Proton Compton Scattering In Unified Proton- + Theory ZHANG Yun - - PowerPoint PPT Presentation

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Proton Compton Scattering In Unified Proton- + Theory ZHANG Yun - - PowerPoint PPT Presentation

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary 2011 Cross Strait Meeting on Particle Physics and Cosmology Proton Compton Scattering In Unified Proton- + Theory ZHANG Yun (Collaboration

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

2011 Cross Strait Meeting on Particle Physics and Cosmology

Proton Compton Scattering In Unified Proton-∆+ Theory

ZHANG Yun (Collaboration With Konstantin G. Savvidy) Physics Department, Nanjing University April 1, 2011

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

1

Background and Motivation

2

The Model

3

Proton Compton Scattering

4

Vertex Structure

5

Summary

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

∆+(1232MeV, JP = 3

2 +) freedom must be taken into

account in proton Compton scattering

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

∆+(1232MeV, JP = 3

2 +) freedom must be taken into

account in proton Compton scattering Previously p is treated as spin 1/2 Dirac spinor and ∆+ in a different theory

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

∆+(1232MeV, JP = 3

2 +) freedom must be taken into

account in proton Compton scattering Previously p is treated as spin 1/2 Dirac spinor and ∆+ in a different theory Our innovation: We treat p and ∆+ in a unified spin 3/2 field theory

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

∆+(1232MeV, JP = 3

2 +) freedom must be taken into

account in proton Compton scattering Previously p is treated as spin 1/2 Dirac spinor and ∆+ in a different theory Our innovation: We treat p and ∆+ in a unified spin 3/2 field theory How Motivated

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SLIDE 7

Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

∆+(1232MeV, JP = 3

2 +) freedom must be taken into

account in proton Compton scattering Previously p is treated as spin 1/2 Dirac spinor and ∆+ in a different theory Our innovation: We treat p and ∆+ in a unified spin 3/2 field theory How Motivated Proton and ∆+ are both comprised of the same quarks.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

∆+(1232MeV, JP = 3

2 +) freedom must be taken into

account in proton Compton scattering Previously p is treated as spin 1/2 Dirac spinor and ∆+ in a different theory Our innovation: We treat p and ∆+ in a unified spin 3/2 field theory How Motivated Proton and ∆+ are both comprised of the same quarks. Three spin 1/2 particles results in 8 spin states, that for a spin 3/2 particle and two spin 1/2 particles.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

L invariant under point transformation: ψµ → ψµ + λγµγνψν ξ′ = ξ(1 − 4λ) − 2λ

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

L invariant under point transformation: ψµ → ψµ + λγµγνψν ξ′ = ξ(1 − 4λ) − 2λ ξ = 2z − 1 = ⇒ pµψµ(p) = 0.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

L invariant under point transformation: ψµ → ψµ + λγµγνψν ξ′ = ξ(1 − 4λ) − 2λ ξ = 2z − 1 = ⇒ pµψµ(p) = 0. mass spectrum: m3/2 = m,m1/2 =

m 6z−2.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

L invariant under point transformation: ψµ → ψµ + λγµγνψν ξ′ = ξ(1 − 4λ) − 2λ ξ = 2z − 1 = ⇒ pµψµ(p) = 0. mass spectrum: m3/2 = m,m1/2 =

m 6z−2.

Original Rarita-Schwinger Theory: z = (1+3ξ)2+3(1+ξ)2

4

.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

L invariant under point transformation: ψµ → ψµ + λγµγνψν ξ′ = ξ(1 − 4λ) − 2λ ξ = 2z − 1 = ⇒ pµψµ(p) = 0. mass spectrum: m3/2 = m,m1/2 =

m 6z−2.

Original Rarita-Schwinger Theory: z = (1+3ξ)2+3(1+ξ)2

4

. no spin 1/2 on shell component.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Generalized Rarita-Schwinger Theory

The Lagrangian Konstantin G. Savvidy, arXiv:1005.3455: L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν,ζ = 3ξ2+2ξ+1

2

L invariant under point transformation: ψµ → ψµ + λγµγνψν ξ′ = ξ(1 − 4λ) − 2λ ξ = 2z − 1 = ⇒ pµψµ(p) = 0. mass spectrum: m3/2 = m,m1/2 =

m 6z−2.

