# Low-Energy Supernovae Bounds on Sterile Neutrinos

Garv Chauhan  0000-0002-8129-8034 Center for Neutrino Physics, Department of
Physics, Virginia Tech, Blacksburg, VA 24061, USA  Shunsaku Horiuchi
0000-0001-6142-6556 Center for Neutrino Physics, Department of Physics,
Virginia Tech, Blacksburg, VA 24061, USA  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo, Chiba 277-8583, Japan
Patrick Huber  Center for Neutrino Physics, Department of Physics, Virginia
Tech, Blacksburg, VA 24061, USA  Ian M. Shoemaker  0000-0001-5434-3744 Center
for Neutrino Physics, Department of Physics, Virginia Tech, Blacksburg, VA
24061, USA

###### Abstract

Sterile neutrinos can be produced through mixing with active neutrinos in the
hot and dense core of a core-collapse supernova (SN). The standard bounds on
the active-sterile mixing ($\sin^{2}\theta$) from SN arise from SN1987A
energy-loss, requiring $E_{\text{loss}}<10^{52}\leavevmode\nobreak\ {\rm
erg}$. In this letter, we discuss a novel bound on sterile neutrino parameter
space arising from the energy deposition through its decays inside the SN
envelope. Using the observed underluminous SN IIP population, this energy
deposition is constrained to be below $\sim 10^{50}\leavevmode\nobreak\ {\rm
erg}$. Focusing on sterile neutrino mixing only with the tau neutrino, for
heavy sterile masses $m_{s}$ in the range $100$-$500$ MeV, we find stringent
constraints on $\sin^{2}\theta_{\tau}$ reaching two orders of magnitude lower
than those from the SN1987A energy-loss argument. Similar bounds will also be
applicable to sterile mixing only with muons ($\sin^{2}\theta_{\mu}$).

Introduction. Although a great deal has been determined about neutrinos, the
origin of neutrino mass remains unknown. A particularly simple possibility is
that the Standard Model (SM) is augmented with at least two right-handed
neutrinos which are singlets under SM interactions. As a result, such states
can generate Dirac masses through nonzero Yukawa couplings to the SM lepton
doublet and the Higgs, as well as Majorana masses at an unknown scale. After
electroweak symmetry breaking, the left- and right-handed neutrinos mass mix,
which effectively endows the “sterile neutrinos” with a suppressed coupling to
the weak force, where the suppression is controlled by the mixing angles. In
addition to accounting for neutrino masses, sterile neutrinos may play a role
in unravelling other mysteries as well, including acting as a dark matter
candidate [1, 2], and providing an origin for the observed baryon asymmetry
[3].

In the face of significant theoretical uncertainty regarding the masses of
sterile neutrinos, a wide range of varying experimental and theoretical probes
have been brought to bear on the existence of sterile neutrinos over many
orders of magnitude in mass (e.g., [4, 5, 6, 7]). Such searches include
colliders, beta decays, accelerators, as well as astrophysical and
cosmological signatures.

In the present paper, we investigate the impact of 10-500 MeV scale sterile
neutrinos on supernovae (SNe). Our work differs from previous studies on
sterile neutrinos produced in SNe in several ways. In Refs. [8, 9] it was
found that $\sim$100 MeV scale sterile neutrinos can transport large
quantities of energy and augment the explosion. Moreover strong limits on
sterile neutrinos have been derived on the argument that excessive energy loss
would catastrophically shorten the neutrino burst, in contradiction with
observations; this has been applied to eV [10, 11], keV [12, 13, 14, 15, 16],
and $\sim 100$ MeV scale sterile neutrinos [17, 18, 8, 9, 19].

Here however we derive strong constraints on sterile neutrinos based on the
argument that they do not deposit more energy in the stellar envelope than
what is observed in underluminous Type IIP SNe. Thus the lowest energy SNe yet
observed will provide the strongest constraints. Similar arguments have been
studied in axions [20].

