D meson semileptonic decays in lattice QCD with Moebius domain-wall quarks

0.22 MB·

by  JLQCD Collaboration

KEK-CP-350
D meson semileptonic decays in lattice QCD with
Möbius domain-wall quarks
7
1
0 JLQCD Collaboration: T. Kaneko∗a,b†, B. Fahya, H. Fukayac, S. Hashimotoa,b
2

n
a aHighEnergyAcceleratorResearchOrganization(KEK),Ibaraki305-0801,Japan
J
bSchoolofHighEnergyAcceleratorScience,SOKENDAI(TheGraduateUniversityfor

4
AdvancedStudies),Ibaraki305-0801,Japan

] cDepartmentofPhysics,OsakaUniversity,Osaka560-0043,Japan
t
a
l
-
p We report on our study of the D meson semileptonic decays in 2+1 flavor lattice QCD. Gauge
e
ensembles are generated at three lattice cutoffs up to 4.5 GeV and with pion masses as low as
h
[ 300MeV.WeemploytheMöbiusdomain-wallfermionactionforbothlightandcharmquarks.

1 Wereportourpreliminaryresultsforthevectorandscalarformfactorsanddiscusstheirdepen-
v denceonthemomentumtransfer,quarkmassesandlatticespacing.
2
4
9
0
0
.
1
0
7
1
:
v
i
X
r
a
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
∗Speaker.
†E-mail:[email protected]
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
DmesonsemileptonicdecaysinlatticeQCDwithMöbiusdomain-wallquarks T.Kaneko
1. Introduction
TheD→πlνandD→KlνsemileptonicdecaysprovideaprecisedeterminationoftheCabibbo-
Kobayashi-Maskawa matrix elements |V | and |V |, and play an important role in the search for
cd cs
new physics in the charm sector [1]. The vector and scalar form factors fDP describe non-
{+,0}
perturbativeQCDeffects,andaredefinedfromtherelevanthadronicmatrixelementsas
(cid:104)P(p(cid:48))|V |D(p)(cid:105) = (p+p(cid:48)) fDP(t)+(p−p(cid:48)) fDP(t), (1.1)
µ µ + µ −
t
fDP(t) = fDP(t)+ fDP(t), (1.2)
0 + M2 −M2 −
D P
where P specifies the light meson (P=π,K) and t = (p−p(cid:48))2 is the momentum transfer. The
current accuracy of |V | and |V | is limited by the theoretical uncertainty of fDP [2]. Lattice
cd cs {+,0}
QCDistheonlyknownmethodtocalculate fDP withcontrolledandsystematically-improvable
{+,0}
uncertainties.
In this article, we report on our calculation of these form factors in N =2+1 QCD with the
f
tree-level improved Symanzik gauge action and the Möbius domain wall quark action [3]. Nu-
merical simulations are carried out at three lattice cutoffs a−1∼2.5, 3.6 and 4.5 GeV. On such
fine lattices, we employ the domain-wall action also for charm quarks. The simulated values
of m , that is the mass of the degenerate up and down quarks, cover a range of the pion mass
ud
300MeV(cid:46)M (cid:46)500MeV.Wetakeastrangequarkmassm closetoitsphysicalvalue. Anaddi-
π s
tional value of m is simulated at certain choices of (a,m ) in order to study the m dependence
s ud s
of the form factors. The charm quark mass is set to its physical value determined from the D me-
son spectrum [4]. The physical charm quark mass extracted from the same set of simulations is
m (3 GeV)=1.003(10)GeV, which is consistent with the present world average [5]. Our simula-
c
tionparametersaresummarizedinTable1. Afterthepreviousreport[6],weextendoursimulation
tothetwofinerlatticesandimproveourmeasurementmethodtoreducethestatisticaluncertainty.
