Chiral behavior of light meson form factors in 2+1 flavor QCD with exact chiral symmetry

0.25 MB·

by  JLQCD Collaboration

KEK-CP-329
Chiral behavior of light meson form factors in 2+1
flavor QCD with exact chiral symmetry
6
1
0 JLQCD Collaboration: T. Kaneko∗a,b†, S. Aokic,d, G. Cossua, X. Fenge, H. Fukayaf,
2 S. Hashimotoa,b, J. Noakia, T.Onogif

n
a
J
aHighEnergyAcceleratorResearchOrganization(KEK),Ibaraki305-0801,Japan
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bSchoolofHighEnergyAcceleratorScience,SOKENDAI(TheGraduateUniversityfor
2
AdvancedStudies),Ibaraki305-0801,Japan

] cYukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan
t
a d CenterforComputationalSciences,UniversityofTsukuba,Ibaraki305-8577,Japan
l
- ePhysicsDepartment,ColumbiaUniversity,NewYork,NY10027,USA
p
e f DepartmentofPhysics,OsakaUniversity,Osaka560-0043,Japan
h
[

We present a study of chiral behavior of light meson form factors in QCD with three flavors of
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v overlapquarks.Gaugeensemblesaregeneratedatsinglelatticespacing0.12fmwithpionmasses
8
down to 300 MeV. The pion and kaon electromagnetic form factors and the kaon semileptonic
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6 formfactorsarepreciselycalculatedusingtheall-to-allquarkpropagator. Wediscusstheirchiral
7
behaviorusingthenext-to-next-to-leadingorderchiralperturbationtheory.
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:
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X
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The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan*
∗Speaker.
†E-mail:[email protected]
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Chiralbehavioroflightmesonformfactorsin2+1flavorQCDwithexactchiralsymmetry T.Kaneko
1. Introduction
TheK→πlν semileptonicdecaysprovideaprecisedeterminationoftheCabibbo-Kobayashi-
Maskawamatrixelement|V |. LatticeQCDplaysanimportantroletoevaluatethenormalization
us
ofthevectorandscalarformfactors f (0)=f (0)intheK→π matrixelement
+ 0
(cid:104)π(p(cid:48))|V |K(p)(cid:105) = (p+p(cid:48)) f (t)+MK2 −Mπ2 q {f (t)− f (t)} (cid:0)t =q2=(p−p(cid:48))2(cid:1). (1.1)
µ µ + µ 0 +
t
The required accuracy is high, typically within 1%. The phase space integral for the decay rate
is estimated from the form factor shape, namely their t dependence, which has been precisely
measured by experiments. Therefore a rigorous comparison of the shape between lattice QCD
and experiments can demonstrate the reliability of the precision calculation of f (0). This article
+
presentsalatticecalculationoftheseformfactorsandanalysisbasedonnext-to-next-to-leadingor-
der(NNLO)chiralperturbationtheory(ChPT).Wealsodiscussthekaonandpionelectromagnetic
(EM)formfactorsdefinedthrough
(cid:104)P(p(cid:48))|J |P(p)(cid:105) = (cid:0)p+p(cid:48)(cid:1) FP(t) (P=π+,K+,K0), (1.2)
µ µ V
whichprovidehelpfulinformationfortheChPTanalysisofthesemileptonicformfactors.
2. Simulationmethod
Wesimulate2+1flavorQCDusingtheoverlapquarkaction, whichexactlypreserveschiral
symmetry and enables us to directly compare the lattice data with ChPT. Numerical simulations
areremarkablyacceleratedbymodifyingtheIwasakigaugeaction[1]andbysimulatingthetrivial
topological sector [1,2]. Notethat effectsof thefixed globaltopology canbe consideredas finite
volume effects suppressed by the inverse lattice volume [2]. Gauge ensembles are generated at a
lattice spacing a=0.112(1) fm and at a strange quark mass m =0.080 close to its physical value
s
m =0.081. Fourvaluesofdegenerateupanddownquarkmasses,m =0.015,0.025,0.035and
s,phys l
0.050, aresimulatedtoexplorearangeofthepionmass290–540MeV.Ateachm , wechoosea
l
latticesize,163×48or243×48,tocontrolfinitevolumeeffectsbysatisfyingaconditionM L(cid:38)4.
