Table Of ContentNucleon mass and strange content from (2+1)-flavor
Domain Wall Fermion
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Chulwoo Jung∗†
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a PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,U.S.A.
J E-mail: [email protected]
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2 for RBC and UKQCD Collaborations
]
t
a We report on our ongoing study of nucleon properties, the nucleon mass and the strange quark
l
- content ((cid:104)N|s¯s|N(cid:105)) in particular. These calculations are done on (2+1)-flavor Domain Wall
p
e Fermion(DWF)ensembleswithIwasakigaugeactionwith2differentlatticespacings(a∼0.08,
h
0.11fm)andDWFwithDislocationSuppressingDeterminantRatio(DSDR)(a∼0.14fm),gen-
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eratedbyRBC/UKQCDcollaborations.
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The30thInternationalSymposiumonLatticeFieldTheory
June24-29,2012
Cairns,Australia
∗Speaker.
†We thank RBC and UKQCD Collaborations, especially Yasumichi Aoki, Tom Blum, Chris Dawson, Taku
Izubuchi,MeifengLin,ShigemiOhta,ShoichiSasakiandTakeshiYamazaki.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Nucleonmassandstrangecontentfrom(2+1)-flavorDomainWallFermion ChulwooJung
1. Introduction
StrangecontentofNucleon((cid:104)N|s¯s|N(cid:105))andrelatedquantities
m (cid:104)N|s¯s|N(cid:105) dm m
s N s
f = = × , σ =m (cid:104)N|s¯s|N(cid:105) (1.1)
Ts m dm m s s
N s N
has been drawing much attention in the last few years, due to its implications to the dark matter
candidatesearch[1]. Therehavebeenanumberofstudiesof(cid:104)N|s¯s|N(cid:105)onthelatticeusingdifferent
approaches such as direct measurement of (cid:104)N|s¯s|N(cid:105) matrix element with various noise reduction
techniquesfordisconnecteddiagrams[2,3,4],indirectmeasurementsfromfittingtoChPTformu-
las for the nucleon mass [5, 6, 7], as well as numerical derivative via Feynman-Hellman theorem
[8,9]. Asurveyoftherecentresultsfromlatticeshowsthereisnoticeabledisagreementsbetween
studieswithrelativelysmallererrorswhileotherstudieswithmultiplelatticespacingsoftenhave∼
100%relativeerrors. Thissuggeststhesystematicerrorsin(cid:104)N|s¯s|N(cid:105)maynotbewellunderstood.
Herewepresenttheresultof(cid:104)N|s¯s|N(cid:105)measurementsfromnumericalderivativeviareweight-
ing[10,11]. Strangequarkreweightinghasbeensuccessfullyusedtoeliminatethesystematicerror
from the discrepancy between dynamical strange quark mass (m ) and the physical strange quark
s
mass,andhasshowntobeusableinshiftingm by∼20%forthelatticevolumesstudied[12,13].
s
2. MeasurementDetails
Table1showsthedetailsoftheensemblesandmeasurementsusedinthisstudy. Weused(2+1)
dynamicalflavorDWFwithIwasakigaugeactiongeneratedbyRBC/UKQCDCollaborationswith
a∼0.08,0.11fm[12],whichwewillrefertoasDWF+I0.08fmand0.11fmensemblerespectively.
We also measured nucleon mass and (cid:104)N|s¯s|N(cid:105) on the DWF ensemble generated with Dislocation
Suppressing Determinant Ratio(DSDR [14]) in addition to the Iwasaki gauge action (DWF+ID).
Quantities necessary for continuum extrapolation of (cid:104)N|s¯s|N(cid:105) such as lattice spacing and mass
renormalization constant were calculated by a combined fit of DWF+I and DWF+ID ensemble
describedin[13].
