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

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. Claim:

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

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  1. 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

  2. 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 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 • 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

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

  4. Convergence: χ2 -Distribution with variation of parameters χ2 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. χ2 C. Alexandrou, University of Cyprus, Lattice 2008, William and Mary

  5. A1, … Ai … A10, χ2 Sensitivity on a parameter For each solution we can project the dependence of a given parameter on χ2 χ2versus Ai Ai is uniformly distributed (varied) Ai C. Alexandrou, University of Cyprus, Lattice 2008, William and Mary

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

  7. Central value remains stable Uncertainty depends on the χ2cut Uncertainty depends on χ2 used for the cut increased events C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

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

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

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

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

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

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

  14. 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

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

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

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

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

  19. 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

  20. 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

  21. 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

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