Robert Edwards Jefferson Lab - Thomas Jefferson National … · 2012-03-12 · Determining the Hadron Spectrum Using Lattice QCD Robert Edwards Jefferson Lab Twin Approaches to Confinement

Determining the Hadron Spectrum Using Lattice QCD

Robert Edwards Jefferson Lab

Twin Approaches to Confinement Physics

2012

Collaborators (Hadron Spectrum Collaboration):

J. Dudek, P. Guo, B. Joo, D. Richards (JLab), S. Wallace (Maryland)

N. Mathur (Tata), L. Liu, M. Peardon, S. Ryan, C. Thomas (Trinity College)

Determining the Hadron Spectrum Using Lattice QCD

Robert Edwards Jefferson Lab

Twin Approaches to Confinement Physics

2012

Collaborators (Hadron Spectrum Collaboration):

J. Dudek, P. Guo, B. Joo, D. Richards (JLab), S. Wallace (Maryland)

N. Mathur (Tata), L. Liu, M. Peardon, S. Ryan, C. Thomas (Trinity College)

Recent publications:

“Hybrid baryons”, in press PRD, 1201.2349

“Helicity operators for mesons in flight”, PRD85, 1107.1930

“Lightest hybrid meson supermultiplet”, PRD84, 1106.5515

“Excited state baryon spectroscopy”, PRD84, 1104.5152

“Isoscalar meson spectroscopy”, PRD83, 1102.4299

“Phase shift of isospin-2 scattering”, PRD83, 1011.6352

“Toward the excited meson spectrum”, PRD82, 1004.4930

“Highly excited and exotic meson spectrum”, PRL103, 0909.0200

Where are the “Missing” Baryon Resonances?

3

• What are collective modes?

• Is there “freezing” of degrees of freedom?

• What is the structure of the states?

• Where is the glue?


4





PDG uncertainty on

B-W mass

Nucleon & Delta spectrum

QM predictions


5





PDG uncertainty on

B-W mass


2 2 1

QM predictions

1 1 0


6





PDG uncertainty on

B-W mass


2 2 1

QM predictions

4 5 3 1

???

1 1 0 2 3 2 1

???

Spectrum from variational method

Two-point correlator

7


Matrix of correlators


8



“Rayleigh-Ritz method”

Diagonalize: eigenvalues ! spectrum

eigenvectors ! spectral “overlaps” Zin


9







10

Each state optimal combination of ©i






Benefit: orthogonality for near degenerate states


11

Each state optimal combination of ©i

Baryon operators

Construction : permutations of 3 objects

12

1104.5152

Baryon operators


13

• Symmetric: •e.g., uud+udu+duu

• Antisymmetric: •e.g., uud-udu+duu-…

• Mixed: (antisymmetric & symmetric) •e.g., udu - duu & 2duu - udu - uud

1104.5152

Baryon operators


14




1104.5152

Multiplication rules: • SymmetricAntisymmetric ! Antisymmetric

• MixedxMixed ! Symmetric©Antisymmetric©Mixed

•….

Baryon operators


15




1104.5152



•….

Color antisymmetric ! Require Space x [Flavor x Spin] symmetric

Baryon operators


16




1104.5152



•….

©JM Ã¡CGC0s

¢i;j;k

h~Dii

h~Dij[ª]k

Space: couple covariant derivatives onto

single-site spinors - build any J,M


Baryon operators


17




Classify operators by permutation symmetries:

• Leads to rich structure 1104.5152



•….

