Anisotropic lattice QCD studies of penta-quarks and tetra-quarks
N. Ishii (Univ. of Tokyo)
in collaboration with
T. Doi (Riken BNL)H. Iida (TITECH)Y. Nemoto (Nagoya Univ.)M. Oka (TITECH)F. Okiharu (Nihon Univ.)H. Suganuma (Kyoto Univ.)K. Tsumura (Kyoto Univ.)
Plan of the talk:1 Introduction2 General Formalisms3 Numerical Results4 Summary/Discussion(5 Tetra-quarks(4Q))
See Phys.Rev.D71,034001(2005); D72,074503(2005) for detail.
START
1.Introduction
One of the most important issues for Θ+(1540) is to understand
its extremely narrow decay width Γ<1 MeV.
Several ideas have been proposed as
a. I=2 assignment
b. Jaffe-Wilczek’s diquark picture ⇒ JP=1/2(+) and 3/2(+)
c. πKN hepta-quark picture ⇒ JP=1/2(+)
d. The string picture
e. JP=3/2(-) assignment ⇒ JP=3/2(-) In this talk, we are mainly interested in JP=3/2(±) possibilities:
1. We first present our numerical results on JP=1/2(±) penta-quarks brieflyemplyoing a diquak-type interpolating fieldusing a flavor dependent boundary condition(HBC)
2. We then present our numerical results on JP=3/2(±) penta-quarksemploying three Rarita-Schwinger interpolating fieldsusing 1000 gauge field configurations for high statistics
Lattice QCD Setup:
1. Gauge Config by standard Wilson gauge action: a. Lattice size : 123×96 [(2.2fm)3×4.4fm in physical unit]
b. β= 5.75
c. Lattice spacing: from Sommer parameter r0.
d. Anisotropic latticeRenormalized anisotropy: as/at=4for accurate measurements of correlators and masses
e. #(gauge config) = 504 for JP=1/2(±) = 1000 for JP=3/2(±)
2. O(a) improved Wilson quark (clover) action.The quark mass covers the region ms < mq < 2 m s
3. Smeared source to reduce higher spectral contributions
2.General Formalism
[GeV]4.41 a
0.1240 0.1230 0.1220 0.1210
656(2) 784(1) 893(1) 1005(1)
1011(5) 1085(4) 1162(3) 1240(3)[MeV]m
[MeV]m
2.2 fm
Finer lattice spacing along the temporal direction
tim
e
The interpolating fields
ddcbTaabcddcb
Taabc usddCudsudCu 55NK*-type
cddbTaabccddb
Taabc usddCudsudCu 55color-twisted NK*-type
geTdb
Tacfgdefabc sCdCudCu 555 diquark-type
★ Three Rarita-Schwinger interplating fields for JP=3/2(±) states:
★ A diquark-type interplating fields for JP=1/2(±) states: T
cgTfe
Tdbfgadeabc sCCdudCu 5
We consider the following iso-scalar interpolating fields:
(scalar) (pseudo scalar)
(scalar) (vector)
Hybrid Boundary Condition(HBC)We utilize a flavor dependent spatial BC (Hybrid BC (HBC)). (We use HBC in addition to the standard periodic BC(PBC))
quark contents
spatial BC minimum momentum
N anti-periodic BC
K,K*
anti-periodic BC
periodic BC
udduud ,sdsu ,suudd
LLLp ,,min
0,0,0min p
LLLp ,,min
Lp /3min
0min p
Hybrid Boundary Condition(HBC)
L
L
L
The spatial BOX Spatial momentum is quantized due to finite volume effect:
1. periodic BC:
2. anti-periodic BC:
L
np ii
2
L
np ii
12
u quark spatially anti-periodic BC
d quark spatially anti-periodic BC
s quark spatially periodic BC
Lp /3min
Cosequence on hadrons
◎ NK and NK* threshold energies(s-wave) are raised due to , ◎ Θ+,if it is a compact resonance, will not be affected so much.
HBC can be used to determine whether a state is a compact resonance or not.
※ In the case of p/d wave, HBC serves as another boundary condition(other than PBC).
0min pWit
h HBC
3.Numerical Results: JP=1/2(±) states (effective mass plots)
“Effective mass” is defined as
which can be considered as an “weighted average” of massesat each time-slice t.
