Stars as Laboratories for Fundamental Physics - MPP Theory Group

128 Chapter 4
The matrix M µν is the nuclear part of the squared matrix element.
It is exactly the same for axion or neutrino interactions because in
Eq. (4.16) the global coupling constants have been explicitly pulled
out. Therefore, one may go one step further and perform the entire
nucleon phase-space integration for both cases directly on M µν ,
S µν ≡ 1
n B
∫
4∏
i=1
d 3 p i
2E i (2π) 3 f 1f 2 (1 − f 3 )(1 − f 4 )
× (2π) 4 δ 4 (P 1 + P 2 − P 3 − P 4 + K) M µν . (4.18)
Here, the P i are the nucleon four momenta, K = −K a for axion emission,
and K = −(K 1 + K 2 ) for neutrino pairs. Thus, K is the energymomentum
transfer from the radiation (axions or neutrino pairs) to
the nucleons. Because S µν knows about the radiation only through
the energy-momentum δ function it is only a function of K = (ω, k),
apart from the temperature and chemical potentials of the medium.
The slightly awkward definition of the sign of K follows the common
definition of the structure function where a positive energy transfer ω
refers to energy given to the medium.
The energy-loss rates by νν or axion emission are then the phasespace
integrals
Q νν =
Q a =
( ) C
N 2 ∫
A G
√ F
n B
2
(
CN
2f a
) 2
n B
∫
d 3 k 1
2ω 1 (2π) 3 d 3 k 2
2ω 2 (2π) 3 S µνN µν (ω 1 + ω 2 ),
d 3 k a
2ω a (2π) 3 S µνK µ a K ν a ω a . (4.19)
Here, it was assumed that both axions and neutrinos can escape freely
from the medium so that final-state Pauli blocking or Bose stimulation
factors can be ignored.
In the nonrelativistic limit the nucleon current in Eqs. (4.1) and
(4.15) reduces to χ † τ i χ where χ is a nucleon two-spinor and τ i (i =
1, 2, 3) are Pauli matrices representing the nucleon spin operator. Put
another way, in the nonrelativistic limit the axial-vector current represents
the nucleon spin density. Therefore, it has only spatial components
so that S µν → S ij (i, j = 1, 2, 3). In order to construct the
most general tensorial structure for S ij in an isotropic medium only δ ij
is available. Recall that in the nonrelativistic limit S µν does not know
about the momentum transfer k because of Eq. (4.5). There is then no