Baryon Spectroscopy and Resonances Robert Edwards Jefferson Lab Hadron 2011 Collaborators: J. Dudek, B. Joo, D. Richards, S. Wallace Auspices of the Hadron Spectrum Collaboration TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAA
Where are the “Missing” Baryon Resonances? What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? 2
Where are the “Missing” Baryon Resonances? What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? PDG uncertainty on B-W mass Nucleon & Delta spectrum 3
Where are the “Missing” Baryon Resonances? What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? PDG uncertainty on B-W mass Nucleon & Delta spectrum QM predictions 2 2 1 1 1 0 4
Where are the “Missing” Baryon Resonances? What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? PDG uncertainty on B-W mass Nucleon & Delta spectrum QM predictions 2 3 2 1 ??? 4 5 3 1 ??? 2 2 1 1 1 0 5
Spectrum from variational method Two-point correlator 6
Spectrum from variational method Two-point correlator Matrix of correlators 7
Spectrum from variational method Two-point correlator Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues ! spectrum eigenvectors ! spectral “overlaps” Zin 8
Spectrum from variational method Two-point correlator Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues ! spectrum eigenvectors ! spectral “overlaps” Zin Each state optimal combination of ©i 9
Spectrum from variational method Two-point correlator Matrix of correlators “Rayleigh-Ritz method” Diagonalize: eigenvalues ! spectrum eigenvectors ! spectral “overlaps” Zin Each state optimal combination of ©i Benefit: orthogonality for near degenerate states 10
Baryon operators Construction : permutations of 3 objects 1104.5152 11
Baryon operators 12 Construction : permutations of 3 objects Symmetric: e.g., uud+udu+duu Antisymmetric: e.g., uud-udu+duu-… Mixed: (antisymmetric & symmetric) e.g., udu - duu & 2duu - udu - uud 1104.5152 12
Baryon operators 13 Construction : permutations of 3 objects Symmetric: e.g., uud+udu+duu Antisymmetric: e.g., uud-udu+duu-… Mixed: (antisymmetric & symmetric) e.g., udu - duu & 2duu - udu - uud Multiplication rules: SymmetricAntisymmetric ! Antisymmetric MixedxMixed ! Symmetric©Antisymmetric©Mixed …. 1104.5152 13
Baryon operators 14 Construction : permutations of 3 objects Symmetric: e.g., uud+udu+duu Antisymmetric: e.g., uud-udu+duu-… Mixed: (antisymmetric & symmetric) e.g., udu - duu & 2duu - udu - uud Multiplication rules: SymmetricAntisymmetric ! Antisymmetric MixedxMixed ! Symmetric©Antisymmetric©Mixed …. Color antisymmetric ! Require Space x [Flavor x Spin] symmetric 1104.5152 14
Baryon operators 15 Construction : permutations of 3 objects Symmetric: e.g., uud+udu+duu Antisymmetric: e.g., uud-udu+duu-… Mixed: (antisymmetric & symmetric) e.g., udu - duu & 2duu - udu - uud Multiplication rules: SymmetricAntisymmetric ! Antisymmetric MixedxMixed ! Symmetric©Antisymmetric©Mixed …. Color antisymmetric ! Require Space x [Flavor x Spin] symmetric Space: couple covariant derivatives onto single-site spinors - build any J,M 1104.5152 15
Baryon operators 16 Construction : permutations of 3 objects Symmetric: e.g., uud+udu+duu Antisymmetric: e.g., uud-udu+duu-… Mixed: (antisymmetric & symmetric) e.g., udu - duu & 2duu - udu - uud Multiplication rules: SymmetricAntisymmetric ! Antisymmetric MixedxMixed ! Symmetric©Antisymmetric©Mixed …. Color antisymmetric ! Require Space x [Flavor x Spin] symmetric Space: couple covariant derivatives onto single-site spinors - build any J,M Classify operators by permutation symmetries: Leads to rich structure 1104.5152 16