ForDWF+I0.11fmensemble,itwasobservedin[12]thatthecombinationofCoulombgauge-
fixed box source of size 16 and the point sink gives a good overlap with the ground state. We
extendedthemeasurementtouse8differentsourcepositionsperconfiguration,givenby
(x,y,z,t)= (0−16,0−16,0−16,n×8)forn=even,
(2.1)
(12−28,12−28,12−28,n×8)forn=odd, n=0···7
Whereperiodicityisimplied,sox=12-28meansx>=12orx<=4. Thesemeasurementswerebinned
perconfiguration,togetthemostofthesignalwhileavoidingthepossibilityofunderestimatingthe
autocorrelation. EigCG[15]wasemployedtodecreaseamountofcomputationneededfor8×12=
96inversionsperconfiguration,resultinginuptoafactorof3decreaseintheiterationnumberfor
thelightestmassm =0.005.
l
ForDWF+I0.08fmensemble,weusedpropagatorsgeneratedbyLHPCwithagaussiansource
fortheirstudyofnucleonmatrixelements[16]. AsitisshowninFigure3,thenucleonpropagator
tothepointsinkexhibitsaveryslowconvergencetotheasymptoticvalueforthenucleonmassand
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Nucleonmassandstrangecontentfrom(2+1)-flavorDomainWallFermion ChulwooJung
this proved to be particularly problematic when combined with strange quark mass reweighting.
To circumvent this, propagators with unitary sink, which give consistent values and similar errors
forunreweightednucleonmasseswithpointsinksbutwithsmallert ,wereused.
min
Figure4showstheeffectivemassandreweightedmassinDWF+ID(a−1∼0.14fm)ensemble.
It is worthwhile to note that effective masses from both point and unitary sink shows a very good
agreement,incontrasttoDWF+I0.08fm.
Figure 1 shows the nucleon mass as a function of pseudoscalar masses squared in physical
units. While the lack of measurements with other quantities such as the masses of other octet
baryons does not allow us to fit the data with ChPT-motivated fitting forms with any confidence,
thesmallnessofdifferencesinmassesbetweendifferentensemblessuggestsa2 dependenceofthe
nucleonmassisrelativelysmall.
am am MDunits measurements t −t am
l π min max N
DWF+I0.11fm:V =243×64×16,a−1=1.75(3)Gev,am ∼0.003,
res
am(cid:48) =0.032,0.033,···0.04,0.041,···0.047
s
0.005 0.1891(10) 1420,1460···8980 1520 0.652(2)
0.010 0.2420(7) 1460,1500···8540 1424 5-12 0.703(2)
0.020 0.322(1) 1900,1920···3600 680 0.794(3)
DWF+I0.08fm:V =323×64×16,a−1=2.31(4)Gev,am ∼0.00067,
res
am(cid:48) =0.0265,0.0270···0.03.
s
0.004 0.1267(4) 590,600···6600 (1996) 0.474(3)
0.006 0.1509(3) 544,552···7600 3528 7-15 0.503(2)
0.008 0.1725(5) 590,600···6600 2064 0.524(2)
DWF+ID:V =323×64×32,a−1=1.37(1)Gev,am ∼0.0018,
res
am(cid:48) =0.045,0.0455···0.05.
s
0.01 0.1250(2) 500,508···2396 (904) 0.718(6)
5-11
0.042 0.1810(2) 608,616···1920 1320 0.769(5)
Table1:Measurementdetailsofensemblesusedintheanalysis.Parenthesesinthenumberofmeasurements
denotesthemeasurementsarenotcomplete. Underlinesforam(cid:48) denotesthesimulatedstrangequarkmass.
s
3. (cid:104)N|s¯s|N(cid:105)viareweighting
NucleonstrangecontentiscalculatedbyapplyingFeynman-Hellmantheorem
dM (m(cid:48))
(cid:104)N|s¯s|N(cid:105)= N s (3.1)
Z dm(cid:48)
m s
tonucleonmassesforreweightedseastrangemassM (m(cid:48)),whichiscalculatedbyfittingtheeffec-
N s
tivemassescalculatedfromreweightednucleonpropagatorP(t,m(cid:48))toaconstantfort=t ···t
s min max
foreachensemble.
