# On the Effects of Quantum Decoherence in a Future Supernova Neutrino
Detection

Marcos V. dos Santos  Universidade Estadual de Campinas, Instituto de Física
Gleb Wataghin, R. Sérgio Buarque de Holanda, 777, Brazil  Pedro C. de Holanda
<EMAIL_ADDRESS>Universidade Estadual de Campinas, Instituto de Física
Gleb Wataghin, R. Sérgio Buarque de Holanda, 777, Brazil  Pedro Dedin Neto
<EMAIL_ADDRESS>Universidade Estadual de Campinas, Instituto de Física
Gleb Wataghin, R. Sérgio Buarque de Holanda, 777, Brazil  Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  Ernesto Kemp
<EMAIL_ADDRESS>Universidade Estadual de Campinas, Instituto de Física
Gleb Wataghin, R. Sérgio Buarque de Holanda, 777, Brazil  Gran Sasso Science
Institute - GSSI, L’Aquila (IT)

###### Abstract

Quantum decoherence effects in neutrinos, described by the open quantum
systems formalism, serve as a gateway to explore potential new physics,
including quantum gravity. Previous research extensively investigated these
effects across various neutrino sources, imposing stringent constraints on
spontaneous loss of coherence. In this study, we demonstrate that even within
the Supernovae environment, where neutrinos are released as incoherent states,
quantum decoherence could influence the flavor equipartition of $3\nu$ mixing.
Additionally, we examine the potential energy dependence of quantum
decoherence parameters ($\Gamma=\Gamma_{0}(E/E_{0})^{n}$) with different power
laws ($n=0,2,5/2$). Our findings indicate that future-generation detectors
(DUNE, Hyper-K, and JUNO) can significantly constrain quantum decoherence
effects under different scenarios. For a Supernova located 10 kpc away from
Earth, DUNE could potentially establish $3\sigma$ bounds of $\Gamma\leq
6.2\times 10^{-14}$ eV in the normal mass hierarchy (NH) scenario, while
Hyper-K could impose a $2\sigma$ limit of $\Gamma\leq 3.6\times 10^{-14}$ eV
for the inverted mass hierarchy (IH) scenario with $n=0$ — assuming no energy
exchange between the neutrino subsystem and non-standard environment
($[H,V_{p}]=0$). These limits become even more restrictive for a closer
Supernova. When we relax the assumption of energy exchange ($[H,V_{p}]\neq
0$), for a 10 kpc SN, DUNE can establish a $3\sigma$ limit of $\Gamma_{8}\leq
4.2\times 10^{-28}$ eV for NH, while Hyper-K could constrain $\Gamma_{8}\leq
1.3\times 10^{-27}$ eV for IH ($n=0$) with $2\sigma$, representing the most
stringent bounds reported to date. Furthermore, we examine the impact of
neutrino loss during propagation for future Supernova detection.

###### Contents

  1. I Introduction
  2. II Quantum decoherence effects in supernova neutrinos
    1. II.1 Formalism
    2. II.2 Selected Models
  3. III Methodology and simulation
    1. III.1 Factorization of the dynamics
    2. III.2 Exploring a future SN-$\nu$ detection
    3. III.3 Role of Earth matter effects
  4. IV Future limits on quantum decoherence
    1. IV.1 MSC${}^{\not}{\epsilon}$
    2. IV.2 MSCϵ
    3. IV.3 Neutrino loss
  5. V Neutrino mass hierarchy measurement
  6. VI Conclusions
  7. A Decoherence inside the SN and matter effects
  8. B Tables with QD bounds

##  I Introduction

Since the supernova SN1987A, the expectation for the next supernova (SN)
neutrino detection has stimulated a number of works proposing tests on new
physics in our galaxy, making this event a promising natural laboratory for
neutrino physics.

Once $\sim 1$ galactic SN is expected per century Rozwadowska et al. (2021),
the next event holds the opportunity to break through many aspects of neutrino
physics, with capabilities of next-generation detectors, such as DUNE Acciarri
et al. (2016); Abi et al. (2018, 2021), Hyper-Kamiokande (HK) Abe et al.
(2018) and JUNO An et al. (2016), leading to a sensible future measurement,
increasing the number of neutrino events from the current few dozen to tens of
thousands or more in a SN explosion 10 kpc away from Earth. A typical Core-
Collapse SN (CCSN) undergoes three main emission phases to be known (see
Mirizzi et al. (2016) for a review): neutronization burst, where a high amount
of $\nu_{e}$ is emitted given a rate of $e^{-}$ capture in the first $\sim 30$
ms after core bounce; accretion, where progenitor mass infall and a high
luminosity are expected during roughly $\sim 1$ s; and cooling, a thermal
phase where a proto-neutron star cools down via neutrino emission, with $\sim
10$ s of duration.

With the possible future sensitivity and increasing sophistication in SN
neutrino simulations Tamborra et al. (2017); Müller et al. (2017); Gar (2022),
a precise description of standard neutrino evolution until Earth is been
pursued. However, in a SN environment, collective oscillations led by
$\nu-\nu$ interactions are a source of high uncertainties, since a definitive
solution for the $\nu$ equation of motion has not been achieved, even with
many ongoing developments in the topic Tamborra and Shalgar (2021). One
critical remark is that for the three mentioned SN emission phases, collective
oscillations are expected to play an important role only in accretion and
cooling, with no significant impact on neutronization burst, given the large
excess of $\nu_{e}$ over other flavors, turning it in a promising environment
to test new physics.

In SN neutrino-mixing, if we disregard collective effects, with the only
relevant neutrino interaction being the MSW matter effect, the neutrino flux
that comes out of the SN can be treated as an incoherent sum of mass states,
and no oscillation is expected111Given the indistinguishability of $\nu_{\mu}$
and $\nu_{\tau}$ ($\bar{\nu}_{\mu}$ and $\bar{\nu}_{\tau}$) in the detection,
they are generally classified as $\nu_{x}$ ($\bar{\nu}_{x}$) in the
literature.. Since $\nu_{\alpha}$ is generated as a mass state in matter
$\nu_{i}^{m}$, it leaves the SN as a mass state in vacuum $\nu_{i}$ (for an
adiabatic conversion in the SN) until reaching Earth. Despite this expected
incoherence, neutrinos coming from a SN could be affected by quantum
decoherence. In this work, we show the impact of quantum decoherence, or the
neutrino evolution from pure to mixed states given a coupling to the
environment, in the case of a future SN neutrino detection.

There are different possible sources of decoherence in neutrino evolution,
such as wave packet decoherence, that comes from different group velocities of
neutrino mass states disentangling the respective wave packets Kersten and
Smirnov (2016); Akhmedov et al. (2017); Akhmedov and Smirnov (2022), or even
Gaussian averaged neutrino oscillation given by uncertainty in energy and path
length Ohlsson (2001). The underlying physics in this work is of a different
type and refers to effects induced by propagation in a non-standard
environment generated by beyond Standard Model physics, and the term
decoherence used in this work refers to the latter.

The idea of inducing pure elementary quantum states into mixed ones was
originally established by Hawking Hawking (1975) and Bekenstein Bekenstein
(1975) and discussed by a number of subsequent works Hawking (1982); Ellis et
al. (1984); Wald (1994); Unruh and Wald (1995); Ellis et al. (1999), being
attributed to quantum (stochastic) fluctuations of space-time background given
quantum gravity effects. Many authors have given a physical interpretation on
the impact of such stochastic quantum gravitational background in neutrino
oscillations Lisi et al. (2000); Alexandre et al. (2008); Mavromatos (2009);
Dvornikov (2019); Stuttard and Jensen (2020); Luciano and Blasone (2021);
Stuttard (2021); Hellmann et al. (2022a, b); Ettefaghi et al. (2022), with
expected decoherence being well described by open quantum systems formalism
through GKSL (Gorini–Kossakowski–Sudarshan–Lindblad) master equation. In
particular, in Stuttard and Jensen (2020), the authors provided a simple and
interesting interpretation of physical scenarios for specific forms of GKSL
equation, then we use a similar terminology along this manuscript to guide our
choices in the analysis.

Phenomenological studies designed to impose bounds on neutrino coupling to the
environment through open quantum systems formalism were investigated in
atmospheric Lisi et al. (2000); Coloma et al. (2018), accelerator Farzan et
al. (2008); Oliveira et al. (2014); Oliveira (2016); Coelho et al. (2017);
Coelho and Mann (2017); Carrasco et al. (2019); Carpio et al. (2019); Gomes et
al. (2019); Romeri et al. (2023), reactor Gomes et al. (2017); Romeri et al.
(2023), and solar Coloma et al. (2018); de Holanda (2020); Farzan and Schwetz
(2023) neutrinos with different approaches. Only upper limits over quantum
decoherence parameters were obtained up to now.

This manuscript is structured as follows: in Section II we show the quantum
decoherence formalism, introducing the models to be investigated. In Section
III we discuss the methods to factorize the neutrino evolution and how to use
them to impose bounds on quantum decoherence with a future SN detection. We
also discuss the role of Earth matter effects. Our results are presented in
Section IV and in Section V we discuss how quantum decoherence could affect
the neutrino mass ordering determination. Finally, in Section VI we present
our conclusions.

##  II Quantum decoherence effects in supernova neutrinos

In this section, we devote ourselves to revisiting quantum decoherence
formalism in neutrino mixing and show the impacts on the (already) incoherent
SN neutrino fluxes.

