B-physics with dynamical domain-wall light quarks and relativistic b-quarks

0.29 MB·English

by  Ruth S. Van de Water

B-physics with dynamical domain-wall light quarks
and relativistic b-quarks
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a Ruth S. Van de Water and Oliver Witzel∗
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BrookhavenNationalLaboratory,DepartmentofPhysics,Upton,NY11973,USA
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2 E-mail: [email protected]

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a WereportonourprogressincalculatingtheB-mesondecayconstantsandB0-B¯0 mixingparam-
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- eters using domain-wall light quarks and relativistic b-quarks. We present our computational
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e method and show some preliminary results obtained on the coarser (a≈0.11fm) 243 lattices.
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[ ThisworkispresentedonbehalfoftheRBCandUKQCDcollaborations.

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TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
B-physicswithdynamicaldomain-walllightquarksandrelativisticb-quarks OliverWitzel
1. Introduction
The study of B-meson physics on the lattice is of special phenomenological interest because
it allows one to obtain constraints on the CKM unitarity triangle. In the standard global unitarity
triangle fit, the apex of the CKM unitarity triangle is constrained using lattice input for neutral
B-meson mixing. Experimentally, B –B¯ mixing is well measured in terms of mass differences
q q
(oscillationfrequencies)∆m ;inthestandardmodelitisparameterizedby[1]
q
G2m2
∆m = F Wη S m f2 B |V∗V |2, (1.1)
q 6π2 B 0 Bq Bq Bq tq tb
wheretheindexqdenoteseitherad-oras-quark,m isthemassoftheB -meson,andV∗ andV
Bq q tq tb
areCKMmatrixelements. TheInami-Limfunction,S [2],andtheQCDcoefficient,η [1],canbe
0 B
computedperturbatively,whereasthenon-perturbativeinputisgivenby f2B : theleptonicdecay
q Bq
constant f and the B-meson bag parameter B . Computing the ratio ∆m /∆m is particularly
Bq Bq s d
advantageous because statistical and systematic uncertainties largely cancel and, moreover, the
ratioofCKMmatrixelementsbecomesaccessible
∆m m |V |2
s = Bs ξ2 ts . (1.2)
∆m m |V |2
d Bd td
Thenon-perturbativeinputisnowsolelycontainedintheSU(3)-breakingratio
(cid:112)
f B
ξ = Bs Bs . (1.3)
(cid:112)
f B
Bd Bd
An alternative way of constraining the CKM triangle has been proposed by Lunghi and Soni
[3] that uses the CKM matrix elementsV orV , thereby avoiding the tension between inclusive
ub cb
and exclusive determinations of both V (3σ) and V (2σ). This alternative method, however,
ub cb
requires precise knowledge of the decay constant f as well as BR(B→τν) and ∆m . Moreover,
B s
Lunghi and Soni point out [4, 5] that already the current data may show signs of physics beyond
thestandardmodelmanifestingthemselvesasdeviationsofexperimentalvaluesforsin(2β)(3.3σ)
and BR(B→τlν) (2.8σ) from the Standard Model. They argue that the most likely sources for
newphysicsareinB mixingandsin(2β).
q
Therefore, it is timely for the lattice community to determine these parameters precisely and
prove by using statistically and systematically independent setups that all sources of errors are
well under control. Currently, there are three determinations for ξ using 2+1-flavor gauge field
configurations: anexploratorystudybytheRBC-UKQCDcollaboration[6]withnon-competitive
errors and two precise determinations by Fermilab-MILC [7, 8] and HPQCD [9], which however
bothrelyonthesamesetofMILCgaugefieldconfigurations. Thelattertwocollaborationsalsouse
the same configurations to obatin the decay constants f and f and find good agreement. Here
Bd Bs
wepresentthefirstresultsofourprojecttodetermineB-mesonparametersusingdomain-walllight
quarksandrelativisticb-quarks.
