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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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