Structure of Nucleon Excitations on the Lattice · PDF fileI CrystalBall@MAMI...
Transcript of Structure of Nucleon Excitations on the Lattice · PDF fileI CrystalBall@MAMI...
Structure of Nucleon Excitations on the Lattice
Finn M. Stokes
Waseem Kamleh, Derek B. Leinweber and Benjamin J. Owen
Centre for the Subatomic Structure of Matter
QCD Downunder 2017
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 1 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSA
I Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.
On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Introduction
In nature, observe negative-parity resonances N∗(1535) and N∗(1650)
Could investigate the structure of the N∗(1535) via
γp → γηp
I Crystal Barrel/TAPS at ELSAI Crystal Ball @ MAMI
In a finite volume these resonances are associated with a tower ofstable energy levels.On the lattice, we can use variational analysis techniques to accessthese energy levels
I Using local operators, we observe two negative parity states in theenergy region of the resonances.
I How are these finite-volume states related to the physical resonances?
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 2 / 28
Negative parity spectrum
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2π/GeV2
1200
1400
1600
1800
2000E/M
eVCSSM
Cyprus
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 3 / 28
Negative parity spectrum
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2π/GeV2
1200
1400
1600
1800
2000E/M
eV
non-int. π-N energy
non-int. η-N energy
CSSM
Cyprus
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 3 / 28
Hamiltonian Effective Field Theory (Liu et al. 2016)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2π/GeV2
1200
1400
1600
1800
2000E/M
eV
non-int. π-N energy
non-int. η-N energy
matrix Hamiltonian model
CSSM
Cyprus
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 4 / 28
Hamiltonian Effective Field Theory (Liu et al. 2016)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2π/GeV2
1200
1400
1600
1800
2000E/M
eV
matrix Hamiltonian model
1st most probable
2nd most probable
3rd most probable
CSSM
Cyprus
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 4 / 28
Lattice ensemble
Middle PACS-CS (2 + 1)-flavour full-QCD ensemble
323 × 64 latticesa = 0.0961(13) fm by Sommer parameterκu,d = 0.13754, corresponding to mπ ≈ 411MeVUse 368 configurations, with two sources on each configuration
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 5 / 28
Lattice ensemble
Middle PACS-CS (2 + 1)-flavour full-QCD ensemble323 × 64 lattices
a = 0.0961(13) fm by Sommer parameterκu,d = 0.13754, corresponding to mπ ≈ 411MeVUse 368 configurations, with two sources on each configuration
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 5 / 28
Lattice ensemble
Middle PACS-CS (2 + 1)-flavour full-QCD ensemble323 × 64 latticesa = 0.0961(13) fm by Sommer parameter
κu,d = 0.13754, corresponding to mπ ≈ 411MeVUse 368 configurations, with two sources on each configuration
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 5 / 28
Lattice ensemble
Middle PACS-CS (2 + 1)-flavour full-QCD ensemble323 × 64 latticesa = 0.0961(13) fm by Sommer parameterκu,d = 0.13754, corresponding to mπ ≈ 411MeV
Use 368 configurations, with two sources on each configuration
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 5 / 28
Lattice ensemble
Middle PACS-CS (2 + 1)-flavour full-QCD ensemble323 × 64 latticesa = 0.0961(13) fm by Sommer parameterκu,d = 0.13754, corresponding to mπ ≈ 411MeVUse 368 configurations, with two sources on each configuration
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 5 / 28
Conventional variational analysis
Start with some basis of operators χi
We use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Conventional variational analysis
Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Conventional variational analysis
Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Conventional variational analysis
Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Conventional variational analysis
Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Conventional variational analysis
Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Conventional variational analysis
Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators
χ1 = εabc [ua>(Cγ5) db] uc
χ2 = εabc [ua>(C ) db] γ5 uc
Apply 16, 35, 100 and 200 sweeps of gauge invariant Gaussiansmearing
Correlation Matrix
G ijαβ(p ; t) ≡∑
xe ip·x 〈Ω|χi
α(x)χjβ(0)|Ω〉
Seek optimised operators φB that couple strongly to a single energyeigenstate
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 6 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
Effective mass
Projected correlation function
GB(p ; t ; Γ−) ≡ Tr Γ−∑
xe ip·x 〈Ω|φB(x)φB(0)|Ω〉
Has an exponential time dependence
GB(p ; t ; Γ−) ≈ zBp zBp e−EB(p) t
Effective energy
EBeff(p, t) ≡ 1
δtln
GB(p ; t ; Γ−)
GB(p ; t + δt ; Γ−)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 7 / 28
First negative parity excitation at p = (0, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 8 / 28
First negative parity excitation at p = (0, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
χ2/dof = 1.142
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 8 / 28
Electromagnetic Structure
0
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
0
t1
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
0
t1
t2
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
0
t1
t2
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
0
t1
t2
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
p
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
p
q
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
Electromagnetic Structure
p
q
p′
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 9 / 28
First negative parity excitation at p = (0, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
χ2/dof = 1.142
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 10 / 28
First negative parity excitation at p = (1, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
χ2/dof = 5.845
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 11 / 28
First negative parity excitation at p = (2, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
χ2/dof = 3.683
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 12 / 28
Dispersion Relation
0.0 0.2 0.4 0.6 0.8
p2 /GeV2
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(p)/
GeV
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 13 / 28
Dispersion Relation
0.0 0.2 0.4 0.6 0.8
p2 /GeV2
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(p)/
GeV
√m2
eff + p2
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 13 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both parities
Terms in unprojected correlation matrix have Dirac structure(
EB±(p)±mB± − σkpkσkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
Parity-Expanded Variational Analysis (PEVA)
Expand basis to simultaneously isolate finite momentum energyeigenstates of both paritiesTerms in unprojected correlation matrix have Dirac structure
(EB±(p)±mB± − σkpk
σkpk − (EB±(p)∓mB±)
)
Define PEVA projector
Γp =14
(I + γ4)(I− iγ5γk pk)
χip ≡ Γpχ
i couples to positive parity states at zero momentum
χi ′p ≡ Γpγ5χ
i couples to negative parity states at zero momentum
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 14 / 28
First negative parity excitation at p = (2, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
χ2/dof = 3.683
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 15 / 28
First negative parity excitation at p = (2, 0, 0)
16 18 20 22 24 26
t/a
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(t)/
GeV
χ2/dof = 0.490
χ2/dof = 3.683
N∗1 (PEVA)
N∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 15 / 28
Dispersion Relation
0.0 0.2 0.4 0.6 0.8
p2 /GeV2
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(p)/
GeV
√m2
eff + p2
N∗1 (Conv.)
0.0 0.2 0.4 0.6 0.8
p2 /GeV2
0.0
0.5
1.0
1.5
2.0
2.5
EB eff
(p)/
GeV
√m2
eff + p2
N∗1 (Conv.)
N∗1 (PEVA)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 16 / 28
Dispersion Relation
0.0 0.2 0.4 0.6 0.8
p2 /GeV2
0.0
0.5
1.0
1.5
2.0
2.5EB eff
(p)/
GeV
√m2
eff + p2
N∗1 (Conv.)
