1. Nucleon Resonances from Lattice QCD
Robert Edwards
Jefferson Lab
GHP
April 2009
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2. Hadron Spectrum Collaboration
University of Pacific
J Juge
JLAB
S Cohen
J Dudek
R Edwards
B Joo
H-W Lin
D Richards
C Thomas
CMU
J Bulava
J Foley
C Morningstar
UMD
E Engelson
S Wallace
Tata (India)
N Mathur

3. Physics Goals Exotic and excited state meson spectrum
E.g., Future JLab Hall D & GSI/Panda experiments
Photo-couplings & electromagnetic transition form-factors
Isoscalar spectrum
Also heavy quark spectrum – cross-over to HEP
Baryon spectrum
Light quark and strange quarks
E.g., many cascade states unknown parity
NP2012: Masses along with ground state and excited state transition form-factors up to 7 GeV2
These are the simplest examples of views
NOTE: spectrum & structure division arbitrary
Also consider 3D structure view of (excited) hadrons

4. Requirements Dynamical quarks: Light quarks (u,d) – can be degenerate
Strange quark
Charm?
Chiral extrapolation: problematic ! physical limit
Multi-volume/decays
Continuum extrapolation
Disconnected contributions:
Isosinglets
3pt & 4-pt
Annihilation diagrams

5. Strange Quark Mass
Decouple strange quark mass & lattice spacing “a” determinations
Consider strange quark determination understood!
Chiral PT extendable in (lX, sX)
SU(3) stable hadron for scale X
Physics observables (lX, sX)
BMW, HadSpec 2008

6. Continuum Extrapolation
Clover action - small discretization effects
Chiral PT works well in ratio method
Ratio method
phys limit
BMW 2008

7. Some Ground State Masses
Some of the ground state masses
Missing negative parity octet and decuplet – much more to do!
Many of these states decay
BMW Collab, Science (2008)

8. No-go Theorem
Scattering matrix elements: only from finite volume Euclidean correlation functions (except at threshold) [Maiani-Testa]
3-d box with size L, spectrum of states discrete – even above threshold
Scattering lengths inferred
Missing negative parity octet and decuplet – much more to do!

9. Decays ½ mass from effective range expansion: finite box QCDSF 2008
physical ½
2¼, n=111
2¼, n=110
2¼, n=100
m¼=390 MeV
m¼=250 MeV
QCDSF 2008

10. Decays
½ mass
QCDSF 2008

11. More complicated decays
Exotic 1-+: cascading decays as mass decreases:

12. More complicated decays
Exotic 1-+: cascading decays as mass decreases :
first ! b1¼ S-wave [Dominant decay in flux-tube models]

13. More complicated decays
Exotic 1-+: cascading decays as mass decreases :
first ! b1¼ S-wave [Dominant decay in flux-tube models]
second b1¼ ! !¼¼ S-wave

14. More complicated decays
Exotic 1-+: cascading decays as mass decreases :
first ! b1¼ S-wave [Dominant decay in flux-tube models]
second b1¼ ! !¼¼ S-wave

15. More complicated decays
Exotic 1-+: cascading decays:
Also ! ½¼ P-wave [P-wave suppressed in flux-tube models]
L=2.0fm
L=1
Need multiple volumes
Theory not well developed
Will get to cost…
L=1

16. Nucleon spectrum (Experimental)½+
5/2+
3/2-
5/2-
3/2+ ½-
NP2012 milestone:
Spectrum & E&M transitions up to Q2 = 7 GeV2
Challenges/opportunities:
Compute excited energies
Compute decays
N¼¼ or ¢¼
N¼ or N´ or N(1440)¼
N¼¼ or ¢¼

17. Strange Quark Baryons Strange quark baryon spectrum poorly known
Future:
Narrow widths: easy(er) to extract (?)
¥ & : unknown spin & parities
Widths are small

18. Variational Method Correlation matrix: Diagonalize
Mass from eigenvalue
Basis complete enough to capture excited states
Small contamination as expected:
Diagonalization gives ~ <state out| Tt | state in>, T transfer matrix
Luscher,Wolff; HadSpec PRD72:074501,2005, PRD72:094506,2005

19. Anisotropic Lattice
Noisy signals – go anisotropic [Hadron Spectrum Collaboration]
Why? COST!!
Lower cost with only one fine lattice spacing instead of all 4.
m~720MeV, as=0.1fm, =3
HadSpec PRD72:074501,2005, PRD72:094506,2005

20. Nf=2 Nucleon Spectrum via Group Theory
HadSpec 2009
Nf=2, m= 416 MeV, as~0.11fm
Nf=2, m= 572 MeV ½+
3/2+, 5/2+
5/2- ½-
3/2+, 5/2+
5/2- ½+
3/2+, 5/2+
5/2- ½-
3/2+, 5/2+
5/2-

21. Nf=2 Nucleon Spectrum via Group Theory
Possible 5/2- state
HadSpec 2009
Nf=2, m= 416 MeV, as~0.11fm
Nf=2, m= 572 MeV
5/2- ½+
3/2+, 5/2+
5/2+ ½-
3/2-, 5/2-
5/2- ½+
3/2+, 5/2+
5/2+ ½-
3/2-, 5/2-
5/2-

22. Nucleon Spectrum Possible 5/2- state: pattern similar to exp: Future:
As expected, most states decaying
Multiple volumes for decay analysis
3/2-, 5/2- ½-
5/2-
3/2-, 5/2- ½-
5/2-

23. Strategy for Excited Decays
Variational results: use in 3-pt
Excited sink: p=0
Ground source: p 0 Q2
 v(n)

24. Nucleon Form-Factors (Ground state)
Nf=0 anisotropic lattices, M¼ = 480, 720, 1080 MeV

25. Nucleon Form-Factors (Ground state)
Nf=2+1 anisotropic lattices, M¼ = 580, 875 MeV

26. Nucleon Radiative Transition
Nf=0 exploratory: P11->Nucleon transition

27. Nucleon Radiative Transition
Excited transition: large “pion cloud” effects ! small mass
m¼ = 480, 720, 1100 MeV
PRD 2008

28. Nucleon Radiative Transition
Nf=2+1 anisotropic lattices, M¼ = 580, 875 MeV
Does not use derivative operators:

29. Multi-hadrons Meson and baryon excited state energies obtainable
2-pt correlators: e.g., 2-mesons
Different than in 3-pt
Two-particle signals suppressed relative to single particle?

30. Current and future work
Some efforts underway
Strange quark spectrum (hybrids) and radiative transitions
Excited light baryon spectrum (N, ¢, ¥, §, ¤) [showed first versions in N and ¥ (Mathur)]
Radiative transitions for P11(1440), S11(1535), D13(1520)
Q2 <~ 5 GeV2
Need to disentangle decay states:
Two-meson states, I=1 & 0
Meson-baryon