A fitting procedure for the determination of hadron excited states applied to the Nucleon

A fitting procedure for the A fitting procedure for the determination of hadron excited determination of hadron excited

states applied to the Nucleonstates applied to the Nucleon

C. Alexandrou, University of Cyprus

with

C. N. Papanicolas, University of Athens and Cyprus Institute

E. Stiliaris, University of Athens

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

The MethodThe Method

Claim:

Provides a scheme of analysis that derives the parameters of the model in a totally unbiased way, with maximum precision.

Test it in the case of lattice data:

Developed initially to address the issue of precision and model error in the analysis of experimental data on the N-Δ transition studies. C.N. Papanicolas and E. Stiliaris, AIP Conf.Proc.904 , 2007

The simplest case is to study excited states from two-point correlators

• Apply it to the ηc correlator - Thanks to C. Davies for providing the data and their results

• Apply it to the analysis of the nucleon local correlators with dynamical twisted mass fermions and NF=2 Wilson fermions - Thanks to the ETM Collaboration for providing the correlators for the twisted mass fermions

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

The MethodThe Method

We assume that all possible values are acceptable solutions, but with varying probability of being true.

Assign to each solution {A1,…,An} a χ2 and a probability.

Construct an ensemble of solutions.

The ensemble of solutions contains all solutions with finite probability.

The probability distribution for any parameter assuming a given value is the solution.

Relies only on the Ergodic hypothesis:

Any parameter of the theory (model) can have any possible value allowed by the theory and its underlying assumptions. The probability of this value representing reality is solely determined by the data.

C. Alexandrou, University of Cyprus, Lattice 2008, William and Mary

χχ22

--- --- 2w2w--- --- 3w3w--- --- 4w4w--- --- 5w5w

χ2

Convergence: χ2 -Distribution with variation of parameters

Using a wider range in the variation of the parameters yields different distributions

After a sufficiently wide range in the variation of parameters a convergence in χ2 is reached.

Random variation of all parameters uniformly

C. Alexandrou, University of Cyprus, Lattice 2008, William and Mary

A1, … Ai … A10, χ2

Sensitivity on a parameter

χ2 versus Ai

Ai is uniformly distributed (varied)

For each solution we can project the dependence of a given parameter on χχ22

Ai

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

PROJECTIONPROJECTION

ALL VALUES

χ2 < 200

χ2 < 80

χ2 < 120

χ2 < 40

Ai Distribution

Apply a χ2 - cut on a sensitive parameter Ai

χ2

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

increased events

Central value remains stable

Uncertainty depends on χ2 used for the cut

Uncertainty depends on the χ2 cut

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

PROJECTION

ALL VALUESALL VALUES

χχ22 < 200 < 200

χχ22 < < 8 800

χχ22 < < 112020

χχ22 < < 4 400

Ai Distribution

Apply a χ2- cut on a parameter Ai that the system is not sensitive on

χ2

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

AA11 vs A vs A99A1 vs A2

Correlations

Correlations in the parameters are automatically included through randomization in the ensemble and can be easily investigated.

Data not sensitive to A9

A9 vs A10

Data not sensitive to A9 and A10Data sensitive to A1 and A2

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Instead of projectingout the “best solutions”

Weigh the significance of each solution by its likelihood to be correct

P=erfc[(χ2 -χ2min)/χ2

min]

Probability Distribution

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Mass spectrum of ηc Mass spectrum of ηc Precise Lattice data: C. T. H. Davies, private Communication 2007, Follana et al. PRD75:054502, 2007;

PRL 100:062002, 2008

C(t) = An e-mnt + e-mn (T-t)( )

n=0

N

∑Fit to:

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

ηc

For the time range chosen determine the number of states N that the correlator is sensitive on

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

c: Probability Distributions c: Probability Distributions

Correlators provided by C. T. H. Davies

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

ηc: Derived Probability Distributions

Analysis by C. Davies et al. using priors (P. Lepage et al. hep-lat/0110175):

1.3169(1) 1.62(2) 1.98(22)

Jacknife errors

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

NucleonNucleon

Qu

ickTim

e™

and

aTIF

F (L

ZW

) decom

pre

sso

rare

nee

de

d to

see th

is pictu

re.

Summary of even parity excitations taken from B. G. Lasscock et al. PRD 76, 054510 (2007)

Quenched results

DWF Overlap

GBR Collaboration

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Positive Parity Positive Parity

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Correlators on a 243x48 lattice, a=0.0855 fm using two dynamical twisted mass fermions, provided by ETMC mπ=484 MeV

Interpolating field: χ1(x) =εabc ua(x)Cγ5d

Tb(x)( )uc(x)

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Negative parityNegative parity

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

mπ=484 MeV

Interpolating field: χ1(x) =εabc ua(x)Cγ5d

Tb(x)( )uc(x)

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Fits to nucleon correlatorsFits to nucleon correlators

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Nucleon Probability distributionsNucleon Probability distributions

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

+ve Parity -ve Parity

x 2.3 GeV

Interpolating field: χ1(x) =εabc ua(x)Cγ5d

Tb(x)( )uc(x)

mπ=484 MeV

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Different Interpolating fieldsDifferent Interpolating fields

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

NF==2 Wilson fermions mπ= 500 MeV

χ2(x) =εabc uTa(x)Cdb(x)( )γ5u

c(x) χ1(x) =εabc uTa(x)Cγ5d

b(x)( )uc(x)

Νο difference for the -ve Parity

Positive Parity

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Dependence on quark massDependence on quark mass

Roper at 1.440 GeV is not observed if we use the interpolating field

χ1(x) =εabc ua(x)Cγ5d

Tb(x)( )uc(x)

χ2(x) =εabc ua(x)CdTb(x)( )γ5u

c(x)If we use

then mass in positive channel close to that of negative parity stateQuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

PRELIMINARY

PRELIMINARY

Qu

ickTim

e™

an

d a

TIF

F (LZ

W) d

ecom

pre

ssor

are

need

ed

to se

e th

is pic

ture

.

C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

• A method that provides a model independent analysis for identifying and extracting model parameter values from experimental and simulation data.

• The method has been examined extensively with pseudodata and shown to produce stable and robust results. It has also applied successfully to analyze pion electroproduction data.

• It has been successfully applied to analyze lattice two-point correlators. Two cases are examined: - The ηc correlator reproducing the results of an analysis using priors with improved

accuracy. - Local nucleon correlators extracting the ground state and first excited states in the

positive and negative parity channels

main conclusion is that our analysis using local correlators is in agreement with more evolved mass correlation matrix analyses

ConclusionsConclusions