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| TU Darmstadt | Kristian König1 Structure of quarkonium states and potential models

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| TU Darmstadt | Kristian König2 Outline Introduction Phenemonological Approach –Positronium –Quarkonium Theoretical Approach –Lattice QM –Lattice QCD Decay of quarkonium

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| TU Darmstadt | Kristian König3 Introduction q q 2 1 q q 1 1 mesonquarkoniumquarks P some sets of quantum numbers are absent -> exotic some occur twice. There is the possibility of, e.g., mixing, as for the deuteron

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| TU Darmstadt | Kristian König4 Phenomenological Approach Positronium Bound e e –system Coulomb potential Solving the Schrödinger equation -> Energy eigenvalues + -

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| TU Darmstadt | Kristian König5 Positronium Schrödinger eq. Ansatz radial eq. energy eigenvalues, 4

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| TU Darmstadt | Kristian König6 Positronium Global Fine structure Hyperfine structure FS and HFS effects of same order B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

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| TU Darmstadt | Kristian König7 Model/potential which describes characteristics reasonable motivation produce concrete results can be directly confirmed or falsified by experiment may guide experimental searches Phenomenological Approach B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

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| TU Darmstadt | Kristian König8 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)

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| TU Darmstadt | Kristian König9 Solving the SE Central potential -> same ansatz as for positronium No analytic solution. But e.g. the Nikiforov-Uvarov method yields approximate analytic formulas whereand S. Kuchin, N. Maksimenko, Analytical Solution the Radial Schrödinger Equation for the Quark-Antiquark System (2013)

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| TU Darmstadt | Kristian König10 Results StatesPresent model Quadratic + Coulomb pot. Linear + Coulomb pot. + Constant Experiment 1S P S D P S S S ± Mass spectra of charmonium (in GeV) m =1.209 GeV, a = 0.2 GeV, b = 1.244, δ = GeV c 2 S. Kuchin, N. Maksimenko, Analytical Solution the Radial Schrödinger Equation for the Quark-Antiquark System (2013)

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| TU Darmstadt | Kristian König11 Mass spectra of cc and bb Similar structure -> model is flavor independent B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)

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| TU Darmstadt | Kristian König12 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)

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| TU Darmstadt | Kristian König13 Hyperfine structure B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009) K. Seth, Hyperfine interaction in heavy quarkonia (2012)

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| TU Darmstadt | Kristian König14 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 Fine structure where and where M is the average triplet mass t

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| TU Darmstadt | Kristian König15 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 Fine structure t ss M / GeVM t / GeV / GeV 1P(cc) P(bb) P(bb) J. Richard, An introduction to the quark model (2012)

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| TU Darmstadt | Kristian König16 More improvements Relativistic corrections Orbital mixing Coupling to decay channel Strong decay of quarkonia

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| TU Darmstadt | Kristian König17 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

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| TU Darmstadt | Kristian König18 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

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| TU Darmstadt | Kristian König19 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

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| TU Darmstadt | Kristian König20 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)

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| TU Darmstadt | Kristian König21 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 congurations

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| TU Darmstadt | Kristian König22 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 24 quark degrees of freedom per flavor ( x 2 particle/antiparticle, x 3 color, x 4 spin), 2 flavors 32 gluon degrees of freedom ( x 8 color, x 4 spin) In total: 32 x (2 x ) 83 x 10 dimensional integral lattice sites 4 6

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| TU Darmstadt | Kristian König23 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

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| TU Darmstadt | Kristian König24 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 B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009) J. Richard, An introduction to the quark model (2012)

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| TU Darmstadt | Kristian König25 Decay of quarkonia J. Richard, An introduction to the quark model (2012)

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| TU Darmstadt | Kristian König26 Thanks for the attention

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| TU Darmstadt | Kristian König27 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) 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)

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| TU Darmstadt | Kristian König28 Back-up

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| TU Darmstadt | Kristian König29 J. Richard, An introduction to the quark model (2012)

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| TU Darmstadt | Kristian König30

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