Figure 1: Bounds on sterile-active mixing $\sin^{2}\theta_{\tau}$ as a
function of sterile mass $m_{s}$. The dashed black curve shows the low-energy
SN bound assuming full energy deposition from decaying sterile inside the SN
envelope ($18.8M_{\odot}$), while the solid blue curves denotes the bound
arising from energy deposition through visible decay channels only (shown for
an Fe core $18.8M_{\odot}$ progenitor and a ONeMg core $8.8M_{\odot}$
progenitor, as labeled). The red-shaded region is the bound from energy loss
in SN1987A [19]. Other displayed constraints include BBN bounds assuming
standard cosmology [21, 22, 23], T2K [24], and BEBC bounds [25].

As such, we extend the SM to include a heavy sterile neutrino $\nu_{s}$ which
interacts with the SM exclusively via the type I seesaw Lagrangian [26, 27,
28],

$\mathcal{L}\supset
y_{\nu}\bar{L}\tilde{H}\nu_{s}+\frac{m_{s}}{2}\bar{\nu}^{c}_{s}\nu_{s},$ (1)

where $H$ and $L$ are the $SU(2)_{L}$ Higgs and lepton doublets respectively,
$y_{\nu}$ is the neutrino Dirac Yukawa matrix, and $m_{s}$ is the Majorana
mass term.

At the low-scales relevant for our analysis, the main effect of Eq. (1) is
encoded in the active-sterile mixing angle. The parameter space SNe are most
sensitive to, in the case of sterile-active mixing with electron
($\sin^{2}\theta_{e}$) is already strongly constrained from the terrestrial
probes. In absence of muons in the SNe, the cases for sterile-active mixing
with muon or tau neutrino are equivalent. In the former case, presence of
muons can lead to extra channels for sterile neutrino production through
charged-current interactions. Therefore in this work, we assume that the
sterile neutrino mixes exclusively only with $\nu_{\tau}$. In terms of mass
eigenstates $\nu_{1}$ and $\nu_{2}$,

$\displaystyle\nu_{\tau}$
$\displaystyle=\cos{\theta_{\tau}}\,\nu_{1}+\sin{\theta_{\tau}}\,\nu_{2}$
$\displaystyle\nu_{s}$
$\displaystyle=-\sin{\theta_{\tau}}\,\nu_{1}+\cos{\theta_{\tau}}\,\nu_{2}$ (2)

where $\sin{\theta_{\tau}}$ denotes the mixing angle between the sterile and
the tau neutrino.

Low-Energy Supernovae. While SN1987A was unique in its promity, SNe are not
rare and can be used to study sterile neutrinos. We focus on a sub-class of
core-collapse SNe with low-explosion energies, sometimes called underluminous
Type IIP SNe, where the Type II represents the presence of hydrogen in the
spectra and the P represents the SN’s light-curve displaying a flat, or
plateau, shape in time. The luminosity and duration of the plateau reflects
the explosion energy, ejecta mass, nickel 56Ni mass and progenitor radius.
Therefore, the explosion energy can be inferred given the spectrum and the
light curves. For example Refs. [29, 30, 31, 32] used fitting formulae,
simulations, and statistical inference along with quantifying the
uncertainties, to infer the most likely explosion energies for a collection of
Type IIP SNe. The inferred explosion energy $E_{obs}$ ranges from $7.4\times
10^{49}$ erg to $4\times 10^{51}$ erg. While the upper end of the
reconstructed energies are larger than typical predictions of numerical
simulations, the lower end is approximately consistent with various simulated
SNe. Here, we are interested in sterile neutrinos produced in the core which
can decay and deposit energies of a similar magnitude, and hence can be
constrained from the observations of low-energy SNe by requiring
$E_{\text{dep}}<10^{50}$ erg. Since numerical simulations are typically under-
powered compared to observations, we expect our constraints to be
conservative, unless some surprising systematic over-evaluation is discovered
in simulations.

