1. Multiquark Systems with Strangeness
Makoto Oka
岡　真
Tokyo Institute of Technology
SNP06
張家界, 中華人民共和国
September, 2006

2. Contents Quark model for multiquark systems Lattice QCD for exotics
ground states, excited states
scalar nonets, pentaquarks
Lattice QCD for exotics
pentaquarks
5-quark components of (1405)
QCD sum rules for exotics
mixings of exotic multi-quark components
Conclusion
岡 真 (M. 張家界 (9/6/06)

3. Constituent quarks in hadrons
Quark model
Constituent quarks in hadrons
mesons qq
baryons qqq
exotics qqqq, qqqqq, . . .
Constituent quarks must be effective degrees of freedom which are valid only in low energy regime. They must have the same "conserved charges" as the QCD quarks: baryon 1/3, spin 1/2, color 3, flavor 3.
The constituent quarks have dynamical masses induced by chiral sym breaking of QCD vacuum
岡 真 (M. 張家界 (9/6/06)

4. Ground states
Ground-state mesons (PS+V nonets) and baryons (8+10) are well described by the quark model based on SU(3) x SU(2) -> SU(6)
mass spectrum
em-weak properties, ex. magnetic moments
Required (minimum) dynamics consists of confinement + color-magnetic interactions.
with a few exceptions
“πρ problem” -> chiral symmetry
“η' problem” -> UA(1) anomaly
岡 真 (M. 張家界 (9/6/06)

5. Excited states: Scalar mesons
“Quark-shell-model” based on SU(6) x O(3) encounters difficulties for excited states.
scalar meson nonets: 3P0: 0++
0+: σ(~600), f0(980), a0(980), κ(900)*
The ordering is not "natural" as a qq nonet.
expected m() ~ m(a0) < m(f0)
observed m() < m(a0) ~ m(f0)
No spin-orbit partners in the vicinity: 3PJ: 1+, 2+
*κ(900) : indicated in K final states of J/ and
D meson decays, but not yet established.
岡 真 (M. 張家界 (9/6/06)

6. Scalar mesons as tetra-quark states
A possible explanation is to consider them as exotic tetra-quark states.
composed of diquarks in flavor 3
Then the expected spectrum from strange quark counting is m() < m(a0) ~ m(f0)
岡 真 (M. 張家界 (9/6/06)

7. Exotic multi-quark systems
Multi-quark components may help to explain anomalies in the excited hadron spectrum.
ex. scalar nonets, (1405), X(3872)
Dynamics: Why is (1405) likely to be 5q?
(1405) J= 1/2-, flavor singlet
☆ uds L=1 orbital excited states with spin 1/2
=> J=1/2- and 3/2-
☆ udsuu, L=0 ground state
(ud)(su) u . .
s=0 s=0 S=1/2 => J=1/2 isolated
diquarks (1520) 3/2-
岡 真 (M. 張家界 (9/6/06)

8. Exotic multi-quark systems
Necessary techniques have been developed in studying purely exotic hadrons.
+ B=1, S=1, Y=2
-- Y=-1, I=3/2 Q+
X-- X+
Nakano et al., (2003)
M 1540 MeV with  1 MeV or less
B=1, S=+1, 5-quark ( u2d2s ) bound state
岡 真 (M. 張家界 (9/6/06)

9. Quark Models for Pentaquark +
How does the known dynamics of the quark model work for +?
 "shell-model" approaches (+ correlations)
Carlson et al, Jaffe-Wilczek, Karliner-Lipkin, Jennings-Maltman, Bijker et al
 variational approaches
Takeuchi-Shimizu, Hiyama et al, Stancu et al, Matsumura et al,
岡 真 (M. 張家界 (9/6/06)

10. Quark Models for Pentaquark +
A variational method calculation
by Takeuchi, Shimizu (2006)
models
Large + mass, small splitting of 1/2 and 3/2
岡 真 (M. 張家界 (9/6/06)

