massive photon and cosmologyhome.kias.re.kr/psec/theoreticalphy/kias_talk_6_12.pdf ·...
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![Page 1: Massive Photon and Cosmologyhome.kias.re.kr/psec/TheoreticalPhy/kias_talk_6_12.pdf · 2014-06-16 · Massive Photon: Introduction Coester, Umezawa, Glauber, Stueckelberg, Boulware](https://reader034.vdocuments.mx/reader034/viewer/2022042418/5f351261d0e7183bac21947a/html5/thumbnails/1.jpg)
Massive Photon and Cosmology
Phillial OhSungkyunkwan University
KIAS 2014. 6.
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Contents
I. Introduction
II. Massive QED and Cosmology
III. Massive Dark Photon and Cosmology
IV. Conslusions
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Massive Photon: Introduction
� Maxwell-Proca Equations
� de Broglie: Photon mass would lead to a largerspeed of violet light than that of the red one
� Schrodinger: Magnetic field of the Earth would beexponentially cut off at distances of the order of thephoton Compton wave length λγ ∼ 1/mγ
� The longitudinal photons do not manifestthemselves in the black body radiation: Bass andSchrodinger
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Massive Photon: Introduction
� Coester, Umezawa, Glauber, Stueckelberg,Boulware and Gilbert: massive QED smoothly goesover to QED. Amplitudes describing longitudinalmodes suppressed as mγ → 0, provided thatelectromagnetic current is conserved
� Deser: Longitudinal mode is transformed into ascalar mode decoupled from the current, remainscoupled to gravitation
� Reviews on massive photon: Goldhaber and Nieto(2010)
� Reviews on Stuckelberg theory: Ruegg andRuiz-Altaba (2003)
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Massive QED and Cosmology
Invariant Action
Linv = −1
4FμνF
μν −1
2m2(Aμ +
1
m∂μφ)(Aμ +
1
m∂μφ)
+ψ(iγμ∂μ − mf )ψ + Aμψγμψ
Invariant under Aμ → Aμ − ∂μΛ, φ → φ + mΛ, ψ → e−iΛψ
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Massive QED and Cosmology
Invariant Action
Linv = −1
4FμνF
μν −1
2m2(Aμ +
1
m∂μφ)(Aμ +
1
m∂μφ)
+ψ(iγμ∂μ − mf )ψ + Aμψγμψ
Invariant under Aμ → Aμ − ∂μΛ, φ → φ + mΛ, ψ → e−iΛψ
Gauge fixed action
Lξ = −1
4FμνF
μν −1
2m2AμA
μ −1
2∂μφ∂μφ −
1
2ξm2φ2
−1
2ξ(∂μA
μ)2 + ψ(iγμ∂μ − mf )ψ + Aμψγμψ
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Massive QED and Cosmology
� Propagator:
A :
[
gμν −(1 − ξ)kμkν
k2 + ξm2
]−1
k2 + m2
φ :−1
k2 − ξm2
� m → 0 exists, gives the gauge-fixed massless propagator
� Good high energy behavior, provided ξ 6= ∞
−1k2 + m2
[
gμν +kμkν
m2
]
+kμkν
m2(k2 + ξm2)
� ξ → ∞ gives Proca propagator
� ξ dependent poles cancel each other: In the limit, ξ → ∞,they disappear
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Massive QED and Cosmology
n� Unitary and Renormalizable
n� Dirac Monopole
n� Infrared cutoff
n� Indistinguishable from QED withphoton mass limit
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Massive QED and Cosmology
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Massive QED and Cosmology
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Massive QED and Cosmology
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Massive QED and Cosmology
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Massive QED and Cosmology
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Massive QED and Cosmology
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Summary
� Massive QED could be a viable model for darkenergy
�
Λ ∼ m2γ
� The vacuum Aμ = fμ breaks Lorentzsymmetry� A small deviation from ΛCDM� Non-minimal coupling of the Stuckelbergscalar field to gravity� ξ-(in)dependence in quantum cosmology?
