Table Of ContentExcited-State Hadrons using the Stochastic
LapH Method
C. Morningstar∗, A. Bell∗, J. Bulava†, J. Foley∗∗, K.J. Juge‡, D. Lenkner∗
and C.H. Wong∗
1
1 ∗Dept.ofPhysics,CarnegieMellonUniversity,Pittsburgh,PA15213,USA
0 †NIC,DESY,Platanenallee6,D-15738,Zeuthen,Germany
2 ∗∗PhysicsDepartment,UniversityofUtah,SaltLakeCity,UT84112,USA
n ‡Dept.ofPhysics,UniversityofthePacific,Stockton,CA95211,USA
a
J
Abstract. ProgressincomputingthespectrumofexcitedbaryonsandmesonsinlatticeQCDisde-
7
scribed.Largesetsofspatially-extendedhadronoperatorsareused.Anewmethodofstochastically
2
estimating the low-lyingeffects of quarkpropagationis utilized which allows reliable determina-
] tions of temporal correlations of both single-hadron and multi-hadron operators. The method is
t testedontheh ,s ,w mesons.
a
l
- Keywords: LatticeQCD,hadronspectroscopy
p PACS: 12.38.Gc,11.15.Ha,12.39.Mk
e
h
[
We are currently carrying out computations of the excitation spectrum of QCD in
1 finitevolumewithabinitioMarkov-chainMonteCarlo path integrationson anisotropic
v
space-timelattices.Ourfirstresultsusingtwoflavorsofdynamicalquarkswerereported
8
9 in Ref. [1], and our most recent results can be found in Ref. [2]. Such calculations are
3 very challenging. Computational limitations cause simulations to be done with quark
5
masses that are unphysically large, leading to pion masses that are especially heavier
.
1
than observed. The use of carefully designed quantum field operators is crucial for
0
accurate determinations of low-lying energies. To study a particular state of interest,
1
1 the energies of all states lying below that state must first be extracted, and as the
:
v pion gets lighter in lattice QCD simulations, more and more multi-hadron states lie
i below the masses of the excited resonances. The evaluation of correlations involving
X
multi-hadronoperators contains new challenges since not only must initial to final time
r
a quark propagation be included, but also final to final time quark propagation must be
incorporated.
Theuseofoperators whosecorrelation functionsC(t)attaintheirasymptoticformas
quicklyaspossibleiscrucialforreliablyextractingexcitedhadronmasses.Animportant
ingredient in constructing such hadron operators is the use of smeared fields. Operators
constructed from smeared fields have dramatically reduced mixings with the high fre-
quency modes of the theory. Both link-smearing[3] and quark-field smearing[4] must
be applied. Since excited hadrons are expected to be large objects, the use of spatially
extendedoperatorsisanotherkeyingredientintheoperatordesignandimplementation.
Amoredetaileddiscussionoftheseissuescan befoundin Ref. [5].
Alargeeffortwasundertakenduringthelasttwoyearstoselectoptimalsetsofbaryon
and meson operators in a large variety of isospin sectors and for zero momentum and
3 3 3
) ) )
V V V
Ge2 Ge2 Ge2
( ( (
s s s
s s s
a a a
M1 M1 M1
I=1 I=1 I=1
(p mesons) (a mesons) (r mesons)
0 - - - - - 0 - - - - - 0 + + + + +
A T E T A A T E T A A T E T A
1u 1u u 2u 2u 1g 1g g 2g 2g 1u 1u u 2u 2u
3 3 3
) ) )
V V V
Ge2 Ge2 Ge2
( ( (
s s s
s s s
a a a
M1 M1 M1
I=1 (kaons) (kaons)
(b mesons) I=½, S=-1, P=-1 I=½, S=- 1, P=+1
0 + + + + + 0 0
A T E T A A T E T A A T E T A
1g 1g g 2g 2g 1u 1u u 2u 2u 1g 1g g 2g 2g
3 3 3
) ) )
V V V
e e e
G G G
(2 (2 (2
s s s
s s s
a a a
M M M
I=0, S=-1 I=1, S=-1 I=½, S=-2
1 (L baryons) 1 (S baryons) 1 (X baryons)
G H G G H G G H G G H G G H G G H G
1g g 2g 1u u 2u 1g g 2g 1u u 2u 1g g 2g 1u u 2u
FIGURE 1. Hadron operator selection: low-statistics simulations have been performed to study the
hundreds of single-hadron operators produced by our group-theoreticalconstruction. A “pruning" pro-
cedurewas followedin eachchannelto selectgoodsets of betweensix to a dozenoperators.The plots
aboveshowthestationary-stateenergiesextractedtodatefromcorrelationmatricesofthefinallyselected
single-hadronoperators. Results were obtained using between 50 to 100 configurationson a 163×128
anisotropiclatticeforN =2+1quarkflavorswithspacinga ∼0.12fm,a /a ∼3.5,andquarkmasses
f s s t
suchthatmp ∼380MeV.Eachboxindicatestheenergyofonestationarystate;theverticalheightofeach
boxindicatesthestatisticalerror.
