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Baryon Resonances from Lattice QCD Robert Edwards Jefferson Lab GHP 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA A A A AA A A A A A Collaborators: J. Dudek, D. Richards, S. Wallace Auspices of the Hadron Spectrum Collaboration
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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 experimentally @ 6GeV 2 PDG uncertainty on B-W mass Nucleon spectrum
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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, 520, 440, 400, 230 MeV 0810.3588 Improved (distilled) operator technology with many operators 0909.0200, 1004.4930
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Baryon spectrum calculation Two-point correlator Overlap onto tower of J P baryon states Some state ! optimal linear combination of operators Baryon creation operator
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Variational Method Matrix of correlators Variational solution = generalized eigenvalue problem Eigenvalue ! spectrum Eigenvectors ! `optimal’ operators Orthogonality needed for near degenerate states
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Connection to QCD Hamiltonian Rewrite: Diagonalized the QCD Hamiltonian in basis of rank dim(C) Eigenvectors provide info on content of state
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Light quark baryons in SU(6) O(3) Conventional non-relativistic construction: 6 varieties of quarks under a SU(6) Baryons : permutations of 3 objects 7 Color antisymmetric ! Require Flavor Spin Space symmetric
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Orbital angular momentum via derivatives 1104.5152 (today!) Couple derivatives onto single-site spinors: Enough D’s – build any J,M 8 Use all possible operators up to 2 derivatives (transforms like 2 units orbital angular momentum) Only using symmetries of continuum QCD
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Relativistic operator construction: SU(12) Relativistic construction: 3 Flavors with upper/lower components Times space (derivatives) Dirac More operators than SU(6): mixes orbital ang. momentum & Dirac spin 9
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Operators are not states Full basis of operators: many operators can create same state States may have subset of allowed symmetries Two-point correlator
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Spin identification Normalized Nucleon correlator matrix C(t=5) Near perfect factorization: Continuum orthogonality arXiv:1104.5152 (today!) Overlaps -> reconstruct spin
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Spin identified Nucleon & Delta spectrum m ¼ ~ 520MeV 12 arXiv:1104.5152 Statistical errors < 2%
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Spin identified Nucleon & Delta spectrum 13 Discern structure: wave-function overlaps arXiv:1104.5152 453123 2 1 221 1 1 SU(6)xO(3) counting No parity doubling m ¼ ~ 520MeV
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Spin identified Nucleon & Delta spectrum 14 Discern structure: wave-function overlaps arXiv:1104.5152 [ 70,1 - ] P-wave [ 70,1 - ] P-wave m ¼ ~ 520MeV [ 56,0 + ] S-wave [ 56,0 + ] S-wave
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Nucleon J - 15 Overlaps Little mixing in each J - Nearly “pure” [S= 1/2 & 3/2] 1 -
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N=2 J + Nucleon & Delta spectrum 16 arXiv:1104.5152 Significant mixing in J + [56’,0 + ], [70,0 + ], [56,2 + ], [70,2 + ], [20,1 + ] 13 levels/ops [56’,0 + ], [70,0 + ], [56,2 + ], [70,2 + ] 8 levels/ops
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N & ¢ spectrum: lower pion mass 17 m ¼ ~ 400 MeV Still bands of states with same counting More mixing in nucleon N=2 J +
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Observations so far No parity doubling Realize full symmetries of SU(6) O(3) –Consistent energies across Nucleon & Delta –Pattern/structure consistent over pion mass –Little mixing N=1, J - sector –Significant mixing in N=2, J + sector –All multiplets appear, including 2 P A [20,1 + ] Inconsistent with quark-diquark model Consistent with non-relativistic quark model What about “Roper”? What about multi-particles?
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Roper?? 19 Near degeneracy in ½ + consistent with full SU(6) O(3) Discrepancies?? Operator basis – spatial structure What else? Multi-particle operators
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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 20
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I=1 ¼¼ : the “ ½ ” p 2 =0 p 2 =1 p 2 =2 p 2 =0 p 2 =1 p 2 =2 p 2 =0 p 2 =1 X. Feng, K. Jansen, D. Renner, 1011.5288 System in flight Extract δ 1 (E) at discrete E g ½¼¼ m ¼ 2 ( GeV 2 ) Extracted coupling stable in pion mass
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Hadronic Decays 22 m ¼ ~ 400 MeV Some candidates: determine phase shift Somewhat elastic ¢ ! [N ¼ ] P S 11 ! [N ¼ ] S
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Prospects Strong effort in excited state spectroscopy –New operator & correlator constructions ! high lying states Results for baryon excited state spectrum: –Realize full SU(6) O(3) symmetries –No “freezing” of degrees of freedom nor parity doubling –Broadly consistent with non-relativistic quark model –Add multi-particles ! baryon spectrum becomes denser 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 more general formalism for multi-channels decays Future: –Form-factors: finite-volume formalism exists for matrix elements (Kim,Sachrajda, Sharpe)
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