Table Of ContentCharmonium, D and D from overlap fermion on
s ∗s
domain wall fermion configurations
∗
4
1
0
2
Y.B.Yang†,1,2 Y.Chen,1 A.Alexandru,3 S.J.Dong,2 T.Draper,2 M. Gong,1,2 F.X.Lee,3
n
a A. Li,4 K.F.Liu,2 Z.Liu,1 M.Lujan,3 andN.Mathur5
J
7 (c QCDCollaboration)
]
t
a 1 InstituteofHighEnergyPhysicsandTheoreticalPhysicsCenterforScienceFacilities,
l
- ChineseAcademyofSciences,Beijing100049,China
p
E-mail: [email protected],[email protected],[email protected],
e
h 2 DepartmentofPhysicsandAstronomy,UniversityofKentucky,Lexington,KY40506,USA
[
E-mail: [email protected]
1 3 DepartmentofPhysics,GeorgeWashingtonUniversity,Washington,DC20052,USA
v 4 InstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195,USA
7
5 DepartmentofTheoreticalPhysics,TataInstituteofFundamentalResearch,Mumbai40005,
8
4 India
1
.
1 Wetakeanewapproachtodeterminethescaleparameterr ,thephysicalmassesofstrangeand
0
0
charmquarksthroughaglobalfitwhichincorporatescontinuumextrapolation,chiralextrapola-
4
1 tionandquarkmassinterpolationtothelatticedata. Thecharmoniumandcharm-strangemeson
:
v spectrumarecalculatedwithoverlapvalencequarkson2+1-flavordomain-wallfermiongauge
i
X configurationsgeneratedbytheRBCandUKQCDCollaboration. We usethemassesofDs, D∗s
r andJ/y asinputsandobtainmMS(2GeV)=1.110(24)GeV,mMS(2GeV)=0.104(9)GeVand
a c s
r =0.458(11)fm. Subsequently,thehyperfine-splittingofcharmoniumand f arepredictedto
0 Ds
be112(5)MeVand254(5)MeV,respectively.
31stInternationalSymposiumonLatticeFieldTheory-LATTICE2013
July29-August3,2013
Mainz,Germany
ThisworkissupportedinpartbytheNationalScienceFoundationofChina(NSFC)underGrantsNo.10835002,
∗
No.11075167,No.11105153andalsobytheU.S.DOEGrantsNo.DE-FG05-84ER40154.Y.C.andZ.L.alsoacknowl-
edgethesupportofNSFCandDFG(CRC110).
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
1. Introduction
In lattice QCD simulations, quark masses and the strong coupling constant are bare param-
eters in the QCD action. The coupling constant is related closely to the dimensional quantity on
the lattice, the lattice spacing a, which should be determined first through a proper scheme to set
the scale. After that, the physical point is approached through extrapolation or interpolation of
the bare quark masses by requiring that the physical values of hadron masses and other physical
quantities are reproduced. In this proceeding, we implement a global fit scheme combining the
continuum limit extrapolation, the chiral extrapolation and the quark mass interpolation to set the
scale parameter r and determine quark masses (at a specific energy scale m ) simultaneously. We
0
adopt the overlap fermion action for valence quarks and carry out the practical calculation on the
2+1-flavordomainwallfermiongaugeconfigurations generatedbytheRBC/UKQCDCollabora-
tion[1,2]. Ithasbeenverifiedthatthecharmquarkregioncanbereachedbytheoverlapfermions
onthesixensembles oftheRBC/UKQCDconfigurations attwolattice spacings a 1.7GeVand
∼
a 2.3 GeV [3], such that by applying the multi-mass algorithm in the calculation of overlap
∼
fermion propagators, we can calculate the hadron spectrum and other physical quantities at quite
a lot of quark masses ranging from the chiral region (u,d quark mass region) to the charm quark
region. This permits us to perform a very precise interpolation to the physical strange and charm
quark masses. In order to compare with experimental values, the lattice values of dimensionful
physical quantities should be converted to the values in physical units through a scale parameter,
forwhichwechoosetheSommer’sparameterr [4]andexpressitintheunitsoflatticespacing at
0
each ensemble through the calculation of static potential. Assuch, all the dimensionful quantities
calculated onthelattice(andtheirexperimentalvalues)canbeexpressed inunitofr . Inorderfor
0
the physical quantities calculated at different bare quark masses and different lattice spacings to
be fitted together, we calculate the quark mass renormalization constant to convert the bare quark
masses to the renormalized quark masses at a fixed energy scale in the MS scheme . With these
prescriptions, the physical quark masses and the r can be determined by some physical inputs,
0
through whichwecanpredictotherquantities atthephysicalpointinthecontinuum limit.