non-zero on-axis, planar-diagonal, and cubic-diagonal momenta. Low-statistics Monte
Carlo computations were done to accomplish these operator selections using between
50 to 100 configurations on a 163×128 anisotropiclatticefor N =2+1 quark flavors
f
with spacing a ∼0.12 fm, a /a ∼3.5, and quark masses such that the pion has mass
s s t
around 380 MeV. Stationary-state energies using the finally selected operator sets are
shown in Fig. 1. The nucleon, D , X , S , and L baryons were studied, and light isovector
and kaon mesons were investigated. Hundreds of operators were studied, and optimal
sets containing eight or so operators in each symmetry channel were found. Future
computationswillfocus solelyontheoperators intheoptimalsets.
A comprehensive picture of resonances requires that we go beyond a knowledge of
j r j r
r j r j
j r j r
r j j r
j r
t t t
t 0 0
FIGURE2. Diagramsofmulti-hadroncorrelatorsthatrequirehavingr noisesourcesonthelatertime
t.Solutionvectorsaredenotedbyj .(Left)Atwo-mesoncorrelator.(Right)Thecorrelatorofabaryon-
mesonsystem.
the ground state mass in each symmetry channel and obtain the masses of the low-
est few states in each channel. This necessitates the use of matrices of correlation
functions. Rather than evaluating a single correlator C(t), we determine a matrix of
†
correlators C (t) = hO(t)O (t )i, where {O ;i = 1,...,N} are a basis of interpolat-
ij i j 0 i
ing operators with given quantum numbers. We then solve the generalized eigenvalue
equation C(t)u = l (t,t )C(t )u to obtain a set of real (ordered) eigenvalues l (t,t ),
0 0 n 0
where l ≥ l ≥ ... ≥ l . At large Euclidean times, these eigenvalues then de-
0 1 N−1
lineate between the different masses l n(t,t0) −→ e−Mn(t−t0)+O(e−D Mn(t−t0)), where
D M = min{| M −M |: i 6= n}. The eigenvectors u are orthogonal with metric C(t ),
n n i 0
andtheeigenvectorsyieldinformationabout thestructureofthestates.
To study a particular eigenstate of interest with this method, all eigenstates lying
below that state must first be extracted, and as the pion gets lighter in lattice QCD
simulations, more and more multi-hadron states will lie below the excited resonances.
A good baryon-meson operator of total zero momentum is typically a superposition of
localinterpolatingfields at allsiteson atimesliceofthelattice.In theevaluationofthe
temporal correlations of such a multi-hadron operator, it is not possible to completely
remove all summations over the spatial sites on the source time-slice using translation
invariance. Hence, the need for estimates of the quark propagators from all spatial sites
on a time slice to all spatial sites on another time slice cannot be sidestepped. Some
correlators will involve diagrams with quark lines originating at the sink time t and
terminating at the same sink time t (see Fig. 2), so quark propagators involving a large
numberofstartingtimest mustalsobehandled.
Finding better ways to stochastically estimateslice-to-slicequark propagators is cru-
cial to the success of our excited-state hadron spectrum project at lighter pion masses.
