Table Of ContentUpdate on Heavy-Meson Spectrum Tests of the
Oktay–Kronfeld Action
Jon A. Bailey, Yong-Chull Jang∗†, Weonjong Lee‡
LatticeGaugeTheoryResearchCenter,CTP,andFPRD,
DepartmentofPhysicsandAstronomy,SeoulNationalUniversity,Seoul,151-747,SouthKorea
E-mail†:[email protected]
6 E-mail‡:[email protected]
1
0 Carleton DeTar
2
DepartmentofPhysicsandAstronomy,UniversityofUtah,SaltLakeCity,UT84112,USA
n E-mail: [email protected]
a
J
Andreas S. Kronfeld
8
TheoreticalPhysicsDepartment,Fermilab,Batavia,IL60510,USA
1
InstituteforAdvancedStudy,TechnischeUniversitätMünchen,85748Garching,Germany
] E-mail: [email protected]
t
a
l Mehmet B. Oktay
-
p
e DepartmentofPhysicsandAstronomy,UniversityofIowa,IowaCity,IA52242,USA
h
[ Fermilab Lattice, MILC, and SWME Collaborations
1
Wepresentupdatedresultsofanumericalimprovementtestwithheavy-mesonspectrumforthe
v
9 Oktay–Kronfeld(OK)action. TheOKactionisanextensionoftheFermilabimprovementpro-
5
gramformassiveWilsonfermionsincludingalldimension-sixandsomedimension-sevenbilinear
7
4 terms. ImprovementtermsaretruncatedbyHQETpowercountingatO(Λ3/m3)forheavy-light
Q
0
systems, and by NRQCD power counting at O(v6) for quarkonium. They suffice for tree-level
.
1
matchingtoQCDtothegivenorderinthepower-countingschemes. Toassesstheimprovement,
0
6 wegeneratenewdatawiththeOKandFermilabactionthatcoversbothcharmandbottomquark
1
massregionsonaMILCcoarse(a≈0.12fm)2+1flavor,asqtad-staggeredensemble. Weup-
:
v
datetheanalysesoftheinconsistencyquantityandthehyperfinesplittingsfortherestandkinetic
i
X
masses. With one exception, the results clearly show that the OK action significantly reduces
r
a heavy-quarkdiscretizationeffectsinthemesonspectrum. Theexceptionisthehyperfinesplitting
of the heavy-light system near the B meson mass, where statistics are too low to draw a firm
s
conclusion,despitepromisingresults.
The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
TestsoftheOktay–KronfeldAction Yong-ChullJang
1. Introduction
Nowadays,latticeQCDachievesveryhighprecisionforcalculationsoflight-quarkprocesses[1].
However,simulatingheavyquarksinlatticeQCDisstillachallenge[2,3]. Amajorchallengeinin-
creasingtheprecisionoflatticeQCDcalculationsofheavy-quark,candb,quantitiesiscontrolling
heavy-quark discretization errors. Because the heavy-quark masses and the accessible ultraviolet
cutoffa−1 arecomparable,specialcareisneededtohandleheavy-quarkdiscretizationeffects.
The Fermilab method was introduced to address these issues [4]. The Oktay-Kronfeld (OK)
action [5] is an improvement (in the Symanzik sense) of the this approach that incorporates the
dimension-six and -seven bilinear operators needed for tree-level matching to QCD. It explicitly
treats corrections through O(λ3), where λ ∼Λ /m or Λ a, in HQET power counting for
QCD Q QCD
heavy-lightmesons,andO(v6),wherevistherelativequark-antiquarkvelocity,inNRQCDpower
countingforquarkonium. Forasmallmassm a(cid:28)1,theimprovementisequivalenttoO(a2)with
Q
some O(a3) terms with Symanzik power counting [6]. Based on semiquantitative arguments, it is
expectedthatthebottomandcharmquarkdiscretizationerrorscouldbereducedbelowthecurrent
1%levelwiththeOKaction[5]. Weaimtotesttheimprovementquantitatively.
