Table Of ContentApplications of the Stochastic LapH Method
J. Bulava
CERN,Geneva23,CH-1211,Switzerland.∗
E-mail:[email protected]
J. Foley
UniversityofUtah,SaltLakeCity,UT84112,U.S.A.
E-mail: [email protected]
K. J. Juge†
3 UniversityofthePacific,Stockton,CA95211,U.S.A.
1
E-mail: [email protected]
0
2
C. J. Morningstar
n
CarnegieMellonUniversity,Pittsburgh,PA15213,U.S.A.
a
J E-mail: [email protected]
2
2 B. Fahy
CarnegieMellonUniversity,Pittsburgh,PA15213,U.S.A.
]
t E-mail: [email protected]
a
l
- Y-C. Jhang
p
CarnegieMellonUniversity,Pittsburgh,PA15213,U.S.A.
e
h E-mail: [email protected]
[
D. Lenkner
2
CarnegieMellonUniversity,Pittsburgh,PA15213,U.S.A.
v
2 E-mail: [email protected]
5
4 C. H. Wong‡
4
UniversityofCalifornia,SanDiego,CA92093,U.S.A.
.
1 E-mail: [email protected]
0
3 ProgressincomputingthehadronspectruminlatticeQCDusingstochasticLapHquarkpropaga-
1 torsisdescribed.ThestochasticLapHalgorithmisaparticularquarksmearingalgorithmthatalso
:
v allowsthecomputationofall-to-allquarkpropagators. All-to-allquarkpropagatorsarerequired
i
X in our approach of using a large set of spatially extended hadron operators and explicit multi-
particleoperatorstoaccessexcitedstates. Wereportontheprogressmadeinthevariousisospin
r
a channelson2+1dynamical,anisotropiclatticesgeneratedbytheHadronSpectrumCollaboration.
CERN-PH-TH/2013-010
TheXXXInternationalSymposiumonLatticeFieldTheory-Lattice2012
June24-29,2012
Cairns,Australia
∗Presentaddress:SchoolofMathematics,TrinityCollege,Dublin2,Ireland.E-mail:[email protected]
†Speaker.
‡Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Appl. ofstochasticLapH C.H.Wong
1. Introduction
One of the goals of our collaboration has been to compute, from first principles lattice QCD
simulations, the low-lying hadron spectrum in various symmetry channels in hopes to aid our un-
derstandingofQCDthroughcomparisonswithlowenergyeffectivetheories. Thereliableextrac-
tionoftheenergiesoftheexcitedstatesineachsymmetrychannelrequiresthatweuseinterpolating
operatorswhichcouplewellwiththegroundstatesinthechannelandalsoreasonablywellwiththe
excitedstatesuptotheenergyscalethatoneisinterestedin. Theprogressmadeinthelastcouple
ofyearsforsingle-particleandsometwo-particleoperators/correlationfunctionshaveappearedin
variouspapersandproceedings([1]-[7]).
Theextractionofexcitedenergylevelsondynamicalconfigurationsmustconfronttheproblem
of mixing of single-particle states with multi-particle states and requires new techniques to be
developed given the finite resources available. The construction of a set of good interpolating
operators which have good signal-to-noise ratio is crucial in computing the excited states through
the variational method as the signal for the heavier excited states decay rapidly into the noise,
destabilizingthediagonalizationprocedure. Inthiscontribution,wereportonourprogresstowards
solvingtheseissues.
2. Lattices
Thedynamicallatticesthatweusearetheclover-improved,anisotropiclattices([8])thathave
been generated by the HadSpec collaboration [9]. The anisotropy not only allows one to extract
energies at early timeslices (as long as the interpolating operators are effective) before the signal-
to-noiseratiodeterioratesbutalsogenerallyhassmallerfluctuations. Therehasonlybeenasingle
lattice spacing generated so far corresponding to r /a =3.221(25). The anisotropy was tuned to
0 s
3.5withameasuredvaluedof3.3usingthepiondispersionrelation. Thequarkmassusedinthis
studycorrespondstopionmassesof240MeV(250configurations).
3. InterpolatingOperators
The interpolating operators that are used to form the correlation matrix for digaonalization is
constructed so that they transform irreducibly under the lattice symmetry group of rotations and
translations and have the appropriate flavor structure and G-parity. Operators for hadrons at rest
can be constructed from a handful of conventional quark propagators as long as the displacement
typesarerestrictedtothesimpledisplacements. Theinterpolatingoperatorsthatwewillbeusing
are based on the irreducible representations of the appropriate cubic point groups and the pruning
of these types of operators have been studied in detail in previous studies. The choice of the
elemental operators to use were based on its simplicity and diversity while maintaining the best
overlapandlownoise-levelsinthesingleparticlecorrelationfunctions. Apreliminarystudyofthe
isovector meson spectrum was presented last year at the lattice conference for the heavier quark
masscorrespondingtom =390MeV.