###  II.1 Formalism

Considering the effects of quantum decoherence, we can write the GKSL equation
in propagation (mass) basis in vacuum Gorini et al. (1976); Lindblad (1976)

$\frac{d\rho}{dt}=-i[H,\rho]+\mathcal{D}(\rho)$ (II.1)

where $\mathcal{D}(\rho)=\sum_{p}^{N^{2}-1}(V_{p}\rho
V_{p}^{\dagger}-\frac{1}{2}\\{V_{p}^{\dagger}V_{p},\rho\\})$ is a dissipation
term, representing the neutrino subsystem coupling to the environment. If
(II.1) is a general equation of motion to describe $\nu$ propagation and a
non-standard effect induces a non-null $\mathcal{D}(\rho)$, we require an
increase of von Neumann entropy in the process, which can be achieved imposing
$V_{p}=V_{p}^{\dagger}$ Benatti and Narnhofer (1988). It is also possible to
write the dissipation term at the r.h.s. of (II.1) expanding in the
appropriated group generators as
$\mathcal{D}(\rho)=\mathcal{D}(\rho)_{\mu}\lambda^{\mu}=D_{\mu\nu}\rho^{\nu}\lambda^{\mu}$,
in which $\lambda^{\mu}$ are the generators of SU($N$) for a system of $N$
neutrinos families. In fact, the same procedure can be done in the Hamiltonian
term of (II.1) in order to get a Lindbladian operator
$\mathcal{L}=-2(H_{\mu\nu}+D_{\mu\nu})$, leading to:

$\ket{\dot{\rho}}=-2\mathcal{L}\ket{\rho}$ (II.2)

that operates in a “vectorized” density matrix $\ket{\rho}$ with dimension
$N^{2}$ (where $N$ is the number of levels of the system). In 3 neutrino
mixing, $\ket{\rho}$ has dimension 9 and $\mathcal{L}$ is a $9\times 9$
matrix.

One of the advantages of this formalism is that, despite a lack of
understanding about the microscopic phenomena we are interested to model, we
are able to infer the resulting damping effects by properly parameterizing
$\mathcal{D}(\rho)$ (or more specifically $D_{\mu\nu}$) in a generic way222For
some forms of $\mathcal{D}(\rho)$ derived from first principles, see Nieves
and Sahu (2019, 2020).

$D=\begin{pmatrix}0&0&0&0&0&0&0&0&0\\\
0&-\gamma_{1}&\beta_{12}&\beta_{13}&\beta_{14}&\beta_{15}&\beta_{16}&\beta_{17}&\beta_{18}\\\
0&\beta_{12}&-\gamma_{2}&\beta_{23}&\beta_{24}&\beta_{25}&\beta_{26}&\beta_{27}&\beta_{28}\\\
0&\beta_{13}&\beta_{23}&-\gamma_{3}&\beta_{34}&\beta_{35}&\beta_{36}&\beta_{37}&\beta_{38}\\\
0&\beta_{14}&\beta_{24}&\beta_{34}&-\gamma_{4}&\beta_{45}&\beta_{46}&\beta_{47}&\beta_{48}\\\
0&\beta_{15}&\beta_{25}&\beta_{35}&\beta_{45}&-\gamma_{5}&\beta_{56}&\beta_{57}&\beta_{58}\\\
0&\beta_{16}&\beta_{26}&\beta_{36}&\beta_{46}&\beta_{56}&-\gamma_{6}&\beta_{67}&\beta_{68}\\\
0&\beta_{17}&\beta_{27}&\beta_{37}&\beta_{47}&\beta_{57}&\beta_{67}&-\gamma_{7}&\beta_{78}\\\
0&\beta_{18}&\beta_{28}&\beta_{38}&\beta_{48}&\beta_{58}&\beta_{68}&\beta_{78}&-\gamma_{8}\\\
\end{pmatrix},$ (II.3)

in 3 neutrino mixing. Although it is not explicit, the entries in matrix
(II.3) can be directly related to the coefficients of expansion of $V_{p}$ in
the generators of SU(3), or $\gamma,\beta=f(v_{p})$, with $v_{p}$ coming from
$V_{p}={v_{p}}_{\mu}\lambda^{\mu}$. Note that the null entries in the first
column of (II.3) are given by the hermiticity of $V_{p}$, which also enables
rewriting the dissipation term as
$\mathcal{D}(\rho)=\frac{1}{2}\sum_{p}^{N^{2}-1}[[V_{p},\rho],V_{p}]$, showing
that terms proportional to identity in the SU(3) expansion vanish, making the
first line of (II.3) also null. It is important to note that the parameters
used to define $D_{\mu\nu}$ are not all independent. They are related to each
other in order to ensure complete positivity, which is a necessary condition
for a quantum state to be physically realizable Benatti and Floreanini (1997,
2000); Carrasco et al. (2019) (see Carrasco et al. (2019) for a set of
relations in a 3-level system).

However, it is not viable to investigate this general format of (II.3) given
the number of parameters. Therefore, in this work, we restrict ourselves to
cases in which $D$ is diagonal as in de Holanda (2020), in order to capture
the effects of interest arising from QD. We tested a non-diagonal version of
$D$ using complete positivity relations and our results are not significantly
affected.

In the context of supernova neutrinos, the neutrino propagates a large
distance inside the supernova ($\sim 10^{8}$ km), then we also investigate the
impact of QD combined with SN matter effects. A possible procedure to cross-
check it is by rotating eq. (II.1) to flavor basis, where the Hamiltonian can
be summed to an MSW potential, i.e. $H_{f}=H_{f}^{\text{vac}}+V_{W}$. However,
as it will be more clear in Section III.1, the probability we are interested
in is between mass eigenstates on both matter and vacuum, which can be
accomplished by diagonalizing the Hamiltonian in flavor basis using a proper
transformation.

###  II.2 Selected Models

Since we analyse diagonal versions of (II.3), $\beta_{\mu\omega}=0$ for all
$\mu$ and $\omega$. In works such as Oliveira and Guzzo (2010); de Holanda
(2020) it is shown that quantum decoherence can give rise to two disentangled
effects when the evolution occurs in vacuum: the pure decoherence, where a
coherent state becomes incoherent along propagation; and the relaxation
effect, responsible to lead the ensemble to a maximal mixing. As decoherence
effects on SN neutrinos are suppressed due to matter effects on the mixing
angle and long propagation lengths333If neutrinos are only affected by MSW
effect, it is possible for $\nu_{\mu}$ and $\nu_{\tau}$ oscillate to each
other. It generally does not affect the analysis of flavor conversion, once
they are indistinguishable in the detection, and therefore generally denoted
as $\nu_{x}$. However, as we will see in Section III, their creation in
coherent states changes one of the tested QD models here., we do not expect
pure decoherence effects to play any role in the propagation, being only
(possibly) affected by relaxation.

Figure 1: Coherence length ($L_{\text{coh}}=1/\gamma$) for values of $n$ in a
power law of decoherence coefficients $\gamma=\gamma_{0}(E/E_{0})^{n}$ for a
“natural” scale of quantum gravity, with $\xi_{\text{Planck}}=1$. The yellow
region corresponds to the solar system edge, while the blue region is the
Milky Way diameter, and the dashed grey line is to respect the observable
universe.

Up to this date, and to the authors’s best knowledge, there is no consistent
theory in which you can get the parameters of $D$ from quantum gravity, or
even if the parameters are constant. Different works Lisi et al. (2000);
Amelino-Camelia (2013); Coloma et al. (2018); Stuttard and Jensen (2020)
suggested the possibility of a dependency on energy as
$\gamma_{i}=\gamma_{0_{i}}(E/E_{0})^{n}$ motivated by quantum space-time
phenomenology, where $E_{0}$ is an arbitrary energy scale. In this work, we
chose $E_{0}=10$ MeV to match the energy scale of supernova neutrinos. As for
the energy dependence, we explore the scenarios with $n=0$ and $n=2$, given
that most of the works check this power law exponents for $\gamma_{i}$, which
enables us to compare SN limits to other sources (and works), and $n=5/2$,
well-motivated by the natural Planck scale for the SN energy range of $0-100$
MeV. By natural scale, we refer to
$\gamma_{0_{i}}=\xi_{\text{Planck}}/M_{\text{Planck}}^{n-1}$ with
$\xi_{\text{Planck}}\sim 1$ Anchordoqui et al. (2005); Stuttard and Jensen
(2020), making $\gamma_{0i}=\xi_{\text{Planck}}M_{\text{Planck}}^{1}$,
$\xi_{\text{Planck}}M_{\text{Planck}}^{-1}$, and
$\xi_{\text{Planck}}M_{\text{Planck}}^{-3/2}$ for our choices of $n=0,2$ and
5/2.

With dimensional analysis (which can be further justified when solving the
evolution equation), we expect that the effects of decoherence would show up
for distances larger than a coherence length, defined by
$L_{\text{coh}}=1/\gamma$. In Fig. 1 we show the expected coherence length for
these values of $n$. We see that if this “natural” scale holds, $n=0$ and 2
would be possibly ruled out by terrestrial and solar experiments, whereas for
$n=3$, $L_{\text{coh}}$ is out of the observable universe for the expected
SN-$\nu$ energy scale. For the mentioned values of $n$, we analyze the
following models:

Mass State Coupling (MSC): The neutrino mass basis is coupled to the
environment and the relaxation effect leads to maximal mixing. In 3-$\nu$
mixing, it means a 1/3 (equal) probability of detecting any state. In this
model, we test two possible scenarios related to energy conservation in the
neutrino subsystem:

  * $i$)

MSC${}^{\not}{\epsilon}$ ($[H,V_{p}]=0$): Here, the neutrino energy is
conserved for any non-standard mixing process in vacuum444In our notation, the
superscript symbol $\not{\epsilon}$ accounts to no exchange of energy to the
environment, while $\epsilon$ has the opposite meaning.. It means that
$V_{p}=\mathbf{v}_{3}\lambda_{3}+\mathbf{v}_{8}\lambda_{8}$, where
$\lambda_{\mu}$ are Gell-Mann matrices and
$\mathbf{v}_{\mu}=\sum_{p=1}^{8}{v_{p}}_{\mu}$, with $\mu$ ranging from 0 to 8
in the SU(3) expansion of $V_{p}$. To simplify the analysis we choose a
diagonal version of the dissipation term in (II.3) with a single parameter
$\Gamma$. Additionally, using complete positivity relations Carrasco et al.
(2019), we can find the special case of
$D=-\text{diag}(0,\Gamma,\Gamma,0,\Gamma/4,\Gamma/4,\Gamma/4,\Gamma/4,0)$,
with $\Gamma=\Gamma_{0}(E/10~{}{\rm MeV})^{n}$. The transition probabilities
amongst mass states in vacuum are null in this case. However, if we look at
the propagation inside the supernova layers, in a diagonalized basis of the
mass state in matter $P_{ij}^{m(\text{SN})}$, this probability could be non-
null for $i\neq j$, i.e. transitions between $\nu_{i}^{m}$ and $\nu_{j}^{m}$
are allowed and would change proportionally to $e^{-\Gamma}$. Therefore, the
coherence length to be investigated is the SN radius, and the matter effects
in addition to quantum decoherence would induce a maximal mixing inside the
SN. In Fig. 2 we show the transition probabilities of mass state in matter
basis calculated using the slab approach with a simulated SN density profile
from Garching group Gar (2022); Serpico et al. (2012), corresponding to a
progenitor of 40 $M_{\odot}$. More details about our solution are in Appendix
A. When the neutrino is released to vacuum, it is no longer affected by
quantum decoherence until detection. Since the length traveled inside the
Earth by the neutrino is much smaller then $L_{\text{coh}}^{\text{SN}}$, we do
not take the quantum decoherence in Earth matter into account in this specific
case, albeit standard non-adiabatic MSW effect could play a role. Note that
this regime essentially depends on $\nu$ matter effects in the SN.

