1. Baryons on the Lattice Robert Edwards Jefferson Lab Hadron 09
December 2009
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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?
Status
Can extract excited hadron energies & identify spins,
Pursuing full QCD calculations with realistic quark masses.
New spectroscopy programs world-wide
E.g., BES III, GSI/Panda
Crucial complement to 12 GeV program at JLab.
Excited nucleon spectroscopy (JLab)
JLab GlueX: search for gluonic excitations.

3. Excited states: anisotropy+operators+variational
Anisotropic lattices with Nf=2+1 dynamical fermions
Temporal lattice spacing at < as (spatial lattice spacing)
High temporal resolution ! Resolve noisy & excited states
Major project within USQCD – Hadron Spectrum Collab.
Extended operators
Sufficient derivatives ! nonzero overlap at origin
Variational method:
Matrix of correlators ! project onto excited states
PRD 78 (2008) & PRD 79 (2009)
PRD 72 (2005), PRD 72 (2005), (PRL)
PRD 76 (2007), PRD 77 (2008), (PRD)

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

5. Relativistic operator construction
Relativistic construction: 3 Flavors with upper/lower components
Times space (derivatives)
Dirac
Color contraction is Antisymmetric
More operators than SU(6): mixes orbital ang. momentum & Dirac spin
Symmetric: 182 positive parity negative parity

6. Orbital angular momentum via derivatives
Derivatives in ladders:
Couple derivatives onto spinors:
Project onto lattice irreducible representations
(PRD) & (PRL)

7. 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 ?

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

9. Correlator matrix: near orthogonality
Normalized Nucleon correlator matrix
C(t=5)
Near perfect factorization:
Continuum orthogonality
Small condition numbers ~ 200
PRL (2007), arXiv: & (PRD)

10. 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)

11. Spin identified Nucleon spectrum
m¼ ~ 580MeV

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

13. Towards resonance determinations
Augment with multi-particle operators
Heavy masses: some elastic scattering
Lüscher finite volume techniques
Phase shifts ! width
Inelastic scattering:
Overlapping resonances
Will need/extend to finite-volume multi-channel
E.g., work by Bonn group

14. Phenomenology: Nucleon spectrum
Looks like quark model?
Compare overlaps & QM mixings
[20,1+]
P-wave
[70,2+]
D-wave
[56,2+]
D-wave
[70,1-]
P-wave

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

16. Current and future work
Some efforts underway
Meson strange & light quark spectrum (hybrids) and radiative transitions
Excited light baryon spectrum (N, ¢, ¥, §, ¤)
Radiative transitions for P11(1440), S11(1535), D13(1520)
Q2 <~ 5 GeV2
Gives understanding of internal structure
Phenomenology
Wave function overlaps:
Compute in models: quark model & large-N

17. Summary Strong effort in excited state spectroscopy
Anisotropy+variational method ! high lying states
Spin assignment possible
Lattice can handle decays (simple ones so far)
Example, ½ & ¢
Early stages
Start at heavy masses: have some “elastic scattering”
Already have smaller masses: move there + larger volumes
Will need multi-particle operators
Will need multi-channel finite-volume analysis for inelastic scattering