2. Computationalsetup
Our project is based on using the RBC-UKQCD 2+1-flavor domain wall lattices with the
Iwasaki gauge action for the gluons. Our first results presented in these proceedings are obtained
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B-physicswithdynamicaldomain-walllightquarksandrelativisticb-quarks OliverWitzel
on the coarser a≈0.11fm lattices with a spatial volume of 243 in lattice units [10], whereas the
fullprojectwillalsoutilizethefinera≈0.08fmlattices(323)[11](fordetailsseeTab.1).
approx. #time
L a(fm) m m m (MeV) #configs. sources
l s π
24 ≈0.11 0.005 0.040 331 1640 1
24 ≈0.11 0.010 0.040 419 1420 1
24 ≈0.11 0.020 0.040 558 350 8
32 ≈0.08 0.004 0.030 307 600 1
32 ≈0.08 0.006 0.030 366 900 1
32 ≈0.08 0.008 0.030 418 550 1
Table1.Overviewofthegaugefieldensemblestobeusedforthisproject.
For the heavy quarks we use a relativistic formulation derived from the “Fermilab action”
[12] in which we tune the three relevant parameters of the action non-perturbatively. As demon-
stragedbyChrist,LiandLin,thisrequiresonlyoneadditionalexperimentalinputcomparedtothe
formulationofFermilab[13]. Therelativisticheavyquark(RHQ)actionisgivenby
  (cid:16) (cid:17)2 
 aD2 a (cid:126)D ic 
S=∑Ψ¯ m +γ D − 0 +ζ(cid:126)γ·(cid:126)D− −a∑ Pσ F Ψ , (2.1)
n 0 0 0 2  2  4 µν µν n(cid:48)
n,n(cid:48)  µν 
n,n(cid:48)
wherethecovariantderivativeisdenotedbyD,F isthefieldstrengthtensorandweneedtotune
µν
thethreeparametersm a,c andζ. Exploratorystudiesonhowtotunetheseparametershavebeen
0 P
performed by Li and Peng [14–16]. The idea is to use experimental values for the spin averaged
mesonmass(m=(mBs+3mB∗s)/4)andthehyperfinesplitting(∆m=mB∗s −mBs)asinputstogether
with the constraint from the dispersion relation that the rest mass m equal the kinetic mass m .
1 2
These three quantities are computed for a set of seven trial parameters determined by making an
initialguessform a,c ,andζ andthenvaryingitbyachosenuncertainty±σ :
0 P {m0a,cP,ζ}
             
m a m a−σ m a+σ m a m a m a m a
0 0 m0a 0 m0a 0 0 0 0
 c , c ,  c , c −σ , c +σ ,  c ,  c  (2.2)
 P   P   P   P cP  P cP  P   P 
ζ ζ ζ ζ ζ ζ−σ ζ+σ
ζ ζ
(seeFig.1). Weiterateovertheparameters{m a,c ,ζ}untilwedeterminethevaluesthatrepro-
0 P
ducetheknownexperimentalmeasurementsform,∆ ,m /m :
m 1 2
 RHQ  PDG 
m a m
0
 c  =J−1×∆  −A (2.3)
 P   m 
m
ζ 1
m
2
with
(cid:20) (cid:21)
Y −Y Y −Y Y −Y
J= 3 2, 5 4, 7 6 and A=Y −J×[m a,c ,ζ]t. (2.4)
1 0 P
2σ 2σ 2σ
m0a cP ζ
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B-physicswithdynamicaldomain-walllightquarksandrelativisticb-quarks OliverWitzel
In(2.4)weusethevectorsY asshorthandnotationfor[m,∆ ,m /m ]t obtainedfortheinputpa-
i m 1 2 i
rametersasgiveninEq.(2.2),labeled1-7fromlefttoright. Eq.(2.3)assumesalineardependence
of the meson masses on the parameters of the action; therefore we must be in a linear regime to
reliably extract {m a,c ,ζ}. We stop the iteration once the tuned values lie within our variation
0 P
range.
