Stars as Laboratories for Fundamental Physics - MPP Theory Group

232 Chapter 6
The decay rates of Eqs. (6.80), (6.83) and (6.88) of a plasmon with
three-momentum k are then expressed as
⎧
ωP 2 ˆπ k
α ν
Millicharge,
4π
Γ k = 4π Z
⎪⎨ k
µ 2 ( ) ω
2 2
P ˆπ k
×
Dipole Moment, (6.90)
3 ω k
2 4π
CV 2 G 2 ( )
F ω
2 3
P ˆπ k
⎪⎩
Standard Model,
α 4π
where for Z k and ˆπ k the T or L value appropriate for the chosen polarization
must be used.
6.5.5 Energy-Loss Rates
It is now an easy task to calculate stellar energy-loss rates for the
plasma process. An integration over the Bose-Einstein distributions of
the transverse and longitudinal plasmons yields for the energy-loss rate
per unit volume
Q T = 2 ∫ ∞
dk k 2 Γ T ω
2π 2 0 e ω/T − 1 ,
Q L = 1 ∫ k1
dk k 2 Γ L ω
2π 2 0 e ω/T − 1 . (6.91)
In Q T the factor 2 counts two polarization states. In Q L the integration
can be extended only to the wave number k 1 where the L dispersion
relation crosses the light cone—for k > k 1 decays are kinematically
forbidden. In either case Γ T,L and ω are functions of k, the latter given
by the dispersion relation.
For the specific neutrino interaction models discussed in the previous
section one obtains with the decay rates of Eq. (6.90)
⎧
ωP
2 α ν Q 1 Millicharge,
4π
Q = 8ζ 3
3π T 3 ×
⎪⎨
⎪⎩
µ 2
2
C 2 V G 2 F
α
( ) ω
2 2
P
Q 2
4π
Dipole Moment,
( ) ω
2 3
P
Q 3
4π
Standard Model,
(6.92)
where ζ 3 ≈ 1.202 refers to the Riemann Zeta function. The dimensionless
emission rates Q n for the three cases are each a sum of a transverse