Stars as Laboratories for Fundamental Physics - MPP Theory Group

242 Chapter 6
Fig. 6.15. Amplitudes contributing to forward scattering: (a) Neutralcurrent
scattering for any ν or ν on any f or f as target. (b) ν e -ν e scattering.
(c) ν e -e charged-current scattering. (d) ν e -e + charged-current scattering.
(e) Effective four-fermion vertex in the low-energy limit.
Sect. 6.7.2.) This amounts to reducing the weak interaction to the
low-energy Fermi effective Hamiltonian represented by graph (e) in
Fig. 6.15. For the neutral-current processes (a) and (b) it is explicitly
H int = G F
√ ψ f γ µ (C V − C A γ 5 )ψ f ψ νl
γ µ (1 − γ 5 )ψ νl , (6.105)
2
where ψ νl is a neutrino field (l = e, µ, τ) while ψ f represents fermions
of the medium (f = e, p, n, or even neutrinos ν l ′). Here, G F =
1.166×10 −5 GeV −2 is the Fermi constant. The relevant values of the
vector and axial-vector weak charges C V and C A are given in Appendix
B. In the low-energy limit the charged-current reactions (c)
and (d) can also be represented as an effective neutral-current interaction
of the same form with C V = C A = 1.
It is now straightforward to work out the forward scattering amplitudes.
The axial-vector piece represents the spin of f and so it averages
to zero if the medium is unpolarized. Then one finds for the refractive
index of a neutrino (upper sign) or antineutrino (lower sign) with energy
ω
n refr − 1 = ∓ C ′ V G F
n f − n f
ω √ 2 , (6.106)
where n f and n f
are the number densities of fermions f and antifermions
f, respectively. The effective weak coupling constants C V
′ are
identical with the C V given in Appendix B except for neutrinos as