Stars as Laboratories for Fundamental Physics - MPP Theory Group

158 Chapter 4
The gluon in Fig. 4.13 couples to the quarks with the strong finestructure
constant α s and the gluon propagator involves an effective
mass for which Anand, Goyal, and Iha (1990) used m 2 g = (6α s /π) p 2 F
where p F is the Fermi momentum of the quark sea. The axion coupling
to quarks is the usual derivative form (C q /2f a ) ψ q γ µ γ 5 ψ q ∂ µ ϕ with
q = u, d, s. After a cumbersome but straightforward calculation Anand
et al. found for the energy-loss rate in the degenerate limit
Q a = 62π2 α 2 s
945
p F T 6
(2f a ) 2 × {
C
2
q (I 1 + 2I 2 ) for qq → qqa,
(C 2 q + C 2 q ′) I 1 for qq ′ → qq ′ a,
(4.84)
where the angular integrals are
I 1 ≡ 1 ∫ π ∫ 2π
dθ dϕ s5 θ (1 + c 4 ϕ)
2π 2 0 0 (s 2 θ s2 ϕ + κ2 ) , 2
I 2 ≡ 1 ∫ π
2π 2 0
∫ 2π
dθ dϕ
0
s 5 θ
(s 2 θ s2 ϕ + κ2 )(s 2 θ c2 ϕ + κ2 ) , (4.85)
where s θ ≡ sin(θ/2), s ϕ ≡ sin(ϕ/2), c ϕ ≡ cos(ϕ/2), and κ 2 ≡ m 2 g/2p 2 F =
3α s /2π. Numerically, I 1 = 12.3, 4.23, and 2.25 for α s = 0.2, 0.4, and
0.6, while I 2 = 2.34, 1.30, and 0.86, respectively.
The total emission rate involves three processes with equal quarks
(qq = uu, dd, ss), and three with different ones (qq ′ = ud, us, ds) so
that Q a is the prefactor of Eq. (4.84) times (C 2 u + C 2 d + C 2 s )(3I 1 + 2I 2 ),
to be compared with Eq. (4.10) for degenerate neutron matter. The
ratio between the rates is
Q qq
a
Q nn
a
= C2 u + C 2 d + C 2 s
C 2 n
p F,q
p F,n
αs
2
απ
2
2π (3I 1 + 2I 2 )
G(m π /p F,n ) , (4.86)
where the function G(u) was given in Eq. (4.12). The ratio of coupling
constants is model dependent (Sect. 14.3.3), the Fermi momenta are
approximately equal for equal densities.
For a typical value α s = 0.4 one has α 2 s/α 2 π = 0.6×10 −3 while the last
factor is about 140 with G = 0.7 (Fig. 4.3) so that the last two factors
together are about 0.08. Hence, the emission rate from quark matter
is much smaller than that from neutron matter because α π is so large
relative to α s . In the inevitable presence of protons in a neutron star,
the np process will be even more important than nn (Fig. 4.4), further
enhancing the emission rate of nuclear matter relative to quark matter.
For neutrino pairs the ratio of the emission rates is about the same
as for axions and so quark matter is much less effective at νν emission