Baryon spectroscopy from lattice QCD › conferences › Nstar2011 › Tuesday...Baryon spectroscopy...

HADRON SPECTRUM COLLABORATION

Baryon spectroscopy from lattice QCD

• Goal: Determine the hadron mass spectrum of QCD

• New feature: Spin identification for N∗ and ∆ states

– R. G. Edwards, J. J. Dudek, D. G. Richards andS. J. Wallace, [arXiv:1104.5152].

• Comparisons with SU(6)⊗O(3)

• Conclusions

Nstar Workshop May 2011 1

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Lattice parameters• Nf = 2+1 QCD

– Gauge action: Symanzik-improved– Fermion action: Clover-improved Wilson

• Anisotropic: as= 0.122 fm, at = 0.035 fm

ensemble 1 2 3m` −.0840 −.0830 −.0808ms −.0743 −.0743 −.0743

Volume 163× 128 163× 128 163× 128Physical volume (2 fm)3 (2 fm)3 (2 fm)3

Ncfgs 344 570 481tsources 8 5 7mπ 0.0691(6) 0.0797(6) 0.0996(6)mK 0.0970(5) 0.1032(5) 0.1149(6)mΩ 0.2951(22) 0.3040(8) 0.3200(7)

mπ (MeV) 396 444 524


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H.-W. Lin et al. Phys. Rev. D79, 034502 (2009).

0

400

800

1200

1600

MeV

K*

a0

a1b1

N

*

*

Tuning of m` and ms yields a good account of hadron masses


3


Limitations

• Three-quark operators:

– No multiparticle operators– No clear evidence for multiparticle states: πN, etc.

• One (small) volume and one total momentum P = 0: Noextrapolations or δ’s

• mπ = 396, 444, 524 MeV : Energies generally are high

• The three-quark states essentially are stable; decays aresuppressed.


4


Computational Resources

• USQCD allocations

• Jefferson Laboratory GPUs and HPC clusters

• and the Chroma software system ( Edwards et al.)


5


Standard recipe for lattice spectra

• Use interpolating field operators B†j(x, t) to create three-quark baryons.

• Construct operators so that they transform as irreps of cubic group

• Make smooth operators i.e., smear them over many lattice sites

– Project operators to low eigenmodes of covariant lattice Laplacian– Peardon, et al., Phys. Rev. D80, 054506 (2009)

• Matrices of correlation functions: Cij(t) =∑x < 0|Bi(x, t)B†j(0, 0)|0 >

– Cij(t) ∼< i|e−Ht|j >

• Diagonalize matrices to get principal eigenvalues: λn(t, t0)

– Principal eigenvalues separate the decays of N eigenstates: e−mn(t−t0)

• Fit them & extract masses, mn.


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Contamination from states outside thediagonalization space

Expect λn(t) = e−mn(t−t0) +∑

k>N Bke−mk(t−t0) + · · ·

Two-exponential fits to principal eigenvalues

λfit(t) = (1−An)e−mn(t−t0) +A′ne−m′n(t−t0)

Ratio plots to show the goodness of fits

λfit(t)e−mn(t−t0)

Ratio tends to constant at large t


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0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0

1

2

3

4

5

6

7

8

9

Contaminations are fit well by the 2nd exponential


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0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

N∗ spectrum in irreps of cubic group: mπ = 396 MeV


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Results of standard recipe

• Lots of states and lots of degeneracies

• Spins are ambiguous

– Degenerate states in G1, H, G2 irreps imply a J = 72 state

– or accidentally degenerate J = 12 and J = 5

2 states

• Spin identification fails because:

– there are too many degenerate states to identify thesubductions of high spins

– lattice energies don’t provide sufficient information


10


New recipe to identify spins

• Use operators with known spins in continuum limit

– Incorporate covariant derivatives to realize orbital angular momenta

• Subduce the operators to irreps of cubic group

• Use spectral representation of matrices: Cij(t) =∑

nZn∗i Z

ni e−mnt

• Zni =< n|B†i (0, 0)|0 > is the overlap of operator i with state n

• Use Zni to identify spin: spin of state n is J when largest Z’s are for

operators subduced from spin J

– The different lattice irreps give approximately the same overlaps– En is the energy of a state of good J .


