Anisotropic lattice QCD studies of penta-quarks and tetra-quarksN. Ishii (Univ. of Tokyo)in collaboration withT. 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 Introduction 2 General Formalisms 3 Numerical Results 4 Summary/Discussion (5 Tetra-quarks(4Q))See Phys.Rev.D71,034001(2005); D72,074503(2005) for detail.START
- 1.IntroductionOne of the most important issues for +(1540) is to understand its extremely narrow decay width
Spin of + has not yet been determined experimentally.JP=3/2(-) assignment can solve the puzzle of the narrow decay width. (proposed by A.Hosaka et al., PRD71,074021(2005).) Advantage: (a) It allows the configuration of (0s)5. (b) It only decays into a d-wave KN state. Suppressed overlap between these two states The decay width is expected to be significantly narrow. Disadvantage: (a) The constatituent quark picture suggests such a 5Q state is quite heavy. ( due to the color-magnetic interaction) Since it is not apriori clear whether such a conventional framework can be applied to a new exotic 5Q system as +(1540) or not, it is desirable to perform a direct lattice QCD calculation. JP=3/2(+) is also interesting, which is suggested by the diquark picture.JP=3/2() possibilitiesHOWEVERThe total spin(parity) is 1/2(+) or 3/2(+). 3/2(+) penta-quark may have a narrow decay width !p-wave
Lattice QCD studies of the penta quarks There are several lattice QCD calculations of penta-quarks available. (published one only)SPIN 1/2SPIN 3/2Most of them are devoted to spin 1/2 states except for the recent two.
F.Scikor et al.JHEP11('03)070Wilsonfused NK(2x2 diagonalize)volume dep.(mass?)negative parity
S.SasakiPRL93('04)15200Wilsondiquarkstandard analysisnegative parity
T.-W.Chiu et al.PRD72('05)034505.domain wall3x3 diagonalize(diquark,NK,fused NK)standard analysispositive parity
N.Mathur et al.PRD70('04)0745008overlapNKvolume dep.(mass,spec. weight)not observed
N.Ishii et al.PRD71('05)034001improved Wilsondiquarkhybrid boundary conditionnot observed
C.Alexandrou et al.hep-lat/0503013Wilson2x2 diagonalize(diquark,NK)volume dep.(mass,spec. weight)negative parity
T.T.Takahashi et al.PRD71('05)114509.Wilson2x2 diagonalize(NK, fused NK)volume dep.(mass,spec. weight)negative parity
B.G.Lasscock et al.PRD72('05)014502.FLICNK, fused NK, diquark(3x3 diagonalize)mass spliting analysisnot observed
F.Scikor et al.hep-lat/0503012Wilson"14x14" diagonalize(spatially non-trivial one)volume dep.(mass)not observed
K.Holland et al.hep-lat/0504007fixed pt action2x2 diagonalize(fused NK, NK)quark mass dependencenot observed
B.G.Lasscock et al.hep-lat/0504015(PRD accepted)FLICNK^* (spin 3/2)mass spliting analysis(low-lying) JP=3/2^+
N.Ishii et al.PRD72('05)074503improved Wilsondiquark, N*K, twisted N*K(spin 3/2)hybrid boundary conditionnot observed
Lattice QCD Setup:Gauge Config by standard Wilson gauge action:Lattice size : 12396[(2.2fm)34.4fm in physical unit]= 5.75Lattice spacing: from Sommer parameter r0.Anisotropic lattice Renormalized anisotropy: as/at=4 for accurate measurements of correlators and masses#(gauge config) = 504 for JP=1/2() = 1000 for JP=3/2() O(a) improved Wilson quark (clover) action. The quark mass covers the region ms < mq < 2 m s
Smeared source to reduce higher spectral contributions2.General Formalism
The interpolating fieldsNK*-type color-twisted NK*-type diquark-type Three Rarita-Schwinger interplating fields for JP=3/2() states: A diquark-type interplating fields for JP=1/2() states: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))Hybrid Boundary Condition(HBC)Cosequence on hadrons NK and NK* threshold energies(s-wave) are raised due to , +,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).With HBC
quark contentsspatial BCminimum momentumNanti-periodic BCK,K*anti-periodic BCperiodic BC
u quarkspatially anti-periodic BCd quarkspatially anti-periodic BCs quarkspatially periodic BC
3.Numerical Results: JP=1/2() states (effective mass plots)Effective mass is defined as
which can be considered as an weighted average of masses at each time-slice t. JP=1/2(-) state: A state appears slightly above the NK threshold (mN+mK).JP=1/2(+) state: A state appears above the raised NK threshold (due to the finite box). rather massive !Excited state contributions are reducingA single state dominate the correlator G(t) in this region.
