Stars as Laboratories for Fundamental Physics - MPP Theory Group

162 Chapter 4
The dispersion relation of neutrinos in a SN differs markedly from
the vacuum form; in the core the “effective m ν ” is several 10 keV. However,
m ν in Eq. (4.91) is the vacuum mass which couples left-handed
to right-handed states and thus leads to spin flip while the mediuminduced
“mass” only affects the dispersion relation of left-handed states.
This view is supported by a detailed study of Pantaleone (1991). Of
course, for nonrelativistic neutrinos the situation is more complicated
because an approximate identification of helicity with chirality is not
possible.
Next consider a specific process which involves a left- and a righthanded
neutrino such as “spin-flip scattering” ν L (K L ) → ν R (K R ).
Then, one needs to construct N µν from K 1 = (ω L , k L ) and K 2 =
(m ν /2ω R ) 2 (ω R , −k R ). The contraction with S µν leads to (Raffelt and
Seckel 1995)
˜W (K L , K R ) = G2 Fn B
4
( mν
2ω R
) 2 [(1
− cos θ)S1 + (3 + cos θ)S 2
+ 4ω 2 R(1 − cos θ)S 3 − 4ω R (1 − cos θ)S 4 + 2(ω R − ω L )(1 + cos θ)S 5
]
.
(4.92)
Following Gaemers, Gandhi, and Lattimer (1989) it must be emphasized
that this expression differs in more than the factor (m ν /2ω R ) 2
from the nonflip case Eq. (4.90). This difference is due to the changed
angular momentum budget of reactions with spin-flipped neutrinos.
This angular-momentum difference between spin-flip and no-flip
processes is nicely illustrated by virtue of the pion decay process π ◦ →
νν. If both final-state neutrinos are left-handed, i.e. if ν has negative
and ν positive helicity, this decay is forbidden by angular momentum
conservation. This is seen most easily if one recalls that it is the
pion current ∂ µ π ◦ that interacts with the left-handed neutrino current.
The squared matrix element of the pion current is thus proportional
to K µ π ◦Kν π◦. For a pion decaying at rest only the 00-component contributes.
Contraction with N µν leads to identically zero if one recalls
that for the final-state neutrinos ω 1 = ω 2 and k 1 = −k 2 . For the spinflip
process one must use the reversed momentum instead so that in
N µν one must use ω 1 = ω 2 and k 1 = k 2 , leading to a nonvanishing
contribution. The decay of thermal pions is an important process for
populating the right-handed states of massive Dirac neutrinos in the
early universe (Lam and Ng 1991). Contrary to a discussion by Natale
(1991), however, pion decays do not seem to provide a particularly
strong contribution in SN cores (Raffelt and Seckel 1991).