Figure 2: Survival probabilities of mass state in matter basis inside the SN
for the MSC／ϵ model (no exchange of energy from neutrinos and environment in
vacuum) and $n=0$ (and then $\Gamma=\Gamma_{0}$). The SN matter density
profile used is from a Garching simulation of a 40 $M_{\odot}$ (LS180-s40.0)
progenitor Gar (2022); Serpico et al. (2012), shown in Fig. 22 in Appendix A.

  * $ii$)

MSCϵ ($[H,V_{p}]\neq 0$): In this model, we relax the above assumption,
allowing some exchange of $\nu$ energy with the “non-standard” environment. We
choose the most general diagonal version to the dissipation term from (II.3):
$D=-\text{diag}(0,\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5},\gamma_{6},\gamma_{7},\gamma_{8})$.
In Stuttard and Jensen (2020), this choice of $D$ is intrinsically related to
mass state selected scenario to be impacted by quantum gravitational effects.
To quantify the effects of this model, we solve analytically (II.1) to get the
probabilities of interest in mass basis in vacuum555The expected (adiabatic
MSW) solution for the probabilities is a Kronecker delta, i.e.
$P_{ij}=\delta_{ij}$.:

$\begin{split}&P_{11}=\frac{1}{3}+\frac{1}{2}e^{-\gamma_{3}x}+\frac{1}{6}e^{-\gamma_{8}x}\\\
&P_{12}=\frac{1}{3}-\frac{1}{2}e^{-\gamma_{3}x}+\frac{1}{6}e^{-\gamma_{8}x}\\\
&P_{13}=\frac{1}{3}-\frac{1}{3}e^{-\gamma_{8}x}\\\
&P_{33}=\frac{1}{3}+\frac{2}{3}e^{-\gamma_{8}x}\end{split}\quad\quad\begin{split}&P_{22}=P_{11}\\\
&P_{23}=P_{13}\,\,,\end{split}$ (II.4)

with $x$ as the propagated distance. For other possible probabilities on this
basis, we use $P_{ij}=P_{ji}$. It should be noted that on this basis the
probabilities depend only on $\gamma_{3}$ and $\gamma_{8}$. The reason is that
when solving the set of differential equations in (II.2), the equations
corresponding to $\gamma_{3}$ and $\gamma_{8}$, i.e. $\mathcal{L}_{3\nu}$ and
$\mathcal{L}_{8\nu}$ are the only decoupled ones, independent of Hamiltonian
terms.

If we look at $\gamma_{i}$ parameters in terms of $\mathbf{v}_{\mu}$
coefficients of the SU(3) expanded $V_{p}$ we find

$\begin{split}\gamma_{3}&=\mathbf{v}_{1}^{2}+\mathbf{v}_{2}^{2}+\frac{\mathbf{v}_{4}^{2}}{4}+\frac{\mathbf{v}_{5}^{2}}{4}+\frac{\mathbf{v}_{6}^{2}}{4}+\frac{\mathbf{v}_{7}^{2}}{4}\\\
\gamma_{8}&=\frac{3\mathbf{v}_{4}^{2}}{4}+\frac{3\mathbf{v}_{5}^{2}}{4}+\frac{3\mathbf{v}_{6}^{2}}{4}+\frac{3\mathbf{v}_{7}^{2}}{4}\,.\end{split}$
(II.5)

Equation (II.5) shows that $\gamma_{3}$ and $\gamma_{8}$ are not independent.
In order to compare our results to solar limits de Holanda (2020), we can use
the same notation to define:

$\begin{split}\Gamma_{3}&=\mathbf{v}_{1}^{2}+\mathbf{v}_{2}^{2}\\\
\Gamma_{8}&=\frac{3\mathbf{v}_{4}^{2}}{4}+\frac{3\mathbf{v}_{5}^{2}}{4}+\frac{3\mathbf{v}_{6}^{2}}{4}+\frac{3\mathbf{v}_{7}^{2}}{4}\end{split}$
(II.6)

leading to $\gamma_{3}=\Gamma_{3}+\Gamma_{8}/3$ and $\gamma_{8}=\Gamma_{8}$,
resulting in pure (independent) relaxation $\Gamma$ parameters, that will be
the ones effectively inducing the maximal admixture in this scenario. The
energy dependence is explicitly written as $\Gamma_{i}=\Gamma_{0i}(E/10~{}{\rm
MeV})^{n}$ with $i=\\{3,8\\}$. Note that the effective distance of this
particular case is the total neutrino propagation, i.e. vacuum propagation is
also affected and it can be split into the regime in the SN and outside its
surface until Earth, or $L=L^{\text{SN}}+L^{\text{Vac}}$. Similarly as in
$i$), we solve the probabilities associated with possible transitions in
supernova layers only numerically. However, as we discuss in Section III.1,
given that $L^{\text{Vac}}\gg L^{\text{SN}}$, the approximation of $L\sim
L^{\text{Vac}}$ is assumed in our calculations.

Neutrino Loss: As mentioned in Stuttard and Jensen (2020), it is possible to
have a scenario with neutrino loss, where neutrinos are captured by effects of
quantum gravity during propagation, and re-emitted to a different direction,
never reaching the detector at Earth. In this picture, the authors made a
choice of $D_{00}\neq 0$. Looking at the most general form of
$\mathcal{D}(\rho)$, it is possible to say that this choice is completely out
of open quantum systems formalism, i.e. naturally $\mathcal{D}(\rho)_{0\mu}=0$
when the master equation (II.1) is assumed to describe the evolution of the
reduced quantum system, with trace-preserving all times. Even though, to
explore such an interesting physical situation, we test this non-unitary case
that matches the choice $\gamma_{i}=\gamma$ with $i$ from 1 to 8, then
$D=-\text{diag}(\gamma,\gamma,\gamma,\gamma,\gamma,\gamma,\gamma,\gamma,\gamma)$,
with $\gamma=\gamma_{0}(E/10~{}{\rm MeV})^{n}$. The solution of (II.1) gives:

$\begin{split}&P_{ii}=e^{-\gamma x}\\\ &P_{ij}=0\end{split}$ (II.7)

for any $i,j$ from 1 to 3 with $i\neq j$. Note that in this result, in
contradiction to conventional unitary models, one state does not go to
another, i.e. $\sum_{i}P_{ij}\neq 1$, once neutrinos are lost along the way.

In the solutions of the equation of motion shown above, we absorbed a factor
of 2 in the quantum decoherence parameters, i.e.
$-2\gamma_{i}\rightarrow-\gamma_{i}$, with no loss of generality, since what
matters in our results is the intensity of a deviation from a standard
scenario.

##  III Methodology and simulation

To test the QD models discussed in the context of a future SN detection, we
use the neutrino flux coming from supernovae simulations from the Garching
group Gar (2022). For MSC／ϵ described in item $i$) of MSC in Section II.2, we
exploit a 40 $M_{\odot}$ progenitor simulation (LS180-s40.0) Serpico et al.
(2012), since it has detailed matter density profiles, essential to explore
such scenario. For all other cases investigated (MSCϵ and $\nu$-loss), we use
simulations with 27 $M_{\odot}$ (LS220s27.0c) and 11.2 $M_{\odot}$
(LS220s11.2c) progenitor stars, detailed in Mirizzi et al. (2016).

To avoid the large uncertainties of collective effects, we only use the flux
from the neutronization burst phase (first 30 ms) in our analysis, in which
effects induced by $\nu-\nu$ interaction are expected to not play a
significant role. In Fig. 3 we show the luminosity of all flavors along the
time window of this phase.

Next, we explain in more detail how to include non-standard physics of eqs.
(II.4) and (II.7) in SN neutrino evolution and our methods to use a future SN
detection to impose limits on QD parameters.

Figure 3: Simulated $\nu$ luminosity for neutronization burst phase of the
emission models of 27 $M_{\odot}$ (solid) and 11.2 $M_{\odot}$ (dashed)
progenitor stars from Garching group Gar (2022); Mirizzi et al. (2016).

###  III.1 Factorization of the dynamics

Our analysis only takes into account the MSW effect in the neutronization
burst through the standard matter effect on $\nu$ mixing. To combine QD
effects and MSW through the $\nu$ generation, propagation, crossing through
Earth, and detection, it is possible to factorize the flavor probabilities as

$P_{\alpha\beta}=\sum_{i,j,k=1}^{3}P_{\alpha
i}^{m(\text{SN})}P_{ij}^{m(\text{SN})}P_{jk}P_{k\beta}^{m(\text{Earth})}\quad\quad\quad\bar{P}_{\alpha\beta}=\sum_{i,j,k=1}^{3}\bar{P}_{\alpha
i}^{m(\text{SN})}\bar{P}_{ij}^{m(\text{SN})}\bar{P}_{jk}\bar{P}_{k\beta}^{m(\text{Earth})}\,\,,$
(III.8)

where $P_{\alpha\beta}$ ($\bar{P}_{\alpha\beta}$) are the transition
probabilities from flavor $\alpha$ to $\beta$. The meaning of each term in
(III.8) can be summarized as: $P_{\alpha i}^{m(\text{SN})}$ is the probability
of creating a $\nu_{\alpha}$ as a $i$ state in matter $\nu_{i}^{m}$;
$P_{ij}^{m(\text{SN})}$ is the probability of converting
$\nu_{i}^{m}\rightarrow\nu_{j}^{m}$ inside supernova layers; $P_{jk}$ the
probability of converting $\nu_{j}\rightarrow\nu_{k}$ during propagation in
vacuum until Earth; and by the end, $P_{k\beta}^{m(\text{Earth})}$ is the
probability of detecting a $\nu_{\beta}$ given a $\nu_{k}$ state considering
(or not) Earth matter effects. The index $m$ regards that the creation or
propagation is in matter. It is worth remembering that $\nu_{e}$ and
$\bar{\nu}_{e}$ are created as a single mass eigenstate in matter. In this
scenario, the sum over $i$ vanishes, since we have
$P^{m(\text{SN})}_{ei}=\delta_{i3}$ and
$\bar{P}^{m(\text{SN})}_{ei}=\delta_{i1}$ for NH, and
$P^{m(\text{SN})}_{ei}=\delta_{i2}$ and
$\bar{P}^{m(\text{SN})}_{ei}=\delta_{i3}$ for IH. As for $\nu_{x}$, although
it is created in a coherent superposition of the other two mass eigenstates,
the interference phase would be averaged out, and therefore eq. (III.8) is
valid. In the context of a SN flux conservation, the simplest flavor
conversion scheme could be described by just $P_{ee}$ and $\bar{P}_{ee}$, and
in standard neutrino mixing, the factorized probabilities in (III.8) become
$P_{ij}^{m(\text{SN})}=\delta_{ij}$, $P_{jk}=\delta_{jk}$ and
$\bar{P}_{ij}^{m(\text{SN})}=\delta_{ij}$, $\bar{P}_{jk}=\delta_{jk}$ for
adiabatic evolution. Such a scenario can be changed by quantum decoherence,
allowing for the conversion among mass eigenstates in vacuum and matter.

