Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 127
Fig. 4.5. Bremsstrahlung emission of neutrino pairs in nucleon-nucleon collisions.
There is a total of eight amplitudes, four with the neutrinos attached
to each nucleon line, and an exchange graph each with N 3 ↔ N 4 .
of the medium, quantities which are of utmost importance to understand
the general properties of the emission, absorption, and scattering
rates independently of phase-space details of the neutrinos or axions.
Contrary to the νν bremsstrahlung emission in electron-nucleus collisions
(Sect. 3.5.3), nonrelativistically only the nucleon axial-vector
coupling contributes in nucleon-nucleon collision (Friman and Maxwell
1979). This difference originates from the interaction potential of the
colliding particles which involves a spin-dependent force between nucleons
so that the spin fluctuations caused by collisions are more dramatic
than those of the velocity—see Sect. 4.6.5 below. Hence, the part of the
interaction Hamiltonian relevant for neutrino pair bremsstrahlung is
H int = CN A G
√ F
ψ N γ µ γ 5 ψ N ψ ν γ µ (1 − γ 5 )ψ ν , (4.15)
2
where C p A ≈ −CA n ≈ 1 ; see Appendix B for a discussion of the appropriate
values in a nuclear medium.
2
The interaction Hamiltonian Eq. (4.15) has the same structure as
that for axions Eq. (4.1). The squared matrix elements are then of the
general form
⎧
∑ ⎨ (C
|M| 2 A N G F / √ 2) 2 M µν N µν for neutrinos,
=
(4.16)
⎩ (C N /2f a ) 2 M µν K a µ Ka ν for axions.
spins
Here, K a is the axion four-momentum while
N µν = 8 ( K µ 1 K ν 2 + K µ 2 K ν 1 − K 1 · K 2 g µν − iϵ αβµν K 1α K 2β
)
(4.17)
with the neutrino and antineutrino four momenta K 1 and K 2 (Gaemers,
Gandhi, and Lattimer 1989). N µν and K µ a K ν a are the squared matrix
elements of the neutrino and axion current, respectively.