4.4
4.1 σζ
σ
ζ cP
3.8
σ
m0a
3.5
4.25
7.55 4.22
7.45 7.35 4.19
m0a 7.25 4.16 cP
Figure1.InitialguessfortheRHQparametersandtheiruncertainties.
In order to finally compute B-meson decay constants and mixing parameters as precisely as
possible we repeated the original tuning presented at Lattice 2008 [14] with increased statistics
andoptimizedsmearedwavefunctionparametersusedforgeneratingtheheavyquarkpropagators.
Moreover,performingthetuningonthesameconfigurationsweintendtouseforcomputingweak
matrix elements such as f will allow for an improved error analysis in which the correlations
B
amongthethreeRHQparameterscanbefullytakenintoaccount.
Our method for computing the ∆B = 2 four-quark operators requires the light quarks to be
generated with a point source and sink, but for the heavy quarks we are free to explore different
smearingchoices. Inadditiontopointsourcesandsinks,wetriedGaussiansmearedsources/sinks
and also varied the radius of the Gaussian smearing. As a first guess we chose the radius of the
Gaussian source to be the rms radius of the bb¯- and cc¯ states [17]. Later we extended the radius
andfoundthatr =0.634fmgivesthebestsignalascanbeseeninFig.2.
rms
UsingthissetupweobtaintheRHQparametersonthethree243ensemblesasgiveninTable2.
Incontrasttoearlierwork[14]thedeterminationusesonlyquantitiesfromtheheavy-lightsystem.
We expect these values to be close to our final determination. Moreover, we observe that within
statisticaluncertaintiesthereisnodependenceonthelightseaquarkmass(ml ).
sea
ml m a c ζ
sea 0 P
0.005 8.41(9) 5.7(2) 3.1(2)
0.010 8.4(1) 5.8(2) 3.1(1)
0.020 8.4(1) 5.6(2) 3.1(1)
Table2.PreliminarydeterminationoftheRHQparametersonthethree243ensembleswitha≈0.11fm.
3. Computationofdecayconstants
UsingthesenewlydeterminedRHQparameterswecomputeasafirstnon-trivialtestthedecay
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B-physicswithdynamicaldomain-walllightquarksandrelativisticb-quarks OliverWitzel
3.15
3.12
Sm−Pt: r ≈ 0.855 fm
3.09 rms
Sm−Pt: r ≈ 0.634 fm
rms
effmBl Sm−Pt: rrcmcs ≈ 0.423 fm
Sm−Pt: rbb ≈ 0.224 fm
rms
3.06
Pt−Pt
3.03
3.00
5 10 15 20 25
time slice
Figure2.B-mesoneffectivemassesfordifferentb-quarkspatialwavefunctions;ineachcasethelightquarkhasapoint
l
source.
constantsoftheunitary(B )andstrange(B )mesonsviatherelation
l s
√
f =Z Φ a−3/2/ m , (3.1)
Bq Φ Bq Bq
where Φ is the lattice decay amplitude, Z is the renormalization factor. For the B meson
B Φ s
(domain-wall valence quark has mass of physical s-quark) we generated data for the set of seven
differentRHQinputparameters. Thereforewecanusesimilarequationsto(2.3)and(2.4)inorder
todeterminethedecayamplitudeΦ atthetunedRHQparameters:
Bs
 RHQ
m a
0
ΦRHQ=J(1×3)× c  +A (3.2)
Φ  P  Φ
ζ
with
(cid:20) (cid:21)
Φ −Φ Φ −Φ Φ −Φ
J = 3 2, 5 4, 7 6 and A =Φ −J ×[m a,c ,ζ]t. (3.3)
Φ Φ 1 Φ 0 P
2σ 2σ 2σ
m0a cP ζ
In Fig. 3 we show the results for Φ obtained on the ensemble with ml = 0.005. To test the
Bs sea
linearity with respect to the input parameters {m a,c ,ζ} we used three different sets of seven
0 P
RHQ parameters all centered around the same point. The vertical black line with the gray error
band indicates the tuned value of m a, c or ζ and allows for a simple estimate of the error in
0 P
Φ due to the uncertainty in each of the three parameters. These plots show that the effect of the
Bs
uncertaintyinm aandc isnegligible,whereasζ contributesanerrorofabout1%. Moreover,we
0 P
seethatΦ dependslinearlyonm a,c andζ intherangeofinterest.