P(t,m(cid:48)) is calculated by taking the weighted average of the nucleon propagator P(t,m ) with
s s
the weightW(m(cid:48),m ), which is calculated by multiplying reweighting factors for smaller steps in
s s
3
Nucleonmassandstrangecontentfrom(2+1)-flavorDomainWallFermion ChulwooJung
1.4 DWF+I 0.11fm
DWF+I 0.08
DWF+ID 0.14
1.3
v) 1.2
e
G
(N
M
1.1
1
0.9
0 0.05 0.1 0.15 0.2 0.25 0.3
2 2
m (Gev )
p
Figure1: Nucleonmassesoftheensemblesusedinthisanalysisinphysicalunits.
mass
(cid:10)P(t,m(cid:48))(cid:11)= ΣiP([Ui],t,ms)Wi(m(cid:48)s,ms), W(m(cid:48),m )=w(m(cid:48) =m ,m )···w(m ,m =m ).
s ΣW(m(cid:48),m ) i s s i s n n−1 i 2 1 s
i i s s
Using smaller steps decreases the stochastic noise in the reweighting factor, and also avoids pos-
sible difficulties with non-gaussian distribution of the estimated reweighting factor. Reweighting
factors w are calculated by estimating the ratio of strange quark determinants with Gaussian ran-
i
domvectorξ.
Ω(m(cid:48),m )=D(cid:48)−1DD†(D(cid:48)†)−1,D =D([U],m ,m ),D(cid:48)=([U],m ,m(cid:48)),
i n n i i i i i i l n i i l n
√
wi(m(cid:48)n,mn)=det(Ωi(m(cid:48)n,mn))−1/2= (cid:82) dξd(cid:82)ξd†ξe−dξe−Ωξi(†mξ(cid:48)n,mn)ξ =(cid:68)e−ξ†(√Ωi(m(cid:48)n,mn)−1)ξ(cid:69)ξ. (3.2)
(cid:112)
Ω(m(cid:48),m )iscalculatedusingrationalapproximation,similartowhatisusedforthesimulation
i n n
ofstrangequarkin[12,13]. UsingrationalapproximationmakesEq.(3.2)unbiasedandallowsus
touserelativelynoisyestimatesofthereweightingfactorsforeachmasssteps.
(cid:104)N|s¯s|N(cid:105)foreachensembleiscalculatedbycorrelatedfitsto
M (m(cid:48),m ,a)=c(cid:48) +(cid:104)N|s¯s|N(cid:105)(m ,a)m(cid:48), (3.3)
N s l 0 l s
thenthecontinuumandchiralextrapolationistakenbyfittingto
(cid:104)N|s¯s|N(cid:105)(m ,a)=c +c m (+c a2). (3.4)
l 0 1 l 2
Weusedfitsbothwithandwithoutc a2 inEq.(3.4),toestimatesystematicerrorfromthecontin-
2
4
Nucleonmassandstrangecontentfrom(2+1)-flavorDomainWallFermion ChulwooJung
uumextrapolation.