2. Quarkmass renormalizationandthe scalesetting
Ourcalculation arecarried out onthe2+1 flavor domain wallfermion configurations gener-
ated by the RBC/UKQCD collaboration with the parameters listed in Tab. 1. For valence quarks
b L3 T msa msa m a
× s l res
2.13 243 64 0.04 0.005 0.01 0.02 0.00315(4)
×
2.25 323 64 0.03 0.004 0.006 0.008 0.00067(1)
×
Table1: TheparametersfortheRBC/UKQCDconfigurations[2]. msaandmsaarethemassparametersof
s l
thestrangeseaquarkandthelightseaquark,respectively.m aistheresidualmassofthedomainwallsea
res
quarks.
weusetheoverlapfermionoperatorD =1+g e (H (r ))todefinetheeffectivemassivefermion
ov 5 W
2
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
operator
r D
ov
D (ma) +ma, (2.1)
c
≡ 1 D /2
ov
−
where H (r )=g D (r )withD (r )the Wilson-Dirac operator witha negative mass parameter
W 5 W W
r , and the parameter ma is purely the bare current quark mass (in lattice units) and free of ad-
−
ditiverenormalization owingtothegood chiralproperty g ,D (0) =0. Through themulti-mass
5 c
{ }
algorithm, quarkpropagatorsS (ma)=D 1(ma)fordozensofdifferentvalencequarkmassesma
F −c
can be calculated simultaneously, such that we can calculate multiple physical quantities at each
valence quark mass and obtain clear observation of the quark mass dependence of these quan-
tities. We estimate the physical strange quark mass to be around m a = 0.056 for the 243 64
s
×
lattice and m a=0.039 for the 323 64 lattice, therefore we choose the m a to vary in the range
s s
×
m a [0.0576,0.077] andm a [0.039,0.047] forthetwolattices,respectively. Figure1showsan
s s
∈ ∈
almostlinearbarequarkmassdependenceofthemassesofJ/y ,h ,D ,andD calculatedfromthe
c s ∗s
sixgaugeensembles. Inaddition totheaboveobservation, wealsofindthatthehyperfinesplitting
ofvector andpseudoscalar mesons, D =M M ,varies withquark masseslike (cid:181) 1/√m[3],as
V PS
−
shown in Fig. 2. Taking into account the sea quark mass dependence, D can be well described by
theformula
C+C m
1 l
M M = (2.2)
V− PS pmq1+mq2+d m
to a high precision. The detailed discussion of this dependence will be presented in Ref. [5].
Finally, theglobalfitformulaforthemesonsystemis
1
M = (A +A m +A m +A m +(A +A m ) )
meson 0 1 c 2 s 3 l 4 5 l √m +m +d
c s
(cid:0)1+B0a2+B1m2ca2+B2m4ca4(cid:1)+A4a2 (2.3)
×
withd aconstant parameter. NotethatA issettozeroforthecharmquark-antiquark system, and
2
A is expected to be close to 1(or 2) for the meson masses of c¯s(c¯c) system. We keep the m a
1 c
correction to the forth order and add explicit O(a2) correction term to account the artifacts due to
the gauge action and other possible artifacts. There should be m a corrections, but they are very
s
smallandareneglected.
