Table Of ContentUsing a new analysis method to extract excited
states in the scalar meson sector
Jacob Finkenrath∗1, Constantia Alexandrou1,2, Joshua Berlin3, Mattia Dalla Brida4,5,
Theodoros Leontiou6, Marc Wagner3
[email protected],[email protected],
7
[email protected],[email protected],
1
0 [email protected],[email protected]
2
n
a 1Computation-basedScienceandTechnologyResearchCenter,TheCyprusInstitute,
J
20KavafiStreet,2121Nicosia,Cyprus
6
2 2DepartmentofPhysics,UniversityofCyprus,P.O.Box20537,1678Nicosia,Cyprus
3Goethe-UniversitätFrankfurtamMain,InstitutfürTheoretischePhysik,
]
t Max-von-Laue-Straße1,D-60438FrankfurtamMain,Germany
a
4NIC,DESY,Platanenallee6,15738Zeuthen,Germany
l
- 5DipartimentodiFisica,UniversitàdiMilano-Bicocca,andINFN,
p
e sezionediMilano-Bicocca,PiazzadellaScienza3,I-20126Milano,Italy
h 6DepartmentofMechanicalEngineering,FrederickUniversity,7Y.FrederickouStr.,
[
Pallouriotisa,Nicosia1036,Cyprus
1
v
3
7 We explore a method to extract energy eigenstates, called Athens Model Independent Analysis
6
Scheme(AMIAS),whichisanalternativetosolvingstandardGeneralizedEigenvalueProblems
7
0 (GEVP). The method is based on statistically sampling the space of fit parameters according to
.
1 the χ2 value of the fit function. The method is particularly suited for correlators or correlation
0
matriceswithstrongcontributionsfromseveralenergyeigenstatesandforrathernoisydata,e.g.
7
1 forcorrelatorswithdisconnectedandpartlydisconnecteddiagrams. Weapplythemethodtothe
:
v analysisoftheJP=0+channelinthecontextofourinvestigationofthea (980)mesonandpoint
0
i
X outadvantagescomparedtotheGEVP.
r
a
The34ndInternationalSymposiumonLatticeFieldTheory
25-30July,2016
UniversityofSouthampton,UK
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Usinganewanalysismethodtoextractexcitedstatesinthescalarmesonsector JacobFinkenrath
1. Introduction
In this work we explore AMIAS, a rather new analysis method to extract energy differences
andamplitudesfromlatticeQCDcorrelatorsandcorrelationmatrices. Thebasicideaistoconsider
a very large number of fits to these correlators using Monte Carlo methods, which will lead to
probabilitydistributionsforthefitparameters,thataretheenergydifferencesandamplitudes. This
avoidsthenecessityofidentifyingplateauregionsforeffectiveenergies. Inparticular,forunstable
systems and resonances identifying plateaus can be very challenging, since the signal-to-noise
ratio is typically rather poor for large temporal separations. Moreover it overcomes the limitation
of extracting at most N energy eigenstates, when N interpolating fields are used (as it is e.g. the
case,whendoingastandardGEVPanalysis)
WeapplyAMIAStoanalyzecorrelatorscomputedforstudyingthea (980)meson,whichhas
0
quantum numbers I(JP)=1(0+) and mass m ≈980MeV [1]. We use interpolating fields
a0(980)
withbothtwo-quarkandafour-quarkstructuresformedbyanup-quarkuandananti-downquark
d¯andexceptforonecaseastrangeandananti-strangequarksands¯. Thefour-quarksarearranged
as a meson-meson interpolating field or in a diquark-anti-diquark combination, which can probe
differentpossibletetraquarkstructuresofa (980). Weconsidersixinterpolatingfields,
0
1 (cid:16) (cid:17)
O1:=Oqq¯= √ ∑ d¯(x)u(x) (1.1)
V
s x
1 (cid:16) (cid:17)(cid:16) (cid:17)
O2:=OKK¯,point= √ ∑ s¯(x)γ u(x) d¯(x)γ s(x) (1.2)
5 5
V
s x
1 (cid:16) (cid:17)(cid:16) (cid:17)
O3:=Oηsπ,point= √ ∑ s¯(x)γ s(x) d¯(x)γ u(x) (1.3)
5 5
V
s x
1 (cid:16) (cid:17) (cid:16) (cid:17)
O4:=OQQ¯ = √ ∑ε s¯ (x)(Cγ )d¯T(x) ε uT(x)(Cγ )s (x) (1.4)
V abc b 5 c ade d 5 e
s x
1 (cid:16) (cid:17)(cid:16) (cid:17)
O5:=OKK¯,2part= ∑ s¯(x)γ u(x) d¯(y)γ s(y) (1.5)
5 5
V
s x,y
1 (cid:16) (cid:17)(cid:16) (cid:17)
O6:=Oηsπ,2part= ∑ s¯(x)γ s(x) d¯(y)γ u(y) , (1.6)
5 5
V
s x,y
whereV denotesthespatiallatticevolumeandCthechargeconjugationmatrix. Allofthemcouple
s
toa (980)andotherstateswiththesamequantumnumbers. Forexample,theinterpolatingfields
0
O andO mainlygeneratethetwo-mesonstatesK+Kandπ+η,respectively,whichareexpected
5 6
tohavemassesclosethatofthea (980). NotethattheinterpolatingfieldsO andO representtwo
0 2 3
mesons located at the same point in space with only total momentum zero, but arbitrary relative
momenta involved (a structure resembling a 4-quark bound state). In contrast to that O and O
5 6
correspondtotwomesonswithbothtotalandrelativemomentumzero.