Bs 0 P
Alternatively, one may simply use the tuned RHQ values to compute the needed correlation
functions. WefollowedthisproceduretocalculatetheB meson(domain-wallvalencequarkmass
l
equalsthelightseaquarkmass)thedecayamplitudeΦ .
Bl
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B-physicswithdynamicaldomain-walllightquarksandrelativisticb-quarks OliverWitzel
0.090 0.089
0.089
0.088
0.088
±0.1%
Bs Bs0.087
Φ ±0.2% Φ
0.087
0.086
0.086
7.7 7.9 8.1 8.3 8.5 8.7 8.9 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2
ma c
0 P
0.096
0.091
ΦBs ±1.0%
0.086
0.081
2.3 2.6 2.9 3.2 3.5 3.8 4.1
ζ
Figure3. DependenceofΦBs onthethreeRHQparameters{m0a,cP,ζ};resultsareshownforthelightestseaquark
massml =0.005onthe243ensembles.
sea
Atfer renormalizing Φ multiplicatively at 1-loop [18] we obtain the decay constants which
B
are given in Tab. 3 and shown in Fig. 4. As mentioned above, the values for f are obtained
Bl
by simulating directly at the tuned RHQ parameters, whereas the f values are extracted using
Bs
Eq. (3.2). The statistical errors on f are therefore larger because they take into account the
Bs
statistical uncertainties in the three RHQ parameters. For a better comparison, we increase the
errors in f in the Fig. 4 by the error due to the statistical uncertainty in the RHQ parameters
Bl
estimated in Fig. 3. Moreover, we emphasize that this computation is performed without O(a)
improvement.
4. Conclusion
WepresentedourfirstresultscomputingB-mesondecayconstantsusingdomain-walllightand
relativisticb-quarkswithallparametersoftheRHQactiontunednon-perturbatively. Currently,our
results still need to be O(a) improved. For f we expect only a mild chiral extrapolation but of
Bs
course need to perform an extrapolation to the continuum and estimate other systematic errors.
Despitethesecaveatsthesmallstatisticalerrorsandthefactthatourcentralvalueslieinthesame
ballparkasresultsofothercollaborationsindicatesthepromiseofourmethod.
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B-physicswithdynamicaldomain-walllightquarksandrelativisticb-quarks OliverWitzel
ml f (MeV) f (Mev)
sea Bl Bs
215
0.005 188(2) 215(3)
0.010 194(2) 214(4) V) 205
e
M
0.020 — 221(2) f (B195
Table 3. Preliminary results for the decay constants 185 fBs
using1-loopmultiplicativerenormalization,butwith- fBd
175
outO(a)improvementoftheaxial-currentoperator. 0 0.005 0.m01l + m0.015 0.02 0.025
sea res
Figure 4. Preliminary decay constants fBl and fBs
computedonourthree243ensembles.
Acknowledgments
WearethankfultoallthemembersoftheRBCandUKQCDcollaborations. Numericalcom-
putations for this work utilized USQCD resources and were performed on the kaon and jpsi clus-
ters at FNAL, in part funded by the Office of Science of the U.S. Department of Energy. This
manuscript has been authored by an employee of Brookhaven Science Associates, LLC under
ContractNo.DE-AC02-98CH10886withtheU.S.DepartmentofEnergy.
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