Stars as Laboratories for Fundamental Physics - MPP Theory Group

228 Chapter 6
The decay width of a plasmon with four-momentum K = (ω, k) in
the medium frame, and with a definite polarization is
∫
Γ =
d 3 p
2E p (2π) 3
d 3 p
2E p (2π) 3 (2π)4 δ 4 (K − P − P ) 1
2ω
Because of Eq. (6.76) one may use Lenard’s (1953) formula
∑
|M| 2 .
spins
(6.78)
∫
d 3 p d 3 p
P α P β δ 4 (K − P − P ) = π (
K 2 g αβ + 2K α K β) .
2E p 2E p 24
(6.79)
With α ν ≡ e 2 ν/4π this leads to
Γ = α ν Z (ω 2 − k 2 )/3ω, (6.80)
where the normalization ϵ ∗ αϵ α = −1 for transverse and time-like longitudinal
plasmons was used as well as ϵ · K = 0. Γ applies to both
transverse and longitudinal plasmons with the appropriate Z T,L . For a
chosen three-momentum k = |k| the quantities Z, ω, and K 2 = ω 2 − k 2
are all functions of k by virtue of the dispersion relation K 2 = π T,L (K).
In the classical limit transverse plasmons propagate like massive
particles with K 2 = ω 2 P and Z T = 1. Then Γ T = 1 3 α ν ω 3 P (ω P /ω) where
the last factor is recognized as a Lorentz time-dilation factor. For a
general dispersion relation which is not Lorentz covariant it makes little
sense, of course, to express the decay rate in the plasmon frame. An
example is the classical limit for the longitudinal mode for which to
zeroth order in T/m e the frequency is ω = ω P , Z L = ω 2 /K 2 , and then
Γ L = 1 3 α νω P with the restriction k < ω P .
6.5.2 Neutrino Dipole Moments
Another direct coupling between neutrinos and photons arises if the former
have electric or magnetic dipole or transition moments (Sects. 7.2.2
and 7.3.2)
L int = 1 2
∑ ( )
µab ψ a σ µν ψ b + ϵ ab ψ a σ µν γ 5 ψ b F µν . (6.81)
a,b
Here, F is the electromagnetic field tensor, σ µν = γ µ γ ν − γ ν γ µ , and ψ a
with a = ν 1,2,3 or ν e,µ,τ are the neutrino fields.