Wecomputethefullsixbysixcorrelationmatrixincludingbothconnectedanddisuconnected
contributions, which we neglect in our previous studies [2, 3, 4]. Moreover here we have in-
creased statistical accuracy of the correlators C (t) = (cid:104)Oj(t)Ok†(0)(cid:105), included propagation of
jk
strange quarks within a timeslice and analyzed the correlators with AMIAS. We use an ensem-
ble of around 500 gauge link configurations generated with 2+1 dynamical Wilson clover quarks
1
Usinganewanalysismethodtoextractexcitedstatesinthescalarmesonsector JacobFinkenrath
andIwasakigaugeactiongeneratedbythePACS-CSCollaboration[5]. Thelatticesizeis64×323
withlatticespacinga≈0.09fmandpionmassm ≈300MeV.
π
2. Correlatorsonaperiodiclattice
AcorrelatoronaperiodiclatticewithtimeextentT canbeexpandedaccordingto
C (t)=(cid:68)Oj(t)Ok†(0)(cid:69)= ∑m,nexp{−Em(T−t)}(cid:104)m|Oj|n(cid:105)exp{−Ent}(cid:104)n|Ok†|m(cid:105) (2.1)
jk ∑mexp{−EmT}
with energy eigenstates |m(cid:105) and corresponding energy eigenvalues E ≤ E ≤ E ... (|0(cid:105) = |Ω(cid:105)
0 1 2
denotesthevaccum). Thiscorrelatorcanalsobeexpressedinamoreconvenientform,
C (t)=(cid:68)Oj(t)Ok†(0)(cid:69)= ∑m,ncmj,n(ckm,n)∗exp{−(Em+En)(T/2)}cosh{∆En,m(t−T/2)} (2.2)
jk ∑mexp{−EmT}
with cj =(cid:104)m|Oj|n(cid:105) and ∆E =E −E . For example, if j=k, if Oj probes the sector, which
m,n n,m n m
containsenergyeigenstate|1(cid:105),ifthequantumnumbersof|1(cid:105)aredifferentfromthoseofthevacuum
|Ω(cid:105),andifthecorrelatorisnotcontaminatedbymulti-particlestates(seediscussionbelow),(2.2)
reducesto
(cid:68) (cid:69) (cid:12) (cid:12)2
C (t)= Oj(t)Ok†(0) ≈2(cid:12)cj (cid:12) exp{−∆E (T/2)}cosh{∆E (t−T/2)} (2.3)
jj (cid:12) 0,1(cid:12) 1,Ω 1,Ω
for sufficiently large t and T. A standard technique to determine the energy difference ∆E =
1,Ω
E −E ,themassofstate|1(cid:105),fromtheasymptotict behaviorofC (t)(Eq.(2.3))istofit
1 Ω jj
C (t)=Acosh{−∆E (t−T/2)} (2.4)
jj 1,Ω
to the lattice QCD results for C (t) with fitting parameters ∆E and A. Alternatively, one can
jj 1,Ω
alsosolvetheequation
C (t) cosh{E (t)(t−T/2)}
jj eff
= (2.5)
C (t−a) cosh{E (t)(t−a−T/2)}
jj eff
with respect to E (t), where E (t)≈∆E . In other words, a plateau-like behavior of E (t)
eff eff 1,Ω eff
indicates the mass E . In practice, however, the temporal extent T of the lattice is limited and
1,Ω
the effective energy E (t) is often very noisy for larget, rendering a reliable extraction of ∆E
eff 1,Ω
difficult.