One can also note in (II.4), (II.5), (II.6) and (III.8) that for the MSCϵ
model, $P_{ee}$ is a function of $\Gamma_{3}$ and $\Gamma_{8}$ in IH but only
of $\Gamma_{8}$ for NH. The $\bar{P}_{ee}$ has the opposite dependency and we
can write:

$\displaystyle
P_{ee}^{\text{IH}}=P_{ee}^{\text{IH}}(\Gamma_{3},\Gamma_{8})~{}~{}~{}$ ;
$\displaystyle~{}~{}~{}P_{ee}^{\text{NH}}=P_{ee}^{\text{NH}}(\Gamma_{8}).$
$\displaystyle\bar{P}_{ee}^{\text{IH}}=\bar{P}_{ee}^{\text{IH}}(\Gamma_{3})~{}~{}~{}$
;
$\displaystyle~{}~{}~{}\bar{P}_{ee}^{\text{NH}}=\bar{P}_{ee}^{\text{NH}}(\Gamma_{3},\Gamma_{8}).$

These remarks on the survival probabilities of $\nu_{e}$ and $\bar{\nu}_{e}$
are essential in our results, once the flavor conversion of MSC can be
described using uniquely $P_{ee}$ and $\bar{P}_{ee}$.

Particularly for the MSC${}^{\not}{\epsilon}$ case, considering the
propagation along supernova layers, $P_{ij}^{m(\text{SN})}$ and
$\bar{P}_{ij}^{m(\text{SN})}$ will be affected by QD, nevertheless
$P_{jk}=\delta_{jk}$ and $\bar{P}_{jk}=\delta_{jk}$, since with no exchange of
energy to the environment, quantum decoherence would not play any role in the
vacuum propagation. On the other hand, for MSCϵ, both SN matter and vacuum
would affect the neutrino mixing. However, as shown in Fig. 23 in the Appendix
A, it would be needed a $\Gamma_{3,8}\gtrsim 10^{-18}$ eV or even beyond to
have significant changes over $P_{ij}^{m(\text{SN})}$. As it will be clear in
Section IV, this value is much higher than the possible sensitivity of a
future SN detection with only vacuum effects (given the large coherence length
between the SN and Earth), then we take $P_{ij}^{m(\text{SN})}$ and
$\bar{P}_{ij}^{m(\text{SN})}$ as $\delta_{ij}$ for MSCϵ from now on.

In order to put bounds on QD effects, we statistically analyze it in two
scenarios: without Earth matter effects in neutrino (antineutrino)
propagation, or $P_{ke}^{m(\text{Earth})}=P_{ke}$
($\bar{P}_{ke}^{m(\text{Earth})}=\bar{P}_{ke}$) in (III.8); and then we check
how Earth matter effects would impact our results.

Figure 4 shows both scenarios of $P_{ee}$ and $\bar{P}_{ee}$ as a function of
quantum decoherence parameters for neutrinos and antineutrinos, where neutrino
hierarchy plays a relevant role in the considered scenarios. It is possible to
see that Earth regeneration could enhance or decrease the sensitivity of
standard physics on QD parameters for very specific energies and zenith angles
$\theta_{z}$. However, as we will see later, regeneration becomes more
relevant for higher energies, generally at the end of the SN-$\nu$ simulated
spectrum, limiting its impact on SN flavor conversion.

Figure 4: Survivor probability for electron neutrinos (left) and antineutrinos
(right) as a function of decoherence parameters for $n=0$ (energy independent)
and a 10 kpc propagation, without (upper plots) and with (down plots) Earth
matter effects. Solid lines represent MSCϵ scenario ($\Gamma_{8}$) with
$\Gamma_{3}=10^{-27}$ eV and the dashed, the neutrino loss ($\gamma$). For the
upper plots, quantum decoherence is taken into account only in vacuum in
between SN surface until detection at Earth, with no regeneration considered.
In the down ones, we set the zenith angle of $\theta_{z}=180^{o}$ and
$E_{\nu}=30$ MeV.

It is worth mentioning that for the MSC model asymptotically we expect more
sensitivity on $P_{ee}$ in NH than IH, since for IH the standard probability
is about the maximal admixture (1/3). In contrast, for $\bar{P}_{ee}$, both
hierarchy scenarios are almost equally sensitive to a maximal admixture
scenario. In the case of $\nu$-loss we see the opposite picture for $P_{ee}$,
i.e. IH would be more impacted by an asymptotically null probability, and for
$\bar{P}_{ee}$ NH would be highly affected, with low impact on IH.

As we will see later, the most general scheme of SN-$\nu$ fluxes at Earth can
not be parameterized with just $P_{ee}$ and $\bar{P}_{ee}$ for the $\nu$-loss
scenario, given no conservation of total flux. Therefore it is needed to work
out $P_{\alpha\beta}$ also for $\alpha,\beta=\mu,\tau$ (not shown in Figs. 4
for simplicity). We clarify it in the next section.

###  III.2 Exploring a future SN-$\nu$ detection

Figure 5: $\nu$ total cross-sections for inverse beta decay (IBD) Vogel and
Beacom (1999), $\nu_{e}-$Ar charge current interaction (from SNOwGLoBES)
Scholberg et al. (2021) and elastic $\nu-e^{-}$ interaction De Gouvêa et al.
(2020).

Since the detection of SN1987A through neutrinos, a galactic SN is expected by
the community as a powerful natural $\nu$ laboratory. The SN1987A neutrino
detection greatly impacted what we know about SN physics, but the low
statistics of the available data make predictions on standard $\nu$ admixture
extremely challenging. On the other hand, the next generation of neutrino
detectors promises a precise measurement of a galactic SN, highly increasing
our knowledge of SN-$\nu$ flavor conversion, with different detector
technologies and capabilities. Here, we show the sensitivity of DUNE, HK, and
JUNO on QD. These detectors have the following properties:

  * a)

DUNE will be a 40 kt Liquid-Argon TPC in the USA. We consider only the most
promising detection channel $\nu_{e}+\text{Ar}\rightarrow e^{-}+K^{+}$ Abi et
al. (2021) in our analysis, being sensitive to electron neutrinos and
consequently to most neutronization burst flux666Actually, it depends on the
neutrino mass hierarchy, once for MSW-NH the $\nu_{e}$ flux is highly
suppressed.. We set an energy threshold to $E_{\text{th}}=4.5$ MeV and use the
most conservative reconstruction efficiency reported in Abi et al. (2021).

  * b)

Hyper-Kamiokande will be a water Cherenkov detector in Japan with a fiducial
mass of $\sim 374$ kt with main detector channel as the inverse beta decay
(IBD), sensible to electron antineutrinos: $\bar{\nu}_{e}+p\rightarrow
e^{+}+n$. It is also expected several events from elastic scattering with
electrons, with the advantage of sensitivity to all flavors:
$\nu+e^{-}\rightarrow\nu+e^{-}$. We consider both channels in our analysis. We
set a 60% overall detector efficiency and $E_{\text{th}}=3$ MeV.

  * c)

JUNO will be a liquid scintillator detector with a fiducial mass of 17 kt
situated in China collaboration et al. (2022). Despite the interesting multi-
channel detection technology reported by the collaboration, we take into
account only IBD events. We set an overall efficiency of 50% and
$E_{\text{th}}=3$ MeV in our analysis.