Ateachsimulationpoint,ourlatticesizesatisfiesaconditionM L(cid:38)4tocontrolfinitevolume
π
effects,andweaccumulate5,000MolecularDynamicstime. Chiralsymmetryispreservedtogood
accuracybychoosingthesignfunctionapproximationandthekerneloperatorinthe4-dimensional
Table1: Simulationparameters.
latticeparameters m m M [MeV] M [MeV] N
ud s π K x4,src
β=4.17, a−1=2.453(4), 323×64×12 0.0190 0.0400 499(1) 618(1) 2
0.0120 0.0400 399(1) 577(1) 2
0.0070 0.0400 309(1) 547(1) 4
0.0190 0.0300 498(1) 563(1) 2
β=4.35, a−1=3.610(9), 483×96×8 0.0120 0.0250 501(2) 620(2) 2
0.0080 0.0250 408(2) 582(2) 2
0.0042 0.0250 300(1) 547(2) 4
0.0120 0.0180 499(1) 557(2) 2
β=4.47, a−1=4.496(9), 643×128×8 0.0030 0.0150 284(1) 486(1) 1
1
DmesonsemileptonicdecaysinlatticeQCDwithMöbiusdomain-wallquarks T.Kaneko
effectiveaction[7]. TheresidualmassissuppressedtoO(1MeV)atthecoarsestlattice,andeven
smaller(cid:46)0.2MeVatfinerlatticeswithmoderatesizesinthefifthdimension∼10.
2. Calculationofformfactors
Wecalculatethethree-pointfunction
1
CDP(p,p(cid:48);∆x ,∆x(cid:48)) = ∑ ∑ (cid:104)O (x(cid:48)(cid:48),x +∆x +∆x(cid:48))
Vµ 4 4 N3N P 4,src 4 4
s x4,src x4,srcx,x(cid:48),x(cid:48)(cid:48)
×V (x(cid:48),x +∆x )O†(x,x )(cid:105)e−ip(cid:48)(x(cid:48)(cid:48)−x(cid:48))e−ip(x(cid:48)−x), (2.1)
µ 4,src 4 D 4,src
whereN isthespatiallatticesizeandp((cid:48))representsthemomentumoftheinitial(final)meson. We
s
applyaGaussiansmearingtothemesoninterpolatingoperatorsO . Thetemporalcoordinate
{π,K,D}
of the source operator is denoted by x , and ∆x ((cid:48)) represents the temporal separation between
4,src 4
thesource(sink)operatorandthevectorcurrentV .
µ
In this study, we calculateCDP(p,p(cid:48);∆x ,∆x(cid:48)) by varying ∆x with ∆x +∆x(cid:48) kept fixed. Its
Vµ 4 4 4 4 4
physicallengthisthesameforthethreecutoffsandischosenas∆x +∆x(cid:48)=28aatβ=4.17[6]. The
4 4
D meson is at rest (p=0), and we simulate four different values of the momentum transfert with
lightmesonmomenta|p(cid:48)|2=0,1,2,3inunitsof(2π/L)2. ForP=π(K),theminimumvalueofthe
momentumtransferistypicallyt ≈0.3(0.2)GeV2,whilethemaximumist ≈2.6(1.8)GeV2.
min max
Wealsocalculatetwo-pointfunctionsofπ,KandDmesons. Theamplitudesofthecorrelators
areextractedbythefollowingexponentialfitsintermsof∆x
4
CVDµP(p,p(cid:48);∆x4,∆x4(cid:48)) = AVDµP(p,p(cid:48))e−ED(p)∆x4e−EP(p(cid:48))∆x4(cid:48) (P=π,K), (2.2)
CQ(p;∆x ) = BQ(p)e−EQ(p)∆x4 (Q=π,K,D). (2.3)
4
HerethemesonenergiesE areestimatedfromtheirrestmasses[4]andthedispersionrela-
{π,K,D}
tioninthecontinuumlimit. Thematrixelementsaregivenas
(cid:115)
E (p)E (p(cid:48))|ADP(p,p(cid:48))|2
(cid:104)P(p(cid:48))|V |D(p)(cid:105) = 2Z D P Vµ , (2.4)
µ V BD(p)BP(p(cid:48))
whereweusetherenormalizationfactorZ non-perturbativelycalculatedinRef.[8]. Therelevant
V
semileptonicformfactorsarethenextractedviaEqs.(1.1)and(1.2).