©JM Ã¡CGC0s

¢i;j;k

h~Dii

h~Dij[ª]k

Space: couple covariant derivatives onto

single-site spinors - build any J,M


Baryon operator basis

All possible 3-quark operators up to two covariant derivatives: some JP

18



19

Spatial symmetry classification: JP #ops E.g., spatial symmetries

J=1/2- 32 N 2PM ½- N 4PM ½-

J=3/2- 36 N 2PM 3/2- N 4PM 3/2-

J=5/2- 19 N 4PM 5/2-

J=1/2+ 32 N 2SS ½+ N 2SM ½+ N 2PM ½+

N 4DM½+

N 2PA ½+ N 4PM ½+

J=3/2+

36 N 2DS3/2+

N 2DM3/2+ N 2PA 3/2+

N 2PM 3/2+

N 4SM3/2+ N 4DM3/2+

N 4PM 3/2+

J=5/2+ 19 N 2DS5/2+ N 2DM5/2+

N 4DM5/2+

N 4PM 5/2+

J=7/2+ 4 N 4DM7/2+

e.g., Nucleons: N 2S+1L¼ JP



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Spatial symmetry classification: JP #ops E.g., spatial symmetries

J=1/2- 32 N 2PM ½- N 4PM ½-

J=3/2- 36 N 2PM 3/2- N 4PM 3/2-

J=5/2- 19 N 4PM 5/2-

J=1/2+ 32 N 2SS ½+ N 2SM ½+ N 2PM ½+

N 4DM½+

N 2PA ½+ N 4PM ½+

J=3/2+

36 N 2DS3/2+

N 2DM3/2+ N 2PA 3/2+

N 2PM 3/2+

N 4SM3/2+ N 4DM3/2+

N 4PM 3/2+

J=5/2+ 19 N 2DS5/2+ N 2DM5/2+

N 4DM5/2+

N 4PM 5/2+

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

Hold on, what are those PM ??

Operators featuring explicit glue

At two derivatives & L=1, have operators featuring a chromomagnetic B field

21



22

Occurs only in Mixed spatial symmetry with L=1 ! PM with positive parity

Hybrid operator: is 0 unless in a nontrivial background gauge field A 0



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Occurs only in Mixed spatial symmetry with L=1 ! PM with positive parity

Note: Antisymmetric PA operators not of this form. Is not 0 if A = 0

Hybrid operator: is 0 unless in a nontrivial background gauge field A 0

Just a basis

Recall: two-point correlator matrix

24

Just a basis

Spectral decomposition


25

Just a basis


Choose a random linear combination of operators


26

Just a basis


Choose a random linear combination of operators


27

Spectral “overlaps” Zin change

Energies En unchanged

Spin identified Nucleon & Delta spectrum

m¼ ~ 396MeV

28

arXiv:1104.5152, 1201.2349 Statistical errors < 2%


m¼ ~ 396MeV

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m¼ ~ 396MeV

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2 2 1 1 1


m¼ ~ 396MeV

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2 2 1 1 1

4 5 3 1 2 3 2 1


m¼ ~ 396MeV

32


2 2 1 1 1

4 5 3 1 2 3 2 1


m¼ ~ 396MeV

33


2 2 1 1 1

4 5 3 1 2 3 2 1

??? ???


m¼ ~ 396MeV

34


2 2 1 1 1

4 5 3 1 2 3 2 1

??? ???

SU(6)xO(3) counting

No parity doubling


m¼ ~ 396MeV

35

arXiv:1104.5152, 1201.2349 Discern structure: spectral overlaps


m¼ ~ 396MeV

36


[56,0+]

S-wave

[56,0+]

S-wave


m¼ ~ 396MeV

37


[56,0+]

S-wave

[56,0+]

S-wave [70,1-]

P-wave

[70,1-]

P-wave


38

Discern structure: spectral overlaps


39


2SS 2SM 4SM 2DS

2DM 4DM

2PA [56’,0+], [70,0+], [56,2+],

[70,2+], [20,1+]

13 levels/ops

Significant mixing in J+


40


2SS 2SM 4SM 2DS

2DM 4DM

2PA [56’,0+], [70,0+], [56,2+],

[70,2+], [20,1+]

13 levels/ops

2SM 4SS 2DM

4DS [56’,0+], [70,0+],

[56,2+], [70,2+]