When multi-particle states are present in the investigated sector, the determination of low-
lying masses is even more difficult. We sketch this for a simple non-interacting two-meson sys-
tem, which is generated by an interpolating field O = O(1)⊗O(2). Energy eigenstates of such
a system can be written as |n(cid:105) = |n (cid:105)(1)⊗|n (cid:105)(2), where |n (cid:105)(1) and |n (cid:105)(2) (n ,n = 0,1,2,...)
1 2 1 2 1 2
are eigenstates of the two Hamiltonians describing the individual mesons. Energy eigenvalues
of this two-meson system are E(1+2) =E(1)+E(2), in particular the lowest energy eigenvalue is
n1,n2 n1 n2
E(1+2) =E(1)+E(2). Note that in the expansion (2.2) there is a term, which is proportional to
0,0 0 0
cosh{(E(1)−E(2))(t−T/2)}, i.e. only decaying with the mass difference of the two mesons. In
0 0
the region of t ≈T/2 this unwanted term can be more dominant than the signal term, which is
2
Usinganewanalysismethodtoextractexcitedstatesinthescalarmesonsector JacobFinkenrath
proportional to cosh{E(1+2)−E )(t−T/2)}. Thus, one has to be rather careful, when analyzing
0,0 Ω
correlatorsofsectorscontainingmulti-particlestates. ThisisillustratedbyFig.1,whereweshow
the correlator using the interpolating field Oqq¯ (Eq. (1.1)). Clearly, there are two effective mass
plateaus, aE (t)≈0.60 fort/a<7, corresponding to the mass of a π+η or a K+K two-meson
eff ∼
state,andaE (t)≈0.25fort/a>8,whichdoesnotcorrespondtothemassofanystate,buttothe
eff ∼
massdifferencem −m oftwosingle-mesonstates.
η π
0.8
100 0.7
0.6
10−1
0.5
a)
C(t/a)10−2 E (t/eff0.4
a
0.3
10−3 0.2
0.1
10−4
0
0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
t/a t/a
Figure1: Left: CorrelatorusingtheinterpolatingfieldOqq¯. Right: Correspondingeffectivemass.
3. AMIAS
LatticeQCDresultsforthecorrelatorsC (t)=(cid:104)Oj(t)Ok†(0)(cid:105)withOj, j=1,...,6definedin
jk
Eqs.(1.1)to(1.6)canbeparameterizedandfittedtheexpressiongiveninEq.(2.2). Sincestatistical
accuracy is limited, it is sufficient to consider a rather small number of energy eigenstates, i.e.
∑ ,∑ →∑N−1 ,∑N−1 withN<10.
m,n m m,n=0 m=0 ∼
With AMIAS [6, 7, 8] one can determine probability distribution functions (PDFs) for the fit
parameters∆E (energydifferences)andcj (amplitudes). AMIASisabletodealwitharather
n,m m,n
large number of parameters by using Monte Carlo techniques. In contrast to e.g. the GEVP, it is
notnecessarytoidentifyplateauregionsortospecifytemporalfittingranges.
ThePDFforthecompletesetoffitparametersisgivenby
P(∆En,m,cmj,n)= Z1e−χ22 (3.1)
withappropriatenormalizationZ and
(T−1)/a(Clattice(t)−C (t))2
χ2=∑ ∑ jk jk , (3.2)
(w (t))2
j,k t/a=1 jk
which is the well-known χ2 used in χ2 minimizing fits (Clattice(t) denote lattice QCD results for
jk
correlators with corresponding statistical errors w (t)). To obtain the PDF for a specific fit pa-
jk
rameter, one has to integrate in (3.1) over all other parameters. In particular, the probabilty for
parameter A to be inside [a,b] (A represents either one of the energy eigenvalue differences
j j
∆E oramplitudescj )is
n,m m,n
Π(A ∈[a,b])= (cid:82)abdAj (cid:82)−+∞∞∏k(cid:54)=jdAke−χ2/2. (3.3)
j (cid:82)−+∞∞∏kdAke−χ2/2
3
Usinganewanalysismethodtoextractexcitedstatesinthescalarmesonsector JacobFinkenrath
Thismulti-dimensionalintegralcanbecomputedwithstandardMonteCarlomethods. Weimple-
mentedaparallel-temperingschemecombinedwiththeMetropolisalgorithmasdescribedindetail
inRef.[8]. Theparallel-temperingschemepreventsthatthealgorithmisstuckinaregionaround
alocalminimumofχ2 and,thus,guaranteesergodicityofthealgorithm.