We will examine two massive stars with initial masses $8.8M_{\odot}$ and
$18.8M_{\odot}$ for obtaining bounds on the sterile neutrino parameter space.
There are several reasons for these choices. First, the SN explosion energy is
not a simple monotonic function of the initial mass of the progenitor.
Instead, the explosion energy depends on the structure of the inner few solar
masses of the progenitor’s core, which in turn depends in a complex way to the
progenitor mass, metallicity or even final hydrogen mass [33, 29]. Second,
light progenitors have a systematically different core structure to those of
Fe cored stars [34] which collapse by electron capture on its ONeMg core,
which typically explode more readily [35, 36]. Our adopted $8.8M_{\odot}$
progenitor aims to model this. Similarly, we adopt the $18.8M_{\odot}$ as the
typical massive star which evolves into a supergiant where its Fe core
collapses causing a Type IIP SN. Finally, estimates of the progenitor masses
of underluminous Type IIP supernovae cover a range: for example, SN1997D with
estimated energy $1.0\times 10^{50}$ erg has an estimated initial mass $10\pm
2M_{\odot}$ [37], while SN2003Z has estimated $1.6\times 10^{50}$ erg and
$14.15\pm 0.95M_{\odot}$ [38] and SN2008kb $1.8\times 10^{50}$ erg and
$12.15\pm 0.75M_{\odot}$ [38]. We thus consider the reality of the progenitors
of underluminous Type IIP to lie somewhere between our adopted $8.8M_{\odot}$
and $18.8M_{\odot}$ progenitors.

Energy Deposition. Since the explosion energies of SN IIP are inferred from
the light curves and dependent on the ejecta mass and the amount of
$\prescript{56}{}{\text{Ni}}$ synthesized in the outer envelope, the limit on
the energy deposition by any exotic species is constrained to the mantle
region of the SN; in other words, we consider species which escape beyond the
photosphere will simply present an energy sink for the explosion. The total
energy deposited by sterile neutrino produced in the SN core, decaying outside
the core but inside the SN envelope region ($R_{core}<r<R_{env}$) is given by

$\displaystyle E_{\text{dep}}=\eta_{\text{lapse}}^{2}\int
dt\int_{0}^{R_{\text{core}}}dr\int_{m_{s}}^{\infty}dE_{s}\frac{dL_{s}(r,E_{s},t)}{dr\,dE_{s}}\Theta\left(E_{s}-\frac{m_{s}}{\eta_{\text{lapse}}}\right)\times\left\\{\text{exp}\left[-\frac{(R_{core}-r)}{L_{decay}}\right]-\text{exp}\left[-\frac{(R_{env}-r)}{L_{decay}}\right]\right\\},$
(3)

where $\eta_{\text{lapse}}$ is the gravitational redshift factor, $E_{s}$ is
the sterile neutrino energy, $\frac{dL_{s}(r,E_{s},t)}{dr\,dE_{s}}$ is
gradient of the differential sterile neutrino luminosity, $\Theta(x)$ is the
Heaviside theta function and $L_{decay}$ is the decay length. Note that at
large mixing angles, the decay length becomes comparable to the PNS radius and
our treatment should not be taken as rigorous. We hope to return to this
trapping regime in future work with greater precision.

Sterile neutrinos produced in the SN core will also have to overcome the
strong gravitational attraction arising from such high matter densities to
avoid trapping. If sterile neutrino energy is sufficiently small,
$E_{s}<m_{s}/\eta_{\text{lapse}}$, it will be gravitationally trapped inside
the SN core. The lapse factor $\eta_{\text{lapse}}$ is the conversion factor
relating the energy measured locally in the SN frame to the energy measured at
the same point by an observer at infinity. For example in the weak-field limit
of the Schwarzschild metric, the gravitational lapse factor is given by
$\eta_{\text{lapse}}\simeq m_{s}(1+G\,M(r)/r)$. We also need to correct for
the difference between the local time and the observer time in the energy
emission rate. Both of these effects taken together lead to the inclusion of
$\eta_{\text{lapse}}^{2}$ in the expression for $E_{\text{dep}}$. In our case,
we find that our results are sensitive to the gravitational trapping (inside
the core) only for very heavy steriles ($>300$ MeV). Therefore, the results in
absence of the gravitational trapping factor will be almost similar.