)1(
)(log)(
tG
tGtmeff
plateau
JP=1/2(-)
plateau
JP=1/2(+)
KNth mmE NK-threshold (s-wave)
2min
22min
2 pmpmE KNth
NK-threshold (p-wave)
1. JP=1/2(-) state:A state appears slightly above the NK threshold (mN+mK).
2. JP=1/2(+) state:A state appears above the raised NK threshold (due to the finite box).⇒ rather massive !
Excite
d st
ate
cont
ribu
tion
s
are
redu
cing
A single state dominate the correlator G(t) in this region.
Chiral extrapolation (JP=1/2(±))
At physical point
(1) JP=1/2(+): 2.24(11) GeV
(2) JP=1/2(-): 1.75(3) GeV
NK threshold (p-
wave)
NK threshold (s-
wave)
1. Our data does not support a low-lying JP=1/2(+) penta-quark.
2. For JP=1/2(-) state, the mass(1.75 GeV) is OK !Still, it is necessary to check whether it is not an NK scattering state but a compact resonance.⇒ HBC analysis
HBC analysis (JP=1/2(-) state)
PBC HBC
KNth mmE NK-threshold (PBC)
2min
22min
2 pmpmE KNth
NK-threshold (HBC)
1. NK(s-wave) threshold is raised up by 210 MeV.
2. The best fit mass m5Q is raised up by a similar amount.
★ No compact 5Q resonance exists in the region:
★ The state observed in JP=1/2(-) is an NK scattering state.
2min
22min
2 pmpmmmm KNKN
Numerical Results: JP=3/2(-) state (effective mass plot)
This correlator is too noisy !Fit is not performed.
The plateaus appearabove the NK*-threshold and above the raised NK threshold.
plateau
plateau
twisted
×
“Effective mass” is defined as
which can be considered as an “weighted average” of massesat each time-slice t.
)1(
)(log)(
tG
tGtmeff
Chiral extrapolation (JP=3/2(-))
○(circle) from NK*-type correlator
□(box) from color-twisted NK*-type correlator
Physical quark mass region
In the physical quark mass region
(1) NK*-type: m5Q= 2.17(4) GeV
(2) Color-twisted NK*-type: m5Q= 2.11(4) GeV
No evidence for a low-lying 5Q state
HBC analysis suggeststhese states are NK*(s-wave) scattering states
Due to the limited
time, we cannot
show HBC
analysis.
JP=3/2(+) state (effective mass plot)
The plateaus appearabove the raised NK*-threshold and above the raised NK threshold.
plateau
twisted
plateau
plateau
Chiral extrapolation (JP=3/2(+))
○(circle) from NK*-type correlator
□(box) from color-twisted NK*-type correlator
△(triangle) from diquark-type correlator
In the physical quark mass region,
(1) NK*-type: m5Q= 2.64(7) GeV
(2) Color-twisted NK*-type: m5Q= 2.48(10) GeV
(3) Diquark-type: m5Q=2.42(6) GeV
No evidence for a low-lying 5Q states.
Physical quark mass region
NK*(p-wave) scattering states
N*K*(s-wave) scattering state
HBC analysis suggests:
Due to the limited time, we cannot show HBC analysis.
1. We have studied spin=1/2 and 3/2 penta-quarks by using the anisotropic lattice QCD. For acuracy,(a) renormalized anisotropy as /at = 4(b) O(a) improved Wilson (clover) action for quarks(c) smeared source(d) large number of gauge configurations: Ncf=1000 for JP=3/2(±)
2. JP=1/2(±) [with a diquark-type interpolating field]
i. JP=1/2(-) state: JP=1/2(+) state:
ii. HBC analysis shows that the state at 1.75 GeV is an NK scattering state.
3. JP=3/2(±) [A large statistics as Ncf=1000 has played an important role.]
i. Three interpolating fields (NK*-type, color-twisted NK*-type, diquark-type)
ii. Only massive states after the chiral extrapolation:
JP=3/2(-) state: JP=3/2(+) state:
iii. HBC analysis suggests that these 5Q states are NK* and N*K* scattering states.
4. Following possibilies would be interesting for Θ+(1540):i. Small quark mass effect(and/or elaborate chiral extrapolation)
ii. Large spatial volume
iii. Dynamical quarks
iv. Elaborate interpolating fields to fit the diquark picture
v. πKN hepta-quark picture
4. Summary/discussion
GeV17.2,11.25 Qm GeV42.2,48.2,64.25 Qm
GeV25.25 QmGeV75.15 Qm
Too heavy
to
be identifi
ed
as Θ
+ (1540)
See for detail:
Phys. Rev. D71,034001
(2005)
Phys. Rev. D72,074503
(2005)