Chaos in the N* spectrum Vladimir Pascalutsa European Centre for Theoretical Studies (ECT*), Trento, Italy Supported by Presented @ NSTAR 2007 Workshop

Chaos in the N* Chaos in the N* spectrumspectrumVladimir Pascalutsa

European Centre for Theoretical Studies (ECT*) , Trento, Italy

Supported by

Presented @ NSTAR 2007 Workshop ( Bonn, Germany, 5-7 Sep, 2007)

Sept 7, 2007 V. Pascalutsa "Chaos in the N* spectrum"

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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


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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


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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


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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 …


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Quantum billiards (circular vs hart-shaped)

Nearest-neighbor spacing distribution (NNSD)

– Regular– Chaotic


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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


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Hadron spectrum (PDG 2002)< 2.5 GeV

What about thestatistical properties?NNSD?


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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 makesure that mean spacing is constant over the entire spectrum


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NNSD 1. no distinction on quantum numbers

2. yes distinction on quantum numbers


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Moments of NNSD

VP, EPJA 16 (2003)


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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?


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NNSD of quark models (baryons only) C. Fernandez-Ramirez & A. Relano, PRL 98 (2007).

Capstick–Isgurmodel

Exp.

Bonn (L1)

Bonn (L2)

Loring, Metsch,et al.


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‘C1’ set‘L1’ set‘L2’ set

Quark Model Reanalysis

N.N.S. Distribution:(Nammey & Muenzel, 2007)


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C1

L1 L2

Quark Model ReanalysisN.N.S. Distribution:


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‘C1’ set‘L1’ set‘L2’ set

Quark Model ReanalysisMoment Distribution:


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C1

L1 L2

Quark Model ReanalysisMoment Distribution:


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Conclusion no. 2

The NNSD of quark-model spectra follows Poisson distribution

According to BGS, a signature of regular dynamics

What else?


Statistical errors

0tfxxt1

2

5

10

25

14N t

N

4N N tN 1

N 2nd Moment of Wigner at Various N


2nd Moment of Poisson v. Wigner

0tfxxt

Crossover at N=63

N


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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


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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” …

Documents

Chaos in the N* spectrum Vladimir Pascalutsa European Centre for Theoretical Studies (ECT*), Trento, Italy Supported by Presented @ NSTAR 2007 Workshop