1. Baryon Resonances from Lattice QCD
Robert Edwards
Jefferson Lab
high Q2, 2011
Collaborators:
J. Dudek, B. Joo, D. Richards, S. Wallace
Auspices of the Hadron Spectrum Collaboration
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2. Lattice QCD Goal: resolve highly excited states Anisotropic lattices:
Nf = (u,d + s)
Anisotropic lattices:
(as)-1 ~ 1.6 GeV, (at)-1 ~ 5.6 GeV
, ,

3. 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

4. Operator construction
Baryons : permutations of 3 objects
Permutation group S3: 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
Color antisymmetric ! Require Space­ [Flavor­ Spin] symmetric
Classify operators by these permutation symmetries:
Leads to rich structure 4

5. 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) 5

6. Baryon operator basis Nucleons: N 2S+1L¼ JP 6
3-quark operators with up to two covariant derivatives – projected into definite isospin and continuum JP
Spatial symmetry classification: JP
#ops
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 4DM ½+
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+
Nucleons: N 2S+1L¼ JP
Symmetry crucial for spectroscopy
By far the largest operator basis ever used for such calculations 6

7. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv:
Statistical errors < 2% 7

8. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv: 2 3 1 4 5 3 1 2 1 1
SU(6)xO(3) counting
No parity doubling 8

9. Spin identified Nucleon & Delta spectrum
Discern structure: wave-function overlaps
m¼ ~ 520MeV
arXiv:
[70,1-]
P-wave
[70,1-]
P-wave
[56,0+]
S-wave
[56,0+]
S-wave 9

10. N=2 J+ Nucleon & Delta spectrum
Discern structure: wave-function overlaps
Significant mixing in J+
8 levels/ops
13 levels/ops
2SM 4SS
2DM 4DS
2SS 2SM 4SM
2DS 2DM 4DM 2PA 10

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

12. 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 12

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

14. 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,

15. 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 ( )
Nucleon 1st attempt: “Roper”->N ( )
Range: few GeV2
Limitation: spatial lattice spacing

16. (Very) Large Q2 16 Standard requirements:
Cutoff effects: lattice spacing (as)-1 ~ 1.6 GeV
Appeal to renormalization group: Finite-Size scaling
Use short-distance quantity: compute perturbatively and/or parameterize
“Unfold” ratio only at low Q2 / s2N
For Q2 = 100 GeV2 and N=3, Q2 / s2N ~ 1.5 GeV2
Initial applications: factorization in pion-FF
D. Renner 16

17. Hadronic Decays S11! [N¼]S ¢! [N¼]P
Some candidates: determine phase shift
Somewhat elastic
m¼ ~ 400 MeV
S11! [N¼]S
¢! [N¼]P

18. 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, Q2 ~ few GeV2
Decrease lattice spacing: (as)-1 ~ 1.6 GeV ! 3.2 GeV, then Q2 ~ 10 GeV2
Finite-size scaling: Q2 ! 100 GeV2 ???

19. Backup slides
The end

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

21. Phase Shifts demonstration: I=2 ¼¼
¼¼ isospin=2
Extract δ0(E) at discrete E
No discernible pion mass dependence
(PRD)

22. Phase Shifts: demonstration
¼¼ isospin=2
δ2(E)

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

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

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