# Core-collapse Supernova Constraint on the Origin of Sterile Neutrino Dark
Matter via Neutrino Self-interactions

Yu-Ming Chen<EMAIL_ADDRESS>Department of Physics, Carleton
University, Ottawa, ON K1S 5B6, Canada  Manibrata Sen<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany  Walter Tangarife<EMAIL_ADDRESS>Department of Physics, Loyola
University Chicago, Chicago, IL 60660, USA  Douglas Tuckler
<EMAIL_ADDRESS>Department of Physics, Carleton University,
Ottawa, ON K1S 5B6, Canada  Yue Zhang<EMAIL_ADDRESS>Department
of Physics, Carleton University, Ottawa, ON K1S 5B6, Canada

###### Abstract

Novel neutrino self-interaction can open up viable parameter space for the
relic abundance of sterile-neutrino dark matter (S$\nu$DM). In this work, we
constrain the relic target using core-collapse supernova which features the
same fundamental process and a similar environment to the early universe era
when S$\nu$DM is dominantly produced. We present a detailed calculation of the
effects of a massive scalar mediated neutrino self-interaction on the
supernova cooling rate, including the derivation of the thermal potential in
the presence of non-zero chemical potentials from plasma species. Our results
demonstrate that the supernova cooling argument can cover the neutrino self-
interaction parameter space that complements terrestrial and cosmological
probes.

###### Contents

  1. I Introduction
  2. II The formalism
    1. II.1 Model
    2. II.2 Active-Sterile Neutrino Mixing in Supernova
    3. II.3 Collisional Cooling Rate
  3. III Results
    1. III.1 New supernova cooling limit
    2. III.2 Comment on Resonant Transition
  4. IV Discussion and Conclusion
  5. A $\phi$ mediated thermal potential
  6. B The total luminosity integrals
  7. C Width of MSW resonance inside supernova

##  I Introduction

Self-interaction among active neutrinos is a blind spot among the tests of the
Standard Model (SM) of elementary particles. Laboratory experiments are yet to
measure neutrino-neutrino scattering, due to their elusive nature. Although
LEP has measured the neutrino coupling with the $Z$-boson Schael  _et al._
(2006), the result does not exclude the possibility of stronger neutrino self-
interaction through additional light mediators. Indeed, a viable wide window
is open if the mediator is a SM gauge singlet that couples exclusively to
neutrinos at tree level, known as a lepton number charged scalar Berryman  _et
al._ (2018), including the Majoron Gelmini and Roncadelli (1981). A
neutrinophilic mediator can be probed in various ways, from neutrino
experiments to cosmic microwave background (CMB) and stellar physics (see
Berryman  _et al._ (2022) for a recent overview). The presence of such a new
force can be strongly motivated as solutions to other significant problems,
notably the origin and nature of dark matter that fills our universe.

The intimate relationship between neutrino self-interaction and sterile
neutrino dark matter (${\rm S\nu DM}$) has been pointed out and explored in
several recent works De Gouvêa  _et al._ (2020); Kelly  _et al._ (2020, 2021);
Chichiri  _et al._ (2021); Benso  _et al._ (2022); Alonso-Álvarez and Cline
(2021). Remarkably, with a sufficiently strong neutrino self-interaction, the
${\rm S\nu DM}$ relic target can evade the exclusion limits set by indirect
X-ray searches De Gouvêa  _et al._ (2020). In the early universe, the self-
interaction can keep the active neutrinos in thermal equilibrium among
themselves for longer than the weak interaction allows, thereby increasing the
active-sterile neutrino conversion rate. A complete portrait of the ${\rm S\nu
DM}$ relic-density target parameter space was presented in Ref. Kelly  _et
al._ (2021). It features a neutrinophilic mediator mass from keV to multi-GeV
scale, and the corresponding mediator-neutrino coupling ranges from $\sim
10^{-8}$ to order 1. The viable mass of ${\rm S\nu DM}$ lies between keV and
MeV. Compared to other ways of addressing the relic density problem, such a
mechanism resorts to simple low-scale physics, which makes it highly testable.
Phenomenologically, beam neutrino experiments like DUNE and the forward
physics facility at the LHC can be useful for probing the GeV-scale mediator
mass region of the relic target Kelly and Zhang (2019); Kelly  _et al._
(2022); de Gouvêa  _et al._ (2020). In constrast, cosmological probes with
precision $\Delta N_{\rm eff}$ measurement can cover the region with a very
light, sub-MeV scale mediator Escudero Abenza (2020); Kelly  _et al._ (2021);
Chacko  _et al._ (2004). However, an intermediate-mass window for the mediator
is left open, where the coupling is too small to be relevant for terrestrial
experiments, and the mediator is sufficiently heavy, thus evading the
constraints from CMB and BBN. This part of the parameter space can be
susceptible to cooling bounds arising out of core-collapse supernovae (CCSN).
The present work’s task is to fill such a gap by presenting a new supernova
constraint on the relic target.

Our starting point is simple and natural. In the early Universe, ${\rm S\nu
DM}$ is dominantly produced through secret neutrino self-interactions around a
temperature of $\sim 1-100$ MeV. A CCSN is another natural laboratory with a
dense neutrino background where such temperatures can be reached in the core.
As a result, ${\rm S\nu DM}$ can also be efficiently produced in the core of a
CCSN. Production and emission of such non-interacting particles can result in
excessive energy loss of the supernova (SN), leading to additional cooling.
Observations of the neutrinos from SN1987A can constrain the amount of energy
allowed to be carried away by any new and weakly-coupled particles Raffelt
(1996). Stated simply, the neutrino burst must carry sufficient energy from
the collapsed core to ignite the eventual explosion Janka  _et al._ (2007) and
last long enough to match the observation period of $\mathcal{O}(10)\,$s
Hirata  _et al._ (1987); Bionta  _et al._ (1987); Alekseev  _et al._ (1987).
This leads to an exclusion window in the parameter space of new light degrees
of freedom, where they cool the SN fast, causing a shorter duration of
neutrino burst. In general, the SN constraint can be evaded if the particles
are too weakly coupled to be efficiently produced in the core. On the other
hand, if they are too strongly coupled, they will be reabsorbed back to the
core, thereby not transporting the energy efficiently.

Although the fundamental process for ${\rm S\nu DM}$ production is identical
between SN and early universe, there are important differences in the two
analysis. While the early Universe is largely isotropic and homogeneous, a SN
retains a lot less symmetry. The thermal plasma inside a SN is composed of
neutrons, protons, electrons and neutrinos, which have large matter-antimatter
asymmetries. As a result, the forward scattering potential experienced by
neutrinos picks up a non-trivial contribution from finite density effects, in
addition to finite temperature contribution. Moreover, given the dense and
fast-varying matter background, we need to consider the absorption and non-
adiabatic effects on ${\rm S\nu DM}$ en route out of the SN. We expand on
these points in details in the main text.