Wehavedevelopedandtestedanewschemewhichcombinesanewwayofsmearingthe
quarkfieldwithanewwayofintroducingnoise.Thenewquark-fieldsmearingscheme,
calledLaplacian Heaviside(LapH), has been described inRef. [4]and isdefined by
y (x)=Q s 2+D y (x), (1)
(cid:16) s (cid:17)
e e
where D is the three-dimensional covariant Laplacian in terms of the stout-smeared
gauge fieeld and s s is the smearing parameter. The gauge-covariant Laplacian operator
is ideal for smearing the quark field since it is one of the simplest operators that locally
averages the field in such a way that all relevant symmetry transformation properties
of the original field are preserved. Let VD denote the unitary matrix whose columns
are the eigenvectors of D , and let L D denote a diagonal matrix whose elements are the
eigenvalues of D such theat D =VD L D VD†. The LapH smearing matrix is then given by
S =VD Q s s2+eL D VD†. Leet Vs denote the matrix whose columns are in one-to-one
(cid:0) (cid:1)
correspondence with the eigenvectors associated with the N lowest-lying eigenvalues
v
of −D on each time slice. Then our LapH smearing matrix is well approximated by
the Heermitian matrix S =Vs Vs†. Evaluating the temporal correlations of our hadron
operatorsrequirescombiningDiracmatrixelementsassociatedwithvariousquarklines
Q.Sinceweconstructourhadronoperatorsoutofcovariantly-displaced,smearedquark
fields,each andevery quarklineinvolvesthefollowingproduct ofmatrices:
Q=D(j)SM−1SD(k)†, (2)
where D(i) is a gauge-covariant displacement of type i. An exact treatment of such a
quarklineisverycostly,soweresort tostochasticestimation.
Randomnoisevectorsh whoseexpectationssatisfyE(h )=0andE(h h ∗)=d are
i i j ij
usefulforstochasticallyestimatingtheinverseofalargematrixMasfollows[6].Assume
thatforeach ofN noisevectors,wecansolvethefollowinglinearsystemofequations:
R
MX(r) = h (r) for X(r). Then X(r) = M−1h (r), and E(Xh ∗) = M−1 so that a Monte
i j ij
CarloestimateofMi−j1isgivenbyMi−j1≈limNR→¥ N1R (cid:229) Nr=R1Xi(r)h (jr)∗.Unfortunately,this
equationusuallyproducesstochasticestimateswithvarianceswhicharemuchtoolarge
to be useful. Variance reduction is done by diluting the noise vectors. A given dilution
scheme can be viewed as the application of a complete set of projection operators P(a).
Define h [a] =P(a)h , and further define X[a] as the solution of M X[a] =h [a], then we
k kk′ k′ ik k i
have
M−1 ≈ lim 1 (cid:229)NR (cid:229) X(r)[a]h (r)[a]∗. (3)
ij NR→¥ NR r=1 a i j
TheuseofZ noiseensures zero varianceinthediagonalelementsE(h h ∗).
4 i i
The effectiveness of the variance reduction depends on the projectors chosen. With
LapHsmearing,noisevectorsr canbeintroducedonlyintheLapHsubspace.Thenoise
vectorsr nowhavespin,time,andLaplacianeigenmodenumberastheirindices.Color
andspaceindicesgetreplacedbyLaplacianeigenmodenumber.Again,eachcomponent
ofr isarandomZ variablesothatE(r )=0andE(rr †)=I .DilutionprojectorsP(b)
4 d
arenow matrices in theLapH subspace. In thestochasticLapH method,a quark lineon
agaugeconfigurationis estimatedusing
Q ≈ 1 (cid:229)NR (cid:229) j (r)[b](j) r (r)[b](k)∗, (4)
uv u v
N
R r=1 b
where the subscripts u,v are compound indices combining space, time, color, and spin,
and for a noise vector labelled by index r, displaced-smeared-diluted quark source and
quarksinkvectorscan bedefined by
r (r)[b](j) = D(j)V P(b)r (r), (5)
s
j (r)[b](j) = D(j)SM−1V P(b)r (r). (6)
s
Our dilution projectors are products of time dilution, spin dilution, and Laph eigen-
vector dilution projectors. For each type (time, spin, Laph eigenvector) of dilution, we
studied four different dilution schemes. Let N denote the dimension of the space of
the dilution type of interest. For time dilution, N = N is the number of time slices on
t
the lattice. For spin dilution, N =4 is the number of Dirac spin components. For Laph
eigenvector dilution, N =N is the number of eigenvectors retained. The four schemes
v
westudiedare defined below:
P(a) =d , a=0, (nodilution)
ij ij
P(a) =d d , a=0,...,N−1 (fulldilution)
ij ij ai
P(a) =d d a=0,...,K−1, (block-K)
ij ij a,⌊Ki/N⌋
P(a) =d d a=0,...,K−1, (interlace-K)
ij ij a,imodK
where i, j = 0,...,N−1, and we assume N/K is an integer. We use a triplet (T, S, L)
to specify a given dilution scheme, where “T" denote time, “S" denotes spin, and “L"
denotes Laph eigenvector dilution. The schemes are denoted by 1 for no dilution, F for
full dilution, and BK and IK for block-K and interlace-K, respectively. For example,
full time and spin dilution with interlace-8 Laph eigenvector dilution is denoted by
(TF, SF, LI8). Introducing diluted noisein this way produces correlation functions with
significantly reduced variances, yielding nearly an order of magnitude reduction in the
statisticalerroroverpreviousmethods.Thevolumedependenceofthisnewmethodwas
foundtobeverymild,allowingthemethodtobeusefulonlargelattices.Forallforward-
timequarklines,weusedilutionscheme(TF,SF,LI8),andforallsame-sink-timequark
lines,weuse(TI16, SF, LI8).