π
Thesameapproachistakenindesigning/pruningtheinterpolatingoperatorsforhadronswith
non-zero momenta, k (in units of 2π/L). In this study, we consider those states which have mo-
mentaoftheform,k=(0,0,n),k=(0,n,n)andk=(n,n,n). Thegrouptheoryforthesechoices
2
Appl. ofstochasticLapH C.H.Wong
ofmomentawasdiscussedintheconferencereportfromlastyear[1]andthegeneraldiscussionof
thegrouptheoryisgivenin[10].
3.1 QuarkPropagators
In order to incorporate hadron operators with non-zero momentum and explicit two-particle
operatorsinouroperatorset,weconstructthehadroninterpolatingoperatorsfromstochasticLapH
quark propagators ([11]). The stochastic LapH algorithm is an efficient all-to-all extension of the
distillationalgorithm([12])whichdoesnotsufferfrompoorvolumescalingbehaviour. Themethod
allowsonetoinjectanymomentumintotheinterpolatingoperatorsafterthepropagatorshavebeen
generated, the construction of extended operators and it also separates the source operator from
the sink operator which greatly simplifies the construction of large correlation matrices needed to
extracttheexcitedstatespectrum.
3.1.1 StochasticLapHQuarkPropagators
Inthissection,webrieflyreviewthestochasticLapHalgorithm. Thesourcevector(forquark
labelA)isgivenby
(cid:16) (cid:17)
ρ(A)[d](x,t)= D˜(p¯)V P[d]ρ(A)(x,t)
bβkp¯ k S
bβ
wherebisthecolorindex,β isthespinindex,D˜(p¯)isthecovariantderivativeinthekdirectionwith
k
separation p¯, V is the matrix of eigenvectors of the Laplacian and P[d] is the projection operator
S
forthedilutionschemebeingapplied. Thedilutionschemescanbecategorizedintofourdifferent
types:
[d]
P =δ , d=0, (nodilution)
ij ij
[d]
P =δ δ , d=0,...,N−1 (fulldilution)
ij ij di
[d]
P =δ δ d=0,...,K−1, (block-K)
ij ij d,(cid:98)Ki/N(cid:99)
[d]
P =δ δ d=0,...,K−1, (interlace-K)
ij ij d,imodK
whereN isthedimensionofthespaceofdilutiontypeandN/K isaninteger.
(A)[d]
The solution vector, ϕ , is then computed by inverting Ω = γ M with this source. The
aαjp 4
solutionvectoristhensmearedwiththesmearingoperatoranddisplacedappropriately,
(cid:16) (cid:17)
ϕ(A)[d](x,t)= D(p)V V†Ω−1 ρ(A)[d]
aαjp j S S bβkp¯
aα;bβ
whereΩ=γ M andM istheDiracoperator. Thequarkpropagator(forlineA)isthengivenby
4
Nd
∑ϕ(A)[d](x,t)ρ(A)[d]†(x ,t ).
aαj bβk 0 0
d
3.2 MesonCorrelationFunctions
Mesoncorrelationfunctionsareformedbycorrelatingmesonoperatorsgivenby
Ml[d1d2](ρ1,ϕ2;p,t)=c(αlβ) ∑e−ip·(x+(dα+dβ))ρa[dα1](x,t;ρ1)∗ϕa[dβ2](x,t;ρ2)
x
3
Appl. ofstochasticLapH C.H.Wong
1.5e−06 0.005
G−ππ
S
G−G
S S
1e−06 0
5e−07 −0.005
0 −0.01
−5e−07 −0.015
0 10 20 30 0 10 20 30 40
t t/a
t
Figure1: Preliminaryresultsfortheglueballcorre- Figure 2: Preliminary results for the correlation
lation function on the 243 lattice at 240 MeV pion functionoftheglueballandπ-π mixingtermforthe
mass. 240MeVconfigurations.
whered areintroducedtomaintainG-paritywhenneeded. Thegeneralformofthemesoncorre-
α
latortakestheform(manyoftheindiceshavebeensuppressedinthisexpressionforbrevity),
(cid:68)
C (p,t −t )= −δ δ M[d1d2](ϕ¯ ,ϕ ;p,t )M[d1d2](ρ¯ ,ρ ;p,t )∗ (3.1)
ll¯ f 0 AA¯ BB¯ l 1 2 f l 1 2 0
(cid:69)
+δ δ M[d1d1](ρ ,ϕ ;p,t )M[d2d2](ϕ ,ρ ;p,t )∗
AB A¯B¯ l¯ 1 1 f l¯ 2 2 0
wherethesecondtermappearsforisoscalarmesons.