Excessive energy loss from a CCSN has been considered before in the context of
the minimal ${\rm S\nu DM}$ model without secret neutrino self-interactions.
In the minimal ${\rm S\nu DM}$ model, the cooling constraint on the vacuum
mixing angle is found less competitive than the existing X-ray search limit
due to feedback effects Suliga  _et al._ (2020, 2019); Argüelles  _et al._
(2019); Raffelt and Zhou (2011); Hidaka and Fuller (2006); Dolgov  _et al._
(2000a, b). On the other hand, the neutrinophilic mediator can be directly
produced in the SN, leading to excessive cooling, as has been shown for tiny
couplings Escudero  _et al._ (2020); Brune and Päs (2019); Heurtier and Zhang
(2017); Farzan (2003); Kolb and Turner (1987). Larger couplings of neutrino
self-interactions can be constrained from the down-scattering in energy of SN
neutrinos by their interaction with the cosmic neutrino background Kolb and
Turner (1987); Shalgar  _et al._ (2021). Interactions with a neutrinophilic
scalar can also cause distinct changes in the SN collapse dynamics Fuller  _et
al._ (1987, 1988); Chang  _et al._ (2022). Constraints can also arise from the
decay of the mediating boson to neutrinos, thereby altering the expected SN
flux Akita  _et al._ (2022). However, these constraints are not able to probe
the ${\rm S\nu DM}$ relic target completely. In contrast, our work puts
together production of ${\rm S\nu DM}$ via neutrino self-interaction, in the
SN core, and the excessive cooling due to free-streaming of ${\rm S\nu DM}$.
We show that CCSN allows to explore new regions in the model’s parameter
space. This result is highly complementary to neutrino experimental and
cosmological probes of the relic target.

The structure of this article is as follows. In Sec. II, we write down the
model and the formalism for calculating the SN cooling luminosity via ${\rm
S\nu DM}$ emission. In particular, we derive the thermal potential generated
by neutrino self-interaction with a massive mediator and non-zero chemical
potentials. Next, we present our main result in Sec. III, where we derive a
new SN cooling constraint that can cover part of the dark matter relic target
curve. We discuss the physics behind each edge of the exclusion region and
highlight features unique to the cooling mechanism under consideration. We
also present an overview of the model parameter space that shows where our
main result stands along with other experimental probes of the origin of ${\rm
S\nu DM}$. Our conclusions are placed in Sec. IV.

##  II The formalism

###  II.1 Model

The model we consider involves two ingredients beyond the SM. First, sterile
neutrino dark matter (${\rm S\nu DM}$), defined as a linear combination of
active neutrinos $\nu$ (weak eigenstate) and a SM gauge singlet fermion
$\nu_{s}$,

$\nu_{4}=\nu_{s}\cos\theta+\nu\sin\theta\ ,$ (1)

where $\nu_{4}$ is the mass eigenstate and $\theta$ is the active-sterile
neutrino mixing angle in vacuum. Given the indirect-detection constraint on
$\nu_{4}\to\nu\gamma$ decay, the angle $\theta$ needs to be very tiny. As a
result, the above mixing plays no role in generating the observed neutrino
masses, which need to be generated by other physics and can be either the
Majorana or the Dirac type Kelly  _et al._ (2020).

The second ingredient is new neutrino self-interactions mediated by a massive
scalar boson $\phi$. The relevant interacting Lagrangian takes the form

$\mathcal{L}_{\rm
int}=\frac{(L_{\alpha}H)(L_{\beta}H)\phi}{\Lambda_{\alpha\beta}^{2}}+{\rm
h.c.}\to\frac{1}{2}\lambda_{\alpha\beta}\nu_{\alpha}\nu_{\beta}\phi+{\rm
h.c.}\ .$ (2)

The effective operator in the first step is SM gauge invariant. In the second
step, we turn on the electroweak symmetry breaking where the Higgs vacuum
expectation value makes the $\phi$ coupling neutrinophilic. The flavor indices
$\alpha,\beta=e,\mu,\tau$ are summed over.

Within this simple framework, the correct relic abundance for ${\rm S\nu DM}$
can be produced via active-sterile neutrino oscillation in the early universe
De Gouvêa  _et al._ (2020). In the early universe, the self-interaction can
keep the active neutrinos in thermal equilibrium with themselves. After the
production of an active neutrino from the thermal plasma, it oscillates into
the sterile neutrino state until another neutrino self-interaction occurs.
Because of oscillation, there is a small probability for the coherent state to
be measured as the dark matter $\nu_{4}$. A Feynman diagram summarizing the
above processes is shown in Fig. 1. The same story applies to all neutrinos in
the thermal plasma and can repeat many times until the self-interaction
decouples, leading to a buildup of the ${\rm S\nu DM}$ abundance. Because the
$\phi$-mediated neutrino self-interaction is allowed to be much stronger than
the weak self-interactions, the correct relic abundance can be produced with a
sufficiently small mixing angle $\theta$ that satisfies the indirect detection
X-ray constraint.

Figure 1: Identical fundamental process governs the production of ${\rm S\nu
DM}$ in early universe (for relic density) and the excessive cooling of CCSN.
The blue cross represents the effective active-sterile neutrino mixing in the
corresponding epoch. Time flows from left to right.

###  II.2 Active-Sterile Neutrino Mixing in Supernova

The active-sterile neutrino effective mixing angle inside a supernova (SN) is
calculated in a similar way as its early universe counterpart Dodelson and
Widrow (1994),

$\sin^{2}2\theta_{\rm
eff}(E,r)=\frac{\Delta^{2}\sin^{2}2\theta}{\Delta^{2}\sin^{2}2\theta+\Gamma(E,r)^{2}+(\Delta\cos
2\theta-V(E,r))^{2}}\ ,$ (3)

where $\Delta\simeq m_{4}^{2}/(2E)$, and $\Gamma=\Gamma_{\rm
weak}+\Gamma_{\phi}$ and $V=V_{\rm weak}+V_{\phi}$ are the interaction rate
and effective potential seen by an active neutrino (or antineutrino),
respectively. Because SN features a sizable matter–antimatter asymmetry,
neutrino and antineutrino typically see different $\Gamma$ and $V$ and, in
turn, very different effective mixing angles.

Figure 2: Mass density $\rho$, temperature $T$, chemical potential
$\mu_{\nu_{e}}$, and electron fraction $Y_{e}$ as functions of the radial
coordinate $r$, taken from hydrodynamical simulations Gar . The SN model we
use is s20.0-SFHo, as shown by the solid blue curves. For comparison, we also
show a few other models (in colored dashed curves) used in the literature.

The neutrino interaction rate through weak interactions $\Gamma_{\text{weak}}$
is dominated by its scattering with the ambient neutrons and protons via $Z$\-
and $W$-boson exchange. For neutrino energy $E$ well below the neutron mass,

$\Gamma_{\text{\rm
weak}}(E,r)=\frac{(1+3g_{A}^{2})\,G_{F}^{2}E^{2}}{4\pi}\times\left\\{\begin{array}[]{cl}n_{n}(r),&\quad{\rm
for}\ \nu_{\mu},\bar{\nu}_{\mu},\nu_{\tau},\bar{\nu}_{\tau}\\\
5n_{n}(r),&\quad{\rm for}\ \nu_{e}\\\ n_{n}(r)+4n_{p}(r),&\quad{\rm for}\
\bar{\nu}_{e}\\\ \end{array}\right.$ (4)

where $g_{A}\simeq 1.3$ is the low-energy nucleon axial coupling, and
$n_{i}(r)$ is the number density of particle $i$. Hereafter, we assume the SN
is a spherical symmetric object. We also neglect the mass of electron and
proton-neutron mass difference which are much smaller than the typical energy
scales of the CCSN, e.g. core temperature and densities.

