A Genetic Algorithm Analysis
of N* Resonances
D. G. Ireland
Department of Physics and Astronomy
University of Glasgow
Outline:Analysis of N* contribution to gp → K+L
How does using a Genetic Algorithm help?
How much can an analysis of the data currently tell us?
Conclusions and Outlook
Analysis of “World” data (1999)
With D13
Without D13
“Evidence” of missing D13 resonance
Mart & Benhold, Phys. Rev. C 61 012201(R) (1999)
Hadrodynamical Model of Janssen, et al.
Coupling constants and other
parameters have to be
determined by fits to data.
The “fit” is an optimisation
involving 20 – 30 free
parameters.
[This is a single channel model,
more complete descriptions
require coupled channel
analyses.]
Strategy: Genetic Algorithm
(GA) + Minuit
GA components...
A “population” of
encoded trial
solutions
Each solution has
a “fitness”
Evolution of population, consisting of ...
Selection
Crossover
Mutation
+ iteration towards
convergence...
Comparison: GA vs. MINUIT
Phase 1: Calculation with additional D13
Many sets of fitted
parameters = many
calculations with equal
goodness-of-fit
Janssen, Ireland & Ryckebusch, Phys. Lett. B 562 (2003) 51
Distributions of Fitted Parameters
Each calculation has a different set of fitted coupling constants.
Predictions for Unmeasured Observables
Large ambiguities, even within one model
Phase 2: Systematic Study with more experimental
data points...
To address two questions:• Is there more evidence of an extra resonance in the reaction?
• What are the quantum numbers of this extra resonance?
Each model:-
contains a “core” set of resonances: S11(1650), P11(1710) and P13(1720)
contains an extra resonance of mass 1895 MeV, with different quantum
numbers: S11, P11, P13, D13
used 100 calculations (GA + MINUIT)
New photon beam polarisation (SPRing-8), and electroproduction data
(Jlab Hall C) used in fit.
Results:
Core
S11
P11
Total Cross-Section
P13
D13
Photon Beam
Asymmetry
Occam's Razor
“Pluralitas non est ponenda sine necessitate”
- plurality should not be posited without
necessity
For models A and B, calculate ratio of
posterior probabilities:P  A | D  P  D | A P  A P  D | A


P B | D  P D | B  P B  P D | B 
i
  

i 1
P D | A  exp    M max
 2  i 1 i  imin 
2
Best fit to data
William of Occam
(or Ockham, ca. 1285-1349)
←ratio of likelihoods
M
Occam factor (approximate)
Table of Results
Model
Core
S_11
P_11
P_13
D_13
Raw Chi-Squared
5.14
4.47
3.47
3.75
3.35
Number of free
parameters
4
5
5
6
6
Occam Factor
1.00
3.28
5.56
0.02
1.17
Ratio of Posterior
Probability
1.00
4.50
12.57
0.04
2.79
Raw scores indicate D13 most likely
More sophisticated comparison favours P11
Data support hypothesis of extra resonance
Situation still not clear
Model Predictions – New Measurements
Linear
Core
S11
P11
P13
D13
Circular
Beam – recoil polarisation
Phase 3: Lots more data!
e.g. J.W.C. McNabb et al., PRC 69 (2004) 042201
Two approaches:1) Use parameters obtained in previous fit for Core,
S11, P11, P13, D13 models
2) Re-fit, but with two models: Core and
S11+P11+P13+D13 (all hypothetical resonances
together)
Angular Distributions
Data: J.W.C. McNabb et al.,
PRC 69 (2004) 042201 (CLAS)
Differential Cross Sections
Data: J.W.C. McNabb et al., PRC 69 (2004) 042201 (CLAS)
Beware many parameters!
Model
Core
Full
Raw Chi-Squared
5.37
2.48
Number of free parameters
4
10
Likelihood
0.068
0.289
Occam Factor
5.625e-06 2.233e-15
Posterior Probability
3.825e-07 6.453e-16
Full calculation penalised for many parameters.
Occam factor calculation very approximate!
Situation inconclusive
“Full” evaluation of integrals necessary → MCMC
Conclusions
Genetic Algorithm: potentially powerful
addition to analysis toolbox.
Must do many calculations – study parameter
space.
Current data indicates poor agreement with
(tree-level) model and no extra resonances
Adding resonances does not necessarily
improve agreement…
Outlook
Harness fitting strategy to coupled-channels
calculations.
Improve evaluation of Occam factors.
Monte Carlo integration of likelihoods P(D|A)
over parameter space → theoretical error
bars (c.f. lattice QCD simulations).
Experiment: polarisation observables crucial.
Comparing
Different Models
How do we quantify the
intuitive feeling that
some models are better?
Difficulty of Problem
Typical correlation matrix
for the fitted free parameters
Chi-Squared surface very
complicated
Model Predictions Electroproduction
Polarisation transfer data
from CLAS
D. Carman et al., PRL 90
(2003) 131804
Core
S11
P11
P13
D13
p(e,e’k+)L
Recoil Polarisation
Data: J.W.C. McNabb et al., PRC 69 (2004) 042201 (CLAS)