Sterile Neutrino Production. In a SN core, sterile neutrinos can be produced
by $e^{+}$-$e^{-}$ or neutrino pair annihilation and the inelastic scattering
of (anti-)neutrinos. Since $n,p,e^{+},e^{-},\nu_{e},\bar{\nu}_{e}$ are
degenerate in the hot proto-neutron star core, Pauli-blocking will render
pair-annihilation and inelastic scattering on the non-degenerate neutrino
species ($\nu_{\mu},\nu_{\tau}$) the dominant processes for $\nu_{s}$
production [8, 9, 19, 39]. The amplitude for these relevant processes is given
in Table. 1, in terms of Mandelstam variables. Note that we have explicitly
listed the charge-conjugated processes in order to avoid any confusion.

Process |  $S|M|^{2}$/($8G_{F}^{2}\sin^{2}\theta_{\tau}$)
---|---
$\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+\bar{\nu}_{\tau}$ | $4u(u-m^{2}_{s})$
$\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+{\nu}_{\tau}$ | $4u(u-m^{2}_{s})$
$\nu_{\mu}+\bar{\nu}_{\mu}\rightarrow\nu_{s}+\bar{\nu}_{\tau}$ | $u(u-m^{2}_{s})$
$\nu_{\mu}+\bar{\nu}_{\mu}\rightarrow\nu_{s}+{\nu}_{\tau}$ | $u(u-m^{2}_{s})$
$\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+{\nu}_{\tau}$ | $2s(s-m^{2}_{s})$
$\bar{\nu}_{\tau}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+\bar{\nu}_{\tau}$ | $2s(s-m^{2}_{s})$
$\nu_{\mu}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+{\nu}_{\mu}$ | $s(s-m^{2}_{s})$
$\bar{\nu}_{\mu}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+\bar{\nu}_{\mu}$ | $s(s-m^{2}_{s})$
$\nu_{\tau}+\bar{\nu}_{\mu}\rightarrow\nu_{s}+\bar{\nu}_{\mu}$ | $u(u-m^{2}_{s})$
$\nu_{\mu}+\bar{\nu}_{\tau}\rightarrow\nu_{s}+{\nu}_{\mu}$ | $u(u-m^{2}_{s})$
Table 1: Matrix element squared $S|M|^{2}$ for the dominant processes involved
in sterile neutrino production in terms of Mandelstam variables, in units of
$8G_{F}^{2}\sin^{2}\theta_{\tau}$.

Boltzmann Transport. The evolution of sterile neutrino abundances is governed
by the Boltzmann transport equation. Since solving the exact transport
equation for sterile neutrino is not possible, we simplify the task at hand by
assuming the medium is homogeneous and isotropic [40]. This implies that the
change in phase-space density will only be affected by the scatterings/pair-
annihilation processes in the SN core. In this case, the simplified kinetic
equation for sterile neutrino production is

$\frac{\partial f_{s}}{\partial t}=\mathcal{C}_{coll}(f_{s}),$ (4)

where $f_{s}$ is the sterile neutrino phase-space density distribution and
$\mathcal{C}_{coll}$ is the sum of all possible collisional interactions. In
our case, the collisional term for $2\rightarrow 2$ particle interactions can
be written as

$\displaystyle\mathcal{C}_{coll}=$ $\displaystyle\frac{1}{2E_{s}}\int
d^{3}\tilde{p_{2}}d^{3}\tilde{p_{3}}d^{3}\tilde{p_{4}}\,\Lambda(f_{s},f_{2},f_{3},f_{4})\times$
$\displaystyle S|M|^{2}_{12\rightarrow
34}\,\delta^{4}(p_{s}+p_{2}-p_{3}-p_{4})(2\pi^{4}),$ (5)