For the heavy-light and quarkonium spectra of c- and b-mesons, we present results for the
inconsistency quantity [7, 8] and hyperfine splittings, which test how well the Fermilab and OK
actionsreduceheavy-quarkdiscretizationerrorsinpractice. Fortheworkreportedhere,wegener-
ate data using the tadpole-improved Fermilab and OK actions for a range of heavy-quark masses
encompassingcharmandbottom. WeextendourpreliminaryanalysisonaMILCasqtad-staggered
N =2+1 coarse ensemble with a≈0.12 fm [9]. Near both charm and bottom masses, we gen-
f
eratedatawithfour(two)differentvaluesofthehoppingparameterfortheOK(Fermilab)action,
forcomparison. Weuseanoptimizedconjugategradient(CG)inverter[10]. Tadpoleimprovement
ofalltermsisfullyimplementedinthisprogram,completingearlywork[11].
2. MesonCorrelator
We use the MILC asqtad-staggered N =2+1 gauge ensemble which has dimensions N3×
f L
N =203×64,β =6.79,tree-leveltadpolefactoru =0.8688,andlatticespacinga≈0.12fm[12].
T 0
Theasqtad-staggeredaction[12]isusedforthelightdegenerateseaquarkswithmassam =0.02
l
and strange sea quark with mass am = 0.05. For the tests reported here, we use N = 500
s cfg
configurationsoftheapproximately2000configurationsavailable. Foreachconfiguration,weuse
sixsourcesh(rrr,t)forcalculatingvalencequarkpropagators. Thespatialsourcecoordinatesrrr are
i i i
randomly chosen within the spatial cube. The source times t are evenly spaced along the lattice
i
witharandomizedoffsett ∈[0,20)foreachconfiguration.
0
Wecomputetwo-pointcorrelatorsasdescribedinRef.[9]at10mesonmomenta, ppp=2πnnn/N a,
L
withnnn=(0,0,0),(1,0,0),(1,1,0),(1,1,1),(2,0,0),(2,1,0),(2,1,1),(2,2,0),(2,2,1),(3,0,0)—
includingallpermutationsofthecomponents.
ThedefinitionofthehoppingparameterfortheOKactionisgiveninEq.(2.1)ofRef.[11]. For
theFermilabaction,theEq.(2.2)ofRef.[4]definesthehoppingparameter. Thehoppingparameter
valuesusedtosimulatecandbquarksaregiveninTable1. Wefixthevalencelightquarkmassfor
theheavy-lightmesoncorrelatorstothestrangeseaquarkmassam . Hence,werefertotheheavy-
s
2
TestsoftheOktay–KronfeldAction Yong-ChullJang
Table1: Hoppingparameterκ fortheOKactionandκ fortheFermilabaction. Theκ arevertically
OK FL FL
alignedtotheκ ,whichyieldtheclosestheavy-lightmesonkineticmassM . SeealsoFig.1.
OK 2
Q b c
κ 0.039 0.040 0.041 0.042 0.0468 0.048 0.049 0.050
OK
κ 0.083 0.091 0.121 0.127
FL
light mesons as “B ” or “D ,” depending on the hopping parameter. In anticipation of tuning runs
s s
for the OK action, we generate B and D correlators by using the OK action with four different
s s
valuesforthehoppingparameterκ . Forpurposesofcomparison,wesimulatewiththeFermilab
OK
actionwithtwovaluesforthehoppingparameterκ yieldingquarkmassesinthesameranges.
FL
ThegroundstateenergiesE areextractedfromcorrelatorfitstothefunction
f(t)=Ae−Et(cid:8)1−(−1)tre−∆Et(cid:9)+Ae−E(T−t)(cid:8)1−(−1)tre−∆E(T−t)(cid:9), (2.1)
where A is the ground state amplitude. We also incorporate the staggered parity partner state
with amplitude Ap and energy Ep for the ground state into the fit function. In practice, we take
an amplitude ratio r =Ap/A and energy difference ∆E =Ep−E as fit parameters instead of Ap
and Ep. We set the Bayesian prior ∆E =0.2(5). The parity partner is not involved in the fit for
quarkonium,becausebothheavyquarksaredescribedbyeithertheOKorFermilabaction. Hence,
weusedasimplerfitfunctionforquarkoniumwithr≡0inEq.(2.1).
