1. Lambda(1405) as a 5Q from anisotropic lattice QCD
Noriyoshi ISHII (Univ. Tokyo) with Takumi DOI (Riken BNL) Makoto OKA (TITECH) Hideo SUGANUMA (Kyoto Univ.)
KEK 研究会（進化する原子核）
15＋5 minuites talk

2. Lattice QCD studies of standard 3Q baryons
Λ(1405) has been studied in the conventional lattice QCD framework in
Y. Nemoto et al., PRD68, (2003) already published ones
W. Melnitchouk et al., PRD67, (2003)
F.X. Lee et al., NPB(PS)119, 296(2003) [in proceedings of LATTICE2002]
T. Burch et al., hep-lat/
The mass spectra of low-lying baryons are reproduced rather good except for Λ(1405)
A significant discrepancy for Λ(1405) (Mass is overestimated more than 300 MeV)
This discrepancy is often attributed to two possibilities:
Chiral property may be important.
Λ(1405) may not be a conventional 3Q baryon
We are interested in this possibility !

3. ★ Two pictures of Λ(1405)
Λ(1405) JP=1/2(-), I=0, S=-1, Q=0 m=1406.5±4.0 MeV Γ =50±2 MeV
3 quark picture
5 quark picture
conventional 3Q [SU(3)f singlet] p-wave baryon (uds)
exotic “5Q” Λ(1405) was pointed out to be a 5Q object long time ago. [“bound state” of N and Kbar] (uud)(s ubar)... (s-wave)
The prev. lattice QCD calc. ⇔ 3Q picture:
quenched approx. prevents qqbar pair creations in the intermediate state.
⇒ Once it is created as a 3Q state, it propagates essentially as a 3Q state.
35 MeV is of typical size for a binding energy of a hadronic molecule
35 MeV
qqbar pair creation
Examples of hadronic molecule candidates
a0(980), f0(980) [K and Kbar] binding energy: 20 MeV
Θ+(1540) [N, π and K] binding energy: 40 MeV

4. ★ Two pictures of Λ(1405) (cont’d)
Our aim:
To investigate which picture is dominant for Λ(1405), we compare “5Q” calculation with “3Q” calculation
For 3Q calculation, we use 3Q operator
For 5Q calculation, we use 5Q operator
(3Q operator)
3Q component is domant in quenched QCD
(5Q operator)
5Q component is dominant in quenched QCD
※ We neglect the contribution from the disconnected diagram due to
calculational cost
interpretational “simplicity”.

5. There is a serious obstacle in 5Q calculation
★ Spectrum in Λ(1405) channel
３５ MeV
The gap between Λ(1405) and NKbar is only 35MeV !
A brute force way to solve this is to adopt a significantly finer lattice discretization. (This is unrealistic !)
It is quite difficult to distinguish one peak from the other, if their separation is only 35 MeV or so.
Recently, we find a way out of this obstacle: Hybrid Boundary Condition(HBC) proposed for Θ+(1540)

6. Spatial BC L～２fm is assumed ★ Conventional BC Periodic BC(PBC)
★ twised BC Hybrid BC(HBC)
anti PBC PBC
L～２fm is assumed
★ Spatial momenta of N and Kbar cannot vanish ! ⇒ NKbar threshold is raised by ～200 MeV
★ Λ(1405) remains unchanged !
⇒ gap is extended to ～200 MeV
Still we have to distinguish
Λ(1405) from Σπ

7. Spatial BC (cont’d) (L～２fm is assumed)
★ Conventional BC Periodic BC(PBC)
PBC
★ Further twisted BC “Hybrid BC” ( “HBC” )
anti PBC PBC
★ “ HBC ” can isolate Λ(1405)
★ comments: HBC is a sound BC from every point of view.
However, ”HBC” is valid only in the limit where qqbar anihilation diagram can be neglected.