π
Thestatisticsare2,500HMCtrajectoriesateachsimulationpoint(m ,m ).
l s
We calculate the relevant two- and three-point functions of pion and kaon by using the all-
to-all quark propagator. The form factors are precisely estimated from ratios of the correlation
functions. We employ the twisted boundary condition for the valence quarks to simulate near-
zero momentum transfers |t|(cid:46)(300 MeV)2. The m dependence of the form factors is studied by
s
repeating our calculation at a different m (=0.060) with the reweighting technique. We refer to
s
Refs[3,4]formoredetailsonoursimulationparametersandmethod.
3. EMformfactors
Chiral symmetry constrains the chiral behavior of the EM form factors. In the chiral expan-
sion of these form factors at the next-to-leading order (NLO) in terms of the expansion parame-
ters ξ =M2 /(4πF )2 and t, we have only single free parameter Lr, which is one of the
{π,K} {π,K} π 9
2
Chiralbehavioroflightmesonformfactorsin2+1flavorQCDwithexactchiralsymmetry T.Kaneko
1.00
0.00
m = 0.050, m = 0.080
l s
m = 0.035, m = 0.080
l s
m = 0.025, m = 0.080
l s
0.95 m = 0.015, m = 0.080
+K(t)FV l s +K(t)FV,n,X-0.05 ml = 0.01N5,L mOs (=n 0=. 028)0
0.90 NLO, X = L
NLO, X = B
NNLO (n = 4)
-0.10 NNLO, X = L
NNLO, X = C
0.85 NNLO, X = B
-0.10 -0.05 0.00 -0.10 -0.05 0.00
t [GeV2] t [GeV2]
Figure1: Leftpanel: NNLOChPTfittoK+ EMformfactor. DataatdifferentM ’sareplottedindifferent
π
symbolsasafunctionoft. Rightpanel:NLOandNNLOcontributions(thicklines)andtheirdecomposition
intoLEC-dependentandindependentparts(thinlines)at(m,m )=(0.015,0.080). Theblueandredlines
l s
showtheNLOandNNLOcontributions,respectively.
low-energy constants (LECs) in the NLO chiral Lagrangian. Many more LECs appear at NNLO:
Lr and Cr’s in the NLO and NNLO Lagrangians. Since Lr have been well studied in
{1,...,5} i {1,...,5}
phenomenologicalstudiesofexperimentaldataandtheyappearonlyinthepossiblysmallNNLO
corrections, we fix them to a recent phenomenological estimate in Ref. [5]. In general, Cr’s are
i
poorlydeterminedinphenomenologyandhavetobedeterminedonthelattice.
LetuswritetheNNLOchiralexpansionas
FP(t) = FP +FP (t)+FP (t)+FP (t) (P=π+,K+,K0), (3.1)
V V,0 V,2 V,4 V,6
FP (t) = FP (t)+FP (t), FP (t)=FP (t)+FP (t)+FP (t). (3.2)
V,2 V,2,L V,2,B V,4 V,4,L V,4,C V,4,B
The terms FP , FP and FP are the leading-order, NLO and NNLO contributions, respectively.