4. PreliminaryResults
In this section, we present a few selected preliminary results for particles in different isospin
sectors.
4.1 I=0
The isoscalar channel is one of the most challenging channels as there are disconnected
(“same-time”) diagrams and “box” diagrams appearing in the correlation function. The com-
pletely disconnected diagrams generally suffer from large gauge noise and hence is an ideal case
for stochastic estimation over ‘non-stochastic’ evaluations such as the distillation algorithm. The
“box diagrams” are just as difficult to evaluate but are perhaps not as noisy. The stochastic LapH
algorithmallowsthecomputationofallofthesediagramswhichareinvolvedinthischannelfrom
fourindependentnoisesources. Twonoisesourcesareneededjusttoconstructasinglepionopera-
torandsoattheexpenseofcomputingtwomorenoisevectors,wecanevaluateallofthediagrams
needed for the isoscalar channel. The scalar glueball operator comes for free as well because one
can construct it from the eigenvalues of the LapH operator. A glueball correlation function using
these LapH operators is shown in Fig. 2 on the 243×256 lattice with the 240 MeV pion mass.
We have studied the mixings between the scalar meson and the scalar glueball as well as scalar
4
Appl. ofstochasticLapH C.H.Wong
1000
ππ−f (Total)
0
ππ−f (Smt)
0
ππ−f (Box)
0
500
C(t)
0
−500
0 10 20 30 40
t/a
t
Figure3: Preliminaryresultsforthevariouscontributionstotheππ−f mixingterm.
0
mesonswiththeI =0ππ scatteringstate. Theoff-diagonalcorrelationfunctionsforthesediffer-
entinterpolatingoperatorsfortheI =0channelonthe243 latticesareshowninFig.2andFig.3.
4.2 I=1/2
Therearenodisconnecteddiagramsnoranyboxdiagramsthatneedstobeevaluatedforkaons.
ThegainfromthestochasticLapHalgorithmisthatdifferentmomentavaluesforthekaonscanbe
simulatedbyintroducingsimpleexponentialfactorsintothezeromomentumoperators. However,
therewillbeanextranoisesource/solutionpairneededatthestrangequarkmass. Ourpreliminary
results suggest that the introduction of the stochastic noise solely in the LapH subspace does not
make the determination of energies with finite momentum any harder than zero momentum states
withsimilarenergies. Weshowanexampleforthekaonwithmomentum(1,1,1)inlatticeunitsin
Fig.4.
4.3 I=1
The rho decay width has attracted attention in recent years as several groups have performed
large scale simulations at light quark masses ([13]-[15]). The lattice sizes and the quark masses
we have are limited as larger lattices with lighter masses are still being generated now. We have
generated many of the operators with finite momenta that will be needed for the analysis of this
channel. WeshowoneoftherepresentativeeffectivemassesinthischannelinFig.5.
4.4 I=2
The isospin-2 channel is an exotic channel and does not require any disconnected diagrams
to be evaluated. The correlation function can be evaluated by computing the product of two pion
propagators and the corresponding quark-exchanged diagrams. Since the finite momentum single
pion operators have already been computed, we can use many different ππ operators to reduce
the contamination from excited states. The use of many operators for this state is important in
5
Appl. ofstochasticLapH C.H.Wong
0.4
Kaon effective mass mom=<111>
0.3
eff0.2
M
at
0.1
0
0 10 20
t/a
t
Figure4: Theeffectivemassofthekaonwithmomentum(1,1,1)inlatticeunits.
a large volume because the level spacings decrease with increasing volume. We construct a cor-
relation matrix which is then diagonalized to find the optimized correlation function. We fit the
optimalcorrelatorforvarioust valuestocheckforstabilityinthefits. A“t plot”isshownin
min min
Fig.6afterdiagonalization. Thescatteringlengththatisextractedfromthefitsareconsistentwith
continuum two-loop chiral perturbation theory predictions [16], but the value is sensitive to the
choiceoftheoptimizationtimeslice. Weareexaminingthesystematicerrorsofchoosingdifferent
diagonalizationtimesandalsothesensitivitytot includedinthefits.
max
5. Summary
ThestochasticLapHquarksmearingalgorithmhasbeentestedforsingleparticlemesonstates
withvariousmomentaandisoscalarchannelsincludingthemixingoftheπ-π statewiththescalar
glueball and the scalar meson state. The elastic scattering length was estimated using Lüscher’s
formula and found to be consistent with large errors with continuum chiral perturbation theory.
