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Lattice Calculation of Pentaquark Baryons

Lattice Calculation of Pentaquark Baryons. Nilmani Mathur Department of Physics and Astronomy University of Kentucky Collaborators : Kentucky Lattice QCD Group Members, F. Lee, J.B. Zhang and C. Bennhold. Outline. Multi-quark. How many of them are together? Pentaquark on Lattice

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Lattice Calculation of Pentaquark Baryons

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  1. Lattice Calculation of Pentaquark Baryons Nilmani Mathur Department of Physics and Astronomy University of Kentucky Collaborators : Kentucky Lattice QCD Group Members, F. Lee, J.B. Zhang and C. Bennhold

  2. Outline • Multi-quark. How many of them are together? • Pentaquark on Lattice • Overlap Fermion and Particle Spectrum • Lattice Calculation for Pentaquark • Results • Conclusions

  3. Quarks : Six Flavors

  4. Multi-Quark Two, Three or More? q Single quark has not been observed yet. QCD tells it cannot be observed. All naturally occurring particles are colorless. Each quark and anti-quark has three different colors. q q u Two quarks : One quark + One anti-quark. Possible mesons : (uu,ud,dd,us,sd,cd,cc,bb etc.) Example : Pion, Rho, Eta, Omega etc Meson d Three quarks : Possible three quarks : uud, udd, uds, uus, uds, uss, uuu, sss ,dss, dds, ddd, uss etc. Example : Proton, neutron etc. Baryons

  5. Multi-Quark Two, Three or More? Four quarks : Two quark + two anti- quark. Like molecular state. Example : σ (500-600 MeV : ππ) : a0(980), f0(980) (KK) : ρρ (I=2) [γγρ+ρ¯, ρ0ρ0] : DS (Babar) (CS or DK ?) : B± K+π¯π¯ J/ψ (DD*?) q1 q2 q2 q1 Five quarks : Four same or different quarks + one antiquark Possible configuration : colorless baryon + colorless meson q3 q1 q2 q1 q2

  6. Possible Pentaquark candidates u u Near to Nπ threshold. Decay by strong interaction. d d u • Need : • Weak force between them. • Non-zero overlap between initial • wave-function (threshold state) • and final state u u d d s Possible candidate. Can be observed in KN scattering (Signal observed recently). u u True Pentaquark, not seen so far. Heavier particle, experiment will be difficult d s c

  7. Recently Observed Hadrons Hadrons Experiments • Ds+ (2313)BABAR (PRL 90(2003) 242001) • Ds+ (2463)CLEO,hep-ex/0305100 • D0*0 (2308)BELLE • D0'0 (2427)hep-ex/0307021 • Ψ(3871)/DD*(3817)BELLE, hep-ex/0308029 • ΞCC++ (3460) • ΞCC+ (3520)SELEX, hep-ex/0212029 • ΞCC++ (3780) (lattice results before experiments …PRD66, 014502 (2002); PRD64, 094509 (2001)) • Θ+(1540)T. Nakano et. al (LEPS) CLAS, DIANA, SAPHIR, ZEUS, HERMES • Ξ¯ ¯(1862)NA49/CERN

  8. Experimental evidence for Pentaquarks (summary) θ+ Ξ¯ ¯

  9. Prediction from different models

  10. Quantum Chromodynamics (QCD)The Fundamental Theory of the Strong Interaction • Chiral symmetry and its spontaneous breaking • At high energy, perturbative (asymptotic freedom) • At low energy, non-perturbative (confinement)

  11. u u The proton in QCD: d d u u z y t The proton in the quark model:

  12. How good is the quenched approximation? • Light hadron spectrum from CP-PACS, heplat/0206090. • Lattices: 323x56 to 643x128 • Spacing 0.1 fm to 0.05 fm • M/ M is 0.75 to 0.4 • 1 to 3 % statistical error • 2% systematic error • Took more than a year of running on a dedicated computer sustaining 300 Gflops. The computed quenched light hadron spectrum is within 7% of the experiment. The remaining discrepancy is attributed to the quenched approximation.

  13. Overlap Fermion • Exact chiral symmetry. • No exceptional configurations. • No O(a) error, O(a2) is also small. • Critical slowing down is gentle all the way to pion mass ~180 MeV. • Numerically checked that there is no addative quark mass renormalization. • 163 X 28, a = 0.200(3) fm. La = 3.2 fm (80 configurations) • 123X 28, a = 0.200(3) fm. La = 2.4 fm (80 configurations) • 203X 32, a ~ 0.171 fm. La ~ 3.4 fm (100 configurations, not analyzed yet).

