Stars as Laboratories for Fundamental Physics - MPP Theory Group

Radiative Particle Decays 481
In order to estimate the expected photon flux from Eq. (12.22) one
needs to know the distribution of photon energies and emission angles
in the frame of the parent neutrino. In the absence of a detailed calculation
I follow Oberauer et al. (1993) and assume approximate isotropy
for the photon emission. The soft part of the spectrum dN γ /dω from
a bremsstrahlung process is given by the rate of the primary process
times (α/π) ω −1 (Jackson 1975). Extending this behavior up to photon
energies of 1 2 m ν I use
1
f(ω, x) = α/π 1
τ γ τ e + e − 2ω Θ( 1m 2 ν − ω) , (12.34)
an expression which does not depend on x because of the assumed
isotropy. (α = 1/137 is the fine-structure constant, not the previous
anisotropy parameter.)
After dropping the exponential in Eq. (12.22) because the neutrinos
are long-lived, one integrates over a Boltzmann source spectrum
for nonrelativistic neutrinos, integrates over the GRS energy channels,
and compares the expected fluence with the measured upper limits of
Tab. 12.1. The resulting bound on τ e + e− corresponds with Eq. (7.9)
directly to a limit on |U e3 | 2 . In Fig. 12.18 I show these bounds (transformed
into bounds on the mixing angle) as a function of the assumed
neutrino mass for T ν = 4 and 6 MeV. There remains a strong limit
even for masses far exceeding the temperature because the exponential
Fig. 12.18. SN 1987A limits on sin 2 2θ e3 = 4|U e3 | 2 with ν 3 ≈ ν τ for
T ν = 6 MeV (solid line) and 4 MeV (dashed line). Also shown are the corresponding
laboratory and solar limits from Fig. 12.3.