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

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 2 PDG uncertainty on B-W mass Nucleon spectrum

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, 520, 440, 400, 230 MeV Improved (distilled) operator technology with many operators ,

Baryon spectrum calculation Two-point correlator Overlap onto tower of J P baryon states Some state ! optimal linear combination of operators Baryon creation operator

Variational Method Matrix of correlators Variational solution = generalized eigenvalue problem Eigenvalue ! spectrum Eigenvectors ! `optimal’ operators Orthogonality needed for near degenerate states

Connection to QCD Hamiltonian Rewrite: Diagonalized the QCD Hamiltonian in basis of rank dim(C) Eigenvectors provide info on content of state

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

Orbital angular momentum via derivatives (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

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

Operators are not states Full basis of operators: many operators can create same state States may have subset of allowed symmetries Two-point correlator

Spin identification Normalized Nucleon correlator matrix C(t=5) Near perfect factorization: Continuum orthogonality arXiv: (today!) Overlaps -> reconstruct spin

Spin identified Nucleon & Delta spectrum m ¼ ~ 520MeV 12 arXiv: Statistical errors < 2%

Spin identified Nucleon & Delta spectrum 13 Discern structure: wave-function overlaps arXiv: SU(6)xO(3) counting No parity doubling m ¼ ~ 520MeV

Spin identified Nucleon & Delta spectrum 14 Discern structure: wave-function overlaps arXiv: [ 70,1 - ] P-wave [ 70,1 - ] P-wave m ¼ ~ 520MeV [ 56,0 + ] S-wave [ 56,0 + ] S-wave

Nucleon J - 15 Overlaps Little mixing in each J - Nearly “pure” [S= 1/2 & 3/2] ­ 1 -

N=2 J + Nucleon & Delta spectrum 16 arXiv: 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

N & ¢ spectrum: lower pion mass 17 m ¼ ~ 400 MeV Still bands of states with same counting More mixing in nucleon N=2 J +

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?

Roper?? 19 Near degeneracy in ½ + consistent with full SU(6) ­ O(3) Discrepancies?? Operator basis – spatial structure What else? Multi-particle operators

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

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, System in flight Extract δ 1 (E) at discrete E g ½¼¼ m ¼ 2 ( GeV 2 ) Extracted coupling stable in pion mass

Hadronic Decays 22 m ¼ ~ 400 MeV Some candidates: determine phase shift Somewhat elastic ¢ ! [N ¼ ] P S 11 ! [N ¼ ] S

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)