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. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

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/ ):
1.3169(1) (2) (22)
Error=sigma/sqrt(chi2_min)
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, (2007)
Quenched results
DWF Overlap
GBR Collaboration
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary

15. Positive Parity
Correlators on a 243x48 lattice, a= 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

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
mπ=484 MeV
-ve Parity
+ve Parity
x 2.3 GeV
Interpolating field:
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 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