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

3. Quarks : Six Flavors Quarks : Six Flavors

4. Multi-Quark Two, Three or More? 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. 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 u d q q 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? Multi-Quark Two, Three or More? Four quarks : Two quark + two anti- quark. Like molecular state. Example : σ (500-600 MeV : ππ) : a 0 (980), f 0 (980) (KK) : ρρ (I=2) [γγ  ρ + ρ¯, ρ 0 ρ 0 ] : D S (Babar) (CS or DK ?) : B ±  K + π¯π¯ J/ψ (DD*?) q1q1 q2q2 q2q2 q1q1 Five quarks : Four same or different quarks + one antiquark Possible configuration : colorless baryon + colorless meson q1q1 q2q2 q3q3 q2q2 q1q1

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

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

8. Experimental evidence for Pentaquarks (summary) Experimental evidence for Pentaquarks (summary) Experiments Experiments Mass Width Significance Mass Width Significance (MeV) (MeV) (σ) (MeV) (MeV) (σ) SPRING-8 γn  K ¯ (K + n) DIANA K 0 P (541 events) CLAS (JLab) γd  K + K ¯ n (p) SAPHIR ITEP (ν’s) HERMES R. Arndt et al. (K + N Scattering) 1540±10±5 г < 25 4.6±0.1 1540±10±5 г < 25 4.6±0.1 1539±2±’’few’’ г < 8 4.4 1539±2±’’few’’ г < 8 4.4 1542±2±5 г < 21 5.3±0.5 1542±2±5 г < 21 5.3±0.5 1540±4±2 г < 25 4.8 1540±4±2 г < 25 4.8 1535±5 г < 29 6.7 1535±5 г < 29 6.7 1526±2±2.5 г < 20 5.6 1526±2±2.5 г < 20 5.6 г < 1 (if exists) г < 1 (if exists) World Average World Average 1535±2.5 1535±2.5 θ+θ+ Experiments Experiments Results Results NA49 (CERN) NA49 (CERN) Ξ --  Ξ - π -, Ξ 0  Ξ - π + M = 1862 MeV M = 1862 MeV г < 29 MeV г < 29 MeV Ξ¯¯Ξ¯¯

9. Prediction from different models Prediction from different models Model Model Prediction Prediction Chiral Soliton Model (D. Diakonov et al.) Chiral Soliton Model (D. Diakonov et al.) Naïve Quark Model Naïve Quark Model Isotensor Formulation (S. Capstick et al.) Isotensor Formulation (S. Capstick et al.) qq with π interaction (Stanscu, Riska) qq with π interaction (Stanscu, Riska) Chiral potential (A. Hosaka) Chiral potential (A. Hosaka) qq –qqq Model (M. Karliner et al.) qq –qqq Model (M. Karliner et al.) Di-quark Model (Jaffe and Wilzcek) Di-quark Model (Jaffe and Wilzcek) QCD sum rule (Sugiyama et al.) QCD sum rule (Sugiyama et al.) Lattice QCD (F. Csikor et al.) Lattice QCD (F. Csikor et al.) Lattice QCD (S. Sasaki) Lattice QCD (S. Sasaki) 1/2 +, I = 0 1/2 +, I = 0 1/2¯ 1/2¯ 1/2¯, 3/2¯ 5/2¯, I = 2 1/2¯, 3/2¯ 5/2¯, I = 2 1/2 + 1/2 + 1/2 +, I = 0 1/2 +, I = 0 1/2¯, I = 0 1/2¯, I = 0 1/2 +  1/2¯ 1/2 +  1/2¯ 1/2¯ 1/2¯

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

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

13. How good is the quenched approximation? Light hadron spectrum from CP-PACS, heplat/0206090. Lattices: 32 3 x56 to 64 3 x128 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.

14. Overlap Fermion Overlap Fermion Exact chiral symmetry. Exact chiral symmetry. No exceptional configurations. No exceptional configurations. No O (a) error, O (a 2 ) is also small. No O (a) error, O (a 2 ) is also small. Critical slowing down is gentle all the way to pion mass ~180 MeV. Critical slowing down is gentle all the way to pion mass ~180 MeV. Numerically checked that there is no addative quark mass renormalization. Numerically checked that there is no addative quark mass renormalization. 16 3 X 28, a = 0.200(3) fm. La = 3.2 fm (80 configurations) 16 3 X 28, a = 0.200(3) fm. La = 3.2 fm (80 configurations) 12 3 X 28, a = 0.200(3) fm. La = 2.4 fm (80 configurations) 12 3 X 28, a = 0.200(3) fm. La = 2.4 fm (80 configurations) 20 3 X 32, a ~ 0.171 fm. La ~ 3.4 fm (100 configurations, 20 3 X 32, a ~ 0.171 fm. La ~ 3.4 fm (100 configurations, not analyzed yet). not analyzed yet).

15. Some Lattice Results : Kentucky Group Some Lattice Results : Kentucky Group

16. Pentaquark on the Lattice 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

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

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

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

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

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

22. The  ′ ghost in quenched QCD The  ′ ghost in quenched QCD Quenched QCD Full QCD (hairpin) ….. 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 It becomes a light degree of freedom 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 is present in all hadron correlators G(t). It gives a negative contribution to G(t). It gives a negative contribution to G(t). –It is unphysical (thus the name ghost).