In order to compare the examined scenarios, we will consider only the energy
information, calculating the number of events in the $j$-th energy bin as

$N_{j}=n_{d}^{c}\int_{0}^{\infty}dt\int_{0}^{\infty}dE_{\nu}\frac{d^{2}\phi_{\nu}}{dtdE_{\nu}}\eta(E_{\nu})\int_{E_{i}}^{E_{f}}d\bar{E}_{\nu}R_{j}(\bar{E}_{\nu},E_{\nu})\sigma(E_{\nu}),$
(III.9)

where $n_{d}^{c}$ is the number of targets for each detector $d$, with $c$
accounting for each specific channel, $\phi_{\nu}$ is the neutrino flux,
$\eta(E_{\nu})$ is the efficiency that can eventually depend on $\nu$ energy,
$\sigma$ is the neutrino cross-section (with each channel shown in Fig. 5),
$R_{j}$ is the detector resolution. We analyze the $\nu$ energy from the
threshold of each detector up to 60 MeV. The $\nu$ mixing is encoded in the
flux $\phi_{\nu}$, that can be written as

$\begin{split}&\phi_{\nu_{e}}=\phi_{\nu_{e}}^{0}P_{ee}+\phi_{\nu_{x}}^{0}(1-P_{ee})\\\
&\phi_{\bar{\nu}_{e}}=\phi_{\bar{\nu}_{e}}^{0}\bar{P}_{ee}+\phi_{\nu_{x}}^{0}(1-\bar{P}_{ee})\\\
&\phi_{\nu_{x}}=\phi_{\nu_{e}}^{0}(1-P_{ee})+\phi_{\nu_{x}}^{0}(2+P_{ee}+\bar{P}_{ee})+\phi_{\bar{\nu}_{e}}^{0}(1-\bar{P}_{ee})\end{split}$
(III.10)

for the standard MSW (widely found in literature, see Dighe and Smirnov
(2000); Mirizzi et al. (2016) for a review), where $\phi_{\nu_{\alpha}}^{0}$
refers to initial SN neutrino fluxes and non-standard QD effects are hidden in
$P_{ee}$ and $\bar{P}_{ee}$. In Fig. 6, the expected number of events for the
three detectors are reported in the energy spectrum of simulated progenitors
(11.2 $M_{\odot}$ and 27 $M_{\odot}$) for both hierarchies and are compared to
MSCϵ model. The results translate what is shown in Fig. 4, weighted by
detector capabilities. Expected changes in the spectrum look more prominent
when NH is assumed as a standard solution for DUNE, with an increase of
$\nu_{e}$ events for both hierarchies. On the other hand, for HK and JUNO the
MSCϵ effect results in a decrease of events in IH and an increase in NH and it
is not so clear which hierarchy would be more sensible to the MSCϵ effect,
since the number of QD parameters for each one is different for both $P_{ee}$
and $\bar{P}_{ee}$. For instance, for $\bar{P}_{ee}^{\text{NH}}$, fixing
$\Gamma_{3}$, an increase in $\Gamma_{8}$ is weighted by the factor 1/3 in the
exponential terms, while $\bar{P}_{ee}^{\text{IH}}$ is more sensible to
$\Gamma_{8}$, since the same change is multiplied by a factor 1, but it is
also independent of $\Gamma_{3}$.

Figure 6: Spectrum of events for DUNE, HK and JUNO for NH (solid lines) and IH
(dashed), with $n=0$ for a 10 kpc SN with 11.2 $M_{\odot}$ and 27 $M_{\odot}$
progenitor mass simulations. Each column concerns a detector, while the rows
are related to progenitor masses. The size of bins is at least twice the
resolution at the specific energy and given a minimum threshold in the number
of events per bin established in our analysis. The bands are to respect the
40% of the uncertainty of the flux over standard NH and IH, with details in
the text. For the QD parameters, we used the values $\Gamma_{8}=10^{-27}$ eV
and $\Gamma_{3}=4\Gamma_{8}$.

Note that eq. (III.10) is valid for a conserved total flux, which does not
remain in the $\nu$-loss scenario. To get around this issue we propose a more
generalized form of (III.10)

$\begin{split}&\phi_{\nu_{e}}=\phi_{\nu_{e}}^{0}P_{ee}+\phi_{\nu_{x}}^{0}(P_{\mu
e}+P_{\tau e})\\\
&\phi_{\bar{\nu}_{e}}=\phi_{\bar{\nu}_{e}}^{0}\bar{P}_{ee}+\phi_{\nu_{x}}^{0}(\bar{P}_{\mu
e}+\bar{P}_{\tau e})\\\
&\phi_{\nu_{x}}^{\prime}=\phi_{\nu_{e}}^{0}(P_{e\mu}+P_{e\tau})+\phi_{\nu_{x}}^{0}(P_{\mu\mu}+P_{\mu\tau}+P_{\tau\tau}+P_{\tau\mu})\\\
&\phi_{\bar{\nu}_{x}}^{\prime}=\phi_{\bar{\nu}_{e}}^{0}(\bar{P}_{e\mu}+\bar{P}_{e\tau})+\phi_{\bar{\nu}_{x}}^{0}(\bar{P}_{\mu\mu}+\bar{P}_{\mu\tau}+\bar{P}_{\tau\tau}+\bar{P}_{\tau\mu})\\\
&\phi_{\nu_{x}}=\phi_{\nu_{x}}^{\prime}+\phi_{\bar{\nu}_{x}}^{\prime}\end{split}$
(III.11)

where each probability can be factorized as described in (III.8). For the ones
where $\alpha=\mu,\tau$, since these flavors are generated in a superposition
of mass states in matter, the $\nu_{\mu}-\nu_{\tau}$ mixing should be taken
into account, where $P_{\alpha i}^{m\text{SN}}$ and $\bar{P}_{\alpha
i}^{m\text{SN}}$ would correspond to the proper square module of elements from
$U_{\mu\tau}$ mixing matrix777In the $\mu-\tau$ sector, such probability is
associated to $\theta_{23}$ mixing, being a sub-matrix of $U_{23}$ in the
conventional PMNS decomposition. We also assume in this formula that any
oscillation term is averaged out.. In Fig. 7 we show each probability
$P_{\alpha\beta}$ for a 10 kpc SN for the $\nu$-loss scenario. In Fig. 8 we
show the expected spectrum of events for the $\nu$-loss model.

Figure 7: Probabilities with impact of $\nu$-loss with $n=0$ considering a 10
kpc SN for NH (left) and IH (right).  Figure 8: Spectrum of events for DUNE,
HK and JUNO for NH (solid lines) and IH (dashed) compared to $\nu$-loss model,
with $n=0$ for a 10 kpc SN with 11.2 $M_{\odot}$ and 27 $M_{\odot}$ progenitor
mass simulations. For $\nu$-loss we use different bin sizes in order to
achieve the requirement of minimum number of events per bin of $\sim 5$. Given
the lack of events in this scenario, we decided to use a single bin for JUNO.

###  III.3 Role of Earth matter effects

Since a galactic SN detection can be impacted by Earth matter effects, we also
calculate $P_{ee}$ and $\bar{P}_{ee}$ to each detector given the position of
the SN in the sky. However, as shown in Pompa et al. (2022), it is not
expected to play an important role for the neutronization burst. The reason is
that regeneration would start to be important beyond $E_{\nu}\gtrsim 50$ MeV
or even higher energies, which is close to the end of the expected spectrum.
In Fig. 9 we show the impact of Earth matter effects in $P_{ee}$ for a SN flux
of $\nu_{e}$ in IH and $\bar{P}_{ee}$ for $\bar{\nu}_{e}$ in NH in a range of
zenith angles for only non-adiabatic MSW effect (no quantum decoherence
effects) using the PREM density profile available in Anderson (1989), where
$90^{o}$ is a horizon of an observer at Earth (with no matter effects) and
$180^{o}$ represents a propagation all along Earth diameter. Note that for
$P_{ee}$ in NH and $\bar{P}_{ee}$ in IH, regeneration does not play an
important role.

Figure 9: $P_{2e}$ (left) and $\bar{P}_{1e}$ (right) under Earth matter
effects as a function of neutrino energy and zenith angle. In standard MSW in
supernova mixing, $P_{2e}$ and $\bar{P}_{1e}$ can be used to calculate the
survival probabilities of $\nu_{e}$ (IH) and $\bar{\nu}_{e}$ (NH)
respectively. The lines on the color bar are the adiabatic solutions for
$P_{2e}$ (yellow) and $\bar{P}_{1e}$ (black) without regeneration effects.

In Fig. 10 we also see the QD effects (MSCϵ with $n=0$) combined with Earth
matter effects for a specific energy (similarly as shown in Fig. 4, but for a
wide range of $\theta_{z}$ and the QD parameter). The asymptotic maximal
mixing suppresses regeneration effects beyond $\Gamma_{8}\sim 10^{-27}$ eV,
being a leading effect. Since regeneration is a second-order effect, we impose
bounds on QD in the next section without considering Earth matter effects, and
by the end of Section IV.2, we show its impact on results.

Figure 10: $P_{ee}$ in IH (left) and $\bar{P}_{ee}$ in NH (right) under Earth
matter effects as a function of QD parameter for $E_{\nu}=30$ MeV, considering
a SN 10 kpc away from Earth and $n=0$. It is possible to see that QD
suppresses regeneration effects for $\Gamma_{8}\gtrsim 10^{-27}$ eV, where
$\Gamma_{3}=10^{-32}$ eV was set. The white line on the color bar represents
maximal mixing.

##  IV Future limits on quantum decoherence

In order to impose bounds on QD using simulated data, we perform a binned
$\chi^{2}$ through pull method Fogli et al. (2002) over QD parameters for MSC
and $\nu$-loss scenarios:

$\chi^{2}=\sum_{d}\sum_{j=1}^{m}\frac{(N_{j,d}^{\text{true}}-(1+a)N_{j,d}^{\text{th}})^{2}}{N_{j,d}^{\text{th}}}+\frac{a^{2}}{\sigma_{a}^{2}}$
(IV.12)

where $m$ indicates the number of energy bins, $d$ represents each detector,
$N_{j,d}^{\text{true}}$ represents events predicted by the MSW solution, and
$N_{j,d}^{\text{th}}$ accounts the theoretical number of events of the
marginalized model in our analysis, i.e. MSW + quantum decoherence
respectively and the second term on the right-hand side takes our estimation
in the flux uncertainties of $40\%$ into account De Gouvêa et al. (2020).

We can note in Fig. 7 that since all probabilities vanish for high values of
$\gamma$, $N\rightarrow 0$ for $\nu$-loss. However in order to avoid a bias in
our analysis, we marginalize over $\gamma$ only in a range where the
requirement of at least $\sim 5$ events per bin is achieved (we use the same
rule for MSC). We also take the size of the bins to be twice the detector
energy resolution. Using these requirements, JUNO allows a single bin for
$\nu$-loss, being a counting experiment for this analysis. The bins scheme for
DUNE and HK are also changed for $\nu$-loss compared to MSC in order to match
the established minimum number of events per bin in the tested range of
$\gamma$.

Before imposing limits on MSC and $\nu$-loss with eq. (IV.12), we can treat
$P_{ee}$ and $\bar{P}_{ee}$ as free parameters, which is a reasonable
approximation to an adiabatic propagation at the SN, since these probabilities
are energy independent (see Dedin Neto et al. (2023) for a more detailed
discussion in the context of SN1987A), we perform a marginalization with
$\chi^{2}(P_{ee},\bar{P}_{ee})$ in eq. (IV.12) to understand how far
asymptotically QD scenarios are from the standard $\nu$ mixing and also see
how sensible a combined measurement (DUNE+HK+JUNO) could be, using uniquely
the neutronization burst. Fig. 11 shows how a 10 kpc SN can impose limits to
$P_{ee}$ and $\bar{P}_{ee}$, with NH and IH concerning the true MSW model. The
black dot represents maximal mixing or the asymptotic limit of MSC, which is
closer to the IH solution (given by the corresponding best-fit value) than NH
for $P_{ee}$, but in an intermediary point of hierarchies with respect to
$\bar{P}_{ee}$. In the $\nu$-loss scenario it is not so clear from Fig. 11
which hierarchy would lead to stronger constraints, given the presence of
other probabilities, such as the ones in Fig. 7.