We perform correlated fits. The inverse covariance matrix is estimated with singular value
decomposition (SVD). Before performing SVD, the covariance matrix is normalized by the max-
imum component on the diagonal, so that the largest eigenvalue λ is of O(1). Then singular
max
valuesλ belowthenumericaltoleranceλ/λ ≤10−15 areremoved. Oneortwosingularvalues
i i max
areremovedfromthedatawithb-quarkhoppingparametervaluesgiveninTable1.
Toincreasestatistics, eachcorrelatorisaveragedoverpositiveandnegativetimeseparations.
Then, we take the fit interval [t ,t ], where 0 ≤t <t < T/2, equal to [10,19] for the
min max min max
heavy-lightsystemsand[15,20]forthequarkonia. Wefixtheseintervalsforfitstoallcorrelators,
independentofthehoppingparameterκ,momentum ppp,andaction. Thet arechosenbyrequir-
max
ingthatthenoise-to-signalratiointhetwo-pointcorrelatorbelessthanabout20%forallmomenta,
which,inpractice,issetatthelargermomenta,wherethecorrelatorisnoisier. Thet arechosen
min
byobservingtheeffectivemasscalculatedfromthedefinitionm (t)=(cid:2)ln(cid:8)CM(t)/CM(t+2)(cid:9)(cid:3)/2,
eff
aswellascomparingthefitresultsE withm (t). Toestimatethestatisticalerrors,weuseasingle-
eff
eliminationjackknife.
3. MesonMasses
We fit the ground state energy E in Eq. (2.1) for each momentum ppp to the non-relativistic
dispersionrelationE(ppp),includingtermsuptoO((appp)6),
ppp2 (ppp2)2
E =M + − +E(cid:48) +E +E(cid:48), (3.1)
1 2M 8M3 4 6 6
2 4
a3W (ppp2)3 a5W a5W(cid:48)
E(cid:48) =− 4∑p4, E = , E(cid:48) =− 6∑p6+ 6ppp2∑p4, (3.2)
4 6 i 6 16M5 6 3 i 2 i
i 6 i i
3
TestsoftheOktay–KronfeldAction Yong-ChullJang
toobtaintherestmassM andkineticmassM ofeachmeson. Themismatchbetweentherestand
1 2
kinetic meson masses can be exploited to test the improvement of nonrelativistically interpreted
actions. TheM aregeneralizedmasses. TheO(3)rotationsymmetrybreakingtermsareE(cid:48) and
4,6 4
E(cid:48). Inthecontinuumlimit,thecoefficientsW ,W((cid:48))→0andM →M . ThefitparametersM ,
6 4 6 1,4,6 2 1
M−1,M−3,M−5,W ,W ,andW(cid:48)areobtainedbywithalinearfit. Weusethefullcovariancematrix
2 4 6 4 6 6
amongallmomenta.
We investigate variations by excluding some or all the higher-order correction terms E(cid:48) and
4
E((cid:48)). WefindthattheE(cid:48) termisnecessary,andE((cid:48)) termsreducetheχ2. Wealsoinvestigatefitsby
6 4 6
droppinghigh-momentumdata. Fortheimprovementtests,weselecttheresultsfromthedispersion
fits using only the lowest eight momenta without Bayesian constraints on the fit parameters. The
rest mass M and the kinetic mass M from the chosen fits are statistically consistent with other
1 2
fits that we performed: the dispersion fits with Bayesian constraints on the fit parametersW(cid:48) and
4
((cid:48))
W , and the dispersion fits with meson spectra which are obtained from uncorrelated fits for the
6
two-pointcorrelators.
