Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 125
Fig. 4.3. Correction of the degenerate bremsstrahlung rate for a nonzero
pion mass. For pseudoscalars, G(m π /p F ) is given explicitly in Eq. (4.12). It
is assumed that only one species of nucleons is present.
to g ψ N ψ N ϕ the results are different. The degenerate bremsstrahlung
energy-loss rate was worked out by Ishizuka and Yoshimura (1990),
Q scalar = α ′ α 2 π
44
15 3 ( T
m N
) 4
p 5 F G scalar (m π /p F ), (4.13)
with α ′ ≡ g 2 /4π. The function G scalar (u) is similar to Eq. (4.12); it is
plotted in Fig. 4.3 (dashed line).
4.2.6 Mixture of Protons and Neutrons
For a mixed medium of protons and neutrons one needs to consider the
individual Yukawa couplings g an = C n m N /f a and g ap = C p m N /f a as
well as the isoscalar and isovector combinations g 0,1 = 1 2 (g an ± g ap ) and
the “fine-structure constants” α j = g 2 j /4π with j = n, p, 0, 1. For equal
couplings α n = α p = α 0 while α 1 = 0.
In the nondegenerate limit, the main difference is that np scattering
benefits from the exchange of charged pions which couple more strongly
by a factor √ 2. Depending on the chemical composition of the medium,
the emission rate will be increased by up to a factor of 2. On the other
hand, some reduction factors have been ignored such as the (ˆk·ˆl) 2 term
and the pion mass. Therefore, ignoring this enhancement essentially
compensates for the previously introduced errors.
In the degenerate limit the changes are more dramatic. The role
of the Fermi momentum is played by p F → (3π 2 n B ) 1/3 . It sets the