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Structure of quarkonium states and potential models

Structure of quarkonium states and potential models. Outline. Introduction Phenemonological Approach Positronium Quarkonium Theoretical Approach Lattice QM Lattice QCD Decay of quarkonium. Introduction. quarks. quarkonium. meson. some sets of quantum numbers are absent -> exotic

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Structure of quarkonium states and potential models

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  1. Structure of quarkonium states and potential models

  2. Outline • Introduction • Phenemonological Approach • Positronium • Quarkonium • Theoretical Approach • Lattice QM • Lattice QCD • Decay of quarkonium

  3. Introduction quarks quarkonium meson • some sets of quantum numbers are absent -> exotic • some occur twice. There is the possibility of, e.g., mixing, as for the deuteron q q 1 1 q q 2 1 P http://en.wikipedia.org/wiki/Standard_Model

  4. Phenomenological Approach Positronium • Bound e e –system • Coulomb potential • Solving the Schrödinger equation -> Energy eigenvalues - + http://en.wikipedia.org/wiki/Positronium

  5. Positronium Schrödinger eq. Ansatz radial eq. energy eigenvalues , 4

  6. Positronium • Global • Fine structure • Hyperfine structure • FS and HFS effects of same order B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

  7. Phenomenological Approach Model/potential which describes characteristics • reasonable motivation • produce concrete results • can be directly confirmed or falsified by experiment • may guide experimental searches B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

  8. Cornell potential • Coulomb-like at small distances -> asymptotic freedom • Increasing linear at large distances -> confinement B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

  9. Solving the SE Central potential -> same ansatz as for positronium No analytic solution. But e.g. the Nikiforov-Uvarov method yields approximate analytic formulas where and S. Kuchin, N. Maksimenko, Analytical Solution the Radial Schrödinger Equation for the Quark-Antiquark System (2013)

  10. Results Mass spectra of charmonium (in GeV) m =1.209 GeV, a = 0.2 GeV , b = 1.244, δ = 0.231 GeV 2 c S. Kuchin, N. Maksimenko, Analytical Solution the Radial Schrödinger Equation for the Quark-Antiquark System (2013)

  11. Mass spectra of cc and bb B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009) Similar structure -> model is flavor independent

  12. Hyperfine structure • Spin-spin interaction causes hyperfine splittings • The interaction is only strong at small distances • Coulomb part is responsible (1 gluon exchange) • Similar to the positronium (1 photon exchange)

  13. Hyperfine structure K. Seth, Hyperfine interaction in heavy quarkonia (2012) B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

  14. Fine structure More interactions are needed to describe the splitting of e.g. the triplet states P , P , P -> Spin orbit coupling and tensor force Calculating the factors of the triplet P-states yield 3 0 3 1 3 2 where and where M is the average triplet mass t

  15. Fine structure This can be inverted as The experiment shows that M is above the naive weighted average -> One can estimate the size and the sign of V t ss J. Richard, An introduction to the quark model (2012)

  16. More improvements • Relativistic corrections • Orbital mixing • Coupling to decay channel • Strong decay of quarkonia

  17. Other potentials • Other simplest choices for the interquark potential: Powerlaw, logarithm, Coulomb+linear+constant, Coulomb+quadratic • More elaborate potentials • the linear part is smoothed by pair-creation effects • the Coulomb term (at short distance) is weakened by asymptotic freedom -> running coupling constant A.M. Badalian, V.D. Orlovsky, Yu.A. Simonov  Microscopic study of the string breaking in QCD

  18. Theoretical approach Lattice: Numeric method for the QM and the QFT Example to understand the basic principle -> 1D quantum mechanical oscillator Euclidean action of the harmonic oscillator Calculate the mean quadratic displacement in the ground state

  19. Lattice QM The path integral formalism is identical to the SE Integral over all possible pathes x(t) -> Integral over a function space Weighting factor which contains the action ->The pathes near to the classical one (minimum of S[x]) have a strong influence to the observable ->The pathes far away have a small influence

  20. Lattice QM Discretize and compactify the time (1D) -> The path integral is reduced to a normal finite dimensional integral M. Wagner, B-Physik mit Hilfe von Gitter-QCD (2011)

  21. Lattice QCD Euclidean action of the QCD field strength tensor quarkfields and gluonfields Ground state / vacuum expectation value Observable (function of the quark- and gluonfields) Weighting factor Integral over all possible quark- and gluonfield configurations

  22. Lattice QCD Discretize the space time with sufficent small lattice spacing Compactify the space time with sufficent large size Typical dimension of a QCD path integral 4 6 32 ≈ 10 lattice sites 24 quark degrees of freedom per flavor (x2 particle/antiparticle, x3 color, x4 spin), 2 flavors 32 gluon degrees of freedom (x8 color, x4 spin) 4 6 In total: 32 x (2 x 24 + 32) ≈ 83 x 10 dimensional integral

  23. Lattice QCD • Verification/falsification of the QCD by comparing the lattice results with the experiment • Predictions for hadrons and other QCD observables which are not seen yet experimentally • Solving the existing conflicts between experimental results and model calculations • Examples: • the mass of the proton has been determined theoretically with an error of less than 2% • Simulation of the forces in hadrons http://de.wikipedia.org/wiki/Gittereichtheorie

  24. Decay of quarkonia • Change of the excitation level via photon emission • Quark-antiquark annihilation into real or virtual photons or gluons (electromagnetic or strong) • Creation of one or more light qq pairs from the vacuum to form light mesons (strong interaction) • Weak decay of one or both heavy quarks J. Richard, An introduction to the quark model (2012) B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

  25. Decay of quarkonia J. Richard, An introduction to the quark model (2012)

  26. Thanks for the attention

  27. References • Special thanks to Prof. Wambach • A.M. Badalian, V.D. Orlovsky, Yu.A. Simonov, Microscopic study of the string breaking in QCD, Phys.Atom.Nucl. 76 (2013) 955-964 • W. Buchmüller, Quarkonia, North Holland, Amsterdam (1992) • E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. -M. Yan, Charmonium: The model, Phys. Rev. D 17, 3090–3117 (1978) • R. Gupta, Introduction to lattice QCD (1998) • S. Kuchin, N. Maksimenko, Analytical Solution the Radial Schrödinger Equation for the Quark-Antiquark System (2013) • B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009) • J. Richard, An introduction to the quark model (2012) • K. Seth, Hyperfine interaction in heavy quarkonia (2012) • M. Wagner, B-Physik mit Hilfe von Gitter-QCD (2011)

  28. Back-up

  29. J. Richard, An introduction to the quark model (2012)

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