4. InconsistencyParameter
Toassesstheimprovement,weusetheinconsistencyquantity[7,8],
2δM −(δM +δM ) 2δB −(δB +δB )
Qq QQ qq Qq QQ qq
I≡ = , (4.1)
2M 2M
2Qq 2Qq
where the mass differences δM ≡M −M ,(X =Qq,QQ), are obtained from the dispersion
X 2X 1X
relationfits. Becauselightquarksalwayshavema(cid:28)1,theO((ma)2)distinctionbetweenrestand
kineticmassesisnegligible. WethereforeomitδM ≡M −M (orδB ).
q¯q 2qq 1qq q¯q
ThemesonmassesM canbewrittenasasumoftheperturbativequarkmassesm andthe
1,2 1,2
bindingenergiesB . Foraheavy-lightsystem,therelationreads
1,2
M =m +m +B , M =m +m +B , (4.2)
1Qq 1Q 1q 1Qq 2Qq 2Q 2q 2Qq
andsimilarlyfora(light)quarkonium. TheseformulasdefineB andB . Substitutingthemintothe
1 2
definitionofI inEq.(4.1),thequarkmassescancelout,andweobtaintherelationamongbinding
energydifferencesδB=B −B inEq.(4.1).
2 1
In a relativistically invariant theory, the binding energies B and B are equal. The “incon-
1 2
sistency”I isolatesthebinding-energydifferenceδB=B −B (cid:54)=0. Attheleadingorder, O(ppp2),
2 1
itisduetodiscretizationerrorsfromthehigher-dimensionoperatorsintheactionofO((appp)4),or
O(v4)inNRQCDpowercounting,whichenterB [8]. Thisleading-orderinconsistencyvanishesat
2
tree-levelfortheOKaction,butnotfortheFermilabaction[8,13,9]. Hence,byconstruction,the
inconsistency quantity I is good for probing how well the OK action removes these discretization
errorsinthemesonspectra.
The results for the inconsistency I from the pseudoscalar meson spectra are shown in Fig. 1.
WefindthatI isclosetothecontinuumlimit,I=0,fortheOKactioneveninthebottommassre-
gion,whereastheFermilabactionproducesaverylargedeviation,I≈−0.6. ThesmallI shownin
Fig.1fortheOKactionresultsmainlyfromthehigherorderkineticoperatorsofO(v4)inNRQCD,
4
TestsoftheOktay–KronfeldAction Yong-ChullJang
0.2 0.1
0.0 50 49 48 47 42 41 40 39
127
121 50
−0.2 D+ 0.0 49 48
s 47
I I
−0.4
127
91
−0.6 −0.1 121
B0
s 83
−0.8
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5
aM aM
2Q¯q 2Q¯q
(a) (b)
Figure 1: Inconsistency I for pseudoscalar mesons. A magnified view of the boxed region in Fig. 1(a) is
given in Fig. 1(b). Data labels denote κ×103 values. The square (green) represents the Fermilab action
data, andthecircle(purple)representsOKactiondata. Fortuningpurposes, wealsoindicatethephysical
B andD masseswithverticallinesanderrorbands[14];theerrorsaredominatedbytheerrorofthelattice
s s
spacing. IalmostvanishesfortheOKaction,butfortheFermilabactionitdoesnot. Thisbehaviorsuggests
thattheOKactionissignificantlyclosertothecontinuumlimit,I=0,whichisrepresentedbythehorizontal
line(red). Theerrorsarefromthejackknife.
or O(λ2) in HQET, power counting [5]. These terms suffice to tune the quark dispersion relation
toO((appp4)). Thisoutcomeprovidesgoodnumericalevidencethattheimprovementexpectedwith
theOKactionisrealizedinpractice.
5. HyperfineSplittings
The hyperfine splitting ∆ is defined to be the difference in the masses of the vector (M∗) and
pseudoscalar(M)mesons:
∆ =M∗−M , ∆ =M∗−M . (5.1)
1 1 1 2 2 2
Spin-independent contributions to the binding energies cancel in the difference of hyperfine split-
tings∆ −∆ =δB∗−δB[9]. Comparingtothecontinuumlimit,∆ =∆ ,diagnosestheimprove-
2 1 2 1
ment of the spin-dependent terms in the OK action [5] of O(v6) in NRQCD, or O(λ3) in HQET
powercounting.