Baryon operator basis 3-quark operators & up to two covariant derivatives: some JP 17
Baryon operator basis e.g., Nucleons: N 2S+1L¼ JP 18 3-quark operators & up to two covariant derivatives: some JP Spatial symmetry classification: JP #ops E.g., spatial symmetries J=1/2- 24 N 2PM ½- N 4PM ½- J=3/2- 28 N 2PM 3/2- N 4PM 3/2- J=5/2- 16 N 4PM 5/2- J=1/2+ N 2SS ½+ N 2SM ½+ N 4DM½+ N 2PA ½+ J=3/2+ N 2DS3/2+ N 2DM3/2+ N 2PA 3/2+ N 4SM3/2+ N 4DM3/2+ J=5/2+ N 2DS5/2+ N 2DM5/2+ N 4DM5/2+ J=7/2+ 4 N 4DM7/2+ e.g., Nucleons: N 2S+1L¼ JP 18
Baryon operator basis e.g., Nucleons: N 2S+1L¼ JP 19 3-quark operators & up to two covariant derivatives: some JP Spatial symmetry classification: JP #ops E.g., spatial symmetries J=1/2- 24 N 2PM ½- N 4PM ½- J=3/2- 28 N 2PM 3/2- N 4PM 3/2- J=5/2- 16 N 4PM 5/2- J=1/2+ N 2SS ½+ N 2SM ½+ N 4DM½+ N 2PA ½+ J=3/2+ N 2DS3/2+ N 2DM3/2+ N 2PA 3/2+ N 4SM3/2+ N 4DM3/2+ J=5/2+ N 2DS5/2+ N 2DM5/2+ N 4DM5/2+ J=7/2+ 4 N 4DM7/2+ e.g., Nucleons: N 2S+1L¼ JP By far the largest operator basis ever used for such calculations 19
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 Statistical errors < 2% 20
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 21
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 2 1 1 22
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 2 3 1 4 5 3 1 2 1 1 23
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 2 3 1 4 5 3 1 2 1 1 24
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 2 3 1 4 5 3 1 2 1 1 25
Spin identified Nucleon & Delta spectrum m¼ ~ 520MeV arXiv:1104.5152 2 3 1 4 5 3 1 2 1 1 SU(6)xO(3) counting No parity doubling 26
Spin identified Nucleon & Delta spectrum Discern structure: spectral overlaps m¼ ~ 520MeV arXiv:1104.5152 [70,1-] P-wave [70,1-] P-wave [56,0+] S-wave [56,0+] S-wave 27
N=2 J+ Nucleon & Delta spectrum Discern structure: spectral overlaps Significant mixing in J+ 8 levels/ops 13 levels/ops 2SM 4SS 2DM 4DS [56’,0+], [70,0+], [56,2+], [70,2+] 2SS 2SM 4SM 2DS 2DM 4DM 2PA [56’,0+], [70,0+], [56,2+], [70,2+], [20,1+] 28
N=2 J+ Nucleon & Delta spectrum Discern structure: spectral overlaps Significant mixing in J+ 8 levels/ops 13 levels/ops 2SM 4SS 2DM 4DS [56’,0+], [70,0+], [56,2+], [70,2+] 2SS 2SM 4SM 2DS 2DM 4DM 2PA [56’,0+], [70,0+], [56,2+], [70,2+], [20,1+] No “freezing” of degrees of freedom 29
Roper?? Near degeneracy in ½+ consistent with SU(6)O(3) counting, but heavily mixed Discrepancies?? Operator basis – spatial structure
Roper?? What else? Multi-particle operators Near degeneracy in ½+ consistent with SU(6)O(3) counting, but heavily mixed Discrepancies?? Operator basis – spatial structure What else? Multi-particle operators
Spectrum of finite volume field The idea: 1 dim quantum mechanics Two spin-less bosons: Ã(x,y) = f(x-y) ! f(z) Solutions Quantization condition when -L/2 < z < L/2 Same physics in 4 dim version (but messier) Provable in a QFT (and relativistic)
Finite volume scattering Scattering in a periodic cubic box (length L) Discrete energy levels in finite volume E.g. just a single elastic resonance e.g. At some L , have discrete excited energies T-matrix amplitudes ! partial waves Finite volume energy levels E(L) $ δ(E) 33
Resonances Scattering of composite objects in non-perturbative field theory isospin=2 ¼¼ δ0(E) δ2(E) 1011.6352 Feng, et.al, 1011.5288
Resonances isospin=1 ¼¼ m¼=481 MeV m¼=421 MeV m¼=330 MeV m¼=290MeV Scattering of composite objects in non-perturbative field theory isospin=2 ¼¼ δ0(E) δ2(E) 1011.6352 Feng, et.al, 1011.5288