8. Lattice QCD parameters
Gauge Config by standard Wilson gauge action:　
lattice size 123×96　　　　[(2.2fm)3×4.4fm in physical unit]
β=6/g2= 5.75
lattice spacing: from Sommer parameter r0.
anisotropic lattice Renormalized anisotropy: as/at=4 Precision measurement of temporal correlator .
#(gauge config) = 500
O(a) improved Wilson (clover) action for quarks
It is difficult to perform the lattice QCD calculation with physical quark mass mu～md～10 MeV.
Instead, we pick up several points in the range
ms ≦ mu, md ≦ 2ms
The results are extrapolated to the physical quark mass region (chiral extrapolation)

9. Numerical Results(effective mass plot)
★ mu=md=ms case
meff(t) serves as a “weighted average” of the energies of the states contributing to the time-slice t.
Σπ and N Kbar thresholds raised by ``HBC’’
5Q state is created at time-slice 0. “ “
N,Kbar,Σ, π cannot be in their rest frame
threshold is raised
plateau This region is saturated by the lowest lying state.
``HBC’’ data appear below the raised thresholds ⇒ This is not a scattering state of N and Kbar, but a compact 5Q state.
More statistics is needed to stabilize the plateau.

10. Numerical results (chiral extrapolation)
N,Kbar,Σ and π cannot be in their rest frame with ``HBC’’
5Q data(“HBC”)
threshold is raised
Σπ and N Kbar thresholds raised by ``HBC’’
m(N) + m(Kbar)
5Q data(PBC)
3Q data
m(Σ) + m(π)
chiral extrap.
The results are extrapolated to the physical quark mass region. (chiral extrapolation)
★ The chiral extrapolation of 3Q state m3Q = 1.79(8) GeV
★ The chiral extrapolation of 5Q state(``HBC’’) m5Q = 1.63(7) GeV ～ m(N)+m(Kbar) [on the lattice] The state appears below the threshold by MeV. ⇒ NOT scattering state, but compact 5Q state.
★ 5Q component is dominant in Λ(1405).
s-wave 5Q can become lighter than p-wave 3Q in the light quark mass retion
★ A level inversion of 3Q and 5Q takes place at mπ2～0.25GeV2. After that, 5Q state is lighter than 3Q state in the light quark mass region

11. Summary
We have studied Λ(1405) with anisotropic lattice QCD employing NKbar type 5Q op. (quenched lattice QCD, neglecting disconnected diagram)
We adopted ``HBC’’ (a spatial BC where N,Kbar, Σ,π have spatial momentum ～500 MeV), and find a compact 5Q state for Λ(1405).
The 5Q state appears significantly below the raised threshold (Σπ and N Kbar) ⇒ This 5Q state is NOT scattering state, but a compact 5Q state.
This 5Q state is lighter than the 3Q state in the light quark mass region. ⇒ 5Q state is dominant in Λ(1405).
Next things to do
More statistics.
What is the nature of this 5Q state ? Further investigation is required.
Contribution from disconnected diagram
mixing with 3Q
the cross crrelator calculations
genuine 5Q
loosly bound N and Kbar

12. Nuclear Force Project
N. Ishii, S. Aoki and T. Hatsuda, “Nuclear Force from Lattice QCD” A talk given at Lattice 2006, Tucson, Arizona USA, July 2006.
Recently, we have arrived at the method (CP-PACS method), which enables us to access the nuclear force in lattice QCD with existing super computer.
Lattice QCD result for NN BS wave func.
The 1st lattice QCD result for NN potential
PRELIMINARY
1S0
repulsive core
Tendency of attraction ? NO! This is consistent with zero.
heavy pion mass: mπ～500 MeV
small lattice volume: L～2.25 fm
CP-PACS method
Vcentral(r) [GeV]
More realistic calculation will appear in the near future.
QCD 1st principle investigation of the repulsive core is now allowed.
Application of CP-PACS method to YN and YY is interesting.