Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 231
present discussion I will not worry any further about the axial-vector
contribution.
The matrix element for the interaction between neutrinos and photons
can then be read from the effective vertex
i C V G F
e √ 2 Aα Π αβ ψ ν γ α (1 − γ 5 )ψ ν . (6.86)
The electric charge in the denominator removes one such factor contained
in Π which was calculated for photon forward scattering. In
the matrix element, Π is to be taken at the four-momentum K of the
photon. Besides plasmon decay γ → νν, this interaction also allows for
processes such as Cherenkov absorption γν → ν or emission ν → γν.
For the decay of a plasmon with a polarization vector ϵ and fourmomentum
K the squared matrix element has the form Eq. (6.76) with
M αβ = 8 G2 F
2e 2 π2 T,L (g αβ + 2ϵ ∗ αϵ β ). (6.87)
Because the plasmon is a propagating mode it obeys its dispersion
relation, i.e. π T,L = ω 2 − k 2 . The decay rate is then
Γ = C2 V G 2 F
48π 2 α Z T,L
(ω 2 − k 2 ) 3
. (6.88)
ω
This result was first derived by Adams, Ruderman, and Woo (1963),
the correct Z for the longitudinal case was first derived by Zaidi (1965).
This equation is understood “on shell” where ω depends on k through
the dispersion relation ω 2 − k 2 = π T,L (ω, k).
6.5.4 Summary of Decay Rates
In order to express the decay rates in a compact form, recall that on
shell the scale for K 2 is set by the plasma frequency ω P . Therefore, it
is useful to define
ˆπ T,L (k) ≡ π T,L(ω k , k)
, (6.89)
ωP
2
where ω k is the frequency related to k as a solution of the dispersion
equation ω 2 − k 2 = π(ω, k). Recall that 1 ≤ ˆπ T < 3 2 while 0 ≤ ˆπ L ≤ 1
for a time-like K 2 (k < k 1 ). At k = k 1 the L dispersion relation crosses
the light cone.