We have again analyzed the correlator of Oqq¯ shown in Fig. 1 (left), this time using AMIAS
withfitfunction
2
C(t)= ∑A cosh{∆E (t−T/2)}. (3.4)
n n
n=1
TheresultingPDFsforthefourparameters∆E , ∆E , A andA areshowninFig.2. AsinFig.1
1 2 1 2
(right)twoenergydifferencescanclearlybeidentified,a∆E ≈0.25(themassdifferencem −m )
1 η π
anda∆E ≈0.60(themassofaπ+η orK+K two-mesonstate).
2
ε
∆ A
2 2
A
1
ε
∆
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.5 A 1
a E
Figure 2: AMIAS analysis of the correlator of Oqq¯. PDFs for the parameters ∆E , ∆E , A and A from
1 2 1 2
(3.4).
4. Analysisofthe6×6correlationmatrix
We now use AMIAS to analyze the 6×6 correlation matrix with interpolating fields (1.1) to
(1.6). Thefitfunctionisobtainedbyrestricting(2.2)toafinitenumberofterms,
N
C (t)= ∑cj(ck)∗cosh{∆E (t−T/2)}. (4.1)
jk n n n
n=1
InafirstanalysisweuselatticeQCDdata,wherethepropagationofstrangequarkswithinthe
sametimeslicehasbeenneglected. TheinterpolatingfieldOqq¯ thendecouplesand, hence, cannot
beconsideredinatetraquarkstudyofa (980)(cf.alsoRefs.[2,3],wherethesamedatahasbeen
0
used). Moreover, we do not consider the diquark-antidiquark interpolating field OQQ¯. In Fig. 3
(top) we show the four lowest masses obtained with N =8 terms in (4.1)1, corresponding to the
expected two-particle states π+η and K+K with both mesons at rest (∆E and ∆E ) as well as
2 3
with one quantum of relative momentum (∆E and ∆E ). Note that AMIAS also finds the energy
4 5
eigenvalue difference m −m (∆E , not shown in Fig. 3), which has already been discussed in
η π 1
previoussections.
Since (cj)∗ = (cid:104)n|Oj†|Ω(cid:105), the amplitudes extracted with AMIAS are the coefficients of the
n
expansionsofthetrialstatesOj†|Ω(cid:105)intermsoftheextractedenergyeigenstates|n(cid:105),i.e.
N
Oj†|Ω(cid:105)≈ ∑(cj)∗|n(cid:105). (4.2)
n
n=1
1AMIASisabletodeterminetheoptimalNautomatically.Fordetailswereferto[8].
4
Usinganewanalysismethodtoextractexcitedstatesinthescalarmesonsector JacobFinkenrath
a(mπ+mη)=0.4988(15) am2K=0.5456(21)
ε ε ε
∆ ε ∆ ∆
∆
1 3 4
2
a(mπ+mη)(p)=0.6514(15) am2K(p)=0.6723(21)
0.45 0.5 0.55 0.6 0.65 0.7 0.75
a E
1 1 1 1
0.8 E 0.8 E 0.8 E 0.8 E
1 2 3 4
0.6 0.6 0.6 0.6
0.4 0.4 0.4 0.4
0.2 0.2 0.2 0.2
0 0 0 0
1 2 4 5 1 2 4 5 1 2 4 5 1 2 4 5
Figure3: AMIASanalysisofthe4×4correlationmatrixwithinterpolatingfieldsO2=OKK¯,point,O3=
Oηsπ,point, O5 =OKK¯,2part and O6 =Oηsπ,2part (propagation of strange quarks within the same timeslice
neglected). Top: PDFsfortheparameters∆E ,...,∆E from(4.1). Bottom: |vj|,thecorrespondingoverlaps
2 5 n
tothetrialstatesOj†|Ω(cid:105).