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HADRON SPECTRUM COLLABORATION∗

Construction of operators with good J in continuum

• Mesons: Dudek, et al., Phys.Rev.D80:054506,2009

• Baryons: Color singlet structure for 3 quarks, symmetric in space & spin

• J = L + S with

– S = 12 or 3

2 from quark spins– L = 1 or 2 from covariant derivatives– J = 1

2, 32, 5

2 and 72

– Upper (ρ = +) and lower (ρ = −) components of Dirac spinors

• Lots of operators O[J,M ] with good spin in continuum limit

• Feynman, Kislinger and Ravndal formalism for quark states applied tooperator construction, except SU(12)⊗O(3)


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Subduction to irreps of cubic group

• Cubic group irreps Λ and rows r provide orthogonal basis on lattice

• In quantum mechanics, subduction is a change of basis |J,M〉 → |Λ, r; J〉.

• |Λ, r; J〉 =∑M |J,M〉〈J,M |Λ, r; J〉

=∑M |J,M〉 S

J,MΛ,r .

• Subduction matrices: SJ,MΛ,r

• Subduced operators: O[Λ,r;J] =∑M O[J,M ]SJ,MΛ,r

• When rotational symmetry is broken weakly,

〈0|O[Λ,r;J](t)O[Λ,r;J ′]†(0)|0 >≈ δJ,J ′ is block diagonal in J .


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Matrix Cij is block diagonal approximately

0.2 0.4 0.6 0.8 1.0

Magnitude of matrix elements in a matrix of correlationfunctions at timeslice 5.


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HADRON SPECTRUM COLLABORATION∗

Reasons for approximate rotational invariance

• Rotational symmetry is broken at O(a2) by lattice action

• Lattice spacing is 0.12 fm

• Typical hadron size is 1 fm

• Smearing makes operators smooth on the hadron size scale

• Estimate: O(a2) ≈(

0.12fm1.0fm

)2

≈ 0.015

• For hadrons, rotational symmetry is broken weakly.


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Spin identification: Zni values show which operators

dominate each state


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Spin identification: Nearly the same Z in eachlattice irrep that belongs to the subduction of J

J = 52 J = 7

2

0.00

0.02

0.04

0.06

0.08

0.10

0.00

0.01

0.02

0.03

0.04


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Joint fits of G1u, Hu, G2u principal correlators to acommon mass determine the J = 7

2−

energies

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8

0 5 10 15 200.8

1.0

1.2

1.4

1.6

1.8


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Spin identification of baryon excited states

• The spin of a lattice excited state is equal toJ when the state is created predominantly byoperators subduced from continuum spin J .

• Approximately the same Z value is obtained ineach lattice irrep that belongs to the subductionof a single J value.

• Z values often are large only for a few operators,allowing interpretation of the states

• Spin identification is reliable at the scale of hadrons


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Spectral test of approximate rotational invariance

• Rotational invariance implies zero couplings between differentJ’s, so C ∝ δJ,J ′ is block diagonal

• We find small violations of block diagonality in C.

• Does the spectrum exhibit approximate rotational invariance?

• Calculate energies including J 6= J ′ couplings

• Calculate energies omitting J 6= J ′ couplingsNstar Workshop May 2011 20

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Approximate rotational invariance in spectrum,

all 48 Hu 28 J=32 16 J=5

2 4 J=72

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

≈ same energies with and without J 6= J ′ couplingsNstar Workshop May 2011 21

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1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

+ parity - parity

Lattice N∗ excited states vs. JP : mπ = 396 MeVNstar Workshop May 2011 22

22

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

+ parity - parity

Lattice N∗ spectrum: bands with + and − parity.Nstar Workshop May 2011 23

23

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

2 2 1 32⊕1→ 1

2, 3

2, 5

2

12⊕1→ 1

2, 3

2

Pauli spinors

SU(6)⊗O(3) : [70,1− ]

