Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 233
and longitudinal term, Q n = Q L,n + Q T,n where
Q L,n = 1 ∫ k1
dk k 2 Z L ˆπ L
n
4ζ 3 T 3 0 e ω/T − 1 = 1 ∫ k1
dk k 2 ω2
4ζ 3 T 3 0
ω 2 P
˜Z L ˆπ n−1
L
e ω/T − 1 ,
Q T,n = 1 ∫ ∞
dk k 2 Z T ˆπ T
n
2ζ 3 T 3 0 e ω/T − 1 . (6.93)
The second equation for Q L,n relies on the definition Eq. (6.53), i.e.
Z L = ˜Z L ω 2 /(ω 2 − k 2 ), and ω 2 − k 2 = π L was used.
The normalization factors were chosen such that Q T,n = 1 if the
plasmons are treated as effectively massless particles for the phase-space
integration. Then Z T = ˆπ T = 1 which is a reasonable approximation
in a nondegenerate, nonrelativistic plasma. In that limit to lowest
order k 1 = ω P , ˜Z L = 1, and π L = ωP 2 − k 2 . Therefore, in this limit
Q L,n ≪ Q T,n . In fact, the longitudinal emission rate is of comparable
importance to the transverse one only in a narrow range of parameters
of astrophysical interest (Haft, Raffelt, and Weiss 1994).
These simple approximations, however, are not adequate for most of
the conditions where the plasma process is important. In Appendix C
the numerical neutrino emission rates are discussed; a comparison between
Fig. C.1 and Fig. 6.3 reveals that the plasma process is important
for 0.3 < ∼ ω P /T < ∼ 30, i.e. transverse plasmons can be anything from
relativistic to entirely nonrelativistic. For a practical stellar evolution
calculation one may use the analytic approximation formula for the
plasma process discussed in Appendix C, based on the representation
of the dispersion relations of Sect. 6.3.
The main issue at stake in this book, however, is nonstandard neutrino
emission from the direct electromagnetic couplings discussed in
Sect. 6.5. Instead of constructing new numerical emission rate formulae
one uses the existing ones for the standard-model (SM) couplings and
scales them to the novel cases. Numerically, one finds
Q charge
Q SM
= α να (4π) 2
C 2 V G 2 Fω 4 P
Q 1
Q 3
= 0.664 e 2 14
( 10 keV
ω P
) 4
Q 1
Q 3
,
Q dipole
Q SM
= µ2 α 2π
C 2 V G 2 Fω 2 P
( ) 2
Q 2
10 keV
= 0.318 µ 2 Q 2
12
, (6.94)
Q 3 ω P Q 3
where e 14 = e ν /10 −14 e and µ 12 = µ/10 −12 µ B with µ B = e/2m e .
Contours for Q 1 /Q 3 and Q 2 /Q 3 are shown in Fig. 6.13 according to
Haft, Raffelt, and Weiss (1994). Replacing these ratios by unity in a
practical stellar evolution calculation introduces only a small error.