Stars as Laboratories for Fundamental Physics - MPP Theory Group

584 Appendix B
The effective charged-current interaction between charged leptons
and their own neutrinos, e.g. between e − and ν e , is written in the same
form with C V = C A = 1. By virtue of a Fierz transformation it is
brought into the form of a neutral current (see below).
Neutral-current interactions between a neutrino ν and a fermion f
are written in the form
H int = G F
√ ψ f γ µ (C V − C A γ 5 )ψ f ψ ν γ µ (1 − γ 5 )ψ ν .
2
(B.4)
If f is the charged lepton corresponding to ν, there is a contribution
with C V = C A = 1 from a Fierz-transformed charged current. The
compound effective C V ’s and C A ’s for various combinations of f and
ν are given in Tab. B.1. (Note that the |C V,A | for neutral currents are
typically 1 , a factor which is sometimes pulled out front so that the
2
global coefficient is G F /2 √ 2 while the couplings are then twice those of
Tab. B.1.) For neutrinos interacting with neutrinos of the same flavor a
factor 2 for an exchange amplitude for identical fermions was included.
The C A ’s for nucleons were thought to be given by isospin invariance
to be ±1.26/2. However, because of the strange-quark contribution to
the nucleon spin there is an isoscalar piece as well giving rise to the
values shown in Tab. B.1—for a discussion and references to the original
literature see Raffelt and Seckel (1995). Moreover, in a nuclear medium
a certain suppression is expected to occur. In analogy to the chargedcurrent
couplings Raffelt and Seckel (1995) suggested the values C p A ≈
1.09/2 and CA n ≈ −0.91/2.
Table B.1. Neutral-current couplings for the effective Hamiltonian Eq. (B.4)
in vacuum.
Fermion f Neutrino C V C A CV 2 CA
2
Electron ν e + 1 + 2 2 sin2 Θ W + 1 2
0.9312 0.25
ν µ,τ − 1 2 + 2 sin2 Θ W − 1 2
0.0012 0.25
Proton ν e,µ,τ + 1 2 − 2 sin2 Θ W +1.37/2 0.0012 0.47
Neutron ν e,µ,τ − 1 2
−1.15/2 0.25 0.33
Neutrino (ν a ) ν a +1 +1 1 1
ν b≠a + 1 2
+ 1 2
0.25 0.25