V,0 V,2 V,4
We add an even higher order correction FP , when necessary. The additional subscripts “L”, “C”
V,6
and “B” represent Lr-dependent,Cr-dependent and LEC-independent parts, respectively. TheCr-
i i i
dependentNNLOanalytictermsFP aregivenas[6]
V,4,C
F4Fπ+ (t) = −4cr M2t−8cr M2t−4cr t2, (3.3)
π V,4,C π+,πt π π+,Kt K t2
F4FK+ (t) = −4cr M2t−4cr M2t−4cr t2, (3.4)
π V,4,C K+,πt π K+,Kt K t2
8
F4FK0 (t) = − cr (M2 −M2)t, (3.5)
π V,4,C 3 K0 K π
wherethecoefficientsarelinearcombinationsofCr’s. Thesearenotindependent: cr =cr +
i K+,πt π+,Kt
cr /3andcr =cr +cr −cr /3. Wethereforetreatthefollowingfourasfittingparame-
K0 K+,Kt π+,πt π+,Kt K0
ters
cr = 4Cr +4Cr +2Cr +Cr +Cr +2Cr , (3.6)
π+,πt 12 13 63 64 65 90
cr = 4Cr +Cr , cr =Cr −Cr , cr =2Cr −Cr . (3.7)
π+,Kt 13 64 t2 88 90 K0 63 65
AnexampleoftheNNLOChPTfitforthechargedkaonEMformfactorFK+ isplottedinthe
V
leftpanelofFig.1. Weobtainχ2/d.o.f=1.8and
L9r = 4.6(1.1)stat(cid:0)+−00..15(cid:1)Lir(0.4)a=0×10−3, ctr2 =−6.4(1.1)stat(0.1)Lir(0.5)a=0×10−5, (3.8)
3
Chiralbehavioroflightmesonformfactorsin2+1flavorQCDwithexactchiralsymmetry T.Kaneko
where the renormalization scale is set to µ=M . The second error is the uncertainty due to the
ρ
choiceoftheinputLr ,whereasthethirdisthediscretizationerrorestimatedbypowercount-
{1,...,5}
ing O((aΛ )2)∼8% with Λ =500 MeV. These results are reasonably consistent with the
QCD QCD
recentphenomenologicalestimateinRef.[5]. Whileotherfitparametersarepoorlyknowninphe-
nomenology,ourresultsareroughlyconsistentwithanaivepowercountingCr=O((4π)−4)[4].
i
An interesting observation in the right panel of Fig. 1 is that the non-trivial chiral correction
FK++FK+ is largely dominated by the NLO analytic term FK+ . Note that this term is not un-
V,2 V,4 V,2,L
expectedly large, because our result for Lr is consistent with the phenomenological estimate as
9
well as a power counting Lr=O((4π)−2). Since this term is independent of the valence quark
i
masses (Fπ+ =FK+ =2Lrt/F2), the NNLO chiral expansion of the charged meson EM form
V,2,L V,2,L 9 π
{π+,K+}
factorsF showsreasonableconvergence. Thisishowevernotthecasefortheneutralkaon,
V
becauseFK0 vanishestosatisfytheconstraintFK0(t)=0atm =m .
V,2,L V l s
WeobtainthefollowingresultsforthechargeradiifromtheNNLOChPTfit
(cid:104)r2(cid:105)π+ = 0.458(15) (cid:0)+9(cid:1) (37) fm2, (cid:104)r2(cid:105)K+ =0.380(12) (cid:0)+7(cid:1) (31) fm2, (3.9)
V stat −1 Lr a(cid:54)=0 V stat −1 Lr a(cid:54)=0
i i
(cid:104)r2(cid:105)VK0 = −0.055(10)stat(1)Lir(4)a(cid:54)=0 fm2. (3.10)
Theseareinreasonableagreementwiththeexperimentalvalues(cid:104)r2(cid:105)π+=0.452(11)fm2,(cid:104)r2(cid:105)K+=
V V
0.314(35)fm2 and(cid:104)r2(cid:105)K0=−0.077(10)fm2 [7].
V
4. Kaonsemileptonicformfactors
InFig.2,wedemonstrateaconventional
determinationofthenormalization f (0). At
+
1.00
simulated M ’s, we fix f (0) assuming the
π +
followingt dependenceof f (t)and f (t)
+ 0
(cid:26) 1 (cid:27) f(0)+0.98
f (t) = f (0) +a t , (4.1)
+ + 1−t/M2 +
K∗ m = 0.080
f (t) = f (0)(cid:0)1+a t+b t2(cid:1), (4.2) ms = 0.060
0 + 0 0 s
0.96 average (N = 3)
f
average (N = 4)
f
basedonthevectormesondominancehypoth-
0.0 0.1 0.2 0.3
esis. The results are well described by the M2 [GeV2]
π
NNLO ChPT formula with χ2/d.o.f.∼0.2. Figure 2: Extrapolation of f+(0) as a function of Mπ2
The extrapolated value f+(0)=0.644(4)stat and MK2 based on NNLO ChPT. Circles and squares
show our data at m =0.080 and 0.060, respectively.
is consistent with the average of recent re- s
The diamond represents the value extrapolated to the
sults [8, 9, 10] by the Flavor Lattice Aver-
physical point (m ,m ). We also plot the pre-
l,phys s,phys
aging Group (FLAG) [11, 12]. We note that
liminaryFLAGaveragesforN =3and4bytriangles.
f
theserecentstudiesarepursuingmoreprecise
determinationbysimulationsnearordirectly
at the reference point t=0 and the physical
point(m ,m ). Notealsothatastudyoftheformfactorshapebasedonaphenomenologi-
l,phys s,phys
calparametrizationisalsoreportedatthisconference[13].