Inordertoimprovethestatisticalaccuracy,weaveragethethree-andtwo-pointfunctionsover
the locations of the source operator. As for the temporal location, we repeat our measurement
over N different values of x . Our choice of N is summarized in Table 1. An important
x4,src 4,src x4,src
improvement from Ref. [6] is to average over the spatial coordinates x as well by putting the
GaussiansourceoperatorassociatedwithaZ noiseateachlatticesiteatagiventime-slicex .
2 4,src
Figure1comparesourresultsfortheamplitudeADπ withdifferentmeasurementsetupsonour
V
4
coarsestlatticeattheheaviestseaquarkmasses. Weobserveabout30%reductionofthestatistical
errorbyaveragingover2t ’sandanadditional30%reductionbyaveragingoverx: aboutafactor
src
oftwoimprovementintotal. Averagingoverpfurtherimprovesthestatisticalaccuracy: at|p|2=2,
for instance, about factor of two improvement by averaging over 12 p’s. The typical statistical
accuracyis1–2%att andM ∼500MeV,and6–9%att andM ∼300MeV.
max π min π
2
DmesonsemileptonicdecaysinlatticeQCDwithMöbiusdomain-wallquarks T.Kaneko
1.50
"
= 10 β = 4.17, mud = 0.0190, ms = 0.0400, |p|2 = 2, |p′|2 = 0
4
x
∆
@
e
u
al
v
"
′) / 4 1.00
x
∆
, 4
x
∆
′p; single Gaussian
p, averaged over source points
πD( averaged over momenta
V4
A 0.50
10 15 20 25
∆x
4
Figure1:EffectivevalueofamplitudeADπ(p,p(cid:48))asafunctionof∆x .Weplotdatawith|p|=2and|p(cid:48)|2=0
V4 4
atβ=4.17and(m ,m )=(0.0190,0.0400). Theopentrianglesshowdatawitha“local”Gaussiansource
ud s
andasinglechoiceofp. Thebluesquaresandredcirclesareobtainedbyaveragingoverthelocationofthe
sourceoperatorandthenover12p’s. Alldataarenormalizedbytheirvalueat∆x =18.
4
3. Momentumtransferdependence
We parametrize the momentum transfer dependence of the form factors in terms of the so-
calledzparameter[9]
√ √
t −t− t −t
+ + 0
z(t,t ) = √ √ , (3.1)
0
t −t+ t −t
+ + 0
where t =(M +M )2 represents the DP threshold (P=π, K). The free parameter t is chosen
+ D P 0
so that our simulated regiont∈[t ,t ] is mapped into a shortest segment z∈[−|z| ,+|z| ]
min max max max
centeredattheorigin. Typicalsizeofthezparameteris|z| (cid:46)0.2.
max
The momentum transfer dependence of the form factors are then parametrized by using this
smallparameteras
N
1 {+,0}
fDP (t) = ∑ a zk. (3.2)
{+,0} B(t) {+,0},k
k=0
In this preliminary analysis, we test two choices of the factor B(t). In the so-called Bourrely-
Caprini-Lellouch(BCL)parametrization[10]with
t
B(t)=1− , (3.3)
M2
pole
possibly small deviation from the lowest pole contribution 1/B(t) is expanded in terms of z. We
alsotestanaivepolynomialexpansionoftheformfactorsthemselveswith
B(t)=1. (3.4)
3
DmesonsemileptonicdecaysinlatticeQCDwithMöbiusdomain-wallquarks T.Kaneko
f , polynomial
+
f , BCL
+
f, polynomial
0
f, BCL
0
π(t) 1.0
D
0
+,
B(t) f
0.5
β = 4.17, m = 0.0190, m = 0.0400
ud s
-0.1 0.0 0.1
z
Figure 2: Plot of B(t)fDπ (t) as a function of z at β=4.17 and (m ,m )=(0.0190,0.0400). Circles
{+,0} ud s
and squares are data for f and f , respectively. Filled symbols show (1−t/M2 )fDπ (t) for the BCL
+ 0 D∗ +(0)
(0)
parametrization,whereasopensymbolsforthepolynomialexpansionarejusttheformfactorsthemselves.