The I = 2 exotic state was computed using various finite momenta pion operators and a result
consistentwithchiralperturbationtheorywasobtained. Thesystematicerrorsduetothedifferent
diagonalizationprocedurehasnotbeenfullyunderstoodyetandisasubjectofcurrentinvestigation.
Acknowledgements
ThisworkhasbeenpartiallysupportedbyNationalScienceFoundationawards
PHY-0510020,PHY-0653315,PHY-0704171,PHY-0969863,andPHY-0970137,andthrough
6
Appl. ofstochasticLapH C.H.Wong
0.18
A2m LSD mom=<011>
0.16
Meff 0.14
at
0.12
0.1
0 10 20
t/a
t
Figure5: Theeffectivemassoftheρ withmomentum(0,1,1).
TeraGrid/XSEDEresourcesprovidedbythePittsburghSupercomputerCenter,theTexas
AdvancedComputingCenter,andtheNationalInstituteforComputationalSciencesundergrant
numbersTG-PHY100027andTG-MCA075017.
References
[1] J.Foley,J.Bulava,Y.-C.Jhang,K.J.Juge,D.Lenkner,C.MorningstarandC.H.Wong,PoS
LATTICE2011,120(2011)[arXiv:1205.4223[hep-lat]].
[2] J.Bulava,PoSLATTICE2011,021(2011)[arXiv:1112.0212[hep-lat]].
[3] J.Bulava,J.Foley,Y.-C.Jhang,K.J.Juge,D.Lenkner,C.MorningstarandC.H.Wong,PoS
LATTICE2011,131(2011)[arXiv:1111.0845[hep-lat]].
[4] J.Bulava,J.Foley,Y.C.Jhang,D.Lenkner,C.J.Morningstar,C.H.Wong,K.J.Jugeand
M.J.Peardon,arXiv:1109.6614[hep-lat].
[5] J.Bulavaetal.[HadronSpectrumCollaboration],PoSLATTICE2010,110(2010)[arXiv:1011.5277
[hep-lat]].
[6] C.Morningstar,A.Bell,C.Y.Chen,D.Lenkner,C.H.Wong,J.Bulava,J.FoleyandK.J.Jugeetal.,
PoSICHEP2010,369(2010)[arXiv:1011.6573[hep-lat]].
[7] J.Foley,C.H.Wong,J.Bulava,K.J.Juge,D.Lenkner,C.MorningstarandM.Peardon,
arXiv:1011.0481[hep-lat].
[8] R.G.Edwardsetal.,Phys.Rev.D78,054501(2008)[arXiv:0803.3960[hep-lat]].
7
Appl. ofstochasticLapH C.H.Wong
0.24
I=2 fit results
0.19
Efit
0.14
at
0.09
0.04
0 10 20 30 40
t /a
min t
Figure6: At plotoftheenergiesextractedintheI=2channel.
min
[9] H.W.Linetal.[HadronSpectrumCollab.],Phys.Rev.D79,034502(2009)[arXiv:0810.3588
[hep-lat]].
[10] D.C.MooreandG.T.Fleming,Phys.Rev.D73,014504(2006)[Erratum-ibid.D74,079905(2006)]
[hep-lat/0507018].
[11] C.Morningstar,J.Bulava,J.Foley,K.J.Juge,D.Lenkner,M.PeardonandC.H.Wong,Phys.Rev.D
83,114505(2011)[arXiv:1104.3870[hep-lat]].
[12] M.Peardonetal.[HadronSpectrumCollab.],Phys.Rev.D80,054506(2009)arXiv:0905.2160
[hep-lat].
[13] S.Aokietal.[CSCollaboration],Phys.Rev.D84,094505(2011)[arXiv:1106.5365[hep-lat]].
[14] C.B.Lang,D.Mohler,S.PrelovsekandM.Vidmar,Phys.Rev.D84,054503(2011)
[arXiv:1105.5636[hep-lat]].
[15] X.Feng,K.JansenandD.B.Renner,Phys.Rev.D83,094505(2011)[arXiv:1011.5288[hep-lat]].
[16] G.Colangelo,J.GasserandH.Leutwyler,Nucl.Phys.B603,125(2001)[hep-ph/0103088].
8
