reason for the small increase in the central value.
centralvalueofB wetaker =0.3117. Wethenrepeat
K 1 It also leads to a significant reduction in the systematic
the analysis on each ensemble using both r = 0.3139
1 errors from the continuum and sea-quark mass extrapo-
and 0.3095. We find that both changes lead, on all fine,
lations. On the other hand, a more careful estimate of
superfine and ultrafine ensembles, to shifts of magnitude
the systematic error in the valence strange-quark mass
≈0.3%inB . Giventhisuniformity,weexpectasimilar
K extrapolation has led to a significant increase in this er-
shift in the final answer and thus take this as our error
ror. All told, the overall error is only slightly reduced,
estimate.
but, more importantly, the methods of estimating errors
For the decay constant we use f =132MeV (a some-
π have been improved.
what outdated approximation to the physical value of
Ourresult(4)isconsistentwiththeworldaveragepre-
sented in Ref. [5], Bˆ = 0.766(10). Our error is larger
K
thanthisaverageprimarilybecauseofouruseofone-loop
3 Thecombinedshiftis0.4±0.5%ontheU1ensemble. Thelarger matchingfactors. Wearepresentlyworkingonobtaining
errorisduetooursmallernumberofmeasurementsonthisen- thematchingfactorsusingnon-perturbativerenormaliza-
semble. tion [21], which should result in substantial reduction of
6
thematchingerror[22,23]. Inaddition,weplantocalcu- 98CH10886. The research of W. Lee is supported by
late the matching factors perturbatively at the two-loop the Creative Research Initiativesprogram (2013-003454)
level using automated perturbation theory. of the NRF grant funded by the Korean government
Oursecond-largesterroristhatfromY-fits. Thiserror (MSIP). W. Lee would like to acknowledge the support
can,however,beessentiallyremovedinastraightforward from KISTI supercomputing center through the strate-
waybyusingvalencestrangequarkstunedtothephysical gic support program for the supercomputing application
value. research [No. KSC-2012-G2-01]. The work of S. Sharpe
issupportedinpartbytheUSDOEgrantno.DE-FG02-
96ER40956. Computationsforthisworkwerecarriedout
ACKNOWLEDGMENTS inpartonQCDOCcomputersoftheUSQCDCollabora-
tion at Brookhaven National Laboratory and in part on
We are grateful to Claude Bernard and the MILC the DAVID GPU clusters at Seoul National University.
collaboration for private communications. C. Jung is The USQCD Collaboration are funded by the Office of
supported by the US DOE under contract DE-AC02- Science of the U.S. Department of Energy.
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