23. Evidence of η’N GHOST State in S 11 (1535) Channel Evidence of η’N GHOST State in S 11 (1535) Channel - - - - ηη W > 0 W<0 Effect of ghost state decreases as pion mass increases Effect of ghost state is first time seen in baryon channel

24. Ghost States in Pentaquark channel Ghost States in Pentaquark channel

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

26. 0.5 1.0 1.5 2.5 2.0 Mass (GeV) N(938) 1/2 + P 11 (1440) 1/2 + S 11 (1535) 1/2 - What is the nature of the Roper (P 11 (1440) 1/2 + ) resonance? N(938)1/2+ N(1440)1/2+ N(1535)1/2- Naïve quark model gives the wrong ordering ħ ħ  ħ ħ  - -Hybrid state (qqqg)? - -Dynamical meson-baryon state? Radial excitation ? q 4 q state? Radial excitation ? q 4 q state?

27. 0.5 1.0 1.5 2.5 2.0 Mass (GeV) N(938) 1/2 + P 11 (1440) 1/2 + S 11 (1535) 1/2 - Roper is seen on the lattice at the right mass with three quark interpolation field..hep ph/0306199 Cross over occurs in chiral doman

28. Radial excitation ? Roper is seen on the lattice with three-quark interpolation field. Roper is seen on the lattice with three-quark interpolation field. Weight : Weight : | | 2 > | | 2 > 0 (point source, point sink) | | 2 > | | 2 > 0 (point source, point sink) ∑ψ(x) ∑ψ(x) ∑O N (x ) ∑ψ(x) ∑O N (x ) ∑ψ(x) ∑ψ(x) ∑ψ(x) Point sink Wall source Point sink Wall source > 0 > 0 However, < 0 2S q 4 q State? 1S1S

29.  (1232) 3/2 +  (1600) 3/2 +  (1700) 3/2 - Cross-over in Deltas Cross-over in Deltas 0.5 1.0 1.5 2.5 2.0 Mass (GeV)  (1232) 3/2 +  (1600) 3/2 +  (1700) 3/2 -

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

31. Hyperfine Interaction of quarks in Baryons Hyperfine Interaction of quarks in Baryons Flavor spin interaction dominates Goldstone boson exchange No spin-orbit potential _ + ++ + + + _ _ Nucleon (938) Roper (1440) S 11 (1535) Δ(1236) Δ(1620-1700) Δ(1600) Λ(1116) Λ(1450-1520) Λ(1600) Glozman & Riska Phys. Rep. 268,263 (1996)

32. Is a 0 (1450) a two quark state? Is a 0 (1450) a two quark state? Ground state : ghost state. First excited state : a 0 Preliminary results shows mass around 1400-1500 MeV, suggesting a 0 (1450) is a two quark state. CorrelationfunctionforScalarchannel

33. Scattering Length and energy shift Scattering Length and energy shift Threshold energies : Threshold energies : Energy shift on the finite lattice : Experimental scattering lengths : Wave Wave I = 0 I = 1 I = 0 I = 1 S P 0.0±0.03 fm -0.32± 0.02 (~8 MeV 3.2 fm Lattice) (~18 MeV 2.4 fm Lattice) (~18 MeV 2.4 fm Lattice) 0.08±0.01 -0.16±0.1

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

35. S-wave (1/2¯) 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 E K (p=p L ) + E N (p=p L ). Ratio of spectral weight for two non-interacting particles W(12)/W(16) = V 3 (16)/V 3 (12) = 2.37

36. P-wave (1/2 + ) 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 E K (p=p L ) + E N (p=p L ) 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.

37. Volume Dependence in 1/2 + channel 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

38. Comparison of Lattice Results Comparison of Lattice Results Lattice Lattice 1/2¯ 1/2¯ 1/2 + 1/2 + Sasaki Sasaki Csikor et. al This Work E th =1.57GeV (threshold state) E th =1.57GeV (threshold state) E(p=1) = 1.96 GeV E(p=1) = 1.96 GeV Observed state : Observed state : 1. Scattering state 1. Scattering state mass (E 0 ) = ? mass (E 0 ) = ? 2. E 1 ~1.76 GeV (θ + ) 2. E 1 ~1.76 GeV (θ + ) E 0 /E(p=0) ~ 0.99 (E th ) E 0 /E(p=0) ~ 0.99 (E th ) E 1 /E(p=0) ~ 1.074 (θ + ) E 1 /E(p=0) ~ 1.074 (θ + ) E 0 /E(p=0) ~ 1 E 0 /E(p=0) ~ 1 Weight shows characteristic volume dependence of scattering state. E 1 : coincides with E(p=1) state. E th = E(p=1) = 1.96 GeV (threshold state) (threshold state) Observed state : 1. 2.62GeV 1. 2.62GeV No overlap with scattering state! E 0 ~ 2.9 GeV E 0 ~ 2.9 GeV Threshold scattering state ?? Threshold scattering state ?? E 0 : ghost state E 0 : ghost state E 1 /E(p=1) ~ 1 E 1 /E(p=1) ~ 1 Weight shows characteristic volume dependence of scattering state. 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

39. Comments on hep-lat/0309090 (Csikor et.al) Comments on hep-lat/0309090 (Csikor et.al) Correlation function from one interpolating field Cross-correlator : + α +α + α 2 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

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

41. Conclusions Conclusions Several experiments reported the discovery of θ + (1540). One experiment 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. 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. 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 Different model predicts different quantum numbers and masses for θ + (1540). They all predict nearby other additional states. θ + (1540). They all predict nearby other additional states. Lattice QCD can help to find out quantum numbers of pentaquark states. Lattice QCD can help to find out quantum numbers of pentaquark states.

42. Conclusions Conclusions We have not seen θ + state on our lattice calculation. We see only scattering states both in positive and negative parity channel. 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. 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. 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, D S ). 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, D S ).  Bottom-line : It will be an exciting time for experimentalists, theorists and Lattice community, and we are fully involved in this game. Lattice community, and we are fully involved in this game.