Baryon Resonances from Lattice QCD Robert Edwards Jefferson

Baryon Resonances from Lattice QCD Robert Edwards Jefferson Lab N* @ high Q 2, 2011 Collaborators: J. Dudek, B. Joo, D. Richards, S. Wallace Auspices of the Hadron Spectrum Collaboration

Lattice QCD Goal: resolve highly excited states Nf = 2 + 1 (u, d + s) Anisotropic lattices: (as)-1 ~ 1. 6 Ge. V, (at)-1 ~ 5. 6 Ge. V 0810. 3588, 0909. 0200, 1004. 4930

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

Operator construction Baryons : permutations of 3 objects 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 & 2 duu - udu - uud Color antisymmetric ! Require Space [Flavor Spin] symmetric Classify operators by these permutation symmetries: • Leads to rich structure 1104. 5152 4

Orbital angular momentum via derivatives Couple derivatives onto single-site spinors: Enough D’s – build any J, M Only using symmetries of continuum QCD Use all possible operators up to 2 derivatives (transforms like 2 units orbital angular momentum) 1104. 5152 5

Baryon operator basis 3 -quark operators with up to two covariant derivatives – projected into definite isospin and continuum JP Spatial symmetry classification: Nucleons: N 2 S+1 L¼ JP Symmetry crucial for spectroscopy By far the largest operator basis ever used for such calculations JP #ops Spatial symmetries J=1/2 - 24 N 2 PM ½- N 4 PM ½- J=3/2 - 28 N 2 PM 3/2 - N 4 PM 3/2 - J=5/2 - 16 N 4 PM 5/2 - J=1/2+ 24 N 2 SS ½+ N 2 SM ½+ N 4 DM ½+ N 2 PA ½+ J=3/2+ 28 N 2 DS 3/2+ N 2 DM 3/2+ N 2 PA 3/2+ N 4 SM 3/2+ N 4 DM 3/2+ J=5/2+ 16 N 2 DS 5/2+ N 2 DM 5/2+ N 4 DM 5/2+ J=7/2+ 4 N 4 DM 7/2+ 6

Spin identified Nucleon & Delta spectrum m¼ ~ 520 Me. V ar. Xiv: 1104. 5152 Statistical errors < 2% 7

Spin identified Nucleon & Delta spectrum m¼ ~ 520 Me. V ar. Xiv: 1104. 5152 4 5 3 2 1 1 2 2 1 1 SU(6)x. O(3) counting No parity doubling 8 1

Spin identified Nucleon & Delta spectrum Discern structure: wave-function overlaps [56, 0+] S-wave [70, 1 -] P-wave m¼ ~ 520 Me. V ar. Xiv: 1104. 5152 [70, 1 -] P-wave [56, 0+] S-wave 9

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

Roper? ? Near degeneracy in ½+ consistent with SU(6) O(3) but heavily mixed Discrepancies? ? Operator basis – spatial structure What else? Multi-particle operators 11

Spectrum of finite volume field theory Missing states: “continuum” of multi-particle scattering states 2 mπ Infinite volume: continuous spectrum 2 mπ Finite volume: discrete spectrum 2 mπ Deviation from (discrete) free energies depends upon interaction - contains information about scattering phase shift ΔE(L) ↔ δ(E) : Lüscher method 12

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

I=1 ¼¼ : the ½” “ Extract δ 1(E) at discrete E Extracted coupling: stable in pion mass g½¼¼ m¼ 2 (Ge. V 2) Stability a generic feature of couplings? ? Feng, Jansen, Renner, 1011. 5288

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 Phase shift Kinematic factor Requires excited level transition FF’s: some experience • Charmonium E&M transition FF’s (1004. 4930) • Nucleon 1 st attempt: “Roper”->N (0803. 3020) Range: few Ge. V 2 Limitation: spatial lattice spacing

(Very) Large Q 2 Standard requirements: Cutoff effects: lattice spacing (as)-1 ~ 1. 6 Ge. V Appeal to renormalization group: Finite-Size scaling Use short-distance quantity: compute perturbatively and/or parameterize “Unfold” ratio only at low Q 2 / s 2 N For Q 2 = 100 Ge. V 2 and N=3, Q 2 / s 2 N ~ 1. 5 Ge. V 2 Initial applications: factorization in pion-FF D. Renner 16

Hadronic Decays Some candidates: determine phase shift Somewhat elastic S 11! [N¼]S m¼ ~ 400 Me. V ¢! [N¼]P 17

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 Ge. V 2 – Decrease lattice spacing: (as)-1 ~ 1. 6 Ge. V ! 3. 2 Ge. V, then Q 2 ~ 10 Ge. V 2 – Finite-size scaling: Q 2 ! 100 Ge. V 2 ? ? ?

Backup slides • The end 19

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

Phase Shifts demonstration: I=2 ¼¼ ¼¼ isospin=2 Extract δ 0(E) at discrete E No discernible pion mass dependence 1011. 6352 (PRD)

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

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

N & ¢ spectrum: lower pion mass Still bands of states with same counting More mixing in nucleon N=2 J+ m¼ ~ 400 Me. V 24

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