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Electromagnetic Form Factors through Parity-Expanded Variational Analysis Finn M. Stokes Waseem Kamleh, Derek B. Leinweber and Benjamin J. Owen Centre for the Subatomic Structure of Matter Finn M. Stokes (CSSM) Parity-Expanded Variational

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SLIDE 1

Electromagnetic Form Factors through Parity-Expanded Variational Analysis

Finn M. Stokes Waseem Kamleh, Derek B. Leinweber and Benjamin J. Owen

Centre for the Subatomic Structure of Matter

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 1 / 35

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SLIDE 2

Introduction

The isolation of excitations of baryons at nonzero momentum is important for the evaluation of baryon form factors and transition moments

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 2 / 35

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SLIDE 3

Introduction

The isolation of excitations of baryons at nonzero momentum is important for the evaluation of baryon form factors and transition moments Existing parity projection techniques are vulnerable to opposite parity contaminations at nonzero momentum

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 2 / 35

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SLIDE 4

Introduction

The isolation of excitations of baryons at nonzero momentum is important for the evaluation of baryon form factors and transition moments Existing parity projection techniques are vulnerable to opposite parity contaminations at nonzero momentum We propose the Parity Expanded Variational Analysis (PEVA) technique to resolve this issue

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 2 / 35

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SLIDE 5

Nonzero momentum

Eigenstates of nonzero momentum are not eigenstates of parity

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 3 / 35

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SLIDE 6

Nonzero momentum

Eigenstates of nonzero momentum are not eigenstates of parity Categorise states by parity in rest frame

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 3 / 35

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SLIDE 7

Nonzero momentum

Eigenstates of nonzero momentum are not eigenstates of parity Categorise states by parity in rest frame Call states that transform positively under parity in their rest frame “positive parity states” (B+)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 3 / 35

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SLIDE 8

Nonzero momentum

Eigenstates of nonzero momentum are not eigenstates of parity Categorise states by parity in rest frame Call states that transform positively under parity in their rest frame “positive parity states” (B+) Call states that transform negatively under parity in their rest frame “negative parity states” (B−)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 3 / 35

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SLIDE 9

Conventional analysis

Conventional baryon operators {χi} couple to states of both parities Ω|χi|B+ = λB+

i

mB+ EB+ uB+(p, s) Ω|χi|B− = λB−

i

mB− EB− γ5 uB−(p, s)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 4 / 35

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SLIDE 10

Conventional analysis

Conventional baryon operators {χi} couple to states of both parities Ω|χi|B+ = λB+

i

mB+ EB+ uB+(p, s) Ω|χi|B− = λB−

i

mB− EB− γ5 uB−(p, s)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 4 / 35

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SLIDE 11

Conventional analysis

Conventional baryon operators {χi} couple to states of both parities Ω|χi|B+ = λB+

i

mB+ EB+ uB+(p, s) Ω|χi|B− = λB−

i

mB− EB− γ5 uB−(p, s) Form correlation matrix Gij(p; t) :=

  • x

eip·x Ω|χi(x) χj(0)|Ω

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 4 / 35

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SLIDE 12

Parity projection

Introduce Γ

± = (γ4 ± I)/2 and define Gij(Γ ±; p; t) := tr (Γ ± Gij(p; t))

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

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SLIDE 13

Parity projection

Introduce Γ

± = (γ4 ± I)/2 and define Gij(Γ ±; p; t) := tr (Γ ± Gij(p; t))

At zero momentum, projected correlators only contain terms for states

  • f a single parity

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

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SLIDE 14

Parity projection

Introduce Γ

± = (γ4 ± I)/2 and define Gij(Γ ±; p; t) := tr (Γ ± Gij(p; t))

At zero momentum, projected correlators only contain terms for states

  • f a single parity

Gij(Γ

+; 0; t) =

  • B+

e−mB+t λB+

i

λB+

j

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

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SLIDE 15

Parity projection

Introduce Γ

± = (γ4 ± I)/2 and define Gij(Γ ±; p; t) := tr (Γ ± Gij(p; t))

At zero momentum, projected correlators only contain terms for states

  • f a single parity

Gij(Γ

+; 0; t) =

  • B+

e−mB+t λB+

i

λB+

j

Gij(Γ

−; 0; t) =

  • B−

e−mB−t λB−

i

λB−

j

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

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SLIDE 16

Parity projection

Introduce Γ

± = (γ4 ± I)/2 and define Gij(Γ ±; p; t) := tr (Γ ± Gij(p; t))