where $d^{3}\tilde{p_{i}}=d^{3}{p_{i}}/((2\pi^{3})\,2E_{i})$,
$\Lambda(f_{s},f_{2},f_{3},f_{4})=(1-f_{s})(1-f_{2})f_{3}f_{4}-f_{s}f_{2}(1-f_{3})(1-f_{4})$
is the phase-space factor including the Pauli blocking of the final states,
$S$ is the symmetry factor, $|M|^{2}$ is the interaction matrix element
element squared, $E_{i}$ and $p_{i}$ are energy and momentum of the $i$-th
particle with subscript label $s$ for sterile neutrino. For the range of
interest for $\sin^{2}\theta_{\tau}$, the sterile neutrino produced will not
be trapped in the SN, hence we can safely assume $f_{s}=0$. After numerically
solving for $f_{s}$, we can calculate the sterile neutrino luminosity
$\frac{dL_{s}}{dE_{s}}$ as [41, 19]

$\frac{dL_{s}}{dE_{s}}=\frac{2E_{s}}{\pi}\int
dr\,r^{2}\frac{df_{s}}{dt}E_{s}\,p_{s}.$ (6)

As an example, we show this differential and integrated sterile neutrino
luminosity in Figs. 2 and 3, respectively for reference sterile neutrino mass
$m_{s}=200$ MeV.

Figure 2: Differential sterile neutrino luminosity for sterile neutrino mass
$m_{s}=200$ MeV and $\sin^{2}{\theta_{\tau}}=10^{-7}$ at different post-bounce
times, see e.g., Eq. (6). Figure 3: Integrated sterile neutrino luminosity
plotted as a function of post-bounce time for sterile neutrino mass
$m_{s}=200$ MeV for different $\sin^{2}{\theta_{\tau}}$, see Eq. (6).

Visible Energy Deposition. The sterile neutrino decays and deposits energy
into the SN envelope. For the mass range of interest and mixing only with
$\nu_{\tau}$, the charged-current processes are kinematically forbidden.
Therefore, we only consider following neutral current decays and their charge-
conjugate processes, assuming $\nu_{s}$ to be Majorana particles [42, 43, 44,
45, 46]. Note that the analytical expressions for $\pi^{0}$ decay mode width
in Refs. [42, 44, 45] differ by a factor of $2$ (in the numerator) compared to
Refs [43, 46]. In this work, we use expressions given in Ref. [45]. The
relevant decay modes for $\nu_{s}$ are

$\displaystyle\nu_{s}$ $\displaystyle\rightarrow\nu_{\tau}\pi^{0},$
$\displaystyle\nu_{s}$ $\displaystyle\rightarrow\nu_{\tau}e^{+}e^{-},$
$\displaystyle\nu_{s}$ $\displaystyle\rightarrow\nu_{\tau}\mu^{+}\mu^{-},$
$\displaystyle\nu_{s}$
$\displaystyle\rightarrow\nu_{\tau}\nu_{x}\bar{\nu}_{x},$ (7)

where $x=(e,\mu,\tau)$. The process involving neutral pion tends to be the
dominant decay mode for higher masses while the electron channel contributes
negligibly to the decay width. We classify the first 3 listed processes as the
visible decay modes while the last decay mode entirely into neutrinos as
invisible. Based on this classification, we will discuss two limiting cases of
energy deposition, $i$) the entire energy from sterile decay is deposited, and
$ii$) only the visible decay modes deposit their energy into the SN envelope.
The energy deposition in the latter case is determined by the branching ratio
into the visible modes, given by

$BR_{vis}=1-\frac{\Gamma(\nu_{s}\rightarrow\nu_{\tau}\nu_{x}\bar{\nu}_{x})}{\Gamma_{tot}},$
(8)

where $\Gamma_{tot}$ is the total decay width calculated from all four decay
modes.

