Massive Photon and Cosmology

Transcription

1 Massive Photon and Cosmology Phillial Oh Sungkyunkwan University KIAS

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3 Contents I. Introduction II. Massive QED and Cosmology III. Massive Dark Photon and Cosmology IV. Conslusions

4 Massive Photon: Introduction Maxwell-Proca Equations de Broglie: Photon mass would lead to a larger speed of violet light than that of the red one Schrödinger: Magnetic field of the Earth would be exponentially cut off at distances of the order of the photon Compton wave length λ γ 1/m γ The longitudinal photons do not manifest themselves in the black body radiation: Bass and Schrödinger

5 Massive Photon: Introduction Coester, Umezawa, Glauber, Stueckelberg, Boulware and Gilbert: massive QED smoothly goes over to QED. Amplitudes describing longitudinal modes suppressed as m γ 0, provided that electromagnetic current is conserved Deser: Longitudinal mode is transformed into a scalar mode decoupled from the current, remains coupled to gravitation Reviews on massive photon: Goldhaber and Nieto (2010) Reviews on Stuckelberg theory: Ruegg and Ruiz-Altaba (2003)

6 Massive QED and Cosmology Invariant Action L inv = 1 4 F μνf μν 1 2 m2 (A μ + 1 m μφ)(a μ + 1 m μ φ) + ˉψ(iγ μ μ m f )ψ + A μ ˉψγ μ ψ Invariant under A μ A μ μ Λ, φ φ + mλ, ψ e iλ ψ

7 Massive QED and Cosmology Invariant Action L inv = 1 4 F μνf μν 1 2 m2 (A μ + 1 m μφ)(a μ + 1 m μ φ) + ˉψ(iγ μ μ m f )ψ + A μ ˉψγ μ ψ Invariant under A μ A μ μ Λ, φ φ + mλ, ψ e iλ ψ Gauge fixed action L ξ = 1 4 F μνf μν 1 2 m2 A μ A μ 1 2 μφ μ φ 1 2 ξm2 φ 2 1 2ξ ( μa μ ) 2 + ˉψ(iγ μ μ m f )ψ + A μ ˉψγ μ ψ

8 Massive QED and Cosmology Propagator: A : [g μν (1 ξ)kμ k ν ] 1 k 2 + ξm 2 k 2 + m 2 φ : 1 k 2 ξm 2 m 0 exists, gives the gauge-fixed massless propagator Good high energy behavior, provided ξ 1 [g μν k 2 + m 2 + kμ k ν ] k μ k ν m 2 + m 2 (k 2 + ξm 2 ) ξ gives Proca propagator ξ dependent poles cancel each other: In the limit, ξ, they disappear

9 Massive QED and Cosmology Unitary and Renormalizable Dirac Monopole Infrared cutoff Indistinguishable from QED with photon mass limit

10 Massive QED and Cosmology

11 Massive QED and Cosmology

12 Massive QED and Cosmology

13 Massive QED and Cosmology

14 Massive QED and Cosmology

15 Massive QED and Cosmology

16 Summary Massive QED could be a viable model for dark energy Λ m 2 γ The vacuum A μ = f μ breaks Lorentz symmetry A small deviation from ΛCDM Non-minimal coupling of the Stuckelberg scalar field to gravity ξ-(in)dependence in quantum cosmology?

17 Dark Massive Vector and Cosmology S inv = { d 4 x 1 4 F μνf μν 1 4 C μνc μν 1 2 m2 (A μ + αs μ + 1 m μφ 1 )(A μ + αs μ + 1 m μ φ 1 ) 1 2 μ2 (S μ + 1 μ μφ 2 )(S μ + 1 } μ μφ 2 ) Invariant under A μ A μ μ Λ 1, S μ S μ μ Λ 2 φ 1 φ 1 + mλ 1 + αmλ 2, φ 2 φ 2 + μλ 2

18 Double Massive Vector and Cosmology Gauge fixing term: 1 2ξ 1 [ μ A μ + α μ S μ + ξ 1 mφ 1 + λ(a μ + αs μ )S μ ] 2 1 2ξ 2 ( μ S μ + ξ 2 μφ 2 ) 2 { S gf = d 4 x 1 4 F μνf μν 1 4 C μνc μν 1 2 m2 (A μ + αs μ )(A μ + αs μ ) 1 2 μ2 S μ S μ 1 2 μφ 1 μ φ ξ 1m 2 φ μφ 2 μ φ ξ 2μ 2 φ ξ 1 [( μ A μ + α μ S μ ) + λ(a μ + αs μ )S μ ] 2 1 2ξ 2 ( μ S μ ) 2 λmφ 1 (A μ + αs μ )S μ }

19 Dark Massive Vector and Cosmology Redefine: A μ + αs μ A μ, 1 + α 2 S μ S μ, α 1+α 2 χ S gf = { d 4 x 1 4 F μνf μν + χ 2 F μνc μν 1 4 C μνc μν 1 2 m2 A μ A μ 1 2 μ2 (1 χ 2 )S μ S μ 1 2 μφ 1 μ φ ξ 1m 2 φ μφ 2 μ φ ξ 2μ 2 φ [ μ A μ + λ ] 2 1 χ 2ξ 2 A μ S μ 1 1 2ξ 2 (1 χ 2 )( μ S μ ) 2 λ 1 χ 2 mφ 1 A μ S μ }

20 Dark Massive Vector and Cosmology S = d 4 x { R g F μνf μν + χ 2 F μνc μν 1 4 C μνc μν 1 2 m2 A μ A μ 1 2 μ2 S μ S μ + 1 2α [ μa μ + ξ (S μ A μ )] } 2β ( μs μ ) 2 Constant solution: G μν = κt μν A μ = f μ, S μ = h μ

21 Dark Massive Vector and Cosmology ds 2 = dt 2 + a(t) 2 dx i dx i T μν = F μα F ν α 1 4 g μνf αβ F αβ m 2 [ A μ A ν 1 2 g μνa α A α ] +C μα C ν α 1 4 g μνc αβ C αβ + μ 2 [ S μ S ν 1 2 g μνs α S α ] g μν 2α = Λg μν [ ( μ A μ ) + ξ (S α A α ) 2] 2 g μν 2β ( μs μ ) 2

22 Dark Massive Vector and Cosmology Λ = α [ 2ξ 2 m2 μ α β Proca limit: β h μ h μ = 3 4 ( m 2 μ 2 Λ = α 2ξ 2 m2 μ 2 ( αm 2 μ 2 ) [ 4ξ 2 ξ 4 3μ 2 1 ± ) )] (1 ξ2 αm h μh μ 2 ] 1 8ξ2 3μ 2 μ ξ2 Λ 4 3 αm2

23 Summary ρ Λ m 2 μ 2 Inclusion of Stuckelberg fields for cosmology Unified description of dark matter and dark energy ( ) M 2 4 Cosmic coincidence: ρ Λ EW M pl (Alkani-Hamed et al, 2000) Photon oscillations? EW theory with a massive photon (Ruegg and Ruiz-Altaba, 2003)

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