N∗1 (PEVA)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 16 / 28
Electromagnetic Structure
p
q
p′
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 17 / 28
Electromagnetic Structure
p
q
p′
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 17 / 28
GE (Q2 ≈ 0.16 GeV2) for first negative parity excitation
16 18 20 22 24 26 28 30
t/a
0.0
0.2
0.4
0.6
0.8
1.0GE
up∗1 (PEVA) up∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 18 / 28
GE for first negative parity excitation
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Q2 /GeV2
0.0
0.2
0.4
0.6
0.8
1.0GE
up∗1 (PEVA) up∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 19 / 28
Charge radius
Dipole ansatz
Gdipole(Q2) =
G0
1 + Q2/Λ2
State Radius
N (Ground state) 0.68(2) fmN∗1 (First negative) 0.68(4) fmN∗2 (Second negative) 0.89(11) fm
Table: Charge radius extracted from dipole fit
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 20 / 28
Charge radius
Dipole ansatz
Gdipole(Q2) =
G0
1 + Q2/Λ2
State Radius
N (Ground state) 0.68(2) fmN∗1 (First negative) 0.68(4) fmN∗2 (Second negative) 0.89(11) fm
Table: Charge radius extracted from dipole fit
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 20 / 28
Charge radius
Dipole ansatz
Gdipole(Q2) =
G0
1 + Q2/Λ2
State Radius
N (Ground state) 0.68(2) fm
N∗1 (First negative) 0.68(4) fmN∗2 (Second negative) 0.89(11) fm
Table: Charge radius extracted from dipole fit
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 20 / 28
Charge radius
Dipole ansatz
Gdipole(Q2) =
G0
1 + Q2/Λ2
State Radius
N (Ground state) 0.68(2) fmN∗1 (First negative) 0.68(4) fm
N∗2 (Second negative) 0.89(11) fm
Table: Charge radius extracted from dipole fit
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 20 / 28
Charge radius
Dipole ansatz
Gdipole(Q2) =
G0
1 + Q2/Λ2
State Radius
N (Ground state) 0.68(2) fmN∗1 (First negative) 0.68(4) fmN∗2 (Second negative) 0.89(11) fm
Table: Charge radius extracted from dipole fit
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 20 / 28
Electromagnetic Structure
p
q
p′
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 21 / 28
Electromagnetic Structure
p
q
p′
Matrix element
〈B ; p′ ; s ′| jµ|B ; p ; s〉 =
uB
(γµF1(Q2)− σµνqν
2mBF2(Q2)
)uB
Sachs form factors
GE (Q2) = F1(Q2)− Q2
2mBF2(Q2)
GM(Q2) = F1(Q2) + F2(Q2)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 21 / 28
GM(Q2 ≈ 0.16 GeV2) for first negative parity excitation
16 18 20 22 24 26 28
t/a
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5GM/µN
up∗1 (PEVA)
dp∗1 (PEVA)
up∗1 (Conv.)
dp∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 22 / 28
GM(0) estimate
Find that GM(Q2) and GE (Q2) have similar Q2 dependence
Consider GM(Q2)/GE (Q2) as estimate for GM(0)
Combine these estimates from each quark sector
+23 +2
3
−13
Proton*
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 23 / 28
GM(0) estimate
Find that GM(Q2) and GE (Q2) have similar Q2 dependenceConsider GM(Q2)/GE (Q2) as estimate for GM(0)
Combine these estimates from each quark sector
+23 +2
3
−13
Proton*
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 23 / 28
GM(0) estimate
Find that GM(Q2) and GE (Q2) have similar Q2 dependenceConsider GM(Q2)/GE (Q2) as estimate for GM(0)
Combine these estimates from each quark sector
+23 +2
3
−13
Proton*
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 23 / 28
GM(0) estimate
Find that GM(Q2) and GE (Q2) have similar Q2 dependenceConsider GM(Q2)/GE (Q2) as estimate for GM(0)
Combine these estimates from each quark sector
+23 +2
3
−13
Proton*
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 23 / 28
GM(0) estimate
Find that GM(Q2) and GE (Q2) have similar Q2 dependenceConsider GM(Q2)/GE (Q2) as estimate for GM(0)
Combine these estimates from each quark sector
+23 +2
3
−13
Proton*
+23
−13 −1
3
Neutron*
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 23 / 28
GM(0) estimate for first negative parity excitation
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Q2 /GeV2
−2
−1
0
1
2
GM
(0)/µN
p∗1 (PEVA)
n∗1 (PEVA)
p∗1 (Conv.)
n∗1 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 24 / 28
GM(0) estimate for second negative parity excitation
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Q2 /GeV2
−2
−1
0
1
2
GM
(0)/µN
p∗2 (PEVA)
n∗2 (PEVA)
p∗2 (Conv.)