A simultaneous fit to f (filled symbols) and f (open symbols) is shown by solid and dashed lines. The
+ 0
verticaldot-dashedlinerepresentszcorrespondingt=0,andthediamondisthevalueextrapolatedtot=0.
Figure2showsz-dependenceofaquantityB(t)fDπ (t)tobeexpandedintermsofz. Namely,
{+,0}
B(t)fDπ (t)=(1−t/M2 )fDπ (t) for the BCL parametrization with Eq. (3.3), whereas it is just
+(0) D∗ +(0)
(0)
theformfactorforthepolynomialexpansionwithEq.(3.4).
For f+DP, we use the vector meson masses MD∗ calculated at the simulation points, which
(s)
are well below the threshold t . We observe that the z dependence of B(t)fDP(t) is significantly
+ +
reducedbyswitchingfromthepolynomialexpansion(3.4)totheBCLparametrization(3.3). This
suggeststhatthevectormesondominance(VMD)hypothesisisareasonablygoodapproximation
of fDP,andwecanexpandthesmalldeviationfromtheVMDintermsofsmallz. Inthisstudy,we
+
testtwoBCLparametrizationincludingthelinear(N =1)andquadraticterms(N =2).
+ +
We have not yet calculated the scalar meson masses MD∗ , and hence it is not clear whether
(s)0
there exist corresponding isolated poles below t at simulated M ’s. In this analysis, we employ
+ π
thesimplelinearexpansion(3.4)for fDP. WealsotesttheBCLparametrization(N =1)withthe
0 0
experimentalvalueofMD∗ byassumingitsmilddependenceonmud.
(s)0
We estimate the normalization fDP(0)= fDP(0) from the simultaneous fit using the BCL
+ 0
parametrization with N =1 for fDP and the linear parametrization (N =1) for fDP. We also
+ + 0 0
test above-mentioned alternative forms to estimate the systematic uncertainty of the extrapolation
tot=0. As shown in Fig. 2, however, the z dependence of our results is mild except the polyno-
mial parametrization for fDP. The systematic uncertainty is not large compared to the statistical
+
accuracy.
4. Continuumandchiralextrapolation
InFig.3,wecompare fDπ atdifferentpionmasses(leftpanel)andatdifferentlatticespac-
{+,0}
4
DmesonsemileptonicdecaysinlatticeQCDwithMöbiusdomain-wallquarks T.Kaneko
β = 4.35, m = 0.0250 MM ~~ 330000 MMeeVV,, mm ~~ ((33//44 -- 11)) mm
2.0 s 2.0 ππ ss ss,,pphhyyss
f+, Mπ ~ 500 MeV f+, β = 4.17
f+, Mπ ~ 400 MeV f+, β = 4.35
f+, Mπ ~ 300 MeV f+, β = 4.47
πD(t) ff00,, MMππ ~~ 540000 MMeeVV πD(t) ff00,, ββ == 44..1375
f+,0 Cf0L, E MOπ- c~ +3 0F0L MAGeV3 f+,0 Cf0L, E βO =- c4 +.4 7FLAG3
1.0 1.0
0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0
t [GeV2] t [GeV2]
Figure3: Comparisonof fDπ amongdifferent M ’s(left panel)and different a’s(right panel). We plot
{+,0} π
dataatβ=4.35andm =0.0250intheleftpanel,whereastherightpanelshowsdataatM ∼300MeVand
s π
larger m . We also plot the Becirevic-Kaidalov parametrization [11] of the CLEO-c data of f Dπ(t) [12]
s +
combinedwithanaverageofrecentlatticeestimatesof f Dπ(0)[13].