  14. SomeLattice Results : Kentucky Group

  15. Pentaquark on the Lattice Interpolating Field : Combination of colorless meson + baryon For θ+ : Interpolating field with I=0 and J=1/2 Color structure is not unique

  16. Pentaquark Correlation Function KN scattering state is part of this correlation function

  17. Anti-periodic boundary condition Correlation Function for Pentaquark 1/2+ 1/2¯ 1/2¯ 1/2+ 1/2+ 1/2¯ 1/2+ 1/2¯

  18. Correlation Function (1/2¯, 3.2 fm)

  19. Correlation Function (1/2+, 3.2 fm)

  20. Correlation Function (1/2+, 2.4 fm)

  21. The ′ ghost in quenched QCD … .. .. • Modeled as part of G(t) as: • weight w is negative • prefactor (1+Et) preserves the double-pole structure of the hairpin diagram • E ′N is treated as fit parameter to account for interactions between ′ and N Quenched QCD Full QCD (hairpin) • It becomes a light degree of freedom • with a mass degenerate with the pion mass. • It is present in all hadron correlators G(t). • It gives a negative contribution to G(t). • It is unphysical (thus the name ghost).

  22. Evidence of η’N GHOST State in S11 (1535)Channel Effect of ghost state is first time seen in baryon channel Effect of ghost state decreases as pion mass increases η η - - - - W > 0 W<0

  23. Ghost States in Pentaquark channel

  24. Ghost states in Pentaquark • 1/2¯ : Parity Negative. S-wave NKπ –parity : (+)(-)(-) = + Total parity : (-1)L P(NKπ) L = 1, therefore, ghost state will be in P-wave. Ground state is KN scattering state or pentaquark state. • 1/2+ : Parity Positive, P-wave NKπ –parity : (+)(-)(-) = + Total parity : (-1)L P(NKπ) L = 0, therefore, ghost will be in S-wave (mass mπ+mK+mN) Ground state is KNπ ghost state (for our lattice)

  25. 2.5 Naïve quark model gives the wrong ordering 2.0 S11(1535) 1/2- Mass (GeV) N(1440)1/2+ 1.5 P11(1440) 1/2+ ħ  1.0 N(1535)1/2- N(938) 1/2+ ħ  0.5 N(938)1/2+ Roper Radial excitation? q4q state? What is the nature of the Roper (P11(1440) 1/2+) resonance? • Hybrid state (qqqg)? • Dynamical meson-baryon state?

  26. 2.5 2.0 S11(1535) 1/2- Mass (GeV) 1.5 P11(1440) 1/2+ 1.0 N(938) 1/2+ 0.5 Roper Roper is seen on the lattice at the right mass with three quark interpolation field ..hep ph/0306199 Cross over occurs in chiral doman

  27. Roper Radial excitation? q4q State? • Roper is seen on the lattice with three-quark interpolation field. • Weight : |<O|ON|R >|2 > |<O|ON|N>|2 > 0 (point source, point sink) ∑ψ(x) ∑ON(x) ∑ψ(x) ∑ψ(x) Point sink Wall source <O|∑ON(x)|N><N| ∑ψ(x) | ∑ψ(x) | ∑ψ(x)|O> > 0 However,<O|∑ON(x)|R><R| ∑ψ(x) | ∑ψ(x) | ∑ψ(x)|O> < 0 1S 2S

  28. Cross-over in Deltas (1700) 3/2- 2.5 (1700) 3/2- 2.0 (1600) 3/2+ Mass (GeV) (1600) 3/2+ 1.5 (1232) 3/2+ 1.0 (1232) 3/2+ 0.5 Preliminary

  29. What about Hyperons? The (1405)? (1600) 1/2+ (1405) 1/2 - (1115) 1/2+ …different story!! Preliminary 2.5 2.0 (1600) 1/2+ Mass (GeV) 1.5 (1405) 1/2 - 1.0 (1115) 1/2+ 0.5

  30. Hyperfine Interaction of quarks in Baryons • Flavor spin interaction dominates • Goldstone boson exchange • No spin-orbit potential Glozman & Riska Phys. Rep. 268,263 (1996) _ S11(1535) _ Δ(1620-1700) + Λ(1600) _ + + Roper (1440) Δ(1600) Λ(1450-1520) + + + Nucleon (938) Λ(1116) Δ(1236)

  31. Is a0 (1450) a two quark state? Correlation function for Scalar channel Ground state : ghost state. First excited state : a0 Preliminary results shows mass around 1400-1500 MeV, suggesting a0(1450)is a two quark state.