Figure 11: Limits on $P_{ee}$ and $\bar{P}_{ee}$ for the 27 $M_{\odot}$
(solid) and 11.2 $M_{\odot}$ (dashed) progenitor stars from simulations,
considering only the neutronization burst. No quantum decoherence effects are
taken into account in this Figure. The distance from Earth considered was 10
kpc. The probability is assumed to be a free parameter as recently proposed in
Dedin Neto et al. (2023). The assumption of a standard adiabatic MSW
conversion at the SN is taken into account (as all along the manuscript),
getting rid of the energy dependency on $P_{ee}$ and $\bar{P}_{ee}$. The black
dot is the maximal mixing scenario (1/3). Note that the 11.2 $M_{\odot}$ line
for IH matches to the 27 $M_{\odot}$, showing that the sensitivity for
simulated progenitors tested is similar.

Using eq. (IV.12) and the procedures described in Sections II and III, we
treat QD parameters as free and perform a $\chi^{2}$ analysis in order to
impose statistical bounds in this effect using a future SN detection. Since
nowadays the neutrino mass hierarchy is not established, we include both
scenarios in our analysis.

We test both MSW-NH versus the marginalized MSW-NH + QD and also the MSW-IH
versus the marginalized MSW-IH + QD in order to understand how restrictive
future detectors will be. The results will show that if QD plays any role in
SN neutrinos, both possible $\nu$ hierarchies could be affected.

###  IV.1 MSC${}^{\not}{\epsilon}$

For the MSC${}^{\not}{\epsilon}$ model, we calculate the
$\sqrt{\Delta\chi^{2}}$ bounds over the parameter $\Gamma$, where
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\text{min}}$ (since we are not including
statistical and systematic uncertainties when producing the “true” data, we
always have $\chi^{2}_{\text{min}}=0$). The results for the 3 experiments are
summarized in Fig. 12, where the true scenario is NH and we marginalize over
NH+QD. Note that bounds reach different significant limits for each SN
distance, with lower distances being more restrictive.

Figure 12: Limits on $\Gamma$ for various SN distances from Earth for DUNE
(left), HK (middle), and JUNO (right) for the 40 $M_{\odot}$ progenitor star
simulation. The true scenario taken into account was NH, and we marginalize
the parameters over the theoretical NH+QD (MSC／ϵ). No Earth matter effect was
considered. Each row means a different value of $n$ in the parameterization
$\Gamma=\Gamma_{0}(E/E_{0})^{n}$.

Since the traveled distance is a fixed feature, the only aspect that the SN
distance from Earth contributes is the number of events detected. Following
Fig. 12, the best performance in NH is for DUNE, with possible $3\sigma$
limits for a 10 kpc SN away from Earth of:

$\Gamma_{0}\leq\left\\{\begin{array}[]{rl}6.2\times 10^{-14}~{}{\rm
eV{}{}{}}&(n=0)\\\ 5.2\times 10^{-14}~{}{\rm eV{}{}{}}&(n=2)\\\ 1.4\times
10^{-13}~{}{\rm eV{}{}{}}&(n=5/2)\end{array}\right.$ (IV.13)

For a SN at a distance of 1 kpc, limits of $\mathcal{O}(10^{-16})$ eV can be
reached. HK has also a good performance and achieves $2\sigma$ bounds for a 10
kpc SN. JUNO is not capable of individually achieving reasonable bounds on QD
for SN distances $\gtrsim 1$ kpc, but would also have a strong signal for a
galactic SN as close as 1 kpc away from Earth, which can be attributed to the
small fiducial mass compared to HK and a single IBD channel considered in this
work (with a significantly lower cross-section than $\nu_{e}$-Ar for energies
above $\sim 15$ MeV). Other channels, such as $\nu$-p elastic scattering could
possibly improve the results, but given the detection challenges associated,
we decided to not include them here.

We also performed the same analysis using IH as the true theory and
marginalizing over IH+QD. The results are shown in Fig. 13. The best
performance is clearly for HK with $2\sigma$ bound of:

$\Gamma_{0}\leq\left\\{\begin{array}[]{rl}3.6\times 10^{-14}~{}{\rm
eV}&~{}~{}~{}(n=0)\\\ 8.0\times 10^{-14}~{}{\rm eV}&~{}~{}~{}(n=2)\\\
2.4\times 10^{-13}~{}{\rm eV}&~{}~{}~{}(n=5/2)\end{array}\right.$ (IV.14)

for a 10 kpc SN from Earth. DUNE is not capable to impose strong bounds in an
IH scenario. JUNO performance is improved for distances $\lesssim 1$ kpc
compared to NH. Results are summarized in Table 1 in Appendix B.

Figure 13: Same as Fig. 12 but with IH as the true theory, marginalized over
the parameters of the IH+QD model.

A 20 kpc SN could not impose strong bounds for individual experiments.
Distances as far as 50 kpc (as Large Magellanic Cloud) were not investigated
in this work, given the lack of events per bin, in which a more refined
unbinned statistical analysis would be required, which is not strongly
motivated by the fact that expected limits are below $2\sigma$.

The bounds and sensitivity of each detector in a given hierarchy shown above
could be associated with the sensitivity to ${P}_{ee}$ and $\bar{P}_{ee}$
shown in Fig. 11. In NH (left plot), limits over $P_{ee}$ are more restrictive
than $\bar{P}_{ee}$ with respect to maximal mixing represented by the black
dot. For IH (right plot), we have an opposite sensitivity, since $P_{ee}\sim
1/3$, while for $\bar{P}_{ee}$ there is a gap between the best fit and 1/3
probability, allowing limits with certain significance to be imposed. Since
DUNE is most sensitive to $\nu_{e}$, via $\nu_{e}$-Ar interaction, it will be
more sensitive to $P_{ee}$ and then more relevant in the NH scenario. As for
HK and JUNO, they are more sensitive to $\bar{\nu}_{e}$ and therefore to
$\bar{P}_{ee}$, which reflects a better performance in the IH scenario. In our
calculations, the elastic scattering considered in HK does not contribute much
to the total $\chi^{2}$.

###  IV.2 MSCϵ

The same procedure described in the section above was performed on the MSCϵ
model, with bounds over the parameter $\Gamma_{8}$. Results are summarized in
Fig. 14 for NH vs NH+QD. SN distance also plays an important role in this
scenario and results and their aspects are similar to MSC${}^{\not}{\epsilon}$
described in the last section. DUNE has the best performance for the tested SN
distances and even for a 10 kpc SN, bounds with $3\sigma$ could be achieved
for $n=0,2$ and $5/2$. Despite the stronger effects caused by MSC for larger
distances, the number of events decrease with $L^{2}$, and stronger limits can
be imposed for a SN happening at shorter distances, reflecting that the larger
number of neutrinos arriving at the detector is a crucial aspect.

Figure 14: Same as 12 but for MSCϵ with simulations of the 27 $M_{\odot}$
(solid) and 11.2 $M_{\odot}$ (dashed) progenitor masses. The bounds are orders
of magnitude more restrictive than for MSC／ϵ.

From Fig. 14, taking the result of a 10 kpc SN (27 $M_{\odot}$), DUNE would
potentially impose $\Gamma_{8}\leq 4.2\times 10^{-28}$ eV for $2\sigma$ and
$\Gamma_{8}\leq 1.7\times 10^{-27}$ eV for $3\sigma$ with $n=0$, whereas the
HK bound is $\Gamma_{8}\leq 4.2\times 10^{-27}$ eV for $2\sigma$. Looking at
limits from various works Lisi et al. (2000); Farzan et al. (2008); Oliveira
et al. (2014); Oliveira (2016); Coelho et al. (2017); Coelho and Mann (2017);
Carrasco et al. (2019); Carpio et al. (2019); Gomes et al. (2017); Coloma et
al. (2018); de Holanda (2020); Gomes et al. (2019), to the best knowledge of
the authors, this is an unprecedented level of sensitivity for testing quantum
decoherence, orders of magnitude more restrictive than any other work in the
subject. Fig. 15 shows bounds from works with different sources and place the
limits from this work for both hierarchy scenarios.

Figure 15: Current bounds on quantum decoherence for a number of works from
many neutrino sources and also the SN limits presented here ($n=0$). Arrows
with longer horizontal bases correspond to current experimental bounds,
whereas minor bases are to respect to possible future limits. Numbers in the
arrows indicate the reference in which the limits were obtained. Thin arrows
indicate bounds equivalent to MSC${}^{\not}{\epsilon}$, while thick-filled
ones are the MSCϵ. White-filled thick arrows correspond to $\nu$-loss bounds.
Supernova limits described in this work are in red, and are to respect to a
distance of 10 kpc from Earth unless distance is indicated, with more
restrictive bounds being possible for closer SNs.

Note that for $n=2$ and 5/2 the bounds are over $\Gamma_{08}$ in
$\Gamma_{8}=\Gamma_{08}(E/10~{}{\rm MeV})^{n}$. For a 10 kpc SN
(27$M_{\odot}$), DUNE $3\sigma$ bounds reach:

$\Gamma_{08}\leq\left\\{\begin{array}[]{rl}7.0\times 10^{-28}~{}{\rm
eV}&~{}~{}~{}(n=2)\\\ 6.2\times 10^{-28}~{}{\rm
eV}&~{}~{}~{}(n=5/2)\end{array}.\right.$ (IV.15)

HK is able to achieve $2\sigma$ bounds as restrictive as $\Gamma_{08}\leq
2.7\times 10^{-28}$ eV and $\Gamma_{08}\leq 1.2\times 10^{-28}$ eV for $n=2$
and 5/2 respectively. All mentioned results are summarized in Table 2 in the
Appendix B.