Original Rarita-Schwinger Theory: z = (1+3ξ)2+3(1+ξ)2

4

. no spin 1/2 on shell component. superluminal propagation

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν Spin 1/2 component wave function:

u2(0, + 1

2) = 1 3z−1(0, 0, 0, 0, 0, 1 2 √ 3, 0, − 1 2 √ 3, 0, i 2 √ 3, 0, − i 2 √ 3, 1 2 √ 3, 0, − 1 2 √ 3, 0)T

u2(0, − 1

2) = 1 3z−1(0, 0, 0, 0, 1 2 √ 3, 0, − 1 2 √ 3, 0, − i 2 √ 3, 0, i 2 √ 3, 0, 0, − 1 2 √ 3, 0, 1 2 √ 3)T

uµ

2 α(k, σ) = Lµναβ(k, M)uν 2 β(0, σ)

Lµναβ = LV µν ⊗ LSαβ LV, LS: boost matrix for vector and dirac spinor fields respectively.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν LV =        

E M k1 M k2 M k3 M k1 M

1 + ( E

M − 1) k2

1

| k|2

( E

M − 1) k1k2 | k|2

( E

M − 1) k1k3 | k|2 k2 M

( E

M − 1) k2k1 | k|2

1 + ( E

M − 1) k2

2

| k|2

( E

M − 1) k2k3 | k|2 k3 M

( E

M − 1) k3k1 | k|2

( E

M − 1) k3k2 | k|2

( E

M − 1) k2

3

| k|2

        LS =

1

√

2M(E+M)

  • E + M −

k · σ E + M + k · σ

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν Electrodynamic Interaction pµ → pµ − Aµ ⇒ LI = AµJµ = e ¯ ψνΓµνρψρAµ, pµJµ = 0 Γµνρ = γµδνρ + ξ(γνδµρ + γρηνµ) + ζγνγµγρ

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

L = ¯ ψµ[Dµν − mΘµν]ψν, Dµν = γρpρδµν + ξ(γµpν + γνpµ) + ζγµγρpργν, Θµν = δµν − zγνγν Electrodynamic Interaction pµ → pµ − Aµ ⇒ LI = AµJµ = e ¯ ψνΓµνρψρAµ, pµJµ = 0 Γµνρ = γµδνρ + ξ(γνδµρ + γρηνµ) + ζγνγµγρ Comparison

  • V. Pascalutsa and O. Scholten, Nucl. Phys. A591, 658 (1995)

L1

I = iG1 2m ¯

ψαΘαµ(zf)γνγ5T3NF νµ + h.c. L2

I = −G2 (2m)2 ¯

ψαΘαµ(zf)γ5T3∂µNF νµ + h.c. L3

I = −G3 (2m)2 ¯

ψαΘαµ(zf)γ5T3N∂νF νµ + h.c.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

In L = ¯ ψµ[Dµν − mΘµν]ψν,we identify spin 3/2 component as ∆+ and spin 1/2 component as proton. Proton Compton Scattering Feynman Diagrams

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

In L = ¯ ψµ[Dµν − mΘµν]ψν,we identify spin 3/2 component as ∆+ and spin 1/2 component as proton. Feynman Rules Out line: u2(k1), ¯ u2(k4) Vertex: Γµνρ = γµδνρ + ξ(γνδµρ + γρηνµ) + ζγνγµγρ Propogator: [Dµν − mΘµν]−1

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

In L = ¯ ψµ[Dµν − mΘµν]ψν,we identify spin 3/2 component as ∆+ and spin 1/2 component as proton. Feynman Rules Out line: u2(k1), ¯ u2(k4) Vertex: Γµνρ = γµδνρ + ξ(γνδµρ + γρηνµ) + ζγνγµγρ Propogator: [Dµν − mΘµν]−1 Two poles in propogator: p2 = m2: ∆+ pole p2 = M2: proton pole(M =

m 6z−2)

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

In L = ¯ ψµ[Dµν − mΘµν]ψν,we identify spin 3/2 component as ∆+ and spin 1/2 component as proton.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

