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
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
χ2
--- 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
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)
C. Alexandrou, University of Cyprus,
Lattice 2008, William and Mary
Ai
Apply a χ2 - cut on a sensitive parameter Ai
χ2
Ai Distribution
ALL VALUES
χ2 < 200
PROJECTION
χ2 < 120
χ2 < 80
χ2 < 40
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
Central value remains stable
increased events
Uncertainty depends on the χ2 cut
Uncertainty depends on χ2 used for the cut
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
χ2
Ai Distribution
ALL VALUES
χ2 < 200
PROJECTION
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.
Data sensitive to A1 and A2
A1 vs A2
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
Data
Datanot
notsensitive
sensitivetotoAA9 9and A10
A
A91 vs
vs A
A10
9
Probability Distribution
Instead of projecting
out the “best solutions”
P=erfc[(χ2 -χ2min)/χ2min]
Weigh the significance of each solution
by its likelihood to be correct
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
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Fit to:
N

C(t) =  An e-mn t + e-mn (T-t)
n=0

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)
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
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Interpolating field: 1 (x) = 
abc
u (x)C
a
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
5

dTb (x) uc (x)
Negative parity
mπ=484 MeV
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Interpolating field: 1 (x) = 
abc
u (x)C
a
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
5

dTb (x) uc (x)
Fits to nucleon correlators
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
Nucleon Probability distributions
mπ=484 MeV
-ve Parity
+ve Parity
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
x 2.3 GeV
Interpolating field: 1 (x) = 
abc
u (x)C
a
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
5

dTb (x) uc (x)
Different Interpolating fields
NF==2 Wilson fermions mπ= 500 MeV
Positive Parity
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
1 (x) =  abc uTa (x)C 5db (x)uc (x)
Νο difference for the -ve Parity
C.Alexandrou, University of Cyprus, Lattice 2008, William and Mary
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
 2 (x) =  abc uTa (x)Cdb (x) 5uc (x)
Dependence on quark mass
Roper at 1.440 GeV is not observed if
we use the interpolating field
1 (x) =  abc ua (x)C 5dTb (x)uc (x)
If we use
 2 (x) =  abc ua (x)CdTb (x) 5uc (x)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
then mass in positive channel close to
that of negative parity state
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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