Stars as Laboratories for Fundamental Physics - MPP Theory Group

Nonstandard Neutrinos 265
Fig. 7.7. Feynman graphs for neutrino radiative decays. There are other
similar graphs with the photon lines attached to the intermediate W boson.
In general the matrix element can be thought of as arising from an
effective interaction Lagrangian of the form
L int = 1 2 ψ iσ αβ (µ ij + ϵ ij γ 5 )ψ j F αβ + h.c., (7.11)
where F αβ is the electromagnetic field tensor, ψ i and ψ j are the neutrino
fields, and µ ij and ϵ ij are magnetic and electric transition moments
which are usually expressed in units of Bohr magnetons µ B = e/2m e .
The decay rate is
1
= |µ ij| 2 + |ϵ ij | 2
τ γ 8π
( m
2
i − m 2 j
m i
) 3
= 5.308 s −1 (
µeff
µ B
) 2
δ 3 m m 3 eV,
(7.12)
where µ 2 eff ≡ |µ ij | 2 + |ϵ ij | 2 , m eV ≡ m i /eV, and δ m ≡ (m 2 i − m 2 j)/m 2 i .
An explicit evaluation of the one-photon amplitude of Fig. 7.7 yields
for Dirac neutrinos (Pal and Wolfenstein 1982)
µ D }
ij
ϵ D ij
= e√ 2 G F
(4π) 2 (m i ± m j )
∑
l=e,µ,τ
U lj U ∗ li f(r l ). (7.13)
For Majorana neutrinos one has instead µ M ij = 2µ D ij and ϵ M ij = 0 or
µ M ij = 0 and ϵ M ij = 2ϵ D ij, depending on the relative CP phase of ν i and ν j .
In Eq. (7.13) r l ≡ (m l /m W ) 2 where the charged-lepton masses are
m e = 0.511 MeV, m µ = 105.7 MeV, and m τ = 1.784 GeV while the
W ± gauge boson mass is m W = 80.2 GeV. Thus for all charged leptons
r l ≪ 1; in this limit
f(r l ) → − 3 2 + 3 4 r l. (7.14)
If one inserts the leading term − 3 into the sum in Eq. (7.13) one finds
2
that its contribution vanishes because the unitarity of U implies that its
rows or columns represent orthogonal vectors. Because the first nonzero