Results for three isoscalar mesons are shown in Fig. 3. Such mesons are notoriously
difficult to study in lattice QCD, but the new method appears to produce estimates
of their temporal correlations with unprecedented accuracy. These plots suggest that
evaluating correlation functions involving our multi-hadron operators will be feasible
withthestochasticLapH method.
We are currently carrying out these spectrum computations on 243×128 and 323×
256 anisotropic lattices with spatial spacing a ∼0.12 fm and aspect ratio a /a ∼3.5,
s s t
where at is the temporal spacing, for pion masses mp ∼380 MeV and mp ∼220 MeV.
The calculations proceed in several steps: (a) generation of gauge-field configurations
using the Monte Carlo method; (b) computation of quark sinks for various noises and
dilution projectors using the configurations from the first step; (c) computation of the
meson and baryon sources and sinks using the quark sinks from the second step; (d)
evaluation of the correlators using the hadron sinks; (e) analysis of the correlators to
extracttheenergies.OurresultsfortheQCDstationary-stateenergiesusing,forthefirst
time,both single-hadronand multi-hadronoperators, shouldappear soon.
ThisworkwassupportedbytheU.S.NationalScienceFoundationunderawardsPHY-
0510020,PHY-0653315,PHY-0704171,PHY-0969863,andPHY-0970137andthrough
TeraGrid resources provided by the Pittsburgh Supercomputer Center, the Texas Ad-
vanced ComputingCenter, and theNational InstituteforComputationalSciences under
30
h w s
80 fswmdt fswmdt 80 fswmdt
total total
20
60
40
) ) )
C(t C(t C(t40
10
0
20
-40 0
0
0 10 20 30 0 10 20 30 0 10 20 30
t t t
0.4 0.4
0.3 h w s
fwd fwd fwd
total total total
0.3 0.3
0.2
) ) )
m(teff m(teff0.2 m(teff0.2
0.1
0.1 0.1
0 0 0
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
t t t
FIGURE3. CorrelatorsC(t)andeffectivemassesm (t)againsttemporalseparationt forsingle-site
eff
operators which produce the isoscalar pseudoscalar h , vector w , and scalar s mesons. Results were
obtainedusing198configurationswithN =2+1flavorsofquarkloopsona243×128anisotropiclattice
f
withspacingas ∼0.12fmandaspectratioas/at ∼3.5fora pionmassmp ∼220MeV.Inthelegends,
“fwd"referstocontributionsfromthediagramcontainingonlyforward-timesource-to-sinkquarklines,
“smt" refers to contributionsfrom the diagram containing only quark lines that originate and terminate
atthesametime.Forthes channel,the“smt"contributionhasavacuumexpectationvaluesubtraction.
Forward-time quark lines use dilution scheme (TF, SF, LI8) and same-time quark lines use (TI16, SF,
LI8).
grant numbers TG-PHY100027 and TG-MCA075017. The USQCD QDP++/Chroma
library[7] was used in developing the software for the calculations reported here. We
thankourcolleagueswithintheHadron Spectrum Collaboration.
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