1. Excited State Spectroscopy from Lattice QCD Robert Edwards Jefferson Lab MENU 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A Work with Hadron Spectrum Collaboration: J. Dudek, B. Joo, M. Peardon, D. Richards, C. Thomas, S. Wallace

2. Spectroscopy Spectroscopy reveals fundamental aspects of hadronic physics –Essential degrees of freedom? –Gluonic excitations in mesons - exotic states of matter? Status –Can extract excited hadron energies & identify spins, –Pursuing full QCD calculations with realistic quark masses. New spectroscopy programs world-wide –E.g., BES III, GSI/Panda –Crucial complement to 12 GeV program at JLab. Excited nucleon spectroscopy (JLab) JLab GlueX: search for gluonic excitations. 2

3. Baryon Spectrum “Missing resonance problem” What are collective modes? What is the structure of the states? –Major focus of (and motivation for) JLab Hall B –Not resolved experimentally @ 6GeV 3 PDG uncertainty on B-W mass Nucleon spectrum

4. Lattice QCD Lattice QCD on anisotropic lattices N f = 2 + 1 (u,d + s) a s ~ 0.12fm, (a t ) -1 ~ 5.6 GeV V s ~ (2.0) 3 fm 3, (2.4) 3 fm 3, (2.9) 3 fm 3, (3.8) 3 fm 3 m ¼ ~ 700, 720, 450, 400, 230 MeV 0810.3588 Improved (distilled) operator technology with many operators 0909.0200, 1004.4930

5. Spectrum from variational method Matrix of correlators Diagonalize: eigenvalues ! spectrum eigenvectors ! wave function overlaps Benefit: orthogonality for near degenerate states Two-point correlator 5

6. Light quark baryons in SU(6) Conventional non-relativistic construction: 6 quark states in SU(6) Baryons 6

7. Relativistic operator construction: SU(12) Relativistic construction: 3 Flavors with upper/lower components Times space (derivatives) Color contraction is Antisymmetric ! Totally antisymmetric operators Dirac More operators than SU(6): mixes orbital ang. momentum & Dirac spin 7

8. Orbital angular momentum via derivatives 0905.2160 (PRD), 0909.0200 (PRL), 1004.4930 Couple derivatives onto single-site spinors: Enough D’s – build any J,M 8 Use all possible operators up to 2 derivatives (2 units orbital angular momentum) Only using symmetries of continuum QCD

9. Spin identified Nucleon spectrum m ¼ ~ 520MeV 9 Statistical errors < 2%

10. Experimental comparison Pattern of states very similar Where is the “Roper”? Thresholds & decays: need multi-particle ops 10

11. Phenomenology: Nucleon spectrum Looks like quark model? [ 70,1 - ] P-wave [ 56,2 + ] D-wave [ 70,2 + ] D-wave [ 20,1 + ] P-wave m ¼ ~ 520MeV 11 Discern structure: wave-function overlaps

12. Spin identified ¢ spectrum [ 70,1 - ] P-wave [ 56,2 + ] D-wave Spectrum slightly higher than nucleon 12

13. Nucleon & Delta Spectrum Lighter mass: states spreading/interspersing 13 m ¼ ~ 400 MeV

14. Nucleon & Delta Spectrum 14 [ 56,2 + ] D-wave [ 70,1 - ] P-wave [ 70,1 - ] P-wave [ 56,2 + ] D-wave Change at lighter quark mass? Decays! Suggests spectrum at least as dense as quark model

15. Isovector Meson Spectrum 15

16. Isovector Meson Spectrum 16 1004.4930

17. Exotic matter Exotics: world summary 17

18. Exotic matter Suggests (many) exotics within range of JLab Hall D Previous work: charmonium photo-production rates high Current work: (strong) decays 18

19. Spectrum of finite volume field theory Missing states: “continuum” of multi-particle scattering states Infinite volume: continuous spectrum 2m π Finite volume: discrete spectrum 2m π Deviation from (discrete) free energies depends upon interaction - contains information about scattering phase shift ΔE(L) ↔ δ(E) : Lüscher method 19

20. Finite volume scattering Reverse engineer Use known phase shift - anticipate spectrum E.g. just a single elastic resonance e.g. Lüscher method - essentially scattering in a periodic cubic box (length L) - finite volume energy levels E(δ,L) 20

21. Finite volume scattering: Lϋscher method Excited state spectrum at a single volume Discrete points on the phase shift curve Do more volumes, get more points energy levels L ~ 2.9 fm e.g. L ~ 2.9 fm 21

22. The interpretation “non-interacting basis states” DOTS: Finite volume QCD energy eigenvalues LINES: Non-interacting two-particle states have known energies Level repulsion - just like quantum mechanical pert. theory 22

23. The interpretation energy levels 23

24. Hadronic decays Plot the non-interacting meson levels as a guide 24 Current spectrum calculations: no evidence of multi-particle levels Require multi-particle operators (lattice) helicity construction annihilation diagrams Extract δ(E) at discrete E

25. Phase Shifts: demonstration 25

26. Prospects Strong effort in excited state spectroscopy –New operator & correlator constructions ! high lying states –Finite volume extraction of resonance parameters – promising Initial results for excited state spectrum: –Suggests baryon spectrum at least as dense as quark model –Suggests multiple exotic mesons within range of Hall D Resonance determination: –Start at heavy masses: have some “elastic scattering” –Use larger volumes & smaller pion masses (m ¼ ~230MeV) –Now: multi-particle operators & annihilation diagrams (gpu-s) –Need multi-channel finite-volume analysis for (in)elastic scattering Future: –Transition FF-s, photo-couplings (0803.3020, 0902.2214) –Use current insertion probes: TMD’s

27. Backup slides The end 27

28. probability, Regularization of QCD on a lattice quark field gluon field Discretize on a finite grid - in Euclidean space-time (t→i t) quark fields on the sites gauge-fields on the links do the fermion integralMonte Carlo

29. Orbital angular momentum via derivatives 0905.2160 (PRD), 0909.0200 (PRL), 1004.4930 Derivatives in ladders: Couple derivatives onto single-site spinors: Project onto lattice irreducible representations 29

30. Determining spin on a cubic lattice? Spin reducible on lattice Might be dynamical degeneracies H G 2 mass G 1 H G 2 Spin 1/2, 3/2, 5/2, or 7/2 ?

31. Spin reduction & (re)identification Variational solution: Continuum Lattice Method: Check if converse is true

32. Nucleon spectrum in (lattice) group theory PRD 79(2009), PRD 80 (2009), 0909.0200 (PRL) N f = 2 + 1, m ¼ ~ 580MeV Units of  baryon 3/2 + 5/2 + 7/2 + 3/2 - 5/2 - 7/2 - 1/2 + 7/2 + 5/2 + 7/2 + 5/2 - 7/2 - 1/2 - 7/2 - 5/2 - J = 1/2 J = 3/2 J = 5/2 J = 7/2

33. Interpretation of Meson Spectrum Future: incorporate in bound-state model phenomenology Future: probe with photon decays

34. Exotic matter? Can we observe exotic matter? Excited string QED QCD 34

35. Where are the Form Factors?? Previous efforts –Charmonium: excited state E&M transition FF-s (0909.0200) –Nucleon: 1 st attempt: E&M Roper->N FF-s (0803.3020) Spectrum first! –Basically have to address “what is a resonance’’ up front –(Simplistic example): FF for a strongly decaying state: linear combination of states 35 energy levels