Stars as Laboratories for Fundamental Physics - MPP Theory Group

122 Chapter 4
Because the matrix element has been assumed to be a constant
it can be pulled out of the integral which reduces to a phase-space
volume. Nonrelativistically, d 3 p i /[2E i (2π) 3 ] = d 3 p i /[2m N (2π) 3 ] while
E i = p 2 i /2m N in the energy δ function. The axion momentum is ignored
according to Eq. (4.5). One uses CM momenta p 1,2 = p 0 ±p and p 3,4 =
p 0 ± q where p 0 = 1 2 (p 1 + p 2 ) = 1 2 (p 3 + p 4 ) and defines u 2 ≡ p 2 /m N T
and y ≡ v 2 ≡ q 2 /m N T . Then one finds explicitly
Γ σ = 4 √ π απ 2 n B T 1/2 m −5/2
N ,
∫
s(x) = 4 du dv u 2 v 2 e |x|−u2 δ(u 2 − v 2 − |x|)
=
∫ ∞
0
dy e −y ( |x|y + y 2) 1/2
≈
√1 + |x| π/4 . (4.7)
The analytic form is accurate to better than 2.2% everywhere; it has
the correct asymptotic behavior s(0) = 1 and s(|x|≫1) = (|x|π/4) 1/2 .
We will see in Sect. 4.6.3 that for |x| ≫ 1 the nondegenerate s(x)
must actually decrease, in conflict with this explicit calculation. This
problem reveals a pathology of the OPE potential which is too singular
at short distances. For the present discussion this is of no concern so
that I stick to the explicit OPE result in order to facilitate comparison
with the existing literature.
To determine the total emission rate one uses the first representation
of s(x) and performs ∫ dx first to remove the δ function; the remaining
integrals are easily done. Explicit results for s n ≡ ∫ ∞
0 x n s(x) e −x dx are
given in Tab. 4.1. The normalized axion energy spectrum dN a /dx =
x s(x)e −x /s 1 is shown in Fig. 4.2 (solid line). The average axion energy
is ⟨ω a ⟩/T = s 2 /s 1 = 16/7. The total nondegenerate energy-loss rate is
Table 4.1. s n = ∫ ∞
0
x n s(x) e −x dx for nondegenerate (ND) and degenerate
(D) conditions.
n s n (ND) s n (D)
1 8/5 2ζ 3 + 6ζ 5 /π 2
2 128/35 31π 4 /315
3 256/21 24ζ 5 + (180/π 2 ) ζ 7
4 4096/77 82π 6 /315