In L = ¯ ψµ[Dµν − mΘµν]ψν,we identify spin 3/2 component as ∆+ and spin 1/2 component as proton. Amplitude and Differential Cross Section Mσ1,σ4,λ2,λ3 = ie2(¯ u2η(k4, σ4)ΓµηρSργ(k1 − k3)Γνγκu2

κ(k1, σ1)

+¯ u2η(k4, σ4)ΓνηρSργ(k1 + k2)Γµγκu2

κ(k1, σ1))

ǫµ(k2, λ2)ǫ∗

ν(k3, λ3)

dσ dΩ = 1 64π2 (ω′ ω )2

  • σ1,σ4,λ2,λ3

|M|2

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Low energy expansion agrees with F .E.Low,PR96,1428(1954): dσ dΩ = α2(1+cos2 θ)

2M2

− α2(1−cos θ)(1+cos2 θ)

M3

ω + O(ω2)

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Low energy expansion agrees with F .E.Low,PR96,1428(1954): dσ dΩ = α2(1+cos2 θ)

2M2

− α2(1−cos θ)(1+cos2 θ)

M3

ω + O(ω2) Difference at O(ω2) from Dirac theory affects the extraction of polarizability parameters(¯ α,¯ β) from experiment(δ¯ α,δ ¯ β = O(1)) dσ dΩ = ( dσ dΩ)Born − αω2

M ( ¯ α+¯ β 2 (1 + cos θ)2 + ¯ α−¯ β 2 (1 − cos θ)2)

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Low energy expansion agrees with F .E.Low,PR96,1428(1954): dσ dΩ = α2(1+cos2 θ)

2M2

− α2(1−cos θ)(1+cos2 θ)

M3

ω + O(ω2) Difference at O(ω2) from Dirac theory affects the extraction of polarizability parameters(¯ α,¯ β) from experiment(δ¯ α,δ ¯ β = O(1)) dσ dΩ = ( dσ dΩ)Born − αω2

M ( ¯ α+¯ β 2 (1 + cos θ)2 + ¯ α−¯ β 2 (1 − cos θ)2)

50 100 150 200 250 Ω 2.1011 4.1011 6.1011 8.1011

θ = 0 Blue: current theory Red: Dirac theory

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Low energy expansion agrees with F .E.Low,PR96,1428(1954): dσ dΩ = α2(1+cos2 θ)

2M2

− α2(1−cos θ)(1+cos2 θ)

M3

ω + O(ω2) Difference at O(ω2) from Dirac theory affects the extraction of polarizability parameters(¯ α,¯ β) from experiment(δ¯ α,δ ¯ β = O(1)) dσ dΩ = ( dσ dΩ)Born − αω2

M ( ¯ α+¯ β 2 (1 + cos θ)2 + ¯ α−¯ β 2 (1 − cos θ)2)

50 100 150 200 250 Ω 5.1012 1.1011 1.51011 2.1011 2.51011 3.1011 3.51011

θ = π

2

Blue: current theory Red: Dirac theory

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Low energy expansion agrees with F .E.Low,PR96,1428(1954): dσ dΩ = α2(1+cos2 θ)

2M2

− α2(1−cos θ)(1+cos2 θ)

M3

ω + O(ω2) Difference at O(ω2) from Dirac theory affects the extraction of polarizability parameters(¯ α,¯ β) from experiment(δ¯ α,δ ¯ β = O(1)) dσ dΩ = ( dσ dΩ)Born − αω2

M ( ¯ α+¯ β 2 (1 + cos θ)2 + ¯ α−¯ β 2 (1 − cos θ)2)

50 100 150 200 250 Ω 1.1011 2.1011 3.1011 4.1011 5.1011 6.1011

θ = π Blue: current theory Red: Dirac theory

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Low energy expansion agrees with F .E.Low,PR96,1428(1954): dσ dΩ = α2(1+cos2 θ)

2M2

− α2(1−cos θ)(1+cos2 θ)

M3

ω + O(ω2) Difference at O(ω2) from Dirac theory affects the extraction of polarizability parameters(¯ α,¯ β) from experiment(δ¯ α,δ ¯ β = O(1)) dσ dΩ = ( dσ dΩ)Born − αω2