The neutrino interaction rate through $s$-channel $\phi$-exchange is Kelly
_et al._ (2021)

$\Gamma_{\phi}(E,r)=\frac{|\lambda|^{2}\,m_{\phi}^{2}\,T(r)}{16\pi\,E^{2}}\left[\log\left(e^{\frac{\mu_{\nu}(r)}{T(r)}}+e^{\frac{m_{\phi}^{2}}{4ET(r)}}\right)-\frac{m_{\phi}^{2}}{4E\,T(r)}\right]\
,$ (5)

where $T(r)$ is the matter temperature and $\mu_{\nu}(r)$ is the neutrino
chemical potential inside the SN. The latter is nonzero for $\nu_{e}$ but
neglected for $\nu_{\mu}$ and $\nu_{\tau}$. For the sake of simplicity, we
have dropped the flavor indices of the coupling $\lambda$. In the following
analysis, we will assume only one such coupling is around each time, i.e., the
same active neutrino that mixes with $\nu_{s}$ also participates in the novel
self-interaction introduced in Eq. (2).

Because SN is made of asymmetric matter, the leading contribution to the
matter potential through weak interactions occurs at order
$\mathcal{O}(G_{F})$ Abazajian  _et al._ (2001),

$V_{\rm
weak}(E,r)=\left\\{\begin{array}[]{cl}\sqrt{2}G_{F}\left[2(n_{\nu_{e}}-n_{\bar{\nu}_{e}})+(n_{e^{-}}-n_{e^{+}})-\frac{1}{2}(n_{n}-n_{\bar{n}})\right],&\quad{\rm
for}\ \nu_{e}\\\
\sqrt{2}G_{F}\left[(n_{\nu_{e}}-n_{\bar{\nu}_{e}})-\frac{1}{2}(n_{n}-n_{\bar{n}})\right],&\quad{\rm
for}\ \nu_{\mu,\tau}\\\ \end{array}\right.$ (6)

where the number densities depend on $r$. The potentials for antineutrino have
opposite sign. The temperature-dependent terms occur at order
$\mathcal{O}(G_{F}^{2})$ and are negligible. It is useful to note that $V_{\rm
weak}$ depends on a lower power of $G_{F}$ than $\Gamma_{\rm weak}$, which
leads to an interesting interplay when neutrino self-interaction is turned on
as will be discussed in Sec. III.

We employ the SN profile titled s20.0-SFHo from hydrodynamical simulations by
the Garching group Gar . Fig. 2 shows the $r$ dependence of temperature $T$,
chemical potential for the electron-type neutrino $\mu_{\nu_{e}}$, mass
density $\rho$, and electron fraction $Y_{e}$. Our calculations are based on
the profiles at time $t=1\,$s., which is characteristic of the cooling phase
of the neutrino emission. For a given $r$, the particle asymmetries in neutron
and electron are related to the electron fraction, $Y_{e}$, as

$n_{n}-n_{\bar{n}}=\frac{\rho}{m_{n}}(1-Y_{e})\ ,\quad\quad
n_{e^{-}}-n_{e^{+}}=\frac{\rho}{m_{n}}Y_{e}\ .$ (7)

On the other hand, the particle asymmetry in ultra-relativistic neutrino is
derived assuming a thermal distribution,

$n_{\nu_{e}}-n_{\bar{\nu}_{e}}=-\frac{3T^{3}}{\pi^{2}}\left[{\rm
Li}_{3}\left(-e^{\mu_{\nu_{e}}/T}\right)-{\rm
Li}_{3}\left(-e^{-\mu_{\nu_{e}}/T}\right)\right]\ ,$ (8)

where ${\rm Li}_{3}$ is the third-order polylogarithm function.

Figure 3: Self-interaction contribution to the self-energy potential, $V$, in
Eq. (9).

Last but not least, we need $V_{\phi}$, the thermal potential generated by
neutrino self-interaction. In the past, the integral form of self-interaction
contribution to the potential $V_{\phi}$ was derived for a generic mediator
mass but assuming vanishing chemical potential in Ref. Quimbay and Vargas-
Castrillon (1995). In the presence of a chemical potential, the potential was
derived assuming a massless mediator in Ref. Erdas  _et al._ (1993). In this
work, we derive the most general form of the thermal potential for a neutrino
with energy $E$, taking into account both the chemical-potential and mediator-
mass effects. We find that the potential arising from the neutrino self-energy
diagram, as shown in Fig. 3, is given by

$\begin{split}V_{\phi}(E,r)&=\frac{|\lambda|^{2}}{16\pi^{2}}{\rm
Re}\int_{0}^{\infty}dk\left[\left(\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}}{m_{\phi}^{2}+4Ek}+\frac{2k}{E}\right)\frac{1}{e^{(k+\mu_{\nu}(r))/T(r)}+1}\right.\\\
&\hskip
85.35826pt\left.+\left(-\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}}{m_{\phi}^{2}-4Ek}+\frac{2k}{E}\right)\frac{1}{e^{(k-\mu_{\nu}(r))/T(r)}+1}\right]\\\
&+\frac{|\lambda|^{2}}{16\pi^{2}}{\rm
Re}\int_{0}^{\infty}\frac{kdk}{\epsilon_{1}}\left[\left(\frac{2k}{E}-\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}+2Ek+2E\epsilon_{1}}{m_{\phi}^{2}-2Ek+2E\epsilon_{1}}\right)\frac{1}{e^{(\epsilon_{1}-2\mu_{\nu}(r))/T(r)}-1}\right.\\\
&\hskip
85.35826pt+\left.\left(\frac{2k}{E}-\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}+2Ek-2E\epsilon_{1}}{m_{\phi}^{2}-2Ek-2E\epsilon_{1}}\right)\frac{1}{e^{(\epsilon_{1}+2\mu_{\nu}(r))/T(r)}-1}\right]\
,\end{split}$ (9)

where $\epsilon_{1}\equiv\sqrt{k^{2}+m_{\phi}^{2}}$. The first integral
corresponds to cutting the neutrino propagator whereas the second one
corresponds to cutting the $\phi$ propagator. For the thermal distribution of
$\phi$, we have imposed the chemical potential relation
$\mu_{\phi}=2\mu_{\nu}$. For the case of antineutrino, one simply needs to
flip the sign of chemical potential, i.e.,
$\mu_{\nu}\to\mu_{\bar{\nu}}=-\mu_{\nu}$.

We provide the details of the derivation of Eq. (9) in Appendix A. We have
also verified that, in the $\mu\to 0$ or $m_{\phi}\to 0$ limits,
$V_{\phi}(E,r)$ reduces to the results in Refs. Quimbay and Vargas-Castrillon
(1995) or Erdas  _et al._ (1993), respectively.