At zero momentum, projected correlators only contain terms for states

  • f a single parity

Gij(Γ

+; 0; t) =

  • B+

e−mB+t λB+

i

λB+

j

Gij(Γ

−; 0; t) =

  • B−

e−mB−t λB−

i

λB−

j

Can analyse states of each parity independently

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

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SLIDE 17

Nonzero momentum

O(|p|) opposite parity contaminations at nonzero momentum

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 6 / 35

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Nonzero momentum

O(|p|) opposite parity contaminations at nonzero momentum Could remove single opposite parity state with Γ

±(p) = 1

2 mB∓ EB∓(p)γ4 ± I

  • Finn M. Stokes (CSSM)

Parity-Expanded Variational Analysis Lattice 2016 6 / 35

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SLIDE 19

Nonzero momentum

O(|p|) opposite parity contaminations at nonzero momentum Could remove single opposite parity state with Γ

±(p) = 1

2 mB∓ EB∓(p)γ4 ± I

  • Better to use variational analysis to remove all contaminating states

simultaneously

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 6 / 35

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SLIDE 20

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Finn M. Stokes (CSSM)

Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 21

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Finn M. Stokes (CSSM)

Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 22

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Finn M. Stokes (CSSM)

Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 23

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Finn M. Stokes (CSSM)

Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 24

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Define PEVA projector

Γ

p = 1

4(I + γ4)(I − iγ5γk ˆ pk)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 25

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Define PEVA projector

Γ

p = 1

4(I + γ4)(I − iγ5γk ˆ pk) χi

p := Γ pχi couples to positive parity states at zero momentum

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 26

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Define PEVA projector

Γ

p = 1

4(I + γ4)(I − iγ5γk ˆ pk) χi

p := Γ pχi couples to positive parity states at zero momentum

χi′

p := Γ pγ5χi couples to negative parity states at zero momentum

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 27

Parity-Expanded Variational Analysis (PEVA)

Terms in unprojected correlation matrix have Dirac structure EB±(p) ± mB± − σkpk σkpk − (EB±(p) ∓ mB±)

  • Define PEVA projector

Γ

p = 1

4(I + γ4)(I − iγ5γk ˆ pk) χi

p := Γ pχi couples to positive parity states at zero momentum

χi′

p := Γ pγ5χi couples to negative parity states at zero momentum

Both couple to states with consistent Dirac structure Γ

puB(p, s)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

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SLIDE 28

Lattice results

Second lightest PACS-CS (2 + 1)-flavour full-QCD ensemble

◮ 323 × 64 lattices ◮ a = 0.0951(14) fm by Sommer parameter ◮ κu,d = 0.1377, corresponding to mπ = 280(5) MeV Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 8 / 35

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Lattice results

Second lightest PACS-CS (2 + 1)-flavour full-QCD ensemble

◮ 323 × 64 lattices ◮ a = 0.0951(14) fm by Sommer parameter ◮ κu,d = 0.1377, corresponding to mπ = 280(5) MeV

Using conventional spin-1/2 nucleon operators χ1 = ǫabc [ua⊤(Cγ5) db] uc χ2 = ǫabc [ua⊤(C) db] γ5 uc Apply 16, 35, 100 and 200 sweeps of gauge invariant gaussian smearing in creating the propagators

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 8 / 35

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Eigenvector components

Ground state

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i (p)

χ+ Interpolators

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i′(p)

χ− Interpolators

16 sweeps 35 sweeps 100 sweeps 200 sweeps χ+

1 = Γ p χ1

χ+

2 = Γ p χ2

χ−

1 = Γ p γ5 χ1

χ−

2 = Γ p γ5 χ2

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 9 / 35

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Effective energy

Ground state - p2 ≃ 0.166 GeV2

16 18 20 22 24 26 28 30 32 34 t/a 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Eeff

α (t) (GeV)

Ground State

  • Conv. (χ2/dof = 0.598)

PEVA (χ2/dof = 0.367)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 10 / 35

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SLIDE 32

Effective energy

Ground state

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) 0.0 0.5 1.0 1.5 2.0 2.5 Eeff

α (GeV)

Ground State

  • Conv. Γ+ 8 × 8

PEVA 16 × 16

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 11 / 35

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SLIDE 33

Eigenvector components

First negative parity excitation

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i (p)

χ+ Interpolators

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i′(p)

χ− Interpolators

16 sweeps 35 sweeps 100 sweeps 200 sweeps χ+

1 = Γ p χ1

χ+

2 = Γ p χ2

χ−

1 = Γ p γ5 χ1

χ−

2 = Γ p γ5 χ2

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 12 / 35

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SLIDE 34

Effective energy

First negative parity excitation - p2 ≃ 0.166 GeV2

16 18 20 22 24 26 28 t/a 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Eeff

