Stars as Laboratories for Fundamental Physics - MPP Theory Group

156 Chapter 4
where N p
± is a plane-wave nucleon state with energy E p and spin orientation
± relative to k π , and κ 0 is a coupling constant.
In the nonrelativistic limit the axion energy-loss rate corresponding
to the decay Ñ1 → Ñ2 a is written as
∫
Q a =
d 3 k
2ω(2π) 3 ω ∫ d 3 p 1
(2π) 3 ∫ d 3 p 2
(2π) 3 f 1(1 − f 2 )
× 2π δ(E 1 − E 2 − ω) ∑
|M| 2 , (4.79)
spins
where k is the axion momentum, p 1,2 the nucleon momenta, and f 1,2
their occupation numbers. The matrix element, averaged over axion
emission angles (the condensate is not isotropic!) is found to be
⟨ ∑
|M| 2⟩ =
spins
(
C0 − ˜g A C 1
2f a
) 2
4
3 A2 κ 2 0
× (2π) 3 [ δ 3 (∆p + k π ) + δ 3 (∆p − k π ) ] , (4.80)
where ∆p = p 1 − p 2 − k. The isoscalar and isovector axion coupling
constants are C 0 = 1(C 2 p + C n ) and C 1 = 1(C 2 p − C n ) in terms of their
couplings to protons and neutrons. The isovector current carries a
renormalized axial-vector coupling constant for which Muto, Tatsumi,
and Iwamoto (1994) found ˜g A ≈ 0.43 × 1.26 = 0.54.
The phase-space integration was carried out by neglecting the axion
momentum in δ 3 (∆p ± k π ) and taking the degenerate limit. Then,
Q a = π 45
( ) 2
C0 − ˜g A C 1 A 2 κ 2 0m 2 NT 4
, (4.81)
2f a |k π |
an emission rate with a relatively soft temperature dependence.
For a numerical estimate Muto, Tatsumi, and Iwamoto (1994) found
that in cold neutron-star matter near the critical density (about 2.1
times nuclear) k π ≈ 410 MeV, the neutron and proton Fermi momenta
are 410 and 150 MeV, respectively (corresponding to Y p = 0.04), and
the coupling strength to the condensate is κ 0 ≈ 105 MeV. Therefore,
Q a = α a A 2 π2 κ 2 0 T 4
45 |k π |
= α a A 2 1.03×10 44 erg cm −3 s −1 T 4 9 , (4.82)
where α a ≡ [2m N (C 0 − ˜g A C 1 )/2f a ] 2 /4π and T 9 ≡ T/10 9 K.