4
Chiralbehavioroflightmesonformfactorsin2+1flavorQCDwithexactchiralsymmetry T.Kaneko
11..0005 mmmmllll ==== 0000....000053550500,,,, mmmmssss ==== 0000....000088880000 1.05 mmmmllll ==== 0000....000055550000,,,, mmmmssss ==== 0000....000088880000
f(t)+0.95 f(t)0
1.00
0.90
0.85
0.95
-0.10 -0.05 0.00 0.05 -0.10 -0.05 0.00 0.05
t [GeV2] t [GeV2]
Figure3: NNLOChPTfitto f (t)(leftpanel)and f˜(t)(rightpanel)asafunctionoft. Differentsymbols
+ 0
showdataatdifferentM ’s.
π
Weemployadifferentstrategyexploitingexactchiralsymmetry: wefitthelatticedataof f
+
and f totheirNNLOChPTformulaasafunctionofM2 andt. SimilartotheEMformfactors,
0 {π,K}
the chiral expansion of the vector form factor f [14, 15] has only Lr at NLO and other Lr’s at
+ 9 i
NNLO.WefixLr tothevalueobtainedinouranalysisoftheEMformfactors,whereasothersare
9
set to the phenomenological estimate [5]. The analysis of the EM form factors provides helpful
information also for the NNLO LECs. The coefficient cr in Eqs. (3.3)–(3.4) also appears in the
t2
NNLOanalytictermof f as
+
F4f = cr (M2 −M2)+cr M2t+cr M2t−4cr t2. (4.3)
π +,4,C +,πK K π +,πt π +,Kt K t2
OthertwocoefficientscanbewrittenintermsofthosefortheEMformfactors
(cid:16) (cid:17)
cr = −4(2Cr +4Cr +Cr +Cr +Cr )=−2 cr +cr −cr , (4.4)
+,πt 12 13 64 65 90 π+,πt π+,Kt K0
(cid:16) (cid:17)
cr = −4(2Cr +8Cr +2Cr +2Cr +Cr )=−2 cr +3cr +cr . (4.5)
+,Kt 12 13 63 64 90 π+,πt π+,Kt K0
The remaining one cr =−8(Cr +Cr ) describes SU(3) breaking effects at t=0, and hence
+,πK 12 34
is absent in the EM form factors. Therefore we have only one free parameter cr in the chiral
+,πK
extrapolationof f .
+
The scalar form factor f has many additional NNLO LECs. In order to carry out a chiral
0
extrapolationwithlessfreeparameters,weconsiderthefollowingquantityproposedinRef.[15]
(cid:18) (cid:19)
t F
f˜(t) = f (t)+ 1− K . (4.6)
0 0 M2 −M2 F
K π π
TheDashen-Weinsteinrelation[16]suggestsalargecancellationbetweentheNNLOanalyticterms
of f andF /F . Infact,theNNLOanalytictermof f˜ isgiveninarathersimpleform
0 K π 0
F4f˜ (t) = cr (cid:0)M2 −M2(cid:1)+(cid:0)8Cr −cr (cid:1)(cid:0)M2 +M2(cid:1)t−8C t2, (4.7)
π 0,4,C +,πK K π 12 +,πK K π 12
andasimultaneousfitto f and f˜ hasonlytwofittingparameterscr andCr .
+ 0 +,πK 12
Figure 3 shows the NNLO ChPT fit to f (t) and f˜(t) as a function of t. Our data are well
+ 0
describedwith χ2/d.o.f.∼0.7. SimilartotheEMformfactors,thet dependenceof f isroughly
+
5
Chiralbehavioroflightmesonformfactorsin2+1flavorQCDwithexactchiralsymmetry T.Kaneko
approximatedbytheNLOanalyticterm f =FK+ . Ontheotherhand, f˜ hasnoNLOanalytic
+,2,L V,2,L 0
termduetothecancellationbetween f andF /F ,andshowsarathermildt dependence.
