A fitting procedure for the determination of hadron excited states applied to the Nucleon C. Alexandrou, University of Cyprus with C. N. Papanicolas, University of Athens and Cyprus Institute E. Stiliaris, University of Athens

Test it in the case of lattice data: The Method 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 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: 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 Method 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. 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. C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Convergence: χ2 -Distribution with variation of parameters Random variation of all parameters uniformly Using a wider range in the variation of the parameters yields different distributions --- 2w --- 3w --- 4w --- 5w After a sufficiently wide range in the variation of parameters a convergence in χ2 is reached. Histogram χ2 C. Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Ai is uniformly distributed (varied) Sensitivity on a parameter For each solution we can project the dependence of a given parameter on χ2 χ2 versus Ai A1, … Ai … A10, χ2 Ai is uniformly distributed (varied) Ai C. Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Apply a χ2 - cut on a sensitive parameter Ai Ai Distribution PROJECTION ALL VALUES χ2 < 200 χ2 < 120 χ2 < 80 χ2 < 40 C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Central value remains stable Uncertainty depends on χ2 used for the cut Uncertainty depends on the χ2 cut increased events C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Apply a χ2- cut on a parameter Ai that the system is not sensitive on Ai Distribution PROJECTION ALL VALUES χ2 < 200 χ2 < 120 χ2 < 80 χ2 < 40 C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Correlations Correlations in the parameters are automatically included through randomization in the ensemble and can be easily investigated. A9 vs A10 Data not sensitive to A9 and A10 Data sensitive to A1 and A2 Data not sensitive to A9 A1 vs A2 A1 vs A9 C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

out the “best solutions” Instead of projecting out the “best solutions” Probability Distribution Weigh the significance of each solution by its likelihood to be correct P=erfc[(χ2 -χ2min)/χ2min] P=erfc[(χ2 -χ2min)/χ2min] C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Mass spectrum of ηc Precise Lattice data: C. T. H. Davies, private Communication 2007, Follana et al. PRD75:054502, 2007; PRL 100:062002, 2008 ηc Fit to: 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

hc: Probability Distributions Correlators provided by C. T. H. Davies C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

ηc: Derived Probability Distributions Jacknife errors Analysis by C. Davies et al. using priors (P. Lepage et al. hep-lat/0110175): 1.3169(1) 1.62(2) 1.98(22) Error=sigma/sqrt(chi2_min) C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Nucleon 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 Correlators on a 243x48 lattice, a=0.0855 fm using two dynamical twisted mass fermions, provided by ETMC mπ=484 MeV Interpolating field: C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Negative parity Interpolating field: mπ=484 MeV C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Fits to nucleon correlators C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Nucleon Probability distributions mπ=484 MeV -ve Parity +ve Parity x 2.3 GeV Interpolating field: C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Different Interpolating fields NF==2 Wilson fermions mπ= 500 MeV Positive Parity Νο difference for the -ve Parity C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

Dependence on quark mass Roper at 1.440 GeV is not observed if we use the interpolating field If we use PRELIMINARY then mass in positive channel close to that of negative parity state C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

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