α (t) (GeV)

First Negative Parity Excitation

  • Conv. (χ2/dof = 6.618)

PEVA (χ2/dof = 0.570)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 13 / 35

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SLIDE 35

Effective energy

First negative parity excitation

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) 0.0 0.5 1.0 1.5 2.0 2.5 Eeff

α (GeV)

First Negative Parity Excitation

  • Conv. Γ− 8 × 8

PEVA 16 × 16

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 14 / 35

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SLIDE 36

Eigenvector components

Second negative parity excitation

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i (p)

χ+ Interpolators

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i′(p)

χ− Interpolators

16 sweeps 35 sweeps 100 sweeps 200 sweeps χ+

1 = Γ p χ1

χ+

2 = Γ p χ2

χ−

1 = Γ p γ5 χ1

χ−

2 = Γ p γ5 χ2

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 15 / 35

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SLIDE 37

Effective energy

Second negative parity excitation

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) 0.0 0.5 1.0 1.5 2.0 2.5 Eeff

α (GeV)

Second Negative Parity Excitation

  • Conv. Γ− 8 × 8

PEVA 16 × 16

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 16 / 35

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SLIDE 38

Eigenvector components

First positive parity excitation

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i (p)

χ+ Interpolators

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) −1.0 −0.5 0.0 0.5 1.0 uα

i′(p)

χ− Interpolators

16 sweeps 35 sweeps 100 sweeps 200 sweeps χ+

1 = Γ p χ1

χ+

2 = Γ p χ2

χ−

1 = Γ p γ5 χ1

χ−

2 = Γ p γ5 χ2

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 17 / 35

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SLIDE 39

Effective energy

First positive parity excitation

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV2) 0.0 0.5 1.0 1.5 2.0 2.5 Eeff

α (GeV)

First Positive Parity Excitation

  • Conv. Γ+ 8 × 8

PEVA 16 × 16

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 18 / 35

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SLIDE 40

Effective energy

Nucleon spectrum

0.0 0.2 0.4 0.6 0.8 1.0 p2 (GeV 2) 0.0 0.5 1.0 1.5 2.0 2.5 Eeff

α (GeV)

Nucleon Spectrum

  • Conv. Γ+ 8 × 8
  • Conv. Γ− 8 × 8

PEVA 16 × 16

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 19 / 35

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SLIDE 41

Form Factors

Construct three point correlator for a single energy eigenstate G µ

ij (p′, p; Γ; t2, t1) := tr

  • Γ
  • x,y

eip·x ei(p′−p)·y Ω|χi(x) Jµ(y) χj(0)|Ω

  • G µ

α(p′, p; Γ; t2, t1) := vα i (p′) G µ ij (p′, p; Γ; t2, t1) uα j (p)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 20 / 35

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SLIDE 42

Form Factors

Construct three point correlator for a single energy eigenstate G µ

ij (p′, p; Γ; t2, t1) := tr

  • Γ
  • x,y

eip·x ei(p′−p)·y Ω|χi(x) Jµ(y) χj(0)|Ω

  • G µ

α(p′, p; Γ; t2, t1) := vα i (p′) G µ ij (p′, p; Γ; t2, t1) uα j (p)

Define ratio Rµ

α(p′, p; t1) :=

  • G µ

α(p′, p; Γ; t2, t1) G µ α(p, p′; Γ; t2, t1)

Gα(p′; t2) Gα(p; t2)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 20 / 35

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SLIDE 43

Form Factors

Construct three point correlator for a single energy eigenstate G µ

ij (p′, p; Γ; t2, t1) := tr

  • Γ
  • x,y

eip·x ei(p′−p)·y Ω|χi(x) Jµ(y) χj(0)|Ω

  • G µ

α(p′, p; Γ; t2, t1) := vα i (p′) G µ ij (p′, p; Γ; t2, t1) uα j (p)

Define ratio Rµ

α(p′, p; t1) :=

  • G µ

α(p′, p; Γ; t2, t1) G µ α(p, p′; Γ; t2, t1)

Gα(p′; t2) Gα(p; t2) Extract GE(Q2) and GM(Q2) from ratio

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 20 / 35

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SLIDE 44

Form Factors

Fits to ground state form factors

16 18 20 22 24 26 28 30 t/a −1.0 −0.5 0.0 0.5 1.0 1.5 GE & GM

Ground State Form Factors

GE (up) GE (dp) GM (up) GM (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 21 / 35

