1. Chaos in the N* spectrum Vladimir Pascalutsa European Centre for Theoretical Studies (ECT*), Trento, Italy Supported by Presented @ NSTAR 2007 Workshop ( Bonn, Germany, 5-7 Sep, 2007)

2. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 2 Outline A brief intro into (quantum) chaos Stat. analysis of empirical (PDG) N* spectrum VP, EPJA 16 (2003) Stat. analysis of theoretical (quark-model) spectra Fernandez-Ramirez & Relano, PRL 98 (2007) Cross-checks, statistical significance Nammey, Muenzel & VP, in preparation

3. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 3 Chaos cha·os [from Latin, from Greek khaos.] n. 1. A condition or place of great disorder or confusion. 2. A disorderly mass; a jumble: The desk was a chaos of papers and unopened letters. 3. often Chaos The disordered state of unformed matter and infinite space supposed in some cosmogonic views to have existed before the ordered universe: In the beginning there was Chaos… (Genesis) 4. Mathematics A dynamical system that has a sensitive dependence on its initial conditions. www.thefreedictionary.com

4. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 4 Classical chaos in dynamical systems (described by Hamiltonians): fully chaotic (ergodic) dynamics leads to homogeneous phase space Example: kicked top Other examples: double pendulum, stadium billiard

5. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 5 Define ‘quantum chaos’ “How does chaos lurks into a quantum system?..” (A. Einstein, 1917) Phase space? No, Heisenberg uncertainty… Sensitive initial conditions? No, Shroedinger equation is linear, simple time evolution Spectroscopy? Yes, the spectra of classically chaotic systems have universal properties Bohigas, Giannoni & Schmit, PRL 52 (1984) – BGS conjecture There are other definitions …

6. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 6 Quantum billiards (circular vs hart-shaped) Nearest-neighbor spacing distribution (NNSD) – Regular – Chaotic

7. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 7 Connection to Random Matrix Theory E. Wigner reproduced gross features of complicated (neutron- resonance) spectra by an ensemble of random Hamiltonians, i.e., eigenvalues of matrices filled with normally distributes random numbers. The NNSD of eigenvalues of a random matrix approximately described by the Wigner distribution Another interesting math that leads to the Wigner distribution, zeros of the zeta function (Riemann, 1859): NNSD

8. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 8 Hadron spectrum (PDG 2002)  < 2.5 GeV What about the statistical properties? NNSD?

9. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 9 Level Density Mean level density = inverse mean spacing: mean spacing: spacing: Consider spectrum, N+1 levels Because the NNSDs are normalized to unit mean spacing, one needs to make sure that mean spacing is constant over the entire spectrum

10. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 10 NNSD 1. no distinction on quantum numbers 2. yes distinction on quantum numbers

11. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 11 Moments of NNSD VP, EPJA 16 (2003)

12. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 12 Conclusion no. 1 The NNSD of experimental (low-lying) hadron spectrum is of the Wigner type (GOE class) According to the BGS conjecture, this is a signature of chaotic dynamics What about the quark models?

13. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 13 NNSD of quark models (baryons only) C. Fernandez-Ramirez & A. Relano, PRL 98 (2007). Capstick–Isgur model Exp. Bonn (L1) Bonn (L2) Loring, Metsch, et al.

14. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 14 ‘C1’ set ‘L1’ set ‘L2’ set Quark Model Reanalysis N.N.S. Distribution: (Nammey & Muenzel, 2007)

15. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 15 C1 L1L2 Quark Model Reanalysis N.N.S. Distribution:

16. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 16 ‘C1’ set ‘L1’ set ‘L2’ set Quark Model Reanalysis Moment Distribution:

17. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 17 C1 L1L2 Quark Model Reanalysis Moment Distribution:

18. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 18 Conclusion no. 2 The NNSD of quark-model spectra follows Poisson distribution According to BGS, a signature of regular dynamics What else?

19. Statistical errors 1 2 5 10 25 2nd Moment of Wigner at Various N

20. 2 nd Moment of Poisson v. Wigner Crossover at N=63 N

21. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 21 Conclusion The NNSD of experimental (low-lying) hadron spectrum is of the Wigner type (GOE class) The NNSD of quark-model spectra follows Poisson distribution Statistical significance of these results needs to be studied further

22. Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum" 22 Outlook (speculations) “Missing resonances”, will they be missed? 1. Removing states randomly from the quark-model spectra doesn’t help to reconcile the Wigner, no correlations are introduced (Bohigas & Plato, (2004), Fernandez-Ramirez & Relano (2007) ). 2. Sparsing the spectrum (removing a state if it’s too close to another one) helps – introduces correlation. Plausible, if experiment cannot resolute close states. Regular vs. chaotic quark models? why not a “stadium bag model” …