0 K π
Fromthesimultaneousfitto f and f˜,weobtain
+ 0
f (0) = 0.9636(36) (cid:0)+41(cid:1) (29) =0.9636(cid:0)+62(cid:1) (4.8)
+ stat −45 chiral a(cid:54)=0 −65
at the physical point. The first error is statistical. The second is the systematic uncertainty of the
chiral extrapolation, which is estimated by repeating the fit including higher order corrections or
using different values for the input Lr . The third one is the discretization error estimated by
{1,...,8}
powercounting. Thetotaluncertaintyisatthelevelof≤1%.
An advantage of our analysis method is
4.0
thatwecanstudyboththenormalizationand m = 0.080
s
m = 0.060
shapeoftheformfactorsbyauniquefitbased s
CKM2014
on NNLO ChPT. In Fig. 4, we plot the slope 3.0 NLO
λ+(cid:48) inthequadraticparametrization 2x 10
(cid:40) (cid:41) λ′+
λ(cid:48)
f (t) = f (0) 1+ + t+O(t2) . (4.9) 2.0
+ + M2
π±
Namely,
1.0
0.0 0.1 0.2 0.3 0.4
λ+(cid:48) = Mfπ2+±(,p0h)ysdfd+t(t)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.10) Figure4: Slopeλ+(cid:48) asafMuπn2 c[GtieoVn2]ofMπ2. Thesolidline
t=0 is reproduced from our NNLO ChPT fit, and the sta-
The NNLO contribution turns out to be sig- tistical error is shown by the dotted lines. The circles
nificant even near the physical point. It is and squares represent the values estimated at simula-
tionpointsbyassumingthepolynomialparametrization
therefore important to study the form factor
(4.9). ThedashedlineshowstheNLOcontribution.
shapebytakingaccountoftheirnon-analytic
chiralbehavioratNNLO.
FromtheNNLOChPTfit,weobtainλ(cid:48) =3.08(14) (31) ×10−2andλ(cid:48)=1.98(15) (44) ×
+ stat sys 0 stat sys
10−2, where we add the uncertainty of the chiral extrapolation and the discretization error in
quadrature. This is consistent with recent experimental measurements, λ(cid:48) =2.58(7)×10−2 and
+
λ(cid:48)=1.37(9)×10−2 [17] within 2σ. The largest uncertainty comes from the discretization for λ(cid:48)
0 +
andthechiralextrapolationforλ(cid:48).
0
5. Summary
In this article, we have presented our lattice calculation of the light meson EM and semilep-
tonicformfactors. Theseformfactorsarepreciselycalculatedbyusingtheall-to-allquarkpropa-
gator. TheirchiralbehaviorisdirectorycomparedwithcontinuumChPTbyexploitingexactchiral
symmetrypreservedwiththeoverlapquarkaction.
We observe that the lattice data of the charged meson EM form factors are reasonably well
describedbytheirNNLOChPTformula,andestimatetherelevantNLOandNNLOLECs. These
resultsareusedinthechiralextrapolationforthekaonsemileptonicformfactors,andtheirnormal-
ization f (0)isdeterminedwithsub-%accuracy. Weconfirmreasonableconsistencyoftheform
+
factorshape,namelythechargeradiiandslopesofthesemileptonicformfactors,withexperiment.
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Chiralbehavioroflightmesonformfactorsin2+1flavorQCDwithexactchiralsymmetry T.Kaneko
Oneofthelargestuncertaintyisthediscretizationerroratafinitelatticespacing. Itisimportant
to extend this study to finer lattices and heavy flavor physics. Simulations in these directions are
underway[18]usingacomputationallycheaperfermionactionwithgoodchiralsymmetry[19].
WethankJohanBijnensformakinghiscodetocalculatepionandkaonformfactorsinNNLO
ChPTavailabletous. NumericalsimulationsareperformedonHitachiSR16000andIBMSystem
BlueGeneSolutionatKEKunderasupportofitsLargeScaleSimulationProgram(No.15/16-09),
andonSR16000atYITPinKyotoUniversity. ThisworkissupportedinpartbytheGrant-in-Aid
of the MEXT (No. 25287046, 26247043, 26400259 and 15K05065) and by MEXT SPIRE and
JICFuS.
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