We also performed a combined fit for the three detectors using the same $\nu$
hierarchy scheme shown in Fig. 16, where a $3\sigma$ limit for a 10 kpc SN
would reach:

$\Gamma_{08}\leq\left\\{\begin{array}[]{rl}6.2\times 10^{-28}~{}{\rm
eV}&~{}~{}~{}(n=0)\\\ 1.2\times 10^{-28}~{}{\rm eV}&~{}~{}~{}(n=2)\\\
0.72\times 10^{-28}~{}{\rm eV}&~{}~{}~{}(n=5/2)\end{array}.\right.$ (IV.16)

Even a $4\sigma$ of maximal mixing is possible to be achieved for all values
of $n$, but such significance is achieved only by the 27 $M_{\odot}$ simulated
progenitor. Although a combined analysis reaches high significance, it should
be taken with a grain of salt, since it is not possible to be sure that
experiments would be simultaneously in operation.

Using the same procedure as done in NH, we make the analysis assuming IH as
the true mixing and marginalizing over IH+QD. The results are shown in Fig.
17. HK has the strongest bounds on this scenario but does not reach $3\sigma$
for a 10 kpc SN, even though the potential limits for $2\sigma$ are:

$\Gamma_{08}\lesssim\left\\{\begin{array}[]{rl}1.3\times 10^{-27}~{}{\rm
eV}&~{}~{}~{}(n=0)\\\ 1.4\times 10^{-28}~{}{\rm eV}&~{}~{}~{}(n=2)\\\
4.9\times 10^{-28}~{}{\rm eV}&~{}~{}~{}(n=5/2)\\\ \end{array}.\right.$ (IV.17)

DUNE has a very poor performance in this scenario for any distance $\gtrsim 1$
kpc. JUNO sensitivity is similar to NH marginalization discussed above. In a
combined fit in IH, shown in Fig. 18, the following $3\sigma$ limits can be
obtained:

$\Gamma_{08}\lesssim\left\\{\begin{array}[]{rl}5.4\times 10^{-27}~{}{\rm
eV}&~{}~{}~{}(n=0)\\\ 3.5\times 10^{-27}~{}{\rm eV}&~{}~{}~{}(n=2)\\\
3.3\times 10^{-27}~{}{\rm eV}&~{}~{}~{}(n=5/2)\\\ \end{array}.\right.$ (IV.18)

Figure 16: Combined fit for the true MSW-NH marginalizing over MSW-NH with QD
(MSCϵ) effects. Figure 17: Same as Fig. 14 but for IH versus IH + QD. Figure
18: Same as Fig. 16 but now accounting the IH scenario.

To check the impact of regeneration on the above results, we calculated the
bounds of a combined detection of DUNE, HK, and JUNO including this effect. We
test different $\theta_{z}$, the zenith to respect to DUNE, with the
assumption that the SN flux comes from DUNE longitude. The results are in Fig.
19. We can note in the left plot that the impact of the Earth matter effect is
small but enhances QD bounds for a 10 kpc detection and limits could be
stressed beyond $4\sigma$. The right plot shows the situation where the IH
scenario is assumed to be true and NH+QD is marginalized. We will discuss such
a scenario in Section V, but we also see that regeneration will not change
significantly the results.

Figure 19: Limits on MSC with the impact of Earth matter effects for a SN 10
kpc from Earth and the 27 $M_{\odot}$ simulation for different zenith angles
$\theta_{z}$ ($n=0$). The limits correspond to a combined detection of DUNE,
HK, and JUNO, but $\theta_{z}$ is to respect to DUNE, with SN beam in the
direction of DUNE longitude. The $\theta_{z}=320^{o}$ means that regeneration
effects at HK and JUNO are expected, even if the SN beam does not cross Earth
for reaching DUNE.

###  IV.3 Neutrino loss

Since in $\nu$-loss the spectrum of events decreases asymptotically to zero,
the bounds on this scenario are expected to be as significant or even more
than MSC for all experiments. Since the calculated number of events for NH is
low (mainly for DUNE and JUNO) and $\nu$-loss would decrease it, not
fulfilling our requirement of $\gtrsim 5$ events per bin, we perform here only
the IH (true) versus IH+QD. Fig. 20 shows the $\sqrt{\Delta\chi^{2}}$ for each
individual detector. We see that high values of $\gamma$ are strongly bounded,
even for JUNO. For a SN from 10 kpc away from Earth, DUNE, HK and JUNO are
capable to impose $\gamma\leq 5.2\times 10^{-28}$ eV, $\gamma\leq 4.9\times
10^{-28}$ eV and $\gamma\leq 5.9\times 10^{-28}$ eV respectively with
$3\sigma$ of significance ($n=0$). Note that beyond 10 kpc the number of
events per bin would be significantly small for a $\nu$-loss scenario and we
do not consider it in this analysis.

HK is capable to achieve the best ($3\sigma$) bounds with $\gamma_{0}\leq
2.1\times 10^{-29}$ eV and $\gamma_{0}\leq 1.2\times 10^{-29}$ eV for $n=2$
and $5/2$ respectively, with a 10 kpc SN. Although not shown in the plots, it
is worth mentioning that HK would impose bounds on $\gamma$ even for NH, given
the high statistics associated with this experiment, being the most sensitive
one for the $\nu$-loss model. We detail the bounds and all mentioned results
here in Table 3.

Figure 20: Limits on $\gamma$ for various SN distances from Earth for all
detectors in the $\nu$-loss scenario with true IH marginalized over the
parameters of the IH+QD model.

##  V Neutrino mass hierarchy measurement

In a future supernova detection, the neutronization burst arises as a robust
test of neutrino mass hierarchy, with $\nu$-Ar in DUNE capable to determine
the correct scenario with relatively high confidence. However, although the
possible strong bounds are to be imposed on quantum decoherence, if QD plays a
significant role in $\nu$ mixing, the IH could be mimicked by a NH with the
impact of QD (particularly, in the MSC models). A similar analysis was
performed in the context of $\nu$-decay in Delgado et al. (2022). Therefore,
the question that arises is how much NH and IH are distinguishable if we
compare both hierarchies superposing the standard NH to QD. Fig. 21 shows the
statistical bounds of the scenario where IH is taken as the true theory and
NH+QD is marginalized in a combined detection for $n=0,2,5/2$. The results
show that the significance of hierarchy determination significantly weakens
for the tested SN distances and even a combined detection could not
disentangle the hierarchies if MSC plays an important role.

To check this statement we can compare the values of $\sqrt{\Delta\chi^{2}}$
for $\Gamma_{8}\rightarrow 0$ and $\Gamma_{8}\rightarrow\infty$ in Fig. 21. We
can assume that $\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow 0}$ corresponds
to the distinguishability of hierarchy in a standard scenario since
$\Gamma_{8}$ is small enough to neglect QD effects. The plateau in the limit
of $\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow\infty}$ shows how NH+QD
would differ from IH in a future combined detection, in which has lower values
of $\sqrt{\Delta\chi^{2}}$, resulting in a less significant hierarchy
discrimination. Taking as a reference a SN distance of 10 kpc for the 27
$M_{\odot}$ simulation, with a combined detection of DUNE, HK and JUNO, we
have a $\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow 0}=6.89$ going to
$\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow\infty}=3.13$. For an individual
detection with the same SN distance, DUNE would change from
$\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow 0}=5.70$, which is
statistically significant to determine the hierarchy, to a mere
$\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow\infty}=0.37$. HK also could be
affect with a $\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow 0}=3.36$ going to
$\sqrt{\Delta\chi^{2}}|_{\Gamma_{8}\rightarrow\infty}=2.65$. JUNO can not
distinguish the neutrino hierarchies significantly at 10 kpc. It is important
to mention that for 1 kpc and 5 kpc DUNE could be highly affected by this
hierarchy misidentification, but HK still would provide a distinction of
$\gtrsim 5\sigma$ even with QD effects. For SN distances $>5$ kpc, the
neutrino hierarchies would be hardly disentangled by the tested experiments if
QD effects are significant. As far as we tested, the $\nu$-loss model did not
lead to the same potential hierarchy misidentification found in the MSC.

Figure 21: Statistically comparing the inverted hierarchy (IH) to normal
hierarchy (NH) with the impact of quantum decoherence for a combined detection
using the 11.2 $M_{\odot}$ (dashed) and 27 $M_{\odot}$ (solid) simulations. No
regeneration effects were taken into account.

##  VI Conclusions

In this paper, we have explored the capability of a future SN neutrino
detection in imposing limits in quantum decoherence scenarios. As the
neutrinos are already treated as an incoherent mixture of mass eigenstates
inside the SN, damping effects are not expected, then we explore secondary
quantum decoherence scenarios, such as the relaxation mechanism, which can be
potentially observed in a SN neutrino signal. We limit ourselves to scenarios
where the decoherence matrix $D$ is diagonal in the neutrino vacuum mass
basis. Among the possible models to be investigated, we consider the ones we
denoted as Mass State Coupling (MSC), leading to maximal mixing of states, and
the neutrino loss ($\nu$-loss), associated to the loss of neutrino flux along
propagation. These scenarios are well-motivated by quantum gravity, where a
possible dependency with energy is expected in the form of
$\gamma=\gamma_{0}(E/E_{0})^{n}$, and therefore, we explore the limits on the
decoherence parameters for different $n$.

The analysis was done considering DUNE, HK, and JUNO as possible detectors.
For the neutrino flux data, three progenitor stars were considered, a 40
$M_{\odot}$ (LS180-s40.0), 27 $M_{\odot}$ (LS220s27.0c) and 11.2 $M_{\odot}$
(LS220s11.2c), using the SN simulation data from the Garching group Gar
(2022); Serpico et al. (2012); Mirizzi et al. (2016). To get around the
unsolved problem of neutrino collective effects, only the neutronization burst
was considered, given that collective effects are expected to not play a
significant role in this emission phase.

When considering the neutrino propagation inside the supernova, the relaxation
effect could affect the neutrino flavor conversion, even with the assumption
of no exchange of neutrino energy to the environment, or $[H,V_{p}]=0$
(MSC${}^{\not}{\epsilon}$). We show that in this regime it is possible to get
competitive limits to QD parameters. However, the required values for the
decoherence parameters need to be much larger than the ones in the scenario
where $[H,V_{p}]\neq 0$ (MSCϵ) (see the Appendix A), which would provide the
most restrictive bounds on QD to date. For MSCϵ, we only consider the
decoherence/relaxation acting on neutrino propagation in the vacuum from the
SN until it reaches the detectors at Earth, for which the propagation length
is orders of magnitude larger than the SN size, and therefore, more sensible
to the relaxation effects. We also explore the possible effects of Earth
regeneration due to the neutrino propagation inside the Earth, which has minor
effects in the bounds for the relaxation parameters, being the vacuum
propagation the most relevant coherence length.

