1. 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
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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? 2

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 3

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 4

5. 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
???
??? 5

6. Spectrum from variational method
Two-point correlator 6

7. Spectrum from variational method
Two-point correlator
Matrix of correlators 7

8. Spectrum from variational method
Two-point correlator
Matrix of correlators
“Rayleigh-Ritz method”
Diagonalize:
eigenvalues ! spectrum
eigenvectors ! spectral “overlaps” Zin 8

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 9

10. 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

11. Baryon operators
Construction : permutations of 3 objects 11

12. 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 12

13. 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 …. 13

14. 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 14

15. 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 15

16. 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 16

17. Baryon operator basis
3-quark operators & up to two covariant derivatives: some JP 17

18. 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

19. 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

20. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv:
Statistical errors < 2% 20

21. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv: 21

22. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv: 2 1 1 22

23. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv: 2 3 1 4 5 3 1 2 1 1 23

24. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv: 2 3 1 4 5 3 1 2 1 1 24

25. Spin identified Nucleon & Delta spectrum
m¼ ~ 520MeV
arXiv: 2 3 1 4 5 3 1 2 1 1 25

26. 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 26

27. Spin identified Nucleon & Delta spectrum
Discern structure: spectral overlaps
m¼ ~ 520MeV
arXiv:
[70,1-]
P-wave
[70,1-]
P-wave
[56,0+]
S-wave
[56,0+]
S-wave 27

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+] 28

29. 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

30. Roper??
Near degeneracy in ½ consistent with SU(6)­O(3) counting, but heavily mixed
Discrepancies??
Operator basis – spatial structure

31. 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

32. 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)

33. 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

34. Resonances
Scattering of composite objects in non-perturbative field theory
isospin=2 ¼¼
δ0(E)
δ2(E)
Feng, et.al,

35. 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)
Feng, et.al,

36. 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,

37. 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,

38. 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

39. Isoscalar & isovector meson spectrum
Isoscalars: flavor mixing determined

40. Isoscalar & isovector meson spectrum
Exotics
Isoscalars: flavor mixing determined

41. Isoscalar & isovector meson spectrum
Exotics
Isoscalars: flavor mixing determined
Will need to build PWA within mesons

42. 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

43. 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

44. 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

45. 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

46. 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

47. Backup slides
The end

48. 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
, ,

49. 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
Capstick, Isgur; Capstick, Roberts 49

50. 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

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

52. 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

53. 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,

54. 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