n∗2 (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 25 / 28
Magnetic moments
−2
−1
0
1
2
µ/µN
p∗1n∗1
p∗2n∗2
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 26 / 28
Magnetic moments
CQ
M(2
003)
CQ
M(2
005)
χC
QM
(200
5)
χC
QM
(201
3)
EH
(201
4)
−2
−1
0
1
2
µ/µN
p∗1n∗1
p∗2n∗2
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 26 / 28
Magnetic moments
CQ
M(2
003)
CQ
M(2
005)
χC
QM
(200
5)
χC
QM
(201
3)
EH
(201
4)
Con
v.L
atti
ce
−2
−1
0
1
2
µ/µN
p∗1n∗1
p∗2n∗2
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 26 / 28
Conclusion
The PEVA technique is critical to correctly extracting
I Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitations
I Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitions
I Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitionsI Enables a comparison with experimental results
I This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitionsI Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
Conclusion
The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)
Two low-lying localised 12− nucleon excitations on the lattice
I Are seen to have magnetic moments consistent with quark modelexpectations for the N∗(1535) and N∗(1650)
I Suggests that the Hamiltonian Effective Field Theory study should berepeated with the inclusion of a second bare state
The PEVA technique can be extended to nucleon transitionsI Enables a comparison with experimental resultsI This extension is almost complete
Inclusion of multi-particle scattering operators is important
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 27 / 28
More information
“Parity-expanded variational analysis for nonzero momentum”F. M. Stokes, W. Kamleh, D. B. Leinweber, M. S. Mahbub, B. J. Menadue,B. J. OwenPhys. Rev. D 92 (2015) 11, 114506doi:10.1103/PhysRevD.92.114506arXiv:1302.4152 (hep-lat).“Electromagnetic Form Factors of Excited Nucleons via Parity-ExpandedVariational Analysis”F. M. Stokes, W. Kamleh, D. B. Leinweber, B. J. OwenPoS LATTICE2016 (2016) 161arXiv:1701.07177 (hep-lat).“Hamiltonian effective field theory study of the N∗(1535) resonance inlattice QCD”Z.-W. Liu, W. Kamleh, D. .B. Leinweber, F. M. Stokes, A. W. Thomas,J.-J. WuPhys. Rev. Lett. 116 (2016) 8, 082004doi:10.1103/PhysRevLett.116.082004arXiv:1512.00140 (hep-lat).
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 28 / 28
GE (Q2 ≈ 0.16 GeV2) for ground state
16 18 20 22 24 26 28 30 32
t/a
0.0
0.2
0.4
0.6
0.8
1.0GE
up (PEVA) up (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 29 / 28
GE for ground state
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Q2 /GeV2
0.0
0.2
0.4
0.6
0.8
1.0GE
up (PEVA)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 30 / 28
GM(Q2 ≈ 0.16 GeV2) for ground state
16 18 20 22 24 26 28 30 32
t/a
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5GM/µN
up (PEVA)
dp (PEVA)
up (Conv.)
dp (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 31 / 28
GM(Q2 ≈ 0.16 GeV2) for ground state
16 18 20 22 24 26
t/a
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0.0GM/µN
dp (PEVA) dp (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 31 / 28
Error in GM for ground state
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Q2 /GeV2
0.6
0.7
0.8
0.9
1.0
1.1G
Con
v.
M(Q
2)/G
PE
VA
M(Q
2)
dp (Conv.)
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 32 / 28
Scattering state contaminations
0 1 2 3 4 50 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0N * ( 1 / 2 - )
| < m 0 | E 2 > | 2 = 6 7 . 1 %| < m 0 | E 4 > | 2 = 1 6 . 4 % (G
(t) - Σ
jG j(t))/G
(t)
t ( f m )
mπ = 4 1 1 M e V
j = 2 , 4
0 . 0 0 . 5 1 . 0 1 . 5 2 . 00 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
0 . 2 0
Finn M. Stokes (CSSM) Nucleon Excitations on the Lattice QCD Downunder 2017 33 / 28