With all considerations, we show that the detectors used in the analysis are
capable to impose the limits listed in Tables 1 and 2 for the MSC scenario,
depending on the distance being considered and the neutrino mass hierarchy.
For the NH, the DUNE detector is the most promising one, while HK is the most
sensible in the case of IH. The possible limits on the decoherence parameters
are orders of magnitude stronger than the ones imposed by current terrestrial
and solar experiments, as shown in Fig. 15. For the $\nu$-loss scenario, the
limits are shown in Table 3. Due to the neutrino disappearance, extra care
needed to be taken in this scenario so that the requirement of at least 5
events per bin is fulfilled and the $\chi^{2}$ analysis can be applied.

Finally, we explored the possible degeneracy between the different standard
scenarios of unknown mass hierarchy (NH and IH) without QD and the ones with
QD effects included. As we saw, the IH scenario could be easily mimicked by NH
combined with QD-MSC effects.

## Acknowledgements

We thank Hans-Thomas Janka and the Garching group for providing the SN
simulations used in this work. MVS is thankful to Alberto Gago for pointing
out complete positivity relations. EK is very grateful for the hospitality of
GSSI during the preparation of this manuscript. This study was financed by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance Code 001, and partially by the Fundação de Amparo à Pesquisa do Estado
de São Paulo (FAPESP) grants no. 2019/08956-2, no. 14/19164-6, and no.
2022/01568-0.

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##  Appendix A Decoherence inside the SN and matter effects

The neutrino Hamiltonian in flavor basis affected by the charged current
potential $V_{W}$, i.e. $H_{f}=H_{f}^{\text{vac}}+V_{W}$, can be diagonalized
to $H_{m}$ by a unitary transformation provided by $U_{m}$ as

$\rho_{f}=U_{m}\rho_{m}U_{m}^{\dagger}\hskip
28.45274ptH_{f}=U_{m}H_{m}U_{m}^{\dagger},$ (A.19)

getting the most general form of (II.1) in the effective neutrino mass basis
in matter

$\frac{d\rho_{m}}{dt}=-i[H_{m},\rho_{m}]-[U_{m}^{\dagger}\dot{U}_{m},\rho_{m}]+\sum_{p}^{N^{2}-1}(V_{pm}\rho_{m}V_{pm}-\frac{1}{2}\\{V_{pm}^{2},\rho_{m}\\})$
(A.20)

or following the notation in (II.2)

$\ket{\dot{\rho}_{m}}=-2\mathcal{L}_{m}(t)\ket{\rho_{m}}$ (A.21)

For all purposes of this work, the propagation is adiabatic, or
$\dot{U}_{m}=0$ in (A.20).

Figure 22: Snapshot (at 27 ms after the core bounce) of simulated SN electron
density profile from the $40$ $M_{\odot}$ progenitor mentioned in the text
Serpico et al. (2012); Gar (2022).

We are interested in solving equation (A.21) in a variable matter density in
order to get transition probabilities $P_{ij}^{m(\text{SN})}$ and
$\bar{P}_{ij}^{m(\text{SN})}$. It is straightforward to obtain $\ket{\rho}$ in
(II.1) , but in the case of $\ket{\rho_{m}}$, $V_{pm}$ and $H_{m}$ are time-
dependent and the solution is a time-ordered exponential:

$\mathcal{T}\left\\{e^{-2\int^{t}_{t_{0}}dt^{\prime}\mathcal{L}_{m}(t^{\prime})}\right\\}=1+(-2)\int_{t_{0}}^{t}dt_{1}\mathcal{L}_{m}(t_{1})+(-2)^{2}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}\mathcal{L}_{m}(t_{1})\mathcal{L}_{m}(t_{2})+\cdots.$
(A.22)

Analytical solutions for specific cases in a variable matter density can be
found in Benatti and Floreanini (2005); Coloma et al. (2018). However, instead
of using a cumbersome approximated approach, we analyze the neutrino evolution
into the SN making the limits in the integrals in (A.22) $\Delta
t=t_{n}-t_{n-1}\rightarrow 0$, allowing to solve (A.22) numerically through
slab approach, i.e. we divided the SN matter density profile into small parts,
in which the neutrino Hamiltonian is approximately constant, then we make the
time evolution from each step to another until the neutrino reach the vacuum.
We use the simulated density profile in Fig. 22 to perform this calculation.

Figure 23: Solution for a survival probability of mass state 3 along a SN
radius for the MSCϵ (solid opaque line) and neutrino loss (dashed). The
transparent line shows the same probability but in vacuum. More details about
these models are in the text. As it will be clear in our results, even with
enhancement of the conversion in matter, values of $\Gamma\sim 10^{-19}$ eV
are far higher than the sensitivity of a future SN detection compared to
coherence length in vacuum used in the MSCϵ model.

In Fig. 23 we compare the $P_{33}^{m(\text{SN})}$ to the same probability in
mass basis in vacuum, which is shown as an enhancement of the deviation from
the standard expectation of $P_{33}^{m(\text{SN})}=1$. In Fig. 2 we show the
numerical probabilities of MSC／ϵ for the mass state in matter solved as
described above.

##  Appendix B Tables with QD bounds

Table 1: Constrains for each detector for MSC${}^{\not}{\epsilon}$ scenario with 90%($2\sigma$) C.L. in units of $\Gamma\times 10^{-15}$ (eV). For $n\neq 0$ a representative energy of $E_{0}=10$ MeV was chosen and QD parameters are in eV scale. Values are corresponding to the simulated progenitor of 40 $M_{\odot}$. | NH | IH
---|---|---
Detector | SN distance | $n=0$ | $n=2$ | $n=5/2$ | $n=0$ | $n=2$ | $n=5/2$
DUNE | 1 kpc | $0.89(1.1)$ | $0.76(0.89)$ | $0.65(0.87)$ | $0.88(1.0)$ | $2.5(8.8)$ | $3.2(15)$
| 5 kpc | $5.4(7.0)$ | $4.4(5.9)$ | $6.3(8.7)$ |  |  |
| 7 kpc | $8.3(11)$ | $7.0(9.4)$ | $11(16)$ |  |  |
| 10 kpc | $14(20)$ | $12(17)$ | $22(35)$ |  |  |
HK | 1 kpc | $0.96(1.1)$ | $3.7(4.1)$ | $5.0(5.8)$ | $0.93(1.1)$ | $3.9(4.3)$ | $5.3(6.5)$
| 5 kpc | $4.3(5.7)$ | $16(21)$ | $33(47)$ | $4.9(6.6)$ | $18(23)$ | $38(49)$
| 7 kpc | $7.1(11)$ | $27(38)$ | $53(87)$ | $8.5(13)$ | $28(38)$ | $67(99)$
| 10 kpc | $16(51)$ | $65(120)$ | $150(400)$ | $20(36)$ | $52(80)$ | $140(240)$
JUNO | 1 kpc | $4.2(5.4)$ | $15(19)$ | $30(41)$ | $7.2(8.9)$ | $38(51)$ | $100(180)$
Table 2: Same as Table 1 but for MSCϵ scenario with $2\sigma$($3\sigma$) C.L. in units of $\Gamma_{8}\times 10^{-28}$ (eV). The representative energy of $E_{0}=10$ MeV was taken for $n\neq 0$ and QD parameters are in eV scale. Values are corresponding to the simulated progenitor of 27 $M_{\odot}$. | NH | IH
---|---|---
Detector | SN distance | $n=0$ | $n=2$ | $n=5/2$ | $n=0$ | $n=2$ | $n=5/2$
DUNE | 1 kpc | $2.1(3.3)$ | $0.43(0.67)$ | $0.24(0.37)$ | $490(700)$ | $50(180)$ | $33(110)$
| 5 kpc | $2.8(5.2)$ | $0.58(1.1)$ | $0.34(0.75)$ |  |  |
| 7 kpc | $3.2(7.1)$ | $0.71(1.9)$ | $0.46(1.4)$ |  |  |
| 10 kpc | $4.2(17)$ | $1.1(7.0)$ | $0.80(6.1)$ |  |  |
HK | 1 kpc | $6.8(11)$ | $0.81(1.1)$ | $0.43(0.58)$ | $9.2(14)$ | $0.68(1.0)$ | $0.36(0.54)$
| 5 kpc | $9.6(23)$ | $0.92(1.9)$ | $0.48(1.0)$ | $10(18)$ | $0.80(1.5)$ | $0.44(0.82)$
| 7 kpc | $13$ | $1.2$ | $0.61(2.6)$ | $11(25)$ | $0.94(2.5)$ | $0.51(1.4)$
| 10 kpc | $42$ | $2.7$ | $1.2$ | $13$ | $1.4$ | $4.9$
JUNO | 1 kpc | $51(100)$ | $6.4(13)$ | $4.0(7.8)$ | $47(89)$ | $5.6(11)$ | $3.5(6.9)$
Table 3: Same as Tables 1 and 2 but for $\nu$-loss scenario, with $3\sigma$ bounds over $\gamma\times 10^{-29}$ (eV). | IH
---|---
Detector | SN distance | $n=0$ | $n=2$ | $n=5/2$
DUNE | 1 kpc | $500$ | $4.6$ | $2.1$
| 5 kpc | $100$ | $3.3$ | $1.6$
| 7 kpc | $74$ | $3.2$ | $1.5$
| 10 kpc | $52$ | $3.1$ | $1.5$
HK | 1 kpc | $500$ | $2.6$ | $1.4$
| 5 kpc | $100$ | $2.3$ | $1.2$
| 7 kpc | $70$ | $2.2$ | $1.2$
| 10 kpc | $49$ | $2.1$ | $1.2$
JUNO | 1 kpc | $500$ | $150$ | $110$
| 5 kpc | $100$ | $32$ | $24$
| 7 kpc | $78$ | $24$ | $18$
| 10 kpc | $59$ | $19$ | $14$