8 levels/ops

Significant mixing in J+


41

2SS 2SM 4SM 2DS

2DM 4DM

2PA [56’,0+], [70,0+], [56,2+],

[70,2+], [20,1+]

13 levels/ops

2SM 4SS 2DM

4DS [56’,0+], [70,0+],

[56,2+], [70,2+]

8 levels/ops

No “freezing” of

degrees of

freedom

Hybrid baryons

42

Negative parity structure replicated: gluonic components (hybrid baryons)

[70,1+]

P-wave

[70,1-]

P-wave

Hybrid baryons

43

Negative parity structure replicated: gluonic components (hybrid baryons)

[70,1+]

P-wave

[70,1-]

P-wave

Predicted by Barnes & Close, 1983

See talk by J. Dudek

Hadronic decays

44

Current spectrum calculations:

no evidence of multi-particle levels

Hadronic decays

Plot the non-interacting meson levels as a guide

45



Hadronic decays


46



Require multi-particle operators

• (lattice) helicity construction

• annihilation diagrams

Hadronic decays


47



Require multi-particle operators

• (lattice) helicity construction

• annihilation diagrams

Extract δ(E) at discrete E

Spectrum of finite volume field

Solutions

Quantization condition when -L/2 < z < L/2

Same physics in 4 dim version (but messier)

Provable in a QFT (and relativistic)

Two spin-less bosons: Ã(x,y) = f(x-y) ! f(z)

The idea: 1 dim quantum mechanics

Finite volume scattering

49

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)

Scattering in a periodic cubic box (length L)

• Discrete energy levels in finite volume

Resonances

1011.6352, 1203.????

Scattering of composite objects in non-perturbative field theory

isospin=2 ¼¼

δ0(E) & δ2(E)

Resonances

1011.6352, 1203.????

Scattering of composite objects in non-perturbative field theory m¼=481 MeV

m¼=421 MeV

m¼=330 MeV

m¼=290MeV

isospin=2 ¼¼

δ0(E) & δ2(E)

isospin=1 ¼¼

Feng, et.al, 1011.5288

sin2(δ1(E))

Resonances Scattering of composite objects in non-perturbative field theory m¼=481 MeV

m¼=421 MeV

m¼=330 MeV

m¼=290MeV

isospin=1 ¼¼

Feng, et.al, 1011.5288

Manifestation of “decay” in

Euclidean space

Can extract pole position

Resonances Scattering of composite objects in non-perturbative field theory m¼=481 MeV

m¼=421 MeV

m¼=330 MeV

m¼=290MeV

isospin=1 ¼¼

Feng, et.al, 1011.5288

g½¼¼

m¼2 (GeV2)

Extracted coupling: stable in pion mass

Stability a generic feature

of couplings??

Hadronic decays

54

Some candidates: determine phase shift

Somewhat elastic

S11! [N¼]S

+[N´]S ¢![N¼]P

Hadronic decays – the full job

55

Need a lattice program of “amplitude analysis” - lots of room for help!

Summary & prospects

56

Summary & prospects

Results for baryon excited state spectrum: • No “freezing” of degrees of freedom nor parity doubling

• Broadly consistent with non-relativistic quark model

• Extra bits interpreted as hybrid baryons • Add multi-particle ops ! baryon spectrum becomes denser

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Summary & prospects




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Short-term plans: resonance determination! • Lighter pion masses (230MeV available) • Extract couplings in multi-channel systems (with ¼, ´, K…)

Summary & prospects




59

T-matrix “poles” from Euclidean space?


Summary & prospects




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Yes! [with caveats] • Also complicated

• But all Minkowski information is there

Summary & prospects




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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

Backup slides

• The end

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SU(3) limit

Documents

Robert Edwards Jefferson Lab - Thomas Jefferson National … · 2012-03-12 · Determining the Hadron Spectrum Using Lattice QCD Robert Edwards Jefferson Lab Twin Approaches to Confinement