The dark-photon luminosity is given by a volume integral of the differential power,
dP/dωdV . Before focusing on the dark photon, we describe the luminosity for an arbi-
trary differential power. Of relevance is the luminosity in new particles that cannot be
reprocessed efficiently as neutrino energy. We are thus interested in the luminosity from
the zone of neutrino diffusion to a zone in which neutrinos are not produced efficiently. In
other words, we want to find the energy that escapes from behind the “neutrinosphere”
(the isotherm where Standard-Model neutrinos approximately transition from diffusion to
free streaming) defined by some radius R
ν
. In general, this may be written
L(m
0
, , R
ν
, R
far
) =
R
ν
Z
r=0
∞
Z
ω=m
0
exp
−τ(m
0
, , ω, r, R
far
)
dP (m
0
, , ω, r)
dV dω
dωdV, (2.2)
where τ is the optical depth of a dark photon produced at radius r and the energy integral
starts at
1
m
0
. For a dark photon traveling radially outward, we have
τ(m
0
, , ω, r, R
far
) =
Z
R
far
r
Γ
0
abs
(m
0
, ω, , ˜r)d˜r, (2.3)
for an absorptive width Γ
0
abs
; see appendix B for the technical details of this calculation. As
discussed below, dP/dV dω is proportional to the absorptive width, Γ
0
abs
using the principle
of detailed balance, so we can simply calculate Γ
0
abs
for any given interaction.
Two radii appear in eq. (2.2). The radius R
ν
∼ O(40 km) of the neutrinosphere
is the radius outside of which most neutrinos free stream until arriving at Earth. For
definiteness, we define R
ν
to be the radius where the temperature of the star has fallen
to 3 MeV. This is roughly consistent with the condition for neutrino free streaming. The
radius R
far
∼ O(100 km–1000 km) is some “far radius” outside of which neutrinos no longer
are produced efficiently. The reason these radii enter our calculations in such a fundamental
way is that they provide a measure of the amount of energy that is diverted away from
neutrinos in a manner that can alter the observed neutrino signal. If we were interested
in the ability of terrestrial experiments to detect the new physics particles, R
far
would be
the distance to the Earth, which takes into account interactions with the progenitor star
material outside of R
ν
, interactions in the circumstellar medium, and the contribution from
the column density of the interstellar medium across the remaining distance. However, the
dark photons need not travel very far to have an effect on the evolution of the neutrino
flux from the supernova explosion. Instead, the “Raffelt criterion” in eq. (2.1) says that if
energy is taken away from the core and deposited in a different region of the star, the energy
is effectively lost because it becomes unavailable to neutrinos. This diversion depletes the
fuel of the nuclear “engine” that allows the cooling timescale to be t
cool
∼ O(10 sec).
There are a number of reasonable choices for R
far
; the only strict requirement is
R
far
> R
ν
. If R
far
is too close to R
ν
, we would erroneously conclude that arbitrarily
1
The gravitational potential well of the proto-neutron star can in principle trap dark photons with small
boosts [27]. However, we find that at most only about ? O(10%) of the power is gravitationally trapped
for masses of interest, which is the same order of magnitude as several other effects we have ignored, so we
will henceforth omit this effect.
– 6 –