Stars as Laboratories for Fundamental Physics - MPP Theory Group

266 Chapter 7
contribution is from 3 4 r l, the transition moments are suppressed by
(m l /m W ) 2 , an effect known as GIM cancellation after Glashow, Iliopoulos,
and Maiani (1970). Explicitly, the transition moments are
µ D }
ij/µ B
= 3G ( )
√
Fm e
ϵ D ij/µ (m mτ 2
B 2 (4π)
2
i ± m j )
m W
= 3.96×10 −23 m i ± m j
1 eV
∑
l=e,µ,τ
∑
l=e,µ,τ
U lj U ∗ li
U lj U ∗ li
( ml
m τ
) 2
( ml
m τ
) 2
. (7.15)
These small numbers imply that neutrino radiative decays are exceedingly
slow in the standard model.
Dirac neutrinos would have static or diagonal (i = j) magnetic
dipole moments while the electric dipole moments vanish according to
Eq. (7.13). Their presence would require CP-violating interactions.
Majorana neutrinos, of course, cannot have any diagonal electromagnetic
moments. For µ D ii the leading term of Eq. (7.14) in Eq. (7.13)
does not vanish because the unitarity of U implies that the sum equals
unity for i = j. Therefore,
µ D ii
µ B
= 6√ 2 G F m e
(4π) 2 m i = 3.20×10 −19 m eV , (7.16)
much larger than the transition moments because it is not GIM suppressed.
The two-photon decay rate ν i → ν j γγ is of higher order and thus
may be expected to be smaller by a factor of α/4π. However, it is
not GIM suppressed so that it is of interest for a certain range of neutrino
masses (Nieves 1983; Ghosh 1984). Essentially, the result involves
another factor α/4π relative to the one-photon rate, and f(r l )
in Eq. (7.13) is replaced by (m i /m l ) 2 .
As an example consider the different decay modes for ν 3 → ν 1 ,
assuming that m 3 ≫ m 1 and that the mixing angles are small so that
ν 3 ≈ ν τ and ν 1 ≈ ν e . Then one has explicitly
Φ(m 3 ), ν 3 → ν 1 e
⎧⎪ + e − ,
( )
1
τ ≈ |U e3| 2 G2 Fm 5 ⎨ 27 α mτ 4
3
,
3 (4π) × ν3 → ν 1 γ,
3 8 4π m W (7.17)
( )
1
⎪ α 2 ( ) m3 4
⎩ , ν3 → ν 1 γγ,
180 4π m e
where Φ(m h ) was given in Eq. (7.10) and shown in Fig. 7.6. The γγ
decay dominates in a small range of m 3 just below 2m e .