Baryon Resonances from Lattice QCD Robert Edwards Jefferson Lab N high Q 2, 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 AA Collaborators: J. Dudek, B. Joo, D. Richards, S. Wallace Auspices of the Hadron Spectrum Collaboration

Lattice QCD Goal: resolve highly excited states N f = (u,d + s) Anisotropic lattices: ( a s ) -1 ~ 1.6 GeV, (a t ) -1 ~ 5.6 GeV , ,

Spectrum from variational method Matrix of correlators Diagonalize: eigenvalues ! spectrum eigenvectors ! wave function overlaps Benefit: orthogonality for near degenerate states Two-point correlator 3 Each state optimal combination of © i

Operator construction Baryons : permutations of 3 objects 4 Color antisymmetric ! Require Space ­ [Flavor ­ Spin] symmetric Permutation group S 3 : 3 representations Symmetric: 1-dimensional e.g., uud+udu+duu Antisymmetric: 1-dimensional e.g., uud-udu+duu-… Mixed: 2-dimensional e.g., udu - duu & 2duu - udu - uud Classify operators by these permutation symmetries: Leads to rich structure

Orbital angular momentum via derivatives Couple derivatives onto single-site spinors: Enough D’s – build any J,M 5 Use all possible operators up to 2 derivatives (transforms like 2 units orbital angular momentum) Only using symmetries of continuum QCD

Baryon operator basis 3-quark operators with up to two covariant derivatives – projected into definite isospin and continuum J P 6 Spatial symmetry classification: Nucleons: N 2S+1 L ¼ J P JPJP #opsSpatial symmetries J=1/2 - 24N 2 P M ½ - N 4 P M ½ - J=3/2 - 28N 2 P M 3/2 - N 4 P M 3/2 - J=5/2 - 16N 4 P M 5/2 - J=1/2 + 24N 2 S S ½ + N 2 S M ½ + N 4 D M ½ + N 2 P A ½ + J=3/2 + 28N 2 D S 3/2 + N 2 D M 3/2 + N 2 P A 3/2 + N 4 S M 3/2 + N 4 D M 3/2 + J=5/2 + 16N 2 D S 5/2 + N 2 D M 5/2 + N 4 D M 5/2 + J=7/2 + 4N 4 D M 7/2 + By far the largest operator basis ever used for such calculations Symmetry crucial for spectroscopy

Spin identified Nucleon & Delta spectrum m ¼ ~ 520MeV 7 arXiv: Statistical errors < 2%

Spin identified Nucleon & Delta spectrum 8 arXiv: SU(6)xO(3) counting No parity doubling m ¼ ~ 520MeV

Spin identified Nucleon & Delta spectrum 9 arXiv: [ 70,1 - ] P-wave [ 70,1 - ] P-wave m ¼ ~ 520MeV [ 56,0 + ] S-wave [ 56,0 + ] S-wave Discern structure: wave-function overlaps

N=2 J + Nucleon & Delta spectrum 10 Significant mixing in J + 2 S S 2 S M 4 S M 2 D S 2 D M 4 D M 2 P A 13 levels/ops 2 S M 4 S S 2 D M 4 D S 8 levels/ops Discern structure: wave-function overlaps

Roper?? 11 Near degeneracy in ½ + consistent with SU(6) ­ O(3) but heavily mixed 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 12

Finite volume scattering E.g. just a single elastic resonance e.g. Lüscher method -scattering in a periodic cubic box (length L) -finite volume energy levels E(L) ! δ(E) 13 At some L, have discrete excited energies

I=1 ¼¼ : the “ ½ ” Feng, Jansen, Renner, Extract δ 1 (E) at discrete E g ½¼¼ m ¼ 2 ( GeV 2 ) Extracted coupling: stable in pion mass Stability a generic feature of couplings??

Form Factors What is a form-factor off of a resonance? What is a resonance? Spectrum first! Extension of scattering techniques:  Finite volume matrix element modified E Requires excited level transition FF’s: some experience Charmonium E&M transition FF’s ( ) Nucleon 1 st attempt: “Roper”->N ( ) Range: few GeV 2 Limitation: spatial lattice spacing Kinematic factor Phase shift

(Very) Large Q 2 Cutoff effects: lattice spacing ( a s ) -1 ~ 1.6 GeV 16 Standard requirements: Appeal to renormalization group: Finite-Size scaling D. Renner “Unfold” ratio only at low Q 2 / s 2N Use short-distance quantity: compute perturbatively and/or parameterize For Q 2 = 100 GeV 2 and N=3, Q 2 / s 2N ~ 1.5 GeV 2 Initial applications: factorization in pion-FF

Hadronic Decays 17 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: –No “freezing” of degrees of freedom nor parity doubling –Broadly consistent with non-relativistic quark model –Add multi-particles ! baryon spectrum becomes denser Short-term plans: resonance determination! –Lighter quark masses –Extract couplings in multi-channel systems Form-factors: –Use previous resonance parameters: initially, Q 2 ~ few GeV 2 –Decrease lattice spacing: (a s ) -1 ~ 1.6 GeV ! 3.2 GeV, then Q 2 ~ 10 GeV 2 –Finite-size scaling: Q 2 ! 100 GeV 2 ???

Backup slides The end 19

Baryon Spectrum “Missing resonance problem” What are collective modes? What is the structure of the states? 20 PDG uncertainty on B-W mass Nucleon spectrum

Phase Shifts demonstration: I=2 ¼¼ ¼¼ isospin=2 Extract δ 0 (E) at discrete E No discernible pion mass dependence (PRD)

Phase Shifts: demonstration ¼¼ isospin=2 δ 2 (E)

Nucleon J - 23 Overlaps Little mixing in each J - Nearly “pure” [S= 1/2 & 3/2] ­ 1 -

N & ¢ spectrum: lower pion mass 24 m ¼ ~ 400 MeV Still bands of states with same counting More mixing in nucleon N=2 J +

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