Electromagnetic Form Factors through Parity-Expanded Variational - PowerPoint PPT Presentation

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

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

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

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

Nonzero momentum Eigenstates of nonzero momentum are not eigenstates of parity Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 3 / 35

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

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

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

Conventional analysis Conventional baryon operators { χ i } couple to states of both parities � m B + � Ω | χ i | B + � = λ B + E B + u B + ( p , s ) i � m B − � Ω | χ i | B − � = λ B − E B − γ 5 u B − ( p , s ) i Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 4 / 35

Conventional analysis Conventional baryon operators { χ i } couple to states of both parities � m B + � Ω | χ i | B + � = λ B + E B + u B + ( p , s ) i � m B − � Ω | χ i | B − � = λ B − E B − γ 5 u B − ( p , s ) i Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 4 / 35

Conventional analysis Conventional baryon operators { χ i } couple to states of both parities � m B + � Ω | χ i | B + � = λ B + E B + u B + ( p , s ) i � m B − � Ω | χ i | B − � = λ B − E B − γ 5 u B − ( p , s ) i Form correlation matrix e i p · x � Ω | χ i ( x ) χ j ( 0 ) | Ω � � G ij ( p ; t ) := x Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 4 / 35

Parity projection Introduce Γ ± = ( γ 4 ± I ) / 2 and define G ij (Γ ± ; p ; t ) := tr (Γ ± G ij ( p ; t )) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

Parity projection Introduce Γ ± = ( γ 4 ± I ) / 2 and define G ij (Γ ± ; p ; t ) := tr (Γ ± G ij ( p ; t )) At zero momentum, projected correlators only contain terms for states of a single parity Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

Parity projection Introduce Γ ± = ( γ 4 ± I ) / 2 and define G ij (Γ ± ; p ; t ) := tr (Γ ± G ij ( p ; t )) At zero momentum, projected correlators only contain terms for states of a single parity e − m B + t λ B + λ B + � G ij (Γ + ; 0 ; t ) = i j B + Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

Parity projection Introduce Γ ± = ( γ 4 ± I ) / 2 and define G ij (Γ ± ; p ; t ) := tr (Γ ± G ij ( p ; t )) At zero momentum, projected correlators only contain terms for states of a single parity e − m B + t λ B + λ B + � G ij (Γ + ; 0 ; t ) = i j B + e − m B − t λ B − λ B − � G ij (Γ − ; 0 ; t ) = i j B − Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

Parity projection Introduce Γ ± = ( γ 4 ± I ) / 2 and define G ij (Γ ± ; p ; t ) := tr (Γ ± G ij ( p ; t )) At zero momentum, projected correlators only contain terms for states of a single parity e − m B + t λ B + λ B + � G ij (Γ + ; 0 ; t ) = i j B + e − m B − t λ B − λ B − � G ij (Γ − ; 0 ; t ) = i j B − Can analyse states of each parity independently Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 5 / 35

Nonzero momentum O ( | p | ) opposite parity contaminations at nonzero momentum Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 6 / 35

Nonzero momentum O ( | p | ) opposite parity contaminations at nonzero momentum Could remove single opposite parity state with � m B ∓ ± ( p ) = 1 � Γ E B ∓ ( p ) γ 4 ± I 2 Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 6 / 35

Nonzero momentum O ( | p | ) opposite parity contaminations at nonzero momentum Could remove single opposite parity state with � m B ∓ ± ( p ) = 1 � Γ E B ∓ ( p ) γ 4 ± I 2 Better to use variational analysis to remove all contaminating states simultaneously Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 6 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Define PEVA projector p = 1 Γ 4 ( I + γ 4 )( I − i γ 5 γ k ˆ p k ) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Define PEVA projector p = 1 Γ 4 ( I + γ 4 )( I − i γ 5 γ k ˆ p k ) p χ i couples to positive parity states at zero momentum χ i p := Γ Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Define PEVA projector p = 1 Γ 4 ( I + γ 4 )( I − i γ 5 γ k ˆ p k ) p χ i couples to positive parity states at zero momentum χ i p := Γ p γ 5 χ i couples to negative parity states at zero momentum χ i ′ p := Γ Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Parity-Expanded Variational Analysis (PEVA) Terms in unprojected correlation matrix have Dirac structure � E B ± ( p ) ± m B ± � − σ k p k σ k p k − ( E B ± ( p ) ∓ m B ± ) Define PEVA projector p = 1 Γ 4 ( I + γ 4 )( I − i γ 5 γ k ˆ p k ) p χ i couples to positive parity states at zero momentum χ i p := Γ p γ 5 χ i couples to negative parity states at zero momentum χ i ′ p := Γ Both couple to states with consistent Dirac structure Γ p u B ( p , s ) Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 7 / 35

Lattice results Second lightest PACS-CS (2 + 1)-flavour full-QCD ensemble ◮ 32 3 × 64 lattices ◮ a = 0 . 0951 ( 14 ) fm by Sommer parameter ◮ κ u , d = 0 . 1377, corresponding to m π = 280 ( 5 ) MeV Finn M. Stokes (CSSM) Parity-Expanded Variational Analysis Lattice 2016 8 / 35