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Chaos in the N* spectrum

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). Outline. A brief intro into (quantum) chaos Stat. analysis of empirical (PDG) N* spectrum

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Chaos in the N* spectrum

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  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. 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 V. Pascalutsa "Chaos in the N* spectrum"

  3. Chaos www.thefreedictionary.com 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. 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 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 … V. Pascalutsa "Chaos in the N* spectrum"

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

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

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

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

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

  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? V. Pascalutsa "Chaos in the N* spectrum"

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

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

  15. Quark Model Reanalysis N.N.S. Distribution: C1 L1 L2 V. Pascalutsa "Chaos in the N* spectrum"

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

  17. Quark Model Reanalysis Moment Distribution: C1 L1 L2 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? V. Pascalutsa "Chaos in the N* spectrum"

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

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

  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 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” … V. Pascalutsa "Chaos in the N* spectrum"

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