###  II.3 Collisional Cooling Rate

The new SN cooling process we consider is depicted in Fig. 1, where two active
neutrinos scatter, via $s$-channel, into an on-shell $\phi$ particle, which
quickly decays into an active and a sterile neutrino. The decay of $\phi$
involves the same Yukawa coupling as its production and the active-sterile
neutrino mixing angle inside the SN. The corresponding luminosity is defined
as energy release in the form of S$\nu$DM $\nu_{4}$ per unit time,

$\begin{split}L&=\int
d^{3}\vec{r}\int\frac{d^{3}\vec{p}_{1}}{(2\pi)^{3}}f_{\nu}(E_{1},r)\int\frac{d^{3}\vec{p}_{2}}{(2\pi)^{3}}f_{\nu}(E_{2},r)\frac{1}{4E_{1}E_{2}}\int\frac{d^{3}\vec{p}_{3}}{(2\pi)^{3}2E_{3}}\int\frac{d^{3}\vec{p}_{4}}{(2\pi)^{3}2E_{4}}\\\
&\quad\times(2\pi)^{4}\delta^{4}(\vec{p}_{1}+\vec{p}_{2}-\vec{p}_{3}-\vec{p}_{4})|\mathcal{M}|^{2}\,E_{4}\,e^{-\tau(E_{4},r)}\
,\end{split}$ (10)

where

$f_{\nu}(E,r)=\frac{1}{e^{\frac{E-\mu_{\nu}(r)}{T(r)}}+1}\ $ (11)

is the Fermi-Dirac distribution function for an initial state neutrino inside
core-collapse supernova (CCSN). Both the chemical potential and temperature
depend on the location where the production of $\nu_{4}$ occurs. In the last
factor of Eq. (10), $\tau$ is the optical depth, which measures how likely it
is for $\nu_{4}$ to re-scatter and lose energy back to the SN on its way out,
and is defined as

$\tau(E,r)=\int_{r}^{\infty}dr_{1}\sin^{2}\theta_{\rm
eff}(E,r_{1})\Gamma(E,r_{1})\ .$ (12)

A more accurate derivation of the optical depth involves taking into account
the attenuation of $\nu_{4}$ produced from the source point $\vec{r}_{1}$ in
all the angular directions Caputo  _et al._ (2022). However, as we will show
below (see sec. III), the re-absorption only has a subdominant effect in the
cooling of SN. For simplicity, we use the above approximate form of the
optical depth Chang  _et al._ (2017, 2018).

Under the narrow-width approximation, the squared matrix element is equal to

$|\mathcal{M}|^{2}=32\pi^{2}\lambda^{2}\,m_{\phi}^{2}\,\delta(s-m_{\phi}^{2})\sin^{2}\theta_{\rm
eff}(r,E_{4})\ ,$ (13)

where $\theta_{\rm eff}$ is the effective mixing angle between the active and
sterile neutrinos, defined in Eq. (3), and depends on the energy $E$ of the
state and its position $r$.

Clearly, due to the extra factor of $E_{4}$, the cooling rate defined in Eq.
(10) is not Lorentz invariant. The physical cooling rate is defined in the
rest frame of the CCSN. We first complete the final-state phase-space
integrals in a reference frame with generic $\vec{p}_{1}+\vec{p}_{2}$, and
then simplify the initial-state integral using the $\delta$-function in
$|\mathcal{M}|^{2}$ (Eq. 13). See Appendix B for further details. The energy
loss rate carried away by $\nu_{4}$ is finally reduced to

$\begin{split}L&=\frac{\lambda^{2}m_{\phi}^{2}}{4\pi^{2}}\int_{0}^{4R_{c}}r^{2}dr\int_{0}^{\infty}dE_{1}f(E_{1},r)\int_{m_{\phi}^{2}/(4E_{1})}^{\infty}dE_{2}f(E_{2},r)\frac{1}{\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}}\\\
&\quad\times\int_{\frac{1}{2}\left(E_{1}+E_{2}-\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}\right)}^{\frac{1}{2}\left(E_{1}+E_{2}+\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}\right)}dE_{4}\sin^{2}\theta_{\rm
eff}(r,E_{4})\,E_{4}\,e^{-\tau(E_{4},r)}\ .\end{split}$ (14)

The upper limit of $r$ integral is taken to be $4R_{c}\,=\,40\,$km. The last
three integrals are evaluated numerically. Our main results will be presented
in Sec. III.

##  III Results

###  III.1 New supernova cooling limit

We consider three benchmark choices of the model parameters that are
(marginally) allowed by the present $X$-ray limits Kopp (2022); Abazajian
(2017); Roach  _et al._ (2022),

$\displaystyle{\rm BP1:}\quad m_{4}\,=\,7\,\,{\rm keV}\,,$
$\displaystyle\sin^{2}2\theta\,=\,7\times 10^{-11}\ ,$ $\displaystyle{\rm
BP2:}\quad m_{4}\,=\,21\,\,{\rm keV}\,,$
$\displaystyle\sin^{2}2\theta\,=\,1.4\times 10^{-13}\ ,$ (15)
$\displaystyle{\rm BP3:}\quad m_{4}\,=\,50\,\,{\rm keV}\,,$
$\displaystyle\sin^{2}2\theta\,=\,10^{-14}\ .$

For each benchmark point, we consider two cases where the neutrino chemical
potential is present in the SN (for $\nu_{e}$) and absent (for $\nu_{\mu}$ and
$\nu_{\tau}$). As mentioned earlier, we assume the same active neutrino that
mixes with $\nu_{s}$ also participates in the novel self-interaction
introduced in Eq. (2). We assume one coupling for the self-interaction to be
present in each analysis.

Our main result is shown in Fig. 4, where we required that the luminosity of
S$\nu$DM emission from SN-1987A must satisfy the upper bound111This is often
referred to as the Raffelt bound Raffelt (1996). It is worth noting that when
constraining any new-physics models, the most accurate determination of this
upper bound would require incorporating the new physics in the original SN
simulation. This has to be done point by point in the model parameter space,
which we do not attempt here. Our goal is to work at the order-of-magnitude
level and demonstrate the potential relevance of the SN cooling limit.

$L\lesssim 3\times 10^{52}\,{\rm erg/s}\ .$ (16)

This excludes the blue shaded regions in the $\lambda$-$m_{\phi}$ plane. In
each panel, the black curve is where the correct relic density for the
S$\nu$DM ($\nu_{4}$) can be produced by neutrino self-interaction mediated by
$\phi$, as suggested in De Gouvêa  _et al._ (2020). Our results show that
observations from SN1987A can cover regions with relatively smaller coupling
$\lambda$ compared to the laboratory limits and set a useful constraint on the
origin of dark matter in this model.

Figure 4: Core-collapse supernova constraint derived in this work (blue shaded
regions, labelled by “SN”) on the parameter space of neutrino self-interaction
that can address the relic density of S$\nu$DM (solid black curves). We
consider three benchmark points defined in Eq. (15) and three flavored
coupling choices of the mediator $\phi$ that are labelled correspondingly. In
each panel of the figure, we also show other constraints from precision
measurements of the $Z$-boson, meson decays (purple, orange, green shaded
regions), and big-bang nucleosynthesis (gray shaded regions).