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Dark Massive Vector and Cosmology
Sinv =
∫d4x
{
−1
4FμνF
μν −1
4CμνC
μν
−1
2m2(Aμ + αSμ +
1
m∂μφ1)(A
μ + αSμ +1
m∂μφ1)
−1
2μ2(Sμ +
1
μ∂μφ2)(S
μ +1
μ∂μφ2)
}
Invariant under Aμ → Aμ − ∂μΛ1, Sμ → Sμ − ∂μΛ2
φ1 → φ1 + mΛ1 + αmΛ2, φ2 → φ2 + μΛ2
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Double Massive Vector and Cosmology
Gauge fixing term:
− 12ξ1
[∂μAμ + α∂μS
μ + ξ1mφ1 + λ(Aμ + αSμ)Sμ]2
− 12ξ2
(∂μSμ + ξ2μφ2)
2
Sgf =
∫d4x
{
−1
4FμνF
μν −1
4CμνC
μν
−1
2m2(Aμ + αSμ)(Aμ + αSμ) −
1
2μ2SμS
μ
−1
2∂μφ1∂
μφ1 −1
2ξ1m
2φ21 −
1
2∂μφ2∂
μφ2 −1
2ξ2μ
2φ22
−1
2ξ1
[(∂μAμ + α∂μS
μ) + λ(Aμ + αSμ)Sμ]2
−1
2ξ2
(∂μSμ)2 − λmφ1(Aμ + αSμ)Sμ
}
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Dark Massive Vector and Cosmology
Redefine: Aμ + αSμ → Aμ,√
1 + α2Sμ → Sμ, α√1+α2
≡ χ
Sgf =
∫d4x
{
−1
4FμνF
μν +χ
2FμνC
μν −1
4CμνC
μν
−1
2m2AμA
μ −1
2μ2(1 − χ2)SμS
μ
−1
2∂μφ1∂
μφ1 −1
2ξ1m
2φ21 −
1
2∂μφ2∂
μφ2 −1
2ξ2μ
2φ22
−1
2ξ1
[∂μA
μ + λ√
1 − χ2AμSμ]2
−1
2ξ2
(1 − χ2)(∂μSμ)2 − λ
√1 − χ2mφ1AμS
μ
}
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Dark Massive Vector and Cosmology
S =∫
d4x√−g
{R
2−
14FμνFμν +
χ
2FμνCμν
−14CμνC
μν −12m2AμAμ −
12μ2SμSμ
+12α
[∇μAμ + ξ (SμAμ)]2 +12β
(∇μSμ)2}
Gμν = κTμν
Constant solution:
Aμ = fμ, Sμ = hμ
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Dark Massive Vector and Cosmology
ds2 = −dt2 + a(t)2dxidxi
Tμν = FμαFνα −
14gμνFαβFαβ − m2
[
AμAν −12gμνAαAα
]
+CμαCνα −
14gμνCαβCαβ + μ2
[
SμSν −12gμνSαSα
]
−gμν
2α
[(∇μAμ) + ξ (SαAα)2
]2−
gμν
2β(∇μSμ)2
= −Λgμν
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Dark Massive Vector and Cosmology
Λ =α
2ξ2m2μ2
[
1 − 4α
β
(m2
μ2
)(
1 −ξ2
αm2hμh
μ
)]
� Proca limit: β → ∞
Λ =α
2ξ2m2μ2
hμhμ =34
(αm2μ2
ξ4
)[4ξ2
3μ2− 1 ±
√
1 −8ξ2
3μ2
]
μ2 ≥83ξ2
Λ ≥43αm2
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Summary
�
ρΛ ∼ m2μ2
� Inclusion of Stuckelberg fields for cosmology� Unified description of dark matter and dark
energy
� Cosmic coincidence: ρΛ ∼(
M2EW
Mpl
)4
(Alkani-Hamed et al, 2000)� Photon oscillations ?
� EW theory with a massive photon(Ruegg and Ruiz-Altaba, 2003)
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