InlatticeQCD,thebarequarkmassesareinputparametersinlatticeunits,say,m a. However
q
in the global fit including the continuum extrapolation using Eq. (2.3), one has to convert the
m a to the renormalized current quark mass mR(m ) at a fixed scale m which appears uniformly
q q
in Eq. (2.3) for different lattice spacings. This requires two issues to be settled beforehand. The
first is the renormalization constant Z of the quark mass for a fixed lattice spacing a. Since we
m
use the overlap fermion operator D , if the quark field y is replaced by the chirally regulated
ov
field yˆ =(1 1D )y in the definition of the interpolation fields and the currents, it is expected
− 2 ov
that there are relations Z =Z and Z =Z , where Z , Z , Z , and Z are the renormalization
S P V A S P V A
constants of scalar, pseudoscalar, vector, and axial vector currents, respectively. In addition, Z
m
canbederivedfromZ bytherelationZ =Z 1. InthecalculationofZ andotherZ’softhequark
S m S− S
bilinearcurrents, weadopttheRI-MOMschemetodothenon-perturbative renormalization onthe
lattice first, then convert them to the MS scheme using ratios from continuum perturbation theory
(The numerical details can be found in Ref. [6]). The relations between Z’s mentioned above are
3
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
Mass of Ds mass of Ds*
4 4
2464-005 2464-005
3.5 2464-010 3.5 2464-010
2464-020 2464-020
3264-004 3264-004
M(GeV)PS 2 .235 33226644--000068 M(GeV)V 2 .235 33226644--000068
1.5 1.5
1 1
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
mc+ms(GeV) mc+ms(GeV)
Mass of h c mass of J/y
3.5 3.5
2464-005 2464-005
2464-010 2464-010
3 23426644--002004 3 23426644--002004
M(GeV)PS 2.5 33226644--000068 M(GeV)V 2.5 33226644--000068
2 2
1.5 1.5
0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6
mc(GeV) mc(GeV)
Figure1: ThequarkmassdependenceofthemassesofD ,D ,h ,andJ/y areillustratedintheplotsfor
s ∗s c
thesixRBC/UKQCD configurationensembles,wherethelinearbehaviorsinm aorm a+m aareclearly
c c s
seen.
split of mPS and mV
100000
equal
u/d-s
10000 u/d-c
-2V) 1000 ss--cc*
e
G
-2m( 100
D
10
1
0.1 1
mq1+mq2(GeV)
Figure2:ThequarkmassdependenceofthehyperfinesplittingsD m=m m forc¯sandc¯csystems. For
V PS
−
clarity,the(D m) 2 versusm +m areplottedinthefigure,whereonecanseeanalmostlinearbehavior
− q1 q2
throughouttherangem +m [0.1,3]GeV. ThisbehaviorsuggeststherelationD m 1 .
q1 q2 ∈ ∼ √mq1+mq2+d
verified, andZ atthe scale m =2GeV inthe MSisdetermined tobe ZMS(2GeV)=1.127(9) for
S S
the 243 64 lattice and 1.056(6) for the 323 64 lattice. Besides the statistical error, systematic
× ×
errors from the scheme matching and the running of quark masses in the MS scheme are also
considered in Ref. [6]). The systematic error from the running quark mass in the MS scheme is
negligible small, while the one from scheme matching is at four loops, and has a size of about
1.4%.
The second is the precise determination of the lattice spacing a. This is a subtle question
becauseahasdirectrelationshipwiththecouplingconstantb andtheseaquarkmass. Ontheother
4
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
b =2.13 b =2.25
m a 0.02 0.01 0.005 0 0.008 0.006 0.004 0
l
r /a 3.906(3) 3.994(3) 4.052(3) 4.126(11) 5.421(5) 5.438(6) 5.459(6) 5.504(4)
0
Table2: r /a’sforthesixensemblesweareusing.
0
hand,thecontinuum limitsofphysicalquantities areindependent ofa. Soweneedadimensionful
physicalquantityasascaleparameter,whichisindependentofquarkmasses,suchthatdimensional
quantities such as hadron masses, lattice spacing a, and quark masses can be expressed in units
of this parameter. A proper choice for this purpose is Sommer’s scale parameter r . In each
0
gauge ensemble, the ratio r /a=p(1.65 e )/(s a2) can be determined very precisely through
0 c
−
the derivation of the static potentialV(r/a), where e and s a2 are the parameters inV(r/a), say,
c
aV(r/a)=aV e /(r/a)+s a2(r/a). Table2liststher /a’sforthesixensembles weareusing.