We now discuss the efficacy of the branching ratio method. Two possible issues
can be raised for this approach. Firstly, the presence of a neutrino final
state in the the first 3 decay modes might take away a portion of the (assumed
to-be) deposited energy. Secondly, there might be possible energy deposition
into the mantle from the secondary neutrinos produced in the invisible decay
channel. Before we address both of these issues, a crucial fact to remember is
that neutrinos can get trapped or multiply scatter in the hot and highly dense
SN environment, conditions for which can mainly occur in/near the proto-
neutron star core. In the first case for $m_{s}\geq m_{\pi}$, the decays of
the sterile neutrino are dominated by the pion mode. For mixing angles of
interest, such steriles decay far inside the $R_{env}$, which gives
$\nu_{\tau}$ produced to encounter high densities and undergo at least one
scattering to deposit substantial energy into the envelope111Note that our
discussion only centers around the decay lengths for the average secondary
neutrino energies in a distribution. Some of these neutrinos if not being
trapped/scatter, doesn’t imply all of the produced neutrino in the decay mode
under consideration will free stream. The higher end tail of the energy
spectrum will most likely be deposited.. For $m_{s}\leq m_{\pi}$, the decays
are dominated by the 3$\nu$ mode, with average neutrino energy $\sim 80$ MeV.
Such energetic neutrinos would have been trapped if produced near the core but
for lighter $\nu_{s}$ (and relevant mixings), the average decay length is far
outside the SN. Hence, a major portion of the neutrino energy will not be
deposited and hence considering this channel as “invisible” is a good
approximation. From the above discussion, we can conclude that the branching
ratio method is a good estimator of the actual energy deposition from sterile
neutrino decays. A proper inclusion of secondary neutrino energy deposition is
beyond the scope of the current work and we stress that it should not affect
very strongly the results presented here.

Reference SN Model In this work, we adopt two different SN profiles to obtain
the bounds in the mixing-mass plane as well as to test the robustness of the
bounds to the SN profile used. Since we are concerned with very small mixing
angles, we assume that the $\nu_{s}$ production does not affect the standard
SN processes. In the Garching model, we apply our reasoning to obtain
realistic bounds with the SFHo-18.8 model simulated by the Garching group,
which adopts a $18.8\,M_{\odot}$ progenitor and includes six-species neutrino
transport [47, 48, 49]. The SFHo EoS [50] is used and PNS convection is
modeled by a mixing-length treatment [51]. We use the simulated SN evolution
assuming $R_{\text{core}}\sim 20$ km for all post-bounce time sequences up to
$\sim 10$s and envelope extending up to $\sim 5\times 10^{7}$ km.

We also consider a $8.8\,M_{\odot}$ SN profile [52] (which is on the lower end
of the progenitor mass range that finally end up in a neutron star) to study
the effect of progenitor dependence on our bounds. This profile collapses via
electron-capture on its ONeMg core with a final baryonic mass of
$1.366\,M_{\odot}$ using Shen’s stiff baryonic equation of state [53] for hot
dense nuclear matter and a final neutron star radius of about $15$ km. Also
note that unlike the $18.8\,M_{\odot}$ progenitor model, this simulated
profile does not include muons or convection in proto-neutron star. Although
both of these models have peak temperatures of about $30-40$ MeV, we find
quite different bounds in each case (See Fig. 1 and discussion on results).

The sterile neutrino production rate depends mainly on the sterile energy,
sterile mass and the radial distribution of temperature. For SFHo-18.8, the
outer layer of the core is the hottest in the first few time steps, as the in-
falling matter towards the iron core first heats up the core edge. Thus, the
maximum production occurs at edges of the core for the first few ms after
$t_{pb}=0$. But as the inner core gets hotter with time, the maximum
production rate shifts closer to the center. The total production rate grows
with time up to $\sim 3\mbox{ -- }4$ s after which it falls off rapidly.

It is easy to see that for masses $m_{s}\lesssim T$, the high energy sterile
states can be produced easily. For heavier steriles, although mainly produced
at rest, energies up to $\sim(6\mbox{ -- }8)\,T$ are feasible (since
production is through $2\rightarrow 2$ scattering processes). However, some
higher energy states for heavier steriles can still be produced with modest
Boltzmann suppression.