1st N∗− band: SU(6)⊗O(3) statessee also R. Edwards, I-C, 15:45


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1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

12⊕0→1

2

12⊕1→1

2⊕3

2

12⊕2→ 3

2⊕5

2

32⊕0→ 3

2

32⊕1→1

2⊕3

2⊕5

2

32⊕2→1

2⊕3

2⊕5

2⊕7

2

Pauli spinors

SU(6)⊗O(3) : [56,0+ ], [56,2+ ]

[70,0+ ], [70,2+ ], [20,1+ ]

4 5 3 1

1st N∗+ band: SU(6)⊗O(3) statesNstar Workshop May 2011 25

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Overall pattern of N∗ states

Expt. **** *** ** Lattice

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

Experimental Nucleon Spectrum vs. J

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

+ parity - parity

Many more states in the lattice spectrum.Nstar Workshop May 2011 26

26

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

+ parity - parity

Lattice ∆ excited states vs. JP : mπ = 396 MeVNstar Workshop May 2011 27

27

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

+ parity - parity

Lattice ∆ spectrum : bands of + and − parity statesNstar Workshop May 2011 28

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1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

1 1 0 12⊕1→ 1

2, 3

2

Pauli spinors SU(6)⊗O(3) : [70,1− ]

1st ∆− band : SU(6)⊗O(3) statesNstar Workshop May 2011 29

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1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

12⊕0→1

2

12⊕2→ 3

2⊕5

2

32⊕0→ 3

2

32⊕2→1

2⊕3

2⊕5

2⊕7

2

SU(6)⊗O(3) : [56,0+ ], [56,2+ ]

[70,0+ ], [70,2+ ]

2 3 2 1

1st ∆+ band : SU(6)⊗O(3) statesNstar Workshop May 2011 30

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Patterns of ∆ states

Expt. **** *** ** Lattice

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

Experimental ∆ Spectrum vs. J

1+

23+

25+

27+

2

1−

23−

25−

27−

2

0

1000

2000

3000

E (

MeV

)

+ parity - parity

Many more states in the lattice spectrum.Nstar Workshop May 2011 31

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Comparison of lattice results for Roper resonancesee also D. Leinweber talk in session I-C today at 17:05

1.0

1.5

2.0

2.5

3.0

0 0.05 0.1 0.15 0.2 0.25

CSSMBGR

this paper

Does the Roper resonance have a complex structure?Nstar Workshop May 2011 32

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Conclusions

• Spins are identified reliably up to J = 72

– Covariant derivatives provide orbital angular momenta– Approximate rotational invariance is realized at the scale

of hadrons– Spectral overlaps Z identify which J values dominate a

state

• Low N∗ and ∆ bands: same states as SU(6)⊗O(3) based onρ = + Dirac spiors

• Patterns of lattice baryonic states are similar to patterns ofphysical resonance states.

• Lots of lattice states; no signs of chiral restoration


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The path forward

• No multiparticle states have been identified so far usingthree-quark operators

• Multiparticle operators (e.g, πN , ππN) must be added torealize significant couplings of three-quark states and theirdecay products.

• Moving operators and larger volumes will allow determinationof elastic phase shifts using Luscher’s formalism.

• Much remains to be learned as mπ is lowered toward thephysical limit


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1+

2

7+

2

392 438 521500

1000

1500

2000

2500

3000

E (

MeV

)

G1g

1−

2

7−

2

392 438 521500

1000

1500

2000

2500

3000

E (

MeV

)G1u

Nstar Workshop May 2011

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3+

2

5+

2

7+

2

392 438 521500

1000

1500

2000

2500

3000

E (

MeV

)

Hg

3−

2

5−

2

7−

2

392 438 521500

1000

1500

2000

2500

3000

E (

MeV

)Hu


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5+

2

7+

2

392 438 5211000

1500

2000

2500

3000

3500

E (

MeV

)

G2g

5−

2

7−

2

392 438 5211000

1500

2000

2500

3000

3500

E (

MeV

)G2u


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Baryon spectroscopy from lattice QCD › conferences › Nstar2011 › Tuesday...Baryon spectroscopy...

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