As one can see in Fig. 2(a), the OK action shows clear improvement for quarkonium. The
data points from the OK action lie much closer to the continuum limit ∆ =∆ (the red line) for
2 1
all simulated values of κ . In addition, the deviation is smaller for the charmonium region, near
OK
κ =0.049andκ =0.127,thanforthebottomoniumregion,nearκ =0.041andκ =0.083.
OK FL OK FL
The heavy-light results in Fig. 2(b) also show clear improvement in the region near the D mass.
s
TheresultswiththeOKactionremainconsistentwiththecontinuumexpectationthroughouttheB
s
massregion, buttheimprovementisnotsignificantforκ ≤0.041, becausethestatisticalerrors
OK
arelarge. Evenhere,however,theresultsaresuggestiveofimprovement.
Forbothquarkoniaandheavy-lightmesons,thehyperfinesplittingofthekineticmass(∆ )has
2
a larger error than that of the rest mass (∆ ), mainly because the kinetic mass requires correlators
1
5
TestsoftheOktay–KronfeldAction Yong-ChullJang
0.60 0.60
83
0.40 0.40 127
83
127
∆2 0.20 ∆2 0.20 49
r1 49 r1
0.00 0.00
41 41
−0.20 −0.20
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30
r ∆ r ∆
1 1 1 1
(a) Quarkonium (b) Heavy-lightmeson
Figure2:Hyperfinesplitting∆ obtainedfromthekineticmassesvs.∆ ,thatobtainedfromtherestmasses.
2 1
The square (green) represents the Fermilab action data, and the circle (purple) represents OK action data.
Thelabelsareκ×103, correspondingtokineticmassesclosetothephysicalB (83, 41)andD (49, 127)
s s
masses,asshowninFig.1.Thecontinuumlimitisrepresentedbytheline(red)∆ =∆ .Errorsareestimated
2 1
withthejackknifemethod.
with ppp(cid:54)=000, which are noisier than those with ppp=000. The statistical errors shown in Fig. 2 are
comparable for the OK and Fermilab action, except for ∆ for heavy-light mesons with κ =
2 OK
0.041,0.042,whichareinterestingly30–50%smallerthanthosewithκ =0.083,0.091.
FL
6. Conclusion
The inconsistency quantity shows that the OK action improves the O(ppp2) effects in the bind-
ingenergy,becauseitimprovestheO(ppp4)partoftheeffectiveLagrangian,inpracticeaswellasin
theory. Thehyperfinesplittingsclearlyshowtheimprovementfromthehigher-dimensionchromo-
magneticinteractionterms,exceptintheB massregion,wherestatisticsareatpresentinsufficient
s
toreachanydefiniteconclusion.
7. Acknowledgments
J.A.B.issupportedbytheBasicScienceResearchProgramoftheNationalResearchFounda-
tionofKorea(NRF)fundedbytheMinistryofEducation(2015024974). C.D.issupportedinpart
by the U.S. Department of Energy under grant No. DE-FC02-12ER-41879 and the U.S. National
Science Foundation under grant PHY10-034278. A.S.K. is supported in part by the German Ex-
cellenceInitiativeandtheEuropeanUnionSeventhFrameworkProgrammeundergrantagreement
No. 291763 as well as the European Union’s Marie Curie COFUND program. Fermilab is oper-
ated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the U.S.
DepartmentofEnergy. TheresearchofW.L.issupportedbytheCreativeResearchInitiativesPro-
gram(No.2015001776)oftheNRFgrantfundedbytheKoreangovernment(MEST).W.L.would
like to acknowledge the support from KISTI supercomputing center through the strategic support
programforthesupercomputingapplicationresearch[No.KSC-2014-G3-002]. Thecomputations
werecarriedoutinpartontheDAVIDGPUclustersatSeoulNationalUniversity.
6
TestsoftheOktay–KronfeldAction Yong-ChullJang
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