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

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Spectroscopy Spectroscopy reveals fundamental aspects of hadronic physics –Essential degrees of freedom? –Gluonic excitations in mesons - exotic states of matter? New spectroscopy programs world-wide –E.g., BES III (Beijing), GSI/Panda (Darmstadt) –Crucial complement to 12 GeV program at JLab. Excited nucleon spectroscopy (JLab) JLab GlueX: search for gluonic excitations. 2

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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 6GeV 3 PDG uncertainty on B-W mass Nucleon spectrum

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Lattice QCD Lattice QCD on anisotropic lattices N f = (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 Improved (distilled) operator technology with many operators ,

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

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

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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

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Orbital angular momentum via derivatives (PRD), (PRL), 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

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

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

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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

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

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

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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

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Isovector Meson Spectrum 15 World comparisons: significant improvement Exotics m ¼ ~ 520 MeV

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Exotic matter Exotics: world summary 16

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Exotic matter Suggests (many) exotics within range of JLab Hall D Previous work: charmonium photo-production rates high Current work: (strong) decays 17

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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 18

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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) 19

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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 20

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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 21

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The interpretation energy levels 22

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Hadronic decays Plot the non-interacting meson levels as a guide 23 Current spectrum calculations: no evidence of multi-particle levels Require multi-particle operators (lattice) helicity construction annihilation diagrams Extract δ(E) at discrete E

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Phase Shifts: demonstration 24 Isospin-2 ¼¼ Finite volume methods seem practical(?)

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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 ( , ) –Use current insertion probes: TMD’s

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Backup slides The end 26

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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 ?

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

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Nucleon spectrum in (lattice) group theory PRD 79(2009), PRD 80 (2009), (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

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

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Exotic matter? Can we observe exotic matter? Excited string QED QCD 31

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Where are the Form Factors?? Previous efforts –Charmonium: excited state E&M transition FF-s ( ) –Nucleon: 1 st attempt: E&M Roper->N FF-s ( ) Spectrum first! –Basically have to address “what is a resonance’’ up front –(Simplistic example): FF for a strongly decaying state: linear combination of states 32 energy levels

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