Systematic Study of Multiquark states with group theory method

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Systematic Study of Multi-quark states with
group theory method

• Fan Wang
• Dept. of Phys., Nanjing Univ.
• J.L. Ping and H.X. Huang
• Dept. of Phys., Nanjing Normal Univ.

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Outline

• I. Introduction.
• II. Fractional parentage expansion
• coefficients of symmetry bases
• and transformation coefficients between
• physical bases and symmetry bases.
• III. An example, penta-quark calculation
• six quark system had been calculated in
• a similar manner.

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I. Introduction

• Hadron spectroscopy only detects QCD interaction
  in color singlet states.
• Hadron interaction provides hidden color channel
  information of QCD interaction in principle.
• Multi-quark states detect QCD interaction in
  hidden color channel directly.
• Hidden color channel includes new physics which
  is hard to be studied with the hadron degree of
  freedom if it is not
• impossible.

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QCD quark benzene

• QCD interaction should be able to form a quark
  benzene consisted of six quarks

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Why multi-quark is still interested

• Penta quark might be died, but unquenched quark
  model has been born where one has the penta quark
  components within a baryon.
• Meson-baryon, baryon-baryon scatterings are
  there, its five, six quark systems.
• The multi quark states search will be continued,
  such as four quark states are hot now instead of
  penta quark.

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Lattice QCD results of the quark interaction PRL
86(2001)18,90(2003)182001,hep-lat/0407001
Suppose these lattice QCD results are
qualitatively correct, then multi-quark system is
a many body interaction multi-channel coupling
problem.
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• However lattice QCD seems to be impossible
• To provide the transition interaction between
• colorless channel and hidden color channel
• right now and
• this interaction is essential for mixing
• hidden color channels to colorless ones.
• So we could not but make model assumption
• and what is our quark delocalization color
• screening model (QDCSM) did.

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• To study multi-quark states one meets multi
• channel coupling with many body interaction,
• so one needs a powerful method to deal with.
• Group theory method is a power one.
• The fractional parentage expansion method
• reduces the matrix elements calculation of a
• many body Hamiltonian to be two body
• matrix elements, if only two body interaction
• is included, and overlap calculations.

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II.Fractional parentage expansion coefficientsof
symmetry bases and transformation coefficients
between physical bases and symmetry bases

• To use the FPE method, the many body states must
• be the group chain classified states, the
  symmetry
• bases (SB). And the corresponding FPE
  coefficients
• of SB should be calculated and can be calculated
  by
• group theory method The physical bases (PB) are
• usually not the SB and should be transformed to
  SB,
• the transformation coefficients should be
  calculated
• and can be calculated by group theory method.

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What does systematic mean

• Physics input is included in the Hamiltonian.
• Equipped with the FPEC and TC, different
• physics can be treated with the same set of
• FPEC and TC.
• In this sense one has a systematic method
• to do quark model calculation with non-
• relativistic and even relativistic quark models.
• F.Wang, J.L.Ping, T.Goldman, Phys.Rev.C51,1648,(19
  95).

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Flow Chart

• physical bases
• 
  with TC
• symmetry bases
• 
  with FPEC
• Hamiltonian matrix elements in symmetry bases
• 
  with TC
• Hamiltonian matrix elements in physical bases
• diagonalization the Hamiltonian in physical
  space
• 
  stored the TC and FPC
• computer programized

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Hard job has been done

• A new group theory method for calculating
• the FPEC and TC had been developed in
• the end of 1970s and the beginning of 1980s.
• J.Q.Chen, J.L.Ping and F. Wang, Group
  Representation Theory
• for physicists, (World Scientific, Singapore,
  2002).
• Comprehensive FPEC had been calculated
• and published.
• J.Q.Chen et al.,Tables of the Clebsch-Gordan,
  Racah and
• Subduction Coefficients of SU(n) Groups (World
  Sci., Singapore, 1987)
• Tables of the SU(mn) SU(m)xSU(n) Coefficients
  of Fractional
• Parentage (World Sci., Singapore, 1991).

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III.An example, penta quark calculation

• Four quark calculation, only SU(2) and
• SU(3) CGC is needed. The transformation
• coefficients between symmetry bases and
• physical bases are simple.
• For systems with 5 quarks and more one need the
  full machine of FPE and T methods.

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Physical bases

• Jaffe-Wilczek model states
• will be taken as the physical bases in this
• discussion,
• the baryon-meson model states
• has been taken as the physical bases as well,
• a different transformation coefficients between
  this
• new physical states and symmetry bases
• has been calculated too.
• If the space is large enough the results are
• the same

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di-quark states
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physical sates
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Symmetry bases
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Transformation
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FP expansion

• 4-gt22

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• 4-gt31

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A sample table
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Summary

• I. We have developed a powerful group theory
  method for multi-quark studies.
• 2. In general, penta-quark resonances are
  possible due to hidden color channels coupling.
• 3. For quark models, which fit the NN
  experimental data, the parity of ground state of
  penta-quark is negative. The lowest resonance is
  around 1.8 Gev. The SU(3) flavor symmetry is
  broken by large s quark mass.

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• 4. QDCSM and chiral quark models both fit the NN
  experimental data, they give similar penta-quark
  spectrum.
• This shows that the smeson in the meson
  exchange model can be replaced by QDCS mechanism.
• The spectrum of chiral soliton model is
  different from QDCSM and chiral qurak model ones,
  where the SU(3) flavor symmetry is broken by
  large s quark mass.

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• Thanks