More interesting, however, are the opposite expansions, i.e. the extracted energy eigenstates in
termsofthetrialstates,
|n(cid:105)≈ ∑ vjOj†|Ω(cid:105). (4.3)
n
j=2,3,5,6
Itiseasy toshowthatthe matrixformedbythe coefficientsvi istheinverse ofthematrixformed
n
by the coefficients (cj)∗ (j = 2,3,5,6, n = 2,...,5) up to exponentially small corrections, i.e.
n
∑jvmj(cnj)∗=δm,n. Alternatively,onecanalsousetheAMIASsamplesfor∆Enandcnj andEq.(4.1)
toreconstructacorrelationmatrixforeachsample. SolvingstandardGEVPsforeachreconstructed
j j
correlation matrix yields eigenvectors with components identical to v . The coefficients |v | are
n n
shown in Fig. 3 (bottom) (not the PDFs, but the most likely values). Clearly, ∆E and ∆E corre-
2 3
j
spond to two-particle states π+η and K+K with both mesons at rest, since the coefficients v
n
showalmostexclusivelyoverlaptothetrialstatesOηsπ,2part†|Ω(cid:105)andOKK¯,2part†|Ω(cid:105). ∆E4 and∆E5
areclosetotheexpectedtwo-particlestateswithonequantumofrelativemomentumandtheover-
j
laps shown by v are consistent with this interpretation. There is no indication for any additional
n
stateintheregionof1000MeV,whichcouldcorrespondtoa (980). Thesefindingsareconsistent
0
withourpreviousstudyusingETMCgaugelinkconfigurations[9].
Whenincludingthepropagationofstrangequarkswithinthesametimeslice,theinterpolating
fieldOqq¯ couplestothefour-quarkinterpolatingfieldsgiveninEqs.(1.2)to(1.6). Thisintroduces,
however, a lot of statistical noise and it is rather difficult to resolve energy differences precisely.
In Fig. 4 we show preliminary AMIAS results for the full 6×6 correlation matrix, where N =12
terms in Eq. (4.1) have been used. The two lowest energy eigenstates exhibit rather clear signals
andcorrespondtotheπ+η andK+K two-mesonstates. Eventhoughhigherexcitationsareless
prominent, we find strong indication for an additional state with mass ∆E ≈0.6/a≈1300MeV,
3
whichisbelowtheexpectationforthetwo-particlestateswithonequantumofrelativemomentum
5
Usinganewanalysismethodtoextractexcitedstatesinthescalarmesonsector JacobFinkenrath
and,hence,mightbeacandidateforthea (980).
0
ε
∆
1 ∆ε ∆ε
ε 3 5
∆
2 ε
∆
4
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
a E
Figure 4: AMIAS analysis of the full 6×6 correlation matrix (propagation of strange quarks within the
sametimeslicetakenintoaccount). PDFsfortheparameters∆E ,...,∆E from(4.1)generatedbyAMIAS.
2 6
5. Conclusions
WithAMIASonecanextractenergydifferencesandamplitudeswithoutthenecessitytoiden-
tify pleateau regions or to specify temporal fitting ranges. We have shown that AMIAS can suc-
cessfullybeusedtoanalyzecorrelatorswithstrongcontributionsfromseveralenergyeigenstatesas
wellasrathernoisycorrelationmatrices. Anexampleforthelatterise.g.thefull6×6correlation
matrix for the a (980) channel. Another major advantage of AMAIAS might be, that it does not
0
requireallelementsofacorrelationmatrix,i.e.particularlynoisyelementsorelements,whichare
verytime-consumingtocompute,canbeomitted. Thisweplantoaddressinafuturepublication.
Acknowledgements: M.W.acknowledgessupportbytheEmmyNoetherProgrammeoftheDFG
(German Research Foundation), grant WA 3000/1-1. This work was supported in part by the
HelmholtzInternationalCenterforFAIRwithintheframeworkoftheLOEWEprogramlaunched
by the State of Hesse. This work was cofunded by the European Regional Development Fund
and the Republic of Cyprus through the Research Promotion Foundation (Project Cy-Tera NEA
YΠO∆OMH/ΣTPATH/0308/31)bythegrandcypro914.
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6
