novel analysis method for excited states in lattice qcd
TRANSCRIPT
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Novel analysis method for excited states inlattice QCD
Theodoros Leontiouin collaboration with C. Alexandrou, C. N. Papanicolas and E. Stiliaris
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Outline
1 Motivation
2 Basic Formulation
3 Application to the Nucleon Spectrum
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Nucleon Excited States
Nucleon resonances are well studiedexperimentally
Simulations from LQCD still require improvement
Nucleon Resonances (I=1/2)Symbol Jp
N(1440) P1112
+
N(1520) D1132−
N(1535) S1112−
N(1650) S1112−
N(1675) D1152−
The first excited states of the nucleon
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2 [GeV]
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
mN
[GeV
]
JP = 12+
Twisted Mass (this work)Clover (this work)CSSM
JLABBGRExperiment
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2 [GeV]
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
mN
[GeV
]
JP = 12Twisted Mass (this work)Clover (this work)CSSM
BGRExperiment: N (1535)S wave: +N
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Nucleon Excited States:The variational method
Choose a set of interpolating operators Oi , i = 1, ...,N that couples to the quarkstructure of interest and have the quantum numbers for the state of interest.
Build the correlation matrix Cij = 〈0|Oi (t)O†j (0)|0〉
Cij (t) =∑
nA(ij)
n e−En t
Solve the generalized eigenvalue problem:
C(t)vn = λnC(t0)vn, n = 1, . . . ,N, t > t0
Energies are extracted from the long time-limit of the eigenvaluesEn = limt→∞−∂t logλn(t , t0).→ look for a plateau
The contribution of each quark interpolating operator is extracted from theeigenvectors
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Nucleon Excited States:The variational method
2 4 6 8 10 12 14
t/a0
1
2
3
aEneff (t)
n=0n=1n=2n=3
1.022(65)
0.509(11)
The corrections to En decrease exponentially like e−∆En t where∆En = minm 6=n |Em − En|.Identification of the excited states is limited by pure statistics
motivation Basic Formulation Application to the Nucleon Spectrum Summary
AMIAS
Standard least-squares algorithms are numerically unstable and will fail for lowsignal-to-noise ratio
The variational method works but is limited due to the long-time limit
AMIAS(Athens Model Independent Analysis Scheme)
Relies on statistical concepts
Gives probability distributions of parameters
Insensitive parameters are fully accounted and do not bias the results
Can access a large number of parameters by using Monte Carlo techniques
A novel method of data analysis for hadronic physics, C. Papanicolas and E. Stiliaris, arXiv:1205.6505.
Multipole Extraction: A Novel, Model Independent Method , E. Stiliaris and C. Papanicolas, AIP Conf. Proc. 904,
257 (2007)
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Basic Formulation:The Central Limit Theorem
We have to deal with expectations values obtained from some ProbabilityDensity Function (PDF)The expectation value is approximated as an average of samples taken from thePDF: 〈X̂〉 ∼ 1/N
∑Ni=1 Xi = X̄
Central Limit Theorem
P(X̄) = k exp
−(〈X̂〉 − X̄
)2
2(σ/√
N)2
,P(X̄) is the probability that thesampled average has a valueequal to X̄
〈X̂〉 can be modeled
X̂ = Oi (t)O†j (0), i, j = 1, ...,N and X̄ = C̄ij (simulations)
〈X〉 =∑
n A(ij)n e−En t , i, j = 1, ...,N (model)
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Basic Formulation: PDF for a correlation matrix
We can apply the central limit theorem to all time slices 1, ...,Nt of the lattice andto all matrix elements i, j = 1, ...,N of the correlation matrix
P(C̄ij (tn); ∀i, j, n) = e−χ22 ,
χ2 =∑i,j
Nt∑k=1
(C̄ij (tk )−∑∞
n=0 A(ij)n e−En tk )2
(σ(i,j)tk
/√
N)2.
Each value assigned to the model parameters has a statistical weight
proportional to e−χ22
Model parameters are sampled from the PDF
The probability that the parameter Ai assumes a specific value ai in the range (bi , ci ) isequal to
Π(Ai = ai ) =
∫ cibi
dAi∫∞−∞
∏j 6=i dAj ,Ai e−χ̃
2/2∫∞−∞
(∏j dAj
)Ai e−χ̃
2/2.
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Determination of the number of contributingparameters
The averaged nucleon correlationfunction C̄(t) has a spectraldecomposition as an infinite summationof exponential terms.
