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
meson
quarkonium
quarks
http//en.wikipedia.org/wiki/Standard_Model

• some sets of quantum numbers are absent -gt
  exotic
• some occur twice. There is the possibility
  of, e.g., mixing, as for the deuteron

4
Phenomenological Approach

• Positronium
• Bound e e system
• Coulomb potential
• Solving the Schrödinger
• equation
• -gt Energy eigenvalues

-

http//en.wikipedia.org/wiki/Positronium
5
Positronium

• Schrödinger eq.
• Ansatz
• radial eq.
• energy eigenvalues

,
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
• -gt asymptotic freedom
• Increasing linear at large distances
• -gt confinement

B. Povh, Teilchen und Kerne, Springer-Verlag,
Berlin Heidelberg (2009)
9
Solving the SE

• Central potential -gt 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
States Present model Quadratic Coulomb pot. Linear Coulomb pot. Constant Experiment
1S 3.096 3.096 3.068 3.096
1P 3.255 3.433 3.526
2S 3.686 3.686 3.676 3.686
1D 3.504 3.767 3.829
2P 3.779 3.910 3.993 3.773
3S 4.040 3.984 4.144 4.040
4S 4.269 4.150 4.263
5S 4.425 4.421 0.004
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 -gt 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
• -gt Spin orbit coupling and tensor force
• Calculating the factors of the triplet P-states
  yield

15
Fine structure

• This can be inverted as
• The experiment shows that M is above the naive
  weighted average
• -gt One can estimate the size and the sign of V

t
ss
M / GeV Mt / GeV ltVlsgt / GeV ltVtgt / GeV
1P(cc) 3.520 3.525 0.035 0.01
1P(bb) 9.890 9.900 0.014 0.003
2P(bb) 10.260 10.260 0.009 0.002
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, Coulomblinearconstant,
  Coulombquadratic
• More elaborate potentials
• the linear part is smoothed by pair-creation
  effects
• the Coulomb term (at short distance) is weakened
  by asymptotic freedom -gt 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
• -gt 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)
• -gt Integral over a function space
• Weighting factor which contains the action
• -gt The pathes near to the classical one
  (minimum of Sx) have a strong influence to
  the observable
• -gt The pathes far away have a small influence

20
Lattice QM

• Discretize and compactify the time (1D)
• -gt 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 con?gurations

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

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)
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, 30903117 (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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Back-up
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J. Richard, An introduction to the quark model
(2012)
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