Stars as Laboratories for Fundamental Physics - MPP Theory Group

134 Chapter 4
Fig. 4.6. Inelastic neutrino-nucleon scattering. 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 .
Fig. 4.6. In this process, neutrinos can give or take energy even though
recoil effects for heavy nucleons are small in the elastic scattering process
νN → Nν.
This reaction is identical with pair absorption with the antineutrino
line crossed into the final state. The corresponding mfp is given by the
same expression as for pair absorption, except that the index 2 now
refers to the final-state ν, the initial-state occupation number f 2 is to
be replaced with a final-state Pauli blocking factor (1 − f 2 ), and the
energy transfer is ω = ω 2 − ω 1 ,
λ −1 = 3C2 AG 2 Fn B
2π 2
∫ ∞
0
dω 2 ω 2 2 (1 − f 2 ) S σ (ω 1 − ω 2 ). (4.35)
However, because the energy transfer can be zero, and because S σ (ω) ∝
ω −2 , this expression diverges. The ω −2 behavior seemed harmless before
because it was moderated by powers of ω from the phase space of axions
or neutrinos.
The occurrence of this divergence could have been predicted without
a calculation by inspecting Fig. 4.6. If one cuts the intermediate-state
nucleon line, this graph falls into two sub-processes (nucleon-nucleon
scattering and nucleon-neutrino scattering) which are each permitted
by energy-momentum conservation, allowing the intermediate nucleon
in the compound process to go “on-shell.” Therefore, the pole of the
propagator which corresponds to real particles causes a divergence of
the cross section. Physically, the divergence reflects a long-range interaction
which occurs because the intermediate nucleon can travel arbitrarily
far when it is on its mass shell.
Still, the inelastic scattering process is an inevitable physical possibility.
For nonzero energy transfers its differential rate is given by the
unintegrated version of Eq. (4.35). For a vanishing energy transfer it is
elastic and then its rate should be given by Eq. (4.30).