Model
C̄(t) ∼∑nmax
i Ai e−Ei t
4 8 12 16 20 24 28 32t
1e-14
1e-13
1e-12
1e-11
1e-10
1e-09
1e-08
log(C(t)) 1 10
E (GeV)
nmax=2
nmax=3
nmax=4
nmax=5E4E3
E2E1E0
Pro
bab
ilit
y
The distributions of En for different values of nmax
The distributions of An have a similar behavior
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Determination of the number of contributingparameters
Model parameters that contribute to thesolution have well defined distributions
We can get values for the parametersby fitting the distribution
2 3 4 5 6 7
E (GeV)
E2
E1
Pro
bab
ilit
y
The performance of AMIAS is not biased when ‘insensitive’ parameters areinserted in the modelAMIAS is in agreement with standard χ2 minimization when the latter isapplicable
AMIAS Standard Least Squaresnmax E0 E1 E2 E0 E1 E2
2 1.161(13) 2.8990(32) – 1.162(11) 2.92(132) –3 1.1430(32) 2.0453(98) 5.0220(53) 1.1439(23) 2.03(89) 5.022(2.7)4 1.1430(32) 2.050(11) 4.8850(64) – – –5 1.1432(32) 2.052(11) 4.8394(64) – – –
motivation Basic Formulation Application to the Nucleon Spectrum Summary
We can suppress the contribution fromthe higher exponents by varying thestarting value t0 in order to eliminateunnecessary correlations and tovalidate the dominant exponents.
4 8 12 16 20 24 28 32t
1e-14
1e-13
1e-12
1e-11
1e-10
1e-09
1e-08
log(C(t))
1 2 3 4 5 6 7
E (GeV)
E0E1
E2
t0=1
t0=2
t0=3
t0=4
Pro
bab
ilit
y
0 7e-08 1.4e-07
A
t0=1
t0=2
t0=3
t0=4A0
A1A2
Probability
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Proper handling of correlations
A central issue that is properly treated in AMIAS, is the handling of correlations,since all possible correlations are accounted for
2 , 5 x 1 0 - 8 3 , 0 x 1 0 - 8
0 , 4 5
0 , 5 0
0 , 5 5
0 , 6 0
E 0
A 0
2 , 5 x 1 0 - 8 3 , 0 x 1 0 - 8
0 , 4 5
0 , 5 0
0 , 5 5
0 , 6 0
E 0A 1
2 , 5 x 1 0 - 8 3 , 0 x 1 0 - 8
0 , 8
0 , 9
1 , 0
E 1
A 0
0 , 5 0 , 6
7
8
9E 4
E 0
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Excited States from AMIAS
2 4 6
E (GeV)
C11
(1,1)
C11
(5,5)
E0
E0E2
E1
E1
E0
C11
(i,j)
i,j=1,2,4
C11
(i,j)
i,j=1,2,3
E1
E0C11
(i,j)
i,j=1,..,5
Pro
bab
ilit
y
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Parallel Tempering Monte Carlo
-0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003
A(i)n
Probability
The distributions are multi-modal
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Variational vs AMIAS
5 10 15t
1
2
3
4
Eef
f (G
eV)
C11
(1,2,4)
5 10 15t
C11
(1,2,3)
Results from AMIAS are compatible with the variationalmethodExited States plateaus can be more easily identified fromAMIAS
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Variational vs AMIAS
0 0.02 0.04 0.06 0.08 0.1
mπ
2 (GeV)
0.5
1
1.5
2
2.5
E(G
eV
)
clover (GEVP)
clover(AMIAS)
twisted mass (GEVP)
twisted mass (AMIAS)
N+
(1440)
N
0 0.05 0.1
mπ
2 (GeV)
0.5
1
1.5
2
2.5
3
E (
GeV
)
clover (GEVP)
clover (AMIAS)
twisted mass (GEVP)
twisted mass (AMIAS)
N+π
N-(1535)
motivation Basic Formulation Application to the Nucleon Spectrum Summary
Summary
Model fitting with a large number of parameters is possiblewith the help of statistical methodsThe method described can capture mutli-modal parameterdistributionsWe could obtain the nucleon spectrum with reliableaccuracy:Novel analysis method for excited states in lattice QCD: The nucleon case, C. Alexandrou, T. Leontiou,
C.,N. Papanicolas and S. Stiliaris, Phys. Rev. D 91, 014506 (2015), e-Print Archive hep-lat/1411.6765