For all three benchmark points, the SN constraint applies to $\phi$ masses up
to several hundred MeV. Because the cooling channel considered in this work
relies on the production of on-shell $\phi$ (that promptly decays into the
$\nu_{4}$) and the typical temperature of the SN core is at most $\sim
60\,$MeV, the luminosity gets strongly Boltzmann suppressed for $\phi$ heavier
than several hundred MeV and the cooling constraint goes away.

Figure 5: The $\lambda$ dependence of key parameters in the SN cooling rate.
From left to right: $\sin^{2}2\theta_{\rm eff}/\sin^{2}2\theta$, $e^{-\tau}$
and $\lambda^{2}\sin^{2}2\theta_{\rm eff}e^{-\tau}$. An important message here
is that the effective mixing suppression plays the dominant role in large
coupling region, $\lambda>10^{-3}$. This effect applies generically to all
mediator masses and overshadows the re-absorption of $\nu_{4}$. Here we show
the plots for model point BP1 defined in Eq. (15). The corresponding plots are
similar for BP2 and BP3, and when the chemical potential in switched on (for
the $\nu_{e}$ case) .

Regarding the Yukawa coupling $\lambda$ between $\phi$ and active neutrino
$\nu$, the SN constraint is the strongest for $\lambda$ around $10^{-3}$. To
understand this feature, we present Fig. 5, which shows the $\lambda$
dependence of $\lambda^{2}\sin^{2}2\theta_{\rm eff}\,e^{-\tau}$, a key factor
in the total luminosity calculation (see Eq. (14)). Here, for clarity, we fix
the energy and production radius of the final state $\nu_{4}$ to the typical
values for SN 1987A, $E_{4}\sim T\sim 30\,$MeV and $r\sim R_{c}$.

  * •

For very small values of $\lambda$, the neutrino self-interaction is too weak
to modify the effective active-sterile neutrino mixing angle anywhere inside
the SN, thus $\theta_{\rm eff}\sim\theta$. In this case, the energy loss rate
$L$ is simply proportional $\lambda^{2}$. As $\lambda$ grows, the effective
potential energy and scattering rate for neutrino generated by $\phi$ exchange
exceed those due to the weak interaction, and the effective active-sterile
neutrino mixing becomes strongly suppressed for larger $\lambda$. This is
similar to the “quantum Zeno” effect that hampers the S$\nu$DM production in
the very early universe Dodelson and Widrow (1994). The transition from
“small” to “large” $\lambda$ regime occurs when the potential $V_{\phi}$,
given in Eq. (9), is comparable to $V_{\rm weak}$ in Eq. (6) 222Here, for a
back-of-the-envelope estimate, we use the approximations, $V_{\rm weak}\sim
G_{F}n_{n}$ and $V_{\phi}\sim\lambda^{2}E_{4}/(16\pi^{2})$,

$\lambda\sim 4\pi\sqrt{G_{F}n_{n}/E_{4}}\sim 10^{-3}\ .$ (17)

  * •

In the large-$\lambda$ regime, the neutrino self-interaction rate due to
$\phi$ exchange, $\Gamma_{\phi}\propto\lambda^{2}$, is also much higher than
the weak interaction counterpart, $\Gamma_{\rm weak}$. As a result, the
effective mixing is approximately $\sin^{2}\theta_{\rm
eff}\sim\Delta^{2}\sin^{2}{\theta}/\Gamma_{\phi}^{2}\sim\Delta^{2}\sin^{2}{\theta}/\lambda^{4}$,
and in turn, the luminosity is proportional to
$L\sim\lambda^{2}\sin^{2}\theta_{\rm eff}\sim\lambda^{-2}$. The latter is thus
suppressed for increasing $\lambda$. The above features are clearly displayed
by the plots in the first column of Fig. 5. The second column of Fig. 5 plots
another factor in the cooling luminosity, $e^{-\tau}$, which describes the
survival probability of $\nu_{4}$ against re-absorption on its way out of the
SN. We find the absorption effect to be a subdominant effect. It only occurs
in the large $\lambda$ regions where $\sin^{2}2\theta_{\rm eff}$ is already
strongly damped. The above anatomy reveals an important distinction between
the SN cooling effect derived in this model and many others – in the large
coupling regime, the cooling rate is small mainly due to the dynamical
suppression in the effective active-sterile neutrino mixing parameter rather
than the absorption effect. Finally, the overall $\lambda$ dependence in the
quantity $\lambda^{2}\sin^{2}2\theta_{\rm eff}e^{-\tau}$ is displayed by the
plots in the third column of Fig. 5.

Our result in Fig. 4 shows that the $\lambda_{ee}$ coupling receives a much
stronger constraint from SN compared to $\lambda_{\mu\mu}$ and
$\lambda_{\tau\tau}$. This is mainly because of the large chemical potential
of $\nu_{e}$ in the SN (see Fig. 2). In the core region, the ratio
$\mu_{\nu_{e}}/T$ is much larger than 1. As a result, the Fermi-Dirac
distribution for $\nu_{e}$ remains not exponentially suppressed in the region
$T<E<\mu_{\nu_{e}}$. Effectively, the large chemical potential creates a
“Fermi surface” below which the occupation number of each energy state is
order 1. The integral over the above energy window contributes and enhances
the total cooling rate, Eq. (14), and allows the SN coverage to expand
significantly.

In Fig. 4, we also depict other existing constraints on the model parameter
space. They include precision measurements of the invisible $Z$-boson decay
(new contribution from $Z\to\nu\nu\phi^{*}$), leptonic decay of charged pion,
kaon and $D$ meson (new contribution channel
$\mathfrak{m}^{-}\to\ell^{-}\nu\phi^{*}$) Pasquini and Peres (2016); Berryman
_et al._ (2018); Blinov  _et al._ (2019); Brdar  _et al._ (2020); Lyu  _et
al._ (2021); Esteban  _et al._ (2021). In addition, we also include the
constraint from big-bang nucleosynthesis which sets a lower limit on the
mediator $\phi$ mass. For the range of coupling considered in the figure,
$\phi$ is necessarily thermalized due to its interaction with active neutrinos
before the weak interaction decoupling and can contribute to $\Delta N_{\rm
eff}$ if its mass is around MeV or below.

We want to stress again that the new SN limits derived here rely on the
presence of a light sterile neutrino with certain mixing with the active
neutrinos, as defined in Eq. (15). With the mixing angle and proper neutrino
self-interaction, it can obtain the correct relic density and serve as all the
dark matter in the universe. This differs from any previous works and is
exactly the main motivation and starting point of this work. Our result shows
that observations from a CCSN can be useful for covering a new part on the
S$\nu$DM relic target, especially for the case of $\lambda_{ee}$ coupling.

###  III.2 Comment on Resonant Transition

In this subsection, we comment on an alternative production channel of
S$\nu$DM that could occur in the CCSN, via non-adiabatic transition between
two energy levels due to the Mikheyev-Smirnov-Wolfenstein (MSW) resonance
Mikheev and Smirnov (1986); Wolfenstein (1978), and argue that such an effect
is negligible in the presence of neutrino self-interactions.

