Structure of Nucleon Excitations on the Lattice · PDF fileI CrystalBall@MAMI Inaﬁnitevolumetheseresonancesareassociatedwithatowerof stableenergylevels. Onthelattice,wecanusevariationalanalysistechniquestoaccess

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?


Introduction



γp → γηp






Introduction



γp → γηp

I Crystal Barrel/TAPS at ELSA

I Crystal Ball @ MAMI





Introduction



γp → γηp






Introduction



γp → γηp


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




Introduction



γp → γηp






Introduction



γp → γηp






Introduction



γp → γηp






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


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


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


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


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


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


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


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


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


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



Start with some basis of operators χiWe use local three-quark spin-1/2 nucleon operators




Correlation Matrix


xe ip·x 〈Ω|χi

α(x)χjβ(0)|Ω〉








Correlation Matrix


xe ip·x 〈Ω|χi

α(x)χjβ(0)|Ω〉








Correlation Matrix


xe ip·x 〈Ω|χi

α(x)χjβ(0)|Ω〉








Correlation Matrix


xe ip·x 〈Ω|χi

α(x)χjβ(0)|Ω〉








Correlation Matrix


xe ip·x 〈Ω|χi

α(x)χjβ(0)|Ω〉








Correlation Matrix


xe ip·x 〈Ω|χi

α(x)χjβ(0)|Ω〉



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−)


Effective mass


GB(p ; t ; Γ−) ≡ Tr Γ−∑




Effective energy

EBeff(p, t) ≡ 1

δtln

GB(p ; t ; Γ−)

GB(p ; t + δt ; Γ−)


Effective mass


GB(p ; t ; Γ−) ≡ Tr Γ−∑




Effective energy

EBeff(p, t) ≡ 1

δtln

GB(p ; t ; Γ−)

GB(p ; t + δt ; Γ−)


Effective mass


GB(p ; t ; Γ−) ≡ Tr Γ−∑




Effective energy

EBeff(p, t) ≡ 1

δtln

GB(p ; t ; Γ−)

GB(p ; t + δt ; Γ−)


Effective mass


GB(p ; t ; Γ−) ≡ Tr Γ−∑




Effective energy

EBeff(p, t) ≡ 1

δtln

GB(p ; t ; Γ−)

GB(p ; t + δt ; Γ−)


Effective mass


GB(p ; t ; Γ−) ≡ Tr Γ−∑




Effective energy

EBeff(p, t) ≡ 1

δtln

GB(p ; t ; Γ−)

GB(p ; t + δt ; Γ−)


Effective mass


GB(p ; t ; Γ−) ≡ Tr Γ−∑




Effective energy

EBeff(p, t) ≡ 1

δtln

GB(p ; t ; Γ−)

GB(p ; t + δt ; Γ−)


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.)



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.)


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)



0

t1

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



0

t1

t2

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



0

t1

t2

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



0

t1

t2

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



p

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



p

q

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



p

q

p′

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



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.)



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.)



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.)


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.)


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.)


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



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±)

)


Γp =14


χip ≡ Γpχ









)


Γp =14


χip ≡ Γpχ









)


Γp =14


χip ≡ Γpχ









)


Γp =14


χip ≡ Γpχ









)


Γp =14


χip ≡ Γpχ









)


Γp =14


χip ≡ Γpχ









)


Γp =14


χip ≡ Γpχ






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.)



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.)


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)


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)



p

q

p′

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



p

q

p′

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)


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.)


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.)


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


Charge radius

Dipole ansatz

Gdipole(Q2) =

G0

1 + Q2/Λ2

State Radius




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



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



Charge radius

Dipole ansatz

Gdipole(Q2) =

G0

1 + Q2/Λ2

State Radius





p

q

p′

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)



p

q

p′

Matrix element

〈B ; p′ ; s ′| jµ|B ; p ; s〉 =

uB


2mBF2(Q2)

)uB

Sachs form factors

GE (Q2) = F1(Q2)− Q2

2mBF2(Q2)

GM(Q2) = F1(Q2) + F2(Q2)


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.)


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*


GM(0) estimate

Find that GM(Q2) and GE (Q2) have similar Q2 dependenceConsider GM(Q2)/GE (Q2) as estimate for GM(0)


+23 +2

3

−13

Proton*


GM(0) estimate



+23 +2

3

−13

Proton*


GM(0) estimate



+23 +2

3

−13

Proton*


GM(0) estimate



+23 +2

3

−13

Proton*

+23

−13 −1

3

Neutron*


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.)


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.)


Magnetic moments

−2

−1

0

1

2

µ/µN

p∗1n∗1

p∗2n∗2


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


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


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


Conclusion

The PEVA technique is critical to correctly extractingI Form factors of nucleon excitations

I Precision matrix elements of ground state nucleon (∼ 10% effect)








Conclusion

The PEVA technique is critical to correctly extractingI Form factors of nucleon excitationsI Precision matrix elements of ground state nucleon (∼ 10% effect)








Conclusion









Conclusion









Conclusion









Conclusion









Conclusion





The PEVA technique can be extended to nucleon transitionsI Enables a comparison with experimental results

I This extension is almost complete



Conclusion





The PEVA technique can be extended to nucleon transitionsI Enables a comparison with experimental resultsI This extension is almost complete



Conclusion





The PEVA technique can be extended to nucleon transitionsI Enables a comparison with experimental resultsI This extension is almost complete



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).


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.)


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)


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.)


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.)


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.)


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


Documents

Structure of Nucleon Excitations on the Lattice · PDF fileI CrystalBall@MAMI Inaﬁnitevolumetheseresonancesareassociatedwithatowerof stableenergylevels. Onthelattice,wecanusevariationalanalysistechniquestoaccess