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SLIDE 45

Form Factors

GE(Q2 = 0.15(1) GeV2) for ground state

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

0.0 0.2 0.4 0.6 0.8 1.0 GE

GE for Ground State

Grnd State (up) Grnd State (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 22 / 35

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SLIDE 46

Form Factors

GM(Q2 = 0.15(1) GeV2) for ground state

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 GM (µN)

GM for Ground State

Proton Neutron

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 23 / 35

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SLIDE 47

Form Factors

Fits to GE for first negative parity excitation

16 18 20 22 24 26 28 t/a −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 GE

GE for First Negative Parity Excitation

PEVA (up) PEVA (dp)

  • Conv. (up)
  • Conv. (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 24 / 35

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SLIDE 48

Form Factors

GE(Q2 = 0.15(1) GeV2) for first negative parity excitation

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

0.0 0.2 0.4 0.6 0.8 1.0 GE

GE for First Negative Parity Excitation

Grnd State (up) Grnd State (dp) 1st −ve Par. Exc. (up) 1st −ve Par. Exc. (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 25 / 35

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SLIDE 49

Form Factors

GE(Q2 = 0.15(1) GeV2) for second negative parity excitation

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

0.0 0.2 0.4 0.6 0.8 1.0 GE

GE for Second Negative Parity Excitation

Grnd State (up) Grnd State (dp) 2nd −ve Par. Exc. (up) 2nd −ve Par. Exc. (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 26 / 35

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SLIDE 50

Form Factors

Fits to GM 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 1.5 GM (µN)

GM for First Negative Parity Excitation

PEVA (up) PEVA (dp)

  • Conv. (up)
  • Conv. (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 27 / 35

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SLIDE 51

Form Factors

GM(Q2 = 0.15(1) GeV2) for first negative parity excitation

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 GM (µN)

GM for First Negative Parity Excitation

Proton* Neutron*

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 28 / 35

slide-52
SLIDE 52

Form Factors

GM(Q2 = 0.15(1) GeV2) for second negative parity excitation

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 GM (µN)

GM for Second Negative Parity Excitation

Proton* Neutron*

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 29 / 35

slide-53
SLIDE 53

Form Factors

GM(Q2 = 0.15(1) GeV2) for second negative parity excitation

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 GM (µN)

GM of Individual Quark Sectors

Grnd State (up) Grnd State (dp) 2nd −ve Par. Exc. (up) 2nd −ve Par. Exc. (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 30 / 35

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SLIDE 54

Positive Parity Nucelon Spectrum

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

0.0 0.5 1.0 1.5 2.0 2.5 Eeff

α (GeV)

Ground State First Positive Parity Excitation

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 31 / 35

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SLIDE 55

Form Factors

GE(Q2 = 0.15(1) GeV2) for first positive parity excitation 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

0.0 0.2 0.4 0.6 0.8 1.0 GE

GE for First Positive Parity Excitation

Grnd State (up) Grnd State (dp) 1st +ve Par. Exc. (up) 1st +ve Par. Exc. (dp)

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 32 / 35

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SLIDE 56

Form Factors

GM(Q2 = 0.15(1) GeV2) for first positive parity excitation

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m2

π (GeV2)

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 GM (µN)

GM for First Positive Parity Excitation

Proton* Neutron*

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 33 / 35

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SLIDE 57

Conclusion

Conventional baryon spectroscopy at nonzero momentum contaminated by opposite parity states

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 34 / 35

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SLIDE 58

Conclusion

Conventional baryon spectroscopy at nonzero momentum contaminated by opposite parity states The PEVA technique can effectively remove these opposite parity contaminations

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 34 / 35

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SLIDE 59

Conclusion

Conventional baryon spectroscopy at nonzero momentum contaminated by opposite parity states The PEVA technique can effectively remove these opposite parity contaminations Clear effect on two point function for lowest lying negative parity excitation

Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 34 / 35

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SLIDE 60

Conclusion

Conventional baryon spectroscopy at nonzero momentum contaminated by opposite parity states The PEVA technique can effectively remove these opposite parity contaminations Clear effect on two point function for lowest lying negative parity excitation Has significant effects on three point functions for excited states

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Conclusion

Conventional baryon spectroscopy at nonzero momentum contaminated by opposite parity states The PEVA technique can effectively remove these opposite parity contaminations Clear effect on two point function for lowest lying negative parity excitation Has significant effects on three point functions for excited states This is an important step towards making contact with experiment through calculations such as baryon transition moments

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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. Owen
  • Phys. Rev. D 92 (2015) 11, 114506

doi:10.1103/PhysRevD.92.114506 arXiv:1302.4152 (hep-lat).

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