The MSW resonance condition can be satisfied if the active neutrino scattering
is not very frequent. The effective active-sterile mixing angle defined in Eq.
(3) can reach $\pi/4$ at a radius where $\Delta\cos 2\theta=V(r)$ if the
$\Gamma$ term in the denominator can be neglected. The latter requires the
active neutrino to be free-streaming within a region where the resonance
occurs. To quantify this, it is useful to introduce the spatial width of the
resonance region, $\Delta_{\rm Res}$, and the corresponding neutrino free-
streaming length, $\lambda_{\nu}$. They are defined by

$\displaystyle\Delta_{\rm Res}$ $\displaystyle\equiv
2\tan(2\theta)\left|\frac{V^{\prime}}{V}\right|^{-1}\ ,$ (18)
$\displaystyle\lambda_{\nu}$ $\displaystyle\equiv 1/\Gamma\ .$ (19)

In Appendix C, we present a derivation showing why $\Delta_{\rm Res}$ is the
relevant quantity.

Figure 6: This figures shows that the ratio $\Delta_{\rm Res}/\lambda_{\nu}$
is always larger than 1 in the $\lambda$ versus $m_{\phi}$ plane where we set
the SN constraint (in Fig. 4). This comparison assures that non-adiabatic
transition is unimportant in the region of parameter space of interest to the
present work. The contours are shown for benchmark points BP1 and BP2. For the
case of BP3, the active-sterile neutrino mass difference is so large that
resonance does not occur for $E_{4}=30\,$MeV unless $\lambda$ is sizable
$\gtrsim 10^{-3}$. The corresponding $\Delta_{\rm Res}/\lambda_{\nu}$ is much
larger than 1.

The non-adiabatic transition effect is important only if neutrino passes
through the resonance region while it remains free streaming, i.e.,

$\Delta_{\rm Res}\ll\lambda_{\nu}\ .$ (20)

In order to assess the relevance of this effect, we plot the ratio
$\Delta_{\rm Res}/\lambda_{\nu}$ in the plane of $\lambda$ versus $m_{\phi}$,
as shown in Fig. 6. We first choose a typical SN neutrino energy
$E_{4}=30\,$MeV and solve for the radius $r$ where the resonant condition
$\Delta\cos 2\theta=V$ is satisfied. We then use Eqs. (18) and (19) to obtain
the corresponding $\Delta_{\rm Res}$ and $\lambda_{\nu}$. As we can see in the
figure, the hierarchy in Eq. (20) only occurs for very small values of the
coupling $\lambda$. The $\lambda\to 0$ limit corresponds to the minimal
S$\nu$DM model where neutrino only participates in weak interactions. In that
limit, it has been concluded that the feedback effect must be taken into
account, which substantially weakens the SN cooling constraint Suliga  _et
al._ (2019).

In the region with $\lambda\gtrsim 10^{-5}-10^{-4}$, we always find
$\Delta_{\rm Res}/\lambda_{\nu}>1$. Physically, the introduction of neutrino
self-scattering mediated by on-shell $\phi-$exchange becomes more frequent in
the larger $\lambda$ region which can strongly reduce the neutrino free-
streaming length. As a result, the non-adiabatic resonant transition is
interrupted by the frequent self-scatterings. The new SN constraints presented
in Fig. 4 and 7 (see below) belong to this region. This confirms that the
collisional cooling mechanism taken into account in Sec. II.3 already captures
the dominant cooling effect of CCSN in this model.

##  IV Discussion and Conclusion

keV-mass sterile neutrino dark matter (${\rm S\nu DM}$) can be produced
efficiently in the early Universe through novel secret neutrino self-
interactions. However, the same interactions can also enhance the production
of ${\rm S\nu DM}$ in the cores of core-collapse supernovae (CCSN), resulting
in faster cooling of the SN. This can lead to a shortened neutrino emission
period, which will conflict with observations from SN1987A. This work has
computed the constraints on the ${\rm S\nu DM}$ parameter space using SN
cooling arguments.

Figure 7: Summary of various probes of neutrino self-interaction and the
S$\nu$DM relic target. Existing constraints and future reaches in the figure
are adopted from a recent Snowmass exercise Berryman  _et al._ (2022) and a
very recent work by Chang et al Chang  _et al._ (2022). The S$\nu$DM relic
target is for BP2 defined in Eq. (15), with $m_{4}=21\,$keV and
$\sin^{2}2\theta=1.4\times 10^{-13}$. In the lower-left region
($m_{4}<m_{\phi}\ll 1\,$MeV), the black relic curve can stick out of the gray
shaded region labelled as “Existing Constraints” (mostly from BBN near the
lower edge) only for S$\nu$DM mass below $\sim 5$ keV Kelly  _et al._ (2021).

Considering scalar-mediated neutrino self-interactions, we computed in detail
the luminosity of ${\rm S\nu DM}$ produced inside the SN core, including the
re-scattering effects of the sterile neutrino on its way out. While
calculating the thermal contributions to the neutrino scattering, and the
potential, we included the presence of a non-zero chemical potential, which is
particularly relevant for studying $\nu_{e}$ self-interactions. We found that,
in the presence of strong neutrino self-interactions, the dominant production
mode of ${\rm S\nu DM}$ is through collisional effects. In contrast,
production via non-adiabatic transitions can be neglected. A back-of-the-
envelope estimate shows that the SN constraints are the strongest for
couplings $\sim\mathcal{O}(10^{-3})$, where the thermal potential generated by
the new interactions for $\sim\mathcal{O}({\rm MeV})$ mass mediator is
comparable to that generated by the weak interactions. Interestingly, as we go
to larger couplings, the cooling effect becomes unimportant due to a
suppression of the mixing angle in matter. Our results, shown for three
benchmark points, indicate that cooling constraints from SN allow us to set
novel bounds on the scalar mass-coupling parameter space, which are highly
complementary to laboratory and cosmological probes. We use Fig. 7 to
summarize the interplay among all these frontiers.

One notable assumption in this work is neglecting the dynamical feedback
effect on the effective matter potential, due to the production of the sterile
neutrinos. This effect, studied in detail in Suliga  _et al._ (2019, 2020) can
relax some of the bounds. However, this is most relevant for mixing angles
much larger than what this work considers. A detailed treatment feedback
effects is beyond the scope of this work.

In conclusion, keV-mass ${\rm S\nu DM}$, produced through secret neutrino
self-interactions, can strongly impact the dynamics of a CCSN. Our findings
indicate that a significant part of the model parameter space is susceptible
to faster cooling of the SN, thereby reducing the neutrino luminosity. Using
this, one can set new SN cooling constraints on the parameter space that can
explain the origin of ${\rm S\nu DM}$ relic density. This interplay is more
significant for ${\rm S\nu DM}$ heavier than $\sim 10\,$keV, and when the
active-sterile neutrino mixing and new self-interaction occur through the
$\nu_{e}$ flavor. Our result contributes to the goal of probing the entire
novel neutrino self-interaction window for the S$\nu$DM relic target, drawing
efforts from astrophysical, cosmological and particle physics experiments.