  32. Scattering Length and energy shift • Threshold energies : • Energy shift on the finite lattice : • Experimental scattering lengths :

  33. Scattering state and its volume dependence Normalization condition requires : Lattice Two point function : Continuum And, V For one particle bound state there will be no volume dependence. For two particle state : Fitting function : Therefore, fitted weight (Wi) should be proportional to 1/V for two particle scattering state.

  34. S-wave (1/2¯) • No need to consider ghost • state (propagators are • positive). • Lowest states in 2.4 fm are • higher then those in 3.2 fm • which reflect the volume • dependence of the energy • shift. • The first excited state is • also not the θ+candidate • as it is several hundred • MeV higher near • EK(p=pL) + EN(p=pL). • Ratio of spectral weight for • two non-interacting • particles • W(12)/W(16) = V3(16)/V3(12) • = 2.37

  35. P-wave (1/2+) • Propagators turns negative. • Ground state is S-wave • KNη' ghost state. • In fitting function this ghost • state, pentaquark and KN-P- • wave scattering state are the • first three states. • We find ghost and scattering • state. • The volume dependence in • EK(p=pL) + EN(p=pL) due to • the P-wave nature is seen for • medium and high quark • masses. Near The chiral limit • the scattering length is close • to zero which is consistent • with the experiment.

  36. Volume Dependence in 1/2+ channel • For bound state, fitted weight will not show any volume • dependence. • For two particle scattering state, fitted weight will show • inverse volume dependence Our observed ground state is p-wave scattering state

  37. Comparison of Lattice Results Interpolating field should have overlap with threshold scattering state unless one can show that the used interpolating field cannot be transformed to usual KN interpolating field by Fierz transformation

  38. Comments on hep-lat/0309090 (Csikor et.al) Correlation function from one interpolating field <η1η1> Cross-correlator : <η1η1> + α <η1η2> +α<η2η1> + α2 <η2η2> Claim : One peak for each channel. One is θ+(1/2¯) corresponding to I=0. Observed θ+ peak is not sharp enough and it still could be consistent with the threshold scattering state. Also, 1/2+(I=0) is quite large.Where is the P-wave scattering state?? m(1/2+)/m(1/2-) ~ 2 ~1.5 (Sasaki) Peaks

  39. Ξ¯ ¯ • Diagonal and cross correlators have been calculated • for three lattices. • Analysis will be completed very soon.

  40. Conclusions • Several experiments reported the discovery of θ+ (1540). One experiment reported the discovery of Ξ¯ ¯(1860). However, their existences have not been absolutely established yet. We only know their strangeness. Other important quantum numbers, like spin, parity, isospin need to be established. More experiments (particularly direct KN scattering) and careful analysis are needed. More experiments will be performed soon in various Laboratories (including JLab) around the world. • Width of θ+(1540) found to be very very small (even may be < 1 MeV) which is very different than any other resonance particle. If θ+(1540) exists, theorists must find out new way to explain its width. Its existence will open up entirely new (and richer) hadron spectrum and bring new information about nature of short distance interactions between quarks. • Different model predicts different quantum numbers and masses for θ+ (1540). They all predict nearby other additional states. • Lattice QCD can help to find out quantum numbers of pentaquark states.

  41. Conclusions • We have not seen θ+ state on our lattice calculation. We see only scattering states both in positive and negative parity channel. • To claim convincing evidence for θ+ from lattice calculation, one must see volume dependent scattering states along with the volume insensitive θ+ bound state. For quenched lattice calculation one must consider ghost states in low quark mass region. • Our lattice study for Ξ pentaquark is going on (correlators have already been calculated for three lattices). Analysis will be completed soon. Also study of pentaquark by cross-correlators (a la Csikor et al.) will also be completed soon. • In future, we will carry out similar study using bigger lattices and many more configurations. Furthermore, we will study other exotic states involving four quarks-antiquarks (like, ππ, KK, DS). • Bottom-line : It will be an exciting time for experimentalists, theorists and Lattice community, and we are fully involved in this game.

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