1. Chaos in hadron spectrum Vladimir Pascalutsa European Centre for Theoretical Studies (ECT*), Trento, Italy Supported by Seminar @ JLab ( Newport News, USA, 7 Nov, 2007)

2. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 2 Outline An 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

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

7. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 7 More examples Kicked oscillator, … Compound nuclei Atoms in strong e.m. fields, dots, trapped ions Spectrum of the Dirac operator in lattice QED and QCD [ Berg et al., PRD59:097504,1999; Halasz & Verbaarschot PRL74:3920,1995 ] Free Expensive

8. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 8 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

9. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 9 Hadron spectrum (PDG 2002)  < 2.5 GeV What about the statistical properties? NNSD?

10. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 10 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

11. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 11 NNSD 1. no distinction on quantum numbers 2. yes distinction on quantum numbers

12. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 12 Moments of NNSD VP, EPJA 16 (2003)

13. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 13 Statistical errors 1 2 5 10 25 2nd Moment of Wigner at Various N

14. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 14 2 nd Moment of Poisson v. Wigner N

15. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 15 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?

16. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 16 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.

17. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 17 ‘C1’ set ‘L1’ set ‘L2’ set Quark Model Reanalysis N.N.S. Distribution: (Nammey & Muenzel, 2007)

18. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 18 C1 L1L2 Quark Model Reanalysis N.N.S. Distribution:

19. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 19 ‘C1’ set ‘L1’ set ‘L2’ set Quark Model Reanalysis Moment Distribution:

20. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 20 C1 L1L2 Quark Model Reanalysis Moment Distribution:

21. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 21 Conclusion no. 2 The NNSD of quark-model spectra follows Poisson distribution According to BGS, a signature of regular dynamics

22. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 22 One more quark model [ Markum, Plessas, et al, hep-lat/0505011 ]

23. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 23 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

24. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 24 Hadron spectra from lattice QCD [S. Basak, R.G. Edwards, G.T. Fleming, K.J. Juge, A. Lichtl, C. Morningstar, D.G. Richards, I. Sato, S.J. Wallace, Phys.Rev.D76:074504,2007 ]

25. Nov 7, 2007 V. Pascalutsa "Chaos in hadron spectrum" 25 Outlook (speculations) “Missing resonances”, will they be missed? 1. Removing states randomly from the quark-model spectra doesn’t help to reconcile with the Wigner, no correlations are introduced (Bohigas & Pato, (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” …