14.11.2013 | TU Darmstadt | Kristian König1 Structure of quarkonium states and potential models.

14.11.2013 | TU Darmstadt | Kristian König 1

Structure of quarkonium states and potential models


Outline

• Introduction

• Phenemonological Approach– Positronium– Quarkonium

• Theoretical Approach– Lattice QM– Lattice QCD

• Decay of quarkonium


Introduction

http://en.wikipedia.org/wiki/Standard_Model

q

q2

1q

q1

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


Phenomenological Approach

Positronium

• Bound e e –system

• Coulomb potential

• Solving the Schrödinger

equation

-> Energy eigenvalues http://en.wikipedia.org/wiki/Positronium

+ -


Positronium

Schrödinger eq.

Ansatz

radial eq.

energy eigenvalues ,4


Positronium

• Global

• Fine structure

• Hyperfine structure

• FS and HFS effects of same order

B. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)


Model/potential which describes characteristics• reasonable motivation

• produce concrete results

• can be directly confirmed or falsified by experiment

• may guide experimental searches

Phenomenological Approach



Cornell potential

• Coulomb-like at small distances

-> asymptotic freedom

• Increasing linear at large distances

-> confinement



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)


Results

States Presentmodel

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

Mass spectra of charmonium (in GeV)m =1.209 GeV, a = 0.2 GeV , b = 1.244, δ = 0.231 GeVc

2

S. Kuchin, N. Maksimenko, Analytical Solution the Radial Schrödinger Equation for the Quark-Antiquark System (2013)


Mass spectra of cc and bb

Similar structure -> model is flavor independentB. Povh, Teilchen und Kerne, Springer-Verlag, Berlin Heidelberg (2009)


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)


Hyperfine structure


K. Seth, Hyperfine interaction in heavy quarkonia (2012)


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 masst

3 3 30 1 2


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 / GeV Mt / GeV <Vls> / GeV <Vt> / GeV1P(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 0.002J. Richard, An introduction to the quark model (2012)


More improvements

• Relativistic corrections

• Orbital mixing

• Coupling to decay channel

• Strong decay of quarkonia


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


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


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


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)


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


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 flavors32 gluon degrees of freedom (x8 color, x4 spin)

In total: 32 x (2 x 24 + 32) ≈ 83 x 10 dimensional integral64

32 ≈ 10 lattice sites 4 6


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


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)


Decay of quarkonia

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


Thanks for the attention


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)


Back-up


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

14.11.2013 | TU Darmstadt | Kristian König1 Structure of quarkonium states and potential models.

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