11. Quark Models for Pentaquark +
Variational solution of 5 quark system coupled to NK continuum states
E. Hiyama et al. PLB633 (2006)
L=0
L=1
NK threshold
NK threshold
岡 真 (M. 張家界 (9/6/06)

12. So, . . .
The quark model is based on the chiral symmetry-broken QCD, which contains massive quarks as effective degrees of freedom.
The quark model describes most ground state mesons and baryons fairly well.
The quark model, however, is less effective for excited mesons and baryons. Some of the discrepancies (scalar mesons, (1405), ...) suggest exotic multiquark components.
The pentaquark + is a mystery. No honest calculation reproduces its mass and width simultaneously.
The similar calculations are underway for various multi-quark candidates.
岡 真 (M. 張家界 (9/6/06)

13. QCD
QCD is the theory of hadrons. It does not a priori exclude exotic multi-quark bound (resonance) states (as far as they are color-singlet).
Direct applications of QCD to exotics are desperately needed.
Lattice QCD
QCD sum rules
岡 真 (M. 張家界 (9/6/06)

14. Lattice QCD
Lattice QCD is powerful in understanding non-perturbative physics of QCD:
vacuum structure, phase diagrams, mass spectra of ground state mesons and baryons, interactions of quarks, . . .
Lattice QCD has two (severe) restrictions:
Light quarks are too expensive. It requires (often drastic) extrapolation from large-quark-mass simulations.
No direct access to resonance poles is possible. It is hard to distinguish resonances from hadron scattering states. Real number simulations can not access complex poles.
Applications to exotic hadrons are yet limited.
岡 真 (M. 張家界 (9/6/06)

15. LQCD for Pentaquark + Quenched LQCD for + (J = 1/2)
Csikor, Sasaki, Chiu, Mathur, Ishii, Alexandrou, Takahashi, Lasscock, Holland, Negele, . . .
Quenched LQCD for + (J = 3/2)
Lasscock, Ishii
Anisotropic Lattice QCD studies of + (J = 1/2 and 3/2)
Ishii et al., PRD 71 (2005)
Ishii et al., PRD 72 (2005)
lattice size 123×96, β= 5.75 : (2.2fm)3×4.4fm in phys. unit
anisotropic lattice (as/at=4), 504 configurations
O(a) improved clover Wilson quark
with quark mass mu, d = ms to 2ms
(linearly) extrapolated to the physical mass
岡 真 (M. 張家界 (9/6/06)

16. Summary of LQCD results for +
J =1/2 diquark-operator (DQ)
1/2– 1.75 GeV: consistent with NK(L=0) scatt.
The Hybrid b.c. method rejects 5Q states.
1/ GeV: heavy
J =3/2 DQ, NK*, color-twisted NK* opearators
3/2– 2.11 GeV: consistent with NK(L=0) thres.
3/ GeV: consistent with NK(L=1) thres.
3/ GeV: consistent with NKL=0) thres.
No candidate for compact 5Q state is found.
Most LQCD parties agree with these results.
The QCD sum rules give the consistent results.
岡 真 (M. 張家界 (9/6/06)

17. After the chiral extrapolation
J = 1/2  pentaquark
Ishii et al., PRD 71 (2005)
NK threshold (p-wave)
NK threshold (s-wave)
After the chiral extrapolation
(1) Positive parity: (11) GeV (2) Negative parity: 1.75(3) GeV
岡 真 (M. 張家界 (9/6/06)

18. J =3/2–  pentaquark Physical quark mass NK*-type: m5Q= 2.17(4) GeV
Ishii et al., PRD 72 (2005)
岡 真 (M. 張家界 (9/6/06)

19. Exotic multi-quark components
How do we study the exotic multi-quark components of hadrons in QCD?
Quenched Lattice QCD
(3Q operator)
3Q is dominant in quenched LQCD
(5Q operator)
5Q-dominant diagram
岡 真 (M. 張家界 (9/6/06)

