A Genetic Algorithm Analysis of N* Resonances Outline:- Analysis of N* contribution to  p → K +  How does using a Genetic Algorithm help? How much can an analysis of the data currently tell us? Conclusions and Outlook D. G. Ireland Department of Physics and Astronomy University of Glasgow

Analysis of “World” data (1999) With D 13 Without D 13 Mart & Benhold, Phys. Rev. C (R) (1999) “Evidence” of missing D 13 resonance

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 Evolution of population, consisting of... Each solution has a “fitness” Selection Crossover Mutation + iteration towards convergence...

Comparison: GA vs. MINUIT

Phase 1: Calculation with additional D 13 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: S 11 (1650), P 11 (1710) and P 13 (1720) contains an extra resonance of mass 1895 MeV, with different quantum numbers: S 11, P 11, P 13, D 13 used 100 calculations (GA + MINUIT) New photon beam polarisation (SPRing-8), and electroproduction data (Jlab Hall C) used in fit.

Results: Total Cross-SectionPhoton Beam Asymmetry Core S 11 P 11 P 13 D 13

Occam's Razor William of Occam (or Ockham, ca ) “Pluralitas non est ponenda sine necessitate” - plurality should not be posited without necessity For models A and B, calculate ratio of posterior probabilities:- ←ratio of likelihoods Occam factor (approximate) Best fit to data

Table of Results Raw scores indicate D 13 most likely More sophisticated comparison favours P 11 Data support hypothesis of extra resonance Situation still not clear ModelCoreS_11P_11P_13D_13 Raw Chi-Squared Number of free parameters Occam Factor Ratio of Posterior Probability

Model Predictions – New Measurements Linear Circular Beam – recoil polarisation Core S 11 P 11 P 13 D 13

Phase 3: Lots more data! e.g. J.W.C. McNabb et al., PRC 69 (2004) Two approaches:- 1)Use parameters obtained in previous fit for Core, S 11, P 11, P 13, D 13 models 2)Re-fit, but with two models: Core and S 11 +P 11 +P 13 +D 13 (all hypothetical resonances together)

Angular Distributions Data: J.W.C. McNabb et al., PRC 69 (2004) (CLAS)

Differential Cross Sections Data: J.W.C. McNabb et al., PRC 69 (2004) (CLAS)

Beware many parameters! Full calculation penalised for many parameters. Occam factor calculation very approximate! Situation inconclusive “Full” evaluation of integrals necessary → MCMC ModelCoreFull Raw Chi-Squared Number of free parameters410 Likelihood Occam Factor5.625e e-15 Posterior Probability3.825e e-16

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) Core S 11 P 11 P 13 D 13 p(e,e’k + ) 

Recoil Polarisation Data: J.W.C. McNabb et al., PRC 69 (2004) (CLAS)