0 c 0
−
r /ahasobvious m adependence, soweextrapolate ittothechirallimitm˜ =m a+m a=0by
0 l l l res
a linear fit, r (m˜ a,a)/a=r (0,a)/a+cm˜ a ,for the ensembles with a same b . The extrapolated
0 l 0 l
valuesC(a) r (m˜ a,a)/aarealsolistedinTab.2
0 l
≡
Withtheaboveprescriptions, wecanreplacetherenormalizedquarkmassesandabythebare
quarkmassparametersm a,C(a),r ,andZ (2GeV,a)as
q 0 m
C(a)
mR(2GeV)=Z (2GeV,a)(m a) (2.4)
q m q r
0
In this way r enters into Eq. (2.3) as a new parameter and can be fitted simultaneously with the
0
parameters A andB inEq. (2.3).
i i
Inadditiontothe1Sand1PcharmoniummassesandD /D masses,wealsoobtainpredictions
s ∗s
ofthedecayconstant ofD ,namely, f . f isdefinedas
s Ds PS
Z 0y¯ g g y PS =E f , (2.5)
A a 4 5 b PS PS
h | | i
where E is the energy of the pseudoscalar and Z is the renormalization constant of the axial
PS A
vectorcurrent, oralternatively throughthePCACrelation
E2
0y¯ g y PS = PS f . (2.6)
a 5 b PS
h | | i m +m
qa qb
The later isobviously renormalization independent since Z Z =1 for the overlap fermion. With
m P
the Z calculated in Ref. [6], we find the two definitions are compatible with each other within
A
errors.
Inpractice, wecarryoutacorrelatedglobalfittothefollowingquantitiesinthesixensembles
–themassesofJ/y ,h ,c ,c ,h ,D andD mesonsand f withjackknifecovariancematrices.
c c0 c1 c s ∗s Ds
As a first step, we obtain the parameters A’s and B’s for each quantity. Subsequently we use the
i i
experimental values mJ/y = 3.097GeV, mDs = 1.968GeV, and mD∗s = 2.112GeV to determine
mMS(2GeV),mMS(2GeV),andr . Finally, weusethesephysical parameters topredict themasses
c s 0
of h , c , c , h , and f at the physical pion point and with O(a2) corrections. The results are
c c0 c1 c Ds
illustrated in Tab. 3. Here is the description of the error budget: i) The statistical errors are the
5
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
r0 mMs S(2GeV) mMc S(2GeV) mJ/y −mh c mcc0 mc c1 mhc fDs
(fm) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) (GeV)
PDG – 0.095(5) 1.09(3) 0.117(1) 3.415 3.511 3.525 0.258(6)
thiswork 0.458 0.104 1.110 0.1119 3.411 3.498 3.518 0.2542
s (stat) 0.011 0.006 0.012 0.0054 0.035 0.045 0.028 0.0049
s (r /a) 0.002 0.000 0.000 0.0000 0.003 0.000 0.000 0.0000
0
s (¶ r0) 0.003 0.004 0.005 0.0005 0.003 0.006 0.002 0.0003
¶ a2
s (MR/stat) – 0.003 0.020 0.0000 0.003 0.011 0.004 0.0003
s (MR/sys) – 0.000 0.003 0.0000 0.000 0.000 0.000 0.0000
s (all) 0.011 0.009 0.024 0.0054 0.046 0.047 0.028 0.0049
Table 3: The final results from the global fit in this work. The physical masses of D , D , and J/y are
s ∗s
usedasinputstodeterminer ,mMS(2GeV)andmMS(2GeV). Thephysicalmassesofthe1Pcharmoniacan
0 s c
be reproducedwith these values. The hyperfine splitting mJ/y −mh c and the decay constant fDs are also
predicted. The error budgetis given for each quantity with s (stat), the statistical error, and two kinds of
systematicerrors,onefromr andanotheronefrommassrenormalization. Fortheonefromr ,itincludes
0 0
s (r ) the uncertaintyowingto the statistical errorof r /aon each ensemble, ands ( ¶ r0 ) the systematic
0 0 ¶ (a2)
uncertaintyfromtheadependenceofr /a. Fortheonefrommassrenormalization,itincludess (MR/stat)
0
fromthestatisticaluncertaintiesofZ intheRI/MOMschemewhicharenotcorrelatedonthetwolattices,
m
ands (MR/sys)fromthesystematicuncertaintyinmatchingfromRI/MOMtoMSwhichisthesameonthe
twolatticesandonlycontributestotheuncertaintyofquarkmasses. Thetotalerrors (all)combinesallthe
uncertaintiesinquadrature.