## Acknowledgements

We thank Georg Raffelt and Andrea Caputo for useful discussions. Y.C., D.T.
and Y.Z. are supported by the Arthur B. McDonald Canadian Astroparticle
Physics Research Institute. W.T. is supported by the National Science
Foundation under Grant No. PHY-2013052. M.S and W.T. would like to express
special thanks to the Mainz Institute for Theoretical Physics (MITP) of the
Cluster of Excellence PRISMA+ (Project ID 39083149), for its hospitality and
support. W.T. and Y.Z. are grateful to the organizers of the Pollica Summer
Workshop supported by the Regione Campania, Università degli Studi di Salerno,
Università degli Studi di Napoli “Federico II”, i dipartimenti di Fisica
“Ettore Pancini” and “E R Caianiello”, and Istituto Nazionale di Fisica
Nucleare; and to the townspeople of Pollica, IT, for their generosity and
hospitality. This research was also supported by the Munich Institute for
Astro, -Particle and BioPhysics (MIAPbP) which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s
Excellence Strategy – EXC-2094 – 390783311.

##  Appendix A $\phi$ mediated thermal potential

The exchange of a neutrinophilic scalar $\phi$ at loop level can a generate
thermal potential for neutrino and antineutrino. The contribution comes from
the neutrino self-energy diagram shown in Fig. 3. Using the imaginary time
formalism, it can be calculated as Quiros (1999)

$-i\Sigma(p)=(i\lambda)(i\lambda^{*})\int\frac{d^{4}k}{(2\pi)^{4}}iS_{F}(k)iD_{F}(p-k)\
,$ (21)

where

$\begin{split}S_{F}(p)&=\cancel{p}\left\\{\frac{1}{p^{2}}+2\pi
i\delta(p^{2})\left[\Theta(p^{0})n_{f}^{-}(p^{0})+\Theta(-p^{0})n_{f}^{+}(p^{0})\right]\right\\}\
,\\\ D_{F}(p)&=\frac{1}{p^{2}-m_{\phi}^{2}}-2\pi
i\delta(p^{2}-m_{\phi}^{2})\left[\Theta(p^{0})n_{b}^{-}(p^{0})+\Theta(-p^{0})n_{b}^{+}(p^{0})\right]\
,\end{split}$ (22)

and $\Theta$ is a step function, $\delta$ is a Dirac delta function. We work
in the rest frame of the plasma fluid such that

$n_{f}^{\mp}(p^{0})=\frac{1}{e^{|p^{0}|\mp\mu_{f}}+1},\quad
n_{b}^{\mp}(p^{0})=\frac{1}{e^{|p^{0}|\mp\mu_{b}}-1}\ .$ (23)

Here $f$ stands for neutrino or antineutrino, and $b$ stands for the mediator
$\phi$ or $\phi^{*}$. The temperature-dependent part of the self-energy takes
the form

$\begin{split}\Sigma(p,T)&=-|\lambda|^{2}\int
dk^{0}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\left\\{\frac{\cancel{k}}{(k-p)^{2}-m_{\phi}^{2}}\delta(k^{2})\left[\Theta(k^{0})n_{f}^{-}(k^{0})+\Theta(-k^{0})n_{f}^{+}(k^{0})\right]\right.\\\
&+\left.\frac{\cancel{k}}{k^{2}}\delta\left((k-p)^{2}-m_{\phi}^{2}\right)\left[\Theta(k^{0}-p^{0})n_{b}^{-}(k^{0}-p^{0})+\Theta(-k^{0}+p^{0})n_{b}^{+}(k^{0}-p^{0})\right]\right\\}\
.\end{split}$ (24)

Using the $\delta$-functions and the on-shell condition of the external
neutrino momentum $p^{2}=0$, we can complete the $k^{0}$ integral and the
angular parts of the $\vec{k}$ integral for each term in Eq. (24),

$\displaystyle\Sigma_{n_{f}^{-}}$
$\displaystyle=\frac{|\lambda|^{2}}{16\pi^{2}}\gamma^{0}\int
dk\left(\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}}{m_{\phi}^{2}+4Ek}+\frac{2k}{E}\right)n_{f}^{-}(k)\
,$ (25) $\displaystyle\Sigma_{n_{f}^{+}}$
$\displaystyle=\frac{|\lambda|^{2}}{16\pi^{2}}\gamma^{0}\int
dk\left(-\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}}{m_{\phi}^{2}-4Ek}+\frac{2k}{E}\right)n_{f}^{+}(k)\
,$ (26) $\displaystyle\Sigma_{n_{b}^{-}}$
$\displaystyle=\frac{|\lambda|^{2}}{16\pi^{2}}\gamma^{0}\int\frac{kdk}{\epsilon_{1}}\left(\frac{2k}{E}-\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}+2Ek+2E\epsilon_{1}}{m_{\phi}^{2}-2Ek+2E\epsilon_{1}}\right)n_{b}^{-}(\epsilon_{1})\
,$ (27) $\displaystyle\Sigma_{n_{b}^{+}}$
$\displaystyle=\frac{|\lambda|^{2}}{16\pi^{2}}\gamma^{0}\int\frac{kdk}{\epsilon_{1}}\left(\frac{2k}{E}-\frac{m_{\phi}^{2}}{2E^{2}}\log\frac{m_{\phi}^{2}+2Ek-2E\epsilon_{1}}{m_{\phi}^{2}-2Ek-2E\epsilon_{1}}\right)n_{b}^{+}(\epsilon_{1})\
,$ (28)

where $k=|\vec{k}|$, $\epsilon_{1}=\sqrt{k^{2}+m_{\phi}^{2}}$ and $E$ is the
timelike component of the external momentum $p^{\mu}$. The thermal potential
is related to the self-energy as Weldon (1982)

$V_{\phi}=\frac{1}{4E}{\rm Tr}\left(\cancel{p}{\rm Re}\Sigma\right)\ .$ (29)

Adding up the pieces of contribution from Eqs. (25), (26), (27), (28) gives
the total $\phi$ mediated potential presented in Eq. (9).

##  Appendix B The total luminosity integrals

We present, in this section, some details of the derivation of the total
cooling luminosity Eq. (14). Starting from Eq. (10), where the luminosity is
originally defined, we first complete the final-state phase-space integrals
(F.S.I.) over $\vec{p}_{3}$ and $\vec{p}_{4}$. Because of the extra factor
$E_{4}$ in the integrand, the final state integral is not a Lorentz invariant
quantity. Here, we have to work in a general reference frame where the center
of mass moves with an overall momentum $\vec{P}=\vec{p}_{1}+\vec{p}_{2}$. We
use the $\delta^{4}(\vec{p}_{1}+\vec{p}_{2}-\vec{p}_{3}-\vec{p}_{4})$ function
to reduce the number of integrals down to one,

$\begin{split}{\rm
F.S.I.}&=\int\frac{d^{3}\vec{p}_{3}}{(2\pi)^{3}2E_{3}}\int\frac{d^{3}\vec{p}_{4}}{(2\pi)^{3}2E_{4}}(2\pi)^{4}\delta^{4}(\vec{p}_{1}+\vec{p}_{2}-\vec{p}_{3}-\vec{p}_{4})|\mathcal{M}|^{2}E_{4}e^{-\tau(E_{4},r)}\\\
&=\frac{4\pi\lambda^{2}m_{\phi}^{2}\delta(s-m_{\phi}^{2})}{|\vec{P}|}\int_{\frac{1}{2}(E_{1}+E_{2}-|\vec{P}|)}^{\frac{1}{2}(E_{1}+E_{2}+|\vec{P}|)}dE_{4}\sin^{2}\theta_{\rm
eff}(r,E_{4})E_{4}e^{-\tau(E_{4},r)}\ ,\end{split}$ (30)

where $|\mathcal{M}|^{2}$ is given in Eq. (13) and, in the last line,
$s=(p_{1}+p_{2})^{2}$ is a Mandelstam variable.