+
ings (right panel). The reasonable consistency in both panels suggests a mild dependence on M
π
anda. We note that the decayconstants f also have smalldiscretization errors with ourchoice
D
(s)
ofthelatticeactionandcutoffs[14].
In this preliminary analysis, therefore, we parametrize the a, m and m dependences of
ud s
fDP(0)bythefollowingsimplelinearform
+
fDP(0) = cDP+cDPa2+cDPM2+cDPM2 , (4.1)
+ a π π ηs ηs
where M2 =2M2 −M2. This continuum and chiral extrapolation is plotted in Fig 4. We obtain
ηs K π
χ2/d.o.f∼1.6–1.8. AllthecoefficientscDP have(cid:38) 75%statisticalerror: namely,consistent
{a,π,ηs}
with zero as expected from the good consistency in Fig. 3. This fit is therefore not sensitive to
higher order corrections, and we estimate the systematic uncertainty from three fits in which one
ofthethreecoefficientscDP issettozero. Ourpreliminaryestimates
{a,π,ηs}
fDπ(0)=0.644(49)(27), fDK(0)=0.701(46)(33). (4.2)
+ +
areconsistentwithrecentlatticeaverages fDπ(0)=0.666(29)and fDK(0)=0.747(19)[13].
+ +
5. Summary
Inthisarticle,wereportonourlatticecalculationoftheD→π andD→K semileptonicform
factors. We employ the Möbius domain-wall quark action both for light and charm quarks, and
simulatelatticecutoffsupto4.5GeV.
Our preliminary results for f Dπ(K)(0) have uncertainty of 8 (9)%. We expect significant
+
improvement in the near future by increasing statistics (N ) on the finest lattice and extending
x4,src
ourmeasurementstosmallerM ∼230MeV.
π
WeobservesmalldiscretizationerrorsoftheDmesonformfactorswithoursimulationsetup.
ItisthereforeinterestingtoextendourstudytotheBmesonsemileptonicdecays,whicharebeing
preciselymeasuredatSuperKEKB/BelleIIandLHCbexperiments.
5
DmesonsemileptonicdecaysinlatticeQCDwithMöbiusdomain-wallquarks T.Kaneko
β = 4.47, m = 0.0150 β = 4.47, m = 0.0150
000...999 β = 4.35, ms = 0.0250 111...000 β = 4.35, ms = 0.0250
s s
β = 4.35, m = 0.0250 β = 4.35, m = 0.0180
s s
000...888
0) 0) 000...888
πD( 000...777 DK(
f+ f+
000...666
000...666
β = 4.17, m = 0.0400 β = 4.17, m = 0.0400
s s
000...555 β = 4.17, ms = 0.0300 β = 4.17, ms = 0.0300
FLAG3 FLAG3
000 000...111 000...222 000 000...111 000...222
M2 [GeV2] M2 [GeV2]
π π
Figure 4: Continuum and chiral extrapolation of f Dπ(0) (left panel) and f DK(0) (right panel). Data at
+ +
differenta’sandm ’sareplottedbydifferentsymbolsasafunctionofM2. Thedashedlinesshowthefitline
s π
inthecontinuumlimitandatthephysicalstrangequarkmass. Thevalueextrapolatedtothephysicalpoint
isplottedbythestars. Wealsoplotaveragesofrecentlatticeestimates[13]bythetriangles.
NumericalsimulationsareperformedonHitachiSR16000andIBMSystemBlueGeneSolu-
tionatKEKunderasupportofitsLargeScaleSimulationProgram(No.16/17-14). Thisresearch
is supported in part by the Grant-in-Aid of the MEXT (No. 26247043, 26400259) and by MEXT
as“PriorityIssueonPost-Kcomputer”(ElucidationoftheFundamentalLawsandEvolutionofthe
Universe)andJICFuS.
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