jackknife errors from the global fit. ii) The systematic uncertainty due to the linear a2 continuum
extrapolation cannot be controlled at present since we have only two lattice spacings. iii) For the
chiral extrapolation we only use the linear fit in the u,d sea quark mass and have not considered
a sophisticated fit based on chiral perturbation theory, so this systematic uncertainty has not been
analyzed. iv)Weconsider twopossible systematic uncertainties introduced byr , oneof which is
0
fromthestatisticalerrorofC(a)=r (a)/a(denotedass (r /a)),andtheotherisfromnon-zeroa2
0 0
dependence ofr (a)(denotedass (¶ r0)). Thelatterisnotsostraightforward. Inourglobalfit,we
0 ¶ a2
takether atafinitelattice spacing tobetheoneinthecontinuum limit. Toaddress theissuethat
0
ourpredictionforr (0.458(11)fm)isslightlysmallerthantheonefromRBC/UKQCD(0.48(1)fm),
0
weset ¶ r0 =0.2(whichmakesther (a)atthetwolattice spacings tobearound 0.48astheRBC-
¶ a2 0
UKQCD collaboration so determines) , so as to check the changes of the predictions to estimate
theirsystematicerrors. Itturnsout,thec 2oftheglobalfitisinsensitivetothisdependenceandthe
results do not change much. v) Theerror s (MR)takes into account two kinds of uncertainties of
Z (m ), one ofwhich isthestatistical error ofZ (m )inthe RI/MOMscheme, andthe other isdue
m m
to the systematic error of perturbative matching and the running of the MS masses to the scale of
2GeV. Allof these uncertainties are combined together inquadrature togivethe total uncertainty
s (all)ofeachphysicalquantity.
6
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
1.6
Exp (PDG Live)
Lat (MDs, MDs* and MJ/y as Input)
1.4
0.095(5) 3.52541(16)
1.2 1.09(3) 0.1166(12) 3.51066(7) 0.2575(61)
o 3.41475(31)
ati
R
1
0.458(11)0.104(9) 3.518(28)
0.8 1.110(24) 0.1119(54) 3.498(47) 0.254(5)
3.411(46)
0.6
r0(fm)mc ms MJ/y -Mhc Mcc0Mcc1 Mhc fDs
Figure3: TheratiosbetweenPDGLiveaveragesandoursimulations. NotethatthenumbersofPDGLive
are in italic type, and all the numbersare in unitof GeV, exceptr . For r , we list the one fromHPQCD
0 0
(0.4661(38)fm)andRBC-UKQCD(0.48(1)fm)forreference.
3. Summary
With overlap fermions as valence quarks on domain wall fermion configurations generated
by theRBC/UKQCDCollaboration, wehave undertaken aglobal fitscheme combining thechiral
extrapolation, thephysical quark massinterpolation, and thecontinuum extrapolation. Weusethe
physical masses of J/y , D , and D as the inputs to determine r , the charm and strange quark
s ∗s 0
massestobe
r =0.458(11)fm, mMS(2GeV)=0.104(9)GeV, mMS(2GeV)=1.110(24)GeV (3.1)
0 s c
Our r is smaller than 0.48(1) fm obtained by RBC/UKQCD [7] but close to the HPQCD result
0
0.4661(38) fm [8]. With these results, we can reproduce the physical masses of c , c and h ,
c0 c1 c
andfurther predictthehyperfinesplitting mJ/y −mh c andthedecayconstant fDs,
mJ/y −mh c =112(5)MeV, fDs =254(5)MeV. (3.2)
Theerrorswequoteabovearequadratic combinations ofthestatistical andsystematicerrors.
References
[1] C.Alltonetal.(RBC-UKQCDCollaboration),Phys.Rev.D78,114509(2008).
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[hep-lat]].
[3] A.Lietal.[c QCDCollaboration],Phys.Rev.D82,114501(2010)[arXiv:1005.5424[hep-lat]].
[4] R.Sommer,Nucl.Phys.B411,839(1994).
[5] YingChen,Keh-FeiLiu,ZhaofengLiu,Yi-BoYang,Inpreparation.
[6] Z.Liuetal.[chiQCDCollaboration],PoS(LATTICE2013)307,2013,arXiv:1312.0375[hep-lat].
[7] R.Arthuretal.[RBCandUKQCDCollaborations],arXiv:1208.4412[hep-lat].
7
Charmonium,D andD fromoverlapfermiononDWFconfigurations Y.B.Yang
s ∗s
[8] C.T.H.Daviesetal.[HPQCDCollaboration],Phys.Rev.D81,034506(2010)[arXiv:0910.1229
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8
