Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 163
Right-handed neutrinos can be produced by the interaction with
all medium constituents. For a SN core, the production rate from the
interaction with charged leptons such as e + e − → ν L ν R or ν L e − → e − ν R
was explicitly studied by Pérez and Gandhi (1990) and by Lam and Ng
(1992). Still, in a SN core the dominant contribution to the interaction
between neutrinos and the medium is due to the nucleons.
Therefore, one may simplify the expression Eq. (4.92) by applying
the same approximations that were used earlier in Sect. 4.6 in the context
of purely left-handed neutrinos. Notably, one may use the nonrelativistic
and long-wavelength limits which reveal, again, that only the
terms proportional to S 1 and S 2 contribute, and that these structure
functions can be taken to be functions of the energy transfer ω alone. 25
Moreover, the neutrino phase-space integration will always average the
cos θ term to zero. Then, the spin-flip reaction rate is indeed simply
the nonflip rate times the “spin-flip factor” (m µ /2E ν ) 2 .
Right-handed neutrinos can be produced by spin-flip scattering of
left-handed ones ν L → ν R , or by the emission of pairs ν L ν R or ν R ν L .
In a SN core, left-handed neutrinos are trapped while right-handed
ones can freely escape whence the quantity of interest is the energyloss
rate of the medium in terms of right-handed states. The total
energy-loss rate in ν R is then Q R = Q scat + Q pair where “scat” refers to
spin-flip scattering and “pair” to the pair-emission process. The two
contributions are
∫
Q scat =
∫
Q pair =
d 3 k L
(2π) 3 d 3 k R
(2π) 3 ˜W KL ,K R
f kL ω R ,
d 3 k L
(2π) 3 d 3 k R
(2π) 3 ˜W −KL ,K R
(1 − f kL
)ω R . (4.93)
An analogous expression pertains to the emission of ν R ; if the trapped
left-handed neutrinos are nondegenerate this contribution has the same
magnitude as that for the emission of ν R .
Next, consider the limit where the transition probability ˜W can be
represented in terms of a single structure function 26 S(ω) = S 1 (ω) +
3S 2 (ω). Further, the possibility of neutrino degeneracy is ignored which
allows one to approximate the left-handed neutrino Fermi-Dirac distribution
by a Maxwell-Boltzmann one, (e ω/T + 1) −1 → e −ω/T . Then the
25 In the notation of Sect. 4.6 and for a single species of nucleons one has in this
limit S 1 = CV 2 S ρ and S 2 = CA 2 S σ.
26 In the notation of Sect. 4.6 this is S = CV 2 S ρ + 3CA 2 S σ in a medium consisting
of a single species of nucleons.