Resonances Manifestation of “decay” in Euclidean space isospin=1 ¼¼ m¼=481 MeV m¼=421 MeV m¼=330 MeV m¼=290MeV Scattering of composite objects in non-perturbative field theory Manifestation of “decay” in Euclidean space Can extract pole position Feng, et.al, 1011.5288
Resonances g½¼¼ m¼2 (GeV2) Stability a generic feature of couplings?? isospin=1 ¼¼ m¼=481 MeV m¼=421 MeV m¼=330 MeV m¼=290MeV Scattering of composite objects in non-perturbative field theory g½¼¼ m¼2 (GeV2) Extracted coupling: stable in pion mass Stability a generic feature of couplings?? Feng, et.al, 1011.5288
Hadronic Decays S11! [N¼]S +[N´]S ¢! [N¼]P Some candidates: determine phase shift Somewhat elastic m¼ ~ 400 MeV S11! [N¼]S +[N´]S ¢! [N¼]P
Isoscalar & isovector meson spectrum Isoscalars: flavor mixing determined 1102.4299
Isoscalar & isovector meson spectrum Exotics Isoscalars: flavor mixing determined 1102.4299
Isoscalar & isovector meson spectrum Exotics Isoscalars: flavor mixing determined Will need to build PWA within mesons 1102.4299
Summary & prospects 42 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 42
Summary & prospects 43 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 pion masses (230MeV available) Extract couplings in multi-channel systems (with ¼, ´, K…) 43
Summary & prospects 44 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 pion masses (230MeV available) Extract couplings in multi-channel systems (with ¼, ´, K…) T-matrix “poles” from Euclidean space? 44
Summary & prospects Yes! [with caveats] 45 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 pion masses (230MeV available) Extract couplings in multi-channel systems (with ¼, ´, K…) T-matrix “poles” from Euclidean space? Yes! [with caveats] Also complicated But all Minkowski information is there 45
Summary & prospects Yes! [with caveats] 46 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 pion masses (230MeV available) Extract couplings in multi-channel systems (with ¼, ´, K…) T-matrix “poles” from Euclidean space? Yes! [with caveats] Also complicated But all Minkowski information is there Optimistic: see confluence of methods (an “amplitude analysis”) Develop techniques concurrently with decreasing pion mass 46
Backup slides The end
Lattice QCD Goal: resolve highly excited states Anisotropic lattices: Nf = 2 + 1 (u,d + s) Anisotropic lattices: (as)-1 ~ 1.6 GeV, (at)-1 ~ 5.6 GeV 0810.3588, 0909.0200, 1004.4930
Where are the “Missing” Baryon Resonances? What are collective modes? Is there “freezing” of degrees of freedom? What is the structure of the states? 5 2 3 2 1 1 1 0 4 5 3 1 2 2 1 0 Capstick, Isgur; Capstick, Roberts 49
Operators are not states Two-point correlator Full basis of operators: many operators can create same state Spectral “overlaps” States may have subset of allowed symmetries
Nucleon J- Overlaps Little mixing in each J- Nearly “pure” [S= 1/2 & 3/2] 1-
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 52
I=1 ¼¼ : the “½” g½¼¼ m¼2 (GeV2) Extract δ1(E) at discrete E Extracted coupling: stable in pion mass Stability a generic feature of couplings?? Feng, Jansen, Renner, 1011.5288
Form Factors E What is a form-factor off of a resonance? What is a resonance? Spectrum first! E Extension of scattering techniques: Finite volume matrix element modified Kinematic factor Phase shift Requires excited level transition FF’s: some experience Charmonium E&M transition FF’s (1004.4930) Nucleon 1st attempt: “Roper”->N (0803.3020) Range: few GeV2 Limitation: spatial lattice spacing