M ( ¯ α+¯ β 2 (1 + cos θ)2 + ¯ α−¯ β 2 (1 − cos θ)2)

50 100 150 200 250 Ω 1.1011 2.1011 3.1011 4.1011 5.1011 6.1011

We are not ready to fit experimental data yet, since proton is not a fundamental particle.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Vertex Structure

Reminiscence Dirac spinor electrodynamic interaction: ¯ u(p′)[ (p+p′)µ

2m

F1(q2) + iσµνqν

2m F2(q2)]u(p)Aµ

q = p′ − p F1, F2: form factors.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Reminiscence Dirac spinor electrodynamic interaction: ¯ u(p′)[ (p+p′)µ

2m

F1(q2) + iσµνqν

2m F2(q2)]u(p)Aµ

q = p′ − p F1, F2: form factors. Our Task Find (all) possible Γµνρ(p, p′) in ¯ ψν(p′)Γµνρ(p, p′)ψρ(p)Aµ Gauge Invariance: qµ ¯ ψν(p′)Γµνρ(p, p′)ψρ(p) = 0

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Our Task Find (all) possible Γµνρ(p, p′) in ¯ ψν(p′)Γµνρ(p, p′)ψρ(p)Aµ Gauge Invariance: qµ ¯ ψν(p′)Γµνρ(p, p′)ψρ(p) = 0 Structures We Have Found Γµνρ(p, p′) = Scalar Type ηνρ(p + p′)µ γνγρ(p + p′)µ × ηνργ5(p + p′)µ × γνγργ5(p + p′)µ

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Our Task Find (all) possible Γµνρ(p, p′) in ¯ ψν(p′)Γµνρ(p, p′)ψρ(p)Aµ Gauge Invariance: qµ ¯ ψν(p′)Γµνρ(p, p′)ψρ(p) = 0 Structures We Have Found Γµνρ(p, p′) = Vector Type ηνργµ γνγµγρ γνηµρ + γρηµν × γνηµρ − γρηµν × γ5(· · · )

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Our Task Find (all) possible Γµνρ(p, p′) in ¯ ψν(p′)Γµνρ(p, p′)ψρ(p)Aµ Gauge Invariance: qµ ¯ ψν(p′)Γµνρ(p, p′)ψρ(p) = 0 Structures We Have Found Γµνρ(p, p′) = Tensor Type τ µλνρqλ σµληνρqλ σµλσνρqλ τ µλνκσκρqλ τ µλκρσνκqλ × γ5(· · · )

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Structures We Have Found Scalar Type: ηνρ(p + p′)µ γνγρ(p + p′)µ Vector Type: ηνργµ γνγµγρ γνηµρ + γρηµν Tensor Type: τ µλνρqλ σµληνρqλ σµλσνρqλ τ µλνκσκρqλ τ µλκρσνκqλ Our Claim The scalar, vector and tensor type vertexes we have found comprise the most general set of vertexes that are at most first

  • rder in q, and dominate the low energy Compton scattering

amplitude.

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian;

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution;

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

Proton Compton scattering amplitude and cross section

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

Proton Compton scattering amplitude and cross section

* Low energy limit;

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

Proton Compton scattering amplitude and cross section

* Low energy limit; * comparison with Dirac theory calculation

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SLIDE 45

Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

Proton Compton scattering amplitude and cross section

* Low energy limit; * comparison with Dirac theory calculation

Constructing general electrodynamic interaction vertexes

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SLIDE 46

Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

Proton Compton scattering amplitude and cross section

* Low energy limit; * comparison with Dirac theory calculation

Constructing general electrodynamic interaction vertexes

* Scalar, vector and tensor type vertexes;

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SLIDE 47

Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary Summary

Summary Motivation for unifying proton and ∆+ in a generalized Rarita-Schwinger theory, and the model

* Lagrangian; * spin 1/2 solution; * electrodynamic interaction

Proton Compton scattering amplitude and cross section

* Low energy limit; * comparison with Dirac theory calculation

Constructing general electrodynamic interaction vertexes

* Scalar, vector and tensor type vertexes; * claiming that other vertexes are higher order in q

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Outline Background and Motivation The Model Proton Compton Scattering Vertex Structure Summary

Thank you all!