A plateau in the effective mass plot indicates G(t) is saturated by a single-state contribution.Effective mass plotThe correlator can be expressed as a sum:The Effective Mass is defined as:negligible !If G(t) is dominated by a single state:Then we have,This can be considered as average of masses at each time-slice t(Constant effective mass)
Chiral extrapolation (JP=1/2())At physical point(1) JP=1/2(+): 2.24(11) GeV (2) JP=1/2(-): 1.75(3) GeVNK threshold (p-wave)NK threshold (s-wave)Our data does not support a low-lying JP=1/2(+) penta-quark. 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)PBCHBCNK(s-wave) threshold is raised up by 210 MeV.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.
Combining the results from all the other hopping parameters. data points The best fit value over the plateau. solid lines NK(s-wave) thresholdThe states observed in are NK scattering states !
Numerical Results: JP=3/2(-) state (effective mass plot)This correlator is too noisy ! Fit is not performed.The plateaus appear above the NK*-threshold and above the raised NK threshold.plateauplateautwisted
NK*(JP=3/2(-))NK*(s-wave) threshold is raised up by 179 MeV.NK(d-wave) threshold is lowered down by 66 MeV.Best-fit(m5Q) is raised up by 80 MeV. Its value is almost consistent with NK*-threshold(s-wave). This state is an NK* scattering state. A large number of config. Nconf=1000 has played a crucial role.
color-twisted NK*(JP=3/2(-))The situation is similar to the NK*-correlator.NK*(s-wave) threshold is raised up by 179 MeV.NK(d-wave) threshold is lowered down by 66 MeV.Best-fit(m5Q) is raised up by 90 MeV. Its value is almost consistent with the NK*-threshold. This state is also an NK* scattering state.twistedtwisted
Chiral extrapolation (JP=3/2(-))(circle) from NK*-type correlator(box) from color-twisted NK*-type correlatorPhysical quark mass regionIn the physical quark mass regionNK*-type: m5Q= 2.17(4) GeVColor-twisted NK*-type: m5Q= 2.11(4) GeVNo evidence for a low-lying 5Q stateHBC analysis suggests these states are NK*(s-wave) scattering statesDue to the limited time, we cannot show HBC analysis.
JP=3/2(+) state (effective mass plot)The plateaus appear above the raised NK*-threshold and above the raised NK threshold.plateautwistedplateauplateau
NK*(JP=3/2(+))Changes in the two-particle spectrum are too small in JP=3/2(+) channel.N*K*(s-wave) threshold is raised up by 170 MeV.NK*(p-wave) threshold is lowered down by 57 MeV.NK(p-wave) threshold is lowered down by 66 MeV.Best-fit(m5Q) is raised up by 60 MeV. Its value coincides with N*K*(s-wave) threshold.This state is a N*K*(s-wave) threshold.
Color-twisted NK*(JP=3/2(+))Changes in two-particle spectrum are too small in JP=3/2(+) channel.NK*(p-wave) threshold is lowered down by 57 MeV.NK(p-wave) threshold is lowered down by 66 MeV.Best-fit(m5Q) is raised up by 90 MeV.This state is an NK* scattering state.twistedtwisted
Diquark-type(JP=3/2(+))Changes in the two-particle spectrum are too small in JP=3/2(+) channel.NK*(p-wave) threshold is lowered down by 57 MeV.NK(p-wave) threshold is lowered down by 66 MeV.Best-fit(m5Q) is raised up by 80 MeV.This state is an NK*-scattering state.
Chiral extrapolation (JP=3/2(+))(circle) from NK*-type correlator(box) from color-twisted NK*-type correlator(triangle) from diquark-type correlatorIn the physical quark mass region,NK*-type: m5Q= 2.64(7) GeVColor-twisted NK*-type: m5Q= 2.48(10) GeVDiquark-type: m5Q=2.42(6) GeVNo evidence for a low-lying 5Q states.Physical quark mass regionNK*(p-wave) scattering statesN*K*(s-wave) scattering stateHBC analysis suggests:Due to the limited time, we cannot show HBC analysis.
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()JP=1/2() [with a diquark-type interpolating field]JP=1/