0.85 0.85 0.02
0.02 0.010
0.01 0.005
0.005
0.8 0.8
0.75 0.75
a M a M
N N
0.7 0.7
0.65 0.65
0.6 0.6
0 5 10 0.025 0.03 0.035 0.04 0.045 0.05 0.055
t a ms
Figure2: NucleoneffectivemassandreweightedmassforDWF+I0.11fmensembles
0.008
0.006
0.004
0.55 0.55
a M a M
N N
0.5 0.5
0.008(point)
0.008(unitary)
0.006(point)
0.006(unitary)
0.004(point)
0.45 0.004(unitary) 0.45
5 10 0.024 0.026 0.028 0.03 0.032
t a ms
Figure3: NucleoneffectivemassandreweightedmassforDWF+I0.08fmensembles
Figures2to4showsthereweightednucleonmassM (m(cid:48)),aswellasthefittingrangeusedin
N s
extracting(cid:104)N|s¯s|N(cid:105)(m ,a)forDWF+IandDWF+IDensembles. Apreliminaryextrapolationtothe
l
continuumlimitandphysicalpionmassgives(cid:104)N|s¯s|N(cid:105)=0.33(31)at2GevwithMS whilefitting
withouta2 gives0.09(16),whichinturngives
f =0.035(33)(witha2), 0.009(17)(witouta2). (3.5)
Ts
4. Discussion
Wereportedonthemassandthestrangequarkcontentofthenucleon,measuredonthe(2+1)-
flavor dynamical DWF ensembles generated by RBC/UKQCD collaborations. Having DWF+I
ensembleswith2differentlatticespacingsallowsustodoacontinuumextrapolation. Comparing
the continuum value from fits with and without a2 terms suggests the lattice spacing error could
belargerthanpreviouslyestimated, andcouldexplaintheapparentdiscrepancybetweendifferent
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Nucleonmassandstrangecontentfrom(2+1)-flavorDomainWallFermion ChulwooJung
0.9 0.9
0.8 0.8
a M a M
N N
0.7 0.7
0.001(point) 0.001
0.001(unitary) 0.0042
0.0042(point)
0.0042(unitary)
0.6 0.6
5 10 0.042 0.044 0.046 0.048 0.05 0.052
t a ms
Figure4: NucleoneffectivemassandreweightedmassforDWF+IDensembles
DWF+I 0.11fm
DWF+I 0.08fm
DWF+ID
fit with a^2
1 fit without a^2
>
N
s|
s
N|
<
0
Preliminary
-1
0 10 20 30 40 50
m (MeV)
l
Figure5: NucleonstrangecontentscalculatedviaFeynman-Hellmantheoremandmassreweighting,renor-
malizedat2Gev.
lattice studies. While we did generate new propagators on DWF+I 0.11fm ensembles, this results
presentedinthispaperwereobtainedbymostlyre-usingpropagatorsandreweightingfactorsgen-
eratedforotherstudies[12,13,16].
We also analyzed DWF+ID ensembles generated with Dislocation Suppressing Determinant
Ratio(DSDR),whichcurrentlyhasthelightestpionsandlargestvolumesfordynamicalDWFen-
sembles. The results with the currently available statistics are still too noisy and the results were
6
Nucleonmassandstrangecontentfrom(2+1)-flavorDomainWallFermion ChulwooJung
notincludedinthecontinuumextrapolationof(cid:104)N|s¯s|N(cid:105).
It appears quite possible that further optimization of sources for the measurement of nucleon
mass as well as increased statistics can improve the signal for (cid:104)N|s¯s|N(cid:105) significantly without the
needtoextendtheensemble. Whilegeneratingpropagatorswithdifferentsourcesand/orpositions
traditionallyhasrequiredcompletelyseparateinversionsandmadethenumericalcostprohibitively
expensive,recentdevelopmentsofvarioustechniquessuchasEigCG[15]. Low/Allmodeaverag-
ing(LMA/AMA) [17] makes it possible to generate a large number of propagators per configu-
rations at a numerical cost only a few times the cost for single propagator. Nucleon studies are
particularly well suited to take advantage of this, as the correlation length for nucleons are much
shorterthanthoseforthelightmesons. AnexploratorystudywithAMAisunderway.
C.JwassupportedbytheUSDOEundercontractDE-AC02-98CH10886. Thegaugeconfigu-
rationsusedinthisstudyweregeneratedatQCDOCmachinesatBrookhavenNationalLaboratory
and Edinburgh Parallel Computing Center(EPCC), and IBM BG/P machines at Argonne Leader-
shipComputingFacility(ALCF).ThereweightingfactorsandpropagatorsweregeneratedatALCF,
RIKENIntegratedClusterofClusters(RICC)andNSFTeragrid/XSEDEfacilities.
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