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**Excited State Spectroscopy from Lattice QCD**

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

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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.

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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 Nucleon spectrum PDG uncertainty on B-W mass

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**Lattice QCD Nf = 2 + 1 (u,d + s) Lattice QCD on anisotropic lattices**

Vs ~ (2.0)3 fm3 , (2.4)3 fm3, (2.9)3 fm3, (3.8)3 fm3 m¼ ~ 700, 720, 450, 400, 230 MeV as ~ 0.12fm, (at)-1 ~ 5.6 GeV Improved (distilled) operator technology with many operators ,

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**Spectrum from variational method**

Two-point correlator Matrix of correlators Diagonalize: eigenvalues ! spectrum eigenvectors ! wave function overlaps Benefit: orthogonality for near degenerate states 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) Dirac Color contraction is Antisymmetric ! Totally antisymmetric operators More operators than SU(6): mixes orbital ang. momentum & Dirac spin 7

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**Orbital angular momentum via derivatives**

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

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**Spin identified Nucleon spectrum**

m¼ ~ 520MeV Statistical errors < 2% 9

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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**

Discern structure: wave-function overlaps m¼ ~ 520MeV [20,1+] P-wave [70,2+] D-wave [56,2+] D-wave [70,1-] P-wave Looks like quark model? 11

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**Spin identified ¢ spectrum**

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

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**Nucleon & Delta Spectrum**

Lighter mass: states spreading/interspersing m¼ ~ 400 MeV

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**Nucleon & Delta Spectrum**

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

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**Isovector Meson Spectrum**

World comparisons: significant improvement m¼ ~ 520 MeV Exotics

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Exotic matter Exotics: world summary

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**Exotic matter Current work: (strong) decays**

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

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**Spectrum of finite volume field theory**

Missing states: “continuum” of multi-particle scattering states Infinite volume: continuous spectrum 2mπ 2mπ Finite volume: discrete spectrum 2mπ 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**

energy levels L ~ 2.9 fm e.g. Excited state spectrum at a single volume Discrete points on the phase shift curve Do more volumes, get more points 20

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**The interpretation 21 DOTS: Finite volume QCD energy eigenvalues**

LINES: Non-interacting two-particle states have known energies “non-interacting basis states” Level repulsion - just like quantum mechanical pert. theory 21

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The interpretation energy levels 22

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**Hadronic decays Current spectrum calculations:**

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

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**Phase Shifts: demonstration**

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 (m¼ ~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

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**Determining spin on a cubic lattice?**

Spin reducible on lattice H G2 Might be dynamical degeneracies mass G H G2 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**

Nf= , m¼ ~ 580MeV Units of baryon 5/2- J = 1/2 J = 3/2 J = 5/2 J = 7/2 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- PRD 79(2009), PRD 80 (2009), (PRL)

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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

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**Where are the Form Factors??**

Previous efforts Charmonium: excited state E&M transition FF-s ( ) Nucleon: 1st 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 energy levels 32

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Excited State Spectroscopy from Lattice QCD Robert Edwards Jefferson Lab MENU 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete.

Excited State Spectroscopy from Lattice QCD Robert Edwards Jefferson Lab MENU 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete.

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