Before moving on, we rewrite the remaining $\delta$-function using the
kinematic relation $s=E_{1}^{2}+E_{2}^{2}+2E_{1}E_{2}\cos\theta_{12}$, where
$\theta_{12}$ is the relative angle between the three-momenta $\vec{p}_{1}$
and $\vec{p}_{2}$. This leads to

$\delta(s-m_{\phi}^{2})=\frac{1}{2E_{1}E_{2}}\delta\left(\cos\theta_{12}-1+\frac{m_{\phi}^{2}}{2E_{1}E_{2}}\right)\
.$ (31)

The on-shell condition for the intermediate $\phi$ particle also allows us to
derive, $|\vec{P}|=\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}$.

Among the initial state momentum integrals, the integral over the direction of
$\vec{p}_{1}$ is trivial which allows us to pin it along the $\hat{z}$ axis.
The integral over relative angle $\theta_{12}$ can then be completed with the
above $\delta$-function, Eq. (31). This allows us to finally simplify the
total luminosity to three energy integrals,

$\begin{split}L&=\int
d^{3}\vec{r}\int\frac{d^{3}\vec{p}_{1}}{(2\pi)^{3}}f_{\nu}(E_{1},r)\int\frac{d^{3}\vec{p}_{2}}{(2\pi)^{3}}f_{\nu}(E_{2},r)\frac{1}{4E_{1}E_{2}}{\rm
F.S.I.}\\\
&=\frac{\lambda^{2}m_{\phi}^{2}}{4\pi^{2}}\int_{0}^{4R_{c}}r^{2}dr\int_{0}^{\infty}dE_{1}f(E_{1},r)\int_{m_{\phi}^{2}/(4E_{1})}^{\infty}dE_{2}f(E_{2},r)\frac{1}{\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}}\\\
&\quad\times\int_{\frac{1}{2}\left(E_{1}+E_{2}-\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}\right)}^{\frac{1}{2}\left(E_{1}+E_{2}+\sqrt{(E_{1}+E_{2})^{2}-m_{\phi}^{2}}\right)}dE_{4}\sin^{2}\theta_{\rm
eff}(r,E_{4})E_{4}e^{-\tau(E_{4},r)}\ .\end{split}$ (32)

This is Eq. (14) that we use to proceed with the numerical analysis.

##  Appendix C Width of MSW resonance inside supernova

The Hamiltonian governing active-sterile neutrino oscillation in the SN takes
the form

$H=\begin{pmatrix}\Delta\sin^{2}\theta+V(r)&\Delta\sin\theta\cos\theta\\\
\Delta\sin\theta\cos\theta&\Delta\cos^{2}\theta\end{pmatrix}\ .$ (33)

It can be diagonalized with a unitary transformation

$H=\begin{pmatrix}\cos\theta_{m}&\sin\theta_{m}\\\
-\sin\theta_{m}&\cos\theta_{m}\end{pmatrix}\begin{pmatrix}E_{1}&0\\\
0&E_{2}\end{pmatrix}\begin{pmatrix}\cos\theta_{m}&-\sin\theta_{m}\\\
\sin\theta_{m}&\cos\theta_{m}\end{pmatrix}\equiv U^{\dagger}\hat{H}U\ ,$ (34)

where $E_{1},E_{2}$ and $\theta_{m}$ are functions of $r$, and satisfy

$\displaystyle\tan^{2}2\theta_{m}(r)$
$\displaystyle=\frac{\Delta^{2}\sin^{2}2\theta}{(\Delta\cos
2\theta-V(r))^{2}}\ ,$ (35) $\displaystyle\Delta E(r)=|E_{1}-E_{2}|$
$\displaystyle=\sqrt{\Delta^{2}\sin^{2}2\theta+(\Delta\cos 2\theta-V)^{2}}\ .$
(36)

We first consider an ultra-relativistic neutrino propagating in the radial
direction of the supernova, with $dr=cdt$. The Schrödinger equation takes the
form

$i\hbar c\frac{\partial}{\partial r}\psi=H\psi=U^{\dagger}\hat{H}U\psi\ .$
(37)

Defining $\psi_{m}=U\psi$, the equation for $\psi_{m}$ is

$i\hbar c\frac{\partial}{\partial r}\psi_{m}=\hat{H}\psi_{m}-i\hbar
c\left(U\frac{\partial U^{\dagger}}{\partial r}\right)\psi_{m}\ .$ (38)

In this new basis, although $\hat{H}$ is a diagonal matrix, the additional
term proportional to $U\partial U^{\dagger}/\partial r$ contributes an off-
diagonal element

$\Delta\hat{H}_{12}=-i\hbar c\left(U\frac{\partial U^{\dagger}}{\partial
r}\right)_{12}=i\hbar c\theta_{m}^{\prime}\ .$ (39)

Hereafter ′ means $\partial/\partial r$.

Using Eq. (35), we obtain

$\theta_{m}^{\prime}=\frac{1}{2}\sin 2\theta_{m}\cos
2\theta_{m}\frac{V^{\prime}}{\Delta\cos 2\theta-V}\ .$ (40)

Near the MSW resonance, we have

$\displaystyle\theta_{m}$ $\displaystyle\simeq\pi/4\ ,$ (41)
$\displaystyle\sin 2\theta_{m}$ $\displaystyle\simeq 1\ ,$ (42)
$\displaystyle\cos 2\theta_{m}$ $\displaystyle\simeq\frac{\Delta\cos
2\theta-V}{\Delta\sin 2\theta}\to 0\ .$ (43)

These lead to

$\theta_{m}^{\prime}\simeq\frac{V^{\prime}}{2\Delta\sin
2\theta}\simeq\frac{1}{2\tan 2\theta}\frac{V^{\prime}}{V}\ .$ (44)

The width of resonance is introduced to characterize the importance of
$\Delta\hat{H}_{12}$,

$\Delta_{\rm Res}\equiv\frac{\hbar
c}{|\Delta\hat{H}_{12}|}=\frac{1}{|\theta_{m}^{\prime}|}=2\tan
2\theta\left|\frac{V^{\prime}}{V}\right|^{-1}\ .$ (45)

The regular adiabatic condition corresponds to $|\Delta\hat{H}_{12}|\ll\Delta
E(r)$. In the presence of frequent hard scattering among active neutrinos,
non-adiabatic transition is also negligible if $\Delta_{\rm
Res}\gg\lambda_{\nu}$, where $\lambda_{\nu}=1/\Gamma$ is the neutrino free
streaming length.

For neutrino traveling along non-radial directions, the variations of
background temperature and densities in the CCSN are slower, thus
$|\theta_{m}^{\prime}|$ is reduced, leading to a larger $\Delta_{\rm Res}$. In
that case, it would be safer to neglect non-adiabatic transitions.

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