novel analysis method for excited states in lattice qcd

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


Outline

1 Motivation

2 Basic Formulation

3 Application to the Nucleon Spectrum


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


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


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


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)


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)


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.


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


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) – – –


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


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


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


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


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


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)


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

novel analysis method for excited states in lattice qcd

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