Stars as Laboratories for Fundamental Physics - MPP Theory Group

160 Chapter 4
4.10 Emission of Right-Handed Dirac Neutrinos
So far in this chapter neutrinos were assumed to be massless particles
which interact only by the standard left-handed weak current. It
is possible, however, that neutrinos have a Dirac mass in which case
any reaction with a final-state (anti)neutrino produces both positive
and negative helicity states. Typically, the “wrong-helicity” states will
emerge in a fraction (m ν /2E ν ) 2 of all cases because of the mismatch
between chirality (eigenstates of γ 5 ) and helicity. For E ν ≫ m ν the
wrong-helicity states correspond approximately to right-handed chirality
states and so their interaction-rate with the ambient medium is
weaker by an approximate factor (m ν /2E ν ) 2 . This implies that for a
sufficiently small mass they would not be trapped in a SN core and
thus carry away energy in an “invisible” channel, allowing one to set
constraints on a Dirac neutrino mass from the observed neutrino signal
of SN 1987A (Sect. 13.8.1). Here, the relationship between the production
rate of left- and right-handed neutrinos is explored in some
detail because the simple scaling with (m ν /2E ν ) 2 is not correct in
all cases.
When a neutrino interacts with a medium the transition probability
from a state with four-momentum K 1 = (ω 1 , k 1 ) to one with K 2 =
(ω 2 , k 2 ) is written as W (K 1 , K 2 ). The function W is defined for both
positive and negative energies. The emission probability for a pair
ν(K 1 )ν(K 2 ) is then W (−K 1 , K 2 ), the absorption probability for a pair
is W (K 1 , −K 2 ). The collisional rate of change of the occupation number
f k1 of a neutrino field mode k 1 is then given by
df k1
dt
=
∣ coll
∫ d 3 k 2
(2π) 3 [
WK2 ,K 1
f k2 (1 − f k1 ) − W K1 ,K 2
f k1 (1 − f k2 )
+ W −K2 ,K 1
(1 − f k1 )(1 − f k2
) − W K1 ,−K 2
f k1 f k2
]
, (4.87)
where the variables of W were written as subscripts. The first term
corresponds to neutrino scatterings into the mode k 1 from all other
modes, the second term is scattering out of mode k 1 into all other
modes, the third term is pair production with a final-state neutrino k 1 ,
and the fourth term is pair absorption of a neutrino of momentum k 1
and an antineutrino of any momentum. f k is the occupation number
for the antineutrino mode k; (1−f k ) or (1−f k ) represent Pauli blocking
factors. This collision integral only includes (effective) neutral-current
processes while charged-current reactions were ignored.