Results. We display our main results in Fig. 1 while only assuming non-zero
tau neutrino mixing. For the blue curves (both 8.8 and 18.8 $M_{\odot}$
progenitor), we assume no more than $10^{50}$erg energy deposition, and only
include contributions from the electromagnetic decay products of the sterile.
For 18.8 $M_{\odot}$ progenitor, we see that this constraint can be close to
two orders of magnitude stronger than the energy loss bounds obtained from
SN1987A [19], and provides the leading constraint on the mixing angle for
$15\leavevmode\nobreak\ {\rm MeV}\lesssim m_{s}\lesssim 400$ MeV. This
constraint could be yet stronger if all the sterile decay products including
the active neutrinos deposit energy in the envelope (black dashed curve).
Finally, for the other blue curve we see that the bound is significantly
weakened if the progenitor mass is lowered to 8.8 $M_{\odot}$. It should be
noted however that even in this case, our results provide the leading
constraint in the 100-400 MeV mass window. Also included in Fig. 1 is the
region favored by Type-I seesaw models for neutrino masses (dashed brown
curves), BBN bounds assuming standard cosmology [21, 22, 23], T2K [24] and
BEBC bounds [25]. We note that BBN bounds exclude the region below the green
curve in Fig. 1. However, in the presence of a lepton asymmetry these
constraints can be substantially weakened [54].

We again note that bounds depend on the progenitor model. This is due to the
fact that sterile neutrino production depends sensitively on $T$. The
temperature profiles for $18.8\,M_{\odot}$ SN have consistently higher
temperatures for all $t=(0-10)$s as compared to the $8.8\,M_{\odot}$
progenitor.

We note that while we have fixed to mass-mixing with the tau-neutrino in Fig.
1, we expect qualitatively similar results for mixing with the muon-neutrino.

Future directions. The present paper has examined the implications of heavy
sterile neutrinos for low-energy SNe. This work can be extended in a number of
directions. For one, we have only mixing with one active neutrino flavor at a
time. It would be worthwhile to examine more generic flavor structure
assumptions. Secondly, it is possible that the production and decay of sterile
neutrinos is not controlled by the EW force. For example, sterile transition
magnetic moments have been widely studied [55, 56, 57, 58]. We also anticipate
low-energy SNe placing stringent constraints on transition magnetic moments.

Moreover, even while restricting to mass-mixing, there are additional avenues
to be explored. Given that SNe are efficient production sites of sterile
neutrinos, the impact of the escaping sterile neutrinos (and their decay
products) could be significant. In particular, for axions it is known that
requiring that the axions not produce a photon flux above the diffuse gamma-
ray background leads to strong bounds at low masses [59]. Secondly, the high-
energy neutrino flux produced by the decaying sterile neutrinos may also be
detectable at future neutrino experiments like DUNE and Hyper-K [19], and a
detailed follow-up in light of our results may yield new constraints.

Conclusions. We have found that low-energy SNe can provide leading constraints
on sterile neutrinos in the 15-400 MeV range. In addition, these bounds probe
theoretically well-motivated parameter space predicted by Type-I seesaw models
of neutrino masses. In future work we plan to return to this scenario and
explore implications of the neutrino and gamma-rays emerging from sterile
decays beyond the SN envelope.

Acknowledgements. We are very grateful to Anna Suliga for helpful discussions
and comments. We thank Hans-Thomas Janka and Daniel Kresse for providing the
SN profiles used in this work. The work of GC is supported by the U.S.
Department of Energy under the award number DE-SC0020250 and DE-SC0020262. The
works of SH, PH, and IMS are supported by the U.S. Department of Energy Office
of Science under award number DE-SC0020262 and DE-SC0020250. The work of SH is
also supported by NSF Grant No. AST1908960, No. PHY-1914409 and No.
PHY-2209420; and JSPS KAKENHI Grant Numbers JP22K03630 and JP23H04899. This
work was supported by World Premier International Research Center Initiative
(WPI Initiative), MEXT, Japan.

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