20. (1405) with 3Q operators
Y. Nemoto et al., PRD68, (2003)
W. Melnitchouk et al., PRD67, (2003)
F.X. Lee et al., NPB(PS)119, 296(2003) [LATTICE2002]
T. Burch et al., hep-lat/
Large discrepancy between the lattice prediction and the observed mass of Λ(1405) (about 300 MeV)
Y. Nemoto et al., PRD68, (2003)
岡 真 (M. 張家界 (9/6/06)

21. Λ(1405) with 5Q operator Technical problem to be solved
Spectrum in Λ(1405) channel
３５ MeV
It is difficult to separate (1405) from the NK threshold (scattering states).
岡 真 (M. 張家界 (9/6/06)

22. Hybrid BC method: Twist Spatial BC
N.Ishii, T.Doi, H.Suganuma, MO
★ Twisted BC “Hybrid BC” ( “HBC” )
anti PBC PBC
★ Conventional BC Periodic BC(PBC)
PBC
(L～２fm is assumed)
note: ”HBC” is valid only in the limit where qq anihilation can be neglected.
“HBC” will enable to isolate Λ(1405).
岡 真 (M. 張家界 (9/6/06)

23. HBC method for + Twist spatial BC for u, d quarks
Ishii et al., PRD71 (2005) , PRD72 (2005)
u quark
anti-periodic BC
d quark
s quark
periodic BC
NK(s-wave) threshold
No compact 5q resonance of J=1/2- was found.
岡 真 (M. 張家界 (9/6/06)

24. 5-quark component of Λ (1405)
5Q data(“HBC”)
Σπ and NK thresholds raised by ``HBC’’
m(N) + m(Kbar)
5Q data(PBC)
3Q data
m(Σ) + m(π)
chiral extrapl.
The chiral extrapolation 3Q operator m3Q = 1.79(8) GeV 5Q operator m5Q = 1.63(7) GeV ～ m(N)+m(K) [on lattice]
The preliminary result suggests that Λ(1405) is dominated by 5Q.
岡 真 (M. 張家界 (9/6/06)

25. Exotic multiquark components
How large is the mixing probability of the exotic multi-quark components?
There is NO well-defined mixing prob. in the field theory, because there exists no conserved current corresponding to the number of quarks: N(q)+N(q).
It depends on the choice of the quark operator.
Nevertheless, there are a set of well-defined quantities, which might be useful in connecting the quark model description
although they depend on the definition (normalization) of the operators.
岡 真 (M. 張家界 (9/6/06)

26. "mixing angle" Then one can evaluate the "mixing angle":
☆ QCD sum rule is applied.
a. 4-quark components of a0(I=1; 0+) meson
b. 5-quark components of (singlet; 1/2-) baryon
T. Nakamura, J. Sugiyama, T. Nishikawa, N. Ishii, M.O.
岡 真 (M. 張家界 (9/6/06)

27. Example: scalar a0 A set of interpolating fields
Define a genuine 4-quark operator J4’ so that J2 component of J4 is subtracted.
岡 真 (M. 張家界 (9/6/06)

28. QCD Sum Rule for scalar mesons
qq vs 4-quark in a0 meson
qq + 4-quark
pure 4-quark qq
We conclude that the "4-quark component" is
dominant in the scalar meson a0.
岡 真 (M. 張家界 (9/6/06)

29. Conclusion
A technique using the hybrid boundary conditions seems to work in lattice QCD to distinguish compact states from scattering states.
The quenched LQCD suggests that (1405) is predominantly a 5-quark state.
The mixing of the multi-quark components are not well-defined from the field theoretical viewpoint.
However, one can define a set of useful mixing amplitudes using the well-defined matrix elements of local operators. Whether this definition of the mixing is relevant in the quark model is yet an open problem.
The QCD sum rule indicates a large 4-q components in scalar mesons.
岡 真 (M. 張家界 (9/6/06)