1. Excited State Spectroscopy from Lattice QCD
Robert Edwards
Jefferson Lab
DNP 2010
Collaborators:
J. Dudek, B. Joo, M. Peardon, D. Richards, C. Thomas, S. Wallace
Auspices of the Hadron Spectrum Collaboration
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2. Spectroscopy
Spectroscopy reveals fundamental aspects of hadronic physics
Essential degrees of freedom?
Gluonic excitations in mesons - exotic states of matter?
New spectroscopy programs world-wide
E.g., BES III (Beijing), GSI/Panda (Darmstadt)
Crucial complement to 12 GeV program at JLab.
Excited nucleon spectroscopy (JLab)
JLab GlueX: search for gluonic excitations.

3. 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
Nucleon spectrum
PDG uncertainty on
B-W mass

4. Lattice QCD Nf = 2 + 1 (u,d + s) Lattice QCD on anisotropic lattices
Vs ~ (2.0)3 fm3 , (2.4)3 fm3, (2.9)3 fm3, (3.8)3 fm3
m¼ ~ 700, 720, 450, 400, 230 MeV
as ~ 0.12fm, (at)-1 ~ 5.6 GeV
Improved (distilled) operator technology with many operators ,

5. Spectrum from variational method
Two-point correlator
Matrix of correlators
Diagonalize:
eigenvalues ! spectrum
eigenvectors ! wave function overlaps
Benefit: orthogonality for near degenerate states 5

6. Light quark baryons in SU(6)
Conventional non-relativistic construction:
6 quark states in SU(6)
Baryons 6

7. Relativistic operator construction: SU(12)
Relativistic construction: 3 Flavors with upper/lower components
Times space (derivatives)
Dirac
Color contraction is Antisymmetric ! Totally antisymmetric operators
More operators than SU(6): mixes orbital ang. momentum & Dirac spin 7

8. Orbital angular momentum via derivatives
Couple derivatives onto single-site spinors:
Enough D’s – build any J,M
Use all possible operators up to 2 derivatives (2 units orbital angular momentum)
Only using symmetries of continuum QCD
(PRD), (PRL), 8

9. Spin identified Nucleon spectrum
m¼ ~ 520MeV
Statistical errors
< 2% 9

10. Experimental comparison
Pattern of states very similar
Where is the “Roper”?
Thresholds & decays: need multi-particle ops 10

11. Phenomenology: Nucleon spectrum
Discern structure: wave-function overlaps
m¼ ~ 520MeV
[20,1+]
P-wave
[70,2+]
D-wave
[56,2+]
D-wave
[70,1-]
P-wave
Looks like quark model? 11

12. Spin identified ¢ spectrum
Spectrum slightly higher than nucleon
[56,2+]
D-wave
[70,1-]
P-wave 12

13. Nucleon & Delta Spectrum
Lighter mass: states spreading/interspersing
m¼ ~ 400 MeV

14. Nucleon & Delta Spectrum
Suggests spectrum at least as dense as quark model
[56,2+]
D-wave
[56,2+]
D-wave
[70,1-]
P-wave
[70,1-]
P-wave
Change at lighter quark mass? Decays!

15. Isovector Meson Spectrum
World comparisons: significant improvement
m¼ ~ 520 MeV
Exotics

16. Exotic matter
Exotics: world summary

17. Exotic matter Current work: (strong) decays
Suggests (many) exotics within range of JLab Hall D
Previous work: charmonium photo-production rates high
Current work: (strong) decays

18. 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 18

19. Finite volume scattering
Reverse engineer
Use known phase shift - anticipate spectrum
E.g. just a single elastic resonance
e.g.
Lüscher method
- essentially scattering in a periodic cubic box (length L)
- finite volume energy levels E(δ,L) 19

20. Finite volume scattering: Lϋscher method
energy
levels
L ~ 2.9 fm
e.g.
Excited state spectrum at a single volume
Discrete points on the phase shift curve
Do more volumes, get more points 20

21. The interpretation 21 DOTS: Finite volume QCD energy eigenvalues
LINES:
Non-interacting two-particle states have known energies
“non-interacting basis states”
Level repulsion - just like quantum mechanical pert. theory 21

22. The interpretation
energy
levels 22

23. Hadronic decays Current spectrum calculations:
no evidence of multi-particle levels
Plot the non-interacting meson levels as a guide
Require multi-particle operators
(lattice) helicity construction
annihilation diagrams
Extract δ(E) at discrete E 23

24. Phase Shifts: demonstration
Isospin-2 ¼¼
Finite volume methods seem practical(?)

25. Prospects Strong effort in excited state spectroscopy
New operator & correlator constructions ! high lying states
Finite volume extraction of resonance parameters – promising
Initial results for excited state spectrum:
Suggests baryon spectrum at least as dense as quark model
Suggests multiple exotic mesons within range of Hall D
Resonance determination:
Start at heavy masses: have some “elastic scattering”
Use larger volumes & smaller pion masses (m¼ ~230MeV)
Now: multi-particle operators & annihilation diagrams (gpu-s)
Need multi-channel finite-volume analysis for (in)elastic scattering
Future:
Transition FF-s, photo-couplings ( , )
Use current insertion probes: TMD’s

26. Backup slides
The end

27. Determining spin on a cubic lattice?
Spin reducible on lattice
H G2
Might be dynamical degeneracies
mass
G H G2
Spin 1/2, 3/2, 5/2, or 7/2 ?

28. Spin reduction & (re)identification
Variational solution:
Continuum
Lattice
Method: Check if converse is true

29. Nucleon spectrum in (lattice) group theory
Nf= , m¼ ~ 580MeV
Units of  baryon
5/2-
J = 1/2
J = 3/2
J = 5/2
J = 7/2
3/2+ 5/2+ 7/2+
3/2- 5/2- 7/2-
1/2+ 7/2+
5/2+ 7/2+
5/2- 7/2-
1/2- 7/2-
PRD 79(2009), PRD 80 (2009), (PRL)

30. Interpretation of Meson Spectrum
Future: incorporate in bound-state model phenomenology
Future: probe with photon decays

31. Exotic matter?
Can we observe exotic matter? Excited string
QED
QCD

32. Where are the Form Factors??
Previous efforts
Charmonium: excited state E&M transition FF-s ( )
Nucleon: 1st attempt: E&M Roper->N FF-s ( )
Spectrum first!
Basically have to address “what is a resonance’’